A History of Physics over the Last Two Centuries [1 ed.] 1527581241, 9781527581241

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A History of Physics over the Last Two Centuries

A History of Physics over the Last Two Centuries By

Mario Gliozzi Edited by Alessandra Gliozzi and Ferdinando Gliozzi

A History of Physics over the Last Two Centuries By Mario Gliozzi Edited by Alessandra Gliozzi and Ferdinando Gliozzi This book first published 2022 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2022 by Mario Gliozzi All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-8124-1 ISBN (13): 978-1-5275-8124-1

TABLE OF CONTENTS

FOREWORD................................................................................................ xii 1. FRESNEL’S OPTICS ................................................................................. 1 WAVE THEORY 1.1 - The interference principle .................................................................. 1 1.2 - The polarization of light ..................................................................... 5 1.3 - Fresnel’s wave theory ........................................................................ 8 1.4 - Hamilton-Jacobi optics .................................................................... 14 1.5 - The speed of light ............................................................................. 17 1.6 - Is ether fixed or dragged by bodies in motion? ................................ 19 SPECTROSCOPY 1.7 - Invisible radiation ............................................................................ 23 1.8 - Spectral analysis .............................................................................. 27 OPTICAL INSTRUMENTS 1.9 – Photometry ...................................................................................... 32 1.10 - The camera ..................................................................................... 34 1.11 - The microscope .............................................................................. 36 2. THERMODYNAMICS .............................................................................. 38 THERMAL BEHAVIOUR OF BODIES 2.1 - Thermal expansion ........................................................................... 38 2.2 - Thermal dilation of gaseous bodies.................................................. 41 2.3 – Vapours ........................................................................................... 45 2.4 - The liquefaction of gases .................................................................. 47 2.5 - Specific heats of gases ...................................................................... 50 2.6 - Thermal conductivity ........................................................................ 55 THE PRINCIPLES OF THERMODYNAMICS 2.7 - The crisis at the beginning of the 19th century ................................ 58 2.8 - Carnot’s principle ............................................................................ 61 2.9 - The equivalence principle ................................................................ 65

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2.10 - Conservation of energy .................................................................. 67 2.11 - Thermodynamic temperature scales ............................................... 69 2.12 - The mechanical theory of heat ....................................................... 71 KINETIC THEORY OF GASES 2.13 - The nature of heat .......................................................................... 75 2.14 - Kinetic theory ................................................................................. 76 2.15 - Statistical laws ............................................................................... 78 3. ELECTRIC CURRENT ............................................................................ 84 FIRST STUDIES 3.1 – Galvanism ........................................................................................ 84 3.2 - Chemical phenomena related to current .......................................... 85 3.3 - Premonitions of ionic theories ......................................................... 90 3.4 - Thermal effects of current ................................................................ 91 MAGNETIC EFFECTS OF CURRENT 3.5 - Ørsted’s experiment ......................................................................... 92 3.6 - The galvanometer ............................................................................. 96 3.7 - Ampère’s electrodynamics ............................................................... 99 OHM’S LAW 3.8 - First studies on the resistance of conductors ................................. 106 3.9 - Georg Simon Ohm .......................................................................... 108 3.10 - Electric measurements ................................................................. 111 ELECTRIC CURRENT AND HEAT 3.11 - The thermoelectric effect .............................................................. 114 3.12 - Joule’s law ................................................................................... 117 THE WORK OF MICHAEL FARADAY 3.13 - Rotational magnetism ................................................................... 120 3.14 - Electromagnetic induction ........................................................... 123 3.15 - The nature of force lines and unipolar induction ......................... 128 3.16 - Electrolysis and electrolytic dissociation ..................................... 132 3.17 - Constant batteries ........................................................................ 139 3.18 - Potential Theory ........................................................................... 141 3.19 - Dielectric polarisation ................................................................. 146 3.20 - Magneto-optics ............................................................................. 149 3.21 - Composition of matter .................................................................. 152

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3.22 – Diamagnetism .............................................................................. 154 3.23 - A biographical note on Michael Faraday .................................... 158 3.24 - Some practical applications ......................................................... 160 MAXWELL’S ELECTROMAGNETISM 3.25 - Maxwell: A biographical note ...................................................... 165 3.26 - The first electrical researches ...................................................... 168 3.27 - A description of the electromagnetic field .................................... 172 3.28 - The electromagnetic theory of light .............................................. 176 3.29 - Electromagnetic waves ................................................................. 178 3.30 - A return to magneto-optics ........................................................... 184 3.31 – Metrology .................................................................................... 190 4. THE ELECTRON .................................................................................. 195 4.1 - Cathode rays .................................................................................. 195 4.2 - The nature of cathode rays ............................................................. 199 4.3 - Measuring the charge and mass of the electron ............................. 202 X-RAYS 4.4 - Production of x-rays he nature of x-rays ........................................ 208 4.5 - Measuring the charge and mass of the electron ............................. 214 RADIOACTIVE PHENOMENA 4.6 - Radioactive substances .................................................................. 217 4.7 - The study of new radiation ............................................................. 222 4.8 - The energy of radioactive phenomena ........................................... 225 THE PHOTOELECTRIC EFFECT AND THE THERMIONIC EFFECT 4.9 - New sources of electrons ................................................................ 227 4.10 - Ionization of gases........................................................................ 229 4.11 - The organization of scientific research in the 20th century ......... 231 5. RELATIVITY ....................................................................................... 235 MECHANICS IN THE 19th CENTURY 5.1 - The daily motion of the Earth ......................................................... 235 5.2 - Criticism of Newtonian principles .................................................. 239

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TOWARDS RELATIVITY 5.3 - The electromagnetic theory of Lorentz ........................................... 244 5.4 – Time ............................................................................................... 247 SPECIAL RELATIVITY 5.5 - Relativity of time and space............................................................ 250 5.6 - Special relativity............................................................................. 254 5.7 - Nazi revisionism ............................................................................. 259 GENERAL RELATIVITY 5.8 - Gravitational mass and inertial mass ............................................ 261 5.9 - General relativity ........................................................................... 263 5.10 - Experimental confirmations ........................................................ 264 5.11 - Note on the spread of relativity ................................................... 266 5.12 - Pre-relativistic mechanics after relativity .................................... 267 6. DISCONTINUOUS PHYSICS .................................................................. 269 QUANTA 6.1 - Matter and Energy ......................................................................... 269 6.2 - “Blackbody” radiation ................................................................... 270 6.3 - The absurdities of classical theory ................................................. 273 6.4 - “Quanta” ....................................................................................... 275 6.5 - Difficulties elicited by the quantum hypothesis .............................. 281 AVOGADRO’S CONSTANT 6.6 - The first measurements................................................................... 284 6.7 - The colour of the sky ..................................................................... 285 6.8 - Calculation of N from subatomic phenomena ................................ 285 6.9 - Deducing Avogadro’s constant from quantum theory .................... 288 6.10 - Brownian motion .......................................................................... 288 6.11 - Avogadro’s constant deduced from the theory of fluctuations ..... 296 SPECIFIC HEATS 6.12 - The law of Dulong and Petit......................................................... 297 6.13 - The third principle of thermodynamics ........................................ 300 6.14 - Low temperature phenomena ....................................................... 301 PHOTONS 6.15 - Laws for the photoelectric effect ................................................. 303 6.16 - Quanta of light ............................................................................. 304 6.17 - The Compton effect ...................................................................... 307

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7. THE STRUCTURE OF MATTER ........................................................... 311 RADIOACTIVE DECAY 7.1 - Radioactive transformation ............................................................ 311 7.2 - The Nature of Į-rays ...................................................................... 318 7.3 - The fundamental law of radioactivity ............................................. 321 7.4 - Radioactive isotopes....................................................................... 323 NON-QUANTUM ATOMIC MODELS 7.5 - First ideas regarding the complexity of atoms ............................... 327 7.6 - Thomson’s atom-fragments ............................................................ 330 7.7 - The Nagaoka-Rutherford Atom ...................................................... 333 7.8 - Artificial decay of elements ............................................................ 338 7.9 - Non-radioactive isotopes................................................................ 339 7.10 - Matter and energy ........................................................................ 341 7.11 - Atomic weights and radioactive measures ................................... 344 THE BOHR ATOM 7.12 - Spectral series .............................................................................. 346 7.13 - Bohr’s theory ................................................................................ 348 7.14 - Sommerfeld’s theory ..................................................................... 353 7.15 - The correspondence principle ..................................................... 354 7.16 - The composition of atoms ............................................................. 356 8. WAVE MECHANICS ............................................................................ 358 THE NEW MECHANICS 8.1 - A statistical extension of the radiation law .................................... 358 8.2 - The wave-particle debate ............................................................... 359 8.3 - The wave associated to a particle .................................................. 362 8.4 - Quantum Mechanics....................................................................... 364 8.5 - Wave equations .............................................................................. 366 8.6 - The equivalence of wave and quantum mechanics ......................... 369 8.7 - Experimental verifications ............................................................. 370 8.8 - Quantum statistics .......................................................................... 373 THE INTERPRETATION OF WAVE MECHANICS 8.9 - The position of the particle in the wave.......................................... 375 8.10 - The uncertainty principle. ............................................................ 377 8.11 - Indeterminacy ............................................................................... 379 8.12 - The principle of complementarity................................................. 383 8.13 - Probability waves ......................................................................... 385 8.14 - A movement to return to determinism .......................................... 387

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9. ARTIFICIAL RADIOACTIVITY ............................................................. 393 PARTICLE ACCELERATORS 9.1 - The proton ...................................................................................... 393 9.2 - High voltage devices ...................................................................... 396 9.3 - The cyclotron .................................................................................. 397 1932, THE MARVELLOUS YEAR FOR RADIOACTIVITY 9.4 – Deuterium ...................................................................................... 399 9.5 - Artificial transformations with accelerated particles ..................... 401 9.6 - The neutron .................................................................................... 403 9.7 - Beta decay ...................................................................................... 408 NUCLEAR ENERGY 9.8 - Induced radioactivity...................................................................... 410 9.9 - Neutron bombardment.................................................................... 412 9.10 - Transuranium elements ................................................................ 415 9.11 - Nuclear fission ............................................................................. 416 9.12 - Cosmic rays .................................................................................. 420 9.13 - The nuclear field........................................................................... 426 10. CONTEMPORARY DEVELOPMENTS IN PHYSICS .............................. 428 SOLID STATE PHYSICS 10.1 - Characteristics of solid state physics ........................................... 428 10.2 - Crystal structure........................................................................... 429 10.3 - The electronic structure of atoms in a crystal .............................. 431 10.4 - Magnetic properties ..................................................................... 434 EXTRATERRESTRIAL EXPERIMENTATION 10.5 - Atmospheric exploration .............................................................. 437 10.6 - The ionosphere ............................................................................. 442 10.7 - Rockets and artificial satellites .................................................... 446 10.8 - Microwaves .................................................................................. 448 10.9 - Microwave spectroscopy .............................................................. 450 10.10 - Some applications: radar, maser, laser ..................................... 451

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MATTER AND ANTIMATTER 10.11 - A note on the nature of scientific research after the second world war ............................................................................................... 454 10.12 - Thermonuclear reactions ........................................................... 454 10.13 - The physics of plasma ................................................................ 456 10.14 - Counters and detectors............................................................... 459 10.15 - Particles and antiparticles ......................................................... 461 10.16 - Strange particles ........................................................................ 463 10.17 - Conservation and violation of parity .......................................... 465 11. FUNDAMENTAL INTERACTIONS ....................................................... 469 11.1 - Quantum electrodynamics ............................................................ 469 11.2 – Renormalization........................................................................... 472 11.3 - Gauge symmetries ........................................................................ 474 11.4 - Quantum fluids ............................................................................. 478 11.5 - Quarks and quantum chromodynamics ........................................ 481 11.6 - Electroweak unification................................................................ 485 11.7 - Gravity ......................................................................................... 491 11.8 - String theory ................................................................................. 495 BIBLIOGRAPHY ........................................................................................ 498 INDEX OF NAMES .................................................................................... 504

FOREWORD Mario Gliozzi (our father) worked on A History of Physics right up to his death. To honour his memory, in 2005 we curated its posthumous publication by Bollati Boringhieri. The critical scientific and popular acclaim in Italy prompted us to suggest a version in English, to make the work available to a wider public. We also thought it opportune to complete the text with a final chapter illustrating the complex development of arguments, theories and experimental proofs that have characterised the physics of fundamental interactions. The historian of science Mario Gliozzi was a pupil and friend of the renowned mathematician Giuseppe Peano, who bequeathed him his letters (subsequently donated by his children to the Library of Cuneo), some antique books and his library. As a pupil of Peano, Gliozzi was secretary of the Pro Interlingua Academy, coming into contact with the international historical-scientific world and being elected member of the Académie internationale d’histoire des sciences. In one of his first research projects (presented by Peano to the Turin Academy of Sciences), Mario Gliozzi retraces the definition of the metre by Tito Livio Burattini. The 1934 work “A History of electricity and magnetism from its origins to the invention of the battery” won the Accademia dei Lincei prize and was the start of the “History of electrology up to Volta” issued in 1937. Gliozzi’s historical studies comprise articles, treatises and anthologies of scientific writers. The most challenging and certainly most stimulating work to which Gliozzi dedicated himself is, however, this “A History of Physics". Before leaving the readers to judge the book for themselves, we would like to add a personal note about the author: Mario Gliozzi, our father. We have many childish memories of our family life, but one that, now we ourselves are old, is set in our minds: somewhere in the shadowed study of our old house in Turin, there was the continuous tack-tack of an Olivetti Lettera 22, proof of a dedicated and exemplary life. The English translation is presented in two books. The first one “A History of Physics from Antiquity to the Enlightenment” has been translated by David Climie, M.A. Oxon (English Language and Literature). The present book entitled “A History of Physics over the last two Centuries” has been translated by Jacopo Gliozzi, great-grandson of the author and

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PhD student in Physics at Urbana Champaign University (Illinois, USA), who also added some updating notes; we warmly thank both for their invaluable contribution. We are very grateful to our friends and colleagues, in particular Prof. Matteo Leone, Prof. Roberto Mantovani and Prof. Clara Silvia Roero for advice and suggestions during the different phases of this work. Sincere thanks must also be extended to Prof. Vanni Taglietti for his indefatigable and wide-ranging help in the realisation of the project. Alessandra and Ferdinando Gliozzi

1. FRESNEL’S OPTICS

WAVE THEORY 1.1 The interference principle Thomas Young was born in Milverton on 13 July 1773 in a landowning family and, at a young age, was left in the care of his grandfather, an enthusiast of the classics who directed his precocious grandson to the study of ancient languages (Hebrew, Chaldean, Aramaic, Persian). In 1792, Young left to study medicine, receiving his doctoral degree in 1796; he then practiced as a physician in London, where he was also a professor of physics at the Royal Institution from 1801 to 1804. Young’s work was both vast and varied: an egyptologist of international renown, he made key contributions to medicine and physics, and was also appointed inspector at London Insurance Company; in the last fifteen years of his life, he dedicated himself to actuarial calculations. He was a member of the Royal Society of London and the Académie des sciences of Paris, and died in London on 10 May 1829. In a 1793 paper he showed that the human eye can accommodate to different distances due to changes in the curvature of the crystalline lens. A few years later, he posited that the perception of colour is due to the presence of three different structures in the retina, each of which is sensitive to one of the three primary colours: red, yellow, and blue.1 Both theories were studied and improved by Hermann von Helmholtz in his classic Handbuch der physiologischen Optik (1856-62). Some of Young’s other contributions to physics include the introduction of the elasticity modulus that today carries his name, which he defined as the weight that, when hung at the end of a rectangular prism of unit cross section, causes it to stretch to twice its resting length2; an interpretation of tidal phenomena, which was later expanded by George Aury in 1884; and a theory of capillary action based on the hypothesis that the walls of a tube attract the liquid it contains, as Hauksbee and John 1

Th. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, London 1845, Vol. I, pp. 139, 440. The first edition was published in 1807. 2 Ibid., Vol. I, p. 137.

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Leslie had already noted in 1802. This theory was later independently improved by Laplace3, who transformed Young’s theory into mathematical form. Arguably Young’s most important accomplishment, however, was his work on the wave theory of light. In 1800, he published a paper on sound and light in which, as Euler had already done, he highlighted the analogies between the two types of phenomena: this was the starting point of his theory of interference. In keeping with Young’s nonconformist inclination, Newton’s theory appeared highly unsatisfactory to him. It seemed especially inconceivable that the speed of light particles was constant whether they were emitted from a burning twig or an enormous source like the Sun. Above all, he saw Newton’s theory of “fits”, which attempted to explain the colouring of thin films, as muddy and insufficient. Replicating the passage of light through thin films and reflecting on the effect for a long time, Young had a brilliant idea. He realized that the colouring effect could be explained by a superposition of the light reflected immediately by the top layer of the film with the light that enters the film and then is reflected by its bottom layer. This superposition of the two light rays can either lead to weakening or strengthening in the monochromatic light applied. It is not known how Young arrived at the idea of superposition; perhaps from the study of beats in sound waves, an effect in which there is an audible periodic strengthening and weakening of a sound. In any case, in four papers read to the Royal Society from 1801 to 1803, later reproduced in the Lectures cited above, Young communicated the results of his theoretical and experimental research. Throughout, he repeatedly made reference to a passage from Proposition XXIV in the third book of Newton’s Principia, in which certain anomalous tides observed by Halley in the Philippines are explained as the effect of the superposition of waves. Indeed, Young drew from this same example to introduce the principle of interference: “Suppose a number of equal waves of water to move upon the surface of a stagnant lake, with a certain constant velocity, and to enter a narrow channel leading out of the lake. Suppose then another similar cause to have excited another equal series of waves, which arrive at the same channel, with the same velocity, and the same time with the first. Neither series of waves will destroy the other, but their effects will be combined: if they enter the channel in such a manner that the elevations of one series coincide with those of the other, they must together produce a 3

Laplace’s essays on capillary action were published from 1806 to 1818 in the “Journal de physique”.

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series of greater joint elevations; but if the elevations of one series are situated so as to correspond to the depressions of the other, they must exactly fill up those depressions, and the surface of the water must remain smooth; at least I can discover no alternative, either from theory or from experiment. Now, I maintain that similar effects take place whenever two portions of light are mixed; and this I call the general law of the interference of light.”4 In order to obtain interference, the two rays of light from the same source (ensuring that they have the same period) must be incident on the same point, coming from nearly parallel directions, after having taken different paths. Thus, Young continued, when two rays of the same light reach the eye with nearly identical angles of incidence despite having traversed different paths, the light perceived is strongest when the difference in path length is a whole multiple of a certain characteristic length, while it is weakest when this difference is a half-integer multiple: this characteristic length is different for light of different colours. In 1802, Young confirmed this principle with his now-classic double slit experiment, perhaps inspired by an analogous experiment performed by Grimaldi5. The experiment is well-known: two small slits are adjacently cut in an opaque screen, which is subsequently illuminated by sunlight passing through a small aperture. The two luminous cones that form behind the opaque screen are dilated by diffraction and partially overlap. Instead of increasing uniformly in intensity, the light in the overlapping region forms a series of fringes that alternate between light and dark. If one of the two slits is shut, the fringes disappear and only the ordinary diffraction rings from the other are present. The fringes disappear even if both of the slits are illuminated directly by sunlight or a flame, as Grimaldi had done. Applying the wave theory, Young gave a simple explanation for this phenomenon: the dark fringes appear where the trough of a wave passing through one slit coincides with the crest of a wave passing through the other slit, such that their effects cancel; the bright fringes appear where the waves passing through the apertures have coincident crests and troughs. The experiment also allowed Young to measure the wavelength of various colours, obtaining 1/36,000 of an inch (about 0.7 microns) for red light and 1/60,000 of an inch (about 0.42 microns) for violet, on the other end of the spectrum: these were the first

4

T. Young, An Account of Some Cases of the Production of Colours not Hitherto Described, in “Philosophical Transactions”, 92, 1802, p. 387. 5 § 5.34 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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recorded measurements of the wavelengths of light, and for being the first they were rather accurate as well. Young drew a variety of consequences from this experiment. Using the wave interpretation, he examined the colouring of thin films and explained the effect to the most minute detail, more or less as it is presented in textbooks today; he derived the empirical laws found by Newton and, taking frequency to be invariant for light of a given colour, he explained the tightening of diffraction rings in Newton’s experiments when air was replaced with water by using the fact that the velocity of light is reduced in more refractive media: the hypothesis put forward by Fermat and Huygens thus began to be backed by some experimental evidence. We add in passing that the expression “physical optics” was coined by Young to refer to the study of “the sources of light, the velocity of its motion, its interception and extinction, its dispersion into different colours; the manner in which it is affected by the variable density of the atmosphere, the meteorological appearances in which it is concerned, and the singular properties of particular substances with regard to it.”6 While Young’s works were the most important progress made in optics since Newton, they were met with skepticism by the physicists of his time and even subjected to derision in England. This was partly due to Young’s abuse of the interference principle, as he applied it to phenomena where interference was certainly not present, partly to the scant recognition afforded to novel ideas, which was even greater than that it is now, and partly, as Laplace admonished, because Young had employed rather sloppy and at sometimes simplistic mathematical demonstrations, evidently resulting from his lack of mathematical training as a student. Abuse of the interference principle, for which Young was thoroughly criticised, continued well after his time. Perhaps the best known case is that of Arago, who in 1824 published a theory in which the sparkling of stars was interpreted as an interference effect. This theory was adopted by the scientists of the 19th century and is still today reproduced in certain texts, despite the fact that it is clearly based on an error, as was pointed out in a series of detailed works in the second half of the century that culminated with the research of Lord Rayleigh (John William Strutt). According to Rayleigh, Averroës’ interpretation of the phenomenon (and its later extension by Riccioli) was largely correct: sparkling is due to the light’s refraction in the Earth’s atmosphere and varies because of irregularities and atmospheric motion.

6

Young, A Course of Lectures cit., Vol. I, p. 434.

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Let us return for a moment to the wavelength measurements carried out by Young to point out that the instrument he used was an interferometer, as are mirrors and the Fresnel biprism (§ 1.3): interferometers are instruments with which one can measure very small distances, of the order of half a micron (0.0005 mm). In 1866, Armand-Hyppolyte-Louis Fizeau gave one of the first examples of a practical application of interference, using it to measure extremely small lengths. In particular, he applied an interference-based method to the measurement of the thermal dilation of solids. Since then, interferometers and the application of his interference methods have multiplied so rapidly that we do not have enough space to recount their progression. We make an exception for the interferometer itself, which we will discuss in paragraph 8.6. This instrument was conceived of by Albert Abraham Michelson and later described in an 1881 paper that directly influenced the development of the theory of relativity (§ 5.5). We would also be remiss not to mention the application suggested by Michelson and Edward William Morley in an 1887 paper: establishing a wavelength of visible light as a standard of length. To demonstrate the concreteness of this proposal, in 1893 Michelson measured the length of the international prototype metre in wavelengths of light emitted by cadmium. The idea of changing the definition of the metre was discussed at length in the first half of the 20th century: in 1960, the international scientific community agreed to define an optical metre based on the wavelength of an orange line in the spectrum of an isotope of krypton.

1.2 The polarization of light Huygens discovered a phenomenon that he candidly admitted that he did not know how to interpret7: light that passes through Iceland spar exhibits peculiar properties, such that when it is incident on a second spar with a parallel cross section to the first, it only refracts normally; if the second spar is rotated at an opportune angle, however, double refraction occurs, and the intensity of the refracted rays depends on the angle of rotation. In the early years of the century, this phenomenon was studied by an officer in the French corps of engineers, Etienne-Louis Malus (17751812), who discovered in 1808 that the light reflected by water below an angle of 54 °45’ acquires the same properties as light that has passed 7

§ 6.24 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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through spar, as if the water’s reflecting surface were akin to spar’s cross section. The same effect was later observed with all other types of reflection, occurring below characteristic angles of incidence that vary with the index of refraction of the substance; for reflection of metallic surfaces the effect is a bit more complicated, however. In a paper published later that same year, Malus, through experiments using a polariscope made up of two mirrors at an angle, that today is named after Biot, gave the law that carries his name: if a beam of sparpolarised light of intensity I is normally incident on a second piece of spar, the intensities of the ordinary and extraordinary refracted rays that are produced (I0 and Ie, respectively) are

I0 = I cos2 D ; Ie = I sin2 D where Į is the angle between by the cross sections of the two pieces of spar. Around the same time that Malus conducted these experiments, the Paris Académie des sciences held a contest (1808) for a mathematical theory of double refraction that could be confirmed by experiment. Malus participated and won with a seminal 1810 paper titled Théorie de la double réfraction de la lumière dans les substances cristallisées. In relating his discovery, Malus interpreted it using Newton’s point of view, though not taking it “as an indisputable truth”. For Malus, rather, it was purely a hypothesis that allowed his calculations to work out. Having thus taken the side of the corpuscular theory of light, he sought an explanation of the effect in the polarity of light particles that Newton had discussed in his query XXVI8. “Natural” light, as it is known today, is composed of asymmetrical corpuscles oriented in every direction. When these corpuscles are reflected or pass through birefringent crystals, they orient themselves in the same direction: Malus called such light polarised, a term that is still used today. The polarization studies begun by Malus were continued in France chiefly by Biot and Arago, and in Britain by David Brewster, who had attained fame in his days more for his invention of the kaleidoscope (1817) than for his important contributions to mineral optics. In 1881, Malus, Biot, and Brewster independently discovered that refracted light is also partially polarised, reaching maximum polarization when the reflected and refracted rays are perpendicular to each other. 8

§ 6.20 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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Jean-François-Dominique Arago (1786-1852) demonstrated that moonlight, light from comets, and rainbows are all polarised, confirming that the light emanating from these sources is actually reflected sunlight. The light emitted obliquely by incandescent solids and liquids also proved to be polarised, indicating that it originates from an internal layer of the material and refracts when it comes into contact with the air. The most important and well-known discovery made by Arago, however, was chromatic polarization, which he obtained in 1811. By passing polarised light through a 6 mm thick sheet of rock crystal and observing the refracted rays through a piece of spar, he obtained two images of complementary colours. For instance, if the one of the images was initially red, for different rotation angles of the spar it could turn orange, yellow, or green. Biot repeated these experiments the following year and showed that to obtain a fixed colour, the rotation of the spar had to be proportional to the thickness of the sheet. In 1815, he further discovered the phenomenon of rotational polarization and the existence of dextrorotatory and laevorotatory materials. Incidentally, it was only in 1848 that Louis Pasteur (18221895), who was later immortalized for his work in biology, demonstrated that tartaric acid could exist in both dextrorotatory and laevorotatory form, starting a new chapter in organic chemistry research that was later expanded by Joseph Achille Le Bel (1847-1930) and Jacobus Hendricus Van’t Hoff (1852-1911). We now return to Biot, who discovered, also in 1815, that tourmaline is birefringent with the special property that it absorbs the ordinary ray and emits only the extraordinary one. Based on this effect, Herschel built the tourmaline polariser in 1820, a simple apparatus that has remained more or less the same to this day. The biggest inconvenience of this polariser is its colouration effect, which is avoided by the prism built in 1820 by the British physicist William Nicol (1768-1851); Nicol’s prism also only transmits the extraordinary ray. Nicol combined two such birefringent prisms in 1839, creating a device whose use is still widespread today. In conclusion, the fundamental phenomena of light polarization, a vast and interesting chapter of physics that is outlined in all modern physics texts, were by and large discovered by French physicists between 1808 and 1815. Furthermore, because the discovery of these intriguing phenomena occurred within corpuscular framework, it appeared that this theory was given new life.

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1.3 Fresnel’s wave theory The spike in popularity of the corpuscular theory was short-lived, however. A young civil engineer, Augustin-Jean Fresnel (Fig. 1.1), having joined a small royalist militia as a volunteer to stop Napoleon, who had just returned from Elba, was suspended from his employment and forced to retire to Mathieu (near Caen) during the Hundred Days. Lacking almost any optical training, Fresnel dedicated himself to the study of diffraction throughout his forced leisure, using a makeshift experimental setup. The first fruits of his work were two papers presented on 15 October to the Académie des sciences in Paris. Arago, who was charged with examining them and reporting their results, found them so interesting that he offered for Fresnel to temporarily move to Paris to repeat the experiments in better conditions, as he had been summoned back to work, by the sudden restoration of Louis XVIII.

Fig. 1.1 – Augustin-Jean Fresnel

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Fresnel had already begun to study the shadows produced by small obstacles placed in front of light beams, observing like Grimaldi that fringes appear not only on the exterior of the shadow but also on the interior (a point unaddressed by Newton). Studying the shadow produced by a thin string, he rediscovered the interference principle. He was struck by the fact that when the string was placed directly above the edge of a screen, the internal interference pattern disappeared. He therefore deduced that concurring rays coming from two sides are necessary to produce an internal interference pattern, as the fringes inside the shadow disappear when the light coming from one direction is intercepted. Fresnel described the effect: “The fringes cannot arise from a simple mixture of these rays, since each side of the wire separately casts into the shadow only a continuous light; it is their meeting, the very crossing of these rays which produces the fringes. This consequence, which is only, so to speak, a translation of the phenomenon, totally opposes the hypothesis of Newton and fully confirms the theory of vibrations. One easily sees that the vibrations of two rays that cross at a very small angle can oppose one another when the node of one corresponds to the antinode of the other.”9 The idea was clear, but the statement of the principle was imprecise and subsequently corrected by Fresnel, who specified that all the waves weaken when the “dilated nodes” of one are superposed with the “condensed nodes of the other”; they strengthen, on the other hand, when their motions are “in harmony”. In short, it was an interference principle that, once he had fully grasped it, led Fresnel to retrace Young’s steps and in particular to explain the colouring of thin films. In Paris, Fresnel learned from Arago of Young’s double slit experiments, which seemed to him more than suitable in demonstrating the undulatory nature of light. However, not all physicists agreed. Indeed, the adherents of Newton attributed the phenomenon to an effect at the edges of the slits. To convince even the most obstinate of scientists, it was necessary to devise an experiment in which all possible attraction between matter and light rays was eliminated. Fresnel succeeded in this, in 1816 communicating the well-known “double mirror experiment” and in 1819 the “biprism” experiment, which have now become so standard that we do not waste the reader’s time spelling them out. In 1837, Humphrey Lloyd showed that optical interference can be obtained even with a single mirror, as the light incident on the mirror can interfere with the reflected light. An important advance was made by Julius Jamin (1818-1886), who in 1856, while analysing an observation made by Brewster in 1831, constructed his well9

A.J. Fresnel, Œuvres complètes, Paris 1866, Vol. I, p. 17.

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known Jamin interferometer using two parallel glass plates, which were externally coated in silver by Georg Hermann Quincke (1834-1924) in 1867. As it is known, Jamin’s device produces interference through the path length difference of two rays; only a minute change in the refractive index of medium in which one of the two rays propagates is needed to observe the characteristic fringes and measure the extent of the effect. Interferometers of this type are used to study variations in the index of refraction with changing temperature, pressure, concentration, the presence of a gas, and other variables. We add in passing that John Herschel was inspired by the double mirror experiment in 1833 to conduct the analogous experiment for the interference of sound waves using a two-ended tube. This experiment was later improved in 1866 by Quincke, whom the experimental apparatus is named after. The use of manometric flames for a more objective observation was proposed in 1864 by Karl Rudolph Konig (1832-1901), who substituted Quincke’s rubber tubes with two extendable metallic ones, like a trombone. Let us return once more to the work of Fresnel. Having saved the interference principle from the attacks mounted by Newton’s devotees, wave theory laid out three principles: the principle of elementary waves, the principle of wave envelopes, and the interference principle. These three principles remained disconnected until Fresnel brilliantly united them in his novel description of wave envelopes. For Fresnel, a wave envelope was not a simply a geometrical envelope, as for Huygens. According to the Frenchman, at any given point of the wave, the total effect is the algebraic sum of the pulses produced by all the elementary waves it contains; the sum of all these contributions, which are superposed according the interference principle, can also vanish. Fresnel performed the calculation, though not very rigorously, reaching the conclusion that the effect of a spherical wave on an external point is reduced to that of a small crown of the wave whose centre is aligned with the light source and the illuminated point, there being no other global effects. Thus, was overcome the century-old problem that had always hindered the success of wave theory: reconciling the rectilinear propagation of light with its supposed wave mechanism. Each illuminated point receives light from the small region of the wave that is in its immediate vicinity, and thus the process occurs as if light propagates in a straight line from source to point. While it is true that waves should bypass obstacles, this principle cannot be considered in a purely qualitative manner, as the amount of diffraction around an obstacle is a function of the wavelength and requires quantitive evaluation. Examining diffraction phenomena, Fresnel calculated

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the amount of bending based on his theory, which was seen to agree impressively with experimental results. Because of their lack of mathematical rigour, Fresnel’s first papers on diffraction were not well received by Laplace, Poisson, and Biot, a trio of analytical scientists who placed great importance on firm mathematical underpinnings. After a brief hiatus mandated by his engineering career, Fresnel returned to his theory with an important paper on diffraction that he presented in 1818, entering it in a contest organized by the Académie des sciences. The paper was reviewed by a commission made up of Laplace, Biot, Poisson, Arago, and Joseph-Louis Gay-Lussac: the first three were staunch Newtonians, Arago was partial to Fresnel, and Gay-Lussac was relatively inexperienced in the field but known to be a fair judge. Poisson observed that Fresnel’s theory would lead to conclusions in stark contrast with common sense, as its calculations indicated that light can appear at the centre of the shadow created by an opaque disk of the right dimensions, while the centre of the conical projection of a small circular aperture, at the right distance, can appear dark. The commission invited Fresnel to experimentally demonstrate these consequences of the theory, and Fresnel, unperturbed, methodically confirmed them to the last detail, demonstrating that in this case common sense was wrong and Poisson had put too much faith in it. After these demonstrations, on unanimous recommendation of the commission, the Académie awarded the prize to Fresnel and in 1823 elected him a member. Having established the theory of diffraction, Fresnel moved on to the study of polarization phenomena. Corpuscular theory, in trying to explain the many phenomena that were discovered in the first 15 years of the century, had been forced to introduce a myriad of unfounded and sometimes contradictory hypotheses, rendering the theory incredibly complicated. In the experiment of two mirrors at an angle, Fresnel had obtained from a single light source two virtual sources that were always perfectly coherent. He tried to reproduce this device with the two rays obtained by double diffraction from a single incident beam, suitably compensating the optical path length difference between the two rays. No matter what, however, he could not obtain interference in the two polarised rays. In collaboration with Arago, he continued experimental studies on the possible interference of polarised light. The two scientists experimentally established that two rays of light polarised in parallel directions always interfere, while rays that are polarised in perpendicular directions never interfere (in the sense that they cannot cancel). How could one explain this, and on a larger scale, how could one explain all the other

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polarization phenomena that had nothing in common with acoustic ones? Because light polarised by reflection exhibits two orthogonal planes of symmetry passing through the ray, one could surmise that the vibrations of the ether occur in these planes, transverse to the ray. This idea had been suggested to Fresnel by André-Marie Ampère in 1815, but Fresnel had not thought it important. Young also thought of transverse vibrations when he heard of Fresnel and Arago’s experiments on polarization, but, perhaps due to uncertainty or excessive prudence, spoke of imaginary transverse motion: even to the least conformist of scientists, transverse vibrations seemed mechanically absurd. After using the language of transverse vibrations implicitly for several years, in 1821 Fresnel decided to make the leap and, having found no other avenue to explain polarization phenomena, embraced the theory of transverse vibrations. “It has only been for a few months that,” he wrote in 1821, “in meditating more attentively on this subject, I have recognized that it is very probable that the oscillatory movements of the light waves are executed solely according to the plane of these waves, for direct light as for polarised light௘… I will show that the hypothesis I present includes nothing that is physically impossible and that it can be used to explain the principal properties of polarised light.”10 That the hypothesis could explain the principal properties of light, both polarised and non-polarised, was extensively demonstrated by Fresnel; but showing that the theory included nothing that was physically impossible was another task altogether. The transversality of the vibrations implied that the ether, despite being a very thin, imponderable fluid, had to also be a solid more rigid than iron, as only solids transmit transverse vibrations. Fresnel’s hypothesis appeared very bold, and perhaps even foolhardy. Arago, a physicist who certainly did not let prejudices govern him and had been a friend, advisor, and defender of Fresnel in every occasion, did not want any responsibility for this strange hypothesis and refused to place his signature on the paper presented by Fresnel. Starting in 1821, Fresnel therefore continued on his own path, encountering success after success. The hypothesis of transverse vibrations permitted him to formulate a mechanical model of light. At its basis was the ether that pervades the whole universe and permeates through bodies, shifting its mechanical properties in the presence of matter. Because of these modifications, when an elastic wave propagates from pure ether to ether mingled with matter, part of the wave is reflected at the separation 10

Fresnel, Œuvres complètes, cit., Paris Vol. I, p. 630.

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interface and the other part penetrates inside the matter: thus, the phenomenon of partial reflection, which had remained a mystery for a century, was mechanically explained. Fresnel gave the formulas that carry his name, unchanged since his time: among these we only reproduce here the one describing the reflection coefficient (the ratio between the intensities of the incident and reflected rays) for normal incidence. If n is the relative index of refraction between the second medium and the first, and r is the reflection coefficient, Fresnel’s formula gives:

݊െ1 ଶ ‫=ݎ‬൬ ൰ ݊+1 The propagation velocity of the vibrations that move through matter depends on wavelength and, keeping that constant, is slower in more refractive media: this leads to refraction and scattering of light. In isotropic media, waves are spherical and centred at the light source; in anisotropic media the wavefront is generally of fourth order. In this theory, all of the complicated polarization phenomena are explained in an impressively coherent picture consistent with experimental results, arising as special cases of the general laws of composition and decomposition of velocities. The study of double diffraction led to research on the forces responsible for the microscopic molecular motion in elastic media. This research brought Fresnel to state a few theorems that formed the basis of a new branch of science, as Émile Verdet (1824-1866), the editor of Fresnel’s works, observed. Thus, was born the general theory of elasticity, developing just after Fresnel with the works of Augustin-Louis Cauchy, George Green, Poisson, and Gabriel Lamé. From 1815 to 1823, Fresnel erected his impressive scientific body of work which, as with all human constructions, was not free of inaccuracies. The young engineer approached and solved problems relying more on his powerful intuition than mathematical calculation, therefore he was often mistaken or only hinted at the true solution. Nevertheless, his ideas, despite the opposition of older physicists, quickly captured the support of young researchers, who admired the intuitiveness and simplicity of his theoretical model: Airy, Herschel, Franz Ernst Neumann (1798-1895) and a legion of other physicists corrected and organized the theory and studied its consequences. From 1823 until his death, Fresnel dedicated himself to the study of lighthouses for his work. The latest lighthouses at the time were made up

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of four or five d’Argand lamps11, whose light was directed in more or less a single direction by a metallic parabolic mirror in uniform rotational motion, such that the beam of light was successively emitted radially in all directions. This apparatus was not very efficient: barely half of the light produced was transmitted outwards by the reflector. Fresnel had the idea to substitute it with lenses to horizontally refract the light incident on their focus. An appreciable beam could only be obtained in this way using large lenses, which were very difficult to build at the time. Buffon had already proposed (1748) to build composite lenses, and Condorcet (1773) and Brewster (1811) had reconsidered the idea, but the project could not be practically carried out. Fresnel finally succeeded in making the idea concrete, separately building concentric rings of a small-diameter lens. In this way, he was able to obtain large lenses with a focal length of 92 cm. The composite lenses allowed for nine tenths of the incident light to be transmitted, massively advancing the capacity of lighthouses. Fresnel’s brief life paralleled his scientific work, which remained limited to theoretical and practical optics. He was born on 10 May 1788 in Broglie, near Bernay in the Eure region. Having been judged a boy of modest intellectual capacities, though very ingenious and skilled in manual tasks, at age sixteen he enrolled in the École polytechnique of Paris, where his teacher Adrien-Marie Legendre immediately noticed his uncommon mathematical talent. Having obtained the title of engineer of bridges and roads, he immediately began working as a civil engineer for the state and remained in this line of work, with a few interruptions for study or illness, until his death by tuberculosis on 14 July 1827 in Ville-d’Avray, near Paris. He was a member of the Académie des sciences of Paris and the Royal Society of London, but he was never able to obtain a position at a university, which could have alleviated the suffering caused by his constant ill health and the disease that afflicted him in his last years, which at the time was always fatal. Even then the university system, perhaps unknowingly, followed rules that could be detrimental to scientific research, not to mention unforgiving.

1.4 Hamilton-Jacobi optics William Rowan Hamilton, born in Dublin on 4 August 1805 and died on 2 September 1865, was still a second year student at Trinity College when he read his paper on caustics to the Royal Irish Academy. Further expanding 11

§ 7.15 of M. Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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this idea, he was led a few years later to predict the phenomenon of “conical refraction”: when a ray of light is incident on a rectangular sheet cut by a biaxial crystal perpendicular to an optical axis, and the angle of incidence is parallel to the optical axis, the ray is refracted into a cone of light whose radius depends on the thickness of the sheet. In 1837, on Hamilton’s suggestion, Humphrey Lloyd (1800-1845) experimentally verified this phenomenon in an aragonite sheet. With this theoretical discovery began Hamilton’s brilliant career as a mathematician, although his interests extended beyond science to philosophy, humanism, and poetry. In the first years of Hamilton’s career, wave theory was not unanimously accepted, as we saw earlier. Poisson was still a supporter of Newton’s corpuscular theory; Biot, the most conservative of the great 19th century physicists, maintained his convictions until his death in 1862; Brewster did not accept the wave theory because he could not think “the Creator guilty of so clumsy a contrivance as the filling of space with ether in order to produce light,” and, incredible as it may seem, he claimed that he could not follow Fresnel when he spoke of transverse vibrations as if it was a point in his favour. In light of this general attitude among the prominent scientists of the time, Hamilton set out to construct a formal theory of known optical phenomena that could be interpreted both from an undulatory point of view and from a corpuscular point of view, through the principle of least action. His stated aim was to construct a formal theory of optics that had the same “power, beauty, and harmony” of Lagrangian mechanics. According to Hamilton, the laws of ray propagation could be considered in and of themselves, independently of the theories that interpreted them, to arrive at a “mathematical optics”. Indeed, following this approach, he deduced a doctrine of scientific philosophy. Hamilton distinguished two phases in the development of science: in the first, the scientist generalizes individual facts into laws through induction and analysis; in the second, the scientist goes from general laws to their specific consequences through deduction and synthesis. In short, according to Hamilton, humans gather and group observations until the scientific imagination discovers the inner laws governing them, creating unity out of variety. Then, humans reobtain variety from unity by using the discovered laws to make predictions about the future. This was the method with which Hamilton worked. He observed that the principle of least action, while deduced from the metaphysical concept of the “economy of nature”, was more properly a principle of extremal action, as there were several known cases in which it was the maximum in

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action that described phenomena. He thus spoke of stationary or varied action, depending on whether the endpoints of the rays and trajectories were taken to be fixed or variable. In this way, Hamilton arrived at the formulation of the principle that carries his name, according to which there is a certain optical quantity, defined in a mathematical way, that is stationary in the propagation of light. Through this approach, geometric optics was converted into a formal theory that could explain experimental results without requiring a choice of either the corpuscular hypothesis or wave hypothesis of light. In 1834-35, Hamilton extended his optical theory to dynamics and systematically developed it. In his framework, the solution of a general dynamical problem depends on a system of two partial differential equations. Hamilton’s work was an admirable synthesis of optical and dynamical problems, which Louis De Broglie would rediscover and Erwin Schrödinger would use as inspiration (§ 8.5): it is interesting to see that the most powerful mathematical instruments of quantum mechanics were provided by analytical mechanics, which was developed within the framework of classical physics. It was Carl Gustav Jacobi (1804-1851), however, who with his famous works, beginning in 1842, gave the broadest application of Hamilton’s theory, simplifying it and at the same time generalizing it to a now-classic form: for this reason, the theory is often called Hamilton-Jacobi mechanics. Before moving on, we must also relate another great merit of Hamilton, his introduction of a new mathematical technique involving what he called “quaternions” (a system of four numbers that extends complex numbers). He announced this novel approach in 1843, and developed in in a lengthy treatise (of 872 pages) published in 1853, which appeared incomprehensible to his contemporaries. The theory started from the consideration that the imaginary unit can be thought of as an operator that indicates a 90 degrees rotation, much like a factor of -1 indicates a 180° rotation. The extensive applications of quaternions to many branches of physics are well known, but in the first decades following the introduction of Hamilton’s number system it seemed that physicists did not take the new algebra into consideration (the first example of a noncommutative algebra and the algebraic beginnings of linear algebra), and neither was its study encouraged by Mathematics Departments.

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1.5 The speed of light As we have mentioned many times, corpuscular theory implied a greater speed of light in more refractive media, while wave theory implied a lower speed. To Arago, an opponent of corpuscular theory yet not a convinced devotee of the wave approach, it seemed that measuring the speed of light in materials was the best method, the experimentum crucis, to distinguish between the two theories. As early as 1839, he proposed an experiment with this aim, but, because of his weak eyesight, he was forced to leave its execution to others. Arago had thus given weight to the idea of a critical experiment that could definitively affirm the wave theory, and so measuring the speed of light with terrestrial means acquired urgency and importance in the eyes of young physicists. The first to perform such a measurement was Armand-Hippolyte-Louis Fizeau (1819-1896) in 1849. On a conceptual level, Fizeau’s experiment was similar to the one attempted earlier by Galileo12. Fizeau set up an apparatus in which a ray of light passes through the gaps in a rapidly rotating gear and reflects perpendicularly on a flat mirror 8633 metres away. The reflected ray travels back in the direction of the incident ray and can be observed with a magnifying glass. If the gear is at rest, the reflected ray passes through the same gap that it came from and an observer sees the mirror illuminated. If instead the gear rotates fast enough, in the time taken by the light to travel both legs of the path, the gap will have given way to a tooth of the gear. The ray cannot pass through it, and so there is no light in an observer’s field of vision; if the rotational velocity is increased further such that the reflected ray enters the next gap, then the field of vision is once again illuminated. Fizeau obtained a velocity of light of 313,274,304 m/s. The method was repeated by Marie-Alfred Cornu (1841-1902), who after averaging one thousand trials gave a value of 298,400 km/s in 1873, with an uncertainty of 1/300. The method was improved and applied again in 1882 by James Young (1811-1883) and George Forbes. In 1928, August Karolus and Otto Mittelstaedt replaced the gear with a Kerr cell, a much more precise electro-optical device that allows for the mirror distance to be reduced to only a few metres. With this modification, the experiments were repeated by A. Huttel in 1940 and W.C. Anderson in 1941. Fizeau’s setup did not permit measurements of velocity in different media. In 1834, in order to study the so-called “singing flame” effect and measure the duration of an electric spark, Charles Wheatstone had 12

§ 4.12 of M. Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambdridge Scholars Publ. 2022.

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introduced the rotating mirror, which he later tried to extend to measure the speed of light. He was not successful, however, and his project was reattempted in 1839 by Arago, who devised the very complicated experiment that we mentioned at the beginning of the section. Fizeau and Léon Foucault (1819-1868) set out to simplify it and make it more practical. The two physicists initially worked together (continuing a manyyear propensity for collaboration), but later split and competed with one another to see who could obtain the result first. Foucault won in 1850 with the apparatus that is now described in all physics texts, which functioned through the following mechanism: the time taken by light to travel the distance between two mirrors and back, where the first mirror is rotating rapidly, can be measured by the rotation of the moving mirror, which in turn is calculated based on the deviation of the light ray upon its return. To measure the number of rotations per second of the moving mirror, Foucault, perhaps for the first time, used the stroboscopic method, a technique of seemingly slowing a periodic motion to allow for more convenient observation. Placing a substance different from air between the two mirrors, which were a few metres apart, one could measure the speed of light in the substance. Foucault’s experiments in 1850 were only comparative; by placing a tube of water between the two mirrors he noted that the speed of light in water was 3/4 of that in air. Fizeau arrived at the same result shortly after in collaboration with Louis Breguet (1804-1883). Occupied with other studies for some time, Foucault returned to speed of light experiments in 1862 and measured it to be 298,000 km/s, with an uncertainty of 500 km/s. The measurements were repeated with successive improvements to Foucault’s method by Simon Newcomb (1835-1909) in 1881-82, Albert Michelson from 1878 to 1882, and later W.C. Anderson between 1926 and 1937. Anderson’s measurements gave a speed of light of 299,764 km/s with an uncertainty of 15 km/s: all of these values refer to the speed of light in vacuum. Terrestrial measurements systematically give a higher value than the one obtained from astronomical measurements13 for a reason that is not yet known. All of these measurements, moreover, concurred in finding lower velocities in more refractive media. However, they also brought an important detail to light: the index of refraction of a medium is not exactly equal to the ratio between the speed of light in vacuum and the speed of light in the material, as Fresnel’s theory predicted. A discrepancy that 13

§ 5.35 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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greatly surpassed experimental uncertainty was consistently observed. This disagreement was explained in 1881 by Lord Rayleigh, who introduced the concept of “phase velocity”, namely the velocity of the wave crest resulting from multiple superimposed monochromatic waves: in a dispersive medium, the group velocity, which is the one that can be directly measured, does not coincide with the phase velocity. In 1850, Fizeau and Foucault’s experiments seemed to mark the definitive triumph of the wave theory. Carlo Matteucci, one of the most illustrious Italian physicists of the time, wrote that year: “The direct experimental demonstration of the retarded propagation of light in relation to the density of the media by it traversed, which we now discuss, thoroughly rejects the Newtonian hypothesis and solemnly confirms the truth of the undulatory system.”14 Theories, however, are never definitive. Fresnel’s theory would continue to live untroubled for two decades, after which the problems would arise.

1.6 Is ether fixed or dragged by bodies in motion? The hypothesis of elastic vibrations posed a problem: is ether at rest or in motion? In particular, does the ether accumulated in a body move with it? With a series of subtle experiments, Arago had shown that the motion of the Earth has no appreciable effect on the refraction of light coming from stars. This result was irreconcilable with corpuscular theory, therefore he asked Fresnel if it fits within the framework of wave theory. Fresnel responded in an 1817 letter, writing that the result could be easily explained by wave theory, much like the phenomenon of aberration, by supposing the partial dragging of the ether: a body in motion does not entrain all of the ether contained inside it but only the amount in excess to what would be contained in an equal volume of empty space. With this hypothesis, Fresnel was able to explain all of the effects resulting from the rapid motion of a refractive body. The effects of motion on light or sound-emitting bodies was theoretically studied by the Austrian physicist Christian Doppler (1803-1853) in 1842. He pointed out that if a light source is approaching, an observer sees vibrations of shorter duration than the ones emitted by the source, and therefore the colour of the light shifts towards violet, while it shifts towards red if the source is moving away from the observer. Analogously, 14

C. Matteucci, Lezioni di fisica, Pisa 1850, p. 549

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if a sound-emitting body is approaching, an observer hears a higherpitched sound than the one emitted, while a lower-pitched sound is heard when the source is moving away. This phenomenon can be readily observed by the change in hiss of a moving train or an airplane flying overhead. In 1848, Fizeau proposed to use this phenomenon, called the Doppler Effect (or Doppler-Fizeau effect), to measure the radial velocities of stars based on the shift in their spectral lines, but only in 1860 was Mach able to obtain an experimental demonstration of this shift. Consequently, in 1868, William Higgins was able to apply the effect to measure the radial velocity of Sirius. In 1900, Aristarkh Belopolsky succeeded in measuring a shift in the Fraunhofer solar spectral lines through multiple reflections of a ray of light on rapidly moving mirrors in his laboratory. In 1914, Charles Fabry (1867-1945) and Henric Buisson (1873-1944) experimentally verified the fundamental formula describing the phenomenon. Doppler had already noted that it was possible to employ his effect to measure the velocity of binary stars, but nobody, including Maxwell, had been able to do so. This measurement was obtained only after the introduction of the direct-vision prism in 1860 (§ 1.8). In 1869, Johann Carl Friedrich Zöllner (1834-1882) had the clever idea to couple two Amici direct-vision prisms in opposite directions, such that they gave rise to two opposite spectra. The result was a spectroscope that could take advantage of the Doppler effect in the manner indicated by Fizeau: from then on, the Doppler effect became of considerable importance in astrophysics. The Doppler effect also appeared to confirm Fresnel’s ideas on the partial dragging of ether; nevertheless, this hypothesis was fought by George Gabriel Stokes (1819-1903), one of the most illustrious scientists that continued Fresnel’s work. In an important 1845 paper, Stokes held that the ether was totally dragged in the immediate proximity of the Earth, gradually becoming partially dragged with increasing distance. In 1851 Fizeau attempted to resolve the dispute by looking at the interference between two light rays, the first having passed through a column of water in the direction of the water’s flow, and the second having passed in the opposite direction. If the ether was dragged by objects in motion, one would observe shifted interference fringes compared to when the experiment was performed with the water at rest. Fizeau’s experimental results confirmed Fresnel’s hypothesis, as did the research of Eduard Ketteler (1836-1900) in 1871 and that of Michelson and Edward Morley (1838-1923) in 1886. Five years earlier, however, Michelson had attempted, with a famous experiment, to experimentally detect the motion of the Earth with respect

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to the ostensibly fixed ether, or in other words, the “aether wind”. Michelson’s approach could be called the bifurcation method: a beam of light incident on silver-plated sheet is split into two orthogonal beams, which correspond to the reflected ray and the refracted ray; these are in turn normally reflected by two mirrors, that are equidistant from the sheet, retrace their original trajectories, superpose, and are sent into a telescope. If the Earth was in motion with respect to the fixed ether, the superposed rays would exhibit interference effects due to the time difference between the two orthogonal paths. Furthermore, if the apparatus was rotated by 90 °, one would observe a shift in the fringes that Michelson, because of a calculational error, had predicted to be twice the correct value. This mistake was later rectified by Hendrik Lorentz in a wide-ranging critical paper that reviewed the problem of ether drag from Fresnel’s day to current developments.15 In any case, the experimental apparatus was sensitive enough to detect effects one hundred times smaller than the one calculated by Michelson, notwithstanding the relative smallness of the Earth’s tangential velocity (30 km/s) compared to the speed of light. The experiment, which was repeated many times at different orientations and in different times of the year, gave categorically negative results: in no case was a shift of the fringes observed when the apparatus (called a Michelson interferometer) was rotated. The issues raised by Lorentz in the aforementioned paper led Michelson to repeat his experiments in 1887, this time in collaboration with Morley. The results, however, were identical. Michelson could thus confirm based on his experiments that the ether moves with the Earth. On the other hand, the phenomenon of aberration indicated that the ether is at rest. The two starkly contrasting conclusions were a serious problem for physics. In his public lectures at Trinity College in Dublin, the Irish physicist George Francis Fitzgerald (1851-1901) attempted to overcome this contradiction by supposing that the dimensions of an object were a function of its speed in the ether: this hypothesis was popularized in 1892 by Oliver Lodge in a brief note that appeared in a British scientific magazine16 and later confirmed by Lodge the following year.17

15 H. A. Lorentz, De l’influence du mouvement de la terre sur les phénomènes lumineux, in “Archives néerlandaises des sciences exactes et naturelles”, 21, 1887, pp. 103-76, and in particular pp. 168-76. 16 “Nature”, 46, 1892, p. 165. 17 O.J. Lodge, Aberration Problems, in “Philosophical Transactions”, 184, 1893, pp. 749-50.

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Independently of Fitzgerald, Lorentz proposed the same hypothesis a few years later in a paper that quickly became famous.18 According to him, moving bodies experience a contraction in the direction of their motion that increases with increasing velocity, becoming maximal when the body reaches the speed of light in vacuum: in this limiting case, the length of the body in the direction of motion goes to zero. This effective Lorentz contraction was not experimentally detectable, not so much because of the effect’s small magnitude (the diameter of the Earth, for example, is shortened by 6.5 cm because of its translational motion around the Sun), but more because instruments in motion are also subject to the same contraction. The hypothesis of contraction fully explained the negative results of the Michelson-Morley experiment, as the leg of the apparatus pointing in the direction of the Earth’s motion would be contracted just enough to cancel the time difference between the propagation of light along this direction and along the perpendicular path. Yet Fizeau and Lorentz’s hypothesis seemed too artificial, too tailormade to explain a single specific phenomenon, and without any other theoretical need to justify its introduction. In 1900, the elderly Lord Kelvin was so dissatisfied with the contraction loophole to maintain that the unresolved problem of ether drag hung over physics like an enormous dark cloud. This cloud grew even darker in the first decades of the 20th century, when Michelson’s experiment was upheld by many other optical, electrical, and electromagnetic experiments designed to measure the Earth’s motion with respect to the ether, each giving negative results. These included the work of Lord Rayleigh in 1902, Frederick Trouton and H.R. Noble in 1903, and Dewitt Brace in 1904. In 1905, relativity resolved the enigma, as we will see in § 5.5. The Michelson-Morley experiment, which remained the most famous of the tests because it had inspired all the others, was repeated with additional improvements by Morley and Dayton Miller in 1904, who found the same results. Later, from 1921 to 1925, Miller made a series of observations that led him to conclude that the Earth was in motion with respect to the ether with a velocity of 9 km/s. However, this result was refuted by later experiments, like the ones conducted in 1929 by Michelson with Francis Pease and Fred Pearson.

18

H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Korpern, in “Verslagen en Mededelingen der koninklijke Akademie van Wetenschappen te Amsterdam”, 1892, pp. 74.

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SPECTROSCOPY 1.7 Invisible radiation In the first thirty years of the 19th century, the burgeoning research on polarization and the nature of light made the other discoveries related to luminous phenomena of the time appear secondary. That light rays and caloric effects are connected is an obvious observation that had been made since antiquity; the use of the term “focus” to refer to concave mirrors and lenses, introduced by Kepler and coming from the Latin for hearth, indicates that more attention was paid to the concentration of heat rays than light. The distinction between light rays and heat rays is found, perhaps for the first time, in Porta’s Magiae naturalis (1589), in which he wrote about how marvellous it is that a concave mirror not only concentrates heat but also cold. This observation was the subject of careful experimentation on the part of the Accademia del Cimento, which found a significant cooling at the focus of a concave mirror placed in front of a large block of ice. Paolo Del Buono (16251659), a member of the Accademia, observed that the rays that crossed a lens made of ice did not lose their caloric power at all. The distinction between heat and light is even more clear with Mariotte, who, having built a concave ice mirror, showed that heat rays could reflect off of it without weakening, concentrating to a sufficient degree at the focus that they could light gunpowder. Mariotte was also responsible for the first experiments on the selective nature of the absorption of thermal radiation. In 1662, he observed that the heat reflected by a concave mirror was appreciable at the focus, but if glass was placed between the mirror and focus then no heat was detected.19 In 1777, Lambert showed that heat rays propagate in a straight line like light rays. In 1800, Williams Herschel made a pivotal discovery. He set out to study whether heat was truly uniformly distributed across the solar spectrum, as most at the time supposed purely from a cursory appeal to intuition. Herschel therefore moved a sensitive thermometer across the spectrum, finding not only that the temperature increased in moving from violet to red, but that it reached a maximum in a region beyond red where the spectrum became invisible to the human eye. In natural philosophy, Herschel commented, it is always useful to be suspicious of what is commonly admitted. He interpreted the phenomenon as the result of invisible heat radiation coming from the Sun, which he called infrared 19

“Histoire de l’Académie royale des sciences”, 1, 1733, p. 344.

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because it was deviated by the prism less than red light. Later, he studied this mysterious radiation with a terrestrial source -a heated, but not incandescent, iron cylinder- showing its refraction through lenses. Young understood the importance of this finding and in 1807 Lectures called it the greatest discovery since Newton’s era. On the other hand, John Leslie (1766-1832), a very careful experimenter, attributed Herschel’s phenomenon to air currents; his objections were erroneous and garnered no support, however. More successful were his other experimental studies, detailed in a volume published in 1804 and still cited today in physics texts. Leslie showed, with the aid of the differential thermometer that carries his name (but that was actually first described in 1685 by Johann Christoph Sturm [1635-1703]) and his “cube”, in which some of the faces were blackened and others were covered with mirrors, that the thermal radiation and absorption of a body depend on the nature of its surface. A few years prior to Leslie’s works, the German Johann Ritter (17761810) had made another discovery parallel to Herschel’s and of equal importance. Repeating Herschel’s experiments in 1801, he set out to study the chemical effects of the various types of light radiation. To this end, he used silver chloride, whose darkening due to the effect of light had been demonstrated by Johann Heinrich Schulze (1687-1744) in 1727. In this way, he realized the chemical effect of radiation increased in moving from red to violet on the spectrum, reaching a maximum beyond violet in a region where light was no longer visible to the naked eye: thus, was discovered a new type of radiation present in sunlight, which was called ultraviolet because it was more refracted by the prism than violet light. Young repeated Ritter’s experiments more carefully and also measured intensities, while William Wollaston confirmed the results using a solution of gutta-percha whose yellow colour changed to green by the action of the light. Numerous other studies followed, conducted by De Saussure and Marc-Auguste Pictet, Gay-Lussac and Louis-Jacques Thénard, Thomas Seebeck and Jacques-Étienne Bérard, all of whom gave some contribution to the understanding of the phenomenon being examined, leading to its most popular application of the 19th century, photography, which we will discuss in § 8.10. Connected to the study of ultraviolet radiation was a renewal of research into fluorescence by Stokes (§ 1.14), one of the most well-known British physicists (and mathematicians) of the 19th century. He attempted to study the phenomenon that John Herschel had discovered and called epipolic dispersion in 1845, in which a solution of quinine sulphate shines blue after being illuminated. Stokes soon realized that the effect did not fit within the Newtonian framework of prismatic colours, as the bright blue

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fluorescence was observed in quinine even when the incident light did not contain any blue. In a historic 1852 paper, Stokes demonstrated that the colour exhibited by the solution was mainly caused by the ultraviolet radiation incident upon it, deducing his well-known law: “the refrangibility of the incident light is a superior limit to the refrangibility of the component parts of the dispersed light.”20. This formulation coincides with the modern statement of the law provided that one takes into account that Stokes called the phenomenon dispersive reflection, adding in a footnote: “I confess I do not like this term. I am almost inclined to coin a word, and call the appearance fluorescence from fluor-spar, as the analogous term opalescence is derived from the name of a mineral.”21 In a later paper Stokes further justified his choice as follows: “I intend to use the term fluorescence, which is a singular word not implying the adoption of any theory.”22 Indeed, both his first proposed nomenclature and that of Herschel were based on the hypothesis that the secondary light originated from a superficial layer of the body struck by the primary light. A truly important step forward for research on invisible radiation was made by Macedonio Melloni, born in Parma on 11 April 1798 and died in Portici (Napoli) on 13 August 1854. Melloni, one of Italy’s greatest experimenters, dedicated himself to the study of “radiating heat” by using an instrument that was much more precise than the common thermometers of the time: the thermomultiplier, which was composed of a thermoelectric battery (§ 3.11) coupled to a Nobili galvanometer (§ 3.6). The latter was the most sensitive component of the experimental device, and for a century had been called the “Melloni table”. With the support of Arago, he conducted his critical experiments in Paris, where he took refuge from 1831 to 1839 for political reasons, having praised the Parisian students who had taken part in the 1830 revolution. After having re-examined and corrected the results found by previous physicists in the study of radiating heat, he began his personal research with the study of the absorption of radiated heat by bodies, discovering that rock salt is highly transparent to heat and therefore particularly apt for building prisms and lenses to study infrared radiation. Furthermore, he demonstrated the variation in refrangibility of heat rays, which scientists still denied, and “chemical” (ultraviolet) rays; he showed that radiated heat is polarised; and with an ingenious experiment that today is attributed to John Tyndall (1820-1893), he demonstrated the intensity of radiated heat 20 G. G. Stokes, On the Change of Refrangibility of Light,in “Philosophical Transactions”, 142, 1852, p. 556. 21 Ibid., p. 479. 22 Same title as the previous paper, ibid., 143, 1853, p. 387.

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in air falls like the inverse square of distance. The experiment, which Tyndall himself attributed to Melloni, consisted in showing that if one places the reflector of a thermoelectric battery in front of a radiating wall such that its axis is perpendicular to it and then moves the reflector further or closer to the wall, then the intensity of the generated electric current remains the same.23 Matteucci had already shown that interference phenomena were present for heat rays in 1833, followed by James David Forbes (18091868), who obtained them with Fresnel’s two mirror apparatus. In 1842, Melloni published a seminal paper in Naples, where he was employed at the Academy of Arts and Trades, until he had to abandon his position, once again for political reasons. The brief 47 pages article outlined the idea that radiated heat, light, and chemical (ultraviolet) rays were analogous forms of radiation, differing only in wavelength: this was one of the most important scientific achievements of the time and gave impetus to the unifying theories that characterized late 19th century physics. In a paper published the following year, Melloni showed that the absorption of infrared radiation has the same mechanism as the absorption of visible rays, and that much like bodies can be translucent or opaque to light at different thicknesses, they can also be “diathermic” or “athermic” to heat, to use the terminology that he introduced despite some criticism. Heat, like light, is selectively absorbed by bodies, therefore a translucent body is not always diathermic: glass, for instance, absorbs heat but does not absorb much light. These phenomena and the variation in refrangibility of heat radiation allowed Melloni to speak evocatively of the colour of heat, or “thermochromism”. Later, in 1845, Melloni also showed that heat radiation is not only a surface phenomenon, but receives contributions from internal layers (of variable thickness) of the radiating body. In La thermocrôse ou la coloration calorifique, published in Napoli in 1850 (and republished in Bologna in 1954 as the first volume of his collected works), Melloni expounded his theory of radiating heat and his classic experiments in an organized and captivating manner. After an autobiographical introduction, Melloni first describes the instruments he used to measure thermal radiation and heat sources, then passes to the experimental study of thermal radiation in vacuum and air, and finally treats the propagation of the radiation through various substances. La thermocrôse was a classic work that set off numerous studies on the emissive and absorptive potential of objects (in particular lampblack, forerunning the concept of a blackbody) and demonstrated that the laws 23

J. Tyndall, Heat a Mode of Motion, London 1868, pp. 268-70.

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governing optical phenomena are identical to the analogous ones for thermal radiation. Melloni’s studies were carried on by Tyndall, specifically for absorption in gases. Tyndall showed that dry air absorbs little thermal radiation and, after a lengthy scientific dispute with Heinrich Gustav Magnus, he demonstrated in 1881 that heat rays are strongly absorbed by water vapour, an effect of great importance to meteorology. Much like the thermoelectric battery had enabled Melloni to make his fundamental discoveries, a new and more sensitive thermometer, the bolometer, allowed for further significant progress in the study of radiating energy. This new device was invented by the American physicist and aviation pioneer Samuel Pierpont Langley (1834-1906) in 1881: it consisted of a heat-sensitive component, a thin strip of platinum foil covered in lampblack, inserted in an electric circuit. When radiation was incident on the strip its temperature would increase, thus increasing its electrical resistance; the change in temperature could then be deduced from the change in resistance. The bolometer is an instrument of incredible precision: it can measure temperature variations of up to a millionth of a degree Celsius. With his new thermometer, Langley made several important discoveries: he showed that the maximum intensity of the solar radiation was in the orange region of the spectrum and not in the infrared as it had been believed; that infrared radiation can pass through the atmosphere with relative ease; and that the quantity of energy necessary to produce a visual effect varies with the light’s colour. Lastly, Langley measured very long wavelengths coming from terrestrial sources, up to 5 hundredths of a millimetre.

1.8 Spectral analysis The study of scattering and the construction of achromatic lenses, begun by Dollond, was furthered by Joseph von Fraunhofer (1787-1826), who was both unusually skilled with his hands and in theoretical abilities. Eugenius Lommel (1837-1899), in a foreword to Fraunhofer’s works, summarized his contributions to practical optics as follows: “With his inventions of new and improved methods, of mechanisms and instruments of measure to rotate and clean lenses … he was able to produce pieces of flint and crown of rather large dimensions without streaks; and, in particular, with his discovery of a method to exactly determine the shape of lenses he gave an entirely new direction to practical optics and brought the achromatic telescope to an unexpected level of perfection.”24 24

J. von Fraunhofer, Gesammelte Schriften, München 1888, p. VII

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The son of a master glassmaker and orphaned at a young age, Fraunhofer went to Munich at the age of twelve to work in a mirror factory, where he learned the art of working glass. Having miraculously escaped the collapse of the factory, the young boy attracted the attention and sympathy of a nobleman who provided him with money and books. His scientific work began soon after. To make exact measurements of scattering in prisms, he chose a candle or an oil lamp as a light source and found a light-yellow line in the resultant spectrum that today is known as the sodium line. He quickly realized that this line was always in the same place and thus was very convenient for exact measurements of indices of refraction. After this, Fraunhofer recounts in his first paper published in 1815: “I wanted to find out whether in the colour-image (that is, spectrum) of sunlight, a similar bright stripe was to be seen, as in the colour-image of lamp light. But instead of this, I found with the telescope almost countless strong and weak vertical lines, which however are darker than the remaining part of the colour image; some seem to be completely black.”25 The lines of the solar spectrum had been discovered in 1802 by Wollaston, who had directly observed a fissure illuminated by sunlight in a dark room through a prism. He saw seven lines, of which five were pronounced. Considering them simply as lines separating the different colours of the spectrum, Wollaston left the matter there. Fraunhofer, on the other hand, after careful measurements of the positions of the lines and many variations in experimental conditions, reached the conclusion that “these lines are related to the nature of sunlight and do not arise from diffraction or experimental illusion.” He discovered hundreds of lines (576, to be precise) and carefully studied each of them, distinguishing the brightest with uppercase and lowercase letters (A, B, … Z, a, b, …). Noticing their constant position in the spectrum, Fraunhofer realized the importance of the spectral lines for measurements of index of refraction, as they could function as fixed reference points. In particular, he recognized that the D line in the solar spectrum was in the same position as the bright yellow sodium line from the lamp. His spectroscope was made up of a collimator, a prism, and a telescope, a basic structure that still lies underneath the more complicated modern versions; turning it towards Venus, he found that the spectrum of the light from the planet exhibited the same lines as the solar spectrum, when he examined the spectrum of electric sparks, on the other hand, he found numerous light lines.

25

Ibid., p. 10.

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We add in passing that the ability to conduct exact measurements of index of refraction and the construction of high-quality lenses without streaks allowed scientists to build powerful astronomical telescopes, which made possible the discovery of the annual parallax of Vega (27 light years away from the Earth) in 1837 and, the following year, through an optical instrument devised by Fraunhofer (the heliometer), the parallax of a star in the Cygnus constellation (61 Cygni): thus the problem posed in Copernicus’ time and discussed for centuries had been solved. In the last years of his life, Fraunhofer dedicated himself to the study of diffraction. The popularization of diffraction gratings is due to him, although they had already been used a century earlier by Claude Deschales (1621-1678), who had repeated Grimaldi’s experiments with sheets of pulverized metal on which he traced a series of parallel lines. When Deschales shined a thin beam of light on these sheets in a dark room, he obtained a spectrum of lines on a white screen; the same result was obtained for a striated glass sheet. Fraunhofer made his gratings with very thin parallel wires, and later by tracing parallel lines on a glass sheet with a diamond tip. The making of gratings requires significant skill – at least four lines per millimetre are needed to obtain spectra. Fraunhofer was able to build gratings with up to 300 lines per millimetre, a result that was later surpassed by the American physicist Henry Rowland (1848-1901), who built gratings with 800 lines per millimetre. Diffraction gratings were studied from the theoretical point of view by Ottaviano Fabrizio Mossotti (1791-1863), the most important Italian mathematical physicist of the first half of the 19th century, who suggested that they could be used for simple and precise measurements of wavelength. This is the main application of diffraction gratings today, along with obtaining a pure, or normal, spectrum, in which violet is less deviated than red, contrary to what occurs in the spectrum of a prism. Fraunhofer’s experiments on emission spectra were continued in Britain by Brewster, John Herschel, and William Fox Talbot (1800-1877). The latter, after numerous experiments in which an alcohol flame had melted various salts, concluded in 1834: “When in the spectrum of a flame there appear certain lines, these lines are characteristic of the metal contained in the flame.”26 The following year, Charles Wheatstone (18021875), experimenting on the spectrum of an electric arc, came to the conclusion that the lines depended only on the nature of the electrodes and not the gas that sets off the spark. In 1855, the Swedish physicist Anders 26 “The Philosophical Magazine and Journal of Science”, 3rd series, 4, 1834, p. 114.

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Jones Ångström (1814-1874) demonstrated that with sufficiently low gas pressure, the influence of the electrodes was eliminated and the spectrum was only that of the illuminated gas. The fruitful meeting between Heinrich Geissler (1814-1879), a technician who built physical machines, and Julius Plücker (1801-1868), a German physicist and mathematician, led to the almost simultaneous (1855) construction of Geissler tubes and Plücker tubes, which were particularly suited for the study of gas spectra. A few years earlier, William Allen Miller (1817-1870), continuing experiments started by Herschel, had studied the solar spectrum after making sunlight pass through different kinds of vapour (iodine, bromine, etc.) and had observed dark spectral line: he concluded in 1845 that these were absorption lines and that they were only present in coloured vapours. The latter conclusion was disputed by Pierre Janssen (1824-1907), a French astronomer who was famous for his astrophysical research, when he also observed absorption lines in the spectrum of water vapour. Following a long discussion on the interpretation of his observation of these lines, it was finally recognized that they were indeed caused by absorption. Related to spectroscopic studies was the introduction of the direct vision prism, which was built by Amici around 1860 on the request of his old student Giovanni Battista Donati (1826-1873), a pioneer of astrophysics and stellar spectroscopy who had grown tired of having to observe stellar spectra through an ordinary prism, which forces the observer to stand obliquely to the telescope axis. The new prism built by Amici was composed of two crown prisms with a glass flint prism between them: the angles of the prism were calculated to be such that they would disperse an incident ray of light without deviating it. In reality, this apparatus had already been imagined by Dollond (27), but it had been completely forgotten, while Amici’s reinvention is still in use today. We add in passing that another prism is also named after Amici, the so-called roof prism, formed by a right isosceles prism whose hypotenuse is replaced with two perpendicular faces (making up the roof): it is used to rectify the flipped images that come from the objectives of optical instruments. We will further discuss Amici and his contributions to microscopy in § 1.11. Let us return to spectra. The first relationship between emission and absorption spectra was highlighted by Foucault in 1849, who observed that the spectrum of an electric arc with carbon electrodes contained many 27

§ 7.13 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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bright lines, among which the sodium D line was particularly noticeable. Yet passing a beam of sunlight through the arc and observing its spectrum, the D line became dark: he concluded that an arc that emits the D line also absorbs it when the radiation comes from another source. This observation was the subject of various claims of priority, as it had been also made by Ångström, Stokes, Talbot, and other physicists. The numerous claims, all of which occurred after the importance of spectrum inversion had been recognized, reflect the fact that discovery was in the air, but the real founders of spectral analysis were the Germans Gustav Kirchhoff (1824-1887) and Robert Bunsen. The experimental work of the two scientists between 1859 and 1862 was greatly facilitated by the use of a humble device, the “Bunsen burner”, that Bunsen and the Englishman Henry Roscoe (1833-1915) had invented in 1857 for their innovative research in photochemistry. This appliance provides a hightemperature, low-intensity flame that can vaporize chemical substances, obtaining their spectrum without contamination from the flame’s own lines, which had often been the cause of error in previous experiments. In 1859, Kirchhoff and Bunsen published their first experimental paper, and the following year Kirchhoff reached the conclusion, supported by experimental as well as thermodynamic considerations, that each gas absorbs exactly the same radiation that it can emit – today this is known as the principle of “spectrum inversion” or Kirchoff’s principle, in § 6.2 we will see how he later applied this to the problem of blackbody radiation. Based on their experiments and those of others, the two scientists considered Talbot’s principle that each line in an emission spectrum is characteristic of the element that has emitted it adequately confirmed. Armed with these two principles, they began a series of terrestrial spectral analyses that led them to the discovery of rubidium and caesium in 1861, two metals that they named after the colours of the spectral lines that had allowed for their discovery, respectively red and light blue. That same year William Crookes discovered thallium, in 1863 Ferdinand Reich and Hieronymus Richter discovered indium, and later helium (1886), gallium (1875), and so forth. Kirchhoff also applied spectral analysis to light from celestial bodies, and his explanation of Fraunhofer’s lines as the absorption spectrum of the solar atmosphere (the spectrum of the Earth’s atmosphere was easily distinguishable) marks an important date in the history of physics, the birth of astrophysics. After much work, in 1868 Ångström published what would become a reference book for spectroscopists for many years, in which he provided measured values for the wavelengths of almost two hundred lines of the solar spectrum. The contribution of Ångström’s text

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to the study of spectroscopy was so significant that in 1905 his name was adopted as the international spectroscopic unit of measure: 1 ångström (Å) = 10-7 mm. The first studies of stellar spectroscopy quickly became so successful that Helmholtz wrote that astrophysics had excited and provoked wonder like no other discovery had before, because it had allowed humanity to peer into worlds that had seemed closed off forever. One “peers into” these worlds by comparing the lines in the absorption spectrum of a star with the lines attributed to known elements on Earth, thus deducing what elements are present in the star. From this comparison, Kirchhoff affirmed that the solar atmosphere contained sodium, iron, magnesium, copper, zinc, boron, and nickel. Subsequent investigations led to the much more general conclusion that the elements that exist on Earth are also widespread everywhere else: in short, the universe contains the same kind of matter everywhere. The Moon landings beginning in 1969, which permitted the analysis of samples of lunar matter brought back to Earth, fully confirmed the beliefs of 19th century scientists, at least for our satellite. After Kirchhoff and Bunsen, physicists obtained the spectra of all known elements, measuring the wavelength of the various lines and their relative intensities. The spectrum of elements like iron and neon is truly a sight to behold for the complexity, variety, colour, and intensity of the lines, with a display that rivals the starry sky. Much like stars, the lines also appear randomly distributed; and like astronomer catalogue thousands of stars, recording the identifying characteristics of each, spectroscopists catalogue thousands of lines, labelling each with a wavelength, intensity, and the experimental conditions necessary to obtain it. The use of spectral analysis was limited only by the complexity and variety of spectra, which increased further when at the end of the 19th century it was discovered that many spectral lines were actually split into numerous thinner lines, together making up the “fine structure” of the spectrum.

OPTICAL INSTRUMENTS 1.9 Photometry Alongside theoretical optics, instrumental optics also blossomed in the 19th century. The key factors that led to this progress were the development of technology, mathematical advances, and the evolution of the theory. We have already mentioned a few examples: interferometers and polarimeters, rotating mirrors and spectroscopes, actinometers and gratings. In this

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section we present a few other examples of important optical instruments that were developed in this time period. There were many photometer designs proposed in the course of the century, most based on the inverse square law and only applicable to the visible portion of the spectrum because the measurement was done by eye. A significant improvement over these types of photometers was created between 1889 and 1892 by Otto Lummer (1860-1925) and Eugenius Brodhun (1860-1938), who replaced the stearin ring in Bunsen’s photometer with a double glass prism (two prisms, of which one is totally reflecting, glued to one another along a circular region): from one of the two sources only the light that passes through the circular region is observed, while from the other only the light that is totally reflected is observed. In 1893, the American Agden Nicolas Rood (1831-1902) described a modification of traditional photometers that consisted in illuminating the two surfaces by the sources not simultaneously, but rather with alternating illuminations in very short time intervals. If the surfaces are illuminated in the same way, the alternation is not discernible to the naked eye, but if the sources illuminate the surfaces differently, then an observer sees a flickering effect. In 1833 Arago proposed the first type of photometer that did not use the inverse square law. The device took advantage of Malus’ law of intensity of polarised light and used birefringent prisms as polarisers and analysers. This kind of polarized photometer was chiefly used in astronomy and spectrophotometry. Physical photometers are instruments in which the human eye is replaced with a non-biological device that can detect radiation (thermoelectric battery, bolometer, thermoelectric cell, etc.), and were introduced towards the end of the 19th century. Through much discussion, scientists established the conditions and limits within which physical photometers could act as a replacement for the human eye. The continuous improvement in instruments necessitated a more exact definition of measured values and photometric samples. As it is known, it is enough to establish one photometric standard to determine the others. Scientists chose luminous intensity; the ancient Carcel lamp28 was replaced in 1877 with the pentane lamp, and later (1884) with the amyl acetate lamp: both are free-flame lamps. These standards of measurement were replaced by the violle unit, proposed by the French physicist Jules Violle (1841-1923) in 1881 and 28

§ 7.15 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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defined as the luminous intensity in one square centimetre of pure platinum at its melting point. A series of experimental studies, however, brought to light the inconvenience of this unit and the difficulties in measuring it, so in 1909 an international agreement set the international candela (1 candela = 1/20 violle) as the standard unit. The prototype was a series of special electric light bulbs with a carbon filament. Unusually, the prototype did not remain constant in time but deteriorated with use. Aside from this troublesome detail, the definition and measurement of photometric quantities encountered more and more difficulties as spectral analysis progressed, physiological problems became more complex, and thermodynamic laws were extended to light phenomena. It was precisely the laws of thermodynamics that 9th General Conference for Weights and Measures used to defined the new unit, whose primary prototype is the Burgess blackbody, an oven kept at the melting point of platinum (2043 K) in which a 1 cm2 aperture is cut: the light emitted perpendicularly to the plane of the aperture is given a value of 60 candelas.

1.10 The camera Ritter’s discovery (§ 1.7) had raised the question of reproducing images by means of light. This issue was studied as early as 1802 by Thomas Wedgwood and Humphry Davy, who however did not find any significant results. The introduction of lithography, patented in 1799 and detailed in an 1819 volume by its inventor Aloys Senefelder (1771-1834), had rekindled interest in the problem and spurred two French physicists, Joseph-Nicéphore Niépce (1765-1833) and Jacomus-Mende Daguerre (1787-1851) to dedicate themselves to realizing the coveted invention. After many years of unsuccessful attempts, in 1826 Niépce covered tin foils with Bitumen of Judea (an asphalt found on the shores of the Dead Sea which transforms when struck by light, as alchemists were already aware) and succeeded in fixing camera obscura images (after ten hours of exposure!), immersing the sheet in a solution, that melted the bitumen only in the areas not exposed to light. In 1827 Niépce met up with the painter and stage designer Daguerre, who for years had also pursued the same goal. Following Niépce’s trail, in 1829 Daguerre was finally able to substitute Bitumen of Judea with iodinated silver or copper, on which light formed an invisible image that Daguerre then revealed with mercury vapours and fixed with a solution of table salt: thus, was born the “daguerrotype”, as this difficult to conserve first form of photograph was called. After many unsuccessful attempts to profit economically from their invention, in 1839, on the recommendation

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of an enthusiastic Arago, Daguerre and Niépce’s son ceded their patent to the French government, in exchange for a lifetime pension. Once it entered the public domain, improvements of photographic instruments and techniques quickly followed: from objectives build from a simple achromatic lens it passed to objectives that were double, orthoscopic, triple, aplanar, angular, and astigmatic. The main drawback of the daguerrotype was that it was not reproducible; for each exposure, the instrument gave only one positive copy. Free from this limitation was the procedure devised by the British inventor Talbot, who had worked on the problem independently of Daguerre and around the same time he communicated to the Royal Society in 1839 his invention. Talbot used paper imbued with silver iodide as the photosensitive agent, revealed the latent image with gallic acid, and fixed it with potassium iodide and sodium hyposulphite. This process was modified in 1847 by Blanquart Evrard, who from a single negative phototype on paper obtained up to forty positive copies after development with gallic acid. That same year, Claude-Felix Niépce de Saint-Victor obtained a photographic negative on a sheet of glass covered in albumin. As often happens, especially in technical applications, progress generates appetite for more progress. Once black and white photographs had been achieved, scientists aimed for colour photography. The first attempts date back to 1844, when Alexandre-Edmond Becquerel succeeded in reproducing, but not fixing on a Daguerrotype, the principal colours of the spectrum. Many other attempts followed, none giving practical results. In 1891 the French physicist Gabriel Lippmann (1845-1921) obtained photographs of the solar spectrum using interference. While his technique was not very successful in terms of practical applications, it had enough scientific merit to earn him the 1908 Nobel prize in physics, as the scientific community had long yearned for colour photography. Only with the autochrome plates created by the brothers Louis-Jean and Auguste Lumière in 1907, however, did colour photography become practically possible and commercially successful. We would risk falling into adulation if we attempted to highlight the benefits that scientific research received from photography. Therefore, we direct the reader to more specialized accounts for further information on the history of photography, and limit ourselves to simply listing its first scientific applications. In 1840, the American chemist John William Draper (1811-1882) photographed the Moon, and later in 1842 the solar spectrum, finding Fraunhofer’s lines even in the ultraviolet section. That same year, for the first time, Alessandro Majocchi (1795-1854) photographed the Sun, which was also photographed by Foucault and Fizeau in 1845.

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The Moon was photographed again by William Bond (1789-1859) in 1850, and the phases of a solar eclipse by Warren De La Rue (1815-1889) in 1860.

1.11 The microscope The world’s best compound microscopes around 1800 were built in the Netherlands. Such microscopes had a magnifying power of 150 and 0.005 mm of resolving power: they were still weaker, therefore, than the simple microscope, which consequently had remained practically the only instrument used in scientific research. In 1830, the Englishman Joseph Lister (1786-1869), a self-taught wine merchant, discovered after years of experimental studies that a spherical lens contains a pair of matching aplanatic points. He used this discovery to build a compound microscope made up of multiple lenses that were opportunely chosen and organized. Lister’s microscope, much improved over its predecessors and immediately made commercially available, convinced many experimenters that the compound microscope could compete with the simple microscope and perhaps even surpass it. Around 183629, the astronomer and biologist Amici, also a well-known builder of optical precision instruments, attempted to improve the performance of the Lister compound microscope by bringing the sample closer to the first objective lens. To achieve this, he had the clever idea to place a hemispherical lens over the objective of the microscope to correct its chromatic aberration with consecutive achromatic doublets. Amici’s microscope had a magnifying power of 2000 (one model even reached 6000) and 0.001 mm of resolving power, a marked improvement over the simple microscope. Soon, however, Amici realized that to improve the performance of the microscope one not only had to increase the magnification but also the separating power, which depends on the angular aperture of the objective. To this end, in 1847 he injected a strongly refringent liquid between the central hemispherical lens and the sample, thus beginning the technique of immersive microscopy that is still in use today. The theoretical studies of Lord Rayleigh led to the development of a consistent theory of the compound microscope, which was further 29 For many other details relating to the development of microscopy in the first half of the 19th century see V. Ronchi, Sul contributo di Giovan Battista Amici al progresso della microscopia ottica, in “Atti della Fondazione Giorgio Ronchi”, 24, 1969, pp. 200-15.

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improved by the German physicist Ernst Abbe. Spurred to study microscopy by the optical manufacturer Carl Zeiss, Abbe introduced the idea of a numerical aperture and the homogeneous immersion technique. At the beginning of the 20th century the ultramicroscope seemed to have almost reached the limit of magnifying power, but after a few decades the electron microscope (§ 8.7) pushed this limit unimaginably further.

2. THERMODYNAMICS

THERMAL BEHAVIOUR OF BODIES 2.1 Thermal expansion Eighteenth century experimental studies on thermal expansion had brought about a general confusion that would last almost until the second half of the 19th century. For instance, it was said (simplifying a result found by De Luc30) that “mercury expands uniformly”, but without specifying the sample or the thermometric scale with which the dilation was measured. Scientists tacitly assumed that the mercury in question was always the same sample, because temperature changes were taken to be equal for equal expansions of mercury. With these assumptions, however, saying that mercury expands “uniformly” is devoid of meaning, much like saying that the apparent motion of fixed stars is “uniform” is meaningless when that same motion is used to define equal intervals of time. Defining uniform dilation was thus tantamount to circular reasoning. Nevertheless, if one did not quibble too much over details, things went fairly well from the practical point of view, as the thermometers of the time gave sufficiently comparable measurements. Fourier, for example, defined: “The temperature of a body whose parts are equally heated and that conserves its own heat is the one indicated by the thermometer.”31 To this working definition Fourier added another: letting 0 be the temperature at which ice melts, 1 the temperature of boiling water, C0 the quantity of heat needed to bring the thermometric liquid from the temperature 0 to the temperature 1, and ǻ the corresponding variation in volume, then any temperature is given by the ratio

z

30

ఋ

஼

ο

஼బ

= =

§ 7.17 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 31 J.-B.-J. Fourier, Théorie analytique de la chaleur, Paris 1822; later in Id., Œuvres, edited by G. Darboux, Paris 1888, Vol. I, p. 16.

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where C is the quantity of heat needed to obtain the temperature z and į is the corresponding change in volume. In other words, Fourier postulated that temperature is always proportional to the change in volume and the change in heat provided to the body, regardless of its nature. The postulate, immediately shown to be false by experimental physics, was weakened with the addendum that it did not apply to water and liquids near their critical temperature (the temperature at which they undergo a phase change). A careful comparison between the mercury thermometer and the air thermometer was carried out in 1815 by Pierre-Louis Dulong (1785-1838) and Alexis-Thérèse Petit (1791-1820). Their research was based on the postulate that two different mercury thermometers always give measurements that are in agreement. This postulate was already known to be inadmissible (perhaps not yet to the young French scientists) because as early as 1807 Angelo Bellani (1776-1852) had made a seemingly modest observation that later proved to be important: he observed that the zero in mercury thermometers could shift as a result of the glass bulb’s deformation over time, revealing the cause of many errors in thermometric measurements. In any case, assuming the postulate to be true and, furthermore, that Gay-Lussac’s experiments (§ 2.2) implied that “it is natural to conclude that the expansion of the same type of gas must be constant,”32 the two scientists reached the conclusion that if the expansion of air is taken to be uniform, then the expansion of mercury can no longer be uniform. This was a step forward, but the problem of finding a thermodynamic scale that was independent of the substance utilized in the thermometer would only be resolved in 1848 through the second principle of thermodynamics (§ 2.11). Dulong and Petit, in a later paper that deservedly earned them the physics prize of the Académie des sciences of Paris, noticed that in every mercury thermometer what was observed was only the apparent expansion of the liquid – because solid and liquid phases do not obey the same expansion law, the expansions given by the thermometer cannot be taken to be proportional to the absolute expansion of mercury. There had been no lack of attempts to measure this absolute expansion, all of which reduced to adding to the apparent dilation the unpredictable dilation of the container, obtaining results that were in stark disagreement: from the 1/50 for the fundamental temperature interval given by John Dalton to the 1/67

32

P.-L. Dulong and A.-T. Petit, Recherches sur les lois de la dilatation des solides, des liquides et des fluides et sur la mesure exacte des températures, in “Annales de chimie et de physique”, 2, 1816, p. 243.

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found by Nicolas Casbois. In their paper33, which continued the work of a previous one published two years earlier, Dulong and Petit set out to find the absolute expansion of mercury without making use of its apparent expansion. Their ingenious method has since become classic, and it is still described in physics texts today, allowing an experimenter to obtain absolute expansion with the measurements of two temperatures and two heights. The temperatures were measured with an air thermometer and a mercury thermometer, and the two heights with a special instrument, later called a cathetometer, that has become a valuable tool for determining the height difference (barometric, capillary, etc.) between two offset points. Familiarity with the absolute expansion of mercury allowed Dulong and Petit to experimentally study thermal expansion in other liquids and solids with the same methods. In general, they found that the expansion of mercury varies with temperature from 1/1550 for temperatures between 0 °C and 100 °C to 1/5300 between 100 °C and 300 °C, and that a similar variation can be observed for the other solids tested (glass, iron, copper, platinum). Other studies following in Dulong and Petit’s footsteps showed the same trend for the thermal expansion of liquids. Both in solids and liquids, however, experimenters discovered striking anomalies near phase transitions. These observations made it necessary to define, for each solid and liquid, a coefficient of thermal expansion that theoretically varies with temperature and in practice is (approximately) constant in restricted temperature intervals. It follows that it was also necessary, as Friedrich Wilhelm Bessel (1784-1846) indicated in 1826, to correct the temperatures used in determining specific weights and create a systematic table of corrections for barometric readings: the first such table was produced by Carl Ludwig Winckler in 1820. A new detail regarding the thermal expansion of solids was discovered by Eilhard Mitscherlich (1794-1863) in 1825: crystals, except for monoclinic systems, expand unequally in different directions, and thus change in shape with changes in temperature. The effect was corroborated by Fresnel and thoroughly studied in several papers by Fizeau, ranging from 1865 to 1869, who used a highly refined method based on Newton rings to determine the change in thickness of a layer of air between two surfaces. To this end, one of the faces of the crystal being examined was made slightly convex, a plane-convex lens was placed on top of it, and using a telescope one observed the Newton rings produced by reflection when monochromatic light was shined on the system. Appropriately 33

P.-L. Dulong and A.-T. Petit, Recherches sur la mesure des températures et sur les lois de la communication de la chaleur, in “Annales de chimie er de physique”, 7, 1818, pp. 113-54, 225-64 and 337-67.

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heating the crystal, one then observes a change in the rings from which the change in thickness of the air can be deduced. This method also lends itself to the study of non-crystals, and is so precise that the International Commission for the Metre employed it to study the metal bars of the international prototype metre. With this approach, Fizeau discovered that, in addition to water, other bodies (diamond, emerald, etc.) also have a density maximum and that silver iodide contracts in the temperature interval between -10 °C and 70 °C. The Accademia del Cimento had already recognized the fact that water has a point of maximum density, but the phenomenon was denied by Hooke, accepted by others, and had led De Luc in 1772 to conduct systematic experiments on the irregularities in the dilation of water. De Luc found that water reaches its maximum density at 41 °F and that when it is heated from 32 to 41 degrees Fahrenheit it expands as much as it does when it is heated from 41 to 50 degrees. The same experiments were repeated in 1804 by Benjamin Thompson, the count of Rumford, and in 1805 by Thomas Hope (1766-1844), and subsequently continued for the remainder of the century. In 1868 Francesco Rossetti (1833-1881) placed the density maximum between 4.04 and 4.07 °C, while in 1892 Carl Scheel (1866-1936) placed it at 3.960 °C and, the following year, Pierre Chapuis at 3.98 °C. The 4 °C temperature value referenced in physics texts thus represents a rounded value by convention.

2.2 Thermal dilation of gaseous bodies Amonton’s studies on the thermal expansion of air34 were continued by many other physicists in the course of the 18th century (De la Hire, Stancari, Haukance, De Saussure, De Luc, Lambert, Monge, Claude-Louis Berthollet, Vendermonde, and others), but their conclusions displayed a disheartening lack of agreement. Some held that the expansion was uniform, others that it was variable, and all of the definitional problems mentioned above were still present. An important paper written by Volta in 1793 brought to light that even among the supporters of the first view there was ample disagreement regarding the values measured by different experimenters for the expansion caused by a heating of one degree Réaumur: such values ranged from the 1/85 found by Priestley to the 1/235 of De Saussure.

34 §§ 7.16 and 7.18 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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Fig. 2.1 - The air thermometer, open at point C, is filled to the 100 mark with boiled oil and is immersed in water in the glass container D. Changing the temperature of the bath, measured by the mercury thermometer b, Volta studied the expansion of the air. Source: Volta, Le opere cit., Vol. 7.

The lengthy title of Volta’s article expresses the important conclusion that he reached: On the uniform dilation of air for each degree of heat, starting below from the temperature of ice and reaching above that of boiling water; and on what often makes such expansion appear not equitable, causing an increase out of all proportion in the volume of the air.35 Volta demonstrated, as Vittorio Francesco Stancari (1678-1709) had already suspected in the first years of the 18th century, that the disagreement in the experimental results was due to the fact that the previous experimenters did not work with dry air but rather with humid air, and therefore that the presence of water vapour affected the phenomenon. He 35

Volta, Le opere cit., Vol. 7, p. 345. The article is completely reprinted, with introduction and comments, in Id., Opere scelte cit., pp. 307-44.

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used a very simple device (Fig. 2.1) and had the clever idea to separate the volume of air being examined with a column of pre-emptively boiled linseed or olive oil. After numerous careful experiments, along with counter-evidence gathered by using humid air, Volta declared: “For each degree of the Réaumur thermometer, confined air experiences an increase of about 1/216 in the volume that it has at zero temperature. It experiences an increase in volume equal to the former, both a little over the temperature of ice and when it approaches the boiling point of water.”36 The coefficient found by Volta was thus equal to 1/270 = 0.0037037 per degree Celsius. His paper, however, was published in Luigi Brugnatelli’s (1761-1818) “Annali di chimica”, a journal with a very limited circulation, and was consequently unheard of in scientific circles. Nor did Volta attempt to spread his ideas himself, in part impeded by the little time left over from his public spat with Galvani, and in part because of the slowing down of his scientific production in those turbulent political times (the French revolution and Napoleon’s takeover). Nevertheless, Volta’s conclusion was essentially correct, though his reasoning was flawed, because from his experiments he should have concluded that the changes in air volume were proportional to the quantities of heat added to the mercury thermometer and not to the air itself. Joseph-Louis Gay-Lussac (1778-1850) certainly did not know of Volta’s paper when he also studied the thermal expansion of gases in a now-classic 1803 paper: in the meticulous historical review at the beginning of the paper he does not mention Volta. However, Volta’s work was known to his teacher Claude-Louis Berthollet (1748-1822), who, as Gay-Lussac writes, directed the experimental part of the work. Indeed, on 30 September 1801, in his Paris laboratory, Berthollet agreed “with Volta on the uniformity of gas expansion totalling about 1/214th its volume for each degree of the Réaumur scale,”37 as Brugnatelli attests in the diary of his 1801 journey to Paris with Volta. In any case, from the historical introduction one learns that 15 years earlier Jacques-Alexandre Charles had conducted experimental studies on the same subject, but without publishing anything. Charles was well-known in his time for his spectacular experiments along the lines of those conducted by Abbé Nollet: the most famous was the 1783 launch in Paris of the first hydrogen-filled balloon (the gas had been newly discovered in 1776 by Cavendish), unlike the hot air balloons launched by the brothers JacquesÉtienne and Joseph-Michel Montgolfier, also in 1783. 36 37

Volta, Le opere cit., Vol. 7, p. 370. Volta, Epistolario, cit., p. 487.

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According to what Gay-Lussac writes, Charles found that oxygen, nitrogen, carbon dioxide, and air expand equally between 0 °C and 100 °C. Gay-Lussac extended and completed the work of Charles, arriving at the central theorem: “If one divides total increase in volume by the number of degrees that produced it, one finds, setting the volume at temperature zero equal to unity, that the increase in volume for each degree is of 1/213.33, or 1/266.66 for each degree centigrade.”38 At its heart, therefore, this study was different from the one that Volta had already conducted. While he had demonstrated that the expansion of air is uniform (with respect to the expansion of mercury), Gay-Lussac had instead shown that all gases have equal total expansion between 0 °C and 100 °C and, supposing it uniform, calculated the coefficient of proportionality for all gases. Later, Gay-Lussac realized that his assumption of uniformity was unfounded, and thus proceeded to demonstrate its validity in a series of experiments, which were made famous by Biot in the 1816 edition of his physics textbook, who described them along with the apparatuses they employed. The international physics conference that convened in Como in September 1927 for the centennial of Volta’s death voted to adopt two gas expansion laws to be taught in physics courses – Volta’s law, that the coefficient of isobaric expansion of air is constant; and Gay-Lussac’s law, that all gases have the same coefficient of isobaric expansion. The proposal to remember Volta’s name alongside Gay-Lussac’s, however, was largely ignored by writers and educators. Perhaps it would have been wiser to simply call the law of isobaric expansion of gases the “law of Volta and Gay-Lussac”, though, curiously, it had not been rigorously demonstrated by either, as they had both been tangled up in the confusion of the time. Fortunate circumstances and the numerical values given by the two scientists, which nearly coincided with the true value of the isobaric expansion coefficient of air, allowed the two works to be identified as a single investigation into the behaviour of gases heated at constant pressure. The coefficient 1/266.66 = 0.00375 given by Gay-Lussac, confirmed by Biot and Dalton and accepted by Laplace, was considered one of the most undisputed values in physics for thirty-five years. In 1837, however, Friedrich Rudberg (1800-1839) performed a new measurement and found a lower coefficient than the one given by Gay-Lussac. Consequently, Magnus, attributing the disagreement to the difference in experimental 38 L. Gay-Lussca, Recherches sur la dilatation des gaz et des vapeurs, in “Annales de chimie”, 43, 1802, p. 165.

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methods, repeated Gay-Lussac’s experiments and found a coefficient in agreement with Rudberg’s. According to him, Gay-Lussac’s measurement was erroneous because to trap the mass of air being examined, the French physicist had employed mercury, which is not as well-suited as oil for airtight seals (a mistake that Volta had avoided). The same year that Magnus explained the discrepancy, Regnault published a classic paper giving 0.0036706 for the coefficient, a value that has remained more or less unchanged. One only has to compare the values obtained by Volta and Gay-Lussac with Regnault’s to notice the greater accuracy attained by the scientist from Como, despite the use of more modest equipment. In a later 1842 study Regnault, in line with the results already obtained by Magnus, found that the coefficient of expansion is not exactly equal among gases, but rather that among those that liquefy more easily the coefficient is higher, increasing, as Favy had observed, with increasing gas density. In 1847, correcting the opinion expressed by Dulong and Petit in their aforementioned 1818 paper on the exactness of Boyle’s law, Regnault showed, in an experimental range of up to 30 atmospheres, that at ordinary temperatures all gases except hydrogen are more compressed than what results from Boyle’s law, while hydrogen is less compressed. These conclusions were later confirmed and extended by other physicists (Chappuis, Rayleigh, Gino Sacerdote, and others).

2.3 Vapours Starting in 1789, for fifteen years Volta worked intensely on the behaviour of vapours, as his numerous unpublished manuscripts detail, without ever publishing a complete paper. He gave news of his research to friends –Vassalli, Marsilio Landriani, Mascheroni– and made it the subject of his university courses and talks at various academic events. It was Volta who was responsible for the observation that water at 0 °C has nonzero pressure, meaning that ice can sublimate. Based on measurements conducted at various temperatures of the pressure of the vapour produced inside a barometric leg immersed in thermal bath, Volta thought he could summarize the behaviour of vapours in three laws: the first two (if temperature increases in an arithmetic progression, then vapour pressure increases in a geometric progression; the vapour pressures of all liquids are equal at equal distances from their boiling temperature) were quickly proven false; the third says that the pressure of a vapour is the same whether it occupies empty space or a space filled with air of any density. Aside from these finding, Volta was responsible, along with the Spanish scientist Augustin de Bétancourt (1760-1826), for giving a new direction

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to vapour research: their application to powering engines. In this way, the two scientists replaced the previous empiricism and dissertations on the formation and nature of vapours with quantitative experimental studies. John Dalton (1766-1844) also independently formulated Volta’s third law in 1802, giving his name to it. Dalton deduced, from the same considerations that Volta had already made, that a theory in which evaporation was a chemical phenomenon39, meaning a combination of vapour with air, could not be correct. In 1816, Gay-Lussac extended Dalton’s law also to mixtures of vapour. On the other hand, in 1836, Magnus demonstrated that the law is valid for vapours whose liquids do not mix, for example water and oil; but for vapours whose liquids do mix, like ether and alcohol, the total pressure of the vapour mixture is less than the sum of the pressures of its components: this result was confirmed and extended by Regnault. The increasing spread of vapour-based machines had steered research to the study of the pressure of water vapour at elevated temperatures. In 1813, Johann Arzberger (1778-1825) had already conducted approximate measurements up to pressures of 8 atmospheres, well beyond the limit reached by Augustin de Bétancourt. In 1829, Dulong and Arago, commissioned by the Académie des sciences of Paris, began a systematic measurement of vapour pressures, reaching up to 24 atmospheres. Their results, like their predecessors’, were not sufficiently accurate because they had no way to guarantee uniformity in temperature throughout the vapour sample, and in consequence the measured pressure was that of the colder region, in accordance with the cold wall principle normally attributed to Watt (which in reality was stated by Felice Fontana (17301805) in 1779). The first careful measurements were performed in 1844 by the German physicist Heinrich Gustav Magnus (1802-1870). He experimented with an isolated calorimeter having three air chambers, in which he placed Ushaped tubes filled with vapour and an air thermometer. The study with the most impact, however, was the one conducted by Regnault using innovative methods and described in his historic Report on the experiments undertaken by order of the honourable minister of public works on request of the central commission for vapour machines to determine the principal laws and numerical data that a relevant in the calculation of steam engines (1847). In this comprehensive work, Regnault corrected the results obtained by Dulong and Arago and gave 39

§ 7.20 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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pressure of water vapour for temperatures between -32 °C and 100 °C and between 110 °C and 232 °C. Henri-Victor Regnault was born in Aix-la-Chapelle on 21 July 1810 and after a rough childhood entered the polytechnic school of Paris. Later a student of Justus von Liebig, he began his first research in organic chemistry in 1835, obtaining such success that he was offered a chemistry professorship at the polytechnic school of Paris in 1840, succeeding GayLussac. That year, however, his interests began to shift towards physics and in particular towards thermal measurements for primarily technical ends. Named the director of porcelain manufacture in Sèvres in 1854, he continued his research in the manufacturing laboratory, but during the Franco-Prussian war of 1870 (in which his son, the painter Henri, was killed) his laboratory and all of his manuscripts were destroyed. In 1872 his scientific activity ceased; he died on 19 January 1878. Regnault repeated the experiments of his predecessors, introduced methods of unprecedented precision, and obtained, through an admirable ability in conjunction with great patience, results that still today number among the most reliable measurements in thermodynamics. Aside from the results mentioned above, Regnault also conducted studies on thermometry, on the thermal expansion of solids and liquids, on heats of fusion and evaporation, on the measurement of specific heats, on the measurement of the speed of sound in gases, on the compressibility of water, and on thermoelectric couples. It is said that Regnault was lacking the creative genius to blaze new paths in physics, but his contributions to experimental techniques and applied physics typified an era of scientific thought.

2.4 The liquefaction of gases The first progresses in artificially obtaining low temperatures began with the refrigerated mixtures written about by Porta in his Magia in 1589. Empirical observations on the cooling caused by evaporation had been made and applied for many years, especially in the use of porous vases used to conserve cold water. The first scientific experiments, though, were conducted by Cigna and described in his 1760 paper, De frigore ex evaporatione. Cigna showed that the more an evaporation is rapid, the more pronounced is the associated cooling effect. De Mairan demonstrated that by blowing with a pair of bellows on the wet bulb of a thermometer, one discerns a greater temperature decrease than repeating the same observation with a dry bulb. Antoine Baumé (1728-1803) discovered that cooling was heightened by using the evaporation of sulphuric ether instead of water. Based on these results, Cavallo built the first ice machine in 1800

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and Wollaston his cryophorus in 1810, which is still used today and also allowed for the 1820 introduction of Daniell’s hygrometer. The ice machine became practical only after 1859, when Fernand Carré (18241894) published his method for the production of ice through the evaporation of ether, which was later replaced with ammonia. In 1871, Karl Linde (1842-1934) described his ice machine, in which cooling is obtained through the expansion of a gas. In 1896, he combined this machine with his countercurrent apparatus, described in physics texts, obtaining liquid hydrogen and bringing to the realm of industry the first experimental results that scientists had obtained in the meantime. The problem of gas liquefaction has a long history that begins in the second half of the 18th century with the liquefaction through cooling: with Van Marum using ammonia, Gaspard Monge and Jean-François Clouet using sulphur dioxide, and Thomas Northmore using chlorine in 1805. An important step forward was taken simultaneously and independently by Charles Cagniard de la Tour (1777-1859) and Faraday. The former, in a series of papers published from 1822 to 1823, described his experiments to determine whether, as it intuitively appeared, the expansion of a liquid has a limit beyond which no matter what pressure is applied, it transforms entirely into a vaporous state. To this end he placed a silica sphere in a Papin pot that was one-third filled with alcohol, and heated the ensemble gradually. From the sound that the sphere made rolling around the interior of the pot, he deduced that at a certain temperature all of the alcohol had evaporated. The experiments were repeated with small tubes that had been pre-emptively emptied of air and filled up to the 2/5th mark with a liquid (alcohol, ether, petrol essence), and subsequently heated by a flame: as the temperature increased, the liquid became increasingly mobile and the interface between the liquid and vapour ever more evanescent, until at a certain temperature this surface disappeared completely and the entire liquid was transformed into vapour. Coupling these tubes to a compressed air pressure gauge, Cagniard de la Tour succeeded in measuring the pressure and temperature in the tube at the moment in which the separation between liquid and vapour disappeared. With this method, Cagniard de la Tour was able to determine the critical temperature of several liquids, but contrary to what many have written, he was not able to measure the critical temperature of water, nor did he succeed in evaporating water completely, because the tubes would always break before the desired effect took place. More concrete results were obtained by Faraday’s 1823 experiments, which he conducted using a bent glass tube whose longer branch was closed. In this part of the tube, he placed the substance that when heated would produce the gas to be examined, closed the shorter branch as well,

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and then immersed it in a refrigerated mixture. By then heating the substance in the longer branch, Faraday produced the desired gas with increasing pressure and, in many cases, he obtained its liquefaction in the shorter branch. The tube exploded in two occasions, fortunately with no consequences for the experimenter. Heating bicarbonate in this way, Faraday obtained liquid carbon dioxide, and similarly liquid hydrogen sulphide, hydrochloric acid, sulphur dioxide, and other liquid compounds. The experiments conducted by Cagniard de la Tour and Faraday showed gases could be liquefied by subjecting them to high pressures. Physicists thus directed their efforts to this aim, in particular Johann Natterer (1821-1901). Yet certain gases (hydrogen, oxygen, nitrogen) could not be transformed into a liquid state through this approach. Faraday, still convinced that one day solid hydrogen would be obtained, believed that pressure alone was not enough to obtain the liquefaction (or solidification) of hydrogen, nitrogen, and oxygen, consequently combining high pressures with low temperatures. The temperatures he reached were low, but not low enough to obtain the desired results, though they were sufficient for liquefying other gases for the first time and even solidify a few (ammonia, nitrogen oxide, hydrogen sulphide, etc.). Nevertheless, many physicists continued to believe that high pressures were enough to overcome resistance to liquefaction in all gases. This steadfast conviction pushed Marcellin Berthelot to subject oxygen to a pressure of 780 atmospheres in 1850, but he did not obtain liquefaction, leading him to accept Faraday’s ideas. In 1845, the same year in which Faraday published the results of his experiments and expressed his opinions on the matter, Regnault observed that the compressibility of carbon dioxide showed irregularities at low temperatures and followed Boyle’s law around 100 °C, thus hypothesizing that for each gas there is a range of temperatures for which it follows Boyle’s law. In 1860 Regnault’s idea was adopted and modified by Dmitri Mendeleev (1834-1907), according to whom for all fluids there exists an “absolute boiling temperature” above which they are in a gaseous state no matter what the pressure. The study of this question in a new form was taken up in 1863 by Thomas Andrews (1813-1885). He filled a capillary tube with carbon dioxide and closed it inside a column of mercury. Pushing the mercury down with a screw, the carbon dioxide was subjected to a desired level of pressure, while at the same time the external temperature was slowly varied. Once the partial liquefaction of the gas had been obtained by only adjusting the pressure, Andrews slowly heated the sample, observing the same phenomena that Cagniard de la Tour had studied earlier. When the

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temperature of the carbon dioxide reached 30.92 °C, the interface separating the liquid and gas disappeared, and no change in pressure could bring back the liquefaction of the carbon dioxide. In an important 1869 paper, Andrews proposed to call 30.92 °C the “critical point” of carbon dioxide, and found the critical points for hydrochloric acid, ammonia, sulphuric ether, and nitrous oxide using the same technique. He also proposed to use the term “vapour” for gaseous substances at temperatures below the critical point and “gas” for substances at temperatures above it: thus, an important distinction was made for future physics research. Confirmation of Andrew’s views came from Natterer’s experiments from 1844 to 1855, in which permanent gases were subjected to pressures of 2790 atmospheres without liquefaction, and the many analogous experiments begun in 1870 by Emile Amagat (1841-1915), who reached 3000 atmospheres. These negative results supported the hypothesis advanced by Andrews according to which permanent gases remained in the gaseous state because their critical temperatures were below the temperatures that had hitherto been attained: their liquefaction could only be obtained by a significant cooling, if necessary, followed by compression. This hypothesis was brilliantly confirmed in 1877 by Luigi Cailletet (1832-1913) and Raoul Pictet (1846-1929), who, working independently, liquefied oxygen, hydrogen, nitrogen, and air through a significant pre-emptive cooling. Other physicists continued the work Cailletet and Pictet, but only Linde’s apparatus made the procedure replicable on a practical scale, allowing for the production of large quantities of liquefied gases and their widespread use in scientific research and industry.

2.5 Specific heats of gases The methods to determine specific heat40 were difficult to apply to gaseous matter because of the small specific weights of gases and vapours. For this reason, in the first years of the 19th century the Académie des sciences of Paris held a contest to determine the best method to measure the specific heats of gases. The winners were François Delaroche (17811813) and Jacques Bérard (1789-1869) in 1813, who proposed to place a coil inside the calorimeter along which one could introduce gas at a given temperature, while keeping pressure constant. In reality, this method was not new but had been proposed by Lavoisier twenty years earlier; the 40

§ 7.21 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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results obtained by the two scientists were nevertheless remembered in physics treatises for half a century. The merit of Delaroche and Bérard’s instrument was mainly that it brought attention to the need to distinguish between the specific heat of a gas at constant pressure and at constant volume. The measurement of the latter was very difficult due to the smaller heat capacity of the gas compared to that of the container in which it was held. A few years before the works of Delaroche and Bérard, however, scientists had begun to systematically study an unusual phenomenon noticed by Erasmus Darwin (1731-1802), grandfather of the Charles Darwin, in 1778 and later by Dalton in 1802. The effect is well-known today: the compression of air gives rise to heating and its expansion to cooling, a phenomenon that is particularly pronounced in fire pistons, which became prevalent in physics laboratories around 1803. In 1807, Gay-Lussac’s conducted an experiment to highlight the phenomenon that was later repeated in 1845 by James Joule, to whom it is commonly attributed. Gay-Lussac joined two spheres by their necks, as Guericke had already done41, one full of air and the other empty, and by making the air expand from the full sphere into the empty one, he measured a decrease in temperature of the first balloon and an increase in the second. This thermal behaviour of air made it clear that the specific heat at constant pressure must be greater than the specific heat at constant volume, and that this holds independently of whatever theory describes the nature of heat. Indeed, it is evident that if a gas cools when it expands, when it is heated and allowed to expand, one must provide enough heat to compensate the associated cooling and further heat it. From a knowledge of these experimental results, in 1816 Laplace had the astute idea to attribute the disaccord between the experimentally measured speed of sound propagation and the theoretical speed calculated from Newton’s law42 to the variations in temperature in air layers because of alternating compressions and rarefactions. Starting from these theoretical considerations, Laplace corrected Newton’s formula, introducing a term that is the ratio of specific heats at constant volume and pressure (for air): using the data provided by Delaroche and Bérard, Laplace calculated this ratio to be 1.5, while it resulted as 1.4254 when he compared the experimental value of the speed of sound in air, measured by French academics in 1738, with the

41

§ 5.18 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 42 § 6.9 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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theoretical value from Newton’s formula.43 In the fifth volume of his Celestial Mechanics Laplace returned (1825) to this question and provided a much-improved proof of the theorem, formulated in 1826 by Poisson, which freed it from the questionable hypotheses on the nature of heat that Laplace had introduced earlier. Gay-Lussac’s experiment, which was intended to serve a bastion for the principles of thermodynamics, instead reinforced the belief of Charles Desormes (1777-1862) and his son-in-law Nicolas Clément (1778-1841) in a fluid theory of heat, deceiving them into thinking that they had measured the total “caloric” (fluid quantity of heat) contained in space and bodies, which had already been declared a dead end by De Luc.44 According to Clément and Desormes, though, Gay-Lussac’s experiment was the definitive proof that empty space contained caloric, which is gathered by air falling through it, heating the air; while the empty space left by the moving air, imbued in its own caloric, cools. The two scientists thought of caloric as an elastic fluid whose pressure, which is proportional to the amount contained in a given space, is perceived as temperature. It follows that if Gay-Lussac had been able to measure the mass of the air that entered the empty space and its heating, he would have obtained the total caloric contained in that space. This measurement was impossible, however, both because it was impossible to obtain a perfect vacuum and because some heat was lost to the surroundings during the experiment. It was best, Desormes and Clément continued, to resort to a “partial vacuum” (the difference between the volume of a container and the volume that the air inside it would take up if it were at external pressure) and perform the experiment as rapidly as possible. The ideal apparatus for this, later used by numerous physicists and basically unchanged today, is made up of a large ball AB in which a certain amount of air is evacuated, from which the partial vacuum volume is calculated. With the rapid movement of a tap M, air is inserted into the ball until the internal pressure is equal to the external one: the tap is open for less than 2/5 of a second, and its closing is signalled by a mercury manometer connected to the balloon and the simultaneous cessation of the hiss of the air entering it. Cooling the balloon, the manometer EC indicates a decrease in pressure, from which, through Gay-Lussac’s law, one can deduce the increase in the air’s temperature and thus the caloric given off by the partially empty space. Once the total caloric of the empty space is obtained, by repeating 43

P.-S. Laplace, Sur la vitesse du son dans l’air et dans l’eau, in “Annales de chimie et de physique”, 3, 1816, p. 241. 44 § 7.22 of M.Gliozzi: A History of Physics from Antiquity to the Enlightement, Cambridge Scholars Publ. 2022.

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the experiment at two different temperatures it is then simple to calculate the specific heat of empty space, assuming that it is constant at different temperatures. Yet was it really constant? The scientists were thus led to determine the specific heat of air at constant volume. They obtained it by applying Newton’s law of cooling (the quantity of radiated heat is proportional to the difference in temperatures) to the measuring apparatus and supposing that, at constant temperature differences, the cooling times of various gases are proportional to their respective specific heats. The experimental result indicated that the specific heat capacity of air at constant volume decreases with decreasing temperature and that the specific heat at 18 °C of empty space is 0.4 times the specific heat of air at 18 °C and 763 mm of mercury of pressure. Now, assuming that the specific heat of empty space is proportional to temperature, if a region of space does not contain caloric, its temperature must be zero, and any lower temperature cannot have physical meaning: one finds, in short, absolute zero temperature. Believing to have actually found the specific heat of empty space at 18 °C, Clément and Desormes also determined its value at 98 °C; then, through a simple proportionality relation, they calculated absolute zero to be at -251.8 °C. The scientists also arrived at absolute zero by applying Gay-Lussac’s law, using the same considerations that are employed today, and found a value of 266.66 °C. The fact that such similar values had been found through such ostensibly different methods seemed to the scientists an impressive confirmation of their theory: in reality, the two methods were only apparently different, because at the heart of the calculation of the specific heat of empty space lay Gay-Lussac’s law. Clément and Desormes entered their paper in the competition held by the Académie des sciences of Paris in 1812, which we mentioned earlier, but were beaten by Delaroche and Bérard, chiefly due to the opposition of Gay-Lussac, who (astoundingly) criticized their concept of absolute zero. Unusually, his criticisms were unrelated to the law of gas expansion at constant pressure, but were based on the still unknown law of gas heating by compression. When air is suddenly compressed from a pressure p1 to a pressure p2, its temperature increases by the same amount as it decreases when the gas quickly expands, going from p2 to p1. Now, because the heating increases with increasing compression and, at normal temperatures, air can have any pressure, “the cooling produced by its instantaneous expansion can have no limit”. Against all expectations for a scientist of his status, Gay-Lussac thus based his arguments on unfounded hypotheses and ignored a more credible law that he himself had stated ten years earlier!

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Clément and Desormes did not concede defeat in the face of the negative judgment of the Académie; they repeated the experiments, examined the issues raised by the Académie, refuting them from their point of view (sometimes even without much effort), and in 1819 decided to publish the paper they had presented in the competition in 1812, adding a second section dedicated to the controversy with the Académie and their good friend Gay-Lussac45. On his end, while Gay-Lussac rejected the scientists’ theory, he was so enthralled by the experimental device that he joined Welther to repeat the experiments with a slight modification: instead of producing a decrease in the pressure of the air inside the balloon, Gay-Lussac and Welter compressed it to further reduce the time that the tap had to be open. Perhaps these experiments were commissioned by Laplace, who after having demonstrated his 1816 theorem, turned to Clément and Desormes’ experiment to directly calculate the ratio between specific heat at constant pressure and that at constant volume. According to Laplace, the ratio was given by

‫ ݌‬െ ‫݌‬Ԣ ‫ "݌‬െ ‫݌‬Ԣ where p is the atmospheric pressure, p’ is the pressure in the balloon before the tap is opened, and p’’ is the pressure in the balloon at the end of the experiment46. Readers surprised by this formula should keep in mind that Laplace was not aware of the equations describing reversible adiabatic transformations, given in 1833 by Benoît Clapeyron. With the data found by Clément and Desormes, the ratio turned out to be 1.354, which was subsequently slightly improved to 1.37244 from the data collected by GayLussac and Welter; this ratio remains fairly constant with changes in atmospheric pressure. Despite all the studies detailed above and the ones that followed them, information on the specific heats of gases at constant volume was scant and often unreliable. A critical analysis of the different approaches utilized convinced Dulong that the indirect method suggested by Laplace was the best, provided that the speed of sound in gases could be determined more accurately than when Chladni had first performed this type of measurement. With much sagacity and patience, Dulong measured the 45 Ch. Desormes and N. Clément. Détermination expérimentale de zero absolu et la chaleur du calorique spécifique des gaz, in “Journal de physique, de chimie, d’histoire naturelles et des arts”, 89, 1819, pp. 321-46 and 328-455. 46 P.-S. de Laplace, Traité de mécanique céleste, Paris, 1789, Vol. 5, p. 124.

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speed of sound in various gases, using sound tubes that were excited with a current of the gas being examined. Cagniard de la Tour siren, emitting in unison with the fundamental frequency of the tube, gave the frequency of sound, whose wavelength (and thus velocity) could then be deduced from the length of the tube. The results of the lengthy experimentation were publicized in a paper read to the Académie des sciences of Paris in 1828 and published in 1831.47 For the seven gases tested Dulong gave the following values of the ratio between the two types of specific heat: 1.421 for air, 1.412-1.417 for oxygen, 1.405-1.409 for hydrogen, 1.337-1.340 for carbon dioxide, 1.423-1.422 for carbon monoxide, 1.343 for nitrous oxide, and 1.240 for ethene. Based on the fact that these ratios (and especially those of air, oxygen, hydrogen, and carbon monoxide) were nearly equal, Dulong concluded that elastic fluids (gases and vapours), under the same conditions of volume, pressure, and temperature, give off (or absorb) the same quantity of heat when they are rapidly compressed or dilated by the same factor, and the corresponding changes in temperature are inversely proportional to the specific heat at constant volume. We add that Dulong’s method was improved in 1866 by August Kundt (1839-1894) with the introduction of the Kundt tube, which allows for one of the most effective methods for determining the ratio between specific heat at constant pressure and specific heat at constant volume. The Kundt tube is closed off by a movable piston and filled with a small quantity of dry powder (lycopodium, for instance). When stationary waves are produced in the tube, one can measure the half-wavelength by noting the distance between successive nodes, indicated by small clusters of powder. The difficulties inherent in using the earlier method to measure the frequency of sound are thus bypassed by using the Kundt tube for relative velocity measurements.

2.6 Thermal conductivity Humans have been aware of thermal conductivity, at least on the level of empirical fact, since antiquity. The first experimental studies can be traced back to Richmann, who in two papers published in 1752 described an experimental apparatus (later re-built by many other physicists) made up of an iron bar with small mercury-filled holes, each of which contained a submerged thermometer. Heating one end of the bar, he found that the 47

P.-L. Dulong, Recherches sur la chaleur spécifique des fluides élastiques, in “Mémoires de l’Académie royale des sciences de l’Institut de France”, 10, 1831, pp. 146-82.

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temperatures along the shaft decreased non-linearly. In 1789, using this same device, Ingenhousz gave the following conductivity scale: gold, copper, tin, platinum, iron, lead. A few years earlier, in 1784, the same scientist had conducted another experiment on the suggestion of Franklin: bars of the same diameter but made from different substances were coated with wax and heated equally at one extremity by dipping them in boiling oil or water; the extent of the melted wax was taken as a measure of the relative conductivity of the substances. The conclusions drawn from the use of these two devices were hasty, as important factors like heat capacity and external conductivity were ignored. Nevertheless, due to the simplicity of the devices and their ability to illustrate the phenomenon in question, they are still used as demonstration tools in physics classes. The critical appraisal of the experimental procedure outlined above began with Jean-Baptise-Joseph Fourier, who set out to conduct a theoretical study of the propagation of heat, introducing ideas that also pushed experimental research in new directions. Left an orphan by the death of his father, a tailor, at the age of eight years, Fourier was born in Auxerre on 21 March 1768 and immediately distinguished himself for his intelligence, attracting the attention and tutelage of Lagrange and Monge, who installed him as a teacher at the École polytechnique of Paris. Yet he soon abandoned his teaching career to follow Napoleon in Egypt, along with Monge, where he carried out important political duties. Upon returning to France, he was named prefect and baron by Napoleon in 1808. As the prefect of Isère he dedicated himself to concrete projects: building roads, reclaiming swampland, improving education. Despite these administrative duties, he continued his scientific studies, winning the contest organized by the Académie des sciences of Paris on the mathematical theory of the heat propagation. The first part of his winning article is reproduced almost verbatim in his magnum opus,48 published in 1822. Fourier was elected a member of the Académie française in 1817 and became its permanent secretary after the death of Jean-Baptiste-Joseph Delambre in 1822, before dying suddenly on 16 May 1830. Fourier distinguished between internal and external thermal conductivity. In an infinite solid contained between two parallel planes held at different (constant) temperatures, the flux of heat moves from the higher temperature to the lower temperature. Once a stationary state is reached, occurring when the heat flux going into a layer is equal to the flux going 48 Fourier, Théorie analytique de la chaleur, cit. (English translation by Alexander Freeman)

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out, the temperature is constant in every section of the solid parallel to the planes and decreases linearly from the higher temperature plane to the lower temperature one. In these conditions, the amount of heat that flows uniformly per unit time through an area element of a sheetlike section of the solid is given by

-k

ௗ௩ ௗ௭

where dv is the decrease in temperature from one side of the sheet to the other and dz is its thickness, while k is the coefficient of internal conductivity.49 The coefficient of external conductivity, on the other hand, is defined as the quantity of heat that flows per unit time from an area element of an object’s surface to the surrounding air, where the object is at the boiling point of water while the surrounding air is taken to be at constant velocity and at the melting point of ice.50 If a bar is heated at one end, like in Richmann’s experiment, the heat not only propagates from one parallel layer to another, but is also continually exuded into the environment by radiation, conduction, and convection. Consequently, the parallel sections of the bar cannot have a uniform temperature distribution, and the hypothesis above can only be admitted for thin wires. A stationary state is established when every section receives the same amount of heat that it dissipates per unit time. “The general equations of the propagation of heat are partial differential equations,” wrote Fourier, “and though their form is very simple the known methods do not furnish any general mode of integrating them; we could not therefore deduce from them the values of the temperatures after a definite time. The numerical interpretation of the results of analysis is however necessary… We have been able to overcome the difficulty in all the problems of which we have treated, and which contain the chief elements of the theory of heat… The examination of this condition shows that we may develop in convergent series, or express by definite integrals, functions which are not subject to a constant law, and which represent the ordinates of irregular or discontinuous lines. This property throws a new light on the theory of partial differential equations.”51 Indeed, Fourier’s work shed so much new light that it became a point of reference for the history of mathematics. The fundamental concepts of internal and external 49

Ibid., p. 50. Ibid., p. 21. 51 Ibid., p. 10-11. 50

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conductivity were absorbed by experimentalists, and in consequence the methods of Richmann and Ingenhousz were refined to make the external conditions equal when comparing substances, for example by coating the bars with lampblack or silver. This method was first applied by César Despretz and improved in 1853 by Gustav Heinrich Wiedemann and Rudolph Franz, who, instead of using thermometers like Despretz, had employed thermoelectric batteries. Not only did they find that conductivity was highly variable (for example, if the conductivity of silver is normalized to 100, then that of copper is 74, that of gold 53, and that of iron 12), but also that thermal conductivity was almost equal to electrical conductivity: thus, the parallelism that had been suspected since Franklin’s time was finally experimentally demonstrated. This parallelism was further extended by Ångström, who showed that like electrical conductivity, thermal conductivity decreases with increasing temperature. After studying the thermal conductivity of homogeneous bodies, in the first half of the century scientists moved on to the conductivity of wood (Auguste De la Rive and Alphonse De Candolle, 1828) and crystals (Henri Hureau de Sénarmont, 1847). The study of the conductivity of liquids and gases presented serious difficulties, to the extent that at the beginning of the century many physicists (Rumford and Gren, for example) considered them to be perfectly non-conducting. Yet Despretz in 1839 and Ångström in 1864 showed that liquids are unmistakeably able to conduct heat, though in small quantities. Gas conductivity was demonstrated by Magnus in 1861 and Clausius the following year, while Boltzmann gave a theoretical explanation of the property in 1875. The thermal conductivity of gases, though definitively verified in experiments, is so feeble that a gas at rest, ignoring convection, is practically a perfect thermal insulator.

THE PRINCIPLES OF THERMODYNAMICS 2.7 The crisis at the beginning of the 19th century We have seen how the fluid theory of heat, after having peacefully coexisted with the mechanical theory for centuries, seized the upper hand in the second half of the 18th century; by the end of the century the dispute intensified and entered its final decisive phase. The proponents of the fluid theory at the end of the century included Adair Crawford (1749-1795), Johann Mayer (1752-1830), and Gren, in addition to those already mentioned (Volta, Cavendish, Lavoisier, etc.);

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while the leading proponents of the mechanical theory were Pierre Macquer (1718-1784), Davy, Rumford, Young, and Ampère. It is not true, as often is written, that the American engineer, adventurer, and spy Benjamin Thomson, having sought refuge in Europe where he obtained the title of the count of Rumford, was the first to conceive of heat as the motion of molecules after his famous 1798 experiments in Munich. Before him, Rumford spun a beveled drill at the bottom of a cannon barrel and measured its temperature, which was initially 16.7 °C, with a thermometer placed inside an opening in the cannon. After 360 rotations the drill had scraped off 837 grains and the temperature had risen to 54.4 °C. Later, immersing the cannon in 15.6 °C water, he obtained boiling after two and a half hours of the drill rotating. “By meditating on the results of all these experiments,” wrote Rumford in a paper read to the Royal Society on 25 January 1798, “we are naturally brought to that great question which has so often been the subject of speculation among philosophers; namely, what is heat? Is there any such thing as an igneous fluid? Is there anything that can with propriety be called caloric? We have seen that a very considerable quantity of heat may be excited in the friction of two metallic surfaces, and given off in a constant stream or flux, in all directions, without interruption or intermission, and without any signs of diminution, or exhaustion. And, in reasoning on this subject, we must not forget to consider that most remarkable circumstance, that the source of the heat generated by friction, in these experiments, appeared evidently to be inexhaustible. It is hardly necessary to add, that anything which is any insulated body, or system of bodies, can continue to furnish without limitation, cannot possible be a material substance: and it appears to me to be extremely difficult, if not quite impossible, to form any distinct idea of anything, capable of being excited and communicated, in the manner the heat was excited and communicated in these experiments, except it be motion.”52 The phenomenon of heat production from friction was not new, and Rumford’s experiments were likewise not original: two centuries earlier, Baliani had produced enough heat to boil water by rapidly spinning an iron disk, where the water was contained in a flat iron pot on top of the disk. Yet Baliani’s experiments, which he described in a letter to Galileo dated 4 April 1614 (but published only in 1851), were not well-known, so Rumford’s analogous experiments caused great stir not so much for the mechanism of heat arising from friction, but because of the enormous 52

B. Thompson, An Enquiry Concerning the Source of the Heat Which is Excited by Friction, in Tyndall, Heat as a Mode of Motion, cit. Italics are as in the original paper.

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quantities of heat that this method seemed to produce. Nevertheless, in hindsight they were not as probative as they seemed at the time. The advocates of the existence of caloric countered that in Rumford’s experiments the heat mixed in with solids was partially released, explaining the heating effect when the solids were ground. Rumford’s later experiments, in which he attempted to demonstrate that the metallic powder heated by the scraping had the same heat capacity as the ordinary metal, would have constituted an adequate rebuttal of the counterarguments if heat capacity had meant what it does today, but at the time it referred to the total quantity of heat possessed by a body, and Rumford’s experiments did not indicate anything regarding this quantity. In short, Rumford would have had to show (and he did not) that at least part of the heat produced by friction was not latent heat in the compact metal that was released when it became powder. The phenomena of heating or cooling of gases that are respectively compressed or expanded were also interpreted by the advocates of caloric as further confirmation of their theory. We have already discussed Clément and Desormes’ specious interpretation (§ 2.5). A more widespread view was that gas contains caloric like oranges contain juice: when the orange is squeezed the juice comes out, and similarly when gas is compressed its caloric comes out, giving rise to external thermal effects. This patched theory held for thirty years, and in 1829 Biot wrote in the second edition of his manual, the most comprehensive and authoritative treatise on physics at the time, that the reason for which friction produces heat was still unknown. While Biot was a conservative, Fourier, having kept his distance from the university environment, was anything but, and remained unconvinced by Rumford’s experiments and the arguments put forward by others. He did not advance any hypotheses on the nature of heat: he accepted only the rules and units of measurement passed down from the previous century, dismissing the mechanical nature of heat. “Whatever may be the range of the mechanical theories,” he wrote in the Preliminary Discourse of his magnum opus, “they do not apply to the effects of heat. These make up a special order of phenomena, which cannot be explained by the principles of motion and equilibrium.”53 In writing these words Fourier did not go so far as to state, as was retroactively attributed to him, that the flow of heat is an irreversible phenomenon, and thus not ascribable to reversible mechanical phenomena; he simply meant that thermal phenomena could not be interpreted as hidden motion of the basic particles of matter. The concept of 53

Fourier, Théorie analytique de la chaleur, cit. p. XVI

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irreversibility of thermal phenomena arose with the second principle of thermodynamics after 1850. Despite abstaining from making hypotheses, Fourier embraced the mindset of the supporters of the fluid theory.

2.8 Carnot’s principle We have already noted that the most important studies of heat in the first half of the 19th century were aimed at improving the operation of steam engines. Dalton complained of this direction in scientific inquiry, which to him seemed overly technical. Watt, instead, posed the problem in crudely practical terms: how much coal is required to produce a certain amount of work, and with what means, keeping work constant, can one minimize its consumption. This practical problem was studied by a young engineer, Sadi-Nicolas Carnot (Fig. 2.2), born in Paris on 1 June 1796 to Lazare-Nicolas Carnot, a mathematician who looked after his education until university. The younger Carnot enrolled in the École polytechnique in 1812, and left it in 1814, to take part in the defence of Vincennes. After the fall of Napoleon, Lazare retired to Germany in exile and his son Sadi was sent to the interior of France, where only in 1819 he was promoted to the rank of lieutenant in the French army. The following year, however, the difficulties that his military career faced because of his father’s politics forced him to withdraw from service. In a small apartment left to him by his father in Paris, he dedicated himself to the study of science and arts (he was also an expert violinist) and summarized his studies on the steam engine in a pamphlet published in 1824 with the title Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance.54 Recalled to service in 1827 and sent to Lyon, he once again resigned in 1828 and returned to Paris, where he died of cholera on 24 August 1832, at the age of 36. The appearance of his Réflexions marked an important date in the history of physics, not only for the results he obtained and expounded, but also for his methods, which were imitated and repeated ad infinitum. Carnot based his argument on the impossibility of perpetual motion, which despite not having attained the status of a proper physical principle remained a general mentality among most scientists, though not all (Stevin, for example, used perpetual motion in his theoretical considerations). One can perhaps say that the use of the steam engine had reinforced this idea, 54 Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power

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Fig. 2.2 - Sadi Carnot. Engraving from a portrait by L.L. Boilly (1813)

showing that useful work was inextricably connected with the consumption of something. Carnot, however, did not resort to the example of the steam engine to justify the principle, instead backing it with a series of judicious observations on batteries, which had been subject to some hasty conclusions regarding the possibility of perpetual motion. Carnot began his study with praise for the steam engine; he noted that the theory was rather behind and that, to advance it, it was necessary to disregard the empirical aspects for a bit and instead consider the engine power on an abstract level “independently of any mechanism or special agent”. Using a thought experiment, Carnot showed that assuming the impossibility of perpetual motion, the production of work is only possible if there are two different temperatures in the machine and heat is transferred from the higher one to the lower one. In the same way,

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mechanical work is obtained when water falls from a higher level to a lower one. This analogy is not perfect, added Carnot, because while the work in the mechanical process is proportional to the height difference, the work done by machines certainly increases with increasing temperature difference, “but we do not know if it is proportional”. To find the maximum power of machine he introduced his famous cycle, and a nowfamous argument brought him to the theorem that carries his name: “The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperature of the bodies between which is effected, finally, the transfer of caloric.”55 In short, the yield of a thermal machine is determined by the temperatures of the source and the refrigerant, the latter, as Carnot explicitly affirmed, being as essential as the boiler and becoming the environment when not explicitly built into the machine. All of these observations constitute the essence of “Carnot’s principle”, or the second principle of thermodynamics, as it was later called when this chapter of physics was formulated axiomatically. As Edmond Whittaker56 observed, Carnot’s principle, like his principle on perpetual motion, is a principle of impossibility or “impotence” on which thermodynamics is established as a deductive science, similarly to geometry. Now, other principles of impotence can be encountered in electromagnetism, relativity, and quantum mechanics, so it is reasonable to believe that the sufficiently developed fields of physics could be organized in a deductive form starting from postulates of impotence, as occurred for thermodynamics. Here we make another observation: Carnot did not explicitly address the nature of heat. He claimed to indifferently employ the terms “heat” and “caloric”, from which one may suppose that he believed in the material nature of heat, consequently accepting the conservation principle that had been stated by Black57: according to some 19th century scientists this belief invalidated his argument to derive the principle, while for others the belief did not invalidate his reasoning. In 1911, the British physicist Hugh Callendar advanced the hypothesis that by “caloric” Carnot meant what today is called “entropy”: a bold interpretation that we consider historically

55 S.-N. Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance, Paris 1878, p. 28. 56 E. Whittaker, From Euclid to Eddington. A Study of the Conceptions of the External World, Cambridge University Press, Cambridge 1949, p. 60. 57 § 7.20 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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unfounded, later repeated by other scientists and the subject of a lengthy controversy.58 Having established his seminal theorem, Carnot applied it to solve other problems, for example calculating the work performed during isothermal expansion, for which he reached the conclusion that the ratio Q/T between heat and temperature remained constant during the process. Carnot calculated the work produced in an infinitesimal cycle for a gram of air, vapour, and alcohol, confirming that the work is independent of the substance in the cycle. At some point after the publication of his treatise, Carnot became fully convinced of the mechanical theory of heat, as is evident from the following passage that was found in his manuscripts and published in 1878 in the appendix of a reprinting of his Réflexions: “Heat is simply motive power, or rather motion which has changed form. It is a movement among the particles of bodies. Wherever there is destruction of motive power there is, at the same time, production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power. We can then establish the general proposition that motive power is, in quantity, invariable in nature; that it is, correctly speaking, never either produced or destroyed. It is true that it changes form, sometimes producing one sort of motion, sometimes another, but it is never annihilated.”59 Without explaining how he arrived at it, in a brief footnote Carnot gave the mechanical equivalent of a calorie, which when translated into kilogram-metre units gives a single large calorie of heat. Carnot’s book went almost unnoticed; the lack of interest in his work, in spite of his clear and elegant writing style, can be explained by his scant academic connections and, especially, the novelty of the concepts he wrote about. Ten years later, in 1834, Benoît Clapeyron (1799-1864) drew attention to the work and substituted Carnot’s original cycle with another famous cycle made up of two isotherms and two adiabats, which today is erroneously attributed to Carnot in most texts. In this occasion, Clapeyron also formulated the ideal gas law, which gives a simple relationship between pressure, volume, and temperature in a gaseous body, synthesizing the laws of Boyle and Volta/Gay-Lussac.

58

This controversy can be seen in L. Tisza, Generalized Thermodynamics, The MIT Press, Cambridge (Mass.) 1966. 59 Carnot, Réflexions cit., p. 89.

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2.9 The equivalence principle From Rumfeld’s time to around 1840, there were not major works published on thermodynamics, with the exception of the aforementioned works of Carnot and Clapeyron. The experiments conducted in 1822 by Giuseppe Morosi (1772-1840), taken as the basis for the theory of continuous molecular motion by Domenico Paoli (1782-1853), were a repetition of Rumford’s experiments. They thus did not add much to the field, though perhaps contributing to steering the atmosphere towards a mechanical conception of heat. The true shift in mentality occurred mainly among young scientists that were outside of academic environments, where the weight of tradition and the authority of professors sometimes hinder the development of new ideas. This explains why the idea of equivalence between heat and work was advanced almost simultaneously and independently by several young physicists: the thirty-year-old Carnot, a discharged officer; the twentyeight-year-old Robert Mayer (1814-1878), a medic in the German navy; the twenty-five-year-old James Joule (1818-1889), a beer produced in London, and we could continue with Karl Friedrich Mohor (1805-1879), Ludwig August Colding (1813-1888), and Marc Séguin (1786-1875), all of whom laid claim to scientific priority for the statement, and not gratuitously. There were eleven physicists who more or less had grounds to claim priority, but the ones who have remained most famous are (deservedly) Mayer and Joule. For the first, the law came as a sudden realization in 1840 that was almost akin to a religious conversion, because in developing and defending his idea he dedicated his life, doing so with such fervent mental and physical effort that he ended up in an insane asylum. Mayer wrote a first article expounding his ideas in 1841, but Poggendorff, the director of the “Annalen der Physik”, refused to publish it. Though retroactively ridiculed for this decision by his successors, Poggendorff was probably in the right, since the first paper contained so many errors as to seriously compromise the success of the idea at its heart. A second, corrected paper was published the following year by the journal of chemistry and pharmacy of Liebig. As it was a pivotal document in the history of physics, we now delve a bit deeper into Mayer’s paper. He began by wondering what we mean exactly by “force” and how the different forces are related to each other (the modern reader should replace the word “force” with the word “energy” in Mayer’s work). To study nature, he argued, it is necessary that we have a concept of “force” that is as clear as our concept of matter.

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Mayer continued: “Forces are causes: accordingly, we may in relation to them make full application of the principle causa æquat effectum (the cause corresponds to the effect).”60 Following this line of pseudometaphysical reasoning, Mayer arrived at the conclusion that forces are indestructible, convertible, and imponderable “objects”; and that “if the cause is matter, the effect is matter; if the cause is a force, the effect is also a force”. It follows that “If, for example, we rub together two metal plates, we see motion disappear, and heat, on the other hand, make its appearance, and we have now only to ask whether motion is the cause of heat. In order to come to a decision on this point, we must discuss the question whether, in the numberless cases in which the expenditure of motion is accompanied by the appearance of heat, the motion has not some other effect than the production of heat, and the heat some other cause than the motion.”61 With few considerations Mayer showed that it would be illogical to deny a causal connection between motion (or, according to our modern terminology, work) and heat; as it would be illogical to suppose a cause, the motion, to be without effect, or an effect, the heat, without a cause. Such a position would be like a chemist who sees oxygen and hydrogen disappear and water appear and prefers to say that the gases simply disappeared and water was inexplicably formed. Mayer preferred a more sensible interpretation of the phenomena, assuming that motion transforms into heat and heat is converted into motion: “A locomotive engine with its train may be compared to a distilling apparatus; the heat applied under the boiler passes off as motion, and this is deposited again as heat at the axles of the wheels.”62 Mayer closes his disquisition with a practical deduction: “How great is the quantity of heat which corresponds to a given quantity of motion or falling force?” With a stroke of genius, he deduced this equivalence from a knowledge of the specific heats of gases at constant temperature and pressure. “Mayer’s method” consists in setting equal the difference between the two specific heats and the work needed to overcome external pressure during gas expansion. Using the specific heat values given by Dulong (§ 2.5), Mayer, with a calculation that is barely mentioned in the article, found that a large calorie is equivalent to 365 kilogram-metres, and commented “If we compare with this result the working of our best steam engines, we see how small a part only of the heat applied under the boiler 60 R. Mayer, Bemerkungen über die Krafte der unbelebten Natur, in “Annalen der Chemie und Pharmacie”, 42, 1842, p. 233. 61 Ibid., p. 237. 62 Ibid., p. 239.

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is really transformed into motion or the raising of weights.”63 It was from this very observation of low yield in steam engines that Carnot’s inquiries had also begun. Applying Mayer’s method, Regnault, using the most accurate measurements of the specific heats of gases, calculated that a large calorie is equivalent to 424 kilogram-metres. In 1843, still ignorant of Mayer’s work, Joule experimentally determined the mechanical equivalent of a calorie while he was studying the thermal effect of currents, leading him to the law that carries his name. Joules now-classic setup consisted in hearing the water in a calorimeter by mixing it with a whisk and then measuring the ratio between the work performed and the heat produced. Based on an average of 13 experiments, Joule concluded that “the quantity of heat capable of increasing the temperature of a pound of water (weighed in vacuo, and taken between 55 °and 65 °) by 1 °Fahrenheit requires for its evolution the expenditure of a mechanical force represented by the fall of 772 pounds through the space of one foot.”64 From these numbers it is easy to see that in our ordinary units the mechanical equivalent of one calorie was 460 for Joule. Numerous other experimental measurements of this “constant of the universe”, as Helmholtz called it, were subsequently made. We limit ourselves to noting that of Gustave-Adolphe Hirn (1815-1890), who in 1860-61, experimenting on the collision of two lead blocks and using Joules method, obtained a value of 427 kilogram-metres per calorie, which still today is considered fairly accurate. In 1940 the International Committee for Weights and Measures fixed the conversion factor for one calorie at 15 °C to be 4.18606(1010) erg.

2.10 Conservation of energy Several years passed after the publications of Mayer and Joule before physicists were familiar the principle of equivalence. In 1847, the young physiologist Helmholtz published his famous paper (also rejected by Poggendorff) Über die erhaltung der Kraft without being aware of Mayer’s work (though familiar with Joule’s), which was followed by controversy and accusations, though they had no bearing on Helmholtz, who openly recognized Mayer’s priority in stating the equivalence principle.

63 64

Ibid., p. 240. J.P. Joule, The Scientific Papers, London 1884, Vol. 1, p. 156.

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Helmholtz’s paper was not limited to an analysis of only mechanical and thermal “force” (that is, energy, using the term employed by Young and in 1849 by Lord Kelvin), but also discussed other forms of energy. In essence, Helmholtz, following Mayer, used the word energy (actually “force”) to denote any quantity that could be converted from one form to another and, like Mayer, thought it indestructible in nature and therefore ascribed to it the behaviour of any other substance: it can neither be created not destroyed. Between matter and energy there is then a deep relationship: matter and force are both “abstractions from the real formed in exactly the same way. We can perceive matter only though its forces, never in itself.”65 Helmholtz, like Carnot before him, based his arguments on the impossibility of “continually producing motive force from nothing”. To apply the principle to mechanical phenomena, he followed Carnot’s thought experiment and reasoning: if a system of bodies passes from one state to another, it uses a certain quantity of work, which must be equal to the amount of work needed for the system to go from the second state back to the first, regardless of the kind of motion, velocity, or trajectories involved. Indeed, if the two amounts of work were different, then each cycle would produce (or destroy) motive force from nothing. What is conserved in mechanical processes is the sum of the “tension forces” (potential energy, in modern terminology) and the vis viva (living force; which today we call kinetic energy): this is the principle of conservation of energy. Helmholtz then moved on to demonstrating that the principle applies to mechanical phenomena accompanied and unaccompanied by the production of heat (collisions, friction, compression and expansion of gases), electrostatic and electrodynamic phenomena, magnetic and electromagnetic phenomena, and finally the phenomena of the organic world. This last extension was of particular importance from a historical point of view. The impossibility of perpetual motion was, as we have discussed, merely a mentality among physicists, which struggled to become a general scientific principle because it appeared to be contradicted by a common observation: the work done by animals. Until a relationship was posed between the work done by animals and the food ingested, the production of motive force from nothing appeared possible. In hindsight, now that we are familiar with the concept of energy (or perhaps more with the term than the concept), it may appear that Helmholtz’s paper did not add anything to what Mayer and Joule had asserted earlier. However, to appreciate the novelty of Helmholtz’s conceptions, one only 65

H. von Helmholtz, On the Conservation of Force, in Id., Works cit., p. 51.

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has to reflect on the fact that Mayer and Joule had referred to a particular, though important, case, while Helmholtz introduced a new quantity, previously unknown to physicists or confused with force, that is present in all physical phenomena, mutable but indestructible, imponderable but a regulator of the appearance of matter. All of physics in the second half of the 19th century rested on two concepts, matter and energy, both obeying conservation laws. The distinguishing characteristic between the two was ponderability: matter is ponderable while energy is imponderable. Hermann Ludwig Ferdinand von Helmholtz was born in Potsdam on 31 August 1821. Having obtained his degree in medicine in 1842 in Berlin, he served as a military medic from 1843 to 1848, at the same time conducting physical and medical research. In 1848 he was called to teach anatomy in the Academy of fine arts of Berlin, later moving to Bonn in 1855 and Heidelberg in 1858. In 1870, he transferred to a physics professorship at the University of Berlin, where he remained until his death, on 8 September 1894. Helmholtz’s academic career reveals his wide-ranging scientific interests, which spanned mathematics, physics, physiology, and anatomy. Venturing, like the other physiologists of his time, to introduce the methods of physics and chemistry in the study of physiology, he made important contributions to science, in addition to his 1847 paper, with the Manual of Physiological Optics (1856), The Theory of Acoustic Sensations (1863), and many other papers on mechanics, acoustics, optics, and electricity. Helmholtz’s views were backed and spread by Tyndall in particular, inspiring the school of “energeticists” that arose in Britain based the work of William Rankine (1820-1872). The doctrine of the school was the abandonment of the mechanistic conception of the universe, according to which all phenomena had to be explained through the concepts of matter and force. In place of this conception the energeticists substituted another in which the interplay of energies, actual and potential, present in bodies explains the various phenomena. Energy was the only physically real quantity for the energeticists: matter was merely an apparent support. The excesses of this school were opposed with little success by Planck, and slightly more by Boltzmann, who based his criticism on atomic theory and statistics.

2.11 Thermodynamic temperature scales The issue of devising a thermometric scale, which had preoccupied physicists for two centuries (§ 2.1) found its most logical solution by merit of a twenty-four-year-old scientist, Thomson (later Lord Kelvin), who had

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become acquainted with Carnot’s work through Clapeyron’s exposition. While Carnot had elaborated his theory without the use of calculus, Clapeyron, in the paper cited in § 2.8, had expounded the theory making use of mathematical analysis (thus rendering it more palatable for physicists of the time). At the end of his paper, Clapeyron calculated the amount of work produced by an ideal engine, in which a calorie of heat is transferred between two bodies whose temperature difference is one degree centigrade. The calculation could only be approximative, because the experimental data necessary were missing or also approximate. Nevertheless, Clayperon calculated that this work was not constant, but decreased with increasing temperature, from 1.586 kg-m at 0 °C to 1.072 at 156.8 °C.66 Clapeyron’s calculation attracted the attention of Thomson, who realized that through this approach it was possible to define an absolute temperature scale: absolute in the sense that such a scale would be independent of the thermometric substance employed, while before, as we have seen, the adjective “absolute” referred to a temperature free from algebraic signs. The careful measurements conducted by physicists in the first half of the century had brought about the credence that a thermometry based idea, that equal quantities of heat produce equal expansions in a thermometric substance, was impossible, because it had been definitively established that specific heat varies with temperature. In practice, adjustments were made by thermometer builders through minute prescriptions, based mostly on Regnault’s careful measurements. The most reliable measurements were made by the gas thermometer, against which all other types of thermometers were therefore compared. In an 1847 work, however, Regnault carefully observed that that even gas thermometers did not give reliable measurements if different gases or different pressures were used. In short, to use Thomson’s own words, the thermometric scales used at the time were “an arbitrary series of numbered points of reference sufficiently close for the requirements of practical thermometry.”67 Now, Carnot’s theory established that heat and temperature intervals are the only quantities relevant for the calculation of the corresponding mechanical effect: by measuring both the amount of heat and the work developed in processes, one could attain the precise value of temperature intervals without reference to any thermometric substance. After a few 66 E. Clapeyron, Mémoire sur la puissance motrice de la chaleur, in Journal de l’École royale polytechnique”, 23, 1834, p. 185. 67 W. Thomson, On an Absolute Thermometric Scale Founded on Carnot’s Theory of the Motive Power of Heat, and Calculated from Regnault’s Observations, in “Proceedings of the Cambridge Philosophical Society”, 5 June 1848, p. 68.

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preliminary observations, from which it is clear that Thomson did not know of the principle of equivalence, as the conversion of heat (or caloric, as he wrote) into work seemed to him still cloaked in “mystery”, he observed that, according to Carnot, the flow of heat from a hot object to a cold object can produce a mechanical effect, and vice versa, with an expenditure of mechanical work a certain amount of heat can be transferred from a cold object to a hot object. The mechanical effect can thus act as a parameter in the measurement of temperature, one only has to invert Clapeyron’s problem: instead of calculating the work obtained when a calorie “descends” across a one-degree temperature difference, one should choose the unit of temperature in such a way that “that a unit of heat descending from a body A at the temperature T °of this scale, to a body B at the temperature (T-1) °, would give out the same mechanical effect, whatever be the number T.”68 Descends here refers to a Clapeyron cycle. In this original explanation, the idea of thermodynamic temperature scale is almost unrecognizable compared to our modern treatment. Yet tracing backwards through the mathematical derivation presented in a modern text, one can see that Thomson’s proposition is equivalent to the statement that

ܳଵ ܳଶ = =݇ ܶଵ ܶଶ if Q1 and Q2 are the quantities of heat provided to the heat source and the refrigerant, respectively, while T1 and T2 are the respective absolute temperatures. By assigning an arbitrary value to a temperature, k can be determined and thus every temperature can be obtained from the corresponding quantity of heat. To avoid a break from historical convention, 100 was chosen as the temperature interval between the melting point of ice and the boiling point of water. Using this convention, thermodynamic temperatures (denoted by the symbol K, for Kelvin) coincide with the absolute temperatures of an ideal gas thermometer.

2.12 The mechanical theory of heat Carnot’s principle, deduced in the framework of the fluid theory of heat, appeared to contradict the principle of equivalence because it 68

Ibid., p. 69.

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supposed that the quantity of heat “descended” from a higher temperature to a lower temperature remains constant in an ideal engine, much like the quantity of falling water that produces mechanical work in a waterfall remains constant. Two young scientists, Rudolf Clausius69 and Thomson70, independently and almost simultaneously set out to frame the two principles using the new axioms on which the mechanical theory was based. On this same subject, a more complete and famous paper was published slightly earlier by Rudolf Julius E. Clausius (Fig. 2.3), who was born in Koslin, Pomerania on 2 January 1822. Having finished his gymnasium studies in Szczecin, he enrolled in the University of Berlin, where he obtained his degree in 1848. He began with research in atmospheric optics, but starting in 1850 he dedicated the rest of his scientific career almost exclusively to thermodynamic research. Offered a position at the Zurich polytechnic in 1855, he moved there and remained for twelve years, then spent some time in Würzburg and later transferred definitively to Bonn in 1869, where he died on 24 August 1888. Clausius observed that the ratio between work expended and heat produced is only constant in cyclical transformations, namely those in which the bodies in question, after a series of changes, return to their initial conditions. For instance, this was not the case in Joule’s primitive version of the calorimeter, because it contained cold water at the beginning of the transformation and hot water at the end; to obtain a cyclical process one had to replace Joule’s device with a Bunsen calorimeter. If the transformation is not cyclic, then the ratio is not constant; in other words, the difference between heat expended and work produced (or vice versa), measured in the same units, is nonzero. For example, if a certain amount of water is vaporized at constant temperature, the heat provided is significantly greater than the work done by the expansion of the vapour: where does the energy go?

69

R. Clausius, Über die bewegende Kraft der Warme und die Gesetze, welche sich daraus für die Warmelehre selbst ableiten lassen, in “Annalen der Physik und Chemie”, 79, 1850, pp. 368-97 and 500-24; reprinted and edited by Planck in 1898, for the “Klassiker der exacten Wissenschaften”. 70 W. Thomson, On the Dynamical Theory of Heat, with Numerical Results Deduced from Mr. Joule’s Equivalent of a Thermal Unit, and M. Regnault’s Observations on Steam, in “Transactions of the Royal Society of Edinburgh”, 20, 1853, pp. 261-88. The paper was read in March of 1851.

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Fig. 2.3 – Rudolf Clausius

Clausius had the ingenious idea of fixing the faulty equation by introducing the concept of internal energy: in the example mentioned above, the heat provided to the water in part transforms into the external work needed for the expansion of the vapour (and the water) and in part transforms into internal energy, which becomes heat again when the vapour condenses. With the introduction of internal energy, Clausius gave a precise mathematical form to the equivalence principle that also worked for open systems. The fundamental theorem established by Clausius can be written as

݀ܳ = ‫ ܷ݀(ܣ‬+ ‫)ݒ݀݌‬

where dQ is the amount of heat provided to the body, A is the thermal equivalent of a unit of work, U = f(v,p) is the internal energy, p is the pressure, dv the change in volume, and pdv = dW is the corresponding external work done by the object. Both Clausius and Thomson deduced Carnot’s principle from an equivalent, and more intuitive, statement. Clausius’ formulated the

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principle as follows: “It is impossible for an automatic machine to convey heat from one body to another at a higher temperature if it be unaided by any external agent;” while Thomson analogously affirmed that ““It is impossible, by means of an inanimate material agent, to obtain a mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” More simply, both statements can be summed up by saying that heat cannot spontaneously move from a cold object to a hotter one: the adverb “spontaneously” indicates that when this passage of heat occurs, as it does in refrigerators, chemical solutions, etc., it is in a certain sense “forced”, meaning that it must be accompanied by anther compensating effect. This new ClausiusThomson statement of the principle was rapidly transformed into other equivalent forms: natural phenomena are irreversible; phenomena occur in such a way that the energy involved “degrades”; and other similar ones. All of these statements are in stark opposition with the traditional reversible laws of dynamics: we will return to this point in § 2.15. In 1865, Clausius introduced a new quantity, which assumed a fundamental role in the later development of a thermodynamic framework. This was entropy, a well-defined mathematical quantity but of little intuitiveness from a physical point of view. More precisely, the variation in entropy dS in an infinitesimal transformation in which an infinitesimal quantity of heat dQ is transformed at temperature T is defined as

݀ܵ =

݀ܳ ܶ

Clausius showed that only changes in entropy are well-defined, as opposed to absolute values, and that entropy always increases in thermally isolated irreversible processes, remaining constant only in the ideal case of a reversible process. Entropy is thus one of the characteristic quantities associated with a thermodynamic body, like volume, temperature, and internal energy. The introduction of this novel quantity was met with heated opposition by many physicists, who especially objected to its mysterious nature, mainly due to the fact that it is not perceptible by our senses. Because the entropy change is zero in ideal reversible processes, and positive in real irreversible processes, it measures how much a real process deviates from its idealization: hence the name “entropy”, which has its etymological roots in the Greek words for “inside” and “transformation”. The mechanical theory of heat, for which Rankine claimed priority, citing his presentation of a paper to the Royal Society in 1850 in which he

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only dealt with the principle of equivalence, experienced a tumultuous evolution and was wholly accepted only at the end of the century, in large part due to the work of Planck between 1887 and 1892.

KINETIC THEORY OF GASES 2.13 The nature of heat The founders of classical thermodynamics – Mayer, Joule, Colding, and, in a certain sense, Carnot – for the most part were not interested in the nature of heat. They limited themselves to affirming that heat, under certain conditions, can be transformed into mechanical work, and vice versa. The pioneers of the theory did not believe it necessary to investigate the underlying relationship between mechanical phenomena and their thermal manifestations. It was Helmholtz who first proposed, in the cited 1847 paper, the hypothesis that the fundamental reason for the transformability of heat into work and work into heat could be discovered, with an approach that he did not describe, thus reducing thermal phenomena to mechanical ones, or motion, in other words, as had been attempted the previous century: “the quantity of heat,” wrote Helmholtz, “can be absolutely increased by mechanical forces […], that therefore calorific phenomena cannot be deduced from the hypothesis of a species of matter, the mere presence of which produces the phenomena, but that they are to be referred to changes, to motions, either of a peculiar species of matter, or of the ponderable or imponderable bodies already known, for example of electricity or the luminiferous aether.”71 The suggestion may have been misleading, but traces of the Bernoulli’s fundamental idea72 had remained alive in science and had surfaced several times: we have seen, for example, that Lavoisier and Laplace gave a nod to the concept; in 1848 Joule interpreted gas pressure using Bernoulli’s ideas as well. However, allusions by these and other scientists remained purely qualitative, also because a deeper understanding of atomic theory was necessary for a closer quantitative examination. By the halfway point of the century, atomic theory had made much progress and could already be used with relative confidence by physicists, who therefore combined it with the mechanical theory of heat to develop the 71

Helmholtz, On the Conservation of Force, cit., p. 73. § 7.19 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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framework of the kinetic theory of gases. The foundations were laid by the German physicist and chemist August Krönig (1822-1879) and the following year by Clausius. Before narrating the key points in the history of kinetic gas theory, which we stress is intimately intertwined with the history of atomic theory, we first remind the reader of the law stated in 1811 by Amedeo Avogadro: equal volumes of gas, under equal conditions of temperature and pressure, contain equal numbers of molecules. At the time of the development of kinetic theory, this precise number was not yet known (§ 6).

2.14 Kinetic theory In his paper Foundations of a Theory of Gases, published in 1856 in “Annalen der Physik”, Krönig supposed a gas constituted of a collection of molecules, modelled as perfectly elastic hard spheres in continuous and completely disordered motion. He also held that the volume of the molecules is negligible compared to the total volume of the gas and ignored interactions between molecules. Due to the continuous motion, the molecules collide with each other and the walls of the container, resulting in changes in their velocity. Starting with these hypothesis and Avogadro’s law, Krönig arrived at Boyle’s law. His argument, which is still repeated today in physics texts, led him to conclude that the product of the pressure and the volume of a gaseous mass is equal to 2/3 of the total translational kinetic energy of its molecules. The product, therefore, remains constant as long as the translational kinetic energy of the molecules does too. However, the ideal gas law indicates that the product of pressure and volume varies with temperature, so the kinetic energy depends on the temperature. The idea of defining temperature as the average kinetic energy of the molecules naturally follows, establishing a well-defined mathematical relationship between the two quantities. These bases of the theory were conceived of independently (at least according to him) by Clausius, who, seeing his colleague ahead of him, rushed to develop his ideas first in 1857, and later in a longer paper published in 1862, both in “Annalen der Physik”.73 While Krönig supposed that molecules were only subject to translational motion, Clausius observed that intermolecular collisions necessarily also provoked rotational motion. Furthermore, he noted that the atoms that make up a molecule move

73 Translated into French from the original German in the second volume of R. Clausius, Théorie analytique de la chaleur, Paris 1869.

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internally, perhaps in an oscillatory fashion. On these theoretical bases, Clausius showed that the following relationship holds:

௡௠௨మ

= pkT

ଷ௩

where n is the number of molecules contained in the volume v, m is the mass of each molecule, u is the root mean square velocity, p is the gas pressure, and T is its absolute temperature. Using the data provided by Regnault in the formula, one easily finds that

u = 485ට

்

ଶ଻ଷఘ

m/s

where ȡ is the specific weight of the gas considered. In this way, at 0 °C the velocity of oxygen molecules is calculated to be 461 m/s, that of nitrogen molecules 492 m/s, and that of hydrogen 1844 m/s. These values, almost identical to the ones calculated today, seemed enormous and incompatible with the small diffusion velocity of one gas into another, or the low conductivity of gases. Yet in 1858 Clausius had observed that the process of diffusion does not depend as much on the velocity of the molecules as on the mean free path of a molecule, defined as the average distance a molecule travels between two collisions (calculated by Maxwell in 1860). Consequently, the rate of diffusion depends on the number of collisions in a fixed time interval, which for ordinary materials is an enormous number. Under normal conditions, one finds about 5 billion collisions every second; the first experimental measurements of this kinetic-theoretic quantity were achieved only in 1920 by Otto Stern (1888-1969), who in 1943 received the Nobel prize in physics for his experimental work in thermodynamics (and his discovery of the magnetic moment of the proton). Also in 1920, collaborating with Walter Gerlach in Frankfurt, Stern obtained molecular rays through the electrical heating of an oven, whose walls had a small aperture out of which a gas could escape into a vacuum: the molecules in the so-called molecular (or atomic) ray flowed in a single direction without appreciable collisions, and in Stern and Gerlach’s experiment they were recorded by a detector. The theoretical scheme used by Krönig and Clausius was a bit simplistic, and consequently the conclusions they drew could only be

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experimentally confirmed as a first approximation. In particular, the ideal gas law that the theory stipulated for general circumstances in reality is only obeyed by very rarefied gases; we have already noted the first experimental observations of deviation from the ideal gas law in real gases (§ 2.2). In 1873 the first article by Johannes Van der Waals (1837-1923) appeared, in which he demonstrated that only two points of the previous theory had to be corrected to describe real gases. First of all, one must keep in mind that the molecules have nonzero volume, such that when the gas is subject to an unbounded increase in pressure, the volume remains finite (its limiting value is called the covolume and is tied to the total volume of the molecules). Secondly, the mutual attraction between molecules, that is, the cohesive effect in the gas, cannot be ignored. This attraction reduces the pressure, because in the instant in which a molecule strikes the wall, it is “slowed” by its attraction to the other molecules. Accounting for these two corrections, Van der Waals wrote the gas law that carries his name, which is valid until quantum effects are relevant and is also applicable to vapour-liquid phase transitions (his original article was titled On the Continuity Between the Liquid State and the Gaseous State). For a mole of gas, his formula is

ቀ‫ ݌‬+

ܽ ቁ (‫ ݒ‬െ ܾ) = ܴܶ ‫ݒ‬ଶ

where a and b are constants related to the fluid, and the other variables have their usual meaning.

2.15 Statistical laws We have seen (§ 2.12) that Clausius’ statement of the second principle of thermodynamics was inconsistent with traditional mechanical concepts. Mechanics had always considered mechanical processes as reversible, while the second principle of thermodynamics held them to be irreversible. Kinetic theory transformed this point of conflict into an outright paradox: if heat reduced to motion of individual molecules, governed by reversible dynamical laws, how could one reconcile the reversibility of single processes with the irreversibility of the whole? Perhaps one of the reasons for the bitter opposition of the energeticists – Rankine, Helmholtz, Wilhelm Ostwald, Mach – to atomic theory was precisely this antinomy between microscopic reversibility of dynamical processes and the second principle of thermodynamics. According to them, the paradox would have

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been resolved with the elimination of one of the premises; they therefore proposed to abandon kinetic theory and return to Mayer’s agnostic interpretation. The contradiction, however, was resolved through other means; Maxwell reformulated it, arriving at a precise question in the framework of kinetic theory: if the molecules of a gas are in continuous motion, what is the velocity of a given molecule at a given instant in time? Maxwell began by observing that Bernoulli’s hypothesis, that the speed of all the molecules was the same, was inadmissible. Even if all the molecules of a gas had the same speed at a certain moment, this idealized condition would soon be destroyed by their mutual collisions. For example, if a molecule A is struck by another molecule B perpendicularly to its motion, it is easy to see that A accelerates and B slows down. However, calculating the time evolution of every individual member of the innumerable collection of molecules contained in a gaseous mass is not possible. What can be done, according to Maxwell, is studying the velocities through a statistical approach; that is, not asking what the velocity of a single given molecule is, but rather how many molecules have a given velocity in a given instant. At the basis of his calculation, Maxwell lay down the following intuitive assumptions: no direction of motion is privileged, meaning that a molecule can have any velocity between zero and some maximum; and every isolated gas eventually settles in a stationary state in which the statistical distribution of the velocities remains constant in time. In other words, if two molecules of respective velocities a and b collide and after the collision have different velocities p and q, at the same time two other molecules of velocities p and q collide and acquire the velocities a and b, in such a way that the number of molecules with respective velocities a, b, …, p, and q remains constant. Based on these hypotheses and other less critical assumptions to smooth the creases in his argument, Maxwell gave the formula for the velocity distribution of molecules in a gas. Maxwell’s theory generated ample criticism. We limit ourselves to referring to the fundamental objection expressed by Joseph Loschmidt in 1876. Mechanical phenomena, said Loschmidt, are reversible, so if there exists a series of processes that bring a gas from an abnormal state with a non-Maxwellian velocity distribution to one with a Maxwellian distribution, there must also exist a series of inverse processes that bring the gas to the non-Maxwellian state from the Maxwellian state, which therefore is not a stationary state, contrary to Maxwell’s key hypothesis. The controversy died down only when the aforementioned molecular ray

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method, introduced by Stern, made the experimental verification of the Maxwell distribution possible, which today is widely accepted.

Fig. 2.4 – Ludwig Boltzmann

It should be emphasized that the introduction of statistical laws in physics was a development of momentous importance: there would no longer be causal dynamical laws, replaced by statistical laws that permit one to predict the evolution of a system not with absolute certainty, but only with some probability (though possibly very high). To overcome the difficulties brought up by the second principle of thermodynamics, the concept of probability of physical events, which Maxwell had not explicitly introduced, was formulated in 1878 by Ludwig Boltzmann (Fig. 2.4). Boltzmann was born in Vienna on 14 February 1844 and tragically died by suicide in Duino on 5 September 1906, after a brilliant and eventful academic and scientific career, marked by several controversies in which he defended his theories with impressive intellectual force. Boltzmann set out to place Maxwell’s kinetic theory on firmer conceptual

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bases. In 1871, Maxwell had devised the following (now classic) thought experiment: a gas contained in a box is divided into two compartments by a barrier, in which there is an aperture with a gate that can be opened and closed; a “demon” capable of seeing individual molecules stands near the hole and only opens the gate when molecules are moving in a certain direction, closing it if they move in the opposite sense; after some time, one of the two compartments will be compressed without any work being performed on the system, and the second principle of thermodynamics is violated. Thus, can the second principle of thermodynamics be admitted as a law of physics? Boltzmann, instead of answering the question directly, proposed a radical innovation: the second principle of thermodynamics is not an incontrovertible natural law but rather an extremely probably one. Boltzmann’s argument to illustrate this idea is now standard. Suppose that there are two containers connected by a small aperture that initially only contain a single molecule each. Because of its motion, one of the particles can pass through the aperture and enter the other container. In this case, the filled container experiences spontaneous compression, thus violating the second principle of thermodynamics (equivalently, the configuration of two particles in one container and zero in the other is more ordered than the original configuration, thus representing a spontaneous decrease in entropy). However, if initially there are two molecules in the container, it is clear that this phenomenon is less probable, and decreasingly probable if there are 4, 8, 16, … molecules. Therefore, the second principle of thermodynamics is not an absolute law, but an extremely probable one. Such thermodynamicscale probabilities have been calculated and, as they are quite different from the probabilities that we are familiar with in our daily lives, many illustrative examples have been developed. One of these examples is the following: if a monkey hits keys at random on a typewriter, what is the probability that it will compose the complete works of Shakespeare? The calculation can be done, resulting in a probability on par with those involved in the second law of thermodynamics. Consequently, much like we are practically sure that a monkey will never compose the works of Shakespeare, we can also be fairly confident in the validity of thermodynamic laws. The law of entropy increase in irreversible adiabatic processes is also a probabilistic law. It is thus evident that there must be a relationship between entropy and probability, which Boltzmann brought to light in his 1887 theorem, one of the crowning achievements of theoretical physics in the second half of the 19th century. Because entropy is additive and probabilities are multiplicative, one can intuit a logarithmic law relating

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the two quantities. Indeed, Boltzmann’s theorem established that the entropy of a state is proportional to the logarithm its probability:

S = k logW where W is the probability (proportional to the number of different microstates that give rise to the macrostate in question) and k is the “Boltzmann constant”, first calculated by Planck in 1900. From Boltzmann’s theorem it follows that the increase in entropy in real processes depends on the passage from one state to another more probable. The theorem acted as the bridge between classical thermodynamics and the kinetic theory of heat; allowing for all thermodynamic quantities to be calculated from statistical considerations. By the last decade of the 19th century, the physics community had accepted the revolutionary ideas of Maxwell and Boltzmann, welcoming probabilistic laws in its ideology; but between certainty and probability, even if extremely high, there is a conceptual chasm. Classical physics thus found itself facing an inevitable dualism. For every law purportedly describing a phenomenon, physics had to ask itself: is this a causal, dynamical law or is it a probabilistic, statistical law? Faced with this dualism, physicists split into two camps. A few wanted to overcome it by denying the existence of absolute laws and making all physical laws probabilistic, while the majority wanted to trace statistical laws back to elementary dynamical ones. Statistical laws, this camp argued, are syntheses of individual causal, dynamical laws, which our brains cannot reasonably keep track of all together; the probability that results from statistical laws is simply a measure, according to Jules-Henri Poincaré’s famous definition, of our ignorance. Science cannot stop at statistical laws – it must trace these back to the microscopic dynamics from which they originate, because only in this way can our minds follow the causal connections. Obviously, these physicists believed that natural phenomena are wholly deterministic. This reasoning can be summarized by an oft-cited passage written by Laplace at the beginning of the century: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present

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before its eyes.”74 The physics of the 20th century, however, would overturn Laplace’s determinism, as we will see in Chapter 8.

74

P.-S. Laplace, Théorie analytique des probabilités, Paris 1814, p. 11 of the Introduction. The first edition of the work, which appeared in 1812, did not contain an introduction.

3. ELECTRIC CURRENT

FIRST STUDIES 3.1 Galvanism News of the invention of the electrochemical battery (or voltaic pile) spread rapidly, generating interest the likes of which had not been seen perhaps since Newton. On 17 November 1801, from Paris, where Napoleon had summoned him to repeat his experiments in front of the Institut de France, Volta wrote to his brother with excitement and wonder: “I myself … am amazed at how my old and new discoveries of the socalled galvanism, which demonstrate that this is none other than pure and simple electricity moved by the contact of different metals, have provoked so much enthusiasm. Assessing [the discoveries] dispassionately, I also find them of some importance: they shed new light on electric theory; they open a new field of chemical research, due to certain unique effects that my electro-motor apparatuses produce, namely decomposing water, oxidizing, or calcining metals, etc., and they also allow for applications to medicine… For more than a year, all the newspapers in Germany, France, and England are full of [news regarding the discoveries]. Here in Paris, moreover, one might almost call it a craze, because […] the atmosphere encourages temporary fads.”75 It was not simply a temporary fad, however: the many phenomena he had discovered were truly surprising. Scientific research immediately split into three directions: studies of the nature of the new quantity, construction of ever more powerful batteries; and studies of the new phenomena. While it may appear a trifle today, the controversy regarding the nature of the new quantity introduced by Volta’s battery had a stimulating effect on research at the time. Among the many experiments executed for polemical ends, we mention a few of particular importance: in 1801 Wollaston obtained the decomposition of water, which had already been obtained with a battery, through electric discharges; in 1804 Benedetto Mojon (1784-1849), professor of chemistry in Genoa, and independently Carlo Ludovico Morozzo (1744-1804), professor in Turin, magnetized an 75

Volta, Epistolario cit., pp. 92-93.

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iron needle with the current from a battery, like Beccaria and others had magnetized needles with discharges from electrostatic machines and Leiden jars; in 1800 Wilhelm Cruikshank used a battery to produce sparks that were visible in the daytime and caused the explosion of certain mixtures, as Volta had done with his “gun”; Antoine-François Fourcroy (1755-1809) that same year used the battery to obtain incandescence in an iron coil, which even heated to the point of burning when it was placed in oxygen, as in Lavoisier’s famous experiment; Christian Pfaff (1773-1852) noticed the attraction experienced by a wrought-gold coil placed between two conductors connected to the poles of a battery: the experiment was repeated more accurately by Ritter and inspired Thomas Behrens (17751813) to construct his electrometer (today called a Bohnenberger electrometer), made up of two identical batteries (later called “dry” because they were composed of superimposed disks of zinc, copper, and paper) whose opposite poles were connected through two metal plates placed underneath a glass dome, underneath which hung a thin piece of gold foil; in 1811 De Luc replaced the two batteries with a single one, and only in 1850 did Wilhelm Hankel (1814-1899) adjust the apparatus into its modern form. It was the theory of the galvanic fluid, considered distinct from the electric fluid, and not Galvani himself, as some historians claim, that gave the name to a host of scientific and popular terms introduced in the beginning of the century, like the term “galvanometer”, used by Stephan Robertson (1763-1837) in 1801 to indicate an instrument that measures the intensity of galvanism based on its chemical effect: Ampère took a liking to the term and in 1820 used it in its modern sense.

3.2 Chemical phenomena related to current One of the first phenomena observed by Volta in his electro-motor apparatus (specifically the type also called a Wollaston pile) was the decomposition of the salts in the liquid of the battery and the calcination of the metal plates, in particular of zinc. This effect was confirmed in April of 1800 by his colleague at the University of Pavia Luigi Brugnatelli, the first scientist to whom Volta showed his new apparatus. In his letter to Banks76, however, Volta barely mentioned these phenomena, perhaps because he planned to conduct a more careful study. Therefore Anthony Carlisle (1768-1840) and William Nicholson, to whom Banks had 76

§ 7.36 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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communicated Volta’s letter before the Italian read it to the Royal Society (18 June 1800), did not know of his experiments when, having built a battery, they began their own. After a few months, the British scientists discovered the decomposition of water; to collect the hydrogen and oxygen separately, they put the platinum ends of the circuit inside two closed tubes full of water, which in turn were placed upside down on the basin full of water used in the experiment. In reality, electrochemical phenomena were not new. As early as 1769, Beccaria had reduced metal oxides through electric discharges; in 1785, Cavendish, repeating some experiments that Priestley had not had the patience to see through, obtained the formation of nitrous oxide and nitric oxide by creating a spark in an oxygen-rich sample. Repeating the experiments in 1788, Cavendish was unable to convert only 1/120th of the initial air volume: perhaps this residue had been chiefly composed of argon? This suspicion was not unfounded. With his spectacular electrostatic machine, Van Marum decomposed numerous substances in his Haarlem laboratory starting in 1785, while Paets Van Trostwijk (1752-1827), also from the Netherlands, obtained the decomposition of water in 1790, perhaps not through an electrolytic process, but through the heating provoked by a succession of many rapid discharges (at least six hundred). The previous experiments, however, did not have particular success because of difficulties in executing them and obtaining the desired effects. With the electrochemical battery, the experiments became very simple to carry out and the effects striking; the announcement of Carlisle and Nicholson’s experiments opened the door for a wealth of other analogous studies. Later that same year, William Henry announced that he had decomposed ammonia; a few months before dying, Cruikshank succeeded in building a triangle battery and observing that in solutions of metallic salts hosting electric current, the metal is deposited at the end of the conductor, in the same place where hydrogen forms for acidic solutions. Brugnatelli obtained the first deposits of silver, zinc, and copper: “I have often seen,” he wrote “the silver originating from a conductor collect on platinum or gold and copiously silver it… I have observed in analogous experiments the zincing and coppering of gold and silver with the current of the oxi-electric.”77 In addition, a few years later he announced that he had been able to gild two wide silver medallions, immersing each of them in a saturated gold ammonia solution and connecting them to the negative terminal of a battery.

77

“Annali di chimica e storia naturale” (Pavia), 18, 1800, p. 152.

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Systematic studies on the chemical effects of electric current were also conducted by Humphry Davy, born on 17 December 1778 in Penzance (Cornwall). Davy began his physics and chemistry research as a pharmacist’s apprentice, and gained fame for an experiment that he thought inexplicable by caloric theory: two pieces of ice that are rubbed against each other in vacuum partially melt. In 1797 he was appointed assistant at the pneumatic institute of Bristol, and in 1801 he became a professor of chemistry at the Royal Institution of London in the first year of its existence. His professorial debut, a public lecture on galvanic phenomena, captivated the audience because of his eloquent and evocative oration. He travelled extensively across Europe and died on a trip to Geneva on 29 May 1829. Davy showed that water is not directly decomposed by current, but that instead, current gives rise to the separation of acids and salts dissolved in water. In 1806, after many lengthy attempts, Davy succeeded in decomposing potash with current, obtaining a metal with rather unexpected properties that he baptized potassium: the metal was only slightly heavier than water, whereas at the time, the only known metals were significantly heavier than water, and it spontaneously combusted when placed in contact with water, a property that appeared nothing short of marvellous in those days. Some time later, Davy also decomposed soda, obtaining a new metal – sodium: these results caused a massive stir and had important consequences for the field of chemistry. With Davy was born a new science, electrochemistry, which in the course of the century drifted progressively further away from physics, before converging with it once again, as we will see, towards the end of the century. The quicker oxidation of metals in contact with current than isolated ones was considered the fundamental tenet of scientific electrochemistry by Ostwald, and was discovered by Giovanni Fabbroni (1752-1822), who announced it in a letter read in 1792 to the Accademia dei Georgòfili of Florence, whose proceedings, however, were only published in 1801. This delay in publication is the reason for the error committed by some Italian historians, who attribute the first chemical theory of the battery to Fabbroni. In actuality, he had believed, before the battery existed, that the cause of the contraction in a galvanic frog was not to be sought in the motion of a galvanic or electric fluid, but in a calorific or chemical effect arising from the contact of different metals. Nevertheless, Fabbroni’s work inspired both the Frenchman Nicolas Gautherot (1753-1803) and the Englishman Wollaston, who independently formulated the chemical theory of the battery in 1801, according to which the origin of the battery’s electromotive force lies in a chemical action between the metals and the interposed liquid. Controversy over the electromotive force of the battery

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lasted for the entire century: in the end, the chemical theory gained the upped hand, but it appears that the existence of the “Volta effect”, where a potential difference arises simply between to metals in contact, cannot be denied. With analogous experiments, Ritter also arrived at the fundamental tenet of electrochemistry in 1798. He is also remembered for his important studies on the battery, and in particular on the “charging column” (Ludungssäule) that he described in 1803. He observed that if current is allowed to run for some time through gold conductors inside a tube of water, and the conductors are then detached from the battery and connected to each other, the process of chemical decomposition continues in the tube, but in the opposite direction. At the end of the conductor, where initially hydrogen accumulated, there was now an accumulation of oxygen, and vice versa. The effect became more marked in a column of identical metallic disks separated by wet cardboard. Ritter interpreted the phenomenon as the column of disks absorbing the fluid that flowed from the voltaic pile and then returned it to the external circuit. This explanation was seconded by Biot, who popularized it in his treatise on physics. Consequently, it was repeated by physicists for a few decades, and explains the term secondary pile used by Hans Christian Ørsted to denote Ritter’s device. It was Volta who understood this phenomenon best: observing the chemical phenomena that occur in the secondary pile, he concluded that this was a battery that “changes” and not a battery that charges. Volta’s theory was confirmed by the experiments conducted by Stefano Marianini (1790-1866) in 1826, while Brugnatelli had observed in 1802 that a thin layer of hydrogen is deposited on the conductor connected to the positive terminal. The now-common scholastic experiments in which one demonstrates the polarisation of two platinum electrodes were first described in 1824 by Antoine-César Becquerel (1788-1878). Secondary piles could not be practically employed until scientists discovered how to obtain electric currents with sources other than a battery. This explains why they were only improved much later: only in 1859 did Gaston Planté (1834-1889) propose his famous lead accumulator (capacitor, in modern terminology), and in 1881, after the introduction of the dynamo, Camille Faure (1840-1898) improved it into the form we know today. Until the discovery of electromagnetic induction (§ 3.14), the only devices capable of generating current were Volta’s electrochemical battery and, starting in 1823, the thermoelectric battery (§ 3.11). The simplest method to obtain more powerful batteries seemed to be that of connecting

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an ever-increasing number of elements in series. Yet the Wollaston pile was unwieldy, and the column batteries were, in addition to cumbersome, unreliable, because the weight of the metallic disks made the liquid imbued in the other disks drip, rendering the batteries unusable. Volta had therefore hoped that one day scientists would be able to build entirely solid batteries, an idea looked down upon by modern critics because it implies a violation of the principle of conservation of energy, which however was developed half a century after Volta, and, furthermore, does not necessarily have to be violated, as atomic batteries have shown. In any case, Giovanni Zamboni (1776-1846) had this idea in mind when he began his first attempts at building a battery made of dry conductors in 1812 (which had already been built by Behrens, as we have seen). After numerous efforts, Zamboni was convinced of the necessity of placing a humid body between the two metal plates; it was enough, however, for it to be in its natural state of humidity. He then had the clever idea of replacing the copper and zinc plates with the “gold” and “silver” wrapping used for chocolates, which is made of paper glued to sheets of copper or tin. The natural hygrometric state of the paper is sufficient to ensure the functioning of the battery, which be made up of thousands of individual couples. Zamboni thus obtained the “dry battery”, which is named after him and has since been used in many scientific fields. He immediately realized that with this battery one did not have to account for “chemical effects, but only physical ones, namely pure electric tension.”78 Soon after, he replaced the paper with a paste of pulverized carbon and water and later, on Volta’s suggestion, with magnesium dioxide. In 1831, Zamboni applied his battery to the construction of an electric clock; an 1839 prototype kept at the physics institute of the University of Modena remained in function almost uninterruptedly for over a century, and based on the observations made throughout this period it resulted that Zamboni’s battery, besides deteriorating, became polarised, albeit very slowly. Zamboni’s contribution was not so much having invented the “dry battery”, because before him at least six different types had been proposed –Biot, Behrens (two types), Peter Merchaux, Jean-Nicholas-Pierre Hachette and Desormes, De Luc– but that of having found a practical way of building cheap, high-voltage dry batteries.

78

G. Zamboni, L’elettromotore perpetuo, Verona 1822, Vol. 2, p. 36.

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3.3 Premonitions of ionic theories Since the first experiments on the chemical effects of electric current, scientists wondered: why do oxygen and hydrogen always accumulate on opposite ends of a circuit in a solution of acidic water through which current passes? In the first years of the century, two different answers were given. Some (Monge, Berthollet) held that electric current at one end of the circuit frees the hydrogen from one molecule of water and the resulting oxygen remains dissolved in the water, while at the other end the current frees oxygen and the resulting hydrogen remains in the water. Others (Fourcroy, Thénard) held that galvanic fluid removes the hydrogen from a water molecule at one end of the circuit and transports the resulting oxygen to the other end. The mechanisms described for these schisms and recompositions remained vague – the action of the current was quasisupernatural. However, in those first years of the century the atomic-molecular hypothesis was becoming more and more consistent with the laws of equivalents, constant proportions, and multiple proportions. In this industrious environment surrounding quantitative chemistry, Christian Grotthus (17851822), a twenty-year-old from Courland, made important contributions to the study of the mechanism by which current causes chemical splitting. In 1805, studying abroad in Rome, he published a seminal paper that was later republished the following year in one of the most popular and authoritative scientific journals of the time, the “Annales de chimie” of Paris. In his paper, Grotthus likened Volta’s column battery to an electric magnet, and consequently introduced the terms positive pole and negative pole to indicate the two terminals of the battery. He extended the analogy to “elementary water molecules”, meaning the combination of oxygen and hydrogen atoms in each water particle; when current passes the atoms break away from each other, and perhaps because of friction between the two parts, the hydrogen acquires positive electricity and the oxygen acquires negative electricity. Therefore, for a row of water molecules between the two poles, the O atom of the OH molecule is attracted to the positive pole and gifts it the atom’s own charge, while the H atom, through a mechanism not described by Grotthus, recombines with the O’ of the adjoining molecule, whose hydrogen H’ recombines with the oxygen O’’ of the next molecule, and so on. An analogous phenomenon occurs for the hydrogen atoms of the molecules near the negative pole. With this series of successive separations, electrifications, and recombinations, Grotthus

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explained the fact that hydrogen always forms at one end and oxygen at the other. Grotthus’ theory, despite its rudimentary nature, its lack of clarity in a few points, and a few technical errors, lasted for over half a century, with subsequent improvements, critical among which was the dualistic theory formulated entirely by the Swedish chemist Jöns Jakob Berzelius (17791848) in 1814: all chemical compounds are made up of two oppositely charged parts. Integrated with Berzelius’s hypothesis, the Grotthus’s theory was a cornerstone in the evolution of scientific thought, because it lay the ground for ionic theories.

3.4 Thermal effects of current Among the thermal effects caused by the current from a battery, the most striking is without a doubt the electric arc between carbon conductors. In 1802, the instant in which a battery circuit was closed by an iron conductor placed on a piece of wood coal, Curtet had observed sparks so bright that they illuminated the surrounding objects. A few years later, John Children (1778-1852) found that a few carbon flakes placed in a circuit “emanated light so bright that even the splendour of the Sun pales in comparison.” Davy corroborated these results in 1810, when he obtained truly spectacular effects using a two-thousand component battery that he built on behalf of the Royal Institution. Besides a number of other experiments in which metals quickly became red-hot and melted, to the astonishment of the spectators at the first public lecture after the construction of his colossal battery, Davy also experimented with pieces of charcoal that were about an inch long and one sixth of an inch thick, inserting them in the circuit attached to a battery. Upon closing the circuit, he observed an impressive spark, and each piece of carbon burned white-hot across over half of length: “When pieces of charcoal … were brought near each other, a bright spark was produced, and more than half the volume of charcoal became ignited to whiteness; and, by withdrawing the points from each other, a constant discharge took place through the heated air, in a space equal to at least four inches, producing a most brilliant arch of light, broad and conical in form in the middle.”79 Davy immediately noted the very elevated temperature of the arc, which could melt platinum, “as if it were wax under the flame of a candle.” The length of the arc can be increased by placing it underneath 79

H. Davy, Elements of Chemical Philosophy, 1812.

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the dome of a vacuum pump, which rarefies the surrounding air. In this way, by obtaining a near-vacuum, Davy observed a dazzling arc of purple light six or seven inches long. Clearly, it was not easy to repeat Davy’s experiment, which required a battery as powerful as the one at the Royal Institution, which at the time housed the most powerful battery in the world. Because of this, when ten years later, in July of 1820, Auguste De la Rive (1801-1873) succeeded in repeating the experiment in front of the Geneva Society of Sciences, it appeared so novel to many that still today some historians erroneously attribute the electric arc to the Swiss physicist. If the arc experiment was spectacular, other thermal phenomena appeared capricious; Children, for example, on the suggestion of Wollaston, showed (1815) that when two platinum wires of equal lengths and different diameters were inserted in series in a circuit, only the thinner wire ignited, while when they were placed in parallel only the thicker one ignited; heating one part of a circuit with a lamp to make it red-hot, Davy reduced the temperature of another part, while cooling the first part with ice increased the temperature of the second. Until 1841, all attempts to explain these and other capricious experiments failed, but the widespread conviction was that the heating of conductors is tied to the resistance with which they oppose the passage of current, where a greater resistance results in a greater production of heat. This idea had already been proposed by Kinnersley to explain the heating of a Leiden jar. Davy, based on the experiments above, went further, affirming that “the conducting power of metallic bodies varied with temperature, and was lower in some inverse ratio as the temperature was higher.”80 This law is well-known today, unlike its discoverer. The first exception to the law was found in 1833 by Faraday, who, experimenting on silver sulphide, noticed that its conductivity increased with temperature.

MAGNETIC EFFECTS OF CURRENT 3.5 Ørsted’s experiment A deep relationship between electricity and magnetism had suspected since the first experimenters, who were struck by the analogy between attractive and repulsive phenomena in electrostatics and magnetostatics. The idea was so pervasive that first Cardano and later Gilbert considered it 80 H. Davy. Further researches on the magnetic phenomena produced by electricity, “Philosophical Transactions”, 111, 1821, p. 431.

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a bias and attempted to highlight the differences between the two types of effects. Yet the suspicion arose again, backed by more evidence, in the 18th century, when the magnetizing effect of lighting was discovered, and Franklin and Beccaria succeeded in magnetizing an iron needle through an electric discharge of a Leiden jar. Coulombs laws, formally identical for electrostatic and magnetostatics, further hinted at a connection. With the advent of Volta’s battery, which allowed electric current to flow for a long time, attempts to discover the relationship between electricity and magnetism multiplied and intensified. And yet, despite this feverish push in research, the discovery only came twenty years later. The reasons for this delay can be found in the dominant scientific ideas of the time; researchers only considered forces of Newtonian type, that is, forces that act between matter particles along the straight line joining them. Because of this, experimenters attempted to detect forces of this type, creating apparatuses designed to discover a supposed attraction or repulsion between a magnetic pole and an electric current (or more generally between “galvanic fluid” and “magnetic fluid”), or attempting to magnetize an iron needle by jolting it with current. In his 1802 paper, Domenico Romagnosi (1761-1835) related his experiments to highlight the interaction between galvanic fluid and magnetic fluid. Romagnosi’s paper was later invoked by Guglielmo Libri (1803-1869), Pietro Configliachi (1777-1844), and many others to give credit to the great jurisconsult for the discovery. One only needs to read the article81, however, to realize that Romagnosi’s experiments, set up using an open battery circuit and a magnetic needle, were missing electric current – at most, he could observe an ordinary electrostatic effect. On 21 July 1820, in an extremely succinct four-page paper in Latin titled Experimenta circa effectum conflictus electrici in acum magneticum (Experiments about the effect of an electric shock on a magnetic needle), the Danish physicist Hans Christian Ørsted announced the seminal electromagnetic experiment showing that an ordinary rectilinear current caused the deviation of a magnetic needle. The ensuing interest and astonishment in the scientific community was enormous, not so much for Ørsted’s solution to a problem that had long been pondered, but more because it was immediately clear that the new experiment revealed a nonNewtonian force. Indeed, it appeared from the experiment that the force between a magnetic pole and a current element is not along the line joining them, but normal to it: the effect is, as it was then called, a “force of revolution” (of which only one example was known at the time, the highly 81

G. D. Romagnosi, Opere edite e inedite, Padova 1841, Vol. 1, pp. XI-XII

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discussed rotation of the polarisation plane of light: § 3.20). This was a serious consequence, of which physics became fully aware as the years passed: Ørsted’s experiment represented the first laceration in the Newtonian conception of the world. The unease that permeated science can be seen, for instance, in the predicament of the Italian, French, English, and German translators who attempted to publish Ørsted’s article in their national language, and in many cases, having made a literal translation that they could not understand, elected to maintain the original Latin. Having described the experiment, Ørsted observed that the deviation of the needle decreased with increasing distance from the parallel conductor, did not change when different media were placed between the needle and the conductor, and inverted with the inversion of current. To keep track of the direction of the needle’s deviation, Ørsted gave a rule that lasted less than two months and was replaced by Ampère’s more convenient rule, later called the rule of “personified current”.82 What remains unclear even today about Ørsted’s paper is the explanation he attempted to provide for the phenomena he observed. According to him, the effects were due to two opposite helical motions of the positive and negative “electric matter” about the conductor, and he defined the titular electric conflict, that is, the current, as “the effect that occurs in the conductor and in the surrounding space”: it is a stretch to conflate this vague concept with Faraday’s ideas, as some authors have done. Ørsted was born in Rudkjobing, on the island of Langelande, on 14 August 1777 and died in Copenhagen on 9 March 1851. In 1799, he obtained his degree in medicine in Copenhagen, but immediately dedicated himself to physics, becoming a professor at the University of Copenhagen in 1806. According to his student Christopher Hansteen (1784-1873), his frequent lessons displayed both Ørsted’s remarkable intelligence and his experimental ineptitude. Aside from his critical discovery, which brought him honour and affluence, he was also successful in his studies on the compressibility of water and infrared radiation, and in transforming of a Coulomb scale into a sensitive electrometer, which was further improved at his urging by Johann Friedrich Dellmann (1805-1870) in 1842. The extraordinary nature of Ørsted’s discovery quickly gained the attention of experimenters and theoreticians. Upon returning from Geneva, where he had attended Ørsted’s experiments repeated by e la Rive, Arago spread news of them in Paris, and in September of 1820 built his famous 82

“Annales de chimie et de physique”, 2nd series, 15, 1820, p. 67.

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device in which vertical current passes through a horizontal board sprinkled with iron filings. Arago, however, did not notice the circles made by the filings, which are evident to anyone who repeats the experiment today. Experimenters began to clearly see these lines when Faraday announced his theory of “magnetic curves” or force (field) lines: sometimes the will to see something is enough to see it. Like Davy a month later, Arago only noticed that the conductor, in his words, “charges the iron filings as a magnet would”, from which he deduced that “it develops magnetism in iron that has not been subjected to a pre-emptive magnetization”.83 Still in 1820 (on 20 October and 18 December), Biot read two memories in which he communicated the results of an experimental study conducted with Félix Savart. Biot set out to discover the law that governs the intensity of the electromagnetic force at different distances, which had already been quantitatively given by Ørsted, as we have seen. To this end, Biot applied the method of oscillations, which had already been used by Coulomb. The experimental setup consisted of a large vertical conductor perpendicular to the axis of a magnetic needle: letting current run through the conductor, the needle began to oscillate with a period dependent on the electromagnetic force on the poles of the needle, which in turn varied with the distance between the centre of the needle and the current. From their measurements, Biot and Savart deduced that the electromagnetic force is normal to the plane formed by the wire and the segment connecting it with the magnetic pole, and that it is inversely proportional to the distance between the pole and the wire. In their statement there was no reference to the intensity of the current, as techniques to measure it had not yet been developed. Laplace, informed of the results of Biot and Savart’s experiments, observed that the action of the current on the needle’s poles could be considered the result of the partial actions of an infinite number of infinitesimal elements of current, and further inferred that a current element acts on each pole with a force that is inversely proportional to the square of the distance between the element and the pole. Laplace’s contribution to the problem is discussed by Biot.84 To our knowledge, no trace of this observation can be found in Laplace’s writings; he probably expressed it to Biot in conversation. To complete the description of this elementary force, Biot, this time alone, attempted to experimentally determine if and how the action of a 83

Ibid., p. 82. J.-B. Biot, Précis élémentaire de physique expérimentale, Paris 1821, Vol. 2, p. 122. 84

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current element on a pole varies when one varies the angle between the current direction and the segment joining the current element to the pole. The experiment consisted in comparing the action on the same needle of a straight current and an angulated one; based on the experimental data, Biot, through an unpublished and incorrect calculation (as demonstrated by Félix Savary [1797-1841]), deduced that the force is proportional to the sine of the angle formed between the current and the segment joining the point in question with the midpoint of the current element. In conclusion, what today is called the Biot-Savart law was mostly formulated by Biot, though with contributions from Savart and Laplace. Jean-Baptiste Biot was born in Paris on 21 April 1774 and died there on 3 February 1862, at the end of a long life full of experimentation, observation, geodetic measurements, and historical research on ancient Egyptian, Chinese, and Indian astronomy. He quickly gained the respect of Laplace, who allowed him the honour of reviewing the draft of his Mécanique céleste. In 1800, he became a professor of physics at the Collège de France, and in 1804 he accompanied Gay-Lussac in his first balloon ascent. Biot made numerous geodetic measurements: of particular importance was his measurement of the meridian arc in Spain, performed with the aid of Arago. His greatest contributions were in the fields of light polarisation and electrodynamics, and his treatise on physics, whose first edition was published in 1816, educated generations of physicists despite its highly conservative point of view.

3.6 The galvanometer The result of Arago’s experiment, mentioned in the previous section, which many physicists of the time had interpreted as the effect of the magnetization of a current-carrying wire, was explained on a fundamental level by Ampère, who immediately predicted (and later experimentally showed) that an iron bar placed inside a current-carrying coil becomes permanently magnetized. A new magnetization method was thus found. Ampère’s method was considerably more efficient and convenient than the previous approaches, but above all brought about the construction of a simple yet valuable device, the electromagnet, which has innumerable applications. The first electromagnet had the shape of a horseshoe and was built in 1825 by the American William Sturgeon (1783-1850); the rapidity with which it magnetized and demagnetized when the current in the wire coiled around it was turned on or off left experimenters more than a little surprised. Sturgeon’s device was independently improved by Gereit Moll (1785-1838) and the American Joseph Henry (1797-1878) in 1831: the

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improvement essentially consisted in the fact that instead of wrapping the iron magnet with a single layer of conducting wire covered in silk, they used many layers. Ørsted’s paper in Latin was followed by a second paper in German that did not enjoy widespread circulation. In it, Ørsted demonstrated the reciprocity of the electromagnetic phenomenon that he had discovered. He hung a small battery from a wire, closed the circuit, and observed that it rotated when he brought a magnet close to it: the same experiment was conducted by Ampère, to whom it is commonly attributed. In a simpler experiment, Davy, invited by Arago, demonstrated the action of a magnet on a moving current by bringing the pole of a magnet close to an electric arc. Once he built his electromagnet, Sturgeon modified Davy’s experiment into the form still repeated today in physics courses, in which the arc continuously rotates in a magnetic field. Before Sturgeon, in 1821, Faraday had obtained the rotation of a current-carrying conductor in a magnetic field with a simple setup: the end of a conducting pendulum sits in a container of mercury, where a vertical magnet protrudes from the surface. When current passes through the mercury and the conductor, the pendulum begins to rotate around the magnet. Faraday’s experiment, which Ampère ingeniously modified, was adjusted in countless ways in the course of the century. Among these we only mention the “Barlow wheel” described in 1823, because it functions as a form of electric motor and can be convenient for introducing students to the subject. The setup consists of a metallic wheel whose bottom lies in a pool of mercury, which is itself surrounded by a horseshoe-shaped bar magnet such that the bottom of the wheel lies between the magnet’s poles. When current flows from the axis of the wheel to its edge, passing through the mercury, the wheel rotates. Ørsted’s rules governing the deviation of a magnetic needle and Ampère’s equivalent formulation indicated that the deviation increases if the same current passes above and below the needle. This fact, discovered by Laplace and successfully shown in experiments by Ampère, was applied by Johann Schweigger (1779-1857) in 1820 to construct his multiplier, made up of a rectangular frame enveloped in several layers of coiled current-carrying wire; in the centre of the frame lies a magnetic needle. Nearly at the same time, Avogadro and Vittorio Michelotti built another type of multiplier, without a doubt inferior to Schweigger’s, and described it in 1823. In Avogadro and Michelotti’s multiplier, however, there was an innovation: the magnetic needle, suspended by a cotton string, was allowed to rotate along a marked sector and the whole apparatus was placed inside a glass dome.

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Initially, the multiplier appeared to be an extremely sensitive galvanometer, but scientists soon realized that it could be greatly improved. Already in 1821, Ampère had devised an astatic device, as he called it, identical to the one that had been used by Vassalli Eandi, and even earlier, in 1797, by John Tremery: two parallel magnetic needles rigidly connected with opposite polarities. This system was suspended from a point and rotated by 90 degrees whenever electric current passed through a wire that was parallel and very close to the bottom needle. In this way, Ampère showed that when the needle is unaffected by the Earth’s magnetic field (the actions of the two opposite pieces cancel), the needle aligns itself perpendicularly to the current. Leopoldo Nobili (1784-1835) had the ingenious idea of coupling Ampère’s astatic device with the single string suspension of Avogadro and Michelotti, thus creating his famous astatic galvanometer, first described at a meeting of the Accademia delle Scienze of Modena on 13 May 1825. To convey the sensitivity of his instrument, Nobili said that once the copper ends of the galvanometer are joined to an iron wire, it is enough to heat one of the two joints with one’s fingers to see a 90 degrees deviation in the needle. Nobili’s galvanometer remained the most accurate instrument in scientific laboratories for some decades, and we have already seen its valuable use in Melloni’s research (§ 1.7). In 1828, Ørsted tried to improve it by placing a horseshoe magnet in front of two poles placed at the same end of the moving elements. The attempt was unsuccessful, but it represented the first example of an instrument with an auxiliary field. In 1826, Antoine-César Becquerel built the differential galvanometer by enveloping a frame with a double conductor, which was similar to the wiring in our modern electrical systems. The four free ends are connected to two terminals of a battery such that there are two galvanometric circuits with oppositely flowing currents: if the currents are equal, the net effect on the needle is zero. This was perhaps the first systematic method to cancel electromagnetic effects in the history of instrumentation. Becquerel utilized this instrument to compare the conductivity of metallic wires, as we will discuss later (§ 3.8). A crucial improvement in the galvanometer came with the introduction of the tangent compass by Claude Pouillet (1790-1868) in 1837, as well as the sine compass, also used by Pouillet the previous year, though he likely did not know the exact theory governing it (later given by Wilhelm Weber [1804-1891] in 1840). In 1837, Becquerel introduced the “electromagnetic balance”, a progenitor of the modern “iron-sucking” ammeter that was only successful in the second half of the 19th century. Other forms of

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tangent compasses followed: Helmholtz (1849), T. M. Gaugain (1853), and Friedrich Kohlrausch (1882) all tried their hand at it. In the meantime, Poggendorff had introduced the method of mirror reading in 1826, later developed by Gauss in 1832 and applied to the galvanometer by Weber in 1846. A popular variant was the galvanometer devised by Jacques Arsène D’Arsonval (1851-1940), in which a light movable coil, placed in a magnetic field, is traversed by the current being examined.

Fig. 3.1- André-Marie Ampère

3.7 Ampère’s electrodynamics Physics owes the first chapter of electromagnetism to an exceptional man, a mathematician, philosopher, physicist, biologist, chemist, and poet – a man both fervently religious and deeply humanistic. André-Marie Ampère (Fig. 3.1) was born in Poleymieux-le-Mont d’Or, near Lyon, on 22 January 1775 and was first educated by his parents, who, upon realizing that young boy was gifted, did their best to support his studies and

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encourage his interests. In 1793, his autodidactic upbringing was halted by the tragic death of his father, decapitated by a revolutionary court in Lyon. After two years, the young man was able to recommence his studies and dedicated himself to botany and Latin poets. In 1801 he was appointed professor of mathematics in Bourg, and the following year he published his first mathematical paper on game theory, which attracted the attention of Delambre. In 1804, with the backing of Delambre, Lalande, and Laplace, he became a lecturer at the École polytechnique of Paris, later obtaining a professorship in mechanics in 1809. He published papers on chemistry, mathematics, electrology, and biology; but metaphysics and philosophy were the disciplines that caused both serenity and torment in his lifetime. In 1824, his health began to deteriorate, while his economic difficulties grew. In 1834, he published the first volume of his famous Essay on the philosophy of science; the second unfinished volume was published posthumously by his son Jean-Jacques in 1843: the great Ampère died on 10 June 1836 in Marseille, exhausted by overexertion, during a trip for his job as a secondary school inspector. Ampère came to know of Ørsted’s experiment from two communications made by Arago to the Académie of Paris on 4 and 11 September 1820. He had never specifically concerned himself with electrology, but the Arago’s communications were like a revelation for Ampère, sending him into a state of feverish experimental and theoretical study. Only a week after Arago’s second communication, on 18 September 1820, Ampère announced to the Académie des sciences of Paris that he had experimentally discovered motive actions between currents, which he called electrodynamic actions. More precisely, in his first paper Ampère called these actions “attractions and repulsions of electric currents,” and introduced the term electrodynamics in 1822. Ampère was a prolific and fortunate inventor of neologisms; the terms electrostatic, reophore, solenoid, astatic system were all coined by him. It is said that when Ampère had finished reading his paper on electrodynamic actions, a colleague remarked: “But what is new in what you have told us? It is natural that if two currents act on a needle, they must also act on each other.” According to the anecdote, Ampère, taken by surprise, did not know how to rebut this claim, but Arago came to his rescue. Taking two keys out of his pocket, he said, “these too both act on a needle, yet they do not act on each other.” The crux of this story is probably true, because in his Mémoire sur la théorie mathématique des phénomènes électro-dinamiques uniquement déduite de l’expérience, Ampère felt the need to specify that the action between currents could not

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be deduced from Ørsted’s phenomenon’s, much like the action of two pieces of iron on a needle cannot be deduced from their action on each other. Another episode is also recounted: After Ampère’s first public experimental demonstration (with his “stand”), Laplace waited for his assistant Jean-Daniel Colladon at the door while the crowd dispersed. When Colladon was about to exit, Laplace gave him a slap on the back and, looking him in the eyes, asked “Young lad, you didn’t happen to give the wire a small push?”85 Immediately after Ørsted’s discovery, the following interpretation of the effects appeared natural to physicists: a conductor becomes a magnet with the passage of current. This interpretation was followed by Arago, who was led to his experiment discussed in § 3.5 by this very idea; also Biot accepted this view and in 1820 affirmed that, when a rectilinear current acts on a magnetic molecule, “the nature of its action is the same as that of a magnetized needle placed on the edge of a wire, where its direction is always constant with respect to the direction of voltaic current.” Biot interpreted electrodynamic effects as arising from the mutual action of small elementary magnets generated by the current in a conductor: in substance, each current-carrying conducting wire becomes a magnetic tube. Biot conviction in this interpretation remained firm, despite the fact that in 1820 Gay-Lussac and James Joseph Welter (1763-1852) had devised an experiment in which two tubular rings were strongly magnetized but did not act on each other. This experiment was later repeated by Davy and Georg Erman with the same results. And yet, despite this experimental proof, Davy continued to be of the same mind as Biot; Berzelius, on his part, specified that each transversal section of a current-carrying conductor became a double magnet with opposite polarities.

85

R. de Pictet, Étude critique du matérialisme et du spiritualisme par la physique expérimentale, Geneva 1896, pp. 101-05. The oft-repeated episode is probably fictional, because it is unlikely that Laplace would have had to wait for Ampère’s public demonstration to be aware of his experiments, when Ampère himself remembered Laplace’s help in suggesting to increase the effect observed in Ørsted’s experiment by curving the same wire above and below the needle. Moreover, it is certainly untrue that Ampère held his first public demonstration in 1823 or 1824, with defective equipment that failed to work, disappointing the public and paining Ampère – as Pictet claims. As early as February 1821, Ampère had already commissioned the famous mechanic Hyppolyte Pixii to build his multi-accessory “stand”; this stand was used by Delambre on 2 April 1821 to repeat Ampère’s seminal experiments in front of the Académie des sciences of Paris.

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Ampère, however, was to propose a different brillant interpretation: a current-carrying conductor does not become a magnet but rather a magnet is a collection of currents. If we suppose, said Ampère, that a magnet contains an array of currents lying in planes perpendicular to its axis, all in the same direction, then a current parallel to the magnet’s axis will be at an angle with these currents, causing electrodynamic effects that tend to align the differently oriented currents. If the rectilinear current is fixed and the magnet is not, then the magnet will deviate; if the magnet is fixed and the current is not, then the current will change direction. It is not hard to see that Ampère’s hypothesis was extremely innovative for 1820, explaining the reticence with which it was met; Biot and Arago’s hypothesis appeared much more natural. When Faraday succeeded in rotating currents in a magnetic field in 1821, Ampère noted that such an effect cannot be explained by any possible distribution of magnets, which could generate attractive and repulsive forces, but not rotational torque. More than criticize the competing theory, Ampère tried to experimentally verify his own hypothesis. He thought that if a magnet is thought of as a system of parallel circular currents, all in the same direction, then a coil of metallic wire through which current passes must behave like a magnet, and consequently attain an equilibrium with the Earth’s magnetic field and exhibit two poles. The experiment he conducted confirmed the prediction as far as the behaviour of the coil under the influence of a magnet, but was unclear in regard to the magnetic effect of the Earth on the coil. To study this action, he resorted to using single current-carrying loop, which behaved like a magnetic sheet. These results were puzzling: a single loop behaved like a magnetic sheet, but a coil of many loops, which Ampère thought of as a system of magnetic sheets, did not clearly behave like a magnet. In attempting to clear up this ambiguous detail, Ampère realized, to his surprised, that a helical conductor behaves like a straight conductor with the same ends in electrodynamic phenomena. From this he inferred that in electrodynamic and electromagnetic effects, current elements can be composed and decomposed with the parallelogram rule. Therefore, a current element can be decomposed into two components, one in the direction of the axis and the other perpendicular to it. Integrating the actions of all the current elements of the coil, one sees that the total current is equivalent to a straight-line current along the axis and a system of circulating currents in planes perpendicular to the axis. For the currentcarrying coil to behave exactly like a magnet, the effect of the rectilinear current must be compensated. Ampère obtained this correction using a simple technique: he bent the ends of the wire back along its axis. Despite

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this adjustment, though, there remained an important difference between a current-carrying coil and a magnet: a coil’s poles are precisely at its extremities, while a magnet has internal poles86. With the aim of eliminating this last discrepancy in behaviour, Ampère abandoned his initial hypothesis of currents that are exactly perpendicular to the magnet’s axis and instead postulated that they lie in planes that are differently inclined with respect to the axis. Right after his first electrodynamic experiments, Ampère set out to find the numerical value of the force that acts between two current elements, to then deduce the formula for the force between two sections of conducting wires given their shape and position. Because he could not experiment on the (infinitesimal) current elements themselves, in 1820 Ampère initially attempted to obtain the result through the following method: first one conducts multiple careful measurements of the reciprocal action between two finite currents of different sizes and at different positions; then, one formulates a hypothesis on the action between two current elements and uses it to deduce the resulting action between two finite pieces; this hypothesis must then be adjusted until the theoretical prediction and the experimental result match. This was the classic method that was often used in such situations, but Ampère soon realized that in his case the approach reduced to near-divination, which could be avoided by following a more direct strategy. Noticing that a mobile current always remains in equilibrium between the equal forces produced by two fixed currents, whose shape and dimensions can be varied under certain conditions indicated by experiment, Ampère directly calculated the value of the action between two current elements for which this equilibrium is independent (within the experimentally-defined limits) of the shape and dimensions of the fixed currents. His experimental work allowed him to try to apply this criterion, which was much more restrictive than the first, because he had experimentally found four cases of equilibrium. Of the four, modern textbooks only list two: the equality of the absolute values of the force exerted by a current in opposite directions; equality of the actions on a mobile current by two fixed currents with parallel endpoints, one being linear and the other winding. The third type of equilibrium is obtained in a closed circuit of any shape, which cannot move an arc of current perpendicular to its plane; the fourth condition, given in November 1825, does not appear to have been experimentally demonstrated by Ampère. Riccardo Felici confirmed it by slightly modifying the experimental 86

§ 7.33 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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setup, though maintaining Ampère’s approach. Three circular currents are stacked so that their centres align and their radii increase geometrically; the middle current is not fixed, but remains in equilibrium when the ratio between the distance of the first two centres and the distance of the last two centres is equal to the ratio between the radii of adjacent circles. From these four experimental postulates, through a derivation that was anything but concise, Ampère arrived at the first of a long series of elementary electrodynamic formulas (later versions were given by Grassmann, Weber, Riemann, and others) that give the force between two current elements as a function of their intensity, distance, and positions. All of these formulas were in equal part used and criticized, but Ampère’s original expression was the following:

f = ii’

ௗ௦ௗ௦ ᇲ ௥మ

ଷ

(cosߝ െ cosߜcosߜ ᇱ ) ଶ

where f is the force, i and i’ are the intensities of the currents along the infinitesimal length elements ds and ds’, which are separated by a distance r, with İ being the angle between the two current elements, while į and į’ are the respective angles formed by ds and ds’ with the segment joining their midpoints. In the course of Ampère’s theoretical research, it resulted that the elements of a single current repel each other. This result appeared exceptionally important to him, to the extent that it could be used as the foundation for all of electrodynamics. Therefore, he set out to find a direct experimental demonstration of this fact, which he accomplished in September 1822 in Geneva, in De la Rive’s laboratory. The device employed by Ampère consisted of a container of mercury that was divided by a separator into two compartments, which were electrically connected by a floating, mobile conducting wire. When current passed from one compartment to the other, the wire moved. In reality, this experiment did not demonstrate the self-repulsion of a current-carrying wire but only the tendency of each electric circuit to enclose the largest possible area. Nevertheless, Ampère and many subsequent generations of physicists interpreted the experiment as confirmation of his theory. Using his elementary formula and considering a magnet as a system of molecular currents, Ampère derived Laplace’s first law, and from it, the Biot-Savart law (using modern terminology); he also derived Coulomb’s law for the magnetostatic action between two magnets, considered as two collections of currents.

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Another point in favour of Ampère’s theory was the fact that Poisson’s formula for the force exerted by a magnetic element on an element of austral or boreal fluid (the two forms of the hypothesized magnetic fluid) was identical to his own theory for a very small closed circuit in a plane. Thus, if one assumes that this small, plane, closed circuit is equivalent to a magnetic element, then by decomposing any finite circuit into a mesh, as physicists did for over a century, one immediately obtains that the action of a closed circuit is equivalent to the action of uniformly distributed perpendicular magnetic elements on any finite surface of this circuit. In short, one has Ampère’s famous equivalence theorem. To prove this equivalence, Ampère transformed a double integral over a curved surface into a single integral over its boundary: this theorem was generalized by Stokes, who gave it its name. Ampère realized that the same experimentally verifiable conclusions could also be reached using other elementary laws, so he emphasized another merit of his own theory; that it had traced three apparently different phenomena – magnetostatics, electromagnetism, and electrodynamics – back to a single cause: the action between two elementary current elements. He further noted that his theory had abolished “forces of revolution”, reducing all the natural forces to actions between particles along the straight line that connects them. All of the previous results were collected in his Mathematical theory of electrodynamic phenomena, uniquely deduced from experiment 87, presented to the Académie of Paris in 1825 and published in its “Acts” in 1827 (excerpts were published in 1826). In this impressive paper, which Maxwell described as “perfect in form and irreproachable in precision,” Ampère patched up the mechanistic viewpoint, which had been shaken by Ørsted’s experiment. Yet it was Maxwell’s work that would show that only a simple fix was needed. Weber had based his own theory of electric current, which he considered to be an actual transport of charged particles, on the electromagnetic action of a moving charge, a concept later adopted by Maxwell. On Helmholtz’s suggestion, in 1876 Rowland showed in a classic experiment (which, however, gave rise to a protracted controversy, eventually resolved by Poincaré in 1903) that an electric charge in circular motion produces the same effect on a magnet as a circular current. Moreover, if the velocity of the moving charge increases, then so does the force acting on each pole of the magnet. In short, the intensity of the force depends on the velocity of the charge. However, mechanistic thought had explained all phenomena by using forces that depended only on the 87

The paper can be found in Ampère’s Œevres.

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distance between particles. Rowland’s experiment not only confirmed the existence of “forces of revolution”, but also introduced a new element foreign to the mechanistic theory, which remained shaken thereafter. Yet let us return for a moment to Ampère’s paper to observe, passing over the heated polemic against Biot that was not free of personal attacks, that Ampère attributed, as he had already done in 1821, the Earth’s magnetism to the existence of currents in the interior of the globe: this was simply one of many theories, all flawed, to account for the planet’s magnetism. In 1822, Nobili backed this theory with an instrument made from “a spherical ball enveloped by a metallic wire whose loops are along the sphere’s parallels, where the ends of the wire communicate with the zinc and copper extremities of a voltaic apparatus.”88 This is the instrument described by some physics texts (primarily in the first decades of the 20th century) under the name “Barlow globe”, because Peter Barlow presented it to the Royal Institution on 26 May 1824, two years after Nobili’s publication.

OHM’S LAW 3.8 First studies on the resistance of conductors What is a conductor? It is a purely passive element in an electric circuit, answered the first experimenters to occupy themselves with galvanic phenomena; to mull over conductors is to waste time on abstruse questions, because only the source is an active element. This mindset explains scientists’ relative disinterest in matter, which lasted until at least 1840. For instance, at the second meeting of Italian scientists, held in Turin in 1840 (the first meeting had been held in Pisa in 1839 and had also taken on a political significance), during the discussion of a paper presented by Stefano Marianini, De la Rive affirmed that the ratio between the conductivities of various liquids is not absolute “but rather relative and variable with the force of the current”: Ohm’s law had already been published since fifteen years! More than twenty years passed after the invention of the electrochemical battery before the research on electrical resistance recommenced in the 18th century89, albeit in a climate of disinterest. Only 88

L. Nobili, Sul confronto dei circuiti elettrici coi circuiti magnetitci, Modena 1822; later in Id., Memorie ed osservazioni edite ed inedite, Firenze 1834, Vol. 2, p. 23. 89 § 7.28 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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the demands of telegraphy later pushed the study of resistance to a serious undertaking. It is no coincidence that one of the people who rekindled this research was a telegraph technician: Charles Wheatstone. Among the first to recommence was Davy, who in 1821 experimented on the relative conductivity of various metals, finding that silver is the best conductor. He was followed by Becquerel in 1835, who used the differential galvanometer (§ 3.6), in which each coil is inserted in parallel to the ends of two conductors being examined, which are in turn connected in parallel to the terminals of the same battery: the two conductors have equal or different resistances if, when current passes, the galvanometer needle does not deviate or deviates, respectively. Experimenting with wires of different diameter and length, Becquerel arrived at the conclusion that two wires of the same type have the same conductivity when the ratio of their lengths is equal to the ratio of their cross-sectional areas. This statement is equivalent to the modern statement relating resistance and resistivity., which some historians erroneously attribute to Barlow, who instead explicitly affirmed that the variation of diameter has little effect.90 In the same paper, Barlow noted, by moving a magnetic needle along a conducting wire, that current intensity remains constant along the entire circuit: this was not as trivial an observation as it might appear today, as Gustav Theodor Fechner (1831), Rudolph Kohlrausch (1851), and Wiedemann (1861), to name a few, all felt the need to confirm it with different circuits. Marianini dealt with the internal resistance of the battery. He was led to this by a strange result that he obtained while studying the voltage in large batteries: when he increased the number of elements in a column battery, the electromagnetic effect on the needle did not appreciably increase, causing Marianini to think that each voltaic couple acts as an obstacle to the passage of current. Experimenting on active and inactive (made up of two copper plates separated by wet cloth) couples, he arrived at the empirical relation

d=

௡஽ ௡ା௡ᇱ

where d is the effect on the magnetic needle, D is the average effect of each couple, and n and n’ are the respective numbers of active and inactive couples.91 If one takes into account the fact that the resistance of 90 91

“Philosophical Magazine”, 1825, p. 105. S. Marianini, Saggio di esperienze elettrometriche, Venezia 1825, p. 35.

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Marianini’s external circuit was almost zero, this relation appears like a special case of Ohm’s law to the modern reader. We note, however, that Marianini did not have a clear picture of the quantities at play in an electric circuit.

3.9 Georg Simon Ohm Ohm recognized the significance of Marianini’s research, though he did not draw from it in his own work, which was inspired by Fourier’s analytic theory of heat (§ 2.6). Ohm noticed that “heat flux” mechanism discussed by Fourier could be adapted to describe the flux of electricity in a conductor. Much like in Fourier’s theory the heat flux between two bodies or two points on the same body is attributed to their difference in temperature, Ohm attributed the flux of electricity from one point to another in a conductor to a difference in “electroscopic force” between the two points. To highlight the amount of time it took for new ideas to spread, it is worth noting that Ohm defined electroscopic force as the “density of electricity” at a given point, and only in 1850 did Gustav Robert Kirchoff, observing that this point of view was at odds with the principles of electrostatics, interpret electroscopic force as the “potential function of the total quantity of free electricity.” Guided by the analogy with heat flux, Ohm began his experimental studies with the measurement of the relative conductivities of different conductors. Using a now-classic approach, he inserted threadlike conductors of equal diameter but made from different materials between two points of a circuit, and varied their length until he obtained a given current intensity. The first results obtained by Ohm appear modest by today’s standards. Many historians are surprised that he found silver to be less conductive than copper and gold, though they take Ohm’s word that this was due to his use of a silver wire coated in oil, which according to him altered the apparent diameter. Experiments of the time were rife with causes of error (impurities in the metals, miscalibration of wires, inability to conduct exact measurements of their dimensions), chief among which was the polarisation of batteries, that is, the fact that, because constant (hydroelectric) batteries did not exist yet, the electromotive force in circuits changed in the time necessary to conduct measurements. These were the causes of error that led Ohm to summarize his experimental results on the change in current intensity with resistance using a logarithmic law. After the publication of Ohm’s first article, Poggendorff suggested that he abandon hydroelectric batteries and instead use a copper-bismuth

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couple, which Seebeck had recently introduced (§ 3.11). Ohm followed the advice and built a device with a thermoelectric battery attached to an external circuit, in which he inserted eight copper wires of equal diameter and different lengths. He measured the intensity of the current with a sort of torsional balance made of a magnetic needle hanging from a flattened metallic cable: when current flowing parallel to the needle caused it to deviate, Ohm twisted the cable to bring the needle back to its resting position. According to him, the intensity of the current was proportional to the torsional angle of the cable. Ohm concluded that the experimental results obtained with his eight different wires “can be represented rather well by the equation

ܽ

X = ܾ+‫ݔ‬ where X indicates the intensity of the magnetic effect of the conductor whose length is x, and a and b are constants that respectively depend on the exciting force and the resistance of the remaining parts of the circuit.”92 The experiments were modified by changing the resistors and the thermoelectric battery, but the results could always be summed up by a formula like the one above, which becomes the modern version when X is identified with the current intensity, a with the electromotive force, and b + x with the total resistance of the circuit. Having obtained the formula, Ohm used it to study the effect of a Schweigger multiplier on the deviation of a magnetic needle and the current in a circuit attached to a battery, both for components connected in parallel and in series. In this way, he explained the phenomena involving the external current of a battery that had appeared so capricious to the first experimenters. Ohm hoped that his papers would open university doors for his career, but they went unnoticed; he therefore abandoned teaching at the Cologne gymnasium and went to Berlin to frame his results in a theoretical language. In Berlin, in 1827, he published his magnum opus, a book titled Die galvanische Kette, mathematisch bearbeitet. His theory, inspired by Fourier’s analytic theory of heat, introduced the concepts and definitions of electromotive force (“electroscopic force”), electrical conductivity (Starke der Leitung), electroscopic force drop (voltage drop, in modern terminology), and current intensity. Letting u be the electroscopic force on an area q of a conducting wire, k the inverse resistance of a wire of unit 92

“Journal für Chemie und Physik”, 46, 1826, p. 160.

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length and cross section, and n the direction of the current, the total amount of electricity e that flows through the cross-section q is given by Ohm’s differential law:

݁ = ݇‫ݍ‬

݀‫ݑ‬ ݀݊

from which the ordinary form can easily be obtained:

݅ =

ܽ ‫ݓ‬

where i is the current intensity, a is the electromotive force, and w is the total resistance. Ohm applied his law to the study of a few practical circuits that he had already studies experimentally, in particular to circuits in which there are multiple electromotive forces, formulating the well-known rule for the change in voltage along a circuit. He showed that his law is valid also for liquid conductors: in 1860 Gaugain demonstrated that it also works for bad conductors, and in 1877 Edmund Hoppe (1854-1928) extended it to all types of conductors. After so much original work, the poor Ohm fully deserved a university position. Instead, he was met with nothing: only silence regarding his work. If someone wrote about his work, it was to deride it, faulting it as “incurably delusional, whose only effort is to diminish the dignity of nature.” On top of this, Ohm was not even able to re-obtain his teaching position at the gymnasium, because the scholastic official in charge of the decision was Hegelian, and thus contemptuous of experimental research. Ohm scraped by for six more years in Berlin, teaching at the war school, and later moving to the Nuremberg polytechnic. Finally, the brilliance of his work was brought to light by Poggendorff and Fechner in German, Emily Khristianovich Lenz in Russia, Wheatstone in England, Henry in America, and Matteucci in Italy. In 1841 the Royal Society of London awarded him the Copley medal, and finally in 1852 he was appointed associate professor at the University of Munich. There he died of a stroke two years later, on 7 July 1854; he was born in Erlangen on 16 March 1787.

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3.10 Electric measurements The studies of Antoine-César Becquerel (§ 3.8) were continued by Pouillet, who related them in his later editions of his popular Éléments de physique expérimentale, whose first edition appeared in 1827; the measurement method used therein was that of comparison. In 1825, Marianini demonstrated that the electric current flowing through parallel circuits is divided among all the conducting wires irrespective of their nature, in opposition to Volta, who had held that if a parallel branch is metallic and all the others are liquid, then all the current would flow through the metallic part. Arago and Pouillet spread Marianini’s observation in Franch, and in 1837 Pouillet, still unaware of Ohm’s law, made use of the observation along with Becquerel’s law to show that the conductivity of a circuit equivalent to two circuits in parallel is equal to the sum of the conductivities of the two circuits. Pouillet’s work initiated the study of parallel circuits, which was generalized by Kirchhoff in 1845 with his famous “laws”: at each “node” of a circuit, the algebraic sum of the currents is zero, in each “loop”, the sum of the products of the resistance of each side with its corresponding current is equal to the algebraic sum of the electromotive forces in the loop. The strongest impetus for improving electrical measurements, and in particular those for measuring resistance, came from telegraphy and, especially in its early years, its problems. The earliest idea of using electricity to transmit signals can be traced back to the 18th century. Volta sketched a design for a prototype telegraph and described it in a 1777 letter.93 A long iron cable is stretched across wooden posts from Como to Milan, where it is attached to a “gun” and then continues until it reaches the Naviglio canal. From the canal, the conducting cable reaches the waters of Lake Como, where it connects to its starting point through a Leiden jar. When the bottle at Como is discharged, it causes the “gun” in Milan to fire. As early as 1820, Ampère also suggested to use electromagnetic phenomena for signal transmission. Ampère’s proposal influenced several scientists and engineers: in 1833, Gauss and Weber set up a primitive telegraph line in Göttingen between the astronomical observatory and the physics laboratory, but the telegraph was made practical by the American Samuel Morse (1791-1872). Morse was a founding member of the National Academy of Design of New York, who returning from Europe to the United States in 1932, had the providential 93

Volta, Le opere cit., Vol. 3, p. 192. The letter, addressed to Father Berletti, was published for the first time in 1866, and thus had no influence on the development of the telegraph.

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idea of creating a telegraphic alphabet using only two signals, a dot and a line, obtained by the attraction of a magnet’s armour. After many attempts, he succeeded in making his first rough model work in private at the University of New York in 1835; but only in 1839 was an experimental line stretched between Washington and Baltimore. In 1844, Morse created the first American company to use the new discovery for commercial ends: this was perhaps the first sociologically important application of the study of electrology. In England, Wheatstone, who had been a manufacturer of musical instruments, dedicated himself to the study and improvement of the telegraph. He recognized the importance of resistance measurements and set out to find more reliable methods. The comparison method, which, as we have seen, was the only one in use at the time, gave dubious results, especially because constant voltage batteries did not exist. By 1840, Wheatstone had already found a way to perform resistance measurements independently of the constancy of electromotive force, and presented his instruments to Carl Jacobi. Yet the paper in which he describes them, which could be called the first publication in the field of electrotechnics, only appeared in 1842. In it, he described the “bridge” that later took his name. In reality, this device had already been described in 1833 by Hunter Christie and, independently, by Marianini in 1840: both scientists recommended the use of the “resetting” method, but their theories, which disregarded Ohm’s law, left something to be desired. Wheatstone, on the contrary, was an admirer of Ohm and knew his law well; in consequence, his “bridge” theory is still reported in textbooks today. In addition, to quickly and easily adjust the resistance of one side of the bridge, with the end goal of obtaining zero intensity in the galvanometer placed on the diagonal, Wheatstone devised three types of rheostats (a term he proposed based on the rheophore, introduced by Ampère and imitated by Jean-Claude-Eugène Péclet with his rheometer): a moving slider still in use today, inspired by an analogous device built by Jacobi in 1841; a second, in which the part of a circuit was wrapped around a wooden cylinder, whose effect could be easily altered by unwinding the wire from the wood and winding it around a bronze cylinder; and lastly, a “resistance crate”, later improved and popularized by the scientist and industrialist Ernst Werner Siemens (1816-1892) in 1860. With his bridge circuit, Wheatstone was able to measure resistances and electromotive forces, showing in particular that the width of the plates does not affect the electromotive force of a battery. his results were confirmed by the important works of Rudolph Kohlrausch (1808-1858) in

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1848-49, who also found that the electromotive force of a battery element is proportional to the difference in voltage between its poles as measured by an electrometer. Ohm’s theory, added Kohlrausch in an 1849 paper, is based on the idea that “current consists of a real propagation of electricity from one section to the next along the entire circuit,” but, if the theory is to rest on experimental results, it follows that it would remain valid even if current were not a real movement of electric fluids. Much more famous were the works of Friedrich Wilhelm Kohlrausch (1840-1910), Rudolph’s son, who from 1875 to 1898 carefully measured the conductivity of more than 260 solutions of over 50 substances. To reduce the polarisation effect, he used, as Poggendorff had done in 1840, alternating currents produced by an induction coil. He also found a way to measure the polarisation effect, which also takes place with alternating currents, as he showed by replacing the galvanometer in a Wheatstone bridge with a telephone in 1879. Like aerial telegraphy, underwater telegraphy was a significant and perhaps even stronger catalyst for electric measurements. Underwater telegraphy experiments began in 1837, when of the first important issues was determining the velocity of current propagation. Employing rotating mirrors (§ 1.5), Wheatsone had conducted an initial measurement as early as 1834, the results of which were later contradicted by the experiments of Latimer Clark, which in turn agreed with further research by other experimenters. Thomson (later Lord Kelvin) gave an explanation for these conflicting results in 1855. According to Thomson, the speed of the current in a conductor does not have a fixed value. As the speed of heat propagation in bar depends on the nature of the bar, the speed of current in a conductor depends on the product of its resistance and its electrostatic capacity. Following this theory, which was much debated at the time, Thomson dedicated himself to the problem of underwater telegraphy. The first transatlantic cable linking Britain to the United States remained in service for about a month before it stopped working. Thomson recalculated the conditions for the new cable, made extensive measurements of resistance and capacitance, devised new transmission apparatuses like the astatic reflection galvanometer, which he later replaced with the “syphon recorder” – the new transatlantic cable finally began operation in 1866 with great success. This first large-scale work of electrotechnical engineering was accompanied (and followed) by the formulation of effective units of measurement for electric and magnetic phenomena. We will further discuss electromagnetic metrology later. Here we only note that Thomson was one of the most commendable scientists in the creation of a comprehensive system of electromagnetic

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units of measure, which Weber had predicted in 1851 as an extension of the system proposed by Gauss. It was Thomson who built an absolute electrometer94 and measured the electromotive force of Daniell battery; it was he who established an absolute measure of electrical resistance from the heat produced by a resistor when subject to a known current; it was once again he who prompted the British Association to organize the famous Committee for Electrical Units of Measure. William Thomson was born in Belfast, Northern Ireland on 26 June 1824. Having obtained his academic degrees from Cambridge in 1845, he moved to Paris, where he worked for a year in Regnault’s laboratory. In 1846, he returned to his homeland and became professor of natural philosophy at the University of Glasgow, a position which he held for 53 years. In 1847 he became personally acquainted with Joule and was fascinated by his ideas on the nature of heat. A few years later, he presented his theory of thermodynamics (§ 2.12), which he would always consider his most important scientific work. Besides what we have already discussed, among his principal contributions we mention his study of the oscillatory nature of the discharge in a Leiden jar and his work on the marine compass, tides, and ocean depth probes. Thomson is especially remembered for his insightful pedagogic work; one could say that for the second half of the 19th century his scientific status inspired and dominated British experimental research. In 1892 he was made Baron Kelvin of Lags, a town nearby his habitual residence in Netherhall, Scotland, where he died on 17 December 1907.

ELECTRIC CURRENT AND HEAT 3.11 The thermoelectric effect Since 1794, Volta has repeatedly carried out the following experiment: a frog freshly prepared à la Galvani was placed with its posterior legs dipped in a glass of water, while its back and spinal cord were immersed in a second glass of water; when he closed the circuit with an iron wire, one end of which had been immersed in boiling water for a few minutes, Volta observed violent contractions in the frog, which continued until the end of the wire cooled.95 An analogous experiment was described by Ritter 94

§ 7.33 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambdridge Scholars Publ. 2022. 95 A. Volta. Nuova memoria sull’elettricità animale in alcune lettere al signor Abate Anton Maria Vassalli. Lettera prima, in Id., Le opere cit., Vol. 1, pp. 26566. The letter was first published in “Annali di chimica e storia naturale”, 5, 1794, p. 132. The limited dissemination of the “Annali” was not inconsequential for the

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in 1798. The two experiments went unnoticed by many scientists, perhaps including Thomas Johann Seebeck (1770-1831) when in he read his papers (later collected in a classic paper published in 1825) to the Academy of sciences of Berlin in 1821. The effect discovered by Seebeck is now famous; he described one of his many experiments as follows: a bar of bismuth is welded on both ends to the ends of a copper coil, if one end of the bar is heated by a lamp and the other end is kept cold, a magnetic needle inside the coil rotates, indicating the passage of current from the copper to the cold end of the bar. The phenomenon was publicized in 1832 by Ørsted, who also introduced the terminology still in use today: “It will from now on undoubtably be necessary,” he wrote, “to distinguish this new class of electric circuits by an appropriate term; and as such I propose the expression thermoelectric circuits or perhaps thermometric: at the same time, one would be able to distinguish the galvanic circuit by the name hydroelectric circuit.”96 But because regardless of method used to produce current, the circuits were always the same, Ørsted’s nomenclature was applied to sources and soon became the premier one. Only Seebeck did not accept it and continued to call the phenomenon magnetic polarisation of metals, titling the aforementioned 1825 paper Magnetische polarisation der Metalle und Erze durch Temperatur-Differenz. The title reveals the theory that had led Seebeck to his discovery of the phenomenon, that is, that current is a form of magnetic polarisation, and that the humid conductor is unnecessary as long as the metals in contact are different. It also results from the title that Seebeck was aware that the cooling of a juncture produces a current in the opposite direction of the one produced by heating. As Volta had done for different contact voltages, Seebeck experimentally established several thermoelectric scales: in all of them the first element was bismuth, while antimony was one of the last. It follows that the antimony-bismuth pair is one of the most active for thermoelectric effects. In 1829, Antoine-César Becquerel succeeded in establishing an unambiguous thermoelectric scale, which was completed by his son Alexandre-Edmond in 1864. In reality, these thermoelectric scales were of little value, even less than those for contact voltages. Yet how does current intensity vary with the temperature difference between the two junctures? Seebeck had realized that the two quantities are not proportional; in 1823 James Cumming showed that if the experiment’s lack of fame bordering on oblivion, which remained almost to this day. 96 “Annales de chimie et de physique”, 2nd series, 22, 1823, p. 199.

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difference in temperature is too large, the current can even become inverted. This observation was confirmed in 1856 by Thomson and in 1873 by Peter Tait. Richard Avenarius, based on Thomson’s experiments, gave a theoretical interpretation of the phenomenon in 1863, while in 1892 Wilhelm Wien and Ludwig Holborn found a third-order equation describing the reverse problem: deducing the temperature difference between the junctures from the thermoelectric electromotive force. Let us return, however, to the first decades of the century. In 1823, Fourier and Ørsted showed that the thermoelectric effect is additive and built the first thermoelectric battery, made up of three antimony bars and three bismuth bars welded in alternating order at their ends to form a hexagon. The battery was greatly improved by Nobili in 1829, who instead of arranging the metallic bars polygonally in a plane, placed them almost vertically along a cylindrical surface in a box, which was then flooded with mastic to the point that the bottom junctures were submerged and the top ones protruded slightly from the box. A further improvement was made by Melloni the following year with his construction of the prismatic version. Still in 1830, using a Melloni battery and his galvanometer, Nobili built an incredibly sensitive thermomultiplier for the time: it could detect the heat emanating from a person 18-20 arm’s lengths away (about 10 metres). This vigorous and prolonged study of the phenomenon and thermoelectric batteries can be attributed to the exceptional constancy of thermoelectric current, which does not appreciably change as long as the temperature difference between the two junctures stays the same. Seebeck had suspected that electric current could produce cooling in a juncture if it flowed in an opposite sense to the current generated by the heating of that same juncture. However, he was unable to observe the phenomenon. Jean-Charles-Athanase Peltier (1785-1845), a watchmaker until the age of thirty, and then a scientist and in particular an electrologist, succeeded in this regard. In 1834, in the course of and experimental study on the conductivity of antimony and bismuth, the problem of determining how temperature varies inside a (homogeneous or heterogeneous) currentcarrying conductor arose. Peltier measured the temperature at various did of a thermoelectric circuit using thermoelectric pliers connected to a galvanometer. He discovered in this way that the temperature in the juncture between two different metals can abruptly vary and even decrease; he obtained the greatest effect for the bismuth-antimony couple. In short, electric current could also produce cooling. Becquerel, de la Rive, and other physicists remained unconvinced by Peltier’s experiments, perhaps also because Peltier was considered somewhat of an outsider to scientific research due to his artisan origin. To dispel the doubts, Peltier

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proceeded to demonstrate the phenomenon directly using an air thermometer: the experiment that he did is remembered for in textbooks. In another type of experiment, Peltier welded two different metal shafts into a cross and sent a thermoelectric current through two opposing ends attached to a galvanometer; after some time, he stopped the flow of current and connected the same galvanometer to the other two ends of the cross: the galvanometer indicated a current due to the heating or cooling of the cross juncture. Whether heating or cooling is produced in the juncture was clarified in 1838 by Poggendorff and, independently, in 1840 by Luigi Pacinotti (1807-1889), father of Antonio, the inventor of the continuous current dynamo (§ 3.24). Pacinotti communicated his results at the second meeting of Italian scientists, held in Turin in 1840. Ries measured the decrease in temperature through a Kinnersley thermometer and found it appreciable, especially in the bismuth-antimony couple when the current flows from the former to the latter. A more extensive study was conducted in 1852 by Quintus Icilius, who was able to establish that the amount of heat produced in a juncture traversed by a current i is given by the formula

‫ ݅ܽ = ݍ‬ଶ ± ܾ݅ , where a and b are constants related to each couple (the first term on the right represents Joule heating). Attempts to ascertain whether the Peltier effect also exists in electrolytic solutions did not give conclusive results. Theoretical research on the Peltier effect and the analogous Thomson effect (characterized by the production or absorption of heat in a homogeneous conductor of variable temperature depending on the direction of the current) have continued for several years.

3.12 Joule’s law In forty years following the invention of the battery there was no dearth of attempts, some unsuccessful and some incomplete, to discover the law governing the production of heat by electric current. The failures can be attributed to a lack of clarity in the notions of current intensity and electrical resistance in use at the time, and the consequent lack of welldefined units and appropriate instruments of measurement. In addition, unawareness of Ohm’s law led experimenters to place wires of different resistance in circuits, erroneously believing that in this way only the

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resistance was changed and not the current. For these reasons, and a general lack of coherent experimental objectives, early research into the question was unsuccessful, like the studies conducted in 1834 by William Harris. Harris placed current-carrying conductor in a bottle with a coloured water thermoscope in the nearby air. It appears that he wanted to deduce the amount of heat produced by different types of conductors based on the thermoscopic indications. In 1841, Joule began his experimental work on the heat produced by current. He had the clever idea to pre-emptively calibrate his tangent compass by inserting it in a circuit with a voltameter, as Faraday had explained (§ 3.16), and gave the first scientific definition of a unit of current intensity. The heating device was made up of the conductor in question inside of a glass container, which held a certain amount of water and a sensitive thermometer. In three consecutive experiments, each of which included two resistors (conductors with some resistance) in series immersed in two identical calorimeters, Joule found that at fixed current, the amount of heat produced is proportional to the resistance of the conductor. This first result led him to formulate a hypothesis on the effect of current intensity through the following complicated reasoning: “I thought that the effect produced by the increase of the intensity of the electric current would be as the square of that element; for it is evident that in that case the resistance would be augmented in a double ratio, arising from the increase in the quantity of electricity passed in a given time, and also from the increase of the velocity of the same.”97 Joule probably meant that the heat produced by current is caused by the collisions if the particles in the electric fluid with the particles in the conductor. Now, if the intensity of the current increases, the velocity of the particles in the electric fluid increases and therefore the collisions are more vigorous and frequent, as the quantity of electric fluid passing in a given time through the conductor increases. In any case, Joule subjected his hypothesis to an experimental test, finding that the amount of heat measured by the calorimeter in which the copper coil was immersed was so similar to the calculated value that he could consider his law fully verified, at least in metallic conductors. Rather more original were the experiments carried out by Joule to test the validity of the law inside a battery, in electrolytic conductors, and for induced currents. He began with the heat produced inside a battery. Two main difficulties arose: the comparison of the internal resistance of the

97

Joule, The Scientific Papers cit., Vol. 1, p. 64.

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battery with another sample resistance and the measurement of the heat produced inside the battery based on the metal electrolyte solution. To overcome the first difficulty, having built a sample resistor out of copper, Joule then placed the battery of unknown resistance r in series with a known resistor, first of resistance R and later of R’, and measured the resulting currents respectively i and i’. Ohm’s law allowed him to write

(ܴ + ‫ܴ( = ݅)ݎ‬ᇱ + ‫݅)ݎ‬Ԣ ,

, from which deduced r. To overcome the second problem, Joule preemptively measured the amount of heat produced by p grains of zinc dissolved in sulphuric acid. He then immersed the battery in a calorimeter, made current pass through it for a certain amount of time, and measured the amount of heat produced. The measurement thus included the contributions of all the corrections due to the different specific heats of the components in the battery. Based on the average current that passed through the circuit, he deduced the mass p’ of the dissolved zinc (through Faraday’s electrolytic law) and consequently the amount of heat produced in the solution. The difference between the two heats gave him the amount of heat produced by the passage of current inside the battery. Repeating the experiments with many types of batteries, he concluded that the law he had found for metals also works for the internal resistance of a battery. With the same acumen and experimental meticulousness, Joule extended the law to electrolytes (in which case one has to measure the resistance to electrification, as he called it, or the electromotive polarisation force, according to modern terminology) and induced currents. His work culminated in an 1842 paper, which we already discussed (§ 2.9), where he showed that under any conditions, for any type of conductor, and for any current, the heat produced is proportional to the resistance of the conductor and the square of the current. We have dwelled on the details of Joule’s research to highlight a characteristic aspect of scientific research: the critical importance that he attached to physical measurements and the accuracy with which he carried them out. For the nascent science of electrical metrology, Joule’s work was stimulating example. In 1889, the International Congress of Electricists named the standard unit of work after him; Joule died that same year on 11 October in Sale. He was born on 24 December 1818 in Selford, near Manchester, to a producer of beer. James Prescott Joule was forced to carry out his initial education at home due to his sickly nature, and therefore was effectively an autodidact when he described an

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electromagnetic invention of his in 1838, the modest work with which the twenty-year-old entered the world of science. He became the owner and directory of his father’s beer factory, but after some time abandoned this career to focus on scientific research. Aside from his famous experimental measurement of the mechanical equivalent of heat and the research that we discussed above, Joule conducted other important studies, among which are the experiments in collaboration with Thomson on the cooling produced by expanding a compressed gas, today called the Joule-Thomson effect, the experiments on volume changes in solutions, and those on the temperature variation caused by the straining or compression of solids. His scientific writings, collected when he was still alive by the Physical Society in two volumes (1884-87), inspired generations of experimental physicists. Naturally, many scientists corroborated Joule’s law and shed light on its consequences. Here we only mention a few. In 1844, Lenz (and independently, in 1846, Domenico Botto [1791-1865], professor of physics at the University of Turin) established that the maximum amount of heat that a generator can provide to an external circuit is attained when the resistance of the circuit is equal to the internal resistance of the generator. Lenz then began the difficult study of how the temperature of a conductor changes as a function of the current that flows through it and the type of surrounding medium. A completely general law, however, still does not exist. This research was obliquely related to the experimental work of Heinrich Wiedemann (1826-1899) and Franz, who in 1853 established the constancy of the ratio between thermal and electrical conductivity, as well as its proportionality to the absolute temperature: the Widemann-Franz law, valid for all metals, was explained through the quantum theory of electrons in metals in 1927, and Paul Drude succeeded in finding the proportionality constant. Lastly, we mention that in 1853 Clausius showed that in any electric circuit, the work performed is proportional to the heat produced. After this discovery, scientists preferred to talk about the work done by a current rather than the heat produced, and in consequence employ mechanical units of measurement.

THE WORK OF MICHAEL FARADAY 3.13 Rotational magnetism Ampère wondered whether his hypothesized molecular currents inside a magnet could “by influence” generate analogous currents in magnetized iron, nickel, and cobalt. To answer this question, in 1821 he conducted the

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following experiment: a mobile coil was placed with its edge very close to a fixed coil through which current passed, but the mobile coil did not move under the action of a strong magnet. According to Ampère, this showed that “the proximity of an electric current does not induce, by influence, any excitation in a metallic copper circuit, even in the most favourable of circumstances for this influence.”98 It followed based on his reasoning that molecular currents are pre-existent in magnets. Even Faraday, perhaps unaware of Ampère’s experiment, had attempted to detect the “influence” of one current on another, setting up experiments in 1822 and repeating them, still without success, in 1825. He failed because the phenomenon did not occur in the manner expected by him. Indeed, Faraday thought that like a current modifies the magnetic state of a magnet, so a magnet must modify the state of a current. In short, he expected a static phenomenon, so the real momentaneous effect escaped him in two 1825 experiments: in the first, he placed a magnet inside a metallic rotor connected to a galvanometer; in the second, imitating Ampère’s experiment, he ran current through a coiled wire very close to another coil attached to a galvanometer. Because of all these failures, physicists did not even suspect that electromagnetic effects were at play in the phenomenon observed by Arago in 1824: the oscillations of a magnetic needled in a compass with a sturdy metallic frame were strongly damped. Arago initially suspected that the phenomenon was due to iron particles present in the copper, and thus asked the young chemist Jean-Baptiste Dumas (1800-1884) to analyse the sample. From Dumas’ analysis, it resulted that there were no traces of iron whatsoever inside the copper. Arago then concluded that he was faced with a new phenomenon and immediately began further experimental studies, soon establishing that a metallic disk rotating underneath a magnetic needle shifts it from its equilibrium position, even dragging it into rotational motion when the disk’s rotational speed is large enough. Physicists thus called the effect “rotational magnetism”, and in a five years period the effect generated numerous papers. Arago’s experiment was repeated, adjusted, and theoretically explained by the preeminent physicists of the time: Herschel, Charles Babbage, Christie, Nobili, Pierre Prevost, and Poisson. An interesting modification was made by Sturgeon: when a metallic disk is oscillated between two opposite magnetic poles, the oscillations are quickly damped. Herschel also observed that the dragging effect of the rotating disk on the magnetic needle diminished significantly

98

Letter from Ampère to Van Beck, in Ampère, Œuvres, cit. p. 195.

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if one cut radial incisions into the disk, or better yet, replaced the disk with metallic powder spread and fixed on top of an insulator. Nobili and Liberato Baccelli experimentally concluded that rotational magnetism only occurs with conducting disks. However, Arago contested their conclusion with experiments in which the oscillations of a magnetic needle on a glass plate were increasingly damped when the needle approached the plate, becoming very pronounced when the needle was 0.91 mm away from the plate. Obviously, the effect observed by Arago was almost entirely due to air resistance, but only with Maxwell was the nature and importance of friction in gases understood. Ampère replaced the magnetic needled with a current-carrying coil, obtaining the same results, which to him indicated a “definitive confirmation” of his theory.99 Poisson formulated a mathematical theory of the phenomenon in 1826, though it was too complicated and replete with stopgap hypotheses to be of lasting interest. Jean-Marie-Constant Duhamel gave a qualitative theory in 1824 that earned the backing of many physicists. According to Duhamel, for every pole of a magnetic needle, a pole of the opposite sign forms in the plane below it, caused of the decomposition of the neutral magnetic fluid in the plate. During its rotation, the needle spreads these poles on the disk, and because they require some time to dissolve, they precede the needle and drag it in rotational motion. This clever theory, which lacked an experimental basis, was shattered by Arago in 1826, who observed the repulsion between the pole of a vertical magnetic needle and a rotating disk nearby. The discovery of rotational magnetism, aside from his project to measure the speed of light (§ 1.5), has been the last important contribution to scientific research of Jean-François-Dominique Arago. One of the most brilliant French physicists of the first half of the 19th century, Arago was born on 26 February 1786 in Estagel, in the Pyrenees, and died in Paris on 2 October 1852. After leaving the École polytechnique, he entered the astronomical observatory of Paris and, through Laplace’s support, was sent to Spain to conduct a measurement of the meridian arc. The scientific expedition coincided with the French invasion of Spain, and Arago consequently experienced a series of adventures that ended with him being imprisoned in Algeria. Returning to Paris, in 1809 he succeeded Lalande 99 Ampère, Œuvres cit., p. 434. Ampère mentions the mutual attraction between a magnet and a disk of “any substance”. A few scholars believe that the phrase included an accidental omission, and that Ampère intended to say “any conducting substance”. We, on the other hand, believe that there was no omission, and Ampère wrote exactly what he wanted to say, because he shared Arago’s opinion in opposition to Nobili and Baccelli.

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as professor of analytic geometry at the École polytechnique while maintaining his position as an astronomer at the Paris observatory. His scientific fame was a result of his work in optics and electromagnetism, which little by little we have discussed here and in the previous chapters. Dedicating his later career to politics, in 1830 he became a republican member of parliament and in this capacity worked for the expansion of railroads, telegraphs, and other projects for scientific and technical development. After the 1848 revolution, he became a minister of war and the navy. In his brief time in office, Arago introduced important liberal reforms, including the abolition of slavery in French colonies. Arago was also a prolific and talented popularizer of scientific thought, his works were famously collected and published after his death in 17 volumes (1854-62).

3.14 Electromagnetic induction One of the greatest, if not the greatest discovery made by Faraday (Fig. 3.2) was that of electromagnetic induction in 1831, after his many failed attempts (§ 3.13) to detect the phenomenon that he believed existed because of the symmetry of nature. Faraday obtained the most marked effect with a device (Fig. 3.3) consisting of an iron ring inside two different wire coils, one connected to a battery and the other to a galvanometer. When the first circuit was closed, he observed a deviation of the galvanometer’s needle; the deviation occurred in the opposite direction when the circuit was opened. This critical experiment, later modified in many ways, allowed him to obtain “electricity from magnetism,” the fundamental goal of his research: one only has to insert a magnet inside a solenoid connected to a galvanometer to see deflection in the needle, and remove the magnet to see the opposite deflection. Six years earlier, when he expected a permanent effect, the same experiment (§ 3.13) had appeared a failure to Faraday. At this point, Faraday acutely connected the phenomenon that he had discovered to rotational magnetism, which he thought could be explained by currents induced in the disk. To verify this conjecture, he rotated a copper disk between the polar ends of a magnet, attaching a galvanometer to the axis and edge of the disk. While the disk rotated, the galvanometer indicated a current with constant direction and intensity that varied with the velocity of the disk. Along with the physical principle, Faraday thus created the first example of a non-battery powered generator of electric current, launching the great modern electrical industry and paving the way for the wealth of applications we see today.

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Fig. 3.2 – Michael Faraday (portrait by T. Philips)

Fig. 3.3 - Faraday device for induced currents. Source: “Philosophical transactions”, 1812.

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However, Faraday was not interested in applications; the experiment instead allowed him to uncover the qualitative laws of the induction phenomenon. Indeed, from his experiment he deduced the rule that gives the direction of the current in a rectilinear conductor moving near a magnetic pole. It was in this occasion that Faraday, for the first time, spoke of “magnetic curves”. “By magnetic curves,” he wrote, “I mean the lines of magnetic forces, however modified by the juxtaposition of poles, which would be depicted by iron filings; or those to which a very small magnetic needle would form a tangent.”100 The experiments were modified in several ways, using conducting wires and disks, moving the magnets while keeping the circuits fixed, and moving the circuits around fixed magnets (or the Earth). Faraday was eld to the conclusion that the induced electromotive force is independent of the nature of the conductor. He explained the phenomenon with a theory that remains essentially unchanged today: “When an electrical current is passed through a wire, that wire is surrounded at every part by magnetic curves, diminishing in intensity according to their distance from the wire, and which conceptually may be likened to rings situated in planes perpendicular to the wire or rather to the electric current within it. These curves, although different in form, are perfectly analogous to those existing between two contrary magnetic poles opposed to each other; and when a second wire, parallel to that which carries the current, is made to approach the latter, it passes through magnetic curves exactly of the same kind as those it would intersect when carried between opposite magnetic poles.”101 If there is no relative motion between inductee and inductor, then current does not arise because force lines are not cut; when the inductee moves away from the inductor, it cuts through force lines in the opposite 100

M. Faraday, Experimental Researches in Electricity, § 114. This book contains Faraday’s most important scientific works on electrology, which he presented in thirty series to the Royal Society of London, from 24 November 1831 to 24 October 1855, and were contemporaneously published in the “Philosophical Transactions” of the society. Faraday himself collected them in three volumes: the first, published in 1839, contains the first fourteen series; the second and the third, published in 1855, contain the rest. The third volume also includes other writings on electromagnetism that were published at different times. The series are divided into numbered sections denoted by §§ that span all of his works; we will thus indicate the sections instead of the pages, following common usage. The three volumes were republished many times and translated into German and Russian. We will follow the edition Lavoisier, Fourier, Faraday, Encyclopaedia Brittanica, Chicago-London-Toronto 1952. 101 Ibid., § 232.

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sense as it would if it moved towards it, thus the induced current has the opposite sign. If the two are at rest but a current passes through the inductor, it is as if the magnetic curves move “across the wire under induction, from the moment at which they begin to be developed until the magnetic force of the current is at its utmost; expanding as it were from the wire outwards, and consequently being in the same relation to the fixed wire under induction as if it had moved in the opposite direction across them, or towards the wire carrying the current.”102 Few words, but an important idea. Of particular importance were Faraday’s studies on the induced currents produced by the Earth’s magnetism. Hanging a metallic solenoid with its axis in the same direction as a magnetic needle, he rotated it by half a revolution and the galvanometer connected to the ends of the solenoid measured a current whose direction depended on the direction of rotation.103 The effect increases appreciably if an iron core was placed inside the solenoid. In 1844, Luigi Palmieri (1807-1896) obtained strong induced currents from terrestrial magnetism, using an apparatus made up of a coiled conductor rotating in the earth’s magnetic field. Many physicists got to work on Faraday’s trail. Here we only mention a few names and details. In 1832, Nobili and Vincenzo Antinori experimentally studied the distribution of induced currents in Arago’s disk and noted that the current lines in the two halves of the disk were symmetric, allowing them to deduce the normal component of the current that Arago had already observed. The issue was also studied by Matteucci, who showed in 1857 that current curves could be much more complex than previously thought. Arago’s disk can be replaced with a cylinder or any other conducting body that rotates in a magnetic field. Faraday and Henry had clearly stated that induced currents have the same characteristics of any other electric current; in particular, therefore, they produce heat, as Foucault demonstrated in 1855. Nevertheless, it is not fair to call the currents induced in metallic masses “Foucault currents”, as French physicists did. Joseph Henry, who Americans consider to have preceded Faraday in discovering the phenomenon of induction, discovered self-induction, which was independently re-discovered in 1833 by Salvatore Dal Negro (1768-1839) and the following year by William Jenkin and Antoine Masson (1806-1860) at the same time. Henry’s research on higher-order induced current, that is, induced currents produced by other induced 102 103

Ibid., § 238 Ibid., § 148

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currents (experimentally observed by Marianini the year before), was especially important. Although it may seem obvious today, this existence effect was far from clear a priori. Studying higher-order current led Henry in 1842 to show that the discharge in a Leiden jar is not a single passage of electricity from one end to the other, but a series of rapidly damped electric oscillations. Based on clues from other works, though perhaps not Henry’s, Helmholtz also presented this hypothesis in his paper on the conservation of force (§ 2.10). Potential theory, which we will discuss in § 3.18, allowed Thomson to establish the scientific basis of the oscillatory hypothesis and study the discharge process from a theoretical point of view; in 1855 he gave the necessary conditions for continuous discharge and oscillating discharge, whose duration, t, he wrote as

‫ = ݐ‬2ߨξ‫ ܮܥ‬, where C is the capacitance of the Leiden jar and L is the length of the discharge circuit. No substantial details have been added to Thomson’s theory, which was experimentally verified through the study of images produced by discharge in a rotating mirror by Wilhelm Feddersen (18321918), who dedicated almost all of his scientific activity to the phenomenon, beginning with his doctoral thesis in 1857. In 1834, Emil Khristianovich Lenz (1804-1865) observed that the practical rules given by Faraday and Nobili to determine the direction of induced currents involved too many different cases, while they could all be reduced to a single case using Ampère’s electrodynamic law. He thus formulated the law that carries his name and succinctly stated is the fact that induced current always tends to oppose the effect that has caused it. Franz Neumann (1798-1895) based his theory of induction, presented in two momentous papers in 1845 and 1847, on Lenz’s law and the use of Ohm’s law for induced currents, as well has his own principle: induction occurs in a brief, fixed time interval, proportional to the velocity of the moving conductor. Solving a problem posed by Ampère, Gauss gave the following unpublished explanation: the action between two electric charges does not only depend on their distance, but also on their relative velocity. Coulomb’s law is only valid for two charges at rest. Weber repeated the idea of his teacher in 1846 and gave the corrected version of Coulomb’s formula for two charges in motion. Through this formula, it can be seen that the action between two current elements obeys Ampère’s elementary law, and the entire theory of induction thus follows, in complete agreement with Neumann’s theory.

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Slightly more original was Helmholtz’s theory, contained in the oftcited On the Conservation of Force (1847), later expanded by Thomson. Helmholtz showed that the induction of electric currents could be mathematically deduced from Ørsted’s electromagnetic and Ampère’s electrodynamic phenomena with the assumption that energy is conserved. Yet the laws of Neumann,Weber, Helmholtz, and the analogous works of Jérémie Abria and Henry seemed to contain theoretical hypotheses that were not fully grounded in experiment, so uncovering the laws of electromagnetic induction relying “only on experimental data and the method used by Ampère to find the elementary formula” was the stated goal of a careful theoretical-experimental investigation conducted between 1851 and 1856 by Riccardo Felici (1819-1902). The results were collected in an important paper titled Sulla teoria matematica dell’induzione elettrodinamica that was published in 1854 and 1857. In it, Felici gave a theoretical explanation of the induction phenomena observed when a principal circuit is opened, when two inducting circuits are in relative motion, when one inducting circuit is in motion in a magnetic field, and when two parts of a circuit are in relative motion. Felici’s theory was the subject of much discussion for the entire 19th century, but at the end scientists concluded, especially because of Maxwell’s clarifying intervention, that it was equivalent to the theories of Neumann and Weber, though more grounded in experiment.

3.15 The nature of force lines and unipolar induction Faraday, after having employed the “lines of force” terminology for two decades, thought it important to return to their study to better clarify their nature and associated effects, both new and old. The new description remained phenomenological: lines of force, the scientist explained, are those traced by a small magnetic needle; or equivalently they are the lines along which a coiled wire can move without experiencing any tendency to generate current, while the coil will experience this tendency when moving in any other direction.104 Yet what physical mechanism transmits magnetic action along these lines? Is transmission another action at a distance, like gravity, or is it mediated by a special agent like the case of light, heat, or electric current? Faraday favoured the second hypothesis; the agent could be the ether, which in addition to transmitting radiation, could also propagate magnetic actions. Nevertheless, he made this brief remark in passing, and prudently did not pursue the matter further. 104

Ibid., § 3071

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Much of Faraday’s work, completed in 1851 and described in series XVIII and XXIX of Experimental Researches, was dedicated to the study of the shape and distribution of force lines, or as we would call them, magnetic field lines. To understand how field lines are positioned both inside and outside a magnet, Faraday built the first prototype of a device that he later simplified by replacing two magnets with a single one, which was also electrically connected to two ends of a solenoid. Faraday carried out the experiment in 1831 using this version of the device. If the magnet and coil rotate together, no current is produced in the coil, but if one of the two rotates and the other remains still, then the current is generated in the coil. European physicists called the phenomenon unipolar induction, a rather abstruse and certainly imprecise expression. From the wide variety of experiments he carried out, Faraday inferred that lines of force originate inside the magnet; in other words, that they are closed loops. Such lines can be cut both near and far from the magnet. When a current coil connecting one pole of a magnet to the equator completes a full rotation around the magnet, it cuts all the lines of force only once: regardless of the size and shape of the coil, a full rotation always produces the same amount of electricity. Faraday deduced that lines of force represent an immutable quantity of force and expressed his hope to find a unit of measure, like he had earlier for the conventional graphical representation, in which the intensity of force is indicated by “their concentration or separation, i.e. by their number in that space.”105 In a space in which the conventional representation of lines of force has equal density everywhere, the current induced in circuits that are moved in a given way can be taken as a measure of the magnetic forces. Faraday thought that limited regions above the Earth’s surface constituted such a space and therefore built rectangular and circular circuits of equal area, verifying that one half of a full rotation through the terrestrial lines of magnetic force always produced the same amount of current. He concluded that the current depends only on the number of lines of force pierced by the circuit. This was the last great discovery that Faraday made in the field of electromagnetic induction. In Faraday’s last study, a single point remained unclear and was the subject of extended controversy: in the phenomenon of unipolar induction, do the lines of force produced by the magnet remain fixed when it rotates about its own axis, or are they dragged by the motion of the magnet? In the second case, unipolar induction falls within the framework of ordinary 105

Ibid., § 3122.

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induction as the reciprocal effect of magnetic rotations; if, on the other hand, the magnetic field remains fixed as the magnet rotates, then the phenomenon must be attributed to a distribution of electricity on the surface of the magnet. Perhaps it is superfluous to add that this problem is wholly unrelated to the question of ether dragging (§ 1.6). Faraday initially held that the system of magnetic force lines does not move with the magnet, much like rays of light from the Sun do not rotate with the Sun’s rotation: therefore, one could think of the magnet as rotating among its own forces.106 Later that same year, however, he asserted that he had reason to believe that the lines of terrestrial magnetic force remain attached to the Earth and rotate with it.107 Starting in 1839, Weber also studied unipolar induction with a specialized experimental apparatus, deducing some laws and concluding that the Amperian hypothesis of molecular currents did not appear to agree with his own theory of unipolar induction. Julius Plücker built a device, known to physicists in the second half of the 19th century perhaps more due to the fame of its inventor than its intrinsic utility, composed of a rotating metallic disk with two bar magnets fixed normally on its surface in the same direction, parallel to the axis of the disk, with their equator in the plane of the disk. When the disk was rotated, a current was generated between its edge and axis of rotation. Plücker used the device to interpret the electric charge of the Earth, according to him caused by unipolar induction, which accumulated positive charge at the north pole and negative charge at the equator. This theory was adopted and further developed by Erik Edlund (18181888), a famous Swedish physicist of the time. To interpret the phenomenon of unipolar induction, he hypothesized that in any conductor the “molecules of the electric fluid” or the two electric fluids contained in it can also move by mechanical effects; their motion creates a current that interacts with all other currents, like the molecular currents in a magnet. He supported his hypothesis with the following experiment: when placed around a bar magnet, a current-carrying metallic ring moves until reaching the equator, where it finds a point of stable equilibrium if its current is in the same direction as the molecular currents of the magnet. If, on the other hand, the current in the ring is opposite to the molecular currents of the magnet, then the equilibrium is unstable, because when the ring is shifted slightly away from the equatorial position, it continues to move further away and eventually passes the magnet’s pole. 106 107

Ibid., § 3090. “Proceedings of the Royal Institute”, 11 April 1851.

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Another case of unipolar induction was studied by Edlund. A vertical magnet is fixed perpendicularly to the plane of a metallic disk and at its centre. The disk is rotated around its axis, and consequently every electric fluid molecule in the disk acquires a velocity proportional to its distance from the axis and therefore is subject to the action of the magnet. A study of the dynamics of this device reveals that a neutral circle forms inside the disk; the electric molecules between this circle and the edge are pushed towards the edge, while those between the circle and the centre are pushed towards the centre. Placing two movable contacts on the disk can then indicate the passage or lack of current depending on their position. An experimental study conducted by Felici108, which became famous in 1855, confirmed the theoretical consequences deduced by Edlund, who drew the following general conclusion from all of his research: “Unipolar induction ought not to be considered a real induction, but an ordinary electrodynamic phenomenon due to the action of the magnet upon the electric currents produced by the motion of the conductor relatively to it. As for the action of the magnet, whether the currents on which it acts are produced by special electromotive forces, or by the electric fluid being carried along in the direction of motion of the conductor, is quite immaterial.”109 In short, according to Edlund there was no doubt that currents arise in the rotational motion of a conductor because of the motion imparted to the electric fluid present inside it (two years earlier, Rowland had shown that a moving charge is equivalent to a current: § 3.7); the main objection to this theory was that the velocities of the electric molecules appeared too small to obtain detectable currents. Edlund replied that the velocity of electric molecules in galvanic current had never been measured, and thus could not be equated with the velocity of propagation of electric motion as the two had no necessary relation: one could be insignificant while the other enormous. On the specific question of unipolar induction, Edlund held that the magnetic field of a rotating magnet remains at rest, without being dragged into rotational motion by the magnet. The primary aim of Edlund’s research was to interpret atmospheric electricity, according to him caused to the unipolar induction of the Earth, which makes the atmosphere electropositive and the ground electronegative. Near the polar regions, the electropositive fluid flows towards the electronegative Earth and causes the auroras, while at other latitudes 108

R. Felici, Sur les courants induits par la rotation d’un conducteur autour d’un aimant, in “Annales de chimie et de physique”, 3rd series, 44, 1855, pp. 343-46. 109 E. Edlund, Recherches sur l’induction unipolaire, l’électricité atmosphérique et l’aurore boréale, in “Annales de chimie et de physique”, 5th series, 16, 1878, pp. 67-68.

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destructive discharges are produced. Edlund closes his paper with the admission that much is still unknown about auroras and atmospheric electricity, but reasserts that “the unipolar induction of the Earth plays a very important role in the explanation of these phenomena, and should not be overlooked by the scientists who in the future will concern themselves with the subject.”110 This theory, opposed by Hoppe in a series of articles from 1886-1887 and supported by Ernst Lecher (1856-1926) in 1895, is credited with introducing the idea of “molecules of electricity” that can move inside conductors, or as they are known today, free electrons in conductors.

3.16 Electrolysis and electrolytic dissociation To the electricity produced by rubbing, hydroelectric batteries, and thermoelectric batteries, scientists added the electricity produced by electromagnetic induction. Faraday thus thought it opportune to weigh in on the controversy over whether the nature of electricity is independent of its mode of production, which, though not as heated as the beginning of the century, was still very relevant. He observed that electric effects, following a convenient but “philosophically” ungrounded classification, could be separated in two types: tension effects (attraction, repulsion, discharge) and current effects (thermal, chemical, physiological). Using batteries of 100-140 components, he obtained unmistakeable signs of attraction and repulsion, while he obtained current by heating the space between two metallic ends of a circuit with a spirit lamp. Conversely, by employing special techniques, he also obtained current effects with electrostatic machines. For example, he discharged a Leiden jar onto a galvanometer circuit that included a metre-long humid string, and the galvanometer needle deviated. Repeating one of Wollaston’s 1801 experiments with appropriate adaptations, passing current through a solution of copper sulphate with twenty turns of an electrostatic machine, he obtained the deposition of metallic copper on a platinum wire. Faraday ended the debate on the correspondence between the electric fluid and the galvanic fluid by affirming that “electricity, whatever may be its source, is identical in its nature.”111 Once he had demonstrated the equivalency, Faraday thought it opportune to establish a common unit of measurement. To this end, he gave the first ballistic use of a galvanometer and demonstrated that a 110 111

Ibid., p. 107. Faraday, Experimental Researches cit., § 360.

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Leiden jar battery, when charged in a certain way, and a galvanic battery produce the same effect on the galvanometer needle and the same chemical effect. From this result, he deduced the following law: “chemical power, like the magnetic force, is in direct proportion to the absolute quantity of electricity which passes.”112 In the course of these studies, conducted in the first trimester of 1833, Faraday stumbled upon “dry” chemical decomposition. He had observed that a small piece of ice introduced in a battery circuit interrupts the flow of current, which is restored when the ice melts. To investigate whether the phenomenon was particular to ice, he later experimented with lead chloride, potassium chloride, silver chloride, and solids and insulators at ordinary temperatures, establishing that when melted, these substances contribute to current and are decomposed. Experimenting with a variety of compounds, Faraday reached the conclusion that the conductivity of these bodies is tied to their chemical composition, thus dispelling the preconceived notions held by all the experimenters of the time, who believed, based on a statement in Davy’s Elements of Chemistry, that the presence of water was necessary for electrochemical decomposition and thus for the construction of a battery. He confirmed his result by building batteries with nonaqueous liquids, like potassium chlorate, chlorides, and iodides. Having done this, he tackled the problem of explaining the mechanism for electrochemical decomposition. For the same reasons given by textbooks today, he rejected the idea that the forces of the electric field cause the scission of molecules, asking why, for instance, a sheet of platinum has sufficient force to detach a hydrogen particle from its corresponding oxygen particle, but then not enough force to prevent it from escaping. Consequently, a knowledge of the nature of current was necessary to study this issue. In his previous series Faraday displayed a certain perplexity in this regard: “By current, I mean anything progressive, whether it be a fluid of electricity, or two fluids moving in opposite directions, or merely vibrations, or, speaking still more generally, progressive forces.”113 Experiments on electrochemical effect matured his thinking and brought him to conceive of current as “an axis of power having contrary forces, exactly equal in amount, in contrary directions.”114 Thus, the greatest experimental physicist of the 19th century stripped the concept of current of any possibility of a mechanical model and instead abstracted it, rendering it vague and almost devoid of physical meaning. An “axis of

112

Ibid., § 377. Ibid., § 283. 114 Ibid., § 517. 113

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power”, after all, can evoke confused mechanical notions, but concretely only indicates a direction. The following sections of his Researches in part explain the reasons for this choice of definition. Current has the ability to weaken chemical affinity along one direction of its flow and strengthen it along the other. For example, let (ab), (a’b’), …, be adjoining molecules along the axis of power, starting from one pole. The atomic grouping a moves towards the pole not so much because it is attracted by it, but because it is abandoned by the grouping b to which it had been attached; b in turn moves to attach to a’ because the current has weakened the chemical affinity between a and b and strengthened it between b and a’: in short, the substances produced at the poles are emitted and not extracted through attractive forces by the dissolved compounds.115 This theory is similar to that of Grotthus (§ 3.3), though more elaborate and perhaps more contrived, certainly more vague. Compared to Grotthus’ theory it has the advantage of dispensing with the ideas of poles and attraction, but at the expense of introducing a near-magical concept, the ability of current to adjust chemical affinities through a mysterious and undescribed mechanism. The seventh series of Faraday’s researches, presented in 1834, was critical for the study of the chemical phenomena related to current. It began with the proposal of a new nomenclature to describe the phenomenon of chemical decomposition, suggested by the famous historian of science William Whewell (1794-1866), to whom he had turned with the explicit desire to eliminate the existing term because, as he wrote in 1838, “The word current is so expressive in common language, that when applied in the consideration of electrical phenomena we can hardly divest it sufficiently of its meaning, or prevent our minds from being prejudiced by it.”116 On Whewell’s advice, then, the term pole (suggesting an attractive centre) was replaced with electrode, which was further subdivided into anode and cathode. In choosing these terms, the motion of molecules was not taken into account as it was not included in Faraday’s theory. Instead, they were based on the hypothesized terrestrial currents that supposedly caused the Earth’s magnetic field (§ 3.7); consequently, Faraday introduced the terms anion and cation, with ion being the generic name, and electrolyte to indicate the substance that undergoes chemical decomposition in the process of electrolysis. By setting up a circuit with a main branch and two subbranches, each equipped with a voltmeter, Faraday showed beyond reasonable doubt that 115 116

Ibid., §§ 519, 537. Ibid., § 1617.

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the quantity of electrolyte “decomposed [is] exactly proportionate to the quantity of electricity which has passed, notwithstanding the thousand variations in the conditions and circumstances under which it may at the time be placed… the products of the decomposition may be collected with such accuracy, as to afford a very excellent and valuable measurer of the electricity concerned in their evolution.”117 He called the measurement apparatus volta-electrometer, later shortened to voltmeter, describing five different types and proposing the first practical unit of measure for electricity: the quantity of electricity that decomposes one-hundredth of a cubic inch of water. Experimenting with voltmeters connected in series containing different solutions, Faraday realized that when the quantity of electricity is kept fixed, the amount of electrolyte decomposed changes with the type of electrolyte. After many trials, he reached a conclusion (though not entirely confirmed by his experimental results) that can be stated in modern terms as follows: a fixed amount of electricity frees an amount of substance proportional to its chemical valence. The laws of electrolysis led Faraday to reflect on the electric forces that hold atoms together in a molecule. In reality, he observed, we do not know what an atom is, and we know even less about electricity. Nevertheless, a great number of details bring us to regard the atom as a small particle with electric forces arising from its properties. Now, the laws of electrolysis lead to the conclusion that the amount of electricity that passes through an electrolyte is the same amount present in all of its particles that are separated by the passage of current.118 Introduced in this way, the idea of an elementary unit of charge went nearly unnoticed at the time, but was later revisited by scientists in the last twenty years of the century and elevated to a fundamental tenet of electromagnetic theory (§ 4.3). The critical importance of Faraday’s study was immediately recognized by the scientists of the time, as evidenced by dazzling explosion of studies that followed it and the Royal Society’s bestowing of its annual gold medal prize to Faraday for “the seventh series regarding the nature of the electro-chemical action”. Faraday’s theory on electrolysis left much to be desired: it was based on Berzelius’ jumbled ideas on the constitution of salts and exhibited a similar level of clarity. Without delving into particulars, we simply note that, for example, according to Faraday, the metal deposited at the cathode 117 118

Ibid., § 732. Ibid., § 855.

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is always the effect of a secondary reaction; that in the electrolysis of solution of sulphuric acid it is only water that is directly separated and not the acid, etc. A correction to Faraday’s ideas came from a great admirer of his, John Frederic Daniell (1790-1845), who in 1836 had made Faraday’s electrochemical discoveries the subject of his physics course. In 1839, Daniell demonstrated that for the theory to remain consistent, one had to admit that in salts one ion is metallic and the other comes from the remaining atomic group. This new conception led him in 1840 to describe the electrolysis of an aqueous solution of sulphuric acid in the terms we use today. Equally interesting was another phenomenon observed and studied by the British scientist: during the electrolysis of a salt, the concentration of the electrolyte changes near the electrodes in a manner that depends on the nature of the ions. The emphasis placed by Daniell on this phenomenon appeared exaggerated to other physicists, but not to Johann Wilhelm Hittorf (18241914), who repeated his experiments and in 1853 published a lengthy paper in which he observed that it is entirely unnatural to suppose the velocities of anions and cations to be the same, as Daniell had done, and that the different concentrations at the electrodes not only show that they are not equal, but allows one to determine them, as Hittorf himself demonstrated with an experimental tour de force. It follows that, all else being equal, the conductivity of electrolytes depends on the speed of the ions. In the course of his numerous measurements of electrical conductivity (§ 3.10), Friedrich Kohlrausch also measured the velocity of the ions and concluded that the velocity of each ion is independent of the velocity of the other ion to which it is bound in the molecule, arriving at a law nearly opposite to the one considered “natural” by Daniell. Yet how do ions move inside a solution? Both Grotthus’ theory and that of Faraday did not predict particle movement, describing only decomposition and recomposition at rest. In 1857, Clausius re-examined the issue discussed by Faraday: it cannot be the attractive force between the poles that separates ions in a molecule, because in this case it is clear that the electrolytic process should begin only when the electromotive force applied surpasses a certain threshold, while experiment shows that any electromotive force gives rise to current. Clausius therefore supposed that each molecule is formed by two oppositely charged parts, termed “partial molecules”, and relied on kinetic theory, which he had formulated that same year (§ 2.14). Due to thermal excitement of the electrolyte molecules, the positions of the partial molecules inside them can vary, and it can happen that an ion (we use Faraday’s terminology rather than

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Clausius’ for increased clarity) is more attracted by an oppositely charged ion in another molecule than its partner. In this way, scissions and recomposition of molecules arise, while free ions travel through the solution in all directions. The electric field orients the motion of ions “not in such a way that they become entirely regular, but such that, among the great variety of motion that still exists, the two privileged directions are predominant.”119 Clausius’ original idea, which unknowingly to him had already been advanced by the British chemist Alexander Williamson (1824-1904) in 1852, was met with limited support (Quincke, Rudolph Kohlrausch) and was proposed again in 1884, in a different form, by a young Swedish scientist and student of Edmund, Svante Arrhenius (1859-1927). Arrhenius, with a different method than the one used by Friedrich Kohlrausch, measured the electrical conductivity of 36 substances in highly dilute solutions and concluded that a portion of the dissolved salts, which he termed active, takes part in the electrical conductivity, while the other salts, called passive, do not take part in the effect. Furthermore, the active fraction is proportional to the chemical activity: in other words, electrical conductivity can be taken as a measure of chemical activity. Wilhelm Ostwald (1853-1932) soon confirmed Arrhenius’ law, measuring the conductivity of a dozen substances and comparing it with their chemically-measured activity levels. As we have mentioned, Arrhenius’ result was simply another iteration of Clausius’, but still said nothing about the mechanism of the phenomenon. Further progress on this question was tied to the study of solutions. In 1867, Moritz Traube (1826-1894), a German wholesale wine merchant and amateur biochemist, published his research on osmosis, which had already been discovered by Nollet in 1748 and studied in the first half of the 19th century by other physicists (Fischer, Magnus, Philipp Jolly, Thomas Graham). Traube discovered, apparently independently of his predecessors, the existence of semi-permeable membranes, which allow the solvent but not the solute to pass through them. Traube’s studies were continued by the German botanist Wilhelm Pfeffer (1845-1890), who in 1877 introduced the concept of osmotic pressure and found it to be proportional to the concentration and the absolute temperature, and inversely proportional to the volume of the solution in question. Pfeffer’s discovery went unnoticed by physicists until 1886, when one of Pfeffer’s friends brought it to the attention of a young and already famous Dutch chemist, Jacobus Hendricus Van’t Hoff (1852-1911), who was later given 119

Clausius, Théorie analytique de la chaleur cit., Vol. 2, p. 170.

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the 1901 Nobel prize in chemistry. Van’t Hoff noted that osmotic pressure obeys laws that are analogous to those describing gases, and that it is numerically equivalent to the pressure that the solute would have if it were transformed into a gas at the same temperature and volume of the solution. This marked a return to the analogy already proposed by Gay-Lussac in 1802 between the diffusion of liquids and the expansion of gases. Gas laws, and in particular Avogadro’s law, can be applied to some solutions (but not all). More precisely, Van’t Hoff found that, in general, the ideal gas law could be rewritten in the following form:

‫ܴܶ݅ = ݒ݌‬, where i = 1 for some solutions and i > 1 for others. The Dutch scientist went further, demonstrating through thermodynamic theorems that some phenomena outside the range of osmotic effects could be connected to osmotic pressure, making possible its measurement through the (still difficult) extrapolation of data from non-osmotic phenomena. In the same period, the French chemist François-Marie Raoult (18301901) conducted cryoscopic research (a term he coined) on aqueous and non-aqueous solutions and experimentally arrived at the following theorem: equimolecular quantities of different substances dissolved in equal quantities of solvent lower its melting point temperature by the same amount. This theorem, however, did not apply to certain aqueous solutions. Van’t Hoff then showed that Raoult’s theorem could be deduced from the laws governing osmotic pressure, which is also proportional to the temperature decrease. Moreover, the solutions for which Raoult’s theorem does not apply are the ones whose coefficient i > 1 in Van’t Hoof’s formula. The determination of molecular weights through the measurement of their vapour density had also given rise to similar anomalies, which in that case had been explained by the formation of polymers. With the research on solutions reaching this point, it may seem simple to collect all of these details and channel them in support of Clausius’ interpretation. However, aside from the fact that the historical coordination of facts is an entirely different matter than their practical, real-time coordination, Clausius’ theory of dissociation faced a seemingly insurmountable criticism: if at least a part of the electrolyte splits into ions, why do these not exhibit the expected chemical behaviour? For example, if potassium chloride becomes chlorine and potassium in a solution, why does the chlorine not develop into a gas and the potassium not burn in contact with the water?

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On his part, in the meantime, Arrhenius observed that anomalous solutions with respect to osmotic pressure were electrolytic, and a relation could be established between the coefficient of activity that he defined in his 1884 works and the i coefficient of Van’t Hoff. In short, the electrical conductivity of electrolytes and osmotic pressure were connected to each other. Arrhenius was led to consider the dissociation of the molecules in the electrolyte and cleared the hurdle posed by the objection to Clausius’ hypothesis, observing that ions are not the same as their corresponding chemical elements because, unlike them, they are electrically charged. Ultimately, they are simply isomers of the chemical elements. Only when they reach the electrodes and lose their additional charge can they reacquire their ordinary chemical properties. Arrhenius’ seminal paper on electrolytic dissociation was published in 1887; that same year Planck independently arrived at the same conclusions through a different approach. The theory was completed by Ostwald and Walther Nernst in the following years, while in 1893 Friedrich Kohlrausch calculated the displacement velocity and the amount of electricity transported by a gram molecule of a substance; in a later 1898 work he studied the effect of temperature, whose increase on one hand increases the mobility of ions and on the other hand reduces their degree of dissociation, such that for some electrolytes, like phosphoric acid, it can give rise to a negative temperature coupling coefficient. As one can see from story we have just told, the path towards the theory of electrolytic dissociation was long, difficult, and full of obstacles. Success was achieved through a close collaboration between physicists and chemists, marking the beginning of a rapprochement between the two sciences (§ 3.2) that from then on became increasingly manifest. The developments that led to the theory of electrolytic dissociation were of much broader importance than the specific result they achieved, motivating our lengthy digression.

3.17 Constant batteries The first experimenters had immediately realized that when they closed a battery circuit, a short time later the intensity of the current flowing was greatly diminished. In general, they attributed this to an additional resistance of passage, as Marianini called it, that increased as the current passed through the circuit: an explanation that explained nothing. Faraday gave an unorthodox interpretation, attributing the phenomenon to the formation of zinc sulphate, which halted further oxidation. He was led to this interpretation by the experimental observation that the battery could be

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recharged by removing zinc oxide from the electrode, either by dispersing it in the solution without reproducing it, thus interrupting the current for a short time, or by agitating the liquid with a pen to remove it without breaking the circuit.120 Though this was certainly a modest result for a physicist of Faraday’s stature, it motivated further study by Daniell, a chemistry professor at the Royal College of London and his admirer and friend, who, though convinced that “the science of chemistry would date, from [the publication of Faraday’s discoveries], one of its great revolutions and eras of fresh impulse,”121 had open-mindedly examined them and even improved the explanation for the mechanism of electrolytic conduction, as we previously mentioned. Daniell realized that hydrogen bubbles adhere to the copper plate of a battery for some time. Not only did one have to eliminate zinc oxide then, as Faraday had proposed, but one also had to prevent the hydrogen from depositing on the copper. Because other experimenters like Antoine-César Becquerel had observed that battery performance improves if the copper is immersed in a copper salt solution, Daniell thought to build a battery with two liquids separated by a porous barrier. Thus, the constant battery, as Daniell called it, was conceptually born. The initial form of the battery was modified by Daniell, who realized that removing zinc sulphate was useless and therefore did away with the siphon. John Peter Gassiot later replaced the ox oesophagus separator with a porous barrier. Simplified in this manner, the battery acquired great scientific and practical importance: for fifty years it was employed in telegraphy and laboratories as a source of constant electromotive force. After Daniell’s battery, hundreds of similar constant batteries were built; here we mention a few of them. In 1839, William Robert Grove (1811-1896) built a battery with zinc amalgamated inside dilute sulphuric acid and platinum inside nitric acid, obtaining a much greater electromotive force than Daniell (about 1.8 times as much). In 1841, Bunsen made an anode out of agglomerated coke powder and oil and obtained an electromotive force nearly equal to that of the Grove battery. In 1867, Georges Leclanché (1839-1882) built a battery with one electrode made of zinc and the other made of manganese peroxide mixed with powdered coal: both electrodes were immersed in a solution of ammonium salts. In 1878, Latimer Clark (1822-1898) built a constant battery that slowly replaced Daniell’s by merit of its small temperature coefficient composed of a platinum wire immersed in mercury and a zinc bar covered 120 121

Faraday, Experimental Researches cit., §§ 1003-39. J. Frederic Daniell, “Philosophical Transactions”, 126, 1836, p. 107.

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in mercury sulphide paste in a concentrated solution of zinc sulphate. In 1884, Lord Rayleigh built the device in the shape of an “H”; that same year the German Siegfried Czapski (1861-1907) replaced Clark’s amalgam of zinc with an amalgam of cadmium and the zinc sulphate solution with one of cadmium sulphate. Czapski’s battery was popularized by the British-American industrialist Edward Weston, a well-known manufacturer of scientific instruments. Under his name, Czapski’s battery replaced Clark’s in 1905 as the standard for electromotive force.

3.18 Potential theory Historians attribute the introduction of the function that would later, starting with Green, be called potential, to Lagrange, who in 1777 considered it to solve certain mechanical problems. In reality the credit should go to Euler, who had already stated in his 1736 treatise on mechanics that the elementary work of a force is equal to the sum of the elementary work performed by its components along three orthogonal axes. In Theoria motis corporum rigidorum (1765) he developed the concept, using it to derive the laws of motion, now called the EulerLagrange equations because they were popularized in a paper published by the French-Italian scientist in the proceedings of the Berlin Academy of Sciences in 1773. Even Laplace’s equation, discovered by the French mathematician in 1796, had been written down by Euler in his dissertation Principia motus fluidorum, which appeared in the 1756-57 issue of the “Novi commentarii” of the Academy of St. Petersburg. In 1813, Poisson showed that the Laplacian differential operator is nonzero at every point on an attractive mass. The proof is fairly simple. Poisson divided the attractive mass in two parts, of which one was homogeneous, spherical, and contained the point in question. By directly performing the calculation on the sphere, Poisson found the Laplacian equal to -4ʌȡ with ȡ being the density of the sphere; for the other part he found a Laplacian of 0. The Laplacian of the total mass is the sum of the Laplacian of its parts and therefore is equal to -4ʌȡ. The proof was heavily criticized, but Poisson, ever the prolific worker, responded with two other, completely different proofs, which appeared in 1824 and 1829, respectively. Born on 31 June, 1781 in Pithiviers, near Loiret, Siméon-Denis Poisson, was not considered a bright young lad by his family. Upon graduating from the École polytechnique in 1800, he was immediately appointed lecturer of mathematics, and in 1809 became a professor of rational mechanics in the science department of Paris; in 1816 he became an examiner at the École polytechnique and maintained this

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position until his death in Paris on 25 April 1840. Poisson contributed to mathematics (theory of equations, calculus of variations, curvature of surfaces), mathematical physics (capillarity, theory of heat, motion of elastic fluids), and astronomy (invariability of the sidereal day, libration). Soon we will discuss his works on electricity and magnetism. Some modern critics are of the opinion that the fame that accompanied Poisson’s name for the entire 19th century was ill-deserved, going so far as to say that he stole the ideas of others. This judgement is too harsh and perhaps influenced by his disreputable politics: initially posing as a fervent republican and opponent of Napoleon, Poisson then supported the second restoration, from which he earned various honours and the title of Baron (1852); rumour had it that he was chosen to take part in juries rather more frequently than what mathematical probability would have predicted. In a historic paper published in 1811, Poisson extended potential theory to electrostatic phenomena. He admitted the hypothesis of two electric fluids and based his treatment on the postulate already experimentally discovered by Beccaria, Cavendish, and Coulomb: in a system of charged conductors in equilibrium, the electric force at a point inside a conductor is zero. Considering a charged conducting sphere, Poisson showed that at any point on its surface the following equation must hold:

െ

ௗ௏ ௗ௫

= 4ߨߪ

where ı is the electric density at the point considered. The scientist sensed that this theorem could be made more general, but his formulas did not lend themselves to generalization. Laplace came to his aid, finding a general proof and communicating it to Poisson, who inserted it in his paper.122 Today this theorem due to Poisson and Laplace is frequently called Coulomb’s theorem123; but it was Poisson and Laplace who both explained and rigorously demonstrated it. From this theory, Poisson easily deduced that electrostatic pressure (or “electrostatic tension”, as it was called in the XIX century) is proportional to the fourth power of the density, explaining the effect of the discharge between two charged extremities. The theoretical study of the distribution of electricity on the surface of conductors led Poisson to results that agreed 122

“Mémoires de l”Académie des sciences de l’Institut de France”, 12, 1811, p. 34. § 7.22 of M. Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 123

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with Coulomb’s experimental observations; the agreement reassured Poisson, who saw it as a confirmation of his own theoretical structure. In two other papers published in 1824, Poisson further extended the theory of potential to magnetism, basing his work on Coulomb’s hypothesis124. The theory, though criticized in some points, was pivotal because its results remained the same even if the fundamental hypothesis was changed, as Thomson showed in 1851. Furthermore, it directly inspired the theory of dielectrics (§ 3.27). Among the many results of the theory, we note the following: in a hollow magnetic sphere of constant thickness, under certain conditions, the points inside the cavity do not feel the effect of external magnetic masses. Poisson thus discovered magnetic screening, which had been experimentally known since Porta’s time125. The theory also disproved the hasty conclusion reached by Peter Barlow (1776-1862) whose experiments, though carefully conducted, had erroneously shown that magnetism, like electricity, is confined to the surface of a magnet, or, at best, penetrates less than 1/30th of an inch (0.85 mm) into its interior. The magnetic sphere problem led Poisson to return to the behaviour of a conducting hollow sphere in an electric field. He showed that this type of sphere also exhibits the same screening property with one main difference: while for magnetism the screening effect depends on the thickness of the walls of the cavity, the electrical effect is independent of thickness.126 Contemporaries did not realize the importance of this theorem, to the extent that Poisson’s role in discovering it is even ignored by historians. Potential theory caught the attention of George Green, an unusual British mathematician who was born in Sneinton, near Nottingham, on 14 July 1793. Green was a baker and later a miller until the age of forty, and consequently an autodidact; he enrolled in a university college in 1833, five years after he had published his magnum opus. He died in his hometown on 31 May 1841. Green’s seminal work, An Essay on the Application of Mathematical Analysis in the Theories of Electricity and Magnetism, appeared in 1828 and went completely unnoticed, both because its author was an unknown outsider to the field and because the paper was not published in a scientific journal, but rather self-published using funds raised by his fellow townspeople. Only in 1850 did William Thomson discover the work, 124 § 7.33 of M. Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 125 § 3.19 of M. Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 126 “Bulletin des sciences, par la société philomatique de Paris”, 1824, p. 54.

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which he published in instalments from 1850 to 1854 in a scientific journal. Green, despite his meagre bibliography, declared that his work was connected to that of Cavendish, Coulomb, Laplace, and Poisson: few references but good ones. A key role in the work is played by the function which Green called the “potential function” and defined as “the sum of all the electric particles acting on a given point divided by their respective distance from this point.” The research is aimed at the discovery of a series of relations between the potential function and the electric density of the masses that produce it. After demonstrating that the first derivative of the potential function gives the force arising from the particle that produces the potential, Green showed through a brief argument, later repeated by Maxwell and reproduced in textbooks, that if electric charges obey Newtonian laws, then all charge in conductors must be found on their surface. While Coulomb had already demonstrated this theorem, Green’s proof was much simpler. Having also formulated his famous theorem regarding functions, Green applied it to the study of relations between the potential function and the charge density on the surface of a conductor, making use of the previously attained results in the theoretical studies of the Leiden jar, the distribution of electricity in conductors, and magnetism. Mathematical texts report the pivotal theorems discovered by Green; here we limit ourselves to mentioning two. The first is the theoretical proof that the total charge across many identical bottles in series is equal to the charge of a single bottle, provided it is under the same conditions as the others. In short, this is nothing other than the well-known formula for the capacity of capacitors in series. The second theorem concerns the screening effect of a conducting shell. Starting with a perfectly conducting shell, Green called the bodies inside the shell and the inner surface the interior system, and the bodies outside the shell and the outer surface the exterior system, and arrived at the following theorem: “all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as would take place if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth; and all those of the exterior system will be the same, as if the interior one did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity, equal to the whole of that originally contained in the shell itself, and in all the interior bodies.”127 As one can see, it is Green and not Faraday who should be credited with the complete theorem of induction. That Faraday was not 127

G. Green, An Essay on the Application of Mathematical Analysis in the Theories of Electricity and Magnetism, in “Journal für die reine und angewandte Mathematik”, 47, 1854, p. 167 (ed. in facsimile, 1958, p. 27).

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aware of Green’s Essay is certain; on the other hand, he would not have been able to understand papers of a mathematical nature. Faraday experimentally rediscovered Green’s theorem in the course of his researches on electrostatic induction in 1837. He wanted to ascertain whether it was possible, through some instrument, to obtain a single electric charge independently of its opposite partner. He therefore built a cubic box with 13 feet (3.66 m) sides and covered it with a thick copper wire mesh resting on insulating supports. When he electrified the box either from the interior or the exterior, no charge appeared on the exterior or interior, respectively, so Faraday concluded that “non-conductors, as well as conductors, have never yet had an absolute and independent charge of one electricity communicated to them, and that to all appearance such a state of matter is impossible.”128 Gauss too was unaware of Green’s essay when he introduced the idea of a potential in his research on universal gravitation. The concept was an extension of his work on terrestrial magnetism129, and he noted that the same considerations could hence also be extended to electricity and magnetism. Gauss simply called Green’s function the “potential”, and he realized that it can also be defined for non-Newtonian forces, as long as a function whose partial derivatives give the components of the force, exists. Several new theorems and ideas were introduced by Gauss: the potential of a magnetic sheet at a point is equal to the product of the intensity of the sheet and the solid angle it subtends from the point’s perspective; the idea of an equipotential surface; and the inability of Poisson’s formula 'V=2SU to capture the physics of points on an attractive surface. In 1879, the Russian Pavel Somov (1852-1919) gave a complete demonstration of the problem and showed that 'V=4SUH, with İ = 0 for external point, İ = 1 for an internal point, and 0 d İ d 1 for a point on the surface. Potential theory spread in the second half of the century, in particular due to the effort of Clausius. In 1866, he dedicated an entire treatise on the subject, Die Potentialfunction und das Potential, which was published in four editions (until 1885) and in French translation. The title of the work reflects the complicated terminology introduced by Clausius, which was a source of confusion for some scientists at the time. Clausius later returned to Green’s terminology and called the expression r m/r the “potential function”, while he reserved the term “potential”, used by Gauss, to the expression mm’/r defined between two charges m and m’ at a distance r. 128

Faraday, Experimental Researches cit., § 1174. § 7.34 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

129

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From the account we have provided, it is clear that potential theory originated and developed purely in mathematics; the corresponding physical concept was still called “tension”. It was Georg Friedrich Bernhard Riemann (1826-1866) who gave physical meaning to potential with his now-famous definition: “potential is the work done by the field of force in bringing a unit particle from an infinite distance to [its real position].”130

3.19 Dielectric polarisation Does the action between two bodies occur at a distance or through the intervention of a medium? This was problem that physicists and philosophers had grappled with since Newton’s time. For the most part, despite not believing in action at a distance in principle, they chose to avoid the question: mathematical physicists leaned towards action at a distance not only, as some say, because Newton had operated “as if’ mechanical actions occurred at a distance, but also because, in the absence of better alternatives, it appeared to be the simplest mathematical model they could apply. Electric and magnetic phenomena once again brought up the age-old question. The idea of action at a distance had been supported by Aepinus, Cavendish, Coulomb, and Poisson; the great authority of these scientists had prevented Faraday from supposing the hypothesis of a mediated action. However, in 1837 he decided to confront the problem from an experimental point of view. He held that action at a distance can only occur along a straight line, while mediated action can also occurr along curved lines. Furthermore, if a medium does not participate in the propagation mechanism of electric action, the nature of the substance interposed between the two points in question should not affect the phenomenon, while the opposite should occur if the action is mediated. Guided by these ideas, he conducted several ingenious experiments, that showed that electrostatic action is also manifested along curved lines and that the interposed medium strongly influences the phenomenon. One of the first experiments to demonstrate that induction is also present along curved lines was the following: a vertical wax cylinder is built with a hemispherical cavity indented into its top end, where the diameter of the hemisphere is the same as that of the cylinder. A metallic sphere of slightly larger diameter was placed in the cavity, and then the 130

G. F. B. Riemann, Schwere, Elektricität und Magnetismus, Hannover 1876, p. 158.

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cylinder was electrified by rubbing it with a wool cloth. The electric state of the different parts of the metallic sphere was examined, and Faraday found that it was positively charged at every point, This was without a doubt an induced charge, which cannot be imparted to the apex of the sphere along straight lines, but must pass along curved surface. Other analogous experiments confirmed the hypothesis that electrostatic signals can propagate along curved lines and also revealed that these lines tend to repel each other. Experimenting on the well known spherical shell capacitors and using different insulating materials between the conducting plates, Faraday irrefutably showed that the insulator affects the inductive capacity of the device, a phenomenon he called specific inductive capacity (he also introduced the term dielectric, while Thomson used the expression dielectric constant, which was later adopted by Maxwell and became widespread), thus extending the research of Beccaria 80 years earlier131. In an appendix to his eleventh series, Faraday described the famous experiment with three vertical and parallel conducting disks, which demonstrated that the induction exerted by the middle disk on the other two is altered by the introduction of a nonconducting sheet between two of the disks. Following these experiments, Faraday introduced the theory of dielectric polarisation. How could the action of the dielectric in a capacitor be explained? Johann Wilcke in 1758 and Avogadro in 1806 had supposed that the molecules of a nonconducting body subject to the influence of a charged conductor are polarised. Yet Faraday, perhaps unaware of these works, was instead guided by two analogies: Poisson’s theory of magnetism (§ 3.18) and Grotthus’ theory of electrolytic actions (§ 3.3). He was struck by the similarity between a voltmeter and a capacitor: if two faces of a block of ice are covered with charged conductors, one obtains a capacitor; if the ice melts into water, one obtains a voltmeter in which, according to Grotthus’ hypothesis, the polarised molecules are oriented in the direction of current. Now, according to Faraday, polarisation must already be present in the ice molecules; the liquid state simply allows for the passage of current. Therefore, Faraday concluded, ordinary electrostatic induction is an “action of contiguous particles”. The particles of a body, be it insulating or conducting, are perfect conductors that are not polarised in their normal state, but can become so under the influence of charged particles nearby. A charged body in an insulating 131

§ 7.28 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambdridge Scholars Publ. 2022.

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environment then polarises the surrounding particles layer by layer. The degree to which these particles are forced to polarise is called tension (or voltage in modern terms). Faraday carried on his study of dielectric polarisation in his next three research series, from XII to XIV. In series XI he only considered the static case, but in the following ones he also examined the dynamics of the electric fluid, or rather, to exercise more caution given the difficulty of understanding what the word “current” meant for Faraday, of the electric action. In this way, he related conduction to the polarisation of particles that are not able to maintain a state of tension. This argument led him to experiment on conduction in rarefied air and observe the “dark space” that formed in the proximity of the cathode. Induced currents for him could also be effects analogous to electrostatic induction, because “the relation of conductors and non-conductors has been shown to be one not of opposition in kind, but only of degree; and, therefore, for this, as well as for other reasons, it is probable, that what will affect a conductor will affect an insulator also.”132 In a later paragraph, he added: “I have long thought there must be a particular condition of [insulating] bodies corresponding to the state which causes currents in metals and other conductors; and considering that the bodies are insulators one would expect that state to be one of tension. I have endeavoured to make such state sensible […], but have not succeeded […]. Nevertheless, … it may well be that this may be discoverable by some more expert experimentalist.”133 With our current hindsight we can fully appreciate Faraday’s predictions, but to his contemporaries they (rightfully) appeared hazy, baffling, and at times contradictory. Faraday did not propose any model of a polarised particle, limiting himself to describing an unspecified state similar to a magnetic needle. He also abstained from explaining how electrostatic induction acts in vacuum, but did not exclude the possibility that a charged particle in the middle of empty space could act on particles of the material surface delimiting the space,134 though leaving the interpretation of what happens inside this space up to the reader. In contest organized by the Società italiana di Scienze for a mathematical theory of electrostatic induction as described by Faraday, Ottaviano Fabrizio Mossotti, one of the greatest Italian mathematical physicists of

132

Faraday, Experimental Researches cit., § 1661. Ibid., § 1728. 134 Ibid., § 1616. 133

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the 19th century, submitted a historic paper135, in which the scientist set out to apply Coulomb’s hypothesis on the constitution of magnets, which had been later expanded by Poisson, to the constitution of dielectrics. For him, a dielectric was made up of conducting particles in which the electric fluid is divided into two types: vitreous and resinous. “The two electric fluids move without leaving the space belonging to each molecule or each element, but a greater separation is caused by a greater external action, and in dielectric bodies this grows until the excess electricity accumulated at the extremities is released and jumps from a group of molecule to the other, producing sparks.”136 Taking the hypothesis of a single fluid, this “movement” produces condensation of the fluid on one side of the molecules and rarefaction on the other. Mossotti applied Poisson’s theory of magnetism to this hypothesis and derived the fundamental equations from which he deduced his theorems on “the equilibrium conditions of electricity at the surface of multiple electric bodies under the influence of molecular induction due to a nearby dielectric body.” The theorems agreed with experimental results, and thus Faraday’s model of dielectric polarisation acquired greater credibility. Mossotti’s theory was subsequently applied and extended by Clausius in his mechanical theory of heat (§ 2.14); later we will see how Maxwell also employed these ideas (§ 3.27). We also add that the origin of modern theories of dielectrics can be traced to another famous paper by Mossotti, published in Turin in 1836, in which, starting from Aepinus’ theory, he arrived at a new theory of molecular forces and gave its analytical formulation.

3.20 Magneto-optics Faraday had often come to wonder whether there was a connection between electricity and light, and magnetism and light. Other physicists had also considered the problem, which was in tune with the unificatory tendencies of the time. Domenico Morichini (1773-1836) in 1812 and Christie in 1826 believed to have obtained magnetization through light: the 135

O.F. Mossotti, Discussione sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso, in “Memorie della Società italiana delle scienze”, 24/2, 1850; the paper pulled passages that were dated from 1846; a summary of the paper appeared in 1847 in “Archives des sciences physiques et naturelles”, 6, 1847, pp. 193-98. 136 Ibid., p. 3 of the 1846 version.

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first claiming that light can magnetize iron needles; the second asserting that variations in magnetic conditions could explain the increased damping of the oscillations of a magnetic needle when exposed to sunlight. Faraday, however, was not convinced by Morichini’s experiments, who repeated them in Rome during the former’s trip to Italy with Davy. On the other hand, in a 1846 note, he deemed the effects observed by Christie more credible, though these also were hardly probative and probably caused by secondary actions of air currents, since the experiments gave the same results with non-magnetized needles or with inhomogeneity in temperature in the air around the needle. Faraday was struck by the considerations of John Herschel, who noticed a helical symmetry in the deviation of a magnetic needle by a current: the symmetry appeared analogous to the rotation of the polarisation plane of a ray of light when it crosses certain bodies, so he concluded that there is a connection between electric current and polarised light and that “the plane of polarisation of light could be deflected by magneto-electricity.”137 The experiments carried out by Faraday in 1834, and later repeated in 1838, with the aim of discovering an action of the electric field on light gave nothing but negative results. Abandoning these electro-optic attempts, in 1845 Faraday began magneto-optical experiments, and after the first failures, which did not discourage him, obtained a new phenomenon: a heavy glass prism (made by himself from silicon, boric acid, and lead) was placed at one of the poles of an electromagnet and struck by a ray of polarised light parallel to the field lines; when the electromagnet was excited, the polarisation plane of the light rotated slightly. The discovery was announced by Faraday in November 1845 in a paper (the 17th of his Experimental Researches) titled Magnetization of Light, and the Illumination of the Lines of Magnetic Force. Before the paper was published, however, the title was criticized by many, in particular for the use of the expression “illumination of the lines of magnetic force”. In consequences, Faraday added a note before publishing it with the goal of clarifying and justifying this expression: “I believe” Faraday wrote, “that in the experiments I describe in this paper, light has been magnetically affected, i.e. that that which is magnetic in the forces of matter has been affected, and in turn has affected that which is truly magnetic in the forces of light.”138 Perhaps Faraday, using circumspect 137 138

B. Jones, The Life and Letters of Faraday, London 1870, Vol. 2, p. 205. Faraday, Experimental Researches cit., § 2146.

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language because of the state of science at the time, essentially meant to say that light had some magnetic aspect: a concept that to the physicists of 1845 appeared even less clear than the expression he had tried to clarify because in Fresnel’s theory, then the accepted explanation for light phenomena, light had no magnetic character. Having confirmed the phenomenon, Faraday turned to the study of its modalities. He found that many other substances in addition to heavy glass displayed the property, but could not obtain the effect with a gold sheet: many years later, in 1884, Kundt showed that metal sheets have a very high magnetic rotational power, which can explain the magneto-optic effect discovered by John Kerr in 1877, that we will soon discuss. Replacing the magnet with a current-carrying coiled wire, Faraday saw that the polarisation plane of the light rotated in the direction of the current and quickly distinguished between natural rotation power and magnetic rotational power: the symmetry of the first is helical, while the symmetry of the second is cylindrical. Taking advantage of this distinction, Faraday experimentally showed that the rotation of the polarisation plane depends on the nature of the body struck by the ray (in 1858 Verdet demonstrated that several metallic salts of magnetic type are an exception), and is proportional to the body’ thickness and intensity of the magnetic field, where the maximum effect is attained when the direction of the field is parallel to the ray and there is no effect when the two directions are perpendicular. Basically, what today we call “Verdet’s law” was in fact given by Faraday. Verdet confirmed it with numerous experiments, which in 1863 also brought him to discover that magnetic rotational power is approximately proportional to the square of the wavelength. The rotation of the polarisation plane galvanized Alexandre-Edmond Becquerel, who in 1846 equipped the magnetic poles with punctured extensions and let light pass through these cylindrical holes. Even better was Ruhmkorff’s coil, whose punctured nucleus let the light being examined pass through it. More simply, in 1852 Wiedemann studied the behaviour of carbon sulphide by filling a tube placed inside a currentcarrying solenoid. The effect of temperature was studied by De la Rive in 1871: in general, an increase in temperature corresponds to a decrease in the angle of rotation, but at times the opposite occurs. In 1875, John Kerr (1824-1907), professor of physics at Glasgow, discovered that dielectrics placed in a strong electric field acquire the property of double refraction, which completely disappears without the presence of the field. In 1877, continuing with his experimental studies, he discovered the Kerr effect (§ 3.30).

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The magnetic rotational power discovered by Faraday, the Kerr effect, the magneto-optic effect of Cotton and Mouton (birefringence acquired by an isotropic body in a strong magnetic field), the Stark-Lo Surdo effect, and the Zeeman effect, all provided corroboration of the electromagnetic nature of light: in § 3.30 we will discuss these phenomena. Let us return to Faraday, who had keenly observed a delay in the rotation of the polarisation plane compared to the change in field intensity. The phenomenon was re-examined by Emilio Villari (1836-1904) in 1870, using a simple experimental device that confirmed Faraday’s observation and provided the first measurements of the effect.

3.21 Composition of matter Only in 1878 did Kundt and his student Wilhelm Röntgen observe the magnetically induced rotation of the polarisation plane for polarised light traveling through a compressed gas; they demonstrated that this effect depends on the nature of the gas and is proportional, for a fixed gas, to the density. Faraday, on the other hand, held that this rotation could occur only when light passed through solids or liquids, but not if it propagated in vacuum or in gases. The phenomenon seemed to him so intimately connected to matter to him that it could provide precious indications on the age-old question of its composition, a problem that had often arisen in the course of his research on dielectrics. Faraday began by levelling criticism at the atomistic theory. In short, he wrote that if atoms and space are two distinct entities, one must admit that only space is continuous because atoms are, by definition, individual units separated from each other: space therefore, penetrates all bodies and isolates every atom from other nearby atoms. Now consider an insulator like sealing wax. If space is conducting, the insulator should also be conducting, because the space inside it acts as a metallic net connecting one end to the other: therefore, space is insulating. Now consider a conductor. Like before, each atom is surrounded by space; if space is insulating, as we have just concluded, there can be no passage of electricity from atom to atom, but experiments indicate that a conductor conducts electricity: therefore, space must be conducting. A theory that leads to such contradictions must be erroneous it itself. Given that the atomic hypothesis was thus discredited in his eyes, what was its replacement? Reviewing all of the hypotheses on the nature of

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matter, Faraday declared himself a supporter of Boscovich’s139: “We know the powers [forces] and recognize in every phaenomenon of the creation, the abstract matter in none. Why then assume the existence of that of which we are ignorant, which we cannot conceive, and for which there is no philosophical necessity?”140 According to Boscovich and Faraday, matter, that is, systems of force emanating from centres of force, is everywhere, and there is no region of space that lacks it. Faraday continued: “The view now stated of the constitution of matter would seem to involve necessarily the conclusion that matter fills all space, or, at least, all space to which gravitation extends (including the sun and its system); for gravitation is a property of matter dependent on a certain force, and it is this force which constitutes the matter. In that view matter is not merely mutually penetrable, but each atom extends, so to say, throughout the whole of the solar system, yet always retaining its own centre of force.”141 And thus, was brought to its apex perhaps Faraday’s most original idea, which, according to Einstein, was the most important theoretical invention after Newton: the concept of a field. Newton had considered space as an inactive, impassible container that holds bodies and electric charges. On the other hand, Faraday, following Boscovich, attributed to space an active role in phenomena: it is really their venue. According to Einstein, it is necessary to have “great scientific imagination to realize that it is not the charges nor the particles but the field in the space between the charges and the particles which is essential for the description of electrical phenomena.”142 Faraday’s line of thought is best expressed in a memorable letter he wrote in 1846 to Richard Phillips (1778-1851), in which he hypothesized that the vibrations of light are tremors of lines of force. “This notion,” he wrote, “as far as is admitted, will dispense with the aether, which in another view, is supposed to be the medium in which these vibrations take place.”143 However, perhaps fearing to have gone too far, he closed the letter as follows: “I think it likely that I have made many mistakes in the 139

§ 7.6 of M.Gliozzi A History of Physic from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 140 M. Faraday, A Speculation Touching Electric Conduction and the Nature of Matter, in “Philosophical Magazine”, 3rd series, 24, 1844, p. 136. 141 Ibid., p. 143. 142 A. Einstein and L. Infeld, The Evolution of Physics, Cambridge University Press, 1938, p. 258. 143 M. Faraday, Thoughts on Ray-Vibrations, in “Philosophical Magazine”, 3rd series, 28, 1846, p. 447.

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preceding pages, for even to myself, my ideas on this point appear only as the shadow of a speculation, or as one of those impressions on the mind which are allowable for a time as guides to thought and research. He who labours in experimental inquiries knows how numerous these are, and how often their apparent fitness and beauty vanish before the progress and development of real natural truth.” Four years after writing this historic letter, while in the meantime Maxwell had discovered the first roots of the electromagnetic theory of light, Faraday expanded the argument, wondering if there exists a relationship between electromagnetism and gravity. “The long and constant persuasion that all the forces of nature are mutually dependent, having one common origin, or rather being different manifestations of one fundamental power, has made me often think upon the possibility of establishing, by experiment, a connexion between gravity and electricity.”144 The first series of experiments consisted of letting a copper cylinder surrounded by a metallic coil fall freely and observing if the coil, whose ends were connected to a sensitive galvanometer, carried a current. Initial experiments appeared encouraging, but then Faraday realized that the weak currents induced in the coil were due to the Earth’s magnetic field. No effects were also observed in other experiments, in which different bodies were lifted, dropped, and rapidly oscillated inside a fixed coil connected to a galvanometer. A lack of successful results led the scientist to abandon further research, despite the fact that the failures, as Faraday declared at the end of his XXIV series of researches, presented to the Royal Society on 1 August 1850, “do not shake my strong feeling of the existence of a relation between gravity and electricity.”145 With very different theoretical instruments, but equally unsuccessful results, Einstein would later revisit the problem of a unified field theory, as we will later see.

3.22 Diamagnetism Based on magneto-optic phenomena, Faraday had concluded that the inner fabric of a body in a magnetic field is modified. Consequently, he attempted to study what mechanical modifications were experienced by bodies subject to a magnetic field. In the course of his investigations, he made a great discovery in 1845, the last of his fruitful scientific career: a fragment of heavy glass is weakly repelled by the pole of a powerful 144 145

Faraday, Experimental Researches cit., § 2702. Ibid., § 2717.

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electromagnet. To obtain a clearer manifestation of the effect, Faraday hung a heavy glass rod by a string between the poles of a powerful horseshoe electromagnet. Once the electromagnet was excited, the rod rotated with its length perpendicular to the field lines, or equatorially, as Faraday described it. Subject the same conditions, an ordinary magnetic rod, on the other hand, was axially oriented, that is, along the field lines. Wheatstone informed Faraday that the magnetic repulsion phenomenon he reported was not new: Anton Brugmans (1732-1789) had described it in bismuth in 1778, and in 1827 Antoine-César Becquerel had made an analogous observation for antimony. Yet these observations had remained isolated, unknown to most physicists, despite the few conscientious scientists (like Pouillet, in the second edition [1832] of his Éléments de physique) who had included them in their treatises as curiosities. Even the Faraday discovery seemed uninteresting to many. Alexandre-Edmond Becquerel, son of Antoine-César, and Charles-Nicolas de Haldat held that there was nothing new in Faraday’s experiments, because both Coulomb and Becquerel père had found that bodies containing a few particles of iron in their interior, when cut into needles, arrange themselves equatorially in a magnetic field. However, Matteucci sensibly replied (1846) that the essential detail discovered by Faraday was not the equatorial alignment, but the repulsion of certain bodies from a magnetic pole. The criticisms, therefore, were easily dismissed, especially because Faraday showed that the effect was anything but unique: many solids, liquids (experimentally tested in thin glass tubes whose magnetic behaviour was already documented) and tissues of the human body, so that “if a man could be suspended, with sufficient delicacy, … and placed in a magnetic field, he would point equatorially.”146 Many gases also displayed the effect, as Michele Alberto Bancalari (1805-1900) demonstrated in 1847 and Faraday confirmed when he broadened his studies. He called all such bodies diamagnetic and all the bodies attracted by magnetic poles paramagnetic.147 After long and patient study, Faraday established that all bodies are either paramagnetic or diamagnetic, and neutral bodies, that is, bodies that are neither attracted nor repelled by a magnetic pole, do not exist. Extensive experimentation was dedicated to magnetic behaviour under different physical conditions. A small incandescent iron bar is always axially oriented in a magnetic field, and various other compounds of iron, 146 147

Ibid., § 2281. Ibid., § 2790.

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nickel, and cobalt also behave like paramagnetic substances. An important behaviour was observed in a tube filled with a solution of iron sulphate that was suspended in a vase containing a solution of the same salt, all inside a strong magnetic field: when immersed in a solution of the same concentration, the tube did not orient itself in any particular direction, while when immersed in a more dilute solution, it aligned itself axially, and in a more concentrated solution it aligned itself equatorially.148 In short, the behaviour was analogous to that described by Archimedes’ principle, which Plücker in 1848 and Alexandre-Edmond Becquerel in 1850 also observed, noting that the attraction or repulsion of an object by a magnetic pole depends on the ambient medium. Faraday also experimentally observed that the rotation of the polarisation plane of light does not depend on the paramagnetism or diamagnetism of the body that the light passes through. All of these experimental results called for a theoretical framework to interpret them. Developing a theory of diamagnetism, however, quickly proved to be a thorny endeavour. Faraday advanced two different hypotheses: under the action of a magnetic field, the molecules of a diamagnetic body become magnetized in the opposite direction to the molecules of a paramagnetic body; and that the repulsion of diamagnetic bodies by a magnetic pole is merely apparent, and actually arises due to a differential attraction, as the medium in which the body is immersed is more strongly attracted than the body itself. However, in this last hypothesis one also had to assume that the vacuum, or whatever other medium filled empty space, is magnetic. Faraday leaned towards the first hypothesis, because he thought it distasteful to attribute an attractive capability to space or to suppose it filled with a highly hypothetical ether. Physicists were split between these two theories for many decades. The majority, following Weber, who in 1852 build his diamagnetometer, accepted Faraday’s first hypothesis. Yet in 1889 J. Parker, working under this assumption, made a startling discovery. The repulsive force between a magnetic pole and the corresponding pole induced in a diamagnetic body (like a piece of bismuth) increases as the distance between inductor and inductee is reduced, moreover this inductive effect does not occur instantaneously but rather in a time interval depending on the distance. With this as his starting point, Parker then imagined moving the piece of bismuth closer to the pole, from a position P to a position Q, so rapidly that the repulsive force would remain equal to the one at point P for the entire journey. Then, after allowing the bismuth to sit at P long enough to 148

Ibid., § 2367.

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acquire its equilibrium magnetization, he imagined transporting it quickly along the opposite path, back to position Q: on this return path, the bismuth would perform more work than the amount provided to it in the first path. In conclusion, one would have a closed transformation cycle that creates energy. According to Parker, there were three ways to interpret this result:149 rejecting the principle of energy conservation; assuming that the onset of diamagnetism is instantaneous, unlike any other physical phenomenon; or maintaining the principle of energy conservation and considering the energy created in each cycle to be provided by the heat transferred from one body to another at equal or greater temperature, such that Clausius’ form of Carnot’s principle (§ 2.12) is modified in the sense that for reversible cycles it takes the form

and for irreversible ones

The contradiction between Faraday’s first hypothesis on diamagnetism and the second principle of thermodynamics was confirmed in 1889 by Duhem who, struck by Parker’s note, presented the argument in another form. Essentially, Duhem’s proof rested on the observation that a magnetic equilibrium is established for a substance free of coercive force, and thus the internal thermodynamic potential of the system takes its minimum value. However, writing the general conditions for equilibrium and applying them to a diamagnetic body, one arrives at the strange conclusion that it cannot be in thermodynamic equilibrium. Duhem concluded: “the existence of diamagnetic bodies is incompatible with the principles of thermodynamics.”150 Parker and Duhem’s observation, which occurred in the midst of the development of thermodynamics, caused a sudden crisis, bringing physicists at the end of the century to return to the theory of an influencing medium. 149

J. Parker, On Diamagnetism and the Concentration of Energy, in “Philosophical Magazine”, 5th series, 27, 1889, pp. 403-05. 150 P. Duhem, Sur l’impossibilité des corps diamagnétiques, in “Comptes-rendus hebdomadaires des scéances de l’Académie des sciences de Paris”, 108, 1889, p. 1043.

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The theory preferred by Faraday and Weber, on closer inspection, was soon reincarnated in Langevin’s 1905 theory, which among the many prequantum electronic theories was the simplest and most general. According to Paul Langevin (1872-1946), a magnetic field acts on the motion of electrons, inducing in each atom an orbital current that turns the atom into an elementary magnet with magnetic moment equal and opposite to the inducing field. Diamagnetism naturally arises, as a general property of matter. If the electronic orbits are symmetric, the atom, under the action of the field, does not tend to orient itself, and thus the material is simply diamagnetic. However, if the orbits are not symmetric, the action of the field and thermal excitement can produce a preferred orientation in which the magnetic moment of the atom aligns with the field: the material is thus paramagnetic, and paramagnetism arises as a special case, like ferromagnetism. Before moving on, we mention in passing that, in the course of his research on diamagnetism, Faraday realized that a copper rod hanging in a magnetic field does not oscillate even when pushed, as if halted by a strong frictional force. Faraday sensed that the phenomenon is caused by induced currents in the rod, and validated this hypothesis by performing an experiment: he rotated a copper cube between the poles of an electromagnet, and the rotation stopped as soon the field was turned on. Foucault repeated the experiment in 1855, making it more impressive by using an oscillating pendulum placed between the ends of an electromagnet, as Sturgeon had in 1825, to study rotational magnetism (§ 3.6). In short, the study of currents induced in metallic masses, which were also observed by Joule in 1843, only owes to Foucault the (unoriginal) technique (§ 2.34) used to reduce the currents by lamination of the masses.

3.23 A biographical note on Michael Faraday The magnetic behaviour of gases, the nature and properties of magnetic and electric force lines (field lines in modern language), the effect of crystalline structure in magnetic phenomena, applications of electricity (the telegraph, electric light, the electrical loom) and the teaching of science were Faraday’s favourite subjects of study in the last years of his life. They did not lead to ideas of the same calibre as those he had developed in his youth and maturity, but still remained highly esteemed researches, conceptually refined and skilfully executed experiments, and acts of faith in the inexhaustible richness of nature and the duty of humankind to discover and exploit it.

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With age, Faraday’s devotion to scientific research became almost religious in nature, perhaps because he built his own scientific path, and the discoveries he made were fraught with difficulties and hardship. The third son of the smith James Faraday, he was born in Newington Butts, near London, on 22 September 1791. After attending primary school only for a few years, Faraday was withdrawn by his mother, who was concerned by the abuse he suffered at the hand of an irascible teacher (an experience which remained a bitter memory for Faraday). For some times his school was the streets, until at age thirteen he found work in a bookshop and newspaper stand, where he brought papers to clients during the day and collected them at night, according to the usage in London at the time; he was quickly promoted from delivery boy to apprentice bookbinder. This job gave him the opportunity to see many books and read the ones that interested him, like scientific books, and in particular those about chemistry and philosophy. In 1810-11 he was able to take a twelvelesson evening class in chemistry; he became friend with one of his classmates, whose kitchen hosted Faraday’s first experiments. Yet it was Davy’s four lessons, which he attended between February and April of 1812, that really captivated him and inspired him to apply for a job at a scientific institute, first the President’s of the Royal Society and later Davy’s laboratory. The latter hired him on 1 March 1813 to work in his laboratory at the Royal Institution, and in September of the same year, having appreciated his qualities. proposed that Faraday accompany him on his long continental trip, during which he became Davy’s assistant and reluctant servant to his wife, with which he had more than a few altercations. In any case, the trip to France, Switzerland, and Italy was very useful for Faraday, who was able to come into contact with continental scientific environments and their protagonists, improving his own scientific culture, as the accurate diary he kept shows. Returning to England, he continued his collaboration with Davy, in particular in building the safety lamp introduced in mines in 1816, the year that his first paper appeared: a chemical analysis. The former attendant transformed into a skilled and creative experimenter, attracting the attention of European scientific circles with his electrodynamic experiments, his manufacture of heavy glass (1822), the liquefaction of certain gases (§ 2.4), and his discovery of benzene (1825). In 1824, he became a member of the Royal Society despite the opposition of Davy, its president, who was jealous of Faraday’s success. In 1825, he was made the director of the Royal Institution, where he had

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remained out of gratitude for its role in his career beginnings, refusing the offers of better-paid academic positions. In 1839, Faraday’s health began to feel the effects of his excessive work; his memory weakened and with it his productivity. In 1841 the scientist became so sick that he was forced to take a four year break from any scientific activity and sought intellectual and physical restoration in the Swiss Alps. Faraday returned to his work in 1845, but his health never completely recovered, and his memory grew increasingly weak. He gave up his various appointments and finally, on 1 March 1865, wrote a resignation letter to the Royal Institution, where he had worked for 52 years. Him and his wife Sarah Bernard, whom he had married in 1821 and with whom he had had no children, retired to a small house in Hampton Court that had been gifted to him by the Queen: there he “waited”, as he said in several occasions. Faraday waited until 25 August 1867: that afternoon he died peacefully, seated on the chair in his study.

3.24 Some practical applications We have already abundantly discussed the theoretical importance of Faraday’s discoveries and we will later see the developments that they engendered; here we highlight the practical importance, limiting ourselves to the industrial consequences of his discovery of electromagnetic induction. All the machines used in the modern electrical industry – generators, transformers, electric motors – are based on the phenomenon of electromagnetic induction. We have seen that Faraday had himself built the first current generator (§ 3.14). In 1832, Hippolyte Pixii (1776-1861), manufacturer of physical instruments in Paris, built an electromagnetic machine in which a horseshoe magnet was rotated in front of an electromagnet, made up of a U-shaped piece of iron around which a long (30 metres) copper wire covered in silk was coiled. The two ends of the wire were attached to two mercury wells, which acted as the ends of an external circuit crossed by alternating current. Pixii’s machine was historically important because it underlined, using chemical decomposition effects and the sparks that they provoked, the considerable magnitude of the recently discovered phenomenon. The period change in the direction of the current, however, appeared to be a flaw in the machine, so Pixii set out to obtain a unidirectional current by adding the well-known Ampère switch, which automatically inverts the connections to the ends of the external circuit whenever the current is inverted. The tellurium-electric (or circle) machine

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of Luigi Palmieri, professor of physics at the University of Napoli, was described in 1844 and served as a generator of alternating current. For didactic reasons, writers in the second half of the century added a collector to its description, designing the prototype of a direct current generator. The possibility of extracting mechanical work from electric current was recognized before magnetic induction was even discovered. We have already seen (§ 3.6) how Barlow’s wheel transformed electricity into mechanical energy. In 1831 Dal Negro built the first electric motor and in 1838 Moritz Jacobi, in St. Petersburg on the Neva, powered the first boat using an electric motor: in both cases the attraction of fixed and moving electromagnets was applied. Later the moving electromagnets were replaced by anchors, and the alternating motion was transformed into rotational motion, like in steam engines. Nevertheless, these electric motors did not become widespread, as the cost of the work they produced (according to the experiments performed at the Paris exposition in 1855) was nearly twenty times greater than the cost of the work obtained by steam engines. The applications of these motors were therefore limited to small precision instruments that required little power. Returning to Pixii’s generator, we note that it was successively improved by Edward M.Clarke, Charles Page, Mollet and others, and was first applied in the field of galvanoplastics and, starting in 1862, to electrically power British lighthouses. Induction apparatuses that use extra-currents need a rapid current switch: the first type was designed by Antoine Masson (1806-1860) in 1837 and consisted of a rotating gear whose teeth, when in contact with a corresponding spline, closed an attached circuit. Based on the pitch of the sound produced, Masson deduced the frequency with which the switch opened and closed. In this way, he obtained high-voltage induced currents, which he used for therapeutic purposes and to reproduce static electricity effects. In 1851, the German mechanic Heinrich Daniel Ruhmkorff (18031877), a renowned manufacturer of physics instruments in Paris, noted that Masson’s device would produce better results if it was made up of a much longer wire (hosting the induction) and a quicker switch. The induction coil, also called the Ruhmkorff coil, was therefore born. In America, Charles Page (1812-1868) had built and improved induction coils since 1838, but they had remained completely unknown in Europe. The first induction coils produced 2 cm sparks in the air; in 1853 Fizeau proposed to insert a capacitor in the primary circuit to reduce its opening time; in 1859 Edward S. Ritchie obtained 35 cm sparks in air and soon after Ruhmkorff 50 cm; for his coil, he was given a 50,000 francs

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prize, established in France for having built the most powerful electric machine known to man. The induction coil was improved by Poggendorff, Foucault, Wennelt, and many others, and was instrumental in physics laboratories in the 19th century because it produced electromotive forces of the order of hundreds of thousands of volts, among the highest recorded magnitudes at the time. Some time later, it was replaced by transformers equipped with a rectifier and a thermoionic valve. The use of generators was very limited, as we mentioned, especially because the flaws in switches. Eliminating these problems was a major aim of the industry of the time. In 1860, Antonio Pacinotti gave a brilliant solution with his macchinetta (Fig. 3.4): a direct current motor with a collector, described in an 1864 paper, in which he noted that his machine, also called a Pacinotti dynamo, was reversible. Pacinotti’s invention, popularized with a few practical modifications in 1871 by Zénobe Gramme (1826-1901); the transformer, first described by the French chemist Lucien Gaulard (1850-1888) in 1882; the rotating magnetic field motor (Fig. 3.5), invented in 1885 and described in 1888 by Galileo Ferraris (1847-1897), and later improved in 1886 by Nikola Tesla (18561943): all helped set off modern large-scale electric industry.

Fig. 3.4 - A model of Pacinotti’s “macchinetta”, kept in the Domus galileana of Pisa.

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Fig. 3.5 - A model of Galileo Ferraris’ rotating magnetic field motor kept in the Istituto nazionale di Elettrotecnica of Turin.

The cost of electrical energy obtained from dynamos was calculated in 1880 to be around 16 times smaller than the cost of energy obtained from batteries and therefore could compete with the cost of thermal energy. However, the new industry faced great difficulties in long-distance transport. In 1882, a transport experiment that carried electrical energy between Munich and Miesbach, a distance of 57 km apart, resulted in a yield of 22.1 percent; in 1883 the yield was increased to 46 percent in an electrical system connecting the North station in Paris and Bourget, but the distance traversed was barely 8.5 km. Such yields were too low to encourage the spread of the new form of energy. Nevertheless, more for scientific curiosity than for practical utility, two industrial scientists: Ernst Werner Siemens (1816-1892) and his brother Carl Wilhelm (1822-1883) studied electrical locomotion in 1879 and a few years later built a public electric railroad that stretched for 7 km between Berlin and Lichterfelde. At the Paris exposition in 1881, the brothers presented an electric lift; these remained pioneering attempts, however. Until the introduction of alternating currents and the transformer, Lord

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Kelvin’s programme: burning combustible on the spot, transforming its thermal energy into electricity, and using it at a distance, remained a fantasy. Electrical illumination is an interesting chapter in the story of electrical applications. While we leave a more detailed discussion to specialized texts, here we simply note that arc illumination, achieved with batteries, had been improved multiple times but remained very costly and was therefore only used in large offices. With the introduction of incandescent carbon-filament light bulbs (Thomas Edison, 1879), the cost of electrical lighting barely exceeded that of gas lighting, but the advantages led to its rapid spread. On the 1st of January 1885 there were 18,875 Edison bulbs in New York and 75,500 in Europe; two years later these numbers had nearly doubled. Another application of electromagnetic induction also deserves a brief mention: the telephone, whose invention was the subject of heated controversy regarding credit and priority that even extended into the judicial realm. It appears almost certain by now that the inventor of the telephone was Antonio Meucci (1808-1889), who created his prototype in 1849, while the Alexander Graham Bell’s (1847-1922) first telephone only appeared at the Philadelphia exposition in 1876. The Bell receiver has been preserved to this day, while the transmitter was highly defective: the American Thomas Edison (1847-1931) improved it that same year. In 1878 David Hughes (1831-1900), inventor of the printing telegraph, designed a device he called the microphone: a transmitter made up of a vertical carbon rod inserted into the indents of two carbon blocks fixed to a wooden table.151 The term telephone was introduced in 1860 by the Frankfurt scientist Philip Reiss, to indicate a device that transmits sounds through magnetostriction, a phenomenon discovered by the Americans Page and Henry in 1837. In 1878, Augusto Righi presented his “phone that can be listened at a distance” to the Accademia delle Scienze of Bologna. The receiver was like Bell’s, but upgraded to a genuine loudspeaker that could be heard by assemblies of a hundred people or more. The transmitter was completely original, made up of conducting carbon powder, graphite, and silver pressed onto a vibrating membrane (made of parchment, metal, or wood). In 1878, James D’Arsonval realized that if the telephone is connected to an automatic switch and inserted in a circuit, it acts as an excellent detector of current. Yet it was Friedrich Kohlrausch who was perhaps the first to employ the telephone in physical research, specifically in measuring the resistance of electrolytes (§ 3.10). 151

“Annales de chimie et de physique”, 5th series, 13, 1878, p. 571.

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MAXWELL’S ELECTROMAGNETISM 3.25 Maxwell: a biographical note James Clerk Maxwell (Fig. 3.6) was born into a family of Scottish nobility on 13 June 1831 in Edinburgh. He spent his childhood in the countryside and, at the age of ten, after his mother died, was sent to boarding school in Edinburgh; in 1847 he enrolled in the University of Edinburgh, where he was taught by several famous scientists of the time: Philip Kelland (1808-1879) for mathematics, James David Forbes (18091868) for physics, and William Hamilton (1788-1856) for logic, who for Maxwell, as for many other students, was the greatest influence. Forbes held Maxwell in such high esteem that he took the highly unusual step of allowing him to make free use of the library and the laboratory for his personal researches, which seem to have begun with an experiment of polarised light directed by Nicol, whom he knew personally in those years.

Fig. 3.6 – James Clark Maxwell

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In 1850 he moved to the University of Cambridge, where he graduated in 1854 as the second in his class. During his stay at Cambridge his fellow students came to know his poetic side, which was manifested in the humorous and sometimes eccentric verses that he sent to his friends. Lewis Campbell and William Garnett, his classmates and first biographers, published several of his jocose poems. In February of 1856, Maxwell was offered a position as a physics professor at a college in Aberdeen, which he accepted to honour the dying wish of his father, who suddenly died the following April. In 1860, the college where Maxwell taught and the other college in the city fused to form a university; the position of physics professor was given solely to the professor of the other college, who was preferred over the younger Maxwell. The episode also generated controversy regarding the scientist didactic approach. Based on the accounts of his colleagues and students, perhaps the following assessment is the fairest: Maxwell was not a particularly talented lecturer, but in more personal interactions he became a great teacher because of the richness of his suggestions and the enthusiasm for research that he could impart. That same year, a physics position at the University of Edinburgh opened up, but Tait was chosen over Maxwell, so he had to settle for the open position at the real college of London. Maxwell resigned from this position in 1865 because of the excessive teaching load, which also included an evening course for artisans. He retired to his property in Glenlair, where he dedicated himself to the theory of electricity, a study of British literature, and philosophical and teological readings, interrupting his work with a few trips (in 1867, he and his wife took great delight in visiting Italy). In 1871, after three years of study, a special commission proposed to create a new position at Cambridge to enhance the teaching of physics. The right professor had to be chosen: Thomson (later Lord Kelvin) did not accept the offer and Maxwell was reluctant to leave the tranquillity of his countryside residence. However, the insistences of Stokes, Strutt (later Lord Rayleigh), and others, convinced him to change mind: in October of 1871 he held his inaugural lecture, in which he presented a few ideas that were not as common as they may appear today. We attempt to summarize them here: an experimental demonstration serves a different function than a research experiment performed in a laboratory (the reader should keep in mind that student laboratories were a

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relatively recent invention152; working in a laboratory is not necessary for the mathematics student, but it is indispensable for the student of physics, who must acquire from practical laboratory experience the ability to go from the abstract to the concrete and from the concrete to the abstract. Experimentation alone is not enough to train a physicist; she must also be able to attack problems theoretically and treat them using mathematical analysis. Yet, if the physicist does not want to close herself off in her own small world with the illusion that it fully represents the real world, she must maintain close ties with the other liberal arts: philology, history, and philosophy. Under Maxwell’s supervision, a new laboratory of physics was built and outfitted with modern equipment. It was inaugurated in 1874, baptized the Cavendish Laboratory in honour of the physicist from the previous century and to pay homage to the Duke of Devonshire, his descendant, who had financed the construction. In 1877, Maxwell’s health began to deteriorate, but he continued to teach and engage in scientific research: in 1879 he published two papers on the theory of the radiometer and Bolztmann’s theorem. He spent the summer in the countryside to recover, but a cancer in his stomach, which had killed his mother at the same age, continued to progress: he died on 5 November 1879. Maxwell began his scientific activity with a paper on the mechanical tracing of Cartesian ovals; that he published at the age of fifteen – the subsequent influence on scientific through of the time was vast, incisive, and profound. He continued with other papers on pure geometry, physiological optics (the perception of colours), and mechanics. His paper on the equilibrium of elastic solids and especially his 1856 essay on the stability of the rings of Saturn were particularly important and won him the Adams prize at the University of Cambridge: he showed that the rings are stable only if they are made up of unconnected particles, that move at speeds proportional to their distance from the centre of the planet; the particles can be arranged in a series of thin concentric rings or move irregularly with respect to each other. These works would have been enough to secure the fame of a distinguished scientist for their author, but compared to Maxwell’s other achievements they were minor. His greatest contributions to physics were his research on the kinetic theory of gases, almost an extension of his work on the rings of Saturn (§ 2.15), and his electrical theories. 152

§ 7.2 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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3.26 The first electrical researches Maxwell was only twenty-four years old when, in December of 1855, he presented his first paper on electrical theories to the Cambridge Philosophical Society: it was titled On Faraday’s Lines of Force. In an interesting introduction, the young scientist complained that the electric theories of the time were still unable to connect electrostatic phenomena with electrodynamic ones. According to him, one could attempt to tackle this question by examining physical analogies. “By a physical analogy,” wrote Maxwell, “I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other.”153 Having given a few examples of analogies (the refraction of light and the reflection of particle trajectories in a force field; the propagation of light and the propagation of vibrations in an elastic medium; the propagation of heat in a uniform medium and action at a distance of bodies), he declared that he wanted to use this analogical approach to study electrical phenomena, adopting “the processes of reasoning which are found in the researches of Faraday, and which, though they have been interpreted mathematically by Prof. Thomson and others, are very generally supposed to be of an indefinite and unmathematical character, when compared with those employed by the professed mathematician.”154 With this method, he hoped to demonstrate that the application of Faraday’s ideas could establish a connection between electrostatic and electrodynamic phenomena. In other words, his goal was to give a complete mechanical explanation of electrical phenomena, that is, to provide a mechanical model such that all the electrical phenomena would be reduced to the fundamental principles of dynamics. This approach encompasses the famous “theory of models” that so engrossed physicists and philosophers in the second half of the 19th century: a physical phenomenon is considered “explained” when it can be given a mechanical model. Poincaré later showed that once one mechanical model was found to explain a physical phenomenon, infinitely many others could also be found. As a result, instead of bringing science closer to the truth, this method would lead it astray. The supporters of the theory of models replied that it did not matter whether the model intrinsically corresponded to the phenomenon: science provides not the truth but models of it. The 153 154

J. C. Maxwell, The Scientific Papers, Cambridge 1890, Vol. I, p. 156. Ibid., p. 157.

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controversy was reignited in another form in the 20th century, as we will see in § 8.12. Working in this direction, Maxwell was able to provide a complete mechanical model, attributing to the ether a complex structure that, in reality, made the full system seem rather bizarre. Before dealing with the specific model of the ether, however, in the paper cited above and later ones he dealt only with hydrodynamic analogies. He drew a parallel between electric potential and the level of water in a container, and between electric charge and amount of water: through this correspondence it is simple to explain capacitance, current, and electrical resistance. Yet how could he explain electrical attractions and repulsions? Maxwell treated them by observing that these mechanical actions tend to reduce the potential, much like the pressures exerted by liquids on container walls. The inertia of the ether that surrounds two conductors can explain ordinary induction, mutual induction, and ponderomotive actions between currents. The parallel between hydrodynamics and electrodynamics also extends to the fundamental mathematical equations describing them, meaning that hydrodynamic relations can be also be applied to the theory of electricity: one only has to replace fluid velocities with electric forces and pressure differences with potential differences. Nevertheless, a phenomenon in this theory remained shrouded in the age-old mystery: how is force transmitted to from one body to another non-contiguous one? For action at a distance, so dear to the mathematicians of the time, Maxwell, like Faraday, harboured a great disdain. In an 18611862 paper, he overcame this aversion by hypothesizing the existence of a medium able to exert forces of tension or pressure on material bodies. At that time, there was no dearth of similar attempts: in 1847 Thomson had proposed a mechanical representation of electric and magnetic forces “by means of the displacements of particles in an elastic solid in a state of tension”; and later returned to the question in 1857, to interpret magnetic rotational power, assuming the vorticose molecular motions hypothesized by Rankine; in an 1859 paper on the motion of fluids, Helmholtz observed that the lines of fluid motion are arranged like magnetic field lines, such that the path of electric current corresponds to the axial lines of fluid particles in rotational motion. Encouraged by these precedents, Maxwell set out to “examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed. If, by the same hypothesis, we can connect the phenomena of magnetic attraction with electromagnetic phenomena and with those of induced currents, we shall have found a

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theory which, if not true, can only be proved to be erroneous by experiments which will greatly enlarge our knowledge of this part of physics.”155 The scientist implemented his plan, coming up with a bizarre theory that appeared a return to the triumphant Cartesianism of two centuries earlier. He postulated the existence of an elastic medium in which, by action of a magnetic field, vortices that have the same direction at each field point are produced; passing from one point of the field to another, the direction of the vortex axis, its rotational velocity, and the density of its substance all change. The centrifugal force arising from this vortex motion causes an equatorial pressure and a longitudinal tension along of the lines of force (field lines), and these lines indicate the direction of least pressure at each point in the medium. Yet how can two adjoining vortices rotate in the same direction and not oppositely, like two connected gears would? This problem was resolved by resorting to the same artifice used when one wants to transmit the rotational motion of one wheel to another while conserving the direction of rotation: a third “idle” wheel is introduced between the two. Similarly, Maxwell introduced a “layer of particles” that acts as an idle wheel between two adjacent vortices; moreover, the layer of particles was supposed to make up electrical matter and its motion electrical current. The numerous other equally arbitrary hypotheses he introduced made the theory even abhorrent for some, but it was nevertheless able to provide a mechanical model, that is, according to Maxwell’s thinking, “an explanation” of electromagnetic phenomena and the mathematical relations between the quantities that take part in them. In particular, Maxwell calculated the velocity of propagation of a transverse elastic vibration in the hypothesized medium, applying the classical formula

V=ට

௠ ఘ

,

where m is the elastic coefficient and ȡ is the density of the matter in the vortices. Using the electromagnetic formulas that he had already found, this relation became V = E/ȝ, where E is the ratio between the electrostatic and electromagnetic units of electricity and ȝ is the coefficient of magnetic induction. Because ȝ = 1 in air and vacuum, the expression in these media simplifies to V = E. 155

Ibid., p. 452.

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Now, in 1857, Weber and Kohlrausch had found that

E = 310 740 000 m/s, almost equal to the 314 358 300 m/s value given by Fizeau for the speed of light in vacuum. The two values were so close and the inference was so evocative that, despite the litany of arbitrary initial hypotheses, Maxwell could not help but comment: “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”156 This was Maxwell’s first reference to the electromagnetic theory of light: a reference that, being based on a metrological relation deduced from gratuitous hypothesis, could not have been considered convincing by the physicists of the time. It is not surprising, therefore, that this deduction went unnoticed, along with his entire paper, which had the air of a muddled clockwork mechanism, too sophisticated to be plausible. In any case, Maxwell had succeeded in demonstrating, more to himself that to others, that it was possible to imagine a mechanism of transmission for electric and magnetic actions. In his Treatise, Maxwell himself judged this paper serenely, saying that the results found had great value even if the approach used to attain them had to be changed.157 The approach was changed so drastically that Maxwell wholly did away with the original, as he showed in a long 1864 paper -published on a historic date for physics- titled A Dynamical Theory of the Electromagnetic Field. It reflected his most mature thinking and was later reprinted with minor edits in the Treatise. The theory was called dynamical because electromagnetic phenomena were attributed to the motion of the elastic medium in which electric and magnetic bodies are immersed. Yet no assumptions were made regarding these motions, and the tensions present the medium were left unspecified: in short, it was not a mechanical model. Moreover, the mechanics terminology was even avoided, and its few uses were simply “to direct the reader’s mind to mechanical phenomena that will aid him in understanding the electrical one,” and therefore played more of an illustrative role than an explanatory one. An exception to this general rule was the term energy, that Maxwell employed in its mechanical acceptance. However, while his predecessors located energy in electric or magnetic bodies, Maxwell held that it was the electric field that 156

Ibid., p. 500. J.C. Maxwell, A Treatise on Electricity and Magnetism, London 1873, Vol. I, pp. 532-533.

157

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carried the energy, which manifests, in the elastic medium that fills the field, in the two forms of motional (or kinetic) energy or tensional (or potential) energy. More than the 1864 paper, it was Maxwell’s A Treatise on Electricity and Magnetism, published in 1873 and in a second edition in 1881, that made physicists aware of Maxwell’s conception of the electromagnetic field and in particular his electromagnetic theory of light. In the second edition, he was only able to revise the first nine chapters (out of thirtyfive) because of his premature death. We will now take a look at this classic Treatise.

3.27 A description of the electromagnetic field By 1860 electrodynamics, after the works of Neumann, Weber, Helmholtz and Felici (§ 3.7) seemed a definitively completed field with clear borders. The remaining research appeared to be only directed at deducing all the consequences of the established principles and their practical applications, a task that inventive engineers had already begun. Maxwell’s Treatise disturbed this prospect of untroubled, application-based work, opening up a vast new domain of electrodynamics. As Duhem perceptively observed, “No logical necessity pushed Maxwell to imagine a new electrodynamics; as a guide he only had a few analogies, the desire to provide Faraday’s theory with an extension similar to the one that the work of Coulomb and Poisson had received from Ampère’s electrodynamics, and perhaps also an instinctive feeling for the electrical nature of light.”158 Perhaps the main reason that pushed Maxwell to study a topic that was not immediately necessary for the science of the time was his fascination for Faraday’s new ideas, which were so original that the scientific environment had hitherto not been able to digest them. To a generation of mathematical physicists who had been educated on the ideas and mathematical craftsmanship of Laplace, Poisson, and Ampère, Faraday’s ideas seemed too vague, while on the other hand they appeared too abstruse and abstract for experimental physicists. Faraday was a unique case: not a mathematician by training, he had felt the need to employ a penetrating theoretical instrument like mathematical equations in his researches. “As I proceeded with the study of Faraday,” wrote Maxwell, “I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were 158

P. Duhem, Les théories électriques de J. Clerk Maxwell, Paris 1902, p. 8.

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capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians. For instance Faraday, in his mind’s eye, saw lines of force traversing all space, where the mathematicians saw centres of force attracting at a distance; Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids. When I had translated what I considered to be Faradays ideas into a mathematical form, I found that in general the results of the two methods coincided, so that the same phenomena were accounted for, and the same laws of action deduced by both methods, but that Faraday’s methods resembled those in which we begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by synthesis. I also found that several of the most fertile methods of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form.”159 As far as the value of Faraday’s mathematical instrument, in another occasion Maxwell observed that the mathematicians who had found Faraday’s method lacking in scientific rigour had not found a fundamentally different approach themselves, save for recurring to hypotheses on the reciprocal actions of physically inexistent entities, like the current elements, which “arise from nothing, travel across a stretch of wire, and return to nothing.” To give Faraday’s ideas a mathematical form, Maxwell began by formulating an electrodynamics of dielectrics. His theory was directly linked to Mossotti’s (§ 3.19), as he specifically admitted. Here we add that, according to Mossotti, in each conducting particle immersed in an insulating environment, the electric fluid, which he called ether, exhibits a certain density. When the particle is subjected to an inductive force, the ether condenses on one end and rarefies on the other, without leaving the particle, creating a tension on both sides. In 1867, Clausius was able to obtain a formula using Mossotti’s theory (the Mossotti-Clausius law) that connected the density ȡ of simple dielectric media and their dielectric constant İ; the formula is very simple:

159

J. C. Maxwell, A Treatise on Electricity and Magnetism, London 1873, Vol. I, pp. X.

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U

ఢାଶ ఢିଵ

=k

where k is some constant. We close our digression by once again highlighting that Maxwell acknowledged the influence of Mossotti’s theory, describing it as follows: “The electric polarisation of an elementary portion of a dielectric is a forced state into which the medium is thrown by the action of electromotive force, and which disappears when that force is removed. We may conceive it to consist in what we may call an electrical displacement, produced by the electromotive force. When the electromotive force acts on a conducting medium, it produces a current through it, but if the medium is a non-conductor or dielectric, the current cannot flow through the medium, but the electricity is displaced within the medium in the direction of the electromotive force; the extent of this displacement depending on the magnitude of the electromotive force. So that if the electromotive force increases or diminishes, the electric displacement increases and diminishes in the same ratio. The amount of the displacement is measured by the quantity of electricity which crosses unit of area, while the displacement increases from zero to its actual amount. This, therefore, is the measure of the electric polarisation.”160 If a polarised dielectric is made up of a collection of conducting particles arranged in an insulating environment, on which the electricity is distributed in a certain way, each change of polarisation state is accompanied by a modification in the electricity distribution on the particle, and consequently by a genuine electric current that is limited in the space of the conducting particle. In other words, each change in the polarisation state is accompanied by a flux or a displacement current. “The variations of electric displacement evidently constitute electric currents. These currents, however, can only exist during the variation of the displacement, and therefore, since the displacement cannot exceed a certain value without causing disruptive discharge, they cannot be continued indefinitely in the same direction, as do the currents through conductors.”161 Having already introduced the notion of field intensity, the mathematical translation of Faraday’s concept of a field of force, Maxwell applied the previous ideas of electric displacement and displacement current, deducing that what was called the charge of a conductor is actually the surface charge of the ambient dielectric, that energy is 160 161

Ibid., p. 60. Ibid.

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accumulated in the dielectric in the form of a state of tension, and that the motion of electricity obeys the same laws as the motion of a compressible fluid. Maxwell enumerated the “features of the theory” as follows: “That the energy of electrification resides in the dielectric medium, whether that medium be solid, liquid, or gaseous, dense or rare, or even deprived of ordinary gross matter, provided it be still capable of transmitting electrical action. That the energy in any part of the medium is stored up in the form of a state of constraint called electric polarisation, the amount of which depends on the resultant electromotive force at the place […]. That in fluid dielectrics, the electric polarisation is accompanied by a tension in the direction of the lines of force combined with an equal pressure in all directions at right angles to the lines of force, the amount of the tension or pressure per unit of area being numerically equal to the energy per unit of volume at the same place.”162 Let us stop here. The fundamental idea cannot be expressed any more clearly than in Faraday’s statement: electromagnetic effects occur in the medium. Having developed the theory of dielectrics, Maxwell applied its ideas to magnetism, with the necessary adaptations, thus developing the theory of electromagnetic induction; finally, he summarized his whole theoretical framework in a group of now-famous equations called Maxwell’s equations. These equations are very different from the ordinary equations of mechanics: they define the structure of the electromagnetic field. While mechanical laws are applied to the regions of space that contain matter, Maxwells’s equations are valid in all of space, whether bodies or electric charges exist or not; they govern the evolution of the field, while mechanical laws govern the evolution of material particles. Furthermore, while the laws of Newtonian mechanics had abandoned the idea of a continuous action in space and time163, Maxwell’s equations described continuous phenomena, connecting events that are contiguous in space and time: based on the conditions of a field “here” and “now”. we can predict the conditions of the field at the next instant in time in the immediate surroundings. This conception of the field is in perfect harmony with Faraday’s views, but in an intransigent contrast with two centuries of history. It is no wonder, therefore, that the theory was met with opposition. The criticisms raised against Maxwell’s electric theories were numerous, both directed at the fundamental concepts at the base of his arguments and, perhaps even more, at Maxwell’s casual manner of 162

Ibid., pp. 63-64. § 6.6 of M,Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 163

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reaching conclusions. As he went along, Maxwell created his models “with a few thumb strokes”, as Poincaré aptly put it, referring the logical somersaults that scientists sometimes make to formulate their theories. When an obvious contradiction stood in the way of his analytical development, Maxwell did not hesitate to overcome it by taking troubling licenses: making a term in an equation disappear, reversing an inconvenient sign, changing the meaning of a variable. In those who admired the impeccable logical development of Ampère’s electrodynamics, Maxwell’s theory provoked a sense of uneasiness. Physicists had trouble sorting it out, that is, removing the illogicalities and inconsistencies; on the other hand, they could not abandon a theory that so deeply connected optics and electricity. In consequence, towards the end of the century the preeminent physicists adhered to the Heinrich Hertz’s 1890 declaration: since the arguments and calculations that lead to Maxwell’s electromagnetism are riddled with errors that we cannot correct, let us take Maxwell’s six equations as the starting hypotheses and build up the entire framework of electrical theories on these postulates: “The essential piece of Maxwell’s theories is Maxwell’s set of equations”, says Hertz.

3.28 The electromagnetic theory of light In the formula found by Weber for the force between two electric charges in relative motion, there is a coefficient that has the same nature as velocity. Weber and Rudolph Kohlrausch, in a classic paper published in the acts of the Academy of Sciences of Lipsia in 1856 and disseminated in an extensive summary of the Annalen der Physik in 1857, experimentally determined its value (given in § 3.26), which was measured to be somewhat larger than the speed of light. The following year, Kirchhoff derived from Weber’s theory the laws governing the propagation of electromagnetic induction in a conducting wire: if the resistance is zero, the propagation velocity of an electric wave is independent of the cross section of the wire, its nature as well as its electric density and is almost equal to the speed of light in vacuum. In an 1864 theoretical-experimental study, Weber confirmed Kirchhoff’s results and showed that Kirchhoff’s constant is numerically equal to the number of electrostatic units contained in one electromagnetic unit. He further observed that the agreement between the velocity of light and the velocity of electric wave propagation could be considered evidence of a deeper relation between these two physical quantities. However, he cautioned that before making such a connection, one had to uncover the

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true meaning of this velocity in electrical terms: “This meaning,” Weber melancholically concluded, “does not appear to arouse hope.” For Maxwell, on the other hand, this velocity correspondence did not instil hope, but instil certainty: in his 1864 paper (§ 3.26), and even more in his Treatise, there permeates a conviction rooted in his long years of reflection, in spite of the scant and indirect experimental evidence. “In several parts of this treatise,” he wrote at the beginning of the 20th chapter of the fourth part, “an attempt has been made to explain electromagnetic phenomena by means of mechanical action transmitted from one body to another by means of a medium occupying the space between them. The undulatory theory of light also assumes the existence of a medium. We have now to show that the properties of the electromagnetic medium are identical with those of the luminiferous medium […] We therefore obtain the numerical value of some property of the medium, such as the velocity with which a disturbance is propagated through it, which can be calculated from electromagnetic experiments, and also observed directly in the case of light. If it should be found that the velocity of propagation of electromagnetic disturbances is the same as the velocity of light, and this not only in air, but in all other transparent media, we shall have strong reasons for believing that light is an electromagnetic phenomenon, and the combination of the optical with the electrical evidence will produce a conviction of the reality of the medium similar to that which we obtain, in the case of other kinds of matter, from the combined evidence of the senses.”164 As he had already done in the 1864 paper, Maxwell started from his equations and after a series of manipulations he arrived at the conclusion that transverse displacement currents in vacuum propagate at the speed of light, which “constitutes a confirmation of the electromagnetic theory of light,” as Maxwell confidently proclaimed. Following these generalities, he studied the particular case of electromagnetic perturbations in more detail, reaching conclusions that are well-known today: an oscillating electric charge generates a variable electric field that is always accompanied by a variable magnetic field, thus generalizing Ørsted’s experiment. Maxwell’s equations describe the variation in time of the fields at every point in space: applying them, one finds that at each point in space there are concurrent electric and magnetic oscillations, meaning that the intensities of the respective electric and magnetic fields vary periodically and inseparably, and the fields point 164

J. C. Maxwell, A Treatise on Electricity and Magnetism, London 1873, Vol. II, pp. 431.

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perpendicularly to each other. These oscillations propagate in space with a constant velocity, producing a transverse electromagnetic wave, where the oscillations of the fields are perpendicular to the wave’s propagation direction at every point. Among the many striking results derived from Maxwell’s theory, we mention a theorem that was particularly attacked by critics, which states that the dielectric constant of a medium is equal to the square of its index of refraction, and the theorems regarding the pressure exerted by light in its propagation direction as well as the orthogonality of the two polarised (electric and magnetic) waves.

3.29 Electromagnetic waves At the time of Maxwell’s death, the only adherents of his theory were a few of his young and enthusiastic British students, who supported it out of respect and affection for their teacher. Maxwell himself recognized this in the second to last section of his Treatise: “There appears to be, in the minds of these eminent men, some prejudice, or a priori objection, against the hypothesis of a medium in which the phenomena of radiation of light and heat, and the electric actions at a distance take place. It is true that at one time those who speculated as to the causes of physical phenomena, were in the habit of accounting for each kind of action at a distance by means of a special aethereal fluid, whose function and property it was to produce these actions. They filled all space three and four times over with others of different kinds, the properties of which were invented merely to save appearances, so that more rational enquirers were willing rather to accept not only Newton’s definite law of action at a distance, but even the dogma of Cotes, that action at a distance is one of the primary properties of matter, and that no explanation could be more intelligible than this fact. Hence the undulatory theory of light has met with much opposition, directed not against its failure to explain the phenomena, but against its assumption of the existence of a medium in which light is propagated.”165 After having examined the theories of Gauss, Neumann, and Enrico Betti, the following paragraph concludes the Treatise with the words: “Hence all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as a hypothesis, I think it ought to occupy a prominent place in our investigations, and that

165

Ibid., p. 437.

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we ought to endeavour to construct a mental representation of all the details of its action: and this has been my constant aim in this Treatise.”166 Yet the new physical conception proposed by Maxwell and Faraday struggled to gain traction, partly, but not only, as Maxwell observed, because of a general fear of falling back into the 18th century “vice of fluid obsession”. Works on Maxwell’s theory, and in particular his electromagnetic theory of light, which also became a subject of university courses, focused on its mathematical aspects and in particular on Maxwell’s equations and ended up becoming more mathematical exercises than physical research.

Fig. 3.7 – Heinrich Hertz

The physical question was unmistakeably different: do the electromagnetic waves predicted by Maxwell exist? Can they be produced? Can they be detected? Do they have the properties described by the theory? More than a decade had passed since the publication of the first edition of the Treatise and still no physicist had seriously considered these questions. In 1884, the young physicist Heinrich Hertz (Fig. 3.7) began to realize that 166

Ibid., p. 438.

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only an answer to these questions would provide an unbiased evaluation of Maxwell’s entire electromagnetic theory, of which the theory of light constitutes the central point. Hertz revised the theory concurrently with his experimental researches, making the first critical observation: Maxwell’s equations show that electromagnetic waves can arise from electrical oscillations. Now, as we have discussed in § 3.14, in 1870 it was well known that the discharge of a capacitor could be oscillatory, and Kirchhoff in 1864 had formulated a complete theory of the effect, while in 1869 Helmholtz had shown that electrical oscillations can also be obtained in an induction coil whose ends are connected to a capacitor. Hertz first used the coils from his professor Helmholtz, but later turned to a capacitor discharge in air. There were numerous difficulties that he had to surmount before obtaining suitable waves, essentially consisting in the fact that the frequencies measured in Leiden jar discharges are of the order of hundreds of thousands of cycles per second. If the velocity of the waves is really that of light, a simple calculation shows that the wavelength has to be several kilometres: too long to be measured, especially because the radiated energy decreases with the inverse square of distance. If one wanted to make such waves detectable, one had to produce discharges at much higher frequencies than what physicists at the time could obtain. The formula given by Thomson (§ 3.14), if correct, gave the right approach to follow. Hertz therefore dedicated himself to the study of oscillatory discharges for two years, in the end succeeding and designing a capacitor made up of two parallel plates or two concentric spheres connected with a metallic rod broken in the middle; the ends of the gap in the centre of the rod were attached to small spheres a few millimetres apart from each other. This capacitor was then powered by an induction coil. It is rather difficult to calculate the period of oscillations based on the geometry of such a capacitor. Hertz, in his 1888 paper, gave an approximate theory of his capacitor, which he called an oscillator, but because of a calculational error he found a longer period than the actual one, which was 2(10-8)s; nevertheless, the result was good enough to encourage him to continue. The remaining task was to find the possible waves produced by the oscillator, and Hertz initially thought to use the well-known phenomenon of electric resonance. Yet the fact that the resonator, made up of a simple circular metallic conductor with a gap less than a millimetre thick, exhibited sparks in the gap when sparks were also produced in the oscillator, did not prove the existence of the waves, as it this could simply

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have been an induction effect, especially if the two circuits were not sufficiently separated from each other. Hertz then thought to bypass this problem by making the wave reflect on a large metallic surface. Doing this, he obtained stationary waves in the space between the oscillator and the metallic surface, and was thus able to measure the distance between two successive nodes, equal -as known- to one-half the wavelength. Knowing the oscillatory period and the wavelength (measured to be 4.5 m in the first experiments), he calculated the speed of electromagnetic waves: a providential calculational error opposite to the error, in his calculation of the oscillator’s period, gave him a speed of the same order of magnitude as the speed of light, as Maxwell’s theory predicted. Adjusting his device into its more modern form, Hertz also measured the wavelength of waves transmitted along a copper conductor and obtained 2.8 m: in apparent contrast with Maxwell’s theory, according to which the wave does not travel inside a conductor but rather through the dielectric surrounding it, and the conductor only plays a guiding role. This result appeared to support Helmholtz, who had found Maxwell’s theory insufficient and proposed a more general one. However, other experimenters found different results. In later experiments with waves of higher frequency, Hertz himself found that their velocity was the same in conductors and in free space. Perhaps his first experiments had been disturbed by the presence of boundaries, the pavement of the laboratory, and an iron stove near the device. Maxwell’s theory was therefore accepted, without the additional generalizations conceived by Helmholtz. The first experimental results were published by Hertz in 1887: the following year, he demonstrated the reflection, refraction (in a rosin prism), and polarisation of the waves using parabolic cylindrical mirrors. He also showed that the electric vector and the magnetic vector are perpendicular, in agreement with the Maxwellian theory. Heinrich Hertz was born in Hamburg on 22 February 1857. Upon finishing his secondary studies, he pursued a career as an engineer, which he soon abandoned to dedicate himself to physics. In 1878 he moved to Berlin, where he served as Helmholtz’s assistant at the university laboratory. In 1883 he moved to Kiel, and in 1885 he became a professor of physics the polytechnic in Karlsruhe, where he conducted his research on electromagnetic waves. In 1889 he succeeded Clausius as the professor of physics at the University of Bonn. Here he began his research on discharge in rarefied gases and the drafting of Prinzipien der Mechanik in neuem Zusammenhange dargestellt (Principles of mechanics presented in a new context), which we later will discuss shortly: it would be his last work. In

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1892 he was struck by a bone disease that by November of that year had become quite serious. After a surgical procedure, his health briefly recovered and allowed him to continue his didactic and scientific activity for another year. However, on 1 January 1894, at just 37 years of age, Hertz died. Deeply affected, Helmholtz wrote a eulogizing preface for his former student and his work in Prinzipien der Mechanik, published posthumously later than year.167 Many experimenters quickly followed the path opened by Hertz and new devices for the production and detection of waves were proposed: the induction coil, for example, was replaced with the electrostatic machine; the oscillator’s spark in air was replaced with its spark in olive oil or petrol; the resonator was replaced with Galvani’s frog, and its sparks with the bolometer. Among these and other modifications, the method proposed by Lecher in 1890 is worthy of mention. Lecher’s device is composed of two plates parallel and very close to the plates of an oscillator, which are connected with two parallel conducting wires whose ends lie inside a discharge tube without electrodes. The tube becomes illuminated when the induced circuit is in resonance with the oscillating circuit. If the two wires are joined by a metallic bridge, it becomes possible to determine the positions of the nodes and antinodes of the stationary waves generated in the induced circuit, and therefore measure their wavelength and propagation velocity along the wires. The Lecher line (or Lecher wires), as the device is called today, is still used to measure wavelengths ranging from a few metres to some tens of centimetres. The Lecher line allowed for a more thorough study of “multiple resonance”, namely the fact that an oscillator does not emit a single wavelength but a range of wavelengths, among which the resonator detects only the one corresponding to its own vibrations. In 1884-85, before Hertz had explained how to produced and detect electromagnetic waves, Temistocle Calzecchi Onesti (1853-1922) had discovered that “unpacked metallic filings in a small nonconducting pipe becomes conducting by the effect of an extra current or an induced current involving pipe.” The filings lose their conductivity if the pipe containing them is shaken. Edouard Branly rediscovered the phenomenon in 1890, and the following year Oliver Lodge showed that the tube of filings is a more sensitive detector of electromagnetic waves than the ones currently employed: he named it coherer. In conclusion, in the first years after Hertz’s discoveries, the efforts of experimenters were primarily concentrated on detectors and little attention 167

The preface can also be found in Helmholtz, Works cit., pp. 789-807.

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was given to oscillators. Consequently, little had been added to Hertz’s work. Scientists continued to experiment with the same wavelength (around 66 cm) he had used in his later works, so diffraction effects hid any other phenomena. To avoid this, one would have had to use machines that were so large as to be almost unobtainable. An important step in this regard was taken by Augusto Righi (18501920) with a new type of oscillator. His apparatus is made up of two close metallic balls placed in petroleum jelly, surrounded by two other balls, one on each side, all arranged in a straight line, with the outer balls connected to an electrostatic machine. When the machine is turned on, three sparks can be seen: two in the air, between the (interior) oscillator balls and the (exterior) balls connected to the poles of the machine, and one in the insulating liquid. With this “three sparks oscillator”, Righi obtained wavelengths of a few centimetres (more frequently, he worked with waves of 10.6 cm). He was thus able to reproduce all optical phenomena using devices that often imitated the ones employed in optical research; in particular he was the first to obtain the double refraction of electromagnetic waves. Righi’s works, which began in 1893 and were published in notes and papers in scientific journals throughout his career, were later (1897) collected and expanded in a now classic work: Ottica delle oscillazioni elettriche (Optics of electrical oscillations), whose title alone summarizes an era in the history of physics. Hertz had specifically excluded the possibility of using electromagnetic waves for the transmission of signals at a distance, and the majority of physicists shared his scepticism in this regard. It may be difficult for us to understand their reasoning, whose explanation must lie in psychology because, from an objective point of view, there were certainly great technical difficulties but no fundamental scientific impediments. In reality there were some opposing voices, like that of Crookes, who since 1892 had predicted a telegraphy without cords. Even some concrete attempts had been made, albeit with modest results, by William Preece and Tesla. It is said that in the summer of 1894, a very young Guglielmo Marconi (1874-1937) first had the idea of realizing long-distance transmission using signals, inspired by reading an article on the work of Hertz. In any case, in the autumn of 1894 he withdrew to Pontecchio, near Bologna, in a villa owned by his father, and began a series of experiments that he continued for all of 1895, perhaps initially encouraged by advice from Righi, a family friend. In the summer of 1895, the young autodidact and scientist had a brilliant and seminal idea: attaching the oscillator to an antenna, which was initially made up of a metallic sheet on top of a wooden post, electrically connected to one sphere of Hertz’s oscillator,

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while the other sphere was placed on the ground. He adopted an analogous setup for the detector, made up of a coherer whose ends were attached to a circuit with a battery and a galvanometer. How Marconi arrived at the idea for an antenna is not known (perhaps by analogy with a lamp that shines further when raised ?). Nevertheless, his idea was certainly independent of that of the Russian physicist Alexander Stepanovich Popov (1859-1906), who in May 1895 introduced a device (described in an 1896 paper published in a Russian journal) used to study atmospheric electrical discharges that was built by connecting one end of a coherer to a lightning rod, while the other end was connected to the ground. In July of 1896, Marconi’s experiments on telegraphy began in London, and the rapid evolution and admirable results attained by this field greatly influenced the life and future of humankind.

3.30 A return to magneto-optics The work of Maxwell and his school was interwoven with another line of research begun by Faraday (§ 3.20) in 1845. Already in the 1846 the astronomer George Airy provided an adequate, though necessarily incomplete, theoretical explanation based on Fresnel’s hypothesis. The scientist considered linearly polarised light as two different rays that are circularly polarised in opposite directions: a change in the ratio between the two wave velocities would then produce a rotation in the polarisation plane of the ray of light. Airy’s study, however, did not gain much traction as physicists had turned their attention to a deeper phenomenological analysis of Faraday’s discovery. One the first scientists who attempted to widen the experimental field was the Swiss physicist Elias Franz Wartmann (1817-1886). In 1846, he communicated at a conference of the British Association that he had obtained negative results in numerous experiments to establish the possible influence of a magnetic field on the “position, quantity, or visibility” of the spectral lines in a ray of (natural or artificial, polarised or unpolarised) light. Similar negative results were found by Faraday in his 1850 and 1852 experiments, which he repeated in 1861-62 in “truly his last experimental work,” as Maxwell baptized it in a biography of Faraday written for the eleventh edition of Encyclopaedia Britanica and partially reprinted in later editions. Faraday carried out his last experiments on 28 January 1862, introducing a gas flame with various salts (sodium chloride, barium, strontium, lithium chloride) between the poles of a powerful

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electromagnet, “but not the slightest effect on the polarised or unpolarised ray was observed.”168 Faraday’s lack of success, almost certainly due to the insufficient dispersive power of his spectroscope, weighed heavily on further research for thirty-five years because his authority and the quasi-mythicized experimental ability raised a sort of psychological barrier for other experimenters, impeding them from discovering the phenomenon for which Faraday had searched in vain, thought Tait had theoretically predicted it as early as 1865 for circularly polarised light. Two episodes underscore the negative influence of Faraday’s failures. In 1873, the young German physicist Arthur Schuster (1851-1934), at the time working in Maxwell’s Cambridge laboratory, placed a sodium flame in a magnetic field and initially observed a thickening of its spectral lines. After a few days of excited work, however, he realized that the steel spring in the spectroscope slit was stretched by the action of the magnetic field, and therefore he attributed the observed thickening of the spectral lines to the thickening of the slit and abandoned the research. In this way, Schuster let the discovery of the effect, which he had explained using Weber’s theory of diamagnetism and the recent Maxwellian theory of electricity, slip through his fingers. He later realized to have acted hastily and, embittered, confessed his mistake in a commemorative book that appeared forty years later: too late for him and for science.169 The second episode, which involved the Belgian astronomer Charles Fievez (1844-1890), occurred in the experimental research on the effect of magnetism on electric discharges in very rarefied media, begun after Faraday’s magneto-optical discovery. According to de la Rive and Daniell, the increase in the luminous intensity of the discharge and its spectrum under the action of a magnetic field is due to an increase in the local density of the residual gas; Angelo Secchi, on the other hand, attributed this increase to diamagnetism, saying that the repulsion produced by the magnetic field on the residual gas causes a reduction in the gaseous section of the discharge tube, which in turn leads to an increase in temperature; Achille Auguste Cazin (1832-1877) further extended Secchi’s theory, affirming that the magnetic field does not have a direct action on the rays emitted by the luminous source. To clarify the situation, Fievez decided to eliminate electric current from the experiment and only test magnetic fields and light. To this end, 168

M. Faraday, Faraday’s Diary, edited by T. Martin, Bell, London, 1932-36, Vol. 7, p. 465. 169 A. Schuster, The Progress of Physics during 33 Years (1875-1908), Cambridge University Press, Cambridge 1911, pp. 22-23.

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using a powerful dispersive apparatus installed in the astronomical observatory in Bruxelles and a Faraday electromagnet built by Ruhmkorff, Fievez placed a sodium flame between the two conical ends of the electromagnet, which were 10 mm apart, such that the image of the flame was projected onto the spectroscope slit. As soon as the electromagnet was turned on, the D1 and D2 spectral lines of sodium “immediately became wider”, and if they were already wide, with the field they became even wider and “inverted”, that is, a black line appeared in the middle of the widened (bright) line. Analogous phenomena, though of lesser magnitude, were obtained with the red line in the spectrum of potassium or lithium or the green line in the spectrum of thallium. No difference was noted between observations taken longitudinally or transverse to the magnetic field. The thickening of the lines observed by Fizeau was exactly the same phenomenon that Peter Zeeman had observed earlier, as we will soon elucidate. Yet Fievez too, perhaps to the surprise of the modern reader, concluded his paper with the words: “These experiments, which demonstrate the influence of magnetism on luminous waves, without the intermediary of the electric spark, also show that the phenomena that are manifested under the action of magnetism are identical to those produced by an increase in temperature.”170 Zeeman came to know of Fievez’s experiments after the publication of his own seminal work in the proceedings of the Academy of sciences of Amsterdam, which we will soon discuss. Consequently, in the English edition of his paper he added an appendix in which he discussed the experiments performed by the Belgian astronomer. In particular, the phenomenon of “inversion” described by Fievez in particular aroused his suspicion, leading him to doubt that the phenomenon observed by the Belgian could be attributed to a specific action of the magnetic field on the light waves. In reality, Zeeman appears to have been too preoccupied with defending his scientific priority to the discovery, which incidentally nobody had ever doubted. Let us return to our chronological account and the year 1877, when Kerr, in a rather verbose paper, minutely described a discovery that tied directly to Faraday’s work in 1845. He summarized the phenomenon he discovered as follows: “When plane-polarised light is reflected regularly from either pole of an electromagnet of iron, the plane of polarisation is turned through a sensible angle in a direction contrary to the nominal direction of the magnetizing current; so that [….] a true north pole of 170

Ch. Fievez, De l’influence du magnétisme sur les caracteres des rais spectrales, in “Bulletins de l’Académie royale des sciences des lettres et des beaux arts de Belgique”, 3rd series, 9, 1885, pp. 381-85. The italics is in the original text (p. 385).

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polished iron, acting as a reflector, turns the plane of polarisation righthandedly.”171 In other words, the direction of the plane’s rotation is opposite to the one observed by Faraday, i.e. the direction of the magnetizing current. Kerr’s discovery, of little importance by itself and presented as an almost empirical fact, came at a time in which scientists were beginning to discuss Maxwell’s theory and the electromagnetic character of light, which were without a doubt supported by the discovery. The Kerr effect therefore stimulated both theoretical and experimental research. In a later 1878 paper, Kerr himself expressed the conviction that the phenomenon is somehow tied to the wave theory of light, while the previous year George Francis Fitzgerald (1851-1901) had formulated, though not in exact form (as shown by August Kundt), a closer connection between the new phenomenon and Maxwell’s theory. A consequence of the renewed interest in phenomena occurring in a magnetic field was the discovery announced in 1879 by the American physicist Edwin Herbert Hall (1855-1938) of his eponymous effect: if a metallic, current-carrying bar is placed in a perpendicular magnetic field, the equipotential current lines are rotated. The phenomenon, which Rowland related to Faraday’s magneto-optic effect, immediately became an object of careful study for many physicists (Lorentz, Wiedemann, Nernst, Boltzmann), but perhaps Righi was the one who best detailed the mechanism and size of the phenomenon, whose interpretation, however, remained controversial for many years. In 1887, almost at the same time as the French physicist Sylvester-Anatole Leduc (1856-1937), Righi discovered the analogous thermomagnetic phenomenon, the Righi-Leduc effect: the heat flux through a thin sheet changes if a perpendicular magnetic field is applied to it. In the course of his measurements on the Kerr effect in 1892, the Dutchman Pieter Zeeman (1865-1943) was also led to test whether magnetism affects spectral lines, inspired by studying the theoretical considerations of Lord Kelvin, Maxwell, and perhaps Tait: his results were negative. A few years later, reading Faraday’ s biography edited by Maxwell, his attention was newly directed to the problem. “If a Faraday the young scientist told himself- thought of the possibility of the abovementioned relation; perhaps it might be yet worth worthwhile to try the

171

J. Kerr, On Rotation of the Plane of polarisation by Reflection from the Pole of a Magnet, in “Philosophical Magazine”, 5th series, 4, 1877, p. 322.

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experiment again with the excellent auxiliaries of spectroscopy of the present time, as I am not aware that it has been done by others.”172 The light was analysed with a concave Rowland grating of ten feet radius with 14,938 lines per inch; the Ruhmkorff electromagnet employed was powered by 27 amperes that could be increased to 35. If a flam containing a table salt-impregnated piece of asbestos was placed between the poles of an electromagnet, the D lines of sodium were “distinctly thickened” when the current was turned on; the thickening disappeared when the current was stopped. An analogous phenomenon was observed if the sodium was replaced with lithium carbonate. From this experiment, however, one could not immediately deduce that it was the magnetic field that had acted on the spectral lines, whose thickening could also have been due to an increase in density or temperature of the burning substance. New experiments where the possibility of this secondary action of the magnetic field was eliminated were therefore needed. Zeeman thought to repeat the experiment with absorption spectra. He produced sodium vapour in a porcelain tube and shined a beam of electric lamp light inside it: the D lines in the beam were absorbed by the gas. In the absence of a magnetic field, two distinct absorption lines were clearly visible: once the field was turned on, the two lines immediately thickened and darkened. This series of these experiments almost certainly confirmed that the phenomenon suspected by Faraday almost half a century earlier, both for emission spectra and for absorption spectra (called a direct and inverse effect, respectively) actually occurred. Although Zeeman’s initial 1892 experiments had been motivated by the electric theories of Lord Kelvin and Maxwell, in 1896 he held that the experimental results obtained were better interpreted in the framework of Lorentz’s electromagnetic theory (§ 1.6), which had been published in in 1892 in Leiden, where Lorentz was a professor of mathematical physics and Zeeman conducted experimental research in Kamerlingh Onnes’ laboratory. Lorentz not only approved of his young friend’s ideas, but he also showed him how to treat the motion of an ion in a magnetic field: he predicted that the outer edges of the spectral lines in the experiment would be circularly polarised and also realized that the thickening of the lines could be used to calculate the charge to mass ratio of the ion. 172

P. Zeeman, On the Influence of Magnetism on the Nature of the Light Emitted by a Substance, in “Philosophical Magazine”, 5th series, 43, 1897, pp. 226-39. The first communication of his discovery was found in two consecutive papers, written in four weeks from each other in Dutch and presented to the Academy of sciences of Amsterdam in 1896. It is evident from the passage here that, as we said, Zeeman was not yet aware of Fievez’s work.

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Lorentz’s suggestions allowed Zeeman to work out a theory of the observed effect. The now-obsolete theory essentially stated, applying a famous theorem by Joseph Larmor, that the oscillation of a charged matter particle can split into two uniform circular motions with the same period but in opposite directions, and in one oscillatory motion perpendicular to the circular ones. A magnetic field in the direction of the latter component of the motion leaves it unaltered, while it increases the period of one of the circular motions and decreases that of the other. Therefore, an observer that looks at the flame parallel to the direction of the field will see two circularly polarised lines very close to each other, corresponding to the two circular components. On the other hand, if the observer looks at the flame perpendicularly to the direction of the field, she will see a triplet of lines: the central one is linearly polarised and corresponds to the direction of the field, while the surrounding ones are polarised perpendicularly to the field. Based on the thickness of the doublet or triplet, which Zeeman only observed as a thickening of a single spectral line, the ratio e/m between the charge and mass of the “ion” can be deduced: Zeeman found its order of magnitude to be 107 cgs electromagnetic units. The rotating “ion”, according to Zeeman, is positively charged; the scientist will observe the doublets and triplets only in 1897, in the lines of cadmium.173 The Zeeman effect played an important role and had extensive applications in spectroscopy, atomic physics, nuclear physics, and astrophysics. Its importance was officially recognized in 1902, when Zeeman and Lorentz were awarded the Nobel prize in physics. In 1898, Fitzgerald found a close connection between the Zeeman effect, the Faraday effect, and the phenomenon of anomalous dispersion accompanied by selective absorption, that Kundt had discovered since 1871. However, the correct explanation of the Zeeman effect was only given by the quantum theory proposed by Peter Debye and Arnold Sommerfeld in 1916. After many fruitless attempts, a phenomenon analogous to the Zeeman effect, but for an electric field was discovered by the German physicist Johannes Stark (1874-1957) in 1913, and slightly later by Antonio Lo Surdo (1880-1949), who obtained the effect with simpler experimental instruments. The Stark effect (or Stark-Lo Surdo effect, as some Italians call it) was different in nature from the Zeeman effect and played an important role in quantum theory. Its theoretical explanation, given almost 173

P. Zeeman, Doublets and Triplets in the Spectrum produced by External Magnetic Forces, in “Philosophical Magazine”, 5th series, 44, 1897, pp. 55-60.

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simultaneously by Karl Schwarzschild (1873-1916) and Paul Sophus Epstein (1883-1966) in 1916, was one of the first great successes of Bohr’s theory of the atom (§ 7.13). In 1919 Stark was honoured with the Nobel prize in physics for the discovery of his eponymous effect and the earlier discovery of the Doppler effect in channel rays.

3.31 Metrology In the second half of the 19th century, the progress in electrology and its applications, particularly illumination and aerial and submarine telegraphy, brought about the solution of a difficult problem that stretched beyond science: the issues of metrology and measurement, which arose at the dawn of civilization to fulfil the practical necessities of trade, farming measurements, and timekeeping. Here we skip over the first millennia, for which there is no lack of excellent accounts, and return174 to the widespread scientific interest in the 17th and 18th centuries in a universal system of measure. At the beginning of the French revolution, the demands of trade and the scientific environment bode well for a solution to the problem: the confusion inherent the existing systems of measure was nearly inextricable; decimals, which were very useful in calculations, had been known to scientists for two centuries (Stevin had demonstrated their simple use in a famous 1585 booklet); physicists believed to know of natural phenomena immutable in time that could provide prototype samples, elevating the measurement standard beyond regional differences. Jean Picard (preceded by Wren, Huygens, and perhaps others) had already proposed in 1670 to use the mean solar second as a unit of time and from it to deduce a unit of length, defined as the length of a pendulum with a one-second period (at 45 °of latitude and at sea level, as he later added); that same year Gabriel Mouton (1618-1694) proposed to take as the unit of length one-sixtieth of the meridian arc of the Earth and established decimal multiples and submultiples of this unit. A unified metrological system for all of France based on these concepts was proposed in 1790 to the National Constituent Assembly by Charles-Maurice Talleyrand-Périgord (1754-1838), then a young bishop and later a controversial politician. The proposal was approved and a commission composed of Lagrange, Lavoisier, Condorcet, Mathieu Tillet, Borda, and later others was created to present a report, which was 174

§ 7.31 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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published in March of 1791. The report proposed the “mètre” as a unit of length (a term introduced to refer to a unit of measure in 1675 by Tito Livio Burattini [1615-1681]), defined as 1/40,000,000th of the Earth’s meridian. Picard’s old proposal was abandoned because not all countries were on the 45 °parallel and at sea level: a “universal” standard based on specific geographic locations would not have been an excellent beginning. The units of time and distance became independent, as did the unit of weight, initially denominated gravet and later gramme, defined as 1/1,000th of the weight of a cube of distilled water at 4°C with 0.1 m sides: the first relationships between different types of units of measure were thus established. A decree instituted on 1st August 1793 by the national Convention made the new system of standards mandatory starting on 1st June 1794. However, the practical use of this system was met with much resistance, especially after the fall of Napoleon, to the extent that a new 1837 law was once more needed to make it required. In conclusion, until the first half of the 1800s, the quest for a “universal” system of measure had only taken its first steps. Electric metrology then quickened the pace. This discipline was founded in the 18th century by Cavendish and Volta175; Coulomb laid the necessary foundations for a system of magnetic units and Ampère for a system of electromagnetic units. Gauss gave the first model for absolute magnetic units in his famous 1832 paper176. On this occasion, Gauss proposed a system of absolute measures based on three mechanical units: the second as a unit of time, the millimetre as a unit of length, and the milligram as a unit of mass. Gauss’s system was an important advance from the decimal metric system, because it replaced weight with mass and allowed for the possibility of connecting its units with magnetic units of measure. Gauss’ work was continued by Weber, who built measurement instruments that he devised with Gauss. The Magnetischer Verein (Magnetic Club) of Göttingen, created by Gauss and Weber, was the birthplace of the absolute systems of measure that today are used in all electromagnetic fields. Weber not only established absolute measurements of the Earth’s magnetism, but he brought the methods, in collaboration with Gauss, to electromagnetism in a classic work titled Elektrodynamische Maasbetimmungen (Electrodynamic Measurements), published in Lipsia

175

§§ 7.31-32 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 176 § 7.34 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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in 1852. In 1856, he extended the methods to electrostatics in a collaboration with Rudolph Kohlrausch of the same title (§ 3.28). Although Thomson (later Lord Kelvin) had measured the electromotive forces of voltaic couples and the resistances of conductors using absolute electromagnetic units since 1851, only at the annual meeting of the British Association for Advancement of Science in 1861 was a committee formed to seek standard electrical units to replace the many units then in use: this was a necessity, as Charles Bright and Latimer Clark said at the time, “decreed by the progresses accomplished in the electrical telegraphy.” Among others, the committee consisted of Thomson, Maxwell (the director), and Werner Siemens. To the credit of Maxwell and the British Association, he recognized that the framework of decimal subdivisions was “the best system”, completely forgetting his country’s aversion to the decimal system (which still today has difficulties taking root in Britain). Soon after, in the appendix to the committee’s annual report in 1863, Maxwell and Henry Jenkin proposed the definition and measurement of the ohm, practically creating the c.g.s. system of electrostatic and electromagnetic units. Incidentally, the names ohm, volt, farad, and microfarad were proposed by Bright and Clark in the 1861 meeting of the British Association as part of their coordinated system of measure. In 1875, the Convention du mètre, signed in Paris by seventeen countries, ratified the international recognition of the (international) metric system. With this agreement, the international bureau of weights and measures (Bureau international des poids et mesures), based in Paris, was created with the principal goal to conserve the prototypes of the metre and kilogram and compare them to new ones. In 1921 the agreement was further extended to electrical units of measure; later to thermodynamical, photometrical, and radioactive units, and finally, in 1966, to the measurement of the fundamental physical constants. In the meantime, the definition and preparation of prototypes for electromagnetic units had a life of its own. Of particular historical importance was the International Congress of Electricists, held in Paris in 1881, during which the c.g.s. system was fully defined based on Thomson’s proposal, practically reiterating Maxwell’s earlier proposals. At the International Conference for the Determination of Electrical Units, held in Paris in October 1882, after an absolute system was chosen, a practical question was posed: what is the most convenient electrical unit to choose as the fundamental one? The ohm, the unit of resistance, was chosen by unanimous decision. The work of the brothers Wilhelm and Werner Siemens, who were present at the conference, on the specific resistance of mercury convinced

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the attendees that the best prototype for the unit of resistance was the siemens, a unit already in use to refer to the resistance of a metre-long column of mercury at 0 °C with a constant 1 mm2 cross section. The brothers Siemens had not proposed their unit as an absolute unit but rather as a practical one; some time earlier Lord Rayleigh had found that a siemens was equal to 0.9413 units that the conference had defined as ohms: thus, the length of a standard column of mercury for the ohm was easily found, from which the prototypes of the other electric units were then deduced. Based on the c.g.s. system, the derived “practical system” was established (later called “legal”, “absolute practical”, and, starting in 1960, “international”), whose units were defined as multiples and submultiples of c.g.s. units. Contrary to what commonly occurs in similar cases, the deliberations of the 1881 conference were met with a general consensus because of the looming danger of a proliferation of electrical units for physics (there were already twelve units of electromotive force, fifteen units of electrical resistance, and ten units of current intensity), the general consistency of the system, despite the multiplicity of c.g.s. systems (a source of senseless torment for the following generations of physics students) and finally, and perhaps most of all, due to the authority of the national delegates present at the conference. Here we name a few: Helmholtz, Clausius, Kirchhoff, Siemens, Mach, Rowland, Eleuthere Mascart, Despretz, Thomson, William Crookes, Lord Rayleigh, Ferraris, Lenz. The following international conferences and general conferences on weights and measures, which began in 1889 and occurred periodically every six years, researched and solved the problems that arose in the metrology of a continuously developing science. A related concept to systems of measure is that of “dimensional analysis” introduced by Fourier, who observed that “every undetermined magnitude or constant has one dimension proper to itself, and the terms of the same equation could not be compared, if they had not the same exponent of dimensions.”177 Fourier then showed how the dimensions of certain calorific quantities compare with length, time, and temperature: for example, specific conductivity has a dimension of -1 with respect to all three reference quantities; calorific capacitance has dimension -3 with respect to length and time, and -1 with respect to temperature. It follows that the two sides of each equation must be homogeneous in dimension with respect to all units: a lack of homogeneity indicates a calculational 177

J.-B. Fourier, Théorie analytique de la chaleur, Paris 1822, in Id., Œuvres cit., Vol. I, p. 137.

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error or the introduction of an abbreviated expression. Maxwell spread these ideas and fixed the notation for dimensional analysis equations. The metrology of the 19th century, based on the assertion that all phenomena can be explained through mechanical models, placed great weight on dimensional analysis, from which it hoped to learn no less than the secrets of nature. Even a physicist of Thomson’s stature succumbed to this wish, saying in an 1876 talk that “The relationship between electrostatic measures and electromagnetic measures is very interesting, it leads us from the field of meticulous and exact measurements, which are considered uninteresting, right to the heart of science; they lead us to probe the great secrets of nature.”178 The relationship between metrology and general physics was so seductive that it spurred almost aberrant statements (like “electrical resistance is of the same nature as velocity”) and pseudo-dogmatic affirmations: for instance, it was almost an article of faith that there are three fundamental units. Towards the end of the century, however, in the general process of critical revision of physical principles, scientists began to understand that dimensional analysis formulas were simply conventions to ensure agreement on the language used to express formulas, which nonetheless was not a trivial idea. In consequence, interest in dimensional theories slowly diminished. In 1966, the general conference on weights and measures established the international system, uncontroversially based on six fundamental units: the metre for length, the kilogram for mass, the second for time, the ampère for current intensity, the Kelvin for thermodynamic temperature, and the candela for luminous intensity.

178 W. Thomson, Conférences scientifiques et allocutions, translated into French, Paris 1893, p. 286

4. THE ELECTRON

4.1 Cathode rays A heterodox, an independent researcher who in his private London laboratory studied the problems of chemistry, physics, natural sciences, and spiritualism, William Crookes (1832-1919; Fig. 4.1) can be considered the immediate precursor of the subatomic physics that was born at the end of the 19th century and dominates the scientific research of ours. Crookes discovered thallium in 1861, built the radiometer that bears his name in 1875 and the spinthariscope in 1903, and was made baronet for his scientific merits in 1897. Before Crookes, however, there had been no shortage of references to non-atomic structures and, in particular, to a granular structure of electricity. On the contrary, the hypothesis of a discontinuous composition of electricity had spread through the entire 19th century, sometimes encouraged by the parallel development of atomic theory, sometimes contrasted by the harsh criticisms of the energeticists. However, it was an advanced hypothesis put forth by theoreticians, lacking any experimental justification. Since 1801, Ritter had considered the hypothesis of discontinuous composition of electricity. Faraday had introduced the term “ion” for electrolytic carriers of electricity and postulated that they carried a constant charge. Weber, in 1846, hypothesized that to each ponderable atom “there adhered a negative electrical atom”, whose mass resulted much greater than that of the isolated positive electrical atom. Due to its mass, the negative particle could be considered at rest, while the positive one, rotating around its negative counterpart, gave rise to the molecular current of Ampere.179 Perhaps it was this very concept that Zeeman had in mind when he deemed positive the rotating electric charge responsible for his eponymous effect (§ 3.30). Helmholtz, commenting on Faraday’s laws of electrolysis, could not help but conclude that positive and negative electricity was subdivided in distinct elementary particles “which behaved as if they were atoms of electricity”. Even Maxwell accepted the “molecular charge” or “atom of 179

J.J. Thomson, Cathode Rays, In “Philosophical Magazine”, 5th series, 44, 1897

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Fig. 4.1 – William Crookes.

electricity” as a provisional hypothesis that would be abandoned with further study of electrolytic phenomena. In 1874, George Johnstone Stoney (1826-1911), a well-known Irish theoretical physicist of the time, calculated the charge of an “atom of electricity” in electrolysis, dividing the amount of electricity necessary for the process in one cubic centimetre of hydrogen under standard conditions by the number of atoms contained according to Loschmidt’s recent (1866) calculation: he obtained 10-20 Coulombs and suggested choosing this quantity as the fundamental unit of measurement. Crookes did not join this wave of theoretical speculation, but rather tied his work to an experimental tradition: the passage of electricity through rarefied gases. These were extremely difficult experiments to execute due to the still rudimentary vacuum-based techniques, even more difficult to interpret, and most importantly seemed without a future.

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The study of discharge in gases was initiated by another Englishman, Hauksbee180, yet he could make substantial progress only after the introduction of mercury vacuum pumps (Geissler, 1855), which allowed for very high vacuums to be obtained. After the introduction of this pump, experimental studies of discharge were taken up and furthered by Julius Plücker (1801-1868) and Johann Wilhelm Hittorf (1824-1914), who studied the fluorescence produced on the glass of the tube where discharge and that Cromwell Varley, in 1871, explained as due to due impact against the walls of agents originating from the cathode. In 1876, Eugen Goldstein (1850-1930) discovered the first characteristic of these new agents, which he named, believing them to be of the same nature as light, cathode rays: while light is emitted from luminous bodies in all directions, cathode rays are emitted only in the direction normal to the cathode surface. Crookes continued his studies, further pushing the rarefaction of the gas in the tubes, which were fashioned in highly different forms. He placed a radiometer inside and observed it rotate in the path of a cathode beam: from this he deduced that cathode rays had a mechanical action. When he inserted a metal Maltese cross, he noticed that a shadow formed on the fluorescent glass (Fig. 4.2): from this he deduced that cathode rays inside the tube propagated in straight lines. Lastly, Crookes brought a magnet (Fig. 4.3) close to a beam of cathode rays that had been focused through a slit and noticed that the fluorescent spot on the glass moved: from this he deduced that magnetic fields deflected cathode rays. But what were these cathode rays? According to Crookes, “Radiating matter”, a fourth state of matter or an ultra-gaseous state “as far from gas as gas is from liquid.” Accepting the hypothesis put forth in 1871 by Varley, Crookes believed cathode rays to be composed of residual gas molecules in the tube which, having come into contact with the cathode, became negatively charged and were repelled. Their unusual character was then not due to their nature, which was that of other known substances, but rather their extremely rarefied state of aggregation. In those years, with his characteristic infectious optimism, Crookes declared the following prophetic words, which certainly went beyond the experimental results that had been obtained: “studying this fourth state of matter,” he said, “we seem to finally have at our disposal, under our control, the last particles, that we rightfully consider to constitute the physical

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§ 5.18 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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Fig. 4.2 - Crookes demonstrates the rectilinear propagation of cathode rays: projected from cathode a, they are stopped by the aluminum cross b and the shadow of the cross forms on the tube glass.

Fig. 4.3 - Crookes’ rays bend in a magnetic field. The beam is constrained by the mica foil bd, which has a narrow slit (e); the rays that escape produce fluorescence along ef on the special screen. Bringing the magnet closer to the tube, the beam bends and produces the fluorescence in g. Source: “Proceedings of the Royal Institution of Great Britain”, 1879.

basis of the Universe. We have seen that for certain properties radiating matter has a real consistency like this table; while for others it assumes almost the character of radiating energy; we have positively touched the grey zone, where matter and force are created together, the dark realm between the known and the unknown, that has always exercised over me a

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singular seduction. I dare to think that the greatest problems of the future will find their solution in this grey zone, and further yet that here, in my opinion, lie the last thin, significant, and admirable truths.” However, these visions of Crookes appeared to his contemporaries, perhaps biased against the unconcealed metaphysical beliefs of the scientist and polemicist, as those of a starry-eyed visionary and were derided (a true affront for a metaphysician) as “too clumsily material”.

4.2 The nature of cathode rays In opposition to the matter hypothesis of cathode rays was the wave hypothesis, supported by the German physicists Wiedemann, Goldstein, Hertz, and Philipp Lenard. Hertz had not been able to observe the deviation of cathode rays in an electrostatic field; furthermore, he showed in 1892 that cathode rays could pass through thin sheets of aluminium. Lenard used this discovery to bring the rays out of the vacuum tube through a special “window” that carries his name, consisting of a metallic screen placed in front of the cathode with a small aperture (of diameter 1.7 mm) covered by a sheet of aluminium (of thickness 0.0027 mm) strong enough to withstand atmospheric pressure. If cathode rays are not deviated by electrostatic fields, how can they be electrically charged molecules? If they are electrically charged molecules, how can they pass through metals? With these contradictions in mind, these physicists thought it best to adopt the wave paradigm of Goldstein, albeit attributing to these waves different properties from the ordinary ones, for example, being composed of longitudinal vibrations instead of transverse ones like luminous waves. However, the wave hypothesis was incompatible with the fact that cathode rays could be bent by a magnet, as light rays are unaffected by magnetic fields. In short, both the molecular hypothesis of Crookes and the wave hypothesis of Goldstein were insufficient: to move past this obstacle further experimentation was needed. The task was undertaken by a young physicist, Jean-Baptiste Perrin (1870-1942; Fig. 4.4), who at the time worked with Lippmann in the laboratory of the Ècole normale supérieure. In front of the cathode in a discharge tube, Perrin placed a metal cylinder, which was completely sealed save a minuscule aperture 10 cm away from the cathode, and connected it to an electroscope. When the tube was on, a bundle of cathode rays penetrated the cylinder and it invariably became negatively charged. Conversely, it was enough to deviate the cathode rays with a magnet and avoid penetrating the cylinder to find no reading of charge in the connected electroscope. One could conclude that cathode rays were

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negatively charged, from which their material nature appeared much more likely than their undulatory character.

Fig. 4.4 - Jean Baptiste Perrin

The year was 1895 and electronics was born. It was born, yet not without pushback. The supporters of the wave theory did not give up after Perrin’s experiment. Instead of denying that negatively charged particles could be emitted by the cathode, they denied that these were the particles that made up cathode rays, that is, the agents that produced the fluorescence on the tube glass, which Perrin’s experimental setup had largely masked. The bullet that comes out of a gun is different from the flash that accompanies the gunshot: the charged particles could be the bullets, the agents producing the fluorescence the flash. In other words, the passage of charged particles through Perrin’s cylinder could be accompanied by the emission of waves responsible for the fluorescence.

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Fig. 4.5 – Josef John Thomson

This objection, which was not trivial, was dismissed two years later by Joseph John Thomson (1856-1940; Fig. 4.5), who placed the Perrin cylinder not in front of the cathode, but laterally such that it did not impede the production of fluorescence on the glass. The cylinder became negatively charged and the fluorescence on the glass moved simultaneously only when a magnet curved the cathode rays such that they entered the aperture: in this way he demonstrated that charge was indispensably connected to the rays. If cathode rays were in fact negatively charged particles, the laws of electrodynamics required that they be subject to a deviation when moving through an electrostatic field. How then did the corresponding experiment by Hertz fail? Thomson retried it, initially obtaining the same negative result. He then realized, however, that the absence of deflection was due to

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the conductivity imparted on the residual gas by the cathode rays, which rapidly diminished with diminishing internal tube pressure. In short, a much higher vacuum was necessary. At the time, this was easier said than done due to the state of vacuum-creating techniques. Nevertheless, Thomson was able to create a sufficiently empty vacuum, obtaining deviation of the rays each time metal plates were brought to different potentials by connecting them to the terminals of a battery. He was able to observe the deflection, proportional to the potential difference between the two plates, even when this difference was as low as two volts.

4.3 Measuring the charge and mass of the electron Thus, it was experimentally demonstrated: cathode rays carry negative electric charges, they are bent by electric fields, and they are bent by magnetic fields in the same way in which a magnetic field would act on charged bodies moving along the rays. In the face of this evidence, one cannot help but conclude, according to Thomson, that cathode rays are charged with negative electricity carried by particles of matter.

Fig. 4.6

Yet are these particles molecules, atoms, or an even smaller subdivision of matter? To answer this qualitative question some qualitative indication was needed. It was Thomson himself who answered the question, using two rather different methods of measurement. The most famous of the two is referred to in every physics text, here we will limit ourselves to a sketch of its fundamental concepts. Let A and B (Fig. 4.6) be two metal plates of different potential. A particle of electric charge +m,

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placed at point M, repelled by plate A and attracted by B, is pushed into motion by a force that is constant in the space between the plates: one assumes that the electric field between them is uniform. The falling motion of the +m particle from A to B is analogous to that of a grain of sand in the gravitational field of the Earth. Instead, if a particle of electric charge +n enters the electrostatic field with a velocity v from left to right, it is subject to an electric force downwards and will keep moving horizontally at the same velocity due to inertia, finding itself in the same conditions as a horizontally fired projectile, tracing a parabolic path in the electrostatic field. In short, charged particles in an electrostatic field move in the same way as kinematic bodies on the surface of the Earth, and are thus subject to the laws of mechanics. Analogous considerations apply to the movement of a charged particle in a magnetic field. For example, if one creates a magnetic field of strength B normal to the page and pointing out towards the reader, a particle of charge e and velocity v is deflected downwards, moving in a circular trajectory of radius R = vmc/eB, where m is the mass of the particle and c is the speed of light (Fig. 4.7).

Fig. 4.7

Whether or not it was known to Thomson, as early as 1887 Arthur Schuster had used the phenomenon of cathode ray deflection due to a magnet to measure the relationship between charge and mass of what he believed were the constituent particles of the rays. He obtained a value of 1.1(106) electrostatic units. Schuster’s method, especially after Hertz had declared preferable the wave interpretation of cathode rays, was deemed by all physicists to be affected by some error, although nobody specified which. The research of Schuster fell thus into disregard, victim of the

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principle of authority, ever present in scientific progress. Let us return however to Thomson, who subjected his cathode ray particles simultaneously to both an electric field and a magnetic field, varying their strengths until the two effects on the particles compensated each other, such that the cathode rays were undeflected and the luminous spot they produced on the glass remained fixed. In these conditions, simple mathematical considerations allow one to conclude that the velocity of the particles is given by the ratio between the magnitudes of electric and magnetic fields, which is easily measured. By measuring, instead, the deviation of the fluorescent spot due to the simultaneous effects of judiciously chosen electric and magnetic fields and applying the laws of mechanics to the motion of the particles, Thomson was able to measure the ratio e/m between the electric charge of each particle and its mechanical mass. In addition to this now classic method, Thomson utilized another procedure, less accurate and more laborious, yet applicable to a wider range of internal tube pressures. In brief, the second method is as follows. Let N be the number of electrically charged particles, each with charge e, that in a fixed time enter a Faraday cylinder: it is possible to measure the total transported charge Q = Ne. If the particles strike a solid body with velocity v, assuming that the total kinetic energy ଵ

w= NMv2 ଶ

is completely transformed into heat, one can calculate the amount of energy by measuring the increase in temperature of the body with known heat capacity. Furthermore, if the rays are in a uniform magnetic field of magnitude H and ȡ is the radius of curvature of their trajectory, one has mv/e = Hȡ. From the two formulas one can immediately deduce the velocity v of the particles and the ratio m/e from directly measured quantities. Thomson’s experiments, using both methods, yielded the following results: the velocity of cathode rays is extremely elevated and grows with the rarefaction of the tube interior; it is significantly greater than the mean velocity attributed to the residual gas molecules in the tube by kinetic theory. While in 1894 Thomson had obtained 2(107) cm/s by measuring the speed of cathode rays directly, noting the time interval that separated the appearance of fluorescence at two points on the tube walls which were at different distances from the cathode, with his new methods he deduced a velocity of 1/10 the speed of light, and ten years later calculated it to be 1/3. Furthermore, he found the velocity to grow with the accelerating

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potential, that is, with the difference in potential with which the discharge occurs. The ratio e/m does not depend on the nature of the residual gas in the tube, nor does it depend on the shape of the tube, the nature of the electrodes, the speed of the cathode rays (as long as they do not reach values close to the speed of light), or any other physical circumstances: in short, e/m is a universal constant. These results ruled out that the particles emitted by the cathode could be gaseous ions from the residual gas in the tube. Consequently, the “radiating matter” hypothesis of Crookes had to be abandoned, yet the underlying idea remained and was further validated: cathode rays were of a material nature. The constancy of the ratio e/m revealed without a doubt an individual character of these particles, which seemed identical among each other as no distinguishing phenomena could be observed. The value of the ratio was of the order of 107, measuring e in CGS electromagnetic units and m in grams; a few years later Thomson himself gave the value of 1.7 (107), which, if compared to the currently accepted value of (1.760 ± 0.002) (107), gives an idea of the great precision of the methods used by Thomson. The analogous ratio for an ion of hydrogen in electrolysis, as we have mentioned, had already been calculated to be 104. The disparity between the two ratios led to differing interpretations. To be precise, if instead of using Thomson’s “corpuscules” we adopt the modern nomenclature “electrons” for the negatively charged particles that make up cathode rays, which was introduced by Stoney in 1891, we can make three hypotheses: electrons have the same charge as hydrogen ions and thus their mass is 1/1000th that of an ion; electrons have the same mass as hydrogen ions and thus have 1000 times the charge; or the charge and mass of electrons has nothing to do with the analogous attributes of hydrogen ions. The second hypothesis seemed to disagree with a result obtained by Lenard: he had shown that the mean free path of cathode rays in air at ordinary pressure was half a centimeter, while the size of a molecule under the same conditions was barely one hundred millionth of a centimeter: it is intuitive that this comparison is evidence of the extreme minuteness of the electron. In addition, the fact that electrons appeared to be of the same nature as the residual gas in the tube was also not in harmony with the second hypothesis. However, these considerations were not enough for a definitive conclusion. Only a direct measurement of m or e would have resolved the problem for certain, although Thomson, already in a private note from 1897, did not hesitate to give his preference for the first hypothesis, which allowed for a return to the ancient suggestive hypothesis, revived (1816) by William Prout, of the sole primordial element. “The experiment which seems to me to account in the most

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simple and straightforward manner for the facts,” he wrote in his note, “is founded on a view of the constitution of the chemical elements which has been favourably entertained by many chemists: this view is that the atoms of the different chemical elements are different aggregations of atoms of the same kind.” Naturally these atoms could not be of hydrogen, but rather “some unknown primordial substance”181 In this occasion Thomson observed that if one conceives of a chemical atom as “an aggregation of a number primordial atoms,” the problem of finding the equilibrium configuration for identical particles acting on each other with the same law quickly becomes complicated, from an analytical point of view, with an increasing number of particles, so much so that an exact mathematical treatment is hardly possible. It is better to turn to models: the simplest of which appears to be that of floating magnets proposed by the American physicist Alfred Marshall Mayer (1836-1897) in 1878 (§ 7.5). A fortunate circumstance gave Thomson the satisfaction of confirming his conjecture through the direct measurement of e. In 1897 Charles Thomas Rees Wilson (1869-1959) had discovered that in air supersaturated by water vapor each ion becomes a centre of condensation for the vapor, namely it attracts vapor molecules around itself and a droplet of water begins to form, accreting bit by bit until it becomes visible. Condensation occurs more readily around negatively charged particles. This discovery was used in 1911 by Wilson to build the “camera” that carries his name, one of the most precious instruments in atomic physics, said (and with a certain emphasis) to be “a window into the atomic world”. Thomson exploited the discovery in the following way. Suppose that in an ionized gas there are n ions present, each with a charge e, which move with velocity v; rapid expansion causes supersaturation of the gas such that each ion becomes a centre of condensation. Now, the easily measurable electric current magnitude is nev and the speed v can be measured, as we have seen, so if one can determine n then the value of e follows. To this end, on one hand, one measures how many grams of water vapour condense; on the other hand, a formula given by Stokes allows one to calculate, from the free fall velocity of the droplets, the radius and therefore the mass of each droplet. Dividing the number of grams of water vapour produced by the mass of each droplet, one obtains the number of droplets, that is the number of gaseous ions, and thus the charge e of each of them. The determination of e in this manner is laborious and requires many precautions. Thomson obtained a value of e = 6.5(10-10) electrostatic 181

J.J Thomson, Cathode Rays, In “Philosophical Magazine”, 5th series, 44, 1897, p. 311.

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units by averaging many measurements, a value decently in agreement with the charge of a hydrogen ion, and in good agreement also with the theoretical value assigned by Lorentz in the same year and deduced, as we have mentioned (§ 3.30), from a quantitative study of the Zeeman effect. The preceding method was refined by Wilson in 1899 when he introduced the “balanced droplet” technique: a positively charged plate is placed above the negatively charged droplet such that it attracts the droplet, opposing its fall. One can adjust the setup to make the upwards attraction felt by the droplet equal to its weight: in this case the droplet remains suspended in the air, between heaven and earth, like Muhammad’s chest. From the conditions of equilibrium it is then simple to deduce the charge of the condensation nucleus. Are the droplet charges though truly the charges of electrons? Or are these rather the charges of ionized molecules, that have no reason to be a priori equal to electron charges? This was a serious objection. However, Thomson showed that the charge of an ionized molecule is the same as that of an electron, hence the preceding methods, in determining the charge of ionized molecules, also determined the electron charge. Furthermore, this charge appeared to be completely independent of the ionization method and was always the same as that of a monovalent ion in electrolysis. It sufficed to substitute the value of the charge e into the ratio e/m to calculate the mass of the electron, m = 1.2(10-27) grams, or about 1/1700 the mass of a hydrogen atom. Between 1897 and 1900, the measurements of e/m multiplied and improved. In 1898, Lenard, now converted to the corpuscular theory, measured the ratio for cathode rays that had escaped the discharge tube; Thomson carried out the measurement for electrons produced by X-rays and the thermionic effect; Lenard and Thomson for electrons emitted by the photoelectric effect; Becquerel established that ȕ-rays were electrons; and Planck, in 1900, deduced the value of the electron charge from the theory of blackbody radiation. The most accurate technique to measure the electron charge remained that of the oil droplet, suggested in 1907 by the Austrian physicist Felix Ehrenhaft (1879-1952) and utilized by the American Robert Andrew Millikan (1868-1953) in 1913, with enough success to win a Nobel prize in 1923. Between the parallel plates of a capacitor, one observes through a microscope the movement of a well-lit oil droplet, which due to gravity and its viscosity has reached terminal velocity in free fall. From this velocity, using Stoke’s law, the radius of the droplet (about a micron) and consequently the mass can be inferred. An electrostatic field is then

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produced between the plates of the capacitor and, with the aforementioned balanced droplet technique, one finds that the electric charge of the droplet varies in discrete jumps, which are whole multiples of a single charge unit, proportional to the magnitude of the electric field. This charge unit is thus the electron charge, because the phenomenon described can be easily interpreted supposing that the droplet acquires charge due to friction during its free fall, and thus contains a few elementary charges. The experiment is very delicate, but the results obtained are considered to be among the most reliable. It was acutely observed that Millikan’s experiment demonstrated the existence of a lower limit in the subdivision of charge, yet did not demonstrate the granulation of charges in an electrified body, in the same way that water drops squeezed from a pipette do not prove that water is made up of droplets. Nevertheless, physicists, also for other reasons that we shall see, interpreted these experimental studies in harmony with the insights of the great theoreticians of the 19th century: electricity, at least in its negative form, has a discontinuous composition like matter, and in all known phenomena the atom of negative electricity has constant charge and mass. Today the following values are reported: e = (4800 ± 0.005) (10-10) electrostatic CGS units = 1.601 (10-19) Coulombs; m = (9.08 ± 0.02) (10-28) grams, i.e. about 1/1840 the mass of a hydrogen atom. It is not by chance that the term “electron”, introduced by Stoney back in 1891, spread to become general and popular after 1900, that is after no physicist had any doubts on the granulated constitution of electricity182.

X-RAYS 4.4 Production of X-rays Often in the history of physics the contrasting currents of scientific thought are divided according to the nationalities of physicists. This is not a manifestation of nationalism, at least not generally speaking. The phenomenon is better explained through academic ties, personal friendships, 182

Stoney called electron the electrical charge of a monovalent ion; the term had, therefore, a different meaning than the present one, as one immediately finds in reading his paper published in “Scientific Transactions of the Royal Dublin Society” in 1891. After 1900, on the suggestion of Paul Drude, electron was taken to mean the corpuscle carrying an elementary charge. Yet, the use of the term ion to mean electron continued for many years, especially among English physicists, generating confusion and misunderstandings that characterized the writings of the last decade of the century.

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the use of the same or analogous experimental equipment, or the perfect understanding of the language used by one’s compatriot.

Fig. 4.8 - Wilhelm Conrad Röntgen

It comes as no surprise then that Wilhelm Conrad Röntgen (18451923; Fig. 4.8), a half-Dutch, half-German physicist, had initiated an experimental study on cathode rays, following Lenard’s trail, as a supporter of undulatory hypothesis of cathode rays, like all German physicists of the time. Lenard, working with a discharge tube equipped with an aluminium window from which the rays exited, had observed phenomena described in a paper published in 1894 in the “Annalen der Physik und Chemie” of Berlin. “Cathode rays,” he wrote then, “are photographically active. With a long enough exposure, their mark on a photographic plate is perfectly observed. After the experiment, a photographic plate placed underneath a sheet of cardboard displayed blackened areas that were perfectly delineated. Various metallic foils had been placed on top of the cardboard and, according to their degree of permeability for the passage of cathode

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rays, appeared lighter or darker on the photographic plate. Only underneath foils of a certain thickness was the sheet unblemished: thus it is demonstrated that cathode rays pass through cardboard and metals.” It was not easy to find the very thin sheets of aluminium that closed the windows of the discharge tubes used by Lenard, so Röntgen turned to his younger colleague for further information. Lenard responded immediately, sending him two sheets “from his small stock” having a thickness of five thousandths of a millimetre. However, Röntgen was occupied with administrative work as the rector of the university of Würzburg during the 1894-95 year, and only in the autumn was he able to resume discharge experiments. A scrupulous experimenter, already distinguished among the physicists of the time for his research in various fields (liquid compression, specific heat of gases, magnetic effects of moving dielectrics in an electrostatic field, etc.), Röntgen took an experimental approach that he intentionally left shrouded in mystery. We only know that one day he realized that some crystals, which he had left out on his laboratory table, glowed each time a discharge occurred in the tube, which was wrapped in a cardboard casing. It could not have been an effect of the cathode rays, as the tube with which he was working had no aluminium window like the one used by Lenard, thus the they could not have escaped. This was certainly a novel phenomenon, and as he was able ascertain in a few days, it originated from the inside of the discharge tube. The 8th of November 1895, Röntgen observed a surprising new phenomenon: if a discharge tube was wrapped in black cardboard and was in proximity to a paper screen smeared with barium platinocyanide, he observed that the screen glowed fluorescent green at each discharge of the tube, both when it was facing the smeared side of the screen and when it was facing other side. What was immediately striking about the experiment was that the black cardboard, which completely blocked visible and ultraviolet rays, let through an agent that caused fluorescence. Yet this effect was not only obtained with cardboard: other methodically set-up experiments demonstrated that all bodies were more or less transparent to these new agents. To be precise, the transparency diminished with increased density of the bodies and with increasing thickness. “If one holds out a hand,” Röntgen immediately observed, “between the electrical discharge apparatus and the screen, one can see the dark shadow of the bones silhouetted against the much less accentuated shadow of the hand”183: the first radioscopy had been 183 W.C. Röntgen, Über eine neue Art von Strahlen, in “Sitzungsberichte der wurzburger Physikalischen-Medicinischen Gesellschaft”

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performed. The new agents, that Röntgen called X-rays184 for brevity, did not engender fluorescence only in barium platinocyanide, but also in other bodies, like phosphorescent compounds of calcium, uranium glass, ordinary glass, limestone, rock salts, etc. These rays also marked photographic plates, yet they were invisible to the naked eye. Röntgen was unsure if X-rays underwent refraction; he had not observed it with prisms of water and of carbon sulfate, while it seemed to him that some hint of refraction was displayed by X-rays in prisms of ebonite and aluminium. Experiments using finely ground rock salts, powdered silver obtained electrolytically, and zinc powder showed no difference between the transparency of the powder and the transparency of the corresponding compact material for X-rays. From this one should deduce that they undergo neither refraction nor reflection; the absence of these phenomena being confirmed by the fact that X-rays cannot be focused with a lens. X-rays originate in the point where cathode rays strike the glass of the tube; in fact, deviating the cathode rays inside a discharge tube with a magnet, one finds a simultaneous movement of the source of the X-rays, which still lies at the terminal end of the cathode rays. Nor is it necessary for the production of this new radiation that the cathode rays strike glass: the phenomenon occurs even if the discharge tube is built out of aluminium. The nature of these new entities appeared mysterious; only one thing was certain: they were not identified with cathode rays. It is true that they exhibited fluorescence, had a chemical action, and propagated in a straight line, producing shadows: all exactly like cathode rays. However, they did not display the characteristic phenomenon of cathode rays: being subject to deviation by a magnet. Could they be of the same nature as ultraviolet rays? Yet in this case they would have to discernibly reflect, refract, and become polarized. Nevertheless, considering there to be a certain relationship between these rays and luminous radiation, one could imagine that, while visible rays are transversal vibrations of the ether, X-rays are longitudinal vibrations of the same ether, whose existence remained still unproven by physicists. With these questions, a reiteration of the previous years’ attempts at interpreting the nature of cathode rays, the first paper on X-rays closes, 184

“The reason,” Röntgen explains in his memoirs, “for which I felt authorized to give the name “rays” to the agent that emanated from the wall of the discharge apparatus lies partly in the regular production of shadows that one observes when one interposes a more or less transparent body between the apparatus and the fluorescent screen (or photographic plate).”

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dated December 1895, from the physics institute of Würzburg. The second paper is from 5 March 1896 and contains two important facts. The first was discovered by Righi as soon as he had received news of Röntgen: not only do electrically charged bodies discharge under the influence of X-rays, but the air traversed by them gains the ability to discharge charged bodies. The second important fact had already been alluded to by Röntgen in his first paper: X-rays are produced not only from cathode rays striking the glass walls of a discharge tube, but also after striking any body, including liquids and gases. The intensity of the rays produced depends on the nature of the bodies that were struck. In 1897, Antonio Roiti pointed out that metals of greater atomic weight emitted X-rays of higher intensity.185 From this observation, Röntgen was led to construct his focus tube at the end of February 1896, “in which a concave aluminium mirror acts as a cathode, and a platinum plaque, tilted at 45º from the axis of the mirror and placed at the centre of curvature, acts as the anode”. Until the introduction of thermoelectric tubes, which appeared in 1914 thanks to William D. Coolidge (1873-1975), focus tubes were the only apparatuses used for x-ray production in medical practices and physics laboratories. The new discovery, whose potential for medical and surgical application was immediately foreseen, thrilled scientists and the public. Physics laboratories were inundated with doctors and patients; in the innumerable exhibitions, the public demonstration of live human skeletons shocked audiences and scenes of panic or hysteria sometimes followed. Röntgen was in favour of the rapid dissemination of his discovery, renouncing, with his typical disinterest in economic affairs, any possibility of profit. This general favour contributed to the rapid progress in X-ray techniques, although then, as in the wake of every great human achievement, there was no lack of criticisms and apprehension: criticisms of science that fed “spectral images” to the public as memento mori (remembeer that you have to die); apprehension that the new rays could violate every private aspect of human life. We do not believe it is our duty to follow the development of radioscopic techniques. A single detail, however, may give an idea of the growth undergone in this field: in 1896 a radiography of an arm required an exposure of twenty minutes, today it can be done in an infinitesimal fraction of a second. 185

A.Roiti, Se I raggi X esistano già nel fascio catodico che li produce, in “Rendiconti della Reale Accademia dei Lincei. Classe di scienze fisiche matematiche e naturali”, 2/6, 1897, p. 129; also in “Philosophical Magazine”, 5th series, 45, 1897, p. 510.

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The discovery of X-rays had extraordinary consequences for scientific research and practical applications both medical and industrial. One can perhaps say without exaggeration that this discovery “began a new phase in history”. X-rays discharge electrified objects, like ultraviolet light. However, while ultraviolet light must shine on a negatively charged metallic surface in order to cause the discharge, for x-rays it is enough to traverse a gas placed in an electric field to give rise to equal and opposite charges in the gas moving along the electric field lines. This behaviour was demonstrated in 1896 almost simultaneously by Perrin in France and Thomson and Ernest Rutherford in Great Britain. In 1900 though, Lenard showed that ultraviolet rays of very short wavelength behaved in almost the same manner as x-rays. On 10 December 1901, in the great hall of the Stockholm academy of music, in the presence of the crown prince of Sweden representing the king, the Nobel prize committee, bestowing the award on behalf of all scientists and humanity, awarded the first prize in physics to Röntgen.186 Today it may seem symbolic that the first person recognized by the most coveted international scientific honour was the discoverer of x-rays.

186

Alfred Bernhard Nobel, born in Stockholm on 21 October 1853 and died in Sanremo (Italy) on 10 December 1896, dedicated himself to the study of explosives; in 1867 he obtained the patent for dynamite, a name given by him to a mixture of nitroglycerin and inert substances; in 1876 he obtained the patent for another explosive device, gelignite, and in 1889 for ballistite. Exploiting these patents and some petroleum fields in Baku (Azerbaigian), he accumulated an immense fortune, which he dedicated in his will to the establishment of three annual international prizes, destined for those who had made the greatest discoveries in their respective fields of physics, chemistry, and physiology or medicine. He also established a prize for literature and a prize for peace. The conferral of the prizes began in 1901 and happened every year on the tenth of December, the date of Nobel’s death. The Nobel prize became the highest and most coveted scientific distinction worldwide. It is customary for the laureates to read a review of their work (Röntgen did not partake in what was to become the standard practice): the collection of these reviews, published each year, constitutes in and of itself a record of the evolution of physics in our century. So far, six Italians have been awarded the Nobel prize for physics: Guglielmo Marconi in 1909, Enrico Fermi in 1938, his student, friend and biographer Emilio Segrè in 1959, Carlo Rubbia in 1984, Riccardo Giacconi in 2002 and Giorgio Parisi in 2021.

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4.5 The nature of X-rays While the applications of X-rays continually expanded, the study of their genesis and their nature was an increasingly urgent necessity for theoretical physics. Röntgen’s first interpretation of the genesis of X-rays was immediately unanimously accepted: X-rays originate from the striking of cathode rays, i.e. electrons, against bodies, in particular the anticathode of a discharge tube. Yet what was their nature? Röntgen’s hypothesis of longitudinal waves was not acceptable for various reasons, nor was the hypothesis of materiality of X-rays, which had been put forth in the early days, or the hypothesis of their presence inside the bundle of cathode rays, which was supported by some German scientists and Angelo Battelli and Antonio Garbasso in Italy. Electromagnetic theory predicted that the rapid variation in velocity of a charged body would produce an electromagnetic perturbation, therefore if one acknowledged, as it seemed necessary, a cause-and-effect relationship between the brusque halt of the electrons on the anticathode and the production of X-rays, it followed that X-rays were an electromagnetic perturbation. But, how could one then explain the inability to observe common optical phenomena with X-rays – reflection, refraction, polarisation, diffraction? The electromagnetic perturbation produced by the impact of electrons on the anticathode was not periodic, came the answer. A lack of waves could thus explain the anomalous behaviour of X-rays compared to electromagnetic waves. “Röntgen rays,” wrote Thomson, “are not waves of very short wavelength, but rather impulses.”187 For lack of a better idea, physicists patched the theory in this manner until 1912. Crookes may have been one of the first physicists to hold X-rays analogous to luminous radiation, though with no experimental evidence. In his presidential speech on 29 January 1897 for the Society for Psychical Research, he hypothesized, with the fervent imagination that always accompanied his scientific research, that x-rays could be a means to transmit thought, that they moved at the speed of light, and had a frequency between 2.8(1017) and 2.3(1018) vibrations per second. Curiously enough, Crookes correctly guessed the order of magnitude of their frequency; commonly used X-rays have an average wavelength of 1 Å (= 10-8 cm), corresponding to a frequency of 3(1018) vibrations per second. 187

J.J. Thomson, A Theory of the Connextion Between Cathode and Röntgen Rays, in “Philosophical Magazine”, 5th series 45, 1890, p. 183. This article is preceded by one of Battelli, in which the same thesis was reported.

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Even without adventuring into numerical specifications, that only an uninhibited scientist like Crookes could make in a psychic research centre, there was no shortage of physicists that realized that one did not have to do away with the undulatory character of X-rays, which would have made them similar to luminous rays, in order to explain the unsuccessful attempts to observe common optical phenomena. Rather, one simply had to consider extremely small wavelengths to explain the behaviour of X-rays. The concept is made more clear through an analogy which was often used in the first decade of the century for didactical purposes. Sound waves, whose range of wavelengths extends from less than a centimetre to more than twenty metres, reflect off objects of rather large dimensions, a wall for example, but do not reflect off smaller objects, like a pole planted vertically in the ground. The explanation lies in the fact that the formation of a reflected wave requires the addition of many wavelets, originating from each of the points struck by the incident wavefront. Consequently, in order for reflection to occur, the object hit must be of larger dimensions than the wavelength of the incident wave. Much like one cannot deduce the absence of periodicity in sound from the fact that it does not reflect off of a single pole, so one cannot deduce that X-rays are not waves from their missing or dubious reflection. It is enough to suppose that the wavelength of X-rays is lower than the distance between molecules of an object, allowing each molecule to behave like the vertical pole isolated from sonic phenomena, to prevent reflection of X-rays and leave only diffraction. The physicists who followed this line of thinking were naturally led to try to observe not the reflection, but rather the diffraction of X-rays through extremely thin slits, as necessitated by their very short wavelength. As thin as the slits produced were, however, they remained too large to observe anything, and there were clearly no mechanical means to create openings smaller than an intermolecular gap. It was then that a young German physicist, Max von Laue (1879-1960), already a disciple of Max Planck, had a bold idea. An old theory on the constitution of crystals, attributed to Haüy, had supposed, from the characteristic phenomenon of crystal cleavage, that they were constructed from the juxtaposition of extremely small particles in the shape of parallelepipeds, called integrating molecules, that were in perfect contact with each other. Ludwig August Seeber (1824), Gabriel Delafosse (1834), and more systematically Auguste Bravais (1811-1863) modified Haüy’s idea, substituting integrating molecules with point molecules, arranged at a small, constant distance from each other following a perfectly regular scheme. Thus arose the study of spatial lattices, conducted for over sixty years as purely geometric research. A collection of points, supposedly the

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centres of gravity of molecules, is ordered according to lines (rows) and planes (lattices) such that the network can be thought of as composed of elementary parallelepipeds (cells): points on the same row or on parallel rows are therefore separated by a constant distance, which can be different in rows along different directions. The direction of the rows and the position of the planes can be thought of as corresponding to the edges and the faces observed in crystals. Each elementary parallelepiped is characterized by the relative inclinations of three rows which meet at a point and by the lengths of its edges. Bravais, in 1850, studied the types of elementary parallelepipeds that can be considered the constituent cells of crystal systems, in the sense that their successive repetition can give rise to a lattice compatible with the laws of crystallography: he indicated 14 types which today bear his name. In 1879 Leonhard Sohncke (1842-1897) extended the theory, indicating up to 65 spatial lattices, described by substituting simple lattices with a complex of identical lattices interpenetrated in various ways, superimposable on each other through the operations of translation, rotation about a row, and rotation followed by translation, in short, helical movements around an axis. In 1891, the Russian crystallographer Evgraf Fedorov (1855-1919) and the German mathematician Arthur Schoenflies independently described the complete mathematical theory, adding to the preceding operations also those of reflection across a plane and combinations of reflections with translations and rotations. Using the theory they were able to demonstrate, through completely different approaches, the existence of 230 complex systems of points that could be grouped in 32 classes, identical to the known symmetry classes present in natural crystals. Fundamentally, to each type of symmetry in natural crystals there corresponded a particular spatial lattice. The preceding crystallographic theories remained purely mathematical fantasies: beautiful but outside the realm of physical reality. For the majority of physicists at the time, the molecules in crystals were randomly arranged, and anisotropies were due to particular distributions of molecules in privileged directions. For the majority, yet not for all. It was precisely to the aforementioned geometric fantasies that Laue thought it wise to lend credit, at least provisionally. If crystals really had the structure imagined by theorists, a crystal was like a diffraction grating, or better yet, a series of of diffraction gratings placed on parallel planes. If the diffraction of X-rays by a crystal was proven, two birds would have been killed with one stone: on one hand the undulatory nature of X-rays would have been demonstrated, and on the other hand there would be an

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experimental basis for the hypothesis about the nature of crystals. Laue worked out the quantitative theory of the phenomenon, and in 1912 in Munich suggested the verifying experiment to Paul Knipping (1883-1935) and Walter Friedrich (1883-1968). The experimental apparatus was fairly simple: a certain number of parallel lead sheets were placed in front of a small crystal, for instance Rock Salt, shielding it from the direct effect of X-rays; all the sheets contained minuscule holes with centres aligned across the sheets. A beam of X-rays then passed through the holes, struck the crystal, and proceeded towards a photographic plate covered in black paper to shield it from external radiation. After a few hours of exposure, the sheet was developed and a black mark was observed along the axis of the holes, due to the direct effect of the X-rays, surrounded by a great number of marks of varying intensities, all distributed in a regular manner according to the structural symmetry of the crystal. The experiment, which was immediately repeated, altered, and discussed by many physicists, led to the conclusion that the patterns obtained on the photographic plates were diffraction patterns. From the experimental results obtained, William Henry Bragg (1862-1942) and his son William Lawrence (1890-1971) proposed in 1913 to abandon the hypothesis of integrating or point molecules and instead welcome the hypothesis put forth since 1898 by Barlowe for sodium chloride: atoms and not molecules are at the vertices of the crystal lattice, and diffraction occurs on atoms. It is easy to see how Bragg’s theory, once accepted by physicists, fundamentally changed the traditional concept of a molecule. Yet we cannot linger here on the details and the numerous theoretical and experimental consequences of Laue’s theory. We highlight only two concepts: from the study of diffraction patterns one can calculate the wavelength of the x-rays utilized, and, vice versa, knowing the wavelength one can draw information about the crystal structure. The average wavelength of x-rays resulted as one thousandth the average wavelength of visible rays, even shorter than that of ultraviolet rays, and x-rays displayed spectra analogous to those of visible light.

RADIOACTIVE PHENOMENA 4.6 Radioactive substances Cathode rays strike the walls of the glass tube and produce fluorescence; from the fluorescent part of the tube originate X-rays. These two phenomena, coincident in space and time –fluorescence and the

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production of x-rays– are they related to one another? The question, formulated at the beginning of1896 by Poincaré, would not even have been brought up had X-rays been initially produced using a focus tube, yet as they were observed using a simple discharge tube in Röntgen’s first communications, it was both natural and legitimate. Several experimenters believed they could respond affirmatively, asserting to have obtained photographic impressions on sheets wrapped in black paper in the presence of phosphorescent substances (zinc sulfide, calcium sulfide, artificial zinc blende). Yet nobody was able to replicate the effects claimed by these physicists.

Fig. 4.9 – Henri Becquerel

The question attracted the attention of Henri Becquerel (1852-1908, Fig. 4.9), member of a dynasty of illustrious physicists: his grandfather Antoine-César (1788-1878), his father Alexandre- Edmond (1820-1891), and his son Jean (1878-1953). Fluorescence and phosphorescence were household phenomena in the Becquerel family: his father Edmond, in fact,

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had occupied a large part of his career with the spectroscopic study of phosphorescence and in particular the phosphorescence of uranium, and Henri Becquerel himself, from 1882 to 1892, had continued his father’s studies. As Röntgen’s experiments seemed to have a certain air of familiarity, on the first day he received word of them, in the early months of 1896, Becquerel wondered if X-rays could have been emitted by phosphorescent bodies that had been irradiated for a long time by sunlight. Among phosphorescent bodies, uranium salts lent themselves well for research. Thus, slabs of bisulfate of uranium and potassium were placed on photographic plates wrapped in thick black paper and exposed for many hours to sunlight. The developed plates displayed the profiles of the slabs that had been placed on them: a positive experimental result for the working hypothesis and one that prompted further research. Indeed, it was evident that the uranium salts had emitted agents that, through the black paper, had struck the photographic plates. Were these agents connected to the phenomenon of phosphorescence, that is, solar energy transformed by the uranium-containing material? A lucky break soon allowed Becquerel to answer the question. On the 26th and 27th of February 1896 he set-up the phosphorescence experiments, already quite modified compared to the primitive ones, yet could not perform them because the sun had been out only sparingly. The experimental apparatus (a photographic plate wrapped in a chassis of black canvas closed by a sheet of aluminium on which was placed a thin copper cross and, on top of it, a sheet of bisulfate of uranium and potassium) was therefore closed in a drawer. When the plate was developed on 1 March, there appeared, contrary to all expectations, the dark profile of the cross. Becquerel immediately assumed that the effect had continued in the dark, and new, appropriately organized experiments confirmed his conviction. Therefore, the preliminary exposure of the uranium salts to sunlight was not necessary to obtain the photographic effect; furthermore, the phenomenon was not manifested in other phosphorescent bodies, but was observed in other uranium salts that were not phosphorescent. All of these details were enough to conclude that he was in the presence of a new spontaneous phenomenon, the intensity of which showed no appreciable reduction with the passage of time, as resulted from experimenting with uranium salts created at much earlier times. In those same days, news had reached Paris that many physicists had experimentally observed that X-rays discharged electrically charged objects upon impact. Becquerel attempted the same experiment for the new agents and obtained the same result. He now had two methods to

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continue his research: the photographic method, which was mainly qualitative; and the electric method, suitable for obtaining numerical indications and relative measures. Although it may seem strange, for two whole years Becquerel was the only physicist occupied with such study; then, in 1898, the Curies joined, and after the discovery of radium the number of researchers ballooned to become a legion at the end of the century, with Rutherford, André Debierne, Jules Elster, Hans Geitel, Friedrich Giesel, Walter Kauffmann, Crooks, William Ramsay, and Frederick Soddy. At the end of the century, the most important phenomena produced by uranium radiation were established: ionization, fluorescence and luminescence, chemical, photographical, and physiological effects (burning of skin, yellowing and crumbling of leaves, impediment or delay in the development of microbial colonies, paralysis followed by death of animals subjected to prolonged radiation). Yet, as often happens in such circumstances, the direction of study given by Becquerel guided the most active research. Now, one of the first facts that Becquerel brought to light was the following: all uranium salts, phosphorescent or not, crystallized or in powder, dry or in solution, whatever their origin, emitted radiation of the same nature, whose intensity depended only on the quantity of uranium contained in the salt. The property, therefore, was an atomic property, tied to the element uranium, a conclusion confirmed by the fact that metallic uranium resulted three and a half times more active than the salt utilized in the first experiments. Thus, there was a direction to follow in the search for other possible substances that exhibited analogous phenomena to those of uranium. In 1898, almost simultaneously, Marie Curie (Fig. 4.10), née Sklodowska (1867-1934), in France, and Gerhard Karl Schmidt (1865-1949) in Germany, discovered that thorium had analogous properties to those of uranium. Curie undertook systematic studies on the activity of numerous minerals of uranium and thorium, soon reaching such interesting results that her husband Pierre (Fig. 4.10) abandoned his own work to collaborate with his wife). The introduction of the term radioactivity in this time period (1898) is due to them, indicating the property of substances that can emit Becquerel rays, then the name for the bodies emanating from substances containing uranium and thorium. The activity of radioactive substances, that is the intensity of their irradiating, was measured by the conductivity gained by air under the effect of the substance in question, through an apparatus used by Marie Curie.

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Fig. 4.10 – Pierre and Marie Curie.

It seemed very unlikely to Curie that radioactivity, which was manifested as an atomic property, would be peculiar to certain types of materials and exclude all others. She thus examined all the substances that she could obtain (metals, metalloids, rare substances, rocks, minerals). Among the minerals she examined, pitchblende (a mineral of uranium oxide) behaved in an unexpected way: some varieties resulted four times more active than metallic uranium. Curie inferred that these minerals had to contain a small quantity of a very radioactive substance different from uranium, thorium, or any other simple known material. The distinction of this hypothetical new element could be based only on its radioactivity, the only suspected differentiating phenomenon. The verification process was similar to that of spectral analysis: the available pitchblende was subject to a chemical separation; the radioactivity of the products obtained was measured and one established whether the hypothetical element was contained wholly in one of the products or had been divided amongst them, and in the latter case with which proportions. The analogous process was then applied to each of the products obtained by the first separation. After much arduous and costly work, starting from several tons of the mineral, in July of 1898 the Curies obtained a new, highly radioactive substance, which they christened

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polonium in honour of Marie Curie’s homeland. In December of the same year, with the help of Gustave Bémont, they discovered another even more radioactive element and called it radium. In 1899, André Debierne (18741949), a student of Marie Curie, obtained a new element from pitchblende that he estimated to be one hundred thousand times as radioactive as uranium, and which the following year, having better studied its properties, he named actinium.188 The announcement of these discoveries was met with some wariness by many physicists, both due to the small quantities of substance obtained and because many new elements were not obtained in pure form but rather as salts. The doubts, which had arisen even for the Curies, were dispelled when, on their request, Eugène-Anatole Demarçay (1852-1904), a talented spectroscoper and the discoverer of europium in 1901, took on the task of spectroscopically examining the new elements, managing between 1898 and 1900 to find the spectral lines of radium, which resulted reasonably pure. Assuming radium to be divalent, Marie Curie found its atomic weight to be 225.

4.7 The study of new radiations With an increase in the number of known radioactive substances – uranium, thorium, polonium, radium, and actinium – and the number of researchers, followed a second, more strictly physical phase during which research turned towards the study of modalities of the new phenomena. With the first samples of polonium and radium supplied to him by the Curies, Becquerel – whose work had been diverted for two years by a supposed analogy between luminous radiation and uranium radiation – realized that the rays emitted by radium were much more penetrating than those emitted by polonium: the former passed through slabs of mica and aluminium and still left an impression on a photographic plate, while the latter were unable to pass through even the cardboard walls that surrounded the compound. In short, radioactive radiation is heterogeneous. The study of the penetrating abilities of rays was continued and furthered by Rutherford. “These experiments,” he wrote at the end of his investigation, “show that the uranium radiation is complex, and that there are present at least two distinct types of radiation: one that is very readily 188

A. Debierne, Sur une nouvelle matière radio-active, in “Comptes-rendus de l’Académie des sciences de Paris”, 129, 1899, pp. 593-95; Sur un nouvel elément radio-actif: l’actinium, in “Comptes-rendus de l’Académie des sciences de Paris”, 130, 1900, pp. 906-08.

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absorbed, which will be termed for convenience the Į radiation, and the other of a more penetrative character, which will be termed the ȕ radiation.”189 In the autumn of 1899, the first studies on the behavior of radiation in a magnetic field began. The Viennese physicists Stefan Meyer and Egon von Schweidler observed, through a barium platinocyanide screen perpendicular to a uniform magnetic field, that certain radiation that crossed the field lines was “plausibly” deflected into a circular path, while other radiation remained undeviated. In short, this method too confirmed that there existed two distinct types of radiation: one deviated and one not deviated by a magnetic field. Almost at the same time, Giesel and Becquerel also obtained the deviation of a part of the radiation in a magnetic field. In a later paper, Becquerel added that neither him nor Curie had been able to observe deviation in an electric field or demonstrate that the radiation carried charge.190 Nevertheless, deviation in a magnetic field was inexplicable without the transport of charge, whose absence he justified with an insufficient electric field strength.191 At the same time the Curies, having returned to the problem, postulated that the ionization of the air traversed by radiation somehow interfered with the phenomenon. Consequently, one had to operate in a vacuum to avoid it or, as they preferred, immerse a conductor connected to an electrometer in a solid insulator and irradiate it with radioactive barium salts placed outside the insulator. In this way the non-deviating radiation would be absorbed by the insulator and only the deviated portion would reach the conductor, which would then become negatively charged.192 After this experiment, which was labeled “fundamental” by Becquerel, he was able to demonstrate the deviation of the rays in an electric field by sending a thin beam of radiation between two small, oppositely charged plates and towards a photographic plate covered in black paper. Comparing the measurements carried out with those for magnetic deviation, he deduced that the speed of the rays and the ratio m/e were of 189

E. Rutherford, Uranium Radiation and the Electrical Conduction Produced by It, in “Philosophical Magazine” January 1899, p. 116. 190 H. Becquerel, Influence d’un champ magnetique sur le rayonnement des corps radio-actifs, in “Comptes-rendus de l’Académie des sciences de Paris”, 129, 1899, pp. 996-1001 & 1205-07. 191 H.Becquerel, Contribution à l’étude du rayonnement du radium, ibid., 130, 1900, pp. 206-11. 192 P.Curie and M. Curie, Sur la charge electrique des rayons déviables du radium, ibid., pp. 647-50.

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the order of magnitude obtained for cathode rays.193 Also in 1900, a few weeks after the aforementioned publication by Becquerel, Paul Villard (1860-1934) communicated the results of an experimental study on the refraction of the deviable radium rays. He observed that these rays were superimposed with rays that propagated in a straight line, making the interpretation of the images obtained on the photographic sheets (a technique dear to Becquerel and oft-used by the French physicists of the time) more difficult. He therefore advanced the hypothesis that the radiation “deviated” by a magnetic field was composed of two distinct radiations: one that was effectively deviable and another type, much more penetrating, that was not deviable or, at the very least, not deviated.194 Ultimately, in 1900 four years after the discovery of radioactivity, it was definitively known that, qualitatively, radioactive rays were of heterogeneous composition, consisting of, according to the nomenclature proposed by Rutherford –which became in common usage after 1902 and was extended by formal analogy to the radiation discovered by Villard– Į, ȕ, and Ȗ rays. After this primarily qualitative analysis, began the quantitative determinations. In 1901, Walter Kauffmann (1871-1947) carried out very accurate measurements of a thin beam of rays emitted by radium, which were under the simultaneous influence of an electric field and a magnetic field. He measured their velocity v and the ratio e/m, which turned out to be dependent on the velocity: for example, for v = 2.36(1010) cm/s, he obtained e/m = 0.63(107) electromagnetic units. Given the immutability of e, which resulted from the work of Thomson and John Townsend, the necessary conclusion was that the mass increased with velocity. This fact later led Kauffmann to think that the transversal mass of electrons was, in accordance with the views expressed by Max Abraham (1875-1922) in 1903, partly, if not entirely, of electromagnetic origin; that is, a manifestation of the interaction with the electromagnetic field. Indeed Abraham, starting from the hypothesis of the electromagnetic nature of mass, calculated its value for a velocity v and found that the mass of the corpuscle tends to infinity when v tends towards the speed of light, and tends to a constant value when v is much smaller than the speed of light. The results of Kauffman’s experiments, performed using a minuscule quantity of radium chloride supplied by the Curies, were in 193

H.Becquerel, Deviation du rayonnement du radium dans un champ electrique, ibid., pp. 808-15. 194 P.Villard, Sur la réflection et la refraction des rayons cathodiques et des rayons déviables du radium, ibid., pp. 1010- 12; Sur le rayonnement du radium, ibid., pp. 1178-79.

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agreement with Abraham’s theory and indicated that the lower bound of e/m for ȕ rays of low velocity was equal to the value of the same ratio for cathode rays. From here originated theories on the electromagnetic nature of matter, which greatly fascinated physicists for the first quarter of the century, and to which we will briefly return in Chapter 6.

4.8 The energy of radioactive phenomena Although all of these and other properties that were being discovered about the rays, which now can be read about in any elementary treatise on physics, were very important in their own right, they appear particularly technical compared to the fundamental question that the experiments posed to the first researchers. In radioactive phenomena there is a production of energy in chemical form, electric charges, and corpuscle motion: where does this energy originate? In 1899 and 1900, Marie Curie put forth two hypotheses. According to the first, radioactive bodies capture an external radiation undetected by our instruments and then emit it: in practice they are not generators, but transformers of energy. The second hypothesis, however, supposed that radioactive bodies spontaneously generated energy by slowly transforming themselves, despite their apparent (at the time) invariability. The two hypotheses seemed equally plausible or, if one prefers, equally unfounded. The necessity of a choice grew when, in the 16 March 1903 installment of the “Comptes rendus” of the Académie des sciences de Paris, Pierre Curie announced his most important discovery: uranium salts continuously give off heat in a quantity that, compared to the mass of the radioactive substance, immediately appeared to be enormous. In an early qualitative experiment, Curie, in collaboration with Albert Laborde, highlighted the production of heat through a thermocouple with one end was surrounded by radioactive barium chloride and the other by pure barium chloride: the difference in temperature between the two ends was measured to be about one and a half degrees centigrade, a difference that went well beyond the possible experimental uncertainty. Encouraged by this initial result, the two scientists executed a direct measurement of the amount of heat released using two different methods. In the first, the heat gained by a metal block in a cavity filled with a certain quantity of radioactive substance was set equal to the heat generated by a metal coil carrying current, which was placed inside the cavity instead of the radioactive substance and produced an equivalent heating of the metal block. In the second method, a vial containing radioactive or pure barium chloride was placed on a Bunsen calorimeter

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and the heat generated was measured directly. The two measurements gave similar values, which, for a gram of pure radium, appeared of the order of 100 calories per hour, later reduced by successive measurements to 25.5 calories per hour. How was it that such a large quantity of energy was simply captured energy? Was it possible that such enormous currents of energy pervaded the universe yet we could not detect any manifestations of them besides radioactive phenomena? These basic considerations led physicists to scrap the first of Curie’s hypotheses and align themselves with the second. “If it were in fact so,” commented Curie, “we would be led to conclude that the quantities of energy involved in the formation and transformation of atoms are considerable and surpass all expectations.”195 However, supposing radioactive bodies to be sources of energy due to their slow transformation, which had to be deeper than ordinary chemical transformations, meant newly questioning all of atomic theory. To realize the explosive and revolutionary nature of this view, the modern reader must bring herself back to the mentality that was prevalent among physicists in the first years of the century, to their perception of the world, their pride in the science of the time, which had been weaned on. The atomicity of matter, the eternality of atoms, the invariability of mass, the conservation of energy: these were the dominant principles, which to many even seemed to be not hypotheses, but evidence itself. Who would dare oppose this science, faced with a century of continual successes? It would be two young scientists: we will see them at work. The applications of radioactivity in physics, chemistry, geology, meteorology, and medicine were immediate and numerous. The sometimes lethal effects of radioactivity on living organisms impressed the masses, and once again people asked themselves: is scientific research helpful? “In criminal hands,” Pierre Curie said in 1905, at the conclusion of a conference celebrating his 1903 Nobel prize, a year before he would tragically perish in Paris after falling under an horse-drawn cart, “radium can become very dangerous, and we may wonder whether humanity benefits in knowing the secrets of nature, whether it is mature enough to enjoy them or whether this knowledge will be harmful to it. The example of Nobel’s discoveries is characteristic, powerful explosives allowed men to do admirable work, but have also been a terrible means of destruction in the hands of the great criminals that lead people to war. I am among those who believe, like Nobel, that humanity will draw more benefit than harm 195

M.Curie, Recherches sur les substances radioactives, in “Annales de chimie et de physique”, 7th series, 30, 1903, p.195.

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from the new discoveries.”196

THE PHOTOELECTRIC EFFECT AND THE THERMIONIC EFFECT 4.9 New sources of electrons It was known to the first experimenters that air in normal conditions is not a conductor, yet it becomes one (Hauksbee, 1706) at low pressure; on the other hand a flame, which is a conductor, has the ability to discharge electrified conductors that are in its vicinity (Guericke, 1661). Yet attempts to interpret these phenomena began only in the second half of the 19th century and followed the trail of theories on liquid conduction. It was Hittorf that in 1879 put forth the hypothesis that an explanation for gas conductibility had to be analogous to that elaborated for conductibility in electrolytic liquids; namely, in gases too the conductive ability had to be tied to the existence of ions, charged atoms, or groups of atoms that moved and transported electric charge. However, there being no possibility of verifying Hittorf’s hypothesis, experimental observations on the modality of conduction in gases continued, suddenly intensifying in the period between 1887 and 1890 following an experiment performed by Schuster, in which he showed that an electric spark accelerated the discharge of charged conductors placed nearby. In 1880 Hertz, expanding on the research of Schuster, realized that the electric discharge between two electrodes was much more sizeable when the electrodes were illuminated with ultraviolet rays: in the same year Eilhard Ernst Gustav Weidemann and Hermann Ebert established that the negative electrode was the centre of the action and Wilhelm Hallwachs found that the dispersion of negative charges is accelerated in conductors illuminated by ultraviolet light. Arrhenius, continuing the research, placed two platinum electrodes next to each other in a sealed tube containing rarefied air and connected them in circuit to a galvanometer and a battery (this may have been the first photoelectric cell in history): the galvanometer showed a reading as soon as the electrodes were illuminated by electric discharges that occurred on the tube exterior. Arrhenius believed that air was responsible for the phenomenon and supposed that ultraviolet light had the ability to accelerate the ions inside it.

196

Les prix Nobel en 1903, Stockholm 1906, p.7 of the conference of Pierre Curie.

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In an attempt to explain the preceding phenomena, Righi made a discovery in the first months of 1888: a conducting sheet struck by a beam of ultraviolet light became positively charged. Righi called this phenomenon photoelectric, introducing this term, which then remained, to science. Initially the Italian scientist believed it to be a simple transport of electricity mediated by ultraviolet light; yet Wilhelm Hallwachs (18591922), who had suspected of but not confirmed the phenomenon some months before Righi,197 demonstrated a few months later, followed independently by Righi, that it was not transport of electricity, but rather its production. “The radiations,” according to Righi, “act on metals […] and electrify them positively.” As for the mechanism for producing the electricity, Righi held that ultraviolet radiation split the air molecules in contact with the metal, and in the detachment the positive charges remained on the metal and the negative charges, which remained on the molecules, were noticeably brought along electric field lines. This interpretation seemed validated by another fact established by Righi and studied from 1890 onwards by Julius Elster (1854-1920) and Hans Geitel (1855-1923)198: a flux of negative electric charges leaves the electrode illuminated by ultraviolet light. Thomson, encouraged by the success obtained in analogous research on cathode rays two years prior, turned his attention to the analysis of this flux in 1899. Assuming that the electric current found in the photoelectric phenomenon was made up of moving, negatively charged particles and following a similar procedure to that used for cathode rays, one could theoretically establish the details of the motion of one of these particles under the simultaneous action of an electric and a magnetic field: the experimental test of the theory would allow for the acceptance or rejection of the starting hypothesis and, in the former case, also for the calculation of the ratio e/m between the charge and mass of this particle. Experiment confirmed the working hypothesis: the current that arises between two oppositely charged metal plates when the cathode is illuminated by ultraviolet light behaves as a cluster of negatively charged 197

Controversy erupted between Hallwachs and Righi regarding priority for this discovery: each of the two scientists laid out their case in “Il nuovo cimento”, 3rd series, 28, 1890, pp. 59-62. The scientific community resolved the controversy by calling the phenomenon Hallwachs effect or Hertz-Hallwachs effect, with a harshness towards Righi that to us appears excessive. 198 We will often find these two scientists collaborating. It is a singular case: close friends at the University of Heidelberg, Elster and Geitel continued working together for their entire life, both professors at the gymnasium of Wolfenbuttel, having always refused the university post that had been offered to both of them.

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particles in motion. Averaging many experiments, the value of e/m resulted as 7.3(106) electromagnetic units; the analogous value that Thomson had found for a constituent particle of cathode rays had been 5(106), corrected by Lenard to be 6.4(106) electromagnetic units. On the other hand, the measurement of e, obtained with the same method that had allowed him the previous year to measure the charge of an ion produced due to the effect of X-rays, gave an average value of 6.8(10-10) electrostatic units, a value of the same order of magnitude found for cathode rays. Thomson immediately concluded that the carriers of negative charge in this case were of the same nature as cathode rays; they were, that is, electrons. There was yet another case in which this agreement across “electricities” was verified; a phenomenon discovered by Edison in 1879 and then studied by other physicists: an incandescent carbon filament emits a flux of negative electricity. Thomson decided to analyse this flux as well, subjecting it to the usual simultaneous influence of an electric and a magnetic field. The result was analogous to the preceding one: both the ratio e/m and e resulted of the same order of magnitude as the corresponding values for cathode rays. The conclusion was identical: electric current produced in the Edison effect was a flux of electrons. The existence of the electron, which only five years prior had been hypothesized in a theoretical setting, by the end of the century was experimentally demonstrated through cathode rays, Becquerel rays, the photoelectric phenomenon, and the thermionic phenomenon. The picture of the physical world had been constructed on the hypothesis of the atom, the last constituent of matter, immutable and eternal: this picture had to be erased and substituted with one that better adhered to fact.

4.10 Ionization of gases The mechanism for the passage of electricity in gases appeared, after Faraday, to be too murky of a subject of study and of little utility, and thus was neglected by researchers. In 1882, W. Giese, in the course of his research on the conductibility of flames, had put forth the hypothesis of dissociation of neutral molecules into charged centres, analogous to electrolytic ions. The same hypothesis was utilized by Schuster in 1884 to interpret phenomena of destructive discharge. An objection to this hypothesis similar to one already brought up for electrolytic dissociation (§ 3.16) was raised: if the molecules of a gas split, the chemical products of the schism should be present in the gas; furthermore, for the passage of an equal amount of electricity in a

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substance, like hydrochloric acid in a gaseous state or in solution, the mass decomposed should be equal, while experiment revealed that in the case of a gas there was a negligible decomposition compared to electrolytic decomposition. Nevertheless, another objection appeared unsurpassable to the eyes of the atomists of the time: discharge occurred even in monoatomic gases (mercury vapour, argon, helium) for which any splitting was impossible in principle. However, the discovery of an electron as a particle in its own right, somehow present in atoms and in the conductibility of gases brought about by X-rays, ultraviolet rays, and radioactive irradiating, allowed for all difficulties to be surmounted, especially by merit of the fervent Cambridge school of experimenters, led by a wise and illuminated guide in Thomson. In a paper published in March of 1899, Thomson stated the fundamental concept of the new interpretation: “The electrical conductivity possessed by gases under certain circumstances […] can be regarded as due to the presence in the gas of charged ions, the motions of these ions in the electric field constituting the current.”199 The study of the electric field in a gas traversed by current was tackled by Thomson and his collaborators by examining the production of ions, their combination, and their movement. Measurements of ion velocity in discharge brought to light the fact that their velocity varied much with the cause of ionization and, keeping all other factors constant, negative ions had much greater velocities than positive ions. The mathematical treatment of the motion of ions in an electric field, the atomic model that Thomson had devised (§ 7.6), and the experimental results obtained led him that same year to conceive of the ionization of a gas as due to the detachment of an electron from some of its atoms. The detached electrons constituted the negative ions and the atoms deprived of an electron the positive ones, of equal and opposite charge as the negative ones and of much greater mechanical mass. If the atom joined other atoms to form a molecule, the positive ion would then be the molecule deprived of the electron. Thomson also found that the passage of current in a gas sometimes did not obey Ohm’s law. More precisely, given electrodes at a determined separation and supposing constant the cause of ionization, the current between them does not increase with the potential difference past a certain limit. A limiting current, or, as it is more commonly called today, a saturation current is thus reached. Thomson interpreted the phenomenon by supposing that the saturation current was attained when all of the ions 199

J.J. Thomson, On the Theory of the Conduction of Electricity through Gases by Charged Ions, in “Philosophical Magazine, 5th series, 47, 1899, p. 253.

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produced had reached the electrodes; drawing from his conviction that an ion always transports the same electric charge, no matter the gas from which it is produced.200 It became clear immediately that destructive discharge could occur only if ions pre-existed in the gas and, accelerated by an electric field, they gained sufficient velocity to create new ions by colliding with gas molecules or electrodes. Yet, Townsend demonstrated that if only negative ions take part in this process, the discharge will cease as soon as the negative ions and the ions they in turn produce arrive at the anode with equal velocity: the formation of other ions is therefore necessary. The new source of negative ions lies in the collision of positive ions with gas molecules or the cathode. Guided by these ideas, studies on free discharge and discharge in tubes multiplied, and numerous experimenters (Thomson, Townsend, Lenard, Langevin) tried their hand at it, noting the unexpected difficulties. In 1903, Langevin, laying out the state of the theory of ionization in a paper, advanced the hypothesis of recombination of positive and negative ions, confirmed by experiment for gas discharges at high pressure but not low pressure.201 Thomson completed the explanation in 1924, showing the importance of thermal excitement in the phenomenon of recombination, a detail which had been strangely overlooked by the preceding theories, even though it had already been connected to ionization through flame conductibility phenomena, as we had already mentioned earlier. Many difficulties that arose from the intense study of gas ionization, which continued for some decades into the 20th century, could be surmounted only later, with quantum considerations.

4.11 The organization of scientific research in the 20th century The interaction between scientific research and society is evident throughout history and clearly must also be rather powerful, as ideas are developed by the minds of men and men are conditioned by the society within which they live. We have hinted at this here and there, without extending the analysis too far for obvious reasons of limited time and scope. Here, before turning to the scientific developments of the 20th century, we will describe a characteristic aspect of scientific research in this century. 200

P. Langevin. Recombinaison et mobilités des ions dans les gaz, in “Annales de chimie et de physique”, 7th series, 28, 1903, pp. 433-530. 201 J.J. Thomson, The Discharge of Electricity through Gases, London 1898, p. 35.

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Universities arose in the Middle Ages and academies during the Renaissance: all scientific activity was organized around these two institutions and, starting from the second half of the 18th century, centred increasingly around universities. Throughout the 19th century, the figure of the scientist, save for a few rare exceptions, was that of a professor. Officially, the primary duty of the professor was teaching; scientific research became ancillary and was left freely to the professor’s initiative. However, in the course of the 19th century, science, which had so far been conceived of as “natural philosophy” or “natural history”, began to turn ever more frequently to problems that led to the production of economic goods (think, for instance, of Watt’s research on stream engines, which was followed by that of Carnot and the measurements of Regnault, both fundamental for the underlying science; of the production of synthetic colours by the English chemist William Perkin; of Zeiss’s studies on optical instruments, etc.). Moreover, a revolutionary phenomenon for industry of the time was developed: the discoveries of electromagnetism led to new techniques, independent from and less intuitive than the traditional ones, which therefore could not have been transformed into industry without the contribution, or rather the direct intervention of scientists. Yet the collaboration between science and technology had certain organizational requirements, as there were multiple scientific aspects to each industrial process, requiring the joint collaboration of specialists in scientific fields that normally were very distant from one another. For example, at the end of the century it was clear that the petroleum industry, which had become the key industry of the 20th century, could not have developed without the collaboration not only of physicists and chemists, but also (and especially) geologists. Lastly, the proliferation of scientific knowledge led to more and more restricted specialization among scientists, making collaboration between many scientists in different fields necessary for research. Under such conditions, large industry felt the need to set up laboratories and specialized personnel dedicated exclusively to scientific research, organized in teams of specialists in the various branches of science who coordinated their research to serve a common goal. Thus, arose a new form of organization in which individual research was replaced with collective, coordinated research. This new form of organization was implemented in the first decade of the century, first in Germany, then in the United States. In 1911, German industry displayed a prime example of this new form of organization when Kaiser Wilhelm Gesellschaft founded a series of institutes that conducted and still conduct (having changed their names in

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1949 to Max Planck Gesellschaft) research in both pure and applied science. In the United States of America, during the decade preceding the first world war, important industries like the General Electric Company, Bell Telephone, Westinghouse, Eastman Kodak, and Standard Oil had organized research laboratories. The utility of such experimental laboratories was recognized during and after the first world war in England, France, and the small nations of Northern Europe (Sweden, Norway, the Netherlands, and Belgium), where the organization took cooperative forms; even in Italy there were some examples. Nowadays this industrial organization of science has taken on impressive dimensions in certain countries. In the United States, for example, had 4834 industrial laboratories of scientific research in 1960. During the first world war, new institutions of scientific research were constructed, promoted by governments in need of scientific collaboration to resolve numerous problems that had arisen from the war. These new institutions, frequently called “Research Councils,” rendered great services to their respective countries, and were consequently consolidated after the war and extended to other countries. These had the task of promoting, coordinating, and in some cases carrying out scientific research, keeping in mind the needs of the country, its natural resources, and the availability of means and people. Although all research councils had this objective in common, their internal structure varied from country to country in relation to its traditions and political and economic conditions. The first research councils arose in 1916 in Great Britain (Department of Scientific and Industrial Research) and in the United States (National Research Council), and began spreading to all countries interested in scientific research: today research councils exist in all industrialized countries. In Italy, the Consiglio Nazionale delle Ricerche (CNR) was instituted in 1923 (18 November) and until 1945 lived a shadowy and unstable life, shaken by ten complete transformations of structure, hindered in its development by traditional prejudices and preestablished interests. Definitively, scientific research, which until the previous century had been executed only by the “professor”, today is entrusted to three categories of people: university professors, “researchers” (according to the new, globally accepted meaning of this term) employed by industrial scientific organizations, and researchers employed by the state through research councils. Within these categories of people there is no clear-cut separation, rather, there is a continual exchange of people between categories and sometimes a partial superposition of two categories, with all the waste and misuse that can arise.

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In our century, the organization of scientific research has become an imposing phenomenon also because of the number of people involved. While it is not easy to find the latest data that is sometimes kept secret by countries, we know nevertheless that this number is enormous: it surpasses a million people in the United States of America, reaching hundreds of thousands in France, in Germany, and in Britain. The related expenses are obviously enormous. Next to national research centres there are international scientific institutions, like CERN (Conseil Européen pour la Recherche Nucléaire) in Geneva. In Italy, the centres of scientific research are much less abundant, modestly equipped, and thus less active than in other more industrialized countries. They are (or should be) university research centres, the Consiglio Nazionale delle Ricerche, the Comitato Nazionale dell’Energia Nucleare, the Istituto Superiore di Sanità, the Istituto Superiore delle Poste e Telecomunicazioni, the laboratories of the railways and various ministries, and industrial science laboratories. This situation displays why one cannot, when dispassionately observing the picture of global scientific progress, speak of Italian science of the 20th century, even though there was (and is) no lack of bright scientific minds. In the 19th century, the scientist was completely free to decide the object of their research: today this liberty is afforded only to the increasingly rare isolated scientist, given the enormous financial means needed by today’s scientific research. Yet the scientists who belongs to the powerful modern scientific organizations can never have this same liberty. Even the scientific organizations of large industries deep down belong to the state. Ultimately, therefore, it is the state that plans scientific research through the creation of institutes, the choice of their location, the allocation of funds, the choice of training and utilizing personnel, the rules imposed on it, and the decision of which research subjects to study. This “politics of research”, already enacted by some time in states of high scientific level, is imposed by the necessities of modern life and has shown itself to be an incredibly powerful instrument of progress, from which it is easy to foresee that it will extend to all countries. Modern scientific research is therefore characterized by two elements: it is collective, and it is planned by the state.

5. RELATIVITY MECHANICS IN THE 19th CENTURY 5.1 The daily motion of the Earth The analytic expansion of newtonian mechanics amounted to, as we have said202, the analytical mechanics of Lagrange, which dominated physics for the entire 19th century until the introduction of the relativistic and quantum doctrines. In the 19th century, mechanics was enhanced by some particular advances, didactical methods were refined, and its fundamental concepts came to light through a critiquing of its principles which characterized the second half of the century. Among the particular advances, two theorems of Gaspard-Gustave de Coriolis (1792-1843) on the composition of accelerations, announced in 1831 and 1835, and Foucault’s experiment on the rotational motion of the earth about its own axis are of considerable historical importance. Modern authors often combine the centrifugal force of Coriolis and Foucault’s pendulum into a single idea. Historically, however, the two things arose independently: Coriolis’ discovery was of a mathematical nature and certainly did not inspire the second since Foucault, though a very able experimenter, was a mediocre mathematician who had not read the Coriolis’ works when, in 1851, he presented his historic paper on the experimental demonstration of the earth’s rotational motion. Foucault started from the experimental observation that if one makes the string of a pendulum rotate about itself, even rapidly, the plane of oscillation remains invariant. Therefore, if a pendulum was brought to the North or South pole and suspended from a point on the Earth’s axis of rotation, its plane of oscillation would remain fixed in space. “The motion of the Earth,” continued the scientist in his informal journalistic style, “which incessantly rotates from west to east, becomes discernible by contrast with the immobility of the plane of oscillation, whose mark on the ground will appear animated by a motion commensurate with the apparent 202 § 7.7 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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motion of the celestial sphere; and if the oscillations could continue for twenty-four hours, the marks of their plane would trace in the same time an entire revolution around the vertical projection of the point of suspension.”203

Fig. 5.1 - Foucault’s experiment at the Pantheon of Paris: the tip of the oscillating pendulum grazes a ring of ash placed on the ground. One observes that the tip does not pass over the same furrows, but always produces new ones with regularity, like the hands of a watch, as if the ring of ash, rotating under the tip of the pendulum, displayed its various parts. 203

L. Foucault, Démonstration physique du mouvement de rotation de la Terre au moyen du pendule, in “Comptes-rendus de l’Académie des sciences de Paris”, 32, 1851, p. 135.

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Yet, descending from the poles to our lower latitudes, the phenomenon is complicated because the tangent plane of the location is oblique to the axis of the Earth, such that the normal vector, instead of rotating about itself, traces a conical surface that broadens with increasing proximity to the equator. Foucault realized that the phenomenon had to be qualitatively the same even at intermediate latitudes, varying only in its quantitative aspect, which he specified in a law discovered almost instinctively, that was later confirmed by the calculations of mathematicians. At first, Foucault began his experiments in a cellar, then, thanks to the backing of Arago, in a room of the Paris astronomical observatory; and eventually, at the Pantheon of Paris, packed with spectators, as can be seen from the oleographs of the time (Fig. 5.1), using a 28 kilograms ball hung from a 67 meters wire. Foucault’s experiment was immensely successful, and was followed by innumerable articles of a mathematical nature which interpreted all of its details. Nevertheless, according to Foucault, some had not understood the concept of constancy of the pendulum's plane of oscillation. To this end, he substituted it with “the plane of rotation of a body that is freely suspended from its centre of gravity and rotates around a principal axis, in such a way that the plane considered is physically defined and has an absolute fixedness of direction.”204 In other words, he tried to deduce the experimental proof of the diurnal motion of the Earth from the principle of conservation of axis and plane of rotation of a body, which had already been studied by the founders of analytical mechanics (Euler, D’Alembert, Lagrange, Poisson). To this end, he devised an apparatus composed of a bronze ring that could move along an iron axis by traversing it perpendicularly, supported by a gimbal, which in turn was suspended from a torsionless wire. If the ring was set into rapid rotational motion, there would be a slow and continuous rotational motion of the apparatus from east to west. As the axis of rotation conserves a constant direction in space, it follows that it was the Earth that was moving with respect to the axis of rotation. After a few months, Foucault modified this method of observation, placing the axis of rotation horizontally and allowing it to rotate around a vertical axis. The axis of rotation then “became animated with a very slow oscillatory motion around the meridian, where it would end up stopping, if the rotation persisted for long enough”: a behavior so similar to that of a compass needle that it was unexpected, since the equilibrium position of the axis was inclined off the horizon.205 As all of 204

L. Foucault, Sur une nouvelle démonstration expérimentale du mouvement de la Terre, fondée sur la fixité du plan de rotation, ibid., 35, 1852, p. 421. 205 L. Foucault, Sur les phénomènes d’orientation des corps tournants entrainés

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these phenomena were dependent on the motion of the Earth, Foucault proposed to call the instrument that had allowed him to see them a gyroscope 206. The gyroscopic compass was thus born alongside the gyroscope, the first of numerous mechanical applications (projectile stabilizer, ship stabilizer, vibration absorber, automatic pilot, etc.) which were to become even more numerous, so much so that the initial scientific work was forgotten by the end of the century, when the electric motor had ensured perpetual rotational motion. The invention of the gyroscope immediately provoked great excitement; the general public quickly came to know of it and the instrument became part of the experimental toolkit of even modest scientific laboratories. However, challenges regarding its novelty and originality immediately arose. In the same volume in which Foucault’s writings appeared, there were also a dozen other papers on the same subject. In reality, before Foucault there had been no lack of devices constructed in order to demonstrate the constancy of the rotational plane, of which the spinning top was the simplest. In 1817, Johann Gottlieb Friedrich von Bohnenberger built and described a device very similar to that used by Foucault, yet having only the purpose of proving the constancy of the rotational plane; in 1836, instead, the Scotsman Edward Lang had expressed the idea that the constancy of the rotational plane could furnish a method for the demonstration of Earth’s daily rotation, without however attempting concrete proof. Foucault almost certainly did not know of Lang’s suggestion, yet he was aware of Bohnenberger’s instrument, which was widespread in French physics laboratories by merit of Arago, who had received two models from the German physicist. In short, it is difficult to find a proper “precursor” to Foucault, that is, a scientist that had been able to connect the constancy of the oscillatory plane of a pendulum or the rotational plane of an object with the daily motion of the Earth. Gauss had treated the problem in a manuscript that then remained unedited until 1883. Vincenzo Antinori (1792-1865), after having repeated the experiment with Foucault’s pendulum, set out to verify whether there were any precedents. Among the papers of the Accademia del Cimento, he found a note written by Viviani in which it was noted that a pendulum suspended from a single string “insensibly strays from its original trajectory,” and in another note, previously par un axe fixe à la surface de la Terre, ibid., 35, 1852, p. 426. 206 Ibid., p. 427.

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published by Targioni Tozzetti, it is observed that the pendulum “above marble powder traces its path.” The Accademia del Cimento had thus performed Foucault’s experiment, but had not attempted any interpretation of it. The other experimental demonstration of Earth’s daily motion –the eastward deviation of falling masses– much like Foucault’s experiment, requires accounting for the Coriolis’ centrifugal force for a rigorous explanation. Nevertheless, this deviation can be predicted using a simple intuitive argument, which had already been made by Borelli and demonstrated by the experiments of Guglielmini207, repeated in 1802 in St. Michael’s tower in Hamburg and in 1804 in the pit of the Schleebusch mine by Johann Friedrich Benzenberg (1777-1846). More notable and accurate were the experiments carried out by Ferdinand Reich (17991882) in 1833 in the Freiburg mine pits: for a free fall of 158 metres, he obtained an average deviation of 28.3 millimetres across 106 experiments. To deny the daily motion of the Earth, wrote Arago, which many physical and astronomical observations concur to demonstrate, is to deny evidence. Mach, however, acutely observed that all such observations could just as well demonstrate the immobility of the Earth and the rotation of the celestial sphere.

5.2 Criticism of Newtonian principles The second half of the 19th century was characterized, as we have mentioned, by discussions on the fundamental concepts of newtonian mechanics: force, mass, inertia, action, and reaction. Already at the beginning of the century, Lazare Carnot had pointed out the occult and metaphysical nature of newtonian force. Adhémar-Jean-Claude Barré de Saint-Venant (1797-1886) followed up on the criticisms of Carnot against “these problematic entities, or better, substantivized adjectives” in 1851, hoping that they would be progressively expelled from science as primitive concepts and substituted with the connections that exist between bodies in reciprocal motion. In 1861, the French mathematician and economist Antoine Cournot (1801-1877) anthropomorphized the concept of force, connecting it to the muscular sensations felt in the execution of certain actions, like lifting a weight, stretching or compressing an object, etc.: this anthropomorphic aspect of force, which survives to this day, was not explicit for Newton, who had generalized the Galilean concept of 207 § 5.2 in M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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traction or pressure produced by a weight. In a sense, this Galilean conception is connected to the “School of the wire”, founded by Ferdinand Reech (1805-1880) and laid out in the Cours de mécanique d’après la nature généralment flexible et élastique des corps (1852) and having as its greatest advocate Jules-Frédéric Andrade (18571933) with his Leçons de mécanique physique (1897). We can intuit that the fundamental concept of this school was that of a tense, massless wire and its traction. A point particle (we omit reference to the discussions on the concept of a “point particle” and its legitimacy) hanging from a wire stretches it and elicits a force that is directly measurable from the stretching, which supposedly is proportional; this force is balanced by the “force of inertia” (in Euler’s sense) of the point particle. Fundamentally, as Poincaré acutely observed, this “school of the wire” took the principle of action and reaction as the definition of force, instead of an experimental fact: a nonconventional and highly specific choice. If, for example, the Earth was attached to the Sun by an invisible wire, how could we measure its stretching? Still to avoid the construction of mechanics from an anthropomorphic concept, Kirchhoff defined (1876) force through purely analytic means, using only primitive concepts of space, time, and matter. Obeying in essence the nominalistic tendencies of mathematicians, he gave the name of “accelerating force” to a certain mathematical expression, without investigating its physical meaning because he was convinced that experiment was incapable of giving a complete definition to the concept of force. To Hertz too the traditional exposition of Newtonian mechanics, based on concepts of space, mass, force, and movement, did not appear exempt from contradictions. For example, in the rotational motion of a stone tied to a rope, does the centrifugal force truly differ from the stone’s own inertia? In the habitual treatments of the problem does one not count the stone twice, once as mass and the other time as force? In general, declared Hertz (Die Prinzipien der Mechanik in neuem Zusammenhang Dargestellt, 1894), we cannot understand the motion of bodies that surround us, by resorting only to what we can immediately perceive with our senses. If we want to obtain a clear image of a world subject to laws, we must “imagine other invisible things behind those that we see, and look for the hidden actors behind the barrier of our senses.” In their classic expositions, the concepts of force and energy are idealizations of this type. However, we are free to acknowledge that these hidden actors are nothing but masses and motion that we perceive. Hertz therefore constructed a mechanics based only on concepts of space, time, and mass;

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force was introduced as an auxiliary entity and the entire structure rested on a single principle: if a point particle has an acceleration, then it is subject to a frictionless connection independent of time. What resulted was a mechanical system that Hertz believed to be fundamentally more logical than the classical system, yet of lesser practical value. Helmholtz, in the fond preface to the work of his disciple, who had prematurely passed away on the eve of his book publishing, while not sharing the same viewpoint, held it probable that this idea could “gain a greater heuristic value in the future, guiding those who would attempt to discover new, general characteristics of the forces of nature.” It does not seem, however, that his prediction came true. More than these courses of study, the works of Mach had a great influence on the physicists of the end of the century: Einstein recognized that a reading of the philosophical writings of David Hume (1711-1776) and Mach made his critical work “enormously easier”. Mach started from the concept of mass, traditionally introduced as the ratio between the force applied on a body and the acceleration that ensued. To this traditional method, Mach raised the following criticisms: the concept of mass is made to depend on the different accelerations that the same body experiences under the action of different forces, while it seems to be highlighted when a single force, acting on different bodies, gives rise to different accelerations. In this regard, the importance of mass in mechanics is said to be the following: from the knowledge of the way a single body moves under the action of a certain force, we can deduce the kinematic effects of the same force on different bodies. Mach thus proceeded to reconstruct the concept of mass, invoking the principle of symmetry: if an object A experiences an acceleration, it is due to an object B, which, in turn, experiences an acceleration due to A. He illustrated this principle with an experiment (dating back to Newton) using two floating bobs, one carrying a magnet and the other a piece of iron: when they come into contact, they remained still. Mach then moved onto another series of experiments with a centrifugal force machine. Two objects A and B, of different weight, joined by a string and inserted in a pole, can remain in equilibrium whatever the velocity of the machine. In such conditions, as is known, the accelerations a and a’ are proportional to the respective distances from the axis. The inverse ratio of these two accelerations is defined as the ratio between the masses. From this follows the following explicit definition: the ratio of the masses of two bodies is said to be the inverse ratio, in absolute value, of the accelerations that the two bodies induce in one another. In essence, instead of defining the mass of a body, Mach defined the “mass ratio of

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two bodies”, that is, he gave a definition by abstraction to the concept of mass. Obviously, it is not necessary to resort to the centrifugal force machine for this definition. Mach introduced this experimental component perhaps to shield himself from the criticisms that had been laid on his colleague Boltzmann: the preceding definition by abstraction had implied the postulation of action at a distance; a big issue which one would have been prudent to avoid. In the above definition of mass, Mach observed with satisfaction, there was no theory, and the “quantity of matter,” of which Newton had written, was entirely useless. The definition also rendered useless the principle of action and reaction, which would simply have been a restatement of the same fact. Another one of Mach’s biting criticisms was directed at coordinate systems. While all rigid and fixed systems of coordinates (cartesian, polar, etc.) were perfectly equivalent for geometric shapes, for the laws of mechanics it was necessary to use a certain type of coordinate systems for which the principle of inertia held, for this reason called inertial. If there exists one inertial system, then there exist infinitely many others, differing from the first by a uniform translation in any direction and having any velocity. Yet does an inertial system exist? Mach interpreted inertia as an effect of the joint action of fixed stellar masses, which implies, as Einstein correctly observed, assuming Newtonian mechanics a priori. Nevertheless, classical mechanics rested on the assumption of inertial systems, which, then as now, were of very doubtful existence. Ernst Mach was born in Turany, Moravia on 18 February 1838, graduated from Vienna and in 1860 became the professor of physics at Graz, then Prague, and finally held the post of professor of philosophy of science in Vienna, from 1893 to 1901. He died in Haar, near Munich, on 19 February 1916. He made history with his experimental works on the motion of supersonic projectiles in a fluid, which were summarized in a popular article in 1897, and became so important with the development of aerial transportation that several concepts and measurements in aerodynamics were named after him (Mach angle, Mach cone, Mach line, Mach wave, Mach number). As a philosopher and historian of science, he was one of the harshest critics of Newtonian mechanics, always guided by an anti-metaphysical spirit and a particular conception of science, which to him seemed to follow a principle of economy in research. All science, according to Mach, attempts to substitute and economize experience by means of a mental representation of the facts, therefore all science must be

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continually confirmed or invalidated by experience and moves toward the realm of incomplete experience. If one recognizes the function of the laws of nature as rules that economically summarize the sequence of our sensations, “all mysticism” disappears from the scientific field”. More well-known are the criticisms addressed to classical mechanics by Jules-Henri Poincaré. He was above all a great mathematician, born in Nancy the 29th of April 1854 and died the 17th of July 1912 in Paris: a great mathematician who also made important contributions to astronomy, especially in the field of orbits and the three bodies problem. In the first decades of the century, his volumes on the philosophy of science became popular due to their lively tone, and were reprinted and translated in various languages: La science et l’hypothèse (1902), La valeur de la science (1904), and Science et méthode (1908). In the first volume referenced, Poincaré observed that mechanics, although only capable of conceiving relative motions, places them in an absolute space and time, purely by convention. Classical mechanics admits a principle of inertia, which is neither an experimental fact nor imposed a priori to our minds, and unnatural enough that Greek mechanics did without it. Force, on the other hand, as a cause of movement is a metaphysical concept and for its measurement one must resort to the principle of action and reaction, which is then not an experimental law, but a definition. The law of universal attraction, on the other hand, is a hypothesis that experiment could disprove. What remains then of classical mechanics? We recognize, responded Poincaré, that only by definition is force equal to the product of mass and acceleration, and only by definition does action equal reaction: these principles would only be verifiable in isolated systems, in which, however, we cannot conduct experiments. Yet because nearly isolated systems exist, Newtonian principles can be approximately applied to these, explaining how experiment could have served as their foundation. The critical currents which we have just mentioned did not directly give rise to the relativistic revision of classical mechanics, although Einstein alluded many times to his indebtedness towards Mach. Nevertheless, these currents reveal a state of unease and the awareness that the axioms of classical mechanics, despite two centuries of success, could also have been disproven by experiment. Other dark clouds, on the other hand, formed over the theoretical physics of the first years of the 20th century. Brownian motion seemed to contradict the second law of thermodynamics (§ 6.10); the principle of conservation of mechanical mass was called into question by Abraham’s

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electromagnetic theory208. The nature of ether, the imponderable solid through which other objects hypothetically moved without resistance, remained an enigma, seeming at times immobile respect to the Earth (phenomenon of aberration) and at times moving with it in unison (Michelson experiment, § 5.5). Poincaré, on the eve of the formulation of relativity, asked himself: “In ten years, will the aims and methods of physics appear to our immediate successors in the same light as they appear to us, or, on the contrary, are we about to witness a profound transformation?”. And he answered, “Well, yes, there are hints of a serious crisis, as if we should expect an immediate transformation. But let us not excessively worry: the ailing discipline will not die, rather, we can hope that the crisis will be salutary.”209

TOWARDS RELATIVITY 5.3 The electromagnetic theory of Lorentz Hendrik Antoon Lorentz (Fig. 5.2) was born in Arnheim, in the Netherlands, on 18 July 1853, studied at Leiden and became a professor there of mathematical physics in 1878; in 1923 he was appointed the director of research in the Teyler Institute of Haarlem, where he died on 4 February 1928. He was one of the great theoretical physicists of his field, and in 1902 he split the Nobel Prize in physics with Zeeman (§ 3.30). His scientific activity always took place on the frontier of the science of his time, which around 1890 strove for the development of a unified theory of electricity, magnetism, and light. After Hertz’s experimental verifications, the Maxwellian theory of fields began to gradually establish itself in the consciousness of physicists. Now, Maxwell’s equations differed from the ordinary equations of mechanics not only for the reasons mentioned in paragraph 10.27, but also because they did not remain invariant under Galilean transformations. Furthermore, Maxwell’s equations did not apply to bodies in motion respect to the ether. It was therefore necessary to complete them with the study of this case, which, obviously, again raised the issue of the ether’s behavior respect to moving bodies. In 1890, when Stokes’ hypothesis on the total dragging of ether had 208

§ 5.7 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 209 H.Poincaré, La valeur de la science, Flammarion, Paris 1904, 170-71.

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already been accepted (§ 1.6), Hertz was able to find a group of equations that were invariant under Galilean transformations and gave rise to Maxwell’s equations when moving body came to rest. Yet Hertz’s equations did not agree with the experimental results of Fizeau (§ 1.6) and others.

Fig. 5.2 – Hendrik Antoon Lorentz.

Much more fortunate was Lorentz’s attempt, whose first steps appeared in a pamphlet in 1892 and were expanded on in his classic 1897 paper, mentioned in paragraph 1.6. He found two fundamental flaws in the Maxwell-Hertz electromagnetic equations: they left out ether inside matter and they did not account for the granular composition of matter and electricity. Lorentz had the idea, therefore, to introduce the discontinuous structure of electricity and Hertz’s electromagnetic ether into the equations. It was no longer the elastic ether of Fresnel, but the electromagnetic ether of Maxwell and, especially, Hertz: an ether immobile in absolute

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Newtonian space and conceptually identical to it (a subconscious return to Cartesian views). On the other hand, Lorentz postulated that matter was composed entirely of elementary corpuscles of electricity, by him called positive and negative ions, which we will, for greater clarity aimed at the modern reader, generically call electrons (though they have a positive charge). According to Lorentz’s formulation, a body that is positively electrically charged is a body having excess of positive electrons; an electrically neutral body is a body having electrons of the opposite sign in equal numbers. Electric current in a conductor is the movement of the collection of free electrons contained in the conductor: with this hypothesis, the conduction currents of Maxwell were refuted and currents were conceived of as convection currents. The speed of electrons increases with an increase in temperature. Each moving electron creates an electromagnetic field around itself. If the motion is straight and uniform, then the electron drags its own field along and consequently there is no emission of energy in the surrounding area. However, if the motion is varied (that is, accelerated or decelerated), the emission of an electromagnetic wave results from Lorentz’s equations; following which the electron loses energy, and the instantaneous energy lost is proportional to the square of the electron’s acceleration. When a free electron in a metal strikes the surface without being able to penetrate it, it is reflected; the change in direction then generates an electromagnetic wave. Due to this phenomenon and the increase in electron velocity with temperature, metals can become incandescent when heated. In dielectrics, on the contrary, free electrons have less freedom of movement because they are attracted to fixed electrons: they can only oscillate around these fixed centres. It follows that dielectrics are not conductors. They are, however, often transparent and refracting, because light rays, when striking them, oscillate the free electrons, causing an electromagnetic perturbation. The electromagnetic field observed on a macroscopic scale is the statistical result of the combination of innumerable elementary fields created by single electrons. Maxwell’s laws describe macroscopic fields, while Lorentz gave the laws of microscopic fields, formulated in such a way that the statistical result of the combination of innumerable microscopic fields gives rise to the macroscopic field described by Maxwell’s laws. These are thus the statistical laws of “coarse” electromagnetism, which result from the “fine” structure of Lorentz.

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On these foundations Lorentz deduced five fundamental equations, from which one can obtain, as he himself showed, every other known law of electromagnetism. Lorentz’s theory was received with some skepticism in his time; nevertheless, it represented the furthest evolutionary point of classical electromagnetism and, in particular, of ether physics; it was an inspiration and guide for all electronic theories, including the non-classical ones. Much like Maxwell’s equations, those of Lorentz did not remain invariant under Galilean transformations. In 1904, Lorentz discovered that his equations remained invariant under another, different type of transformation, which the following year Poincaré improved and proposed to call a Lorentz transformation210 Unbeknownst to Lorentz, it had already been discovered in 1887 by Woldemar Voigt (1850-1919). Without specifying the mathematical details of a Lorentz transformation, it will suffice here to mention its characteristic detail: while in a Galilean transformation time remains unchanged for two systems in constant, uniform motion respect to each other, in a Lorentz transformation from one system to another there is also a change in time. Consequently, Lorentz called a certain mathematical expression local time, without attributing to it any physical meaning. Maxwell’s equations remain invariant under Lorentz transformations, yet those of classical mechanics do not. In short, Lorentz’s modification did not reconcile the disagreement between the equations of classical mechanics and those of Maxwell. Nevertheless, according to Einstein’s judgement, Lorentz’s conception was “the only one possible” and “at the time was an audacious and surprising step, without which further developments would not have been possible.”211

5.4 Time Even if it lacked physical meaning, Lorentz’s local time (§ 5.3) could not help but bring physicists back, in a time of revision of the principles of mechanics, to a consideration of the concept of time, which had provoked the rumination of philosophers of every era. In bringing this concept back, the work of Henri Bergson (1859-1941) in his Essai sur les données immediates de la conscience (1889) may have also contributed. In 210

H. Poincaré, Sur la dynamique de l’électron, in “Rendiconti del Circolo matematico di Palermo”, 21, 1906, p. 129. 211 P.A. Schlipp (editor), Albert Einstein scienziato e filosofo, translated in Italian by A. Gamba, Einaudi, Torino 1958, p. 20.

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practice, after the initial controversies (Leibniz, Berkeley, Euler), physicists had ended up accepting the absolute time of Newton and subsequently its measurement in kinematic phenomena. When dealing with the memory or prediction of our sensations, the concept of time is clear: this individual perception of time, or “psychological time”, is discontinuous despite the impression of continuity that we have of it. Yet psychological time is not physical time: the former has to do with our individual perception, while the latter is the stage for all other perceptions and physical events that populate space; the first is qualitative, the second is quantitative. Nobody has a direct intuition for the equality of two times intervals, yet it is this very concept of equality that is necessary for physics. This distinction was already foreshadowed in the Confessions of St. Augustine (354-430), when he declares to know what time is if nobody asks him about it; but not to know if someone asks him to explain it. For centuries physicists were content to accept that two complete rotations of the Earth around its axis had the same duration. Yet what ensures that this is the case? Clearly, nothing; on the contrary, according to Mach (§ 5.2), nobody could guarantee that the Earth turned on its own axis: this was simply a convention, challenged by certain astronomers who, accepting the Newtonian law, held that the tides slowed down the daily rotation of the Earth. What can then be a more acceptable criterion for the equality of two intervals of time? Among all the pre-relativistic physicists that asked this question, perhaps Poincaré gave the most advanced analysis. He began by asking himself what we mean when we affirm that two oscillations of the same pendulum have the same duration. We mean, according to his answer, that the same causes take the same amount of time to produce the same effects: with such a postulate the concept of causality is at stake, an even more murky concept than that of time (§ 8.11). However, Poincaré was a physicist of the 19th century for whom the concept of causality was selfevident. He instead observed that if at two distant points in space there occur in a certain instant two simultaneous phenomena, only the principle of sufficient reason ensures us that at any other moment, the effects of the initial simultaneous phenomena also follow simultaneously. The principle of sufficient reason, which since its first formulation (Leibniz) was met with some wariness by physicists, is not enough to guarantee a definition as fundamental as that of equal time intervals.212 212

Poincaré, La valeur de la science cit., pp. 40-41.

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That is not all: a physical effect is never produced by solely a single cause, but rather a multitude of causes, among which we can always find the rotational velocity of the Earth, as August Calinon (1850-1900) had observed in 1897, and supposing that to be constant would imply already knowing how to measure time. Furthermore, before being able to define the equality of two time intervals, one has to define simultaneity and anteriority. Sometimes we say one phenomenon is prior to another (lightning and thunder, for example) because we consider it to be the cause of the other, and thus temporal succession reduces to causal succession; sometimes we hold that the prior event is the cause of the following event, and thus causal succession reduces to temporal succession. Basically, post hoc, ergo propter hoc (after this, ergo because of this) is worth as much as propter hoc, ergo post hoc (propter hoc, ergo post hoc): how to we break free from this vicious cycle? Even if we consider concrete physical examples (measuring the speed of light, determining longitude, etc.), we realize that not only are we missing the intuition for the equality of two times intervals, but also the intuition for simultaneity. We replace the missing intuition with “convenient” rules: “the simultaneity of two events or the order of their succession, the equality of their durations, must be defined in such a way to give rise to the simplest possible statement of the laws of nature. In other words, all these rules, all these definitions are the fruit of unaware opportunism.”213. According to De Broglie, the excessive importance lent to criteria of conveniency together with a certain skepticism had impeded the sharp mind of Poincaré from accomplishing the admirable relativistic synthesis that Einstein reached. Without a doubt Poincaré had foreboded the advent of a new mechanics which had the following pillars: the rule (deduced from Lorentz contraction) that no velocity could exceed that of light; the impossibility of establishing whether an observer was still or in uniform rectilinear motion respect to another; the mass of electromagnetic energy. Yet to fuse these separate facts it was not enough to free oneself of the preconceived notion of the convenience of science: a more uninhibited criticism of the fundamental concepts of classical mechanics was needed, in particular of the concepts of simultaneity and time. Poincaré, as we have seen, remained on the brink.

213

Ibid., pp. 57-58.

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SPECIAL RELATIVITY 5.5 Relativity of time and space The fruitless attempts to demarcate the movement of the Earth with respect to the ether and unify mechanics and electrodynamics, radically changed in nature with the appearance of a famed 1905 paper by Einstein. We clarify, however, that in 1905 Einstein did not know of Michelson’s experiment: the genesis and development of relativity was independent from the experiment, hence one could almost say that history would have more or less taken the same course even had Michelson’s experiment not occurred. The now general habit of connecting relativity with this experiment arises only from didactic convenience in the exposition of relativity. A brief biographical sketch justifies Einstein’s ignorance of the fundamental experiment, which in 1905 had been talked about for almost a quarter of a century. Born in Ulm on 14 March 1879, Albert Einstein (Fig. 5.3) spent his adolescence in Munich, where his father directed a small electrochemical business. Having acquired Swiss citizenship, he enrolled in the Zurich Polytechnic and obtained a diploma in 1900. He immediately went to work at the patent office of Bern, dedicating his free time from his office work to original research and disregarding the contemporary scientific literature to some extent. Nevertheless, he immediately gained prominence among European physicists for his early work on thermodynamics. In 1905, he published three monumental papers: one on quantum theory (§ 6.16), which earned him the Nobel prize in physics in 1921; one on Brownian motion (§ 6.10); and one on relativity, which we will discuss shortly. In 1909 he was given the chair of theoretical physics in the Zurich Polytechnic, and in 1913 he moved to the Preussische Akademie der Wissenschaften in Berlin. He was dismissed from the professorship and stripped of his German citizenship because of his Jewish heritage and pacifism, and in 1933 was named professor at Princeton University, where he died on 18 April 1955. In the 1905 article mentioned above, titled Zur Elektrodynamik bewegter Körper, the young physicist began by indicating the dissymmetries of Maxwell’s electrodynamics applied to moving bodies: for example, according to the usual treatments, the mutual actions between a current and a magnet not only depended on their relative motions, but also from the fact that it was either the current or the magnet that was in motion. Examples like this and the failed attempts to discover motion of the Earth respect to the ether (not referring to the specific example, we

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repeat, of Michelson’s experiment) led him to hold that “for all coordinate systems for which the equations of mechanics hold, the electromagnetic and optical laws must also hold […].

Fig. 5.3 – Einstein in 1921, the year in which he was awarded the Nobel Prize in Physics.

We would like to elevate this assumption (which shall henceforth be known as the “principle of relativity”) to a fundamental postulate, and in addition introduce the postulate, only apparently incompatible with the preceding one, that in free space light always propagates with a fixed velocity, independent of the velocity of the emitting body.”214 Einstein set out then to construct an electrodynamics of moving bodies 214

A.Einstein, Zur Elektrodynamik bewegter Körper, in “Annalen der Physik”, 17, 1905, pp. 891-921.

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exempt from contradictions and which would lead to Maxwell’s equations for bodies at rest. In such a theory, the consideration of an ether as a means of support for waves became superfluous. The article contained a kinematic part (the definition of simultaneity, the relativity of lengths and times, the transformation of space and time coordinates, the composition of velocities), a second electrodynamic part (the conservation of Maxwell’s equations), and finally a part on the dynamics of a slowly accelerated electron. The first long-established concept at which Einstein directed his criticism was that of time, starting from the concept of simultaneity of two events. Let us consider an arbitrary system in which the laws of classical mechanics are valid. At a point A of the system, we can imagine a clock; an observer at A can determine the instant at which a certain event in the immediate vicinity of A occurs; if at B there is an identical clock, an observer at B can determine the instant at which an event in the immediate vicinity of B occurs. In this way, we have defined a “local time at A” and a “local time at B”. Yet because instantaneously propagating physical agents do not exist, we cannot, without ulterior conventions, compare the indications at A and B. Classical mechanics holds that the simultaneity of two events, occurring in the immediate vicinity of A and B, respectively, could be verified by bringing the clock from one point to the other, without even considering whether the motion of the clock changed its ticking rate. This assumption was evidently made because, instinctively, humans had come to consider the speed of light to be infinite. If, instead, the propagating signals available to humans had been slower, of the order of other common speeds in daily life, the question of defining time between two distant events would have been posed earlier. Nevertheless, it is not at all obvious that the motion of a clock does not alter its rate. Such a proposition cannot be assumed a priori. Einstein was able to give a criterion for the simultaneity of two events that occur at distant points A and B in the same frame of reference: two events at A and B are simultaneous if two light signals sent from A and B when the respective events occur meet contemporaneously at the midpoint AB. This is obviously a convention that does not imply anything in particular about the properties of light, and especially about the isotropy of its propagation. However, this relative simultaneity observed in a frame does not occur in another frame in motion respect to the first. If an observer sees two events occur at the same time in her frame, another observer in uniform translational motion respect to the first does not observe them simultaneously. Simultaneity is therefore a concept that is relative to the observer.

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Fig. 5.4

In his elementary exposition of relativity,215 Einstein illustrated this concept with a now classic example: a train track AB on which travels an extremely long train A’B’ at a speed v (Fig. 5.4). Let M’ be the midpoint of the train A’B’ which coincides with the midpoint M of the section of train tracks AB. Two signals sent from A and B, respectively, simultaneously arrive at M; they do not, however, arrive simultaneously at M’, but rather the signal sent from B arrives before the signal sent from A if the train is moving from A to B. Thus, the events are simultaneous for the observer at M on the train tracks, but successive for the observer at M’ on the train. One cannot retort that this conclusion follows from the fact that the observer at M is privileged, being at rest respect to the observer at M’, as no physical experiment can distinguish the absolute motion of A’B’ respect to AB: each of the two observers can say, with equal truth, to be in motion respect to the other. Each frame of reference has its own local time and there is no universal clock that keeps time for all space; in other words, absolute Newtonian time must be replaced with “the times” of different systems. We note that while the local time of Lorentz (§ 5.3) was a mathematical expression, in Einstein’s theory it gained a concrete physical meaning. Responding to the common criticism of relativity as placing too much conceptual importance to the propagation of light, Einstein retorted: “To give physical meaning to the concept of time we need processes that allow us to establish relations between different places. It does not matter what type of process is chosen for such a definition of time, but it is theoretically advantageous to choose only those processes about which we know something for certain. This is the case for the propagation of light in a vacuum to a greater extent than any other process that we can consider,

215

A. Einstein, The meaning of relativity, translated into Italian by L. Radicati, Einaudi, Torino 1950, p. 37; the first English edition is from 1922.

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thanks to the research of Maxwell and Lorentz.”216 The relativity of time led to the relativity of distance between two points as an inevitable consequence. In fact, in trying to determine the length of a moving ruler, an observer that moves with the ruler can conduct a measurement as often as necessary; while an observer at rest will instead have to determine the position of the ends of the ruler at a certain instant and then measure the distance, with her own ruler, between the two marks. The two procedures are different and consequently the two results will be different: the distance between two ends of a ruler also then depends on the frame of reference, that is on the relative motion of the two observers.

5.6 Special relativity As he had indicated in the brief introduction just referenced, Einstein based his new mechanics on two principles that he stated as follows: 5.6.1 - Principle of relativity (later called special relativity): the laws that govern all physical phenomena are the same for two observers in uniform rectilinear motion respect to one another. In other words, this means that with no mechanical or electromagnetic experiment can an observer distinguish whether she is at rest or in uniform rectilinear motion. The classic principle of relativity stated the same thing, yet was limited to mechanical phenomena, supposing that the observer could discern her state of motion through optical or electromagnetic experiments; 5.6.2 - Principle of constancy of the speed of light: light propagates in vacuum with a constant velocity in all directions, independent from the conditions of motion of its source and the observer. The second postulate immediately explains the negative result of Michelson’s experiment, which, we repeat, was unknown to Einstein. These two new principles led Einstein, through a mathematical approach, to derive Lorentz contraction for a body in motion observed from a rest frame and, reciprocally, for a body at rest observed by a moving frame of reference: if the speed of the moving body or frame reaches that of light in 216

A. Einstein, The theory of special and general relativity, translated into Italian by V. Geymonat, Boringhieri, Torino 1967. The first German edition appeared in 1917, followed by a plethora of new editions and translations in every language. The first Italian translation, edited by G.L. Calisse, appeared in Bologna in 1921.

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vacuum, the contraction is maximal and the shape of the body becomes entirely flat. It follows that a speed greater than that of light in vacuum lacks physical meaning; that is to say that the speed of light is the maximum attainable speed in nature. It is worth noting that initially, as we mentioned, the contraction for Lorentz and Poincaré was artificial in character, more a provisional expedient to fix things; later Lorentz believed the contraction to be “real”, considering it demonstrated by the Michelson experiment.217 For special relativity, on the other hand, contraction was neither an artefact nor an “appearance”, but rather an inevitable consequence of the dependence of the length of a ruler on the frame of reference. As a consequence of Lorentz contraction, a clock in motion runs slower than a fixed one; that is, if a phenomenon has a certain duration for a moving observer, it appears lengthened for a fixed observer. Furthermore, this phenomenon would appear to have an infinite duration if the moving observer traveled at the speed of light: this is the famous “clock paradox” on which much was written after it was fleshed out by Langevin in 1910. Langevin supposed that an interstellar traveler was launched from Earth with a velocity equal to one twenty-thousandth of the speed of light. The traveler proceeds in a straight light for a year (counted by her clock and habits), then comes back, arriving on Earth after two years. Upon her return, she finds, applying the relativistic formula for time dilation, that people on Earth have aged by one hundred years (counted on Earth’s clock, meaning that she finds another generation of people). Yet today it has been brought to light that this is a faulty argument. It is true that the interstellar traveler, when she reaches uniform motion on the journey outwards and the return journey, is in a Galilean frame of reference. However, in the initial phase of launching, during the changing of the route, and in landing, she is subject to an acceleration and thus is not in a Galilean frame of reference: in these three phases we cannot apply the theory of special relativity in the same manner. Jean-François Chazy observed (1929), not without a hint of humour, that we can also hold that during the three phases mentioned above the interstellar traveler ages a total of 98 years: everything adds up and we can remain untroubled. It is the case, however, that this new conception of time gives rise to deep philosophical questions, extending as far as that of consciousness, on which we cannot spend time. 217

H.A. Lorentz, The Theory of Electrons and Its Application, Teubner, Leipzig 1916, p. 196.

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The principle that the speed of light is constant is in stark contrast to the principles of classical mechanics. This places an upper limit on velocity, while classical mechanics also considered infinite velocities. As a consequence of the new principle the classical mechanical rules for composition of velocities change: the speed of light, for example, added to the speed of its source, still gives the same speed of light. The classical formula for the composition of two velocities in the same direction was very simple: the resultant velocity was the algebraic sum of the component velocities; the relativistic formula found by Einstein is more complex and such that for low velocities, far from the speed of light, it is practically equivalent to the classical formula, but departs from it more and more as the velocities involved increase.218 The relativistic formula for the composition of velocities was immediately confirmed by the experiments of Fizeau on the partial dragging of ether (§ 1.6): without turning to any hypothesis on the existence of ether, the results of Fizeau’s experiments were exactly those obtained from composing the speed of light (respect to the ground) in the fluid in question and the speed of the fluid according to the relativistic rule: indeed, the agreement was so perfect that some authors found it suspicious! What happened for the composition of velocities also occurred for the other propositions of relativistic mechanics, constructed by Einstein between 1905 and 1907 based on the two postulates: the propositions of relativistic mechanics were different from those of classical mechanics, but reduced to them for low velocities. Classical mechanics then appeared to be a first approximation, applicable to the ordinary occurrences of everyday life. This explains why for more than two centuries it was held to be exact and confirmed by experiment. “It would be equally as ridiculous,” Einstein and Infeld wrote in a popular book, “to apply the theory of relativity to the motion of cars, boats, trains, as trying to use a calculator in the cases in which it suffices to use the so-called pythagorean table”.219 One of the first consequences of the postulates of relativity was that all the laws of physics, or rather their mathematical expressions, remained 218 To be precise, in classical mechanics, if v and v’ are two velocities in the same direction, the resulting velocity from their composition V is given by V = v + v’ ; in the theory of relativity one instead has: ‫ ݒ‬+ ‫ݒ‬Ԣ ܸ= ‫ ݒݒ‬ᇱ 1+ ଶ ܿ with c as the speed of light in vacuum. 219 Einstein and Infeld, The evolution of physics cit., p. 201.

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invariant under Lorentz transformations. In this way, the criterion for determining whether a physical law was part of the relativistic scheme was formulated: it belongs only if its mathematical expression is such that any Lorentz transformation leaves it invariant. In this way, for example, one finds that Maxwell’s equations fit within the relativistic scheme, but the law of universal gravitation does not. Hermann Minkowski, the former unheeded professor of Einstein in Zurich, in a famous theory formulated in 1907-08, started from the assumption that time and space were absolutely inseparable entities and introduced a new formalism in which the mathematical form of a law was enough to guarantee its invariance under a Lorentz transformation. Naturally, the fundamental postulate of classical mechanics –the proportionality of force and acceleration– was deeply modified in the new mechanics. Without resorting to mathematical elaboration, we can get a glimpse of the necessity of this change by reasoning as follows. -

-

-

-

Having assumed the speed of light to be the maximum attainable velocity in nature, if an object reaches the speed of light, no force can increase its velocity; in other words, in such conditions of motion, force no longer gives rise to acceleration. In relativistic mechanics, the greater a body’s velocity is, the more difficult it is to accelerate it; since the resistance to change in velocity is the mass of a body, it results that mass increases with velocity.220 While classical mechanics considered mass to be an intrinsic constant of the body, relativity instead considered it variable and increasing with velocity. The mass from classical mechanics was then the relativistic rest mass. Furthermore, the new mechanics demonstrated that the mass of a body not only depended on its velocity, but also on the direction of the force, thus giving rise to the terms longitudinal mass and transverse mass. Here it is worth noting that as early as 1890, Paul Painlevé (1863-1933), for a purely mathematical generalization of the classical dynamics of points, had already introduced the concepts of longitudinal and transverse mass.

220 If a body of mass m moves with a speed v respect to our frame of reference, its 0 mass for us becomes ݉଴ ݉= ඥ1 െ ‫ ݒ‬ଶ /ܿ ଶ with c the speed of light in vacuum.

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It is not possible to experimentally observe the variation in mass except for elevated velocities approaching that of light. Electrons are then ideal projectiles for such verifications. As a matter of fact, Kauffmann had already verified the dependence of the transverse mass of ȕ particles on their velocity in 1902, providing prescient confirmation of this consequence of relativity, and later corroborated his results with successive studies in 1906. In 1914, Karl Glitscher (and the following year Arnold Sommerfeld), based on certain experiments carried out by Friedrich Paschen on the fine structure of the spectral lines of Helium, demonstrated that the masses of electrons circulating around the nucleus satisfy the relativistic mass rules. In 1935, Nacken, using cathode ray experiments with a potential difference between the electrodes that reached 200,000 volts, confirmed the relativistic formula for mass dilation to a precision of one percent. Further experimental confirmations came from the study of electronic trajectories in a Wilson chamber and from the study of cosmic rays. In any case, nowadays relativistic mass dilation is verified daily in nuclear physics phenomena. In the same year, 1905, Einstein mathematically deduced from the variability of mass with velocity, in a fairly simple way, an extraordinarily important consequence; later, he gave an intuitive explanation which here we lay out. Suppose that there are many balls at rest in a box; if an external force is applied to the box, it acquires a certain acceleration dependent on the rest mass of the balls. However, if the balls begin to move in all directions like the molecules of a gas, with speeds near that of light, would the external force give rise to the same effect? Clearly not, because the velocity of the balls increases their mass. Therefore, the kinetic energy of the balls resists motion, like mass. This particular case was brilliantly generalized by Einstein: energy under any form behaves like matter. In short, according to relativity, there is no essential difference between mass and energy: energy has mass and mass represents energy. “The mass of a body”, wrote Einstein in his second article on relativity, “is a measure of its energy content; if the energy varies by L, then the mass varies in the same way by L / (9Â1020), where energy is measured in ergs and mass in grams.”221 Classical physics had introduced two ingredients, matter and energy, and had asserted two corresponding conservation laws; relativity reduced the ingredient to one and left only the law of conservation of mass-energy. 221 A.Einstein, Ist die Trägheit eines Körpers von seinem Energiehalt abhängig?, in “Annalen der Physik”, 4th series, 1905, p. 641.

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Mass and energy transformed into one another with a fixed ratio, given by the relativistic formula E = m0 c2, where E is energy, m0 is rest mass, and c is the speed of light in vacuum. This formula was first given by Einstein in 1907. The equivalence between mass and energy seemed the most paradoxical proposition of relativity. Nevertheless, in the course of these pages, we will see that this relativistic idea became one of the most producting, as the general public learned in a terrifying event: the bomb dropped on Hiroshima (§ 9.11).

5.7 Nazi revisionism After World War I, the Nazis, led by Philipp Lenard (1862-1947) who in 1905 had received the Nobel prize in physics, disseminated (especially among Germans) the belief that mass-energy equivalence had been already established in 1904, that is, prior to Einstein, by Friedric Hasenöhrl (18741915): a Viennese physicist, a disciple of Josef Stefan and Boltzmann. He was a successor of the latter in the chair of theoretical physics at the Vienna polytechnic, and died during the war at Folgaria, on the Italian front. This was a great exaggeration, prompted by hatred of the Jewish Einstein, which nevertheless allows us to trace a line of thinking that originated from Faraday and reached its peak development in the first years of the 20th century. Hertz’s experiments had shown without a trace of doubt that space or ether was truly the setting for electric and magnetic phenomena. Thus, energy somehow had to be connected to the state of the ether, or even be identified with it, so as to attribute the inertia manifested in self-induction phenomena to the energy of the field from an electric current. In 1881, Thomson had already observed that an electric charge has inertia due to the electromagnetic field that it produces; indeed, he calculated the electromagnetic mass of an electric charge and found that it varied with velocity, while classical mechanics held mass to be constant. In 1903, Abraham likened the electron to a non-deformable sphere and made Thomson’s calculation more precise. He introduced the concept of electromagnetic moment and tied it to the vector that John Henry Poynting (1852-1914) had introduced in 1883 to represent the propagation of electromagnetic energy. Through the Poynting vector one can explain the pressure exerted on a surface struck by radiation, predicted by Maxwell and Adolfo Bartoli (§ 3.28). This theoretical construction allowed Abraham to calculate the electromagnetic mass of the electron and demonstrate that it grew with velocity, becoming infinite when it reached

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the speed of light.222 Hasenöhrl abandoned his research on the mass of the electron and turned to force fields of Hertzian waves, that carried energy regardless of their wavelength. In 1904, he devised a thought experiment that allowed him, thorough a convoluted line of reasoning in which a fundamental role was given to radiation pressure, to develop a relation between mass and electromagnetic energy: it differed from Einstein’s relation by a numerical factor, perhaps introduced following an error in calculation. Einstein was not aware of Hasenöhrl’s work, which had sunk into oblivion until Larmor simplified its logic and corrected its errors in 1922. Lenard rewrote the aforementioned history as follows: Maxwell and Bartoli predicted that if a body absorbs electromagnetic radiation, it is subject to an impulse equal to E/c per unit section of radiation, where E is the flux of energy and c is the speed of light. Now, if radiation strikes a body and is absorbed, the variation in the body’s momentum is mc, and since change in momentum is equal to impulse, one has mc = E/c. In this way one demonstrates –as, according to Lenard, Hasenöhrl had done– not only that radiation has mass, but also the Einsteinian law of mass-energy equivalence223. This is shown, without offending Lenard, fifteen years after Einstein’s demonstration, whose brilliant idea, incidentally, was the inertia of all forms of energy, not only the electromagnetic form. Lenard, however, held that the extension to all other forms of energy was obvious. Consequently, he closed his volume Grosse Naturforscher (The Great Naturalist), of which numerous editions were printed both in its original German and in English translation, with a chapter on Hasenöhrl, in whose work he located the origins, without ever citing Einstein, all the fundamental concepts of relativity: the speed of light as the universal speed limit, the deviation of light rays by a gravitational field, the fact that Newtonian mechanics was only an approximation. Clearly, the one who revolutionized physics in the 20th century was not Einstein, but rather Hasenöhrl. Racial hate went to such ludicrous lengths that it was worth recalling it! Yet in all fairness, and to provide some relief, we must add that post-Nazi Austrian scientists reevaluated Hasenöhrl and his work, placing it where it belonged within the history of physics.224 222 M. Abraham, Prinzipien der Dynamik des Elektrons, in “Annalen der Physik”, 10, 1904, pp. 105-79. 223 Ph.Lenard, Äther und Uräther, Hirzel, Leipzig 1922, pp. 41-42. 224 Österreichises Biographisches Lexikon 1815-1950, Graz-Köln 1959, vol, 2, pp. 200-01.

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GENERAL RELATIVITY 5.8 Gravitational mass and inertial mass Much like classical mechanics, special relativity had also granted a privileged position to Galilean observers, that is, observers belonging to a frame of reference in uniform rectilinear motion. Yet was there a reason for this privilege? Answering this question would prove to be very arduous. In 1907, Einstein began studying the question by critically reconsidering a well-known fact of classical mechanics. Classically, the inertial mass of a body was defined as the constant ratio between the force applied to it and the acceleration which it consequently experienced, while its gravitational mass was defined as the ratio between the body’s weight and the acceleration due to gravity. Clearly there is no a priori reason to say that the two masses defined as such are equal, because gravity has nothing to do with the definition of inertial mass. The equality of the two masses (with an appropriate choice of units) is an experimental fact that Newton had established225 through experiments on pendulums, and before that Galileo. through experiments on falling bodies. In a falling body, the acceleration is proportional to gravitational mass and inversely proportional to inertial mass, and because all bodies fall with the same acceleration, the two masses are equal. This was the line of reasoning followed by Baliani226, who, identifying gravitational and inertial mass, deduced that the accelerations were equal. More recently, with a series of highly precise experiments from 1896 to 1910 which were subsequently continued, the Hungarian physicist Lóránd von Eötvös (1848-1919) demonstrated this equality to a precision greater than one part in twenty-million. Eötvös conducted his experiments with a torsion balance similar to that of Coulomb and Cavendish, but asymmetrical, in which one mass is directly attached to one end of the beam and the other identical mass hangs from a string attached to the other end. The balance is very delicate and complicated, and its use requires highly specialized technicians: it found (and still finds, in certain cases) employment in geophysical studies or prospecting. Eötvös’s balance experiments were based on the fact that the equilibrium of a lead string is dependent on the attraction of the Earth, which varied with gravitational mass, and on the centrifugal force from Earth’s rotation, which varied with inertial mass. If 225 § 6.3 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 226 § 6.3 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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the two masses were not equal, the direction of the lead string would vary with the nature of the ball (lead, gold, iron, glass, etc.) hanging from the balance. Instead, Eötvös confirmed that the lead strings were parallel independently of the nature of the masses suspended on the balance. In conclusion, on the equality of gravitational and inertial mass there could be no doubt. Classical mechanics, however, had not doubted the equality, but had held it to be purely a coincidence, without even considering an interpretation. In the aforementioned 1907 paper by Einstein, he showed with intuitive arguments that the equality between gravitational mass and inertial mass was not by chance, but had a special nature, appearing as an intrinsic property of the gravitational field. Einstein arrived at this conclusion with a though experiment that has since become famous: the experiment of the elevator in free fall. Let’s imagine a physicist in an elevator in free fall, at the top of a gigantic skyscraper thousands of kilometres tall, a physicist enters an elevator in free fall. The physicist drops his handkerchief and pocket-watch and notices that these objects do not fall towards the floor of the elevator; if she nudges the objects, they move in uniform rectilinear motion and hit the walls of the elevator. The physicist concludes that she is in a limited Galilean reference frame, where the limited condition is necessary to be able to assume that all bodies have an equal acceleration. However, another physicist that observes the fall of the elevator from the outside sees things differently. He observes that the elevator and all its contents are accelerated, in agreement with Newton’s law of gravitation. This example demonstrates that one can transform from a Galilean frame to an accelerated frame by taking into account a gravitational field. In other words: a gravitational field (in which gravitational mass is relevant) is equivalent to accelerated motion (in which inertial mass is relevant): gravitational and inertial mass are the same material property seen from two different perspectives. In this way, Einstein arrived at the formulation of the equivalence principle, stated in the following manner in his autobiography: “In a gravitational field (of small spatial extent), everything happens as in a space free of gravitation, as long as one introduces, in the place of an “inertial reference frame”, a frame of reference that is accelerated with respect to an inertial one.”227 The equivalence principle can also be restated as the statement that an observer, through experiments performed inside a frame of reference, 227

Schilpp (editor), Albert Einstein scientist and philosopher, p. 34.

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cannot decide if she is in a gravitational field or if she is subjected to an acceleration. In the thought experiment of the elevator in free fall, one can see that the equivalence principle holds for small portions of space, namely it is local in nature.

5.9 General relativity The equivalence principle was the starting point for a new revision of the theory of relativity that Einstein called general relativity (and by contrast called the previous theory special). The new theory was presented by Einstein, following preparatory papers in 1914 and 1915, in a fundamental 1916 paper titled Grundlage der allgemainen Relativitätstheorie (The Foundation of the Generalized Theory of Relativity). The. The second part of the article was dedicated to the exposition of the mathematical tools necessary for the development of the theory, tools that fortunately had already been provided by the tensor calculus developed in 1899 by Gregorio Ricci Curbastro (1853-1925) and Tullio Levi-Civita (18731941). The fundamental postulate of general relativity was that there are no privileged frames of reference. “The laws of physics -to repeat the Einsteinian expression- must be in a form that can be applied to reference frames in any kind of motion.” The laws describing physical phenomena maintain their form for any observer, and thus the equations of physics must remain invariant not only for Lorentz transformations, but for any transformation. The mathematical consequences derived by Einstein are no less important than those deduced from special relativity, and lead to a further generalization of the concepts of space and time. If a kinematic change modifies or negates gravitation for a certain frame of reference, it is clear that there is a deep connection between gravitation and kinematics. Furthermore, because kinematics is a geometry in which time is added as a fourth variable, Einstein interpreted the effects of gravitation as spacetime geometry. It follows that, according to general relativity, our world is not Euclidean; its geometry is determined by the distribution of masses and their velocities. With a famous and much discussed thought experiment, Einstein highlighted the close relationship between kinematics and geometry. Suppose that an observer stands on a spinning platform in rapid rotational motion with respect to an outside observer. The external observer traces, in his Galilean frame of reference, a circumference equal to the outer circumference of the platform, measures its length and diameter, takes the

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ratio, and finds it to be the ʌ of Euclidean geometry. Then, the observer on the platform takes the same measurements with the same ruler used by the external observer. When along the platform’s radius, although in motion with respect to the outside observer, the ruler does not experience any variation in length, because the motion of the platform is perpendicular to its radius. However, when the observer goes to measure the edge of the platform, the ruler is shorter when seen by the external observer, because in this position it moves in the direction of motion (Lorentz contraction), the platform appears longer, and therefore a greater value of ʌ than the preceding one is obtained. An analogous phenomenon happens for time. If two identical clocks are placed on the platform, one at the centre and the other on the edge, an external observer will see the peripheral clock, which is in motion with respect to the other, tick more slowly than the central clock and thus will truly believe it to run slow. However, according to the equivalence principle, the effects of motion are analogous to those of a gravitational field: in a gravitational field, Euclidean geometry is no longer valid and watches run slowly. The platform example is mainly of didactic value; a gravitational field is algebraically different from the centrifugal field of the platform. In a gravitational field produced by a central lump of matter, radial lengths are contracted and transversal ones remain invariant: the ratio between circumference and diameter becomes less than ʌ. Arthur Eddington calculated its order of magnitude: if a one ton mass is placed at the centre of a circle having a five metre radius, the 24th digit beyond the decimal point of ʌ changes. The equations of gravitation in general relativity are of the same form as Maxwell’s equations, in the sense that they are structural equations that describe the variations in gravitational field and aim to highlight the geometric properties of our non-Euclidean world.

5.10 Experimental confirmations The new gravitational laws brought about some experimentally verifiable consequences. One of these was the following: because energy has mass and inertial mass is also gravitational mass, it follows that gravity acts on energy; therefore, a ray of light that traverses a gravitational field must be deviated. In actuality, this deviation is also a consequence of the Newtonian theory of light, and the deviation of a ray of light that, originating from a star, grazes the sun, had been calculated in 1804 by Johann von Soldner, who found a value half the one calculated by

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the theory of relativity. The first favourable occasion for an experimental test occurred during the total solar eclipse of May 1919. The region of the sky around the sun was photographed during the eclipse, when the stars were visible by day. Several months later, when the sun had moved from its position during the eclipse, the same region of the sky was photographed at night with the exact same instruments and techniques. From the comparison of the two photos there resulted an apparent shift in the stars photographed during the eclipse, which measurements revealed to be of 1.79” for stars whose light rays grazed the solar disk: almost exactly the value predicted by Einstein’s calculations. The test was repeated tens of times during other total solar eclipses, obtaining a mean reflection of 1.97”, which differs from the theoretical value by a margin within the experimental uncertainty. Artificial satellites have allowed for an analogous test in these past years. Satellites sent to other planets (Venus and Mars) also orbit in an elliptical trajectory around the sun. If electromagnetic signals passing by the sun are sent to these satellites, these too experience a deviation due to the mass of the sun, such that their path is curved and the time needed to arrive on the satellite and be reflected back to Earth increases (compared to that which would have elapsed in the absence of the Sun’s gravitational field). Experimenters from the Jet Propulsion Laboratory, making use of this principle, found in 1970 that the electromagnetic signal is delayed by 204 millionths of a second due to the curvature of its path from the Sun’s gravitational field: 4 millionths of a second more than predicted by the theory. Gravity’s established effect on the path of electromagnetic radiation, not predicted by special relativity, demonstrates that special relativity is valid in the absence of gravitational fields: it appears therefore as an approximate theory to general relativity, as classical mechanics was for special relativity. A second experimental confirmation of general relativity came from the study of planetary motion. As a consequence of the theory, the elliptical trajectory of a planet around the Sun should itself precess slowly with respect to the Sun: the effect, not predicted by Newtonian theory, is greater for planets closer to the Sun, for which the gravitational force is stronger. Mercury is the closest planet to the Sun and therefore it is where this relativistic effect can be most easily observed, although it is of such small magnitude that, according to the theory, three million years will pass before the ellipse of Mercury’s orbit makes a complete rotation. Now, the slow precession of Mercury’s trajectory, or more precisely the shift of its perihelion, had been observed by astronomers, who had

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tried to explain it as a perturbation of the planet’s motion due to the all the other planets. However, calculations following this hypothesis had resulted in a smaller perihelion shift than that observed, and this discrepancy between observation and calculation was in no way explained by Newtonian mechanics. The problem was first addressed using the framework of general relativity by Einstein in 1905, and was fully resolved the following year by Karl Schwarzschild. The agreement between general relativity and astronomical observation was of particular historical significance because it was obtained without any subsidiary hypotheses, but rather as a direct consequence of general relativity. A third experimental confirmation of the theory of general relativity, which after initial conflicting results is now indisputable, was the so-called Einstein effect, namely redshift of the spectral lines emitted by stars. As we have mentioned, a clock placed in a gravitational field runs slow, and because the back and forth of an oscillatory motion can be likened to a clock, the theory predicts a decrease in the frequency of light radiation in the presence of a gravitational field. It follows that the spectral lines of light emitted from a star must be redshifted compared to the corresponding lines obtained from a terrestrial source. This was verified in the study of light coming from “dwarf stars”, whose mean density is of the order of 10000 times that of water. In 1925, Walter Adams, photographing the spectrum of Sirius and its companion, Sirius B, observed the redshift: the phenomenon also quantitatively appeared to be in good agreement with the theoretical predictions.

5.11 Note on the spread of relativity Research on relativity by mathematicians, physicists, and philosophers was boundless: a blossoming the likes of which had never been seen in the history of physics. A bibliography compiled by Maurice Lecat in 1924 already recorded around 4000 titles between volumes, pamphlets, and articles. Naturally, highly original ideas could not be introduced without strong opposition, and when the knowledge of relativity spread to the general public in the immediate postwar period, the technical criticisms of the expert s were augmented by the unrestrained emotional reactions of people not sufficiently educated to discuss the topic. The opponents, both competent and incompetent, ultimately appealed to “common sense” to deny relativity any consideration. In Galileo’s time too, common sense had been invoked as the chief judge in the dispute between Ptolemaic and Copernican theories. Yet in both cases common sense, which changes with the times, aligns itself on the side of the innovators.

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Among physicists, at least outside of Italy, relativistic doctrine spread quickly through the work of Eddington in England, Langevin in France, and Laue in Germany; while in Italy dissemination occurred with some delay, effectively only after World War I. Once the heated discussions had been quelled, relativity no longer provoked a reaction from scientists. On the other hand, as Planck and De Broglie acutely observed, today it could be considered a chapter of classical physics, whose laws it does not contradict, but rather a couple conventional notions, like those of an absolute space and time. Merging the concepts of space and time, mass and energy, gravitation and inertia, relativity, like any other classical theory, was governed by the unifying impulse (paragraph 10.27) that inspired 19th century physics. To underline its exact position in the topography of classical physics, we have presented general relativity before quantum theory, the true revolutionary hypothesis of 20th century physics; deviating slightly from chronological order, and perhaps also from order of impact.

5.12 Pre-relativistic mechanics after relativity After the advent of relativity and wave mechanics (§ 8.1), what importance and role did pre- relativistic mechanics, commonly referred to as classical mechanics, play? Was it only a primary school subject, complete and allowing no possibility of further development? In short, was its future that of the Ptolemaic view after the advent of the Copernican system? There is a fundamental difference between the Ptolemaic system and classical mechanics: knowledge of the first is entirely unnecessary in our daily life, while classical mechanics continues to be the mechanics of everyday life. Precisely for this practical importance does classical mechanics continue to play a substantial role in school and university instruction across the world. Classical mechanics also made further progress in many fields tied to a variety of techniques. First of all, we remind the reader of the progress in nonlinear mechanics (thus named because it makes use non-linear differential equations), started by the Dutch Balthasar Van der Pol (18891959) with articles written between 1920 and 1930, and followed up by important contributions from Nikolay Krilov (1879-1955), Nikolay Bogoljubov (1909-1992), and others. Nonlinear mechanics studies oscillatory phenomena of any kind (mechanical, electrical, magnetic, etc.) and therefore is of great interest to every field of physics and even biology, with Volterra’s theory of biological fluctuations.

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In addition, the study of continuous media, from the theory developed by Nikolay Egorovich Zhukovsky and S.A. Tohplygione between 1906 and 1921, was fundamental to the problems of aerial navigation, which engendered studies on compressibility and the propagation of discontinuity as an acceleration or shock wave, and connected to thermodynamical problems. Fluid dynamics underwent extensive development, especially alongside the ever-increasing speeds of vehicles for aerial navigation and spatial exploration, which could reach regions filled with fluids in very different conditions from those in our atmosphere. Experimentation in wind tunnels, under various conditions, necessarily brought about the study of models, which involved numerous classical physics problems, in particular those of dimensional analysis. Another branch of continuous mechanics is elasticity, and in particular plasticity, whose mathematical foundations had been laid by Barré de Saint-Venant back in 1870. Following several results obtained in that same century, study was resumed after 1920, expanding to rheology, a term proposed in 1928 by Eugene Bingham to refer to the science of deformation and flow of viscous or plastic bodies under the influence of external forces. Tensor calculus greatly facilitated the analysis of these new phenomena, which depend on many physical characteristics, such as the macroscopic and microscopic structure of the bodies, intramolecular actions, the temperature distribution, and the calorific behaviour of the bodies. Ultimately, classical mechanics is still not a finished science with nothing left to discover. Thanks to the efforts of Walter Noll, it received a new axiomatic base, including general relativity, which was combined with the characteristic features of classical mechanics. Noll proposed to treat it like a mathematical structure, so as to give precise definitions to its general concepts, which would be applicable to different subfields: mechanics of point masses, rigid bodies, fluids, and elastic media, as well as materials that are neither fluids nor elastic in a classical sense. Noll’s axiomatization was based on an “axiom of objectivity”, which consisted of admitting that for each body, work, as defined by a certain mathematical expression, remained invariant for an arbitrary change of reference frame. The axiom is compatible with physical reality if inertial forces are considered to be real interactions between the bodies in our solar system and the totality of objects in the remaining universe.

6. DISCONTINUOUS PHYSICS

QUANTA 6.1 Matter and Energy Great unifying theories, which had been the leitmotif of research, had reduced physics at the end of the 19th century to two large chapters: the physics of matter and the physics of ether, or rather, radiation. Matter and radiation seemed to be two independent entities, as matter could exist without radiation and radiation could traverse spaces empty of matter. A deeper analysis, which would occur in the last century and will be discussed shortly here, would bring about the idea that everything we observe, and observation itself, is neither matter nor radiation, but interaction between these two entities. Although this subtler analysis was not the primary aim, the interaction between matter and radiation was one of the fundamental issues of the end of the century, because it was clear that if the two entities existed independently of one another, the entire world of phenomena arose from their reciprocal action. In essence, the problem reduced to finding the mechanics or model through which matter is able to emit and absorb radiation. The electromagnetic theory of light resulted in equations that assumed a link between electromagnetic fields, charges, and currents. These described the macroscopic results of experiments very well; but already in the last decade of the century, as we will better explain in the following chapter, scientists began to notice links between spectroscopy and the periodic events inside an atom: Maxwell’s equations were not applicable to the radiation emitted or absorbed by the fundamental particles of matter. It was necessary, therefore, to modify them and apply them to the emerging subatomic world: Lorentz had attempted this, as we have seen (§ 5.3). However, while Lorentz’s theory could very well have lent itself to the study of the interaction between matter and energy, even at the turn of the century physicists preferred to follow the classical route of thermodynamics, which seemed more solid ground to tread, free of the danger of running into quicksand.

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6.2 “Blackbody” radiation It is useful to recall the terms of the thermodynamic problem of interaction between matter and energy, which had been left unsolved by 19th century physics. Let multiple bodies of different temperatures be enclosed in an empty container with thermally insulating walls. Experiments which have yet to be contradicted ensure us that all these bodies soon reach the same temperature, in accordance with the second law of thermodynamics. Thermodynamics explains this fact by admitting an exchange of energy between the bodies without a matter intermediary, that is, only through radiation: each body is assumed to have emitted and absorbed a complex band of electromagnetic radiation, the hottest ones emitting more than they absorb in such a way to obtain in the end equal temperatures in all the bodies. Once this state is reached, it will maintain itself indefinitely as long as the physical conditions of the system remain constant, because each body in the container emits and absorbs an equal quantity of energy. The general problem was to determine the quantity of radiational energy emitted or absorbed by a body for each temperature and each frequency. By 1859 Kirchhoff, using thermodynamic considerations, had established that when all the bodies in the container reach the same temperature, they emit and absorb radiation such that an exact exchange is established between energy absorbed and energy emitted. This equilibrium state is unique and the distribution of radiation depends only on the temperature of the cavity, but not on its dimensions, shape, the properties of the bodies it contains, or the constituents of its walls. Kirchhoff’s famous “law” of thermal radiation can be stated as follows: the emissive power of a body is proportional to its absorptive power; in other words, a body absorbs more radiation the greater is its capacity to emit it. Kirchhoff also introduced the notion of a “blackbody” in 1860 (for which radiative laws become particularly simple), namely a body that absorbs all radiation that strikes it. In nature, perfect blackbodies do not exist; the blackest (like carbon black, platinum black, etc.) reflect and give off a small portion of the energy that they receive. However, Kirchhoff proposed a procedure to obtain a blackbody according to his given definition. Suppose we have a closed cavity. Incident radiation initially reflects off the internal cavity walls, with some of it being absorbed or scattered. The reflected and scattered portions of the radiation then again strike the walls and, in turn, are partly absorbed and partly reflected or

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scattered, and so on. After a few reflections or scatterings, the energy yet to be absorbed will be minuscule, tending to zero. In short, such a cavity has an absorption index of one and constitutes a blackbody. In practice, a blackbody is obtained using a cavity whose walls are made from a good conductor of heat, like copper, that is internally blackened with lampblack: a minute opening connecting it to the outside is bored in the cavity walls. Any radiation that penetrates this cavity, is virtually entirely absorbed by the previously described mechanism of scattering and absorption. A rough picture of this device, a sort of trap for radiation that enters the aperture, is given by a bedroom with a window: seen from the outside, the room appears dark, because the light that enters the window is in large part absorbed by the walls of the room and only a small portion comes back out. Conversely, the energy radiated through the aperture can be considered equal to that emitted by a blackbody at the same temperature: this energy is referred to using the somewhat imprecise expression “black radiation” or “blackbody radiation”, but can even be dazzling white at high enough temperatures. If the blackbody were kept at a constant temperature in a heat bath and the radiation emitted from its aperture is collected in a receiver (bolometer, thermocouple), which absorbs it all and transforms it into heat energy, one could measure the total energy emitted by the blackbody at each temperature. If instead, using various devices (filters, prisms, reticles), only the radiation corresponding to a given wavelength (or better yet, a narrow spectral band around that wavelength) were incident on the receiver, one could obtain, at each possible temperature of the blackbody, the specific intensity of the radiation for the examined wavelength. These measurements are not simple; they require contrivance and advanced experimental techniques. Furthermore, it is evident that measuring the total energy of blackbody radiation is more straightforward than measuring the specific intensity corresponding to each wavelength. This claim is substantiated by the historical course of events, beginning in 1879 with Josef Stefan (1835-1893), an Austrian physicist of Slavic origins, known for his studies of thermal conductivity in gases and a supporter of kinetic theory, then a subject of topical interest in physics research. Stefan, who at the time was a professor of physics in Vienna, based on his own measurements and those of other physicists of the total energy emitted by a blackbody at various temperatures, asserted that it was proportional to the fourth power of absolute temperature. This law was systematically verified in 1880 by Leo Graetz, who experimented between 0oC and 250°C; in 1887 Otto Lummer and Ferdinand Kurlbaum, with much more specialized equipment, verified it in the temperature interval

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between 290°C and 1500°C. In 1884, another Viennese scientist, Boltzmann, one of the greatest mathematical physicists of the 19th century and a former pupil of Stefan, showed (using a derivation that became universally repeated by modern writers) that Stefan’s law was a consequence of thermodynamics. In particular, it followed from radiation pressure, theoretically demonstrated by Maxwell (§ 3.28) and in 1876 by Adolfo Bartoli (1851-1896) through a thermodynamic argument. The impulse of radiation is extremely small, equivalent to E/c per cross sectional area struck, where E is the energy flux and c is the speed of light in vacuum. This is the reason for which the experimental verification of the Maxwell-Bartoli law was obtained much later; only in 1900, thanks to the Russian physicist Pyotr Lebedev (18661911). Stefan’s integral law, now commonly called the Stefan-Boltzmann law, was followed by powerful experimental works in the last years of the century, to determine the specific density of blackbody radiation at each temperature and wavelength. These found that the spectrum of blackbody radiation was continuous across a large interval of wavelengths. For a given temperature, the radiated energy was maximized at a certain wavelength, and diminished rapidly on either side of this value. In short, the graph of the radiated energy density at a given temperature, as a function of wavelength, is in the shape of a bell curve. The works of Lummer and Ernst Pringsheim in 1899-1900 in the field of visible radiation, Beckmann in 1908, Paschen in 1901 on infrared light between 420oC and 1600oC, and Baisch in 1911 on ultraviolet light experimentally confirmed the general shape of the curve, aside from small experimental errors that were later corrected. However, while the experimental phenomenon of radiation was wellknown by the end of the century and general laws like that of StefanBoltzmann and those of Wien, which we will describe shortly, had been formulated, the scientific ideas of the time could neither explain the origin of radiation nor the most common phenomena of matter and energy interaction. Why, for instance, does a piece of iron not emit light at ordinary temperatures? If it contains particles of electronic or any other nature that vibrate with a given frequency, why are the rapid vibrations corresponding to visible radiation absent until a certain temperature is reached? And yet, even when cold, the piece of iron absorbs light radiation that strikes it. It was necessary then to conceive of a mechanism that allowed for the passage of energy from ether to matter in the form of rapid vibrations, but prohibited the reverse.

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6.3 The absurdities of classical theory In 1894, Wilhelm Wien (1864-1928) further developed Boltzmann’s considerations and, based on thought experiments, showed that the second principle of thermodynamics would be contradicted if specific radiation was not proportional to the fifth power of absolute temperature multiplied by an undetermined function of the product of wavelength and temperature.228 From this proposition, he deduced the important “law of displacement”, which says that the product of a blackbody’s absolute temperature and the wavelength corresponding to its maximum emission is constant: in other words, raising the temperature of a blackbody shifts its region of maximum emission toward smaller wavelengths. Wien’s laws were of great importance, both because they were widely confirmed by experiment and due to their technological applications (the measurement of elevated temperatures through spectroscopy, for instance). Furthermore, as the unfolding of history would show, they represented the furthest limit of the laws of thermal and electromagnetic radiation that were attainable through classical approaches. Indeed, each subsequent attempt to find Wien’s undetermined function with classical methods invariably led to a striking contrast with the experimental reality. It was Wien himself who first attempted to specify the function, giving in 1896 an explicit formula for the radiation emitted by a blackbody.229 However, four years later Lord Rayleigh observed, as others had already, that Wien’s explicit formula led to the conclusion that as temperature increases, the radiation density for a given wavelength reaches a limit. This conclusion contradicted experimental results, which indicated that Wien’s law held fairly well for small wavelengths and low temperatures, but was in stark contrast with experiment for long wavelengths and high temperatures. Lord Rayleigh, having posed his criticisms of Wien’s explicit law, tried to find one himself that was more in agreement with experimental 228

One of the many forms in which Wien’s law is written is the following: ‫ݑ‬௩ =ܶ ହ f (ߣT) where ‫ݑ‬௩ is the specific energy density corresponding to the wavelength ߣ, and T is the absolute temperature; f is an unknown function. 229 Wien’s explicit formula was ‫ݑ‬௩ = ܿଵ ߣିହ ݁ ି௖మ Τఒ் dߣ where ܿଵ and ܿଶ are constants, ‫ݑ‬௩ the specific density of radiation in the wavelength interval between ߣ and ߣ+dߣ, while the other symbols have their usual meaning.

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results by applying the principle of equipartition of energy to ether and ponderable matter. According to this principle, which was developed by Maxwell and Boltzmann, in a multiparticle system the energy is distributed equally among all the degrees of freedom of the system. Following this entirely orthodox assumption, Lord Rayleigh obtained a formula in which the specific density was proportional to the square of the frequency and the absolute temperature.230 The appearance of this formula was, more than a disappointment, a disaster for classical physics; never had a formula deduced from classical laws been in sharper contrast with experimental results. For small frequencies, near the extreme infrared zone, the formula was fairly in accordance with experiment, but as frequency increased so did the disagreement, eventually bordering on the absurd. In fact, specific radiation should have increased continuously with frequency, while, as we have mentioned, the experimental graph of radiation density was a bell curve. Furthermore, Lord Rayleigh’s law led to the conclusion that the total energy emitted by a blackbody, at any temperature, was infinite, while the experimentally confirmed Stefan-Boltzmann law assured that this energy was simply proportional to the fourth power of absolute temperature. The paradox becomes even more jarring if this result is restated as the fact that any temperature increase of an arbitrary ethercontaining system requires the transmission of an infinite amount of heat. Yet another paradox results from Rayleigh’s law, deduced by applying Maxwell’s principle of equipartition of energy. Suppose there is a closed cavity containing matter and ether. To be in thermodynamic equilibrium, energy must be distributed between matter and ether proportionally to their respective degrees of freedom. Matter however, supposedly discontinuous, contains a finite number of molecules or other individual constituents that can be considered, and therefore a finite number of degrees of freedom, while the degrees of freedom of ether, whatever its volume, are infinite, because ether is supposedly a continuous medium. It follows that if Rayleigh’s law held, all energy would be contained in the ether, because any finite number of degrees of freedom is negligible compared to an infinite number. In the ether, this energy would acquire the highest frequencies, leading to what was called the ultraviolet catastrophe. Lastly, even the proportionality of specific radiation and temperature leads 230

Lord Rayleigh’s formula was the following: ଼గ ‫ݑ‬௩ = ௖ య Ȟ2kT where c and k are constants, Ȟ is frequency, and the other symbols have their usual meaning.

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directly to a contradiction, as Lorentz discovered. If this law of proportionality were true, a blackbody that emits white light at 1200°C should still be visible in the dark at 15oC (288°K), which corresponds to about a fifth of the original absolute temperature (1200°C = 1473°K). Roughly the same thing should then occur for any relatively absorbent body. In the end, even Rayleigh’s formula failed to explain the aforementioned enigma: why does a cold body that absorbs luminous radiation not emit light? Several other physicists (Hagen and Heinrich Rubens, Drude, Lorentz), before and after the appearance of Planck’s theory, attempted to obtain radiation laws by applying the laws of classical physics to oscillators or other apt substitutes: all of these attempts invariably led to Rayleigh’s law. Important research and critical works were undertaken from 1905 to 1909 by James Jeans (1877-1946), and thus Lord Rayleigh’s law is sometimes called Jeans’ law or the Rayleigh-Jeans law. Based on the resemblance between the curve of the Maxwellian velocity distribution of gas molecules and the curve of blackbody radiation at various temperatures, he applied the principle of equipartition of energy to the number of standing waves that can exist in a cavity, but even this application of classical statistical mechanics led him to an expression quite similar to that of Lord Rayleigh.

6.4 “Quanta” When it seemed that the mystery of the blackbody emission spectrum was becoming ever more nebulous, Planck proposed an interpretation so unconventional that he himself considered it only a provisional “working” hypothesis, as ideas not aligned with the science of the time are modestly and prudently called. Max Planck (Fig. 6.1) was born in Kiel on 23 April 1858 in a family of legal scholars. He studied in Munich and then Berlin, where he came into contact with Helmholtz, Clausius, and Kirchhoff, whose professorship he succeeded in 1889. He began his scientific activity working in thermodynamics, in the footsteps of Clausius and Boltzmann. He reformulated the two principles of thermodynamics, reconstructed the concept of entropy, and studied many problems from a thermodynamic point of view: osmotic pressure, changes of phase, thermochemical equilibrium, electrolytic dissociation, batteries, etc. Having achieved international fame as one of the leading experts of the time in thermodynamics, his interests shifted towards the problem of blackbody radiation, and his contributions in the new field of inquiry were so important that he became known as the “father of modern physics”.

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Planck was a father slightly advanced in age for the generational laws that seemed to govern progress in the field of physics, where the protagonists typically conducted their best work before thirty years of age. Planck, on the other hand, was already 42 when he formulated his “working hypothesis”, for which he won the Nobel Prize in physics in 1910. In 1930 he was elected president of the Kaiser Wilhelm Gesellschaft, which he abandoned in 1935, having clashed with the Nazi authorities, and retired to Göttingen, where he died on 4 October 1947.

Fig. 6.1 – Max Planck

Taking into account all of the theoretical failures mentioned earlier, Planck thought it perhaps wiser to fall back on a more modest strategy: instead of starting from the theory to obtain a radiation law that then would be compared to experiment, he proposed to start from the experimental data, which were already abundant and still growing, and translate them into an empirical formula, from which one could then try to

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formulate a theoretical interpretation. Already in 1899, Paschen (and then Lummer and Pringsheim the following year) had shown that Wien’s law held fairly well for short wavelengths and Rayleigh’s law held fairly well for long wavelengths, so Planck proposed to find an empirical formula that for short wavelengths coincided with that of Wien and for long wavelengths coincided with that of Rayleigh. In a paper titled On an improvement of Wien’s equation for the spectrum, presented to the Academy of Sciences of Berlin in 1900, he showed how one could achieve this and obtain a formula in agreement with experimental results for all wavelengths. From a mathematical point of view, Planck introduced only one fundamental change in Rayleigh’s theory: he replaced an integral that became infinite with diminishing wavelength with a discontinuous sum of elements, grouped in such a way that their sum remains finite. Aside from this liberty, Planck’s work continued to scrupulously adhere to all the laws and formalisms of classical physics. Having found the formula, Planck, in order to interpret it, had to impart a physical meaning to two constants that appeared. For the first constant this task was reasonably simple. For the second, which he called the elementary quantum of action, after “several weeks of work” – the most intense of his life – “the darkness fell away”, and it became clear to him that this constant “was either a fictional scale, in which case the entire derivation of the radiation law was at its root illusory and nothing more than another clever manipulation of formulas without meaning, or the derivation rested on a genuine physical idea, and thus the quantum of action was of fundamental importance for physics, announcing something truly new and unheard of that seemed destined to revolutionize our physical thought, which since Leibniz and Newton had invented infinitesimal calculus had been based on the hypothesis of continuity in all causal relationships.”231 What was truly unheard of was the assumption that each linear oscillator vibrating in the cavity of a blackbody could only emit in a discontinuous manner, in quanta, thus releasing packets of energy equal to hQ into the ether, where Q is the frequency of the oscillator and h a constant, called the quantum of action by Planck, which he calculated based on the available experimental data at the time (1900) to be 6.458 (10-27) erg.s: today’s value is slightly different at 6.77.(10-27) erg.s. The historic paper outlining the interpretation of this empirical formula is titled On the Theory of the Energy Distribution Law of the Normal Spectrum, 231

M. Planck, Wege zur Physikalischen Erkenntnis, 1908-1933.

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and was presented the 14th of December 1900 to the Academy of Sciences of Berlin.232 A complete description of the phenomenon of emission still necessitated a mechanism of propagation of the elements or units of energy, as they were called in the first years of the century. There are two possibilities: the elements of energy, once emitted, can either conserve their individuality and remain concentrated in small portions of space during propagation, or every emitted element can scatter into a larger and larger space as it travels further from the source. The first hypothesis is incompatible with classical physics, which is based on the wavelike propagation of luminous and heat radiation. Now Planck, whose fate it was to propose the most revolutionary physical theory of our time, was not at all a revolutionary at heart. He was essentially of conservative inclination; a man whose deep historical and humanistic studies had also taught him the value of tradition and the danger inherent in creating sudden rifts in the continuity of the historical process. Here he would have had to tear down a theory that from Young to Maxwell to Hertz could claim a century of continuous successes, without knowing what to replace it with. Guided by these patterns of thought, he instinctively chose the second alternative; consequently, the first form of the theory predicted discontinuous emission and absorption in quanta, but continuous radiation: Maxwell’s equations were safe. It was Einstein that in 1905, as we will soon discuss, proposed to break from classical optics and postulated a distinct individuality of energy elements. For many years, however, Planck firmly opposed the hypothesis advanced by Einstein. In his report to the Solvay Council233 of 1911, 232

Planck’s formula is the following: ௛ఔ ଼గ ‫ݑ‬௩ = య ‫ ݒ‬ଶ ೓ഌΤೖ೅ ௘ ିଵ ௖ where c is the speed of light and k is a constant equal to R/N (R is the constant from the ideal gas law, N is the Avogadro’s number) 233 This first “scientific council” (a sort of private conference) was promoted by Ernest Solvay (1838-1922), the famous chemist who, industrializing his process for the production of soda, had amassed enormous wealth that in part he directed towards a push in scientific research. The conference was presided over by Lorentz and included many of the major scientists of the time. It is useful to list them. From Germany: W. Nernst, M. Planck, H. Rubens, A. Sommerfeld, W. Wien, E. Warburg; from England: J.H. Jeans, E. Rutherford, Lord Rayleigh; from France: M. Brillouin, M. Curie, P. Langevin, J. Perrin, H. Poincaré; from Austria: A. Einstein, F. Hasenöhrl; from Denmark: M. Knudsen; from the Netherlands: H. Kamerling Onnes, J. Van der Waals, and the president H.A. Lorentz. The Solvay conferences, continued under the auspices of the international institute of physics

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Planck, referring to Einstein’s quanta of light hypothesis, wrote: “When one thinks of the complete experimental confirmation that Maxwell’s electrodynamics has received from the most delicate of interference phenomena, when one thinks of the extraordinary difficulties that its abandonment would bring about for the entire theory of electric and magnetic phenomena, one feels a certain repugnance in ruining its foundations so lightly. For this reason, we will leave aside, in what will follow, the hypothesis of light quanta, considering also that its development is still very primitive. We will assume that all phenomena which take place in vacuum are exactly governed by Maxwell’s equations, which have no connection to the constant h.”234 This position, however, led to one of the most frequent and pertinent criticisms of the early days of quantum theory: if the radiation incident upon a body was continuous but could only be absorbed in discrete quantities, where and how did the arriving energy accumulate until reaching the necessary threshold to be absorbed? This was a serious objection, corresponding to a technical criticism that the same Planck had levelled at his own mathematical formulation, which he had derived by simultaneously applying two contradictory hypotheses: that of quanta and the continuous electrodynamics of Maxwell. How could physics break free from the predicament of this untenable position? Planck, still unwavering in his commitment to save the edifice of 19th century physics from ruin, backtracked further, once again reducing the revolutionary weight of his hypothesis. In two notes from 1911, and especially in his report for the Solvay conference of the same year, he came up with a new theory. Having criticized discontinuous absorption, Planck declared: “Faced with these difficulties, it seems inevitable that we must abandon the supposition that the energy of an oscillator must necessarily be a whole multiple of the energy element H = hQ, and admit on founded by Solvay, were of particular importance to the historians of 20th century physics, both for their role in stimulating scientific research, and as reference points for results and proposals in the evolution of scientific research during the century. The accurate records kept of the conferences are a fresh source of information, not devoid, in certain sections, of a human touch. Begun as we have mentioned in 1911, the conferences continued, still under the presidentship of Lorentz, in 1913, 1921, 1924, and 1927. Once Lorentz died, these conferences still occurred, under the presidentship of Langevin, in 1930 and 1933. While they were interrupted by the Second World War, they resumed in 1948. 234 La théorie du rayonnement et les quanta. Rapports et discussions de la réunion tenue à Bruxelles du 30 octobre au 3 novembre 1911, edited by P. Langevin and M. De Broglie, Gauthiers-Villars, Paris 1912, p. 101.

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the contrary that the phenomenon of absorption of free radiation is essentially continuous. From this point of view, we can conserve the fundamental idea of the hypothesis of quanta by further supposing that the emission of radiated heat by an oscillator of frequency Q be discontinuous and produced in whole multiples of the energy element H = hQ”235 . The result was a hybrid theory –discontinuous emission, continuous radiation and absorption– that reduced the primitive theory to little more than a technical expedient, a fortuitous manipulation of formulas. From the discussion that followed Planck’s report, in which Einstein, Lorentz, Poincaré, Jeans, Langevin, Wien, Marcel Brillouin, Nernst, the Curies, Heike Kamerlingh Onnes, and others all took part, one can perhaps infer that Planck’s new theory left everyone unhappy; those most favourable to it probably thought along the lines of Sommerfeld, who in one of his comments wrote: “I believe that the quanta of emission hypothesis, much like the initial hypothesis of energy quanta, must be considered more like a form of explanation than as a physical reality.”236 On his part, still with the aim of saving the theory of electromagnetism, Sommerfeld, having observed that the universal constant discovered by Planck in his radiation theory was not an energy element but rather a quantum of action, proposed to replace the energy quanta hypothesis with a new and at first glance very unique principle, that intuitively can be formulated as follows: the time necessary for matter to receive or cede a certain quantity of energy is reduced as this energy is increased, in such a way that the product of energy and time be determined by h. Perhaps because it appeared to offer a way out of the sea of contradictions that physics seemed to have fallen into, Sommerfeld’s hypothesis was carefully discussed by the most important scientists of the time. However, its criticisms were so numerous (Poincaré, in particular, showed that Sommerfeld’s principle, which attempted to save classical physics, was incompatible with the principle of action and reaction) that it was soon forgotten. At the same time as Sommerfeld, Planck also noticed the importance of the quantum of action and placed it at the centre of the new theory of radiation that he presented in 1911. This new theory was much more general than the theory of energy quanta, because it was applicable not only to oscillators, but to all mechanical systems, yet in the case of oscillators it reduced to the hypothesis of energy quanta. The new theory was even more abstract than the preceding one, because action – the 235 236

Ibid., p. 110; the italics is preserved from the original text. Ibid., p. 129.

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product of energy and time – is a non-intuitive physical quantity, does not obey a principle of conservation, and yet assumed a sort of “atomic” role in the theory. The atomicity of action leads to a relation between space, time, and the dynamical phenomena which take place within them: a relation entirely foreign to classical physics and even more revolutionary than the quanta of energy. Clearly Planck was born to initiate inadvertent revolutions!

6.5 Difficulties elicited by the quantum hypothesis The quantum hypothesis had more or less slipped into science as an explanation for a single specific phenomenon, and initially seemed to be little more than a technical stopgap, a provisional expedient that further theoretical progress would have surmounted, because history had time and time again showed that a single experiment was sufficient to disprove a theory, but not enough to confirm one. However, as the hypothesis became more precise and its applications expanded, two only apparently contradictory phenomena developed in scientific circles: on one hand the theory gained further credibility, but on the other concerns grew and criticism abounded. Two criticisms in the first decade of the century were of particular importance. The first was formulated by Planck himself, as we have mentioned. He had arrived at his formula for blackbody radiation by combining two other formulas, the first deduced from the hypothesis that the energy of an oscillator varies in quanta; the second obtained from Maxwell’s electrodynamics, which assumes that energy varies continuously. The same fundamental contradiction arose in all of the quantum theories proposed at the time to described the properties of an oscillator. How could one reconcile these two markedly antithetical premises in a single line of thought? Only in 1909 (and later, more fully in 1917) was Einstein able to show that the antithesis was only apparent, because the relation classically deduced by Planck was instead of a general nature, independent of any physical interpretation. In the meantime, following Planck’s example and somewhat due to the still unclear distinction between the two concepts, physicists simultaneously used classical and quantum concepts in their work, greatly alarming theorists. Poincaré worriedly admonished: “As soon as the use of contradictory premises is allowed, there is no proposition that cannot be easily demonstrated.” Another serious question was brought up by mathematical physicists: physics had until then only considered continuous quantities, and only because of this was the use of differential equations, which were the

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essence of theoretical classical physics, legitimate. Did quantization not invalidate the use of differential equations? Should the new physics not substitute those equations with equations of finite differences or develop a new mathematical formalism?237 These concerns of Poincaré corresponded to Einstein’s perplexities: “We are all in agreement,” he said at the Solvay conference of 1911, “that quantum theory in its present form can have a practical use, but it does not truly constitute a theory in the ordinary sense of the word; in any case not a theory that can currently be developed in a coherent form. On the other hand, it is well known that classical dynamics, encoded in the equations of Lagrange and Hamilton, cannot be considered a sufficient scheme for the representation of all physical phenomena.”238 Brillouin best summarized the frame of mind of the most important and open physicists of the time, declaring at the end of the conference: “I would like to recapitulate the impression that I have from reading the reports, and better yet from the entirety of our discussions. Perhaps my conclusion will appear very timid to the youngest among us; but as it is, it seems to me already quite important. It appears clear that we must introduce into our physical and chemical conceptions a discontinuity, an element that varies in jumps, of which we had no idea a few years ago. How must it be introduced? This I see less clearly. Will it be in the first form proposed by Planck, despite the difficulties it brings up, or in the second? Will it be in the form of Sommerfeld or in another that has yet to be discovered? I do not know; each of these forms is well-suited for a group of phenomena, but less suited to others. Must we go further and disrupt the very foundations of electromagnetism and classical mechanics instead of limiting ourselves to adapting the new discontinuity to the old mechanics? I am a bit doubtful of this, and as important as the phenomena to which we have turned our attention are, I cannot ignore the enormous mountain of physical phenomena to whose description classical mechanics and electromagnetism are so well-suited; this is an important fact that I do not intend to compromise at all, at the cost of appearing conservative to some of our colleagues.” Given the numerous criticisms, perplexities, and uncertainties that quantum theory provoked in even the most open minded, perhaps the reader would expect to know what extraordinary event forced its acceptance. After having so extensively cited the discussions from the 1911 Solvay conference, it is unnecessary to waste time in refuting the thesis of 237 238

H. Poincaré, Dernières pensées, Flammarion, Paris 1913. La théorie du rayonnement et les quanta cit., p. 436.

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those who hold that this extraordinary event was the quantum interpretation of specific heat (§ 6.12), which took place in 1907. In our view, an extraordinary event did not take place. Nothing in particular can be called decisive in the triumph of the theory. Its strength lay in its fertility, that is in its ability to predict new phenomena and explain others, even those apparently unrelated to each other. The acceptance of the physical reality of quanta was slow and gradual as new phenomena were framed in the paradigm, as we will lay out in the following pages. It should come as no surprise, then, that in the years immediately following World War I, many physicists prudently still did not avow the reliability of the theory, and others, even of great renown, denied it on all grounds. Charles Barkla (1877-1944), for example, who had conducted extensive studies on X-ray spectra, as late as 1920, while receiving the Nobel prize in physics, affirmed that it resulted from his experimental works on X-rays that emission and absorption were continuous, and only atoms in certain critical conditions emitted in quanta. Without a doubt tied to this slow affirmation of the theory is a historical detail that we here report: the Nobel prize in physics was awarded to Planck only in 1918, when quantum theory had amassed so many applications as to appear, even regardless its physical validity, a heuristic instrument of exceptional incisiveness. Planck himself converted to the fully unabridged theory, so to speak –emission, propagation, absorption in quanta– only following the great successes it obtained. The fears regarding the ruin of the marvellous edifice erected by classical physics proved to be excessive, because the revolution found accommodating methods, a certain modus vivendi that little by little everyone accepted. The extremely small size of Planck’s constant h leads to the consequence that in phenomena where numerous quanta take part the discontinuity disappears, giving way to an apparent continuity comparable to the apparent continuity of a large mound of finely-grained sand. In other words, the majority of the laws of classical physics maintained their validity, provided that these laws be interpreted as statistical descriptions of various phenomena. At this price, which had already been accepted by Lorentz’s before the theory of quanta, physics was able to keep the bulk of the laws accumulated over the last three centuries mostly intact. Undoubtedly, however, dusting off the old adage of Leibniz239, natura non facit saltus, the new physics had to counter with its opposite, nature 239

§ 7.14 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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moves in leaps. Because of this radical contraposition, the name modern physics is given to the part of natural philosophy in which it is necessary to introduce quantum considerations.

AVOGADRO’S CONSTANT 6.6 The first measurements The pivotal role of Avogadro’s hypothesis (or law, as it is also called) –that equal volumes of gas under the same conditions of temperature and pressure contain an equal number of molecules– is well known in molecular theory and theoretical chemistry in general. It results from the law that a molecular gram of any substance (that is the number of grams equal to its molecular weight) always contains the same number of molecules N, called Avogadro’s number or constant. The number N is therefore an universal constant, valid for any substance. One of the most arduous problems for the chemistry and physics of the second half of the 1800s was the determination of the value of N. In 1866, Joseph Loschmidt (1821-1895) attempted the first calculation of N, starting from considerations due to the kinetic theory of gases. As it is known, this theory attributes the pressure of a gas on the walls of a container to the collisions of the gaseous molecules against the walls. It follows, all other things being equal, that the pressure of a gas will be stronger the more numerous are the collisions and, consequently, the more molecules are contained inside the container: we can then see a relationship between Avogadro’s number and the pressure exerted by a given gaseous mass at a given temperature. This was the relationship utilized by Loschmidt for the first calculation of N, naturally still a rough estimate. Seven years later, the calculation was improved by Van der Waals, who with his famous formula better described the real behaviour of gases, which diverges appreciably from the ideal behaviour described by the laws of Boyle and Volta-Gay-Lussac, especially at high pressures. Van der Waals found that the value of N had to be greater than 45(1022) and later, with more in-depth probing, assigned to N an approximate value of 62(1022). Van der Waals’ theory and this consequence elicited great admiration, but because of the many hypotheses on which both the theory and the applied calculation were based, the numerical value was met with little confidence. However, the belief that hoping to one day find a reliable value of N was not a fantasy, which was of considerable psychological

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importance, had spread. Among the countless methods that fulfilled this expectation, we will discuss the most significant in the following paragraphs.

6.7 The colour of the sky It is known that a beam of light that propagates in the air becomes visible through the lateral scattering that it undergoes inside grains of atmospheric dust. This phenomenon, studied by Tyndall in 1868, today goes by the name of the Tyndall effect. Each grain of dust diffracts light, and the smaller the wavelength of the incident light, the greater the diffraction; it follows that in a beam of white light, diffraction is more apparent for the smallest wavelength rays. In 1871, and then more fully in 1899, Lord Rayleigh, basing himself on the elastic ether theory of Fresnel, calculated the ratio between the intensity of direct sunlight and the intensity of the light scattered by the sky, assuming his hypothesis to be true. The number N appears in Lord Rayleigh’s formula, as he realized that the greater the ratio, the less centres of diffraction were present, meaning less air molecules contained in a given volume of atmospheric air: herein lay the need for N in the formula. The experimental verification of Lord Rayleigh’s formula, which was also confirmed through another approach by Einstein, could not be immediately undertaken. It required mountaineering scientists, because the observations had to be made at high altitudes with a very clear sky, so as to disregard the perturbations due to atmospheric dust and fog droplets. The first experimental data were obtained by Lord Kelvin from old experiments that Quintino Sella had performed at the peak of the Monte Rosa; by simultaneously comparing the brightness of the Sun, which was 40o above the horizon, and the brightness of the sky at its zenith, Lord Kelvin deduced that N had to be between 30(1022) and 150(1022). In 1910, Bauer and Moulin, with the appropriate equipment (and unfavorable weather) gave a value of N between 45(1022) and 75(1022): these experiments were repeated using the same equipment at Monte Rosa by Brillouin, who found values around 60(1022).

6.8 Calculation of N from subatomic phenomena Subatomic phenomena offered other methods to calculate N at the end of the century. We have already discussed the measurement of the electron charge (§ 4.3), which was found equal to the charge of monovalent ions in electrolysis, and the associated granularity of electricity, which was

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conceived of as constituted by elementary charges of quantity e. Now, the elementary charge is related to Avogadro’s number through the simple expression Ne=F, often mistakenly called the Helmholtz relation, in which F (equal to 96550 Coulombs according to turn of the century measurements) indicates the quantity of electricity released by a gram molecule of a substance in electrolysis. This formula had been used since 1874 by Stoney to calculate the elementary unit of electric charge, for which he obtained a value around ten times smaller than the one accepted today, given his poor knowledge of the value of N. Starting from the end of the 19th century, however, the application of the Helmholtz relation was inverted: instead of using the formula to calculate e, supposing known the value of N, the formula was employed to calculate N, having preemptively measured e. This inversion of the relation’s use is indicative of the fact that new, direct measurements of e were trusted much more than the indirect calculation of N. From each measurement of e, therefore, a value of N was obtained; these values were in agreement at least in order of magnitude with each other, and with the values already found through other means. In the first years of the 20th century, radioactive phenomena provided the most direct methods to calculate N. Having identified Į-rays as helium ions produced, according to the audacious views of Rutherford, in atomic explosions, which we will lay out in the next chapter, Crookes built the spinthariscope in 1903, an instrument that would have widespread commercial success. It consisted of a small cylinder containing a small amount of radioactive substance, placed at a distance of about one centimetre from a fluorescent screen, that was observed through a magnifying glass. On this screen, a twinkling of bright lights was visible, as if one was looking at a dark sky in which bright stars intermittently shone and went dark. Crookes interpreted each twinkle as arising from the collision of an Į-particle against the screen. In 1908, Enrich Regener proposed a method for measuring molecular sizes using the spinthariscope. He counted the scintillations produced by a given quantity of polonium at a given solid angle, and from it deduced the total number of Į-particles emitted by polonium in one second. He then measured the saturation current due to the ions produced in the air by Įrays from the same polonium sample, which Rutherford had already estimated to be 94000 times greater than the charge transported by the rays, and through simple arithmetic he obtained the charge of an Įparticle. This charge turned out to be twice the elementary charge, in good agreement with known measurements: from the elementary charge, through the Helmholtz relation, one could arrive at N.

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Yet, notwithstanding the agreement between these measurements and those that had been already performed, some doubts remained regarding the Crookes’ interpretation of the scintillations. Could we be certain that each twinkle was produced by a single Į-particle and that each particle produced one twinkle, such that the number of Į-particles emitted by the radioactive sample in question could be identified with the number of luminous points counted on the screen? All doubts evaporated that same year thanks to Rutherford and Geiger, who used a very clever device in which projectiles entered one by one into an ionization chamber; the ions they produced struck the outer shell of a capacitor and produced completely distinct electrometer impulses that were irregularly distributed in time (for example, 2 to 5 per minute); the number of impulses gave the number of Į-particles, which were then easily counted one by one. A knowledge of this number can lead to a value of N through several methods, as experiments between 1908 and 1910 showed. One can collect Į-rays in a Faraday cylinder and measure their charge, which divided by their number gives the charge of each particle – twice the elementary charge– from which N can be deduced. Rutherford and Hans Geiger found a value of N = 62.(1022). The number of molecules can also be used to measure the volume of individual molecules. The challenge at the time laid in collecting all the molecules and preventing other gas from entering the container. James Dewar (1842-1923), more famous for his vessels that became commercially recast as the thermos, was able to perform this in 1908. He found that, under normal conditions, a gram of radium in radioactive equilibrium emits 164 mm3 of helium per year (from which, N = 60.1022). This measuring technique was also adopted, with some refinements, by Bertram Boltwood and Rutherford and, in 1911, by Marie Curie and Debierne, who found the same results, in order of magnitude, as those obtained by Dewar. Yet, upon further inspection, all these methods of calculation rested on assumptions that, while obvious today, at the time were the centre of scientific discussions. What would have been of this admirable body of experimental work, of all these ingenious devices, of all these subtle critical examinations, if the existence of an elementary charge or the theory of atomic decay had been disproven? The reservation with which these results were met by the physicists of the time is therefore justified. One more detail of exceptional historical importance must be taken into account to understand the scientific evolution of the first quarter of the century: for the entirety of radioactive phenomena, the large majority of physicists were forced to believe in verba magistri (in the words of the

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teacher), as only very few scientific laboratories were equipped to study radioactivity. In the first quarter of the century, Cambridge, Manchester, Paris, Vienna, and Berlin were the only such laboratories in the world.

6.9 Deducing Avogadro’s constant from quantum theory It is impossible to overstate the importance of an accurate knowledge of N, which was not only necessary for chemical theories, but also for the development of the most advanced branches of physics in the early years of the 20th century. Suffice to say that each new value of N adjusted the previous values of molecular sizes and, in particular, the value of the elementary charge, tied to N through the Helmholtz relation. Consequently, it is no small feat that N can be calculated through the blackbody radiation law given by Planck. Yet this initial success of quantum theory is certainly better appreciated today than it was then. At the time, when Planck performed the calculation in 1901, there was scant and uncertain information regarding N. The calculation of N could neither corroborate quantum theory, as there was no established value to compare to, nor give greater assurance to the previous values that had been found, because the legitimacy of the theory that had furnished the value was in question. These circumstances are the reason for which not much was importance was given to Planck’s calculation in the first decade of the century. On the other hand, as the reader may have noticed, to attribute to this first success of quantum theory the credit that it deserves, we have slightly deviated from chronological order in our exposition. Upon first glance, it can seem paradoxical that a radiation law, namely a law describing a phenomenon that occurs in the absence of matter, could contain elements that relate to the structure of matter. The apparent paradox originates from the fact that Planck’s argument uses the laws of statistical thermodynamics, which include a coefficient that must be independent of the system. This also then applies to gases, and it is here that the number N is introduced, as we see in footnote 5 of § 6.4. In reality, Planck did not only use that formula, but also the laws of Stefan and displacement (which can be easily deduced, on the other hand, from his radiation law), obtaining a value of 61.6(1022) for N, in excellent agreement with the Van der Waals’ value, the most reliable at the time.

6.10 Brownian motion The weight of all the preceding results was greatly augmented by the results obtained through a completely different approach, which had the

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advantage of a more concrete evidence base, resting on a phenomenon that had been studied for almost a century and a solid classical theory: brownian motion. In 1828, the English botanist Robert Brown (1773-1858) reported that when he observed minute particles floating in water, even with a modest microscope, he noticed each of them is subject to an incessant excitement; it comes and goes, rises and falls, only to rise again, always in rapid and irregular motion without rest: perpetual motion in the literal sense of the expression. Spallanzani too had observed the phenomenon and believed it tied to phenomena of life, perhaps misled by the vivacity of the motion, which was similar to the restless darting around of a microscopic animal. Brown, on the other hand, recognized that this perpetual motion belonged to inanimate particles. Initially, physicists did not lend much importance to this phenomenon, while still observing it attentively and describing it in minute detail, not without a note of amazement. Some said that the phenomenon was due to the mechanical quivering of the support on which the sample was resting; others that it was analogous to the motion that could be observed in atmospheric dust struck by a beam of light, which was known to be caused by convective currents in the air. In the second half of the century, however, as more and more observations were made, this certainty of interpretation was losing ground to a more serious critical reflection. The most perceptive observers of the phenomenon (Brown, Louis-Georges Gouy, Giovanni Cantoni, Franz Exner, Otto Wiener) established that the motion of each particle is entirely independent of its neighboring particles, that it is truly endless, that it occurs no matter what precautions are taken to ensure the mechanical and thermal equilibrium of the liquid, that the nature or intensity of the light used to observe the phenomenon has no effect on the motion of the particles, and that the phenomenon is unaffected by the nature of the particles floating in the liquid, but affected only by their mass, as it appears faster for smaller particles. This bountiful supply of experimental observations led Wiener to conclude in 1863 that the agitated motion originates neither from the suspended particles, nor from external causes, but has to be attributed to an internal motion characteristic of the fluid state. In brief, brownian motion is due to the collision of the liquid molecules with the particles suspended in the liquid. Wiener’s reflections, however, appeared (and were) confused, weighed down by the introduction of “ether atoms” in addition to material atoms. Yet Ramsay in 1876, the priests Joseph Delsaulx, Ignace Carbonelle, and Thicion in 1877, and especially Georges Gouy (1854-1926) in 1888 –when kinetic theory had already made remarkable

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advances and earned the respect of physicists– stated the thermodynamic origin of brownian motion very clearly. “In the case of a large surface”, wrote Delsaulx and Carbonelle, “molecular collisions, the cause of pressure, do not give rise to any shaking of a suspended body because, as a whole, they push the body equally in all directions. However, if the surface is smaller than the size necessary to ensure that all irregularities are compensated, there will be unequal and continuously variable pressures that vary from point to point; they cannot be homogenized by the law of large numbers, resulting in a nonzero force, which constantly changes in magnitude and direction.”240 Once this interpretation was accepted, the thermal vibration of fluids could be said proven ad oculos. Much like we cannot see the distant waves in the ocean, but can attribute to them the bobbling of the ship at the horizon, we cannot see the motion of molecules, but we can infer it from the jitteriness of particles suspended in a fluid. This interpretation of brownian motion was not only important as evidence for kinetic theory; it also led to notable theoretical consequences. If we want to keep the principle of conservation of energy, we must recognize that each variation in a suspended particle’s velocity must be accompanied by a variation of the liquid’s temperature in the immediate vicinity: to be precise, this temperature increases if the velocity of the particle decreases, and decreases if the velocity increases. The thermal equilibrium of a fluid body, thus, is in statistical equilibrium. Still more important, however, was the observation made by Gouy in 1888: brownian motion does not obey the second principle of thermodynamics in its strict formulation. Indeed, when a particle suspended in liquid spontaneously rises, it transforms part of the surrounding heat into mechanical work; it is not true then that such a transformation cannot occur spontaneously, as the second principle of thermodynamics affirms. Put more simply, a suspended particle spontaneously falls and rises in the liquid: it is therefore false that if a phenomenon occurs spontaneously in one direction, it cannot occur spontaneously in the other, as the second principle states. Gouy’s observations, however, also showed that the rise of a particle grew more improbable with increasing weight; for a grain of matter of appreciable size, the probability was practically zero. In short, the second principle of thermodynamics became a probabilistic law, rather than an absolute (§ 2.8). Clausius, Maxwell, Boltzmann, and 240

Reported by J. Perrin, Les atomes, Alan, Paris 1920, p. 127. The original edition of this work is from 1913. This is an extension and a collection of articles published starting in 1908: Œvres scientifiques, Centre national de la recherche scientifique, Paris 1950.

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Josiah Willard Gibbs had already much insisted on the statistical nature of the second principle of thermodynamics. Yet their interpretation was based on thought experiments, like Maxwell’s daemon, which postulated the existence of molecules. No concrete experiment supported the statistical interpretation. One had only to deny the existence of molecules, like the school of energeticists, which flowered under the direction of Mach and Ostwald (contemporaries of Gouy), to interpret the second principle of thermodynamics as an exact law at all times. Because of the existence of Brownian motion, however, a rigorous interpretation was no longer possible: there was a concrete experiment that demonstrated, independent of any molecular theory, that the second principle of thermodynamics was continuously violated in nature, and furthermore that perpetual motion was not only possible, but transpired continuously in front of our very eyes. This philosophical view shifts if we take the viewpoint of indeterministic statistical physics, in which, by definition, individual exceptions do not constitute a violation of the law, much like it is not a violation of the statistical rule that “the house always wins” if a person wins a one-time large sum playing roulette. Nevertheless, physics in the 19th century was rigorously deterministic, and so this new conception faced a difficult path in the course of the 20th century, as we will expound in chapter 8. It was for this reason that in the first decades of the 20th century it had to be “authoritatively asserted” multiple times that the principle of thermal irreversibility, the foundation of all of thermodynamics, was not true without limit. But let us return to the end of the 19th century, when brownian motion was gaining significant theoretical importance and attracted the attention of many theoretical physicists: in particular Einstein in 1905. Since the first studies on Brownian motion, physicists had tried to establish the mean velocity of suspended particles. But the values obtained were crude approximations, because the trajectory of a particle is so complicated that its motion was impossible to follow; the mean velocity varies in magnitude and direction, without tending toward a limit in the time considered. Nor could one find the tangent to the particle’s trajectory at a point, because the trajectory is less like a continuous curve and more like a nowhere differentiable function, as Ugo Cassina observed in 1950, noting the similarity between Peano’s famous curve and the trajectories of Brownian motion. Einstein became convinced that the continued search for the unmeasurable mean velocity was in vain, and chose as a characteristic parameter the displacement of a suspended particle, i.e. the segment connecting the starting point with the endpoint of the particle in the time

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considered. He also had the providential idea to apply the same hypothesis on which Maxwell had based his calculation of the velocity distribution of gas molecules to Brownian motion, a curious, contradictory-seeming postulate: Brownian motion is perfectly irregular. Assuming this postulate, the same considerations Maxwell took into account for molecular velocities can be applied to Brownian motion, leading Einstein to conclude that liquids with very fine suspended particulates, or as we will call them, emulsions, diffuse like solutions, with a well-defined coefficient of diffusion that depends on the temperature of the emulsion, Avogadro’s number, the radius of the fragments in the emulsion (all taken to be equal), and the viscosity of the liquid. It follows that if one can measure all of the other factors, Einstein’s formula allows for the immediate calculation of N. This was easier said than done, as the experimental difficulties to overcome were quite serious. The first measurement attempts gave unconvincing results; if anything, a cinematographic recording of Brownian motion (perhaps the first example of the use of cinematography in the study of physical phenomena), made in 1908 by Victor Henri, seemed to contradict Einstein’s theory241 and led many impatient physicists to conclude that an inadmissible hypothesis must have somehow inadvertently slipped into it. That same year, however, Jean-Baptiste Perrin, a physicist of extraordinary experimental abilities, set out to solve the problem from a very advantageous starting point: in the same year, having also been inspired by Einstein’s ideas, he had calculated N, allowing him to tackle the study of brownian motion from another point of view, as we will soon describe. Perrin began by verifying the fundamental hypothesis of the Einsteinian theory, the perfect irregularity of motion. With a camera lucida, he observed the successive positions of the same droplet in an emulsion at regular time intervals. For example, the consecutive positions assumed every 30 seconds by a grain of mastic, of diameter not much larger than a micron, are shown in Fig. 6.2.

241

Einstein’s brilliant 1905 paper and his successive works on Brownian motion were collected and annotated by R. Führt in 1920: the volume was translated into English by A.D. Cowper, Investigations on the Theory of Brownian Movement, Meuthen, London 1926.

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Fig. 6.2

Naturally, if the observations had been made once every second, each segment of in the picture would be replaced by a fragmented 30-sided shape, highlighting the continual zigzagging of real-life particle trajectories. If the motion was perfectly irregular, the projections of these segments onto an arbitrary axis would be distributed around their mean following the law of probability, given by a well-known mathematical formula from which the experimental results could be calculated. More simply, Langevin suggested to verify the perfect irregularity by shifting all of the displacements observed to a common origin: if the motion was truly perfectly irregular, governed by the laws of random chance, the external endpoints of the segments would be distributed about their common origin as bullets distributed around a target’s bullseye. Having re-centred 500 observations, each taken every 30 seconds, Perrin obtained the figure reproduced in Fig. 6.3: upon first glance, did the reader not think it a shooting target, riddled with holes from a battalion of soldiers engaging in target practice? Having verified the fundamental postulate, Perrin and his collaborators proceeded to conduct seven series of measurements of diffusion in emulsions, varying the experimental conditions in many ways. The results they obtained, when introduced into Einstein’s formula, yielded 68.5.(1022) as the most probably value of N

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Fig. 6.3

This measurement was preceded by another analogous work by Perrin, who, remembering how Van’t Hoff had extended gas laws to dilute solutions, had thought: “The sugar molecule, which already contains 45 atoms, that of quinine, which contains more than 100, are worth no more and no less than the lithe water molecule that contains 3. Therefore, could one not suppose that there is no size limit for the conjunction of atoms that obey these laws? Could one not assume that even particles visible to the naked eye still obey them exactly, such that an excited granule in Brownian motion counts as much as an ordinary molecule in terms of the effect of its collisions with an enclosing wall? In short, could one not assume that the ideal gas laws also apply to emulsions composed of visible particulates?”242 It appeared to Perrin that the simplest way to put this hypothesis to the experimental test would be to study the height distribution of emulsion particles. Much like air molecules, the particles in an emulsion should become more sparse with height if the laws of gases could truly be extended to emulsions. To be precise, the rarefaction of air increases with altitude due to the fact that the top layers of air press on the bottom ones with their weight243. Laplace, considering that the difference in pressure 242 243

Perrin, Les atomes cit., pp. 128-29. § 5.21 of “A History of Physics from Antiquity to the Enlightenment”, 2022.

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between two sides of a horizontal layer of air is due to the weight of the layer itself, gave the law of atmospheric rarefaction with height through a simple calculation, which also applied to any gas only subject to gravity. Laplace’s law is exponential and essentially says that for equal altitudes there are equal rarefactions, proportional to the molecular mass of gas considered. For example, in oxygen at 0oC, the density is halved for each 5 km rise in altitude; yet in hydrogen one must rise 16 times as much to obtain the same rarefaction, because the molecular mass of hydrogen is 16 times smaller than that of oxygen. If emulsions follow the laws of gases, the height distribution of grains in an emulsion must follow Laplace’s law, mutatis mutandis: precisely, for identical grains in an emulsion, much like for identical gas molecules, molar mass given by Avogadro’s constant multiplied by the mass of a grain; the analogue of gas density is given by the number density of grains in the emulsion, namely the number or grains contained in a given volume. Consequently, in an emulsion, once an equilibrium state is reached, in which the offsetting actions of the particles’ weight pulling them down and Brownian motion scattering them are balanced, at equal elevations there should be equal rarefactions of the particles. If experiment verifies this regime predicted by theory, then Laplace’s formula applied to emulsions provides a method of calculating N from the measurement of the other values that accompany it. Experimental difficulties are evident: having chosen a suitable emulsion (after many trials, Perrin chose emulsions of gamboge or mastic), one must obtain identical grains of very small diameter (less than half a micron), determine their density and mass, study the microscopic observational apparatus, and carry out observations across heights below a tenth of a millimeter by counting particles in different layers. That Perrin’s experimental astuteness met the experimental challenges is clear from the agreement of the series of diverse measurements carried out by him and his collaborators. In the emulsions examined, the theoretical regime of equilibrium was established, and from the subsequent experimental measurements the mean value of N was found to be 68.2.(1022), with an experimental error that Perrin gave as 3 percent. We conclude by noting that Einstein’s work on Brownian motion and Perrin’s experimental corroboration were instrumental in convincing physicists and chemists that had been reluctant to accept the existence of atoms and molecules, the kinetic theory of heat, and the importance of probability in the laws of nature.

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6.11 Avogadro’s constant deduced from the theory of fluctuations The mathematical treatments of physical phenomena are idealizations, based on the assumption that the laws of nature are simple. As such, when we say that a quantity (for example, the pressure of cooking gas) is “constant”, we mean that its variations in time are not detectable by ordinary instruments. Yet with more delicate instruments it would be possible to detect variations in time that are quite irregular: fluctuations around a mean value. This phenomenon is of particular interest to the kinetic theory of gases. When we say that a certain gaseous mass contains n0 molecules, we mean it as a mean value, implying that in a certain instant the number could be not exactly n0, but a certain value n, slightly larger or slightly smaller than n0: brownian motion ensures such fluctuations in density, regulated by the expression (n-n0)/n0. The polish scientist Maryan Ritter Smoluchowski (1872-1917), professor of physics at the university of Krakow, using a simple statistical argument was able to calculate the mean value of (n-n0)/n0 (in absolute value) for a gas or dilute solution. Sometime later, in 1907, Smoluchowski also calculated the mean density fluctuation for any liquid and demonstrated that it becomes observable on scales visible with a microscope when the fluid is near its critical state. Furthermore, the large fluctuation in density allowed him to interpret a phenomenon that had remained an enigma for some time: opalescence, namely the visibility of a beam of light traversing a liquid, present in all fluids near their critical state. According to Smoluchowski, in the proximity of a critical state, various adjoining layers of possibly very different densities form because of the extreme compressibility of the fluid. Molecular jitteriness then facilitates the formation of dense clusters and lumps of molecules, which laterally deviate light like dust in the air. The quantitative theory of opalescence, completed by Willem Keesom, gives an expression that includes only measurable quantities and N for the intensity of light normally scattered by a cubic centimeter of fluid. In the Leiden laboratory, Kamerlingh Onnes and Keesom verified the theory of opalescence for ethylene and found a value of 75.(1022) for N, in fair agreement with the known values at the time.

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Despite not knowing of the preceding works, Einstein, in a 1910 paper on the theory of opalescence, dealt with the same problems with a more comprehensive framework, arriving at similar results. To provide some perspective on this result and the others we have referred to, we add that one of the best methods known today to determine N is based on X-ray diffraction, as is described in relevant papers. With this method, the most recent calculation is that given in 1963 by Richard Cohen and Jesse DuMond: N = (6.02252 r 0.00028).1023. The uncertainty is less than one part in a thousand; returning to the image of Millikan, Avogadro’s constant is known today with more precision that that required to know, in any given instant, the exact population of a city like New York.

SPECIFIC HEATS 6.12 The law of Dulong and Petit In the first decades of the 19th century, as chemists were striving to make their atomic theory of matter acceptable, many physicists were also trying to relate atoms to their own science. Dulong and Petit held that the action of heat and, more precisely, specific heat could be tied to atomic composition: an intuitive concept, if matter was assumed to be made up of atoms. Yet the specific heat tables available were anything but reliable, as it was not uncommon to find that one author reported values that were three or four times those reported by others. The first task of Dulong and Petit was, therefore, to move to new and more accurate measurements. In many cases, however, they did not have enough of a substance to apply the fusion or mixing methods of measurement, and thus they decided to resort to the cooling method. This method had already been developed by Mayer and improved by Leslie, according to whom, all other things held constant, the ratio of the heat capacities of two bodies was equal to the ratio of their cooling times. From meticulously conducted experiments, the scientists obtained the specific heats (relative to water) of 13 elements in the solid state (bismuth, lead, gold, platinum, tin, etc.). To go from these specific heats to those of atoms, it should be enough to divide the value obtained by the number of atoms in the trial masses, which was still unknown, as we alluded to in the previous paragraphs. However, the number of atoms is inversely proportional to their weight, so multiplying the heat capacities of the experimental substances by their respective atomic weights (relative to oxygen) one obtains the specific heat capacities of the various atoms. This

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product gave results that oscillated between 0.383 for bismuth and 0.3675 for tellurium, that is, practically the same value within the limits of the experimental uncertainty of the time, hence the law: “The atoms of all simple bodies have exactly the same heat capacity.”244 Kinetic theory restated the law in another equivalent form: the quantity of heat (called molecular heat) necessary to heat a molecular gram of any solid element and raise its temperature by one degree centigrade is equal to about 6 calories. The law proved to be valid for a great number of solid elements at ordinary temperatures, so much so that it was often used by chemists to determine uncertain molecular masses of elements. Within a decade of the law’s enunciation, however, people began to realize that it was inaccurate for certain solids, generally of a very hard nature, like diamond. Furthermore, in 1875 H.F. Weber, experimenting on boron, carbon, and silicon, demonstrated that the molecular specific heat increases with temperature until reaching a limiting value given by the Dulong-Petit law; for diamond at -50 °C, he found that the molecular specific heat was 0.76. For the entire19th century, each attempt at an explanation of this behavior of certain solids was fruitless; on the contrary, it was quite easy to prove that the law of Dulong and Petit was an immediate consequence of the theorem of equipartition of energy, one of the cornerstones, as we have mentioned, of classical statistical mechanics. Also in contradiction to the theory was the behaviour of certain polyatomic gases, for which Drude had demonstrated in optical experiments that constituent atoms underwent relative oscillations: the number of oscillating degrees of freedom per molecule was greater than 6, and consequently the specific heat should have been greater than that calculated by the law. Einstein noted that it was still inexplicable that, in classical theory, the oscillating electrons of an atom did not contribute to specific heat. The difficulties were overcome when, in 1907, Einstein had the idea to try to apply Planck’s blackbody radiation law, itself based on the refutation of the theorem of equipartition of energy, to molecular specific heats. Fundamentally, Einstein’s considerations were the following. In a solid body one can assume that the atoms are maintained by their mutual actions in fixed positions, about which they can oscillate, with an energy of oscillation equal to their heat capacity. If one supposes that energy varies continuously, a thermodynamics argument already elaborated by 244

P.Petit and P.-L. Dulong, Receherches sur quelques points importants de la Théorie de la Chaleur, in “Journal de physique, de chimie et d’historire naturelle”, 89, 1819, p. 87.

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Boltzmann leads to the Dulong-Petit law, which does not always correspond, as we mentioned, to experimental reality. But, if one supposes that the energy of an oscillating atom can only vary in discrete quantities proportional to the frequency of oscillation, the calculations of classical mechanics and thermodynamics must all be modified. For example, if a gaseous molecule strikes an atom oscillating around its equilibrium position, it can neither impart to it nor remove from it the energy predicted by classical mechanics, but only a whole multiple of the quantum. It follows that if the atom were to have, according to Maxwell’s distribution law, an energy smaller than the quantum, it would remain at rest, and equipartition would not occur. Now, the quantum is fairly small, since for the majority of solids at ordinary temperatures, thermal excitement is enough to supply it: in these conditions we have equipartition and consequently the validity of the Dulong-Petit law. But for very hard bodies, in which the atoms are firmly bound, the quantum of oscillation is too high for thermal excitement to suffice in imparting it to all the atoms: in these cases there is no equipartition, and a deviation from the Dulong-Petit law follows. Likewise, for any body at low temperatures, thermal excitation is insufficient to provide each atom with its quantum of oscillation. For example, if a body is slowly heated from absolute zero, any one of its N oscillators can acquire the energy hȞ; if the body gains two quanta, the number of ways it can acquire the energy increases, because any oscillator can gain both quanta or any two oscillators can each gain one quantum. This pattern follows: increasing the number of quanta given to the body, increases the number of ways in which the oscillators can absorb them; ultimately, the more a body is heated, the easier its energy can increase. It follows that to transfer heat to a body at low temperatures, a greater increase in temperature is needed than if the body could absorb any quantity of energy, however small: in other words, at low temperatures the specific heat is small. Indeed, according to Nernst, for all bodies of temperature below -200oC, the specific heat is basically negligible. In short, Einstein’s theory explains the disagreement between the DulongPetit law and experiment at low temperatures and at ordinary temperatures for very hard bodies with the “freezing” of molecular degrees of freedom, and justifies this by supposing exchanges of energy in quanta. Based on these concepts Einstein, with a straightforward calculation, gave the formula for the variation in atomic heat capacity. In his formula, the atomic heat capacity tends to zero as temperature approaches absolute zero, and tends to a value of 6 calories per degree centigrade with increasing temperature. The Dulong-Petit “constant” of 6 is therefore an

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asymptotic value to which the atomic heat capacities of all elements tend. In a certain sense, the Einsteinian interpretation generalized the rule put forth by Dulong and Petit, which governs all elements without exception, but at variable temperatures from element to element. Einstein’s discovery also had a broader implication: it demonstrated that quantum jumps were not simply a property limited to radiation, but rather inherent in other physical processes. Nernst and his school took on the not too simple task of experimentally verifying Einstein’s formula and toiled for several years, finally (1911) eaching the conclusion that Einstein’s law was qualitatively confirmed for all elements subjected to experiment (silver, zinc, copper, aluminium, mercury, iodine, …), including lead, for which the experimental results obtained in 1905 by Dewar, which found a constant heat capacity at low temperatures, had up until then been taken as valid.

6.13 The third principle of thermodynamics Nernst was particularly interested in Einstein’s theory. Indeed, since 1905 he had held that if quantum theory was correct, the undetermined constant in the ordinary thermodynamic definition of entropy was equal to zero at absolute zero. One consequence of this theorem, now known as the third principle of thermodynamics, concerned precisely the specific heat of solids at low temperatures, because it was easily demonstrated that if Nernst’s theorem was true, the specific heat at absolute zero must be zero. The confirmation of Einstein’s formula also made the third principle of thermodynamics more probable (as Einstein observed to Nernst), which, on the other hand, was also confirmed by many other phenomena. Today, Nernst’s principle is stated in a more concrete formulation, an immediate consequence of the aforementioned original statement: absolute zero cannot be attained with any experimental means. On the contrary, experiments indicated, in Nernst’s words, that “in accordance with quantum theory, for each solid body there exists an interval of temperatures near absolute zero for which the notion of temperature practically disappears.” In simpler terms, in that temperature interval, all properties of the body (volume, thermal dilation, compressibility) are independent of temperature. This range of thermal unresponsiveness varies for different bodies; in diamond, for instance, it stretches no less than 40 degrees from absolute zero, according to Nernst. However, according to the Nernst’s works, Einstein’s formula did not quantitatively agree with experimental results. For example, for copper at 22.5 degrees Kelvin, Einstein’s formula gave a value of 0.023 for the

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atomic heat capacity, while experiments reported a value of 0.223. Based on these results, Nernst and Frederick Alexander Lindemann empirically modified Einstein’s formula, developing another that was much better suited to experimental reality. Few scholars of quantum physics at the time were convinced that the mere qualitative agreement of Einstein’s formula with experiment was an indication of the substantive reliability of the theory, and that only some minor details needed modifying. Einstein himself, on the other hand, asserted that the hypothesis of atoms as point particles was overly simplified. Peter Debye, Max Born, and von Katman, in works beginning in 1912, subsequently supported Einstein’s theory, explained the reasons for the quantitative discrepancies with experiment; extended it to the specific heat of gases, obtaining good agreement with experiment, and established a quantitative theory of the thermodynamics of molecules and crystals. The fundamental modification made by Debye (1914) to Einstein’s theory was the following: Einstein had considered the oscillators in a solid to be independent; Debye more realistically observed that, in a solid, each oscillator is linked to its neighbours, and thus the theory must consider coupled oscillators. A more complicated theory follows, which curiously relates to one of the oldest problems in classical mechanics, the theory of vibrating strings245: a question which is today studied by solid state physics,246 a vast field of modern physics that deals with the properties of matter in the solid, and more specifically crystalline, state, taking into account its composition (§ 10.1).

6.14 Low temperature phenomena Einstein’s theory brought attention to the profound transformations that matter can undergo at low temperatures and gave new impetus to experimental and theoretical research in the laboratories that were particularly well-equipped to produce extremely low temperatures. Those who dedicated themselves to such work included Nernst, Lindemann, Eduard Grüneisen, and others. In particular, Kamerlingh Onnes made a name for himself in this field of study, and starting in 1882, for twelve years he dedicated his time exclusively to the organization of a university 245

§ 7.11 of M.Gliozzi.: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022. 246 In modern usage, the term Solid State Physics has given way to the broader Condensed Matter Physics.

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laboratory in Leiden which he called cryogenic, using an epithet that distinguished it from all others in the world. In this laboratory, Kamerlingh Onnes liquefied hydrogen in 1896 and helium, the last “permanent” gas, in 1908; Keesom, his successor in directing the laboratory, was able to solidify helium in 1926. These physicists, along with others, studied the behaviour of low temperature materials in many respects, like thermal dilation, compression, heat conduction, vapour tension, thermoelectricity and magnetism, discovering a phenomenological world very different from the ordinary one, which are described in noteworthy papers. Yet there is one phenomenon for which it is necessary to at least say a few words: electric conductivity at low temperatures. The production of extremely low temperatures raised the issue of their measurement. Kamerlingh Onnes sought to deduce it from the familiar variation of electric resistance with temperature. After years of work, in 1911 he discovered, to his surprise, that near the temperature of liquid helium (that is, below 4.5 Kelvin), the electric resistance of a platinum wire becomes independent of temperature; on the contrary, at temperatures very close to absolute zero, the electric resistance of certain metals (mercury, lead, tin, thallium, indium) in their purest states virtually disappears. For example, in a lead coil at a temperature below 7.2°K, an electric current, obtained by moving a magnet at its centre, continues to flow for a very long time without appreciable decrease. For the discovery of this phenomenon, which he called superconductivity, he was awarded the Nobel prize in physics. This phenomenon cannot be explained by Lorentz’s electronic theory, which attributes electrical resistance to the collision of moving electrons with atoms; according to Kamerlingh Onnes it is explained, however, by quantum theory. It is enough to assume that obstacles to the movement of electrons in pure metals do not arise from atomic collisions, but rather from the excitement of Planck’s oscillators. Taking this point of view, Kamerlingh Onnes formulated a quantitative theory in fair accord with experimental results. For many years, it was discussed whether superconductivity was a general property of matter or only a characteristic property of certain solid elements. From the experimental work of the Russian physicist Kapitza (1929) on the conductivity of metals in powerful magnetic fields, it seems like one can conclude that superconductivity is a general property of all bodies.

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PHOTONS 6.15 Laws for the photoelectric effect In § 4.9 we saw that, the qualitative nature of the photoelectric effect had been demonstrated since 1899: an expulsion of fast-moving electrons by matter when struck by a beam of radiation. This phenomenon arises as the inverse of X-ray production: schematically, we can say that the collision of electrons with matter produces radiation, and the collision of radiation with matter produces electrons. Owing to its relationship with the most advanced scientific discovery of the time, the study of the photoelectric effect was carefully undertaken by many physicists. In 1902, Lenard experimentally demonstrated that the emission velocity of electrons was independent of the intensity of incident light, but depended only on its frequency, increasing with an increase in frequency; with growing intensity only the number of electrons ejected grew. Furthermore, the phenomenon only occurred when the incident radiation had a frequency greater than a certain threshold, which varied with the nature of the body struck by radiation. These empirical facts were exceptionally difficult to interpret for classical optics. The wave theory of light considered radiated energy to be uniformly distributed in the light wave. Due to conservation of energy, a part of the radiated energy in the photoelectric effect is transformed into kinetic energy of emitted electrons. Consequently, if the incident radiation carries more energy, then emitted electrons should also have more, meaning that they should be faster. On the other hand, how can the energy absorbed by electrons depend on the frequency, or, using an imprecise but more intuitive term, the colour of the light? Classically, it should only be proportional to the intensity of the radiation, and completely independent of its colour. The attempts made in the first years of the century to frame the photoelectric phenomenon within the laws of classical physics were entirely unfruitful; moreover, it was clear that the difficulties were not momentary growing pains that could be surmounted with slight modifications to the theory or some added hypotheses, as had occurred many times in the history of physical thought: they represented a profound and radical clash. Experiment broke the barriers of theory, highlighting the incompatibility between old theory and new experimental facts.

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6.16 Quanta of light The photoelectric effect was among the collection of phenomena unexplained by classical physics that Einstein, in 1905, set out to frame in terms of the recent quantum theory. He demonstrated that the photoelectric effect could be explained in a natural way, down to its most minute details, if quantization, which Planck had limited to emissions, was also extended to radiation: namely, if one supposed that the quantum of energy hQ, once emitted, did not disperse but maintained its own spatially-localized individuality. Einstein’s reflections247 begin with Boltzmann’s relation between entropy and probability, little appreciated at the time and for which Einstein gave a new, simple derivation. He applied it in the opposite sense of its habitual use, considering the probability P as a function of the entropy S: ܲ = ݁ܵ/݇ Taking advantage of the thermodynamic properties of entropy, he was able to demonstrate that at low temperatures the energy of radiation behaves as if it were composed of independent particles of size hQ. “We must imagine,” he explained, “that homogeneous light is composed of […] packets of energy or ‘light quanta’ [Lightquanten], basic particles of energy that travel through empty space at the speed of light.” After the first part of the article, which was of a general nature, Einstein followed with a second part in which the new idea of light quanta was applied to Stoke’s luminescence rule, the photoelectric effect, and gas ionization. In 1923, Compton gave these light quanta the fateful name of photons, which we will henceforth adopt. Many years later, in a volume written in collaboration with Infeld, which became widespread in libraries around the world, Einstein gave a simple interpretation of the photoelectric effect: “It is at once evident that this quantum theory of light explains the photoelectric effect. A shower of photons is falling on a metal plate. The action between radiation and matter consists here of very many single processes in which a photon impinges on the atom and tears out an electron. These single processes are all identical and the extracted electron will have the same energy in every 247

Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, in “Annalen der Physik”, 4th series, 17, 1905, pp. 132-48. The Nobel prize in physics was officially given to Einstein for this work.

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case. We also understand that increasing the intensity of the light means, in our new language, increasing the number of photonic projectiles. In this case, a greater number of electrons would be thrown out of the metal plate, but the energy of any single one would not change. Thus, we see that the theory is in perfect agreement with observation.”248 If to this quantitative explanation we would like to add a quantitative element, we can say that if a packet of light strikes a matter electron, it imparts onto it all of its own energy: the electron uses part of it to overcome the forces that bind it to the matter and transforms the rest into kinetic energy. It follows that the packet of light, or photon, must have at least sufficient energy to free the electron from the matter. Essentially, the following energetic balance must occur: energy of photon = binding energy + kinetic energy of electrons If light is thought of as a collection of photons, with what must we replace the concept of wavelength, the purported cause of our perception of colour? We must replace it, answered Einstein, with the energy of photons. With this substitution, every law of the wave theory is immediately translated into a law of the quantum theory. For example: - Wave theory law: All homogeneous light has a definite wavelength. The wavelength from the red end of the spectrum is twice the wavelength from the violet end. - Quantum theory law: All homogeneous light consists of photons of definite energy. The energy of photons on the red end of the spectrum is half of the energy on the violet end. Two more details of the theory are worth highlighting. The particles of light (photons) are not all of the same magnitude like the particles of electricity, but rather they have variable energy, equal to H=hQ for each photon. Photons are not eternal in time: they are born and they die; they appear in emission and disappear in absorption, therefore the total number of photons in an environment varies with time. It follows that photons are not Newtonian corpuscles, which, though massless, have a material nature. Einstein’s theory was initially met with strong opposition: we have already seen Planck’s heated reaction. Born, who became one of the most astute scholars of quantum mechanics, related that in 1906 in Wroclaw, where Lummer and Pringsheim worked and had contributed ample experimental results to the study of blackbody radiation, “Einstein’s light 248

Einstein and Infeld, The Evolution of Physics, pp. 275-76.

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quanta were not taken seriously.” As a young student, Born himself, influenced by the scientific environment, attempted to classically explain Einstein’s results with a curious mechanical model, which was then picked up by Planck. Suppose there is an apple tree in which each apple has a stem of length l, that is inversely proportional to the square of its height above the ground; the frequency of each apple’s oscillation will then be inversely proportional to the square root of l, and thus proportional to the height of the apple above the ground. If tree is shaken, apples at a certain height go into resonance, detach from the tree, and fall to the ground with a kinetic energy proportional to their height and thus to the frequency of shaking249, in parallel to Einstein’s results. Yet this contrivance was too artificial and convoluted to be convincing. More persuasive, or rather, dissuasive, after a century of fighting against the emission theory of light, was the observation that the theory of light quanta was a return (while only partial) to Newtonian optics. Despite the criticisms, the theory proved to be ever more fruitful. Not only did it explain the photoelectric effect, but it also explained many other classically mysterious phenomena, like Einstein had begun to demonstrate. An experiment repeated by many physicists, and in particular Millikan with great accuracy, was of particular importance: a speck of metallic powder suspended between the plates of a capacitor is struck by X-rays of very weak intensity. Classical electromagnetic optics allows one to calculate the time necessary for the speck to absorb the energy necessary to emit an electron; one can then adjust parameters to set this time to several seconds. Instead, contrary to the prediction of classical physics, the experiment demonstrates that the emission of an electron occurs immediately after the irradiation of the powder with X-rays, where the delay between the two events is certainly below 1/2000th of a second (the immediacy of the photoelectric phenomenon is one of the main reasons for its application in marvellous modern techniques, like sound cinema and television): one must conclude that it is not electromagnetic waves striking the powder, but rather a hail of photons. Yet this experiment had another interesting consequences. Each speck of powder emits, on average, one electron for each interval of time calculated using the classical equations; it follows that the classical theory could still be taken as a statistically valid law when considering the flux of many quanta.

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M. Born, Physics in My Generation, 1956, pp. 354-55.

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6.17 The Compton effect The hypothesis of light quanta, as we have emphasized, greatly embarrassed the theoretical physics of the time. Light particles were certainly not reconcilable with classical optics. Certain optical phenomena could not be explained assuming a granular constitution of light: Young’s double slit experiment, for instance, is inexplicable, as are all interference and diffraction experiments in general. Moreover, Lorentz demonstrated that one cannot explain the laws of angular resolution of optical instruments, supposing light to be made up of quanta. On the other hand, X-ray diffraction experiments and the subsequent successes in the study of crystal composition (§ 4.5) brought new vigour and sway to the wave theory; it was still capable of predicting phenomena that experiment confirmed, the most reliable quality of a physical theory. Why discard it then, when the competing theory was unable to explain the collection of phenomena grouped by classical theory under the name of “physical optics”? By 1920, a curious modus vivendi had been established between the old theory and the new theory. The majority of physicists held that photons were not physically real, but rather a simple heuristic expedient to denote a certain quantity of energy that was perhaps connected with some irregularity of the electromagnetic field. In short, the quantum of light was a unit of measurement, not a sui generis corpuscle. Fundamentally, it was not only the physical reality of the photon that was questioned, but all of quantum theory, which physicists continued to regard with wariness. Some say that the determining factor for the success of the photonic theory was the discovery (1922) of the Compton effect and of the Raman effect the following year. Without a doubt the discovery of these two phenomena had an important role in the theory’s success. Perhaps, though, this judgement is a bit one-sided. It does not take into account the slow maturing of ideas on one hand, and on the other the importance of the (literal) deaths of the old scientists, who remained married, even sentimentally, to the theories of their youth. In our view, the succession of generations is no small factor in scientific evolution. Be that as it may, the Compton phenomenon, or effect, had particular historical importance. It consists of the following. It is known that if radiation strikes a body, part of the energy is generally dispersed in all directions, or scattered, maintaining the same frequency, and therefore colour, of the incident radiation. Lorentz’s theory explained this phenomenon by saying that the electrons of the body struck by radiation

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go into resonance and therefore emit spherical waves of their own that radiate part of the energy of the primary waves in all directions. It is clear that, in this mechanism, the secondary waves must have the same frequency as the primary waves, such that the scattered radiation has the same colour as the incident radiation. This theory proved itself appropriate in describing scattering phenomena of both visible light (in the works of Drude and Lord Rayleigh) and invisible radiation (in the works of Thomson, Debye, and others). In 1922, however, the young American physicist Arthur Compton (1892-1962) demonstrated that in X-ray scattering, alongside the classical occurrence of scattering without a change in frequency, there is also a scattering of radiation with a lower frequency. The two secondary components of the primary radiation have different relative intensities: for longer wavelengths the non-modified portion has more energy, while for short wavelengths the modified component prevails, to the extent that if Ȗrays of very high frequency are used to produce this phenomenon, it is impossible to find any scattered radiation having the original frequency. A careful examination of the modalities of the phenomenon allowed Compton to conclude that it was not caused by fluorescence, as Thomson, Debye, and others had predicted. Here we have another phenomenon that cannot be explained with wave theory, but is given an immediate explanation by photonic theory, as Compton and Debye instantly showed. We can think of X-rays as a shower of light corpuscles, first called photons by Compton. A photon strikes an electron and an exchange of energy ensues. But because the electron can be considered at rest compared to the high velocity photon, it follows that in the collision it is the photon that loses energy, imparting it to the electron. Yet, due to the fundamental hypothesis, the energy of the photon is (inversely) analogous to the wavelength of radiation in the wave theory, and so after the impact the photon loses energy and thus has a greater wavelength. Furthermore, the electron struck, called the recoil electron by Compton, changes its velocity, having acquired part of the energy of the photon. The mathematical theory of the phenomenon, which is treated in the scheme of special relativity, leads to a formula that ties the scattering of the photon to its initial and altered frequencies: experiment, repeated in countless conditions by many physicists, has always confirmed this theory. Indeed, when Compton proposed it, he had not yet been able to observe recoil electrons. But a few months later, Charles Wilson in England and Walther Bothe in Germany experimentally observed them, and later other experimenters confirmed that their number, energies, and

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spatial distributions were in full agreement with the predictions of photonic theory. In his theory, Compton treated photons like real corpuscles and held the principles of conservation of energy and momentum were valid for their collisions with electrons. In addition, the Compton effect involves a photon in the midst of a transformation, while previously it was only known that photons were born and died, and no transformation in their individual lives had been observed. In short, the Compton effect stressed the individual existence of photons, their physical reality. New evidence confirming photonic theory came from the discovery of the Raman effect, named after the Indian physicist Chandrasekhara Venkata Raman (1888-1970), who first observed it, publishing his research in 1928. The first idea, as he recounts, came to him from observing the blue opalescence of the Mediterranean during a journey in Europe in 1921. Why this colour? The study of the laws of diffusion in liquids, which he began the year of his return to Calcutta, could provide an answer to his question. Yet it soon became clear that the question had exceeded the narrow field from which Raman began. The molecular diffraction of light, to which the blue colour of the sky was attributed, had to be studied not only in gases and vapours, but also in liquids and crystalline or amorphous solids. Raman tasked his able collaborators with the study of all of these problems, interpreting the results in light of the theory of electromagnetism and thus under the influence of Lord Rayleigh’s works, which we discussed earlier. In 1923, the meteorologist Kalpathi Ramanathan, a collaborator of Raman, observed that in addition to the molecular diffraction of RayleighEinstein type, there was another, weaker diffraction that differed from the classical one in that the diffracted light did not have the same wavelength as the incident light. Experimental efforts in this direction multiplied, and in 1923, the first unmistakable result was obtained: sunlight scattered inside a sample of highly purified glycerine was bright green instead of the usual blue. An analogous phenomenon was observed in organic vapors, compressed gases, frozen crystals, and also in optical glass. The occurrence was analogous to the Compton effect, but differed in two fundamental details: in the Raman effect, the change in frequency can also be an increase; and the nature of the scattering body critically influences the phenomenon, while it does not affect Compton scattering whatsoever. Yet much like the Compton effect, the Raman effect was also could not be explained by classical theory, while it had a relatively simple explanation in photonic theory: one must simply assume that a photon striking a molecule or atom, depending on the type of collision, will either split into two photons or reflect with a different energy. Intuitively, the

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modalities of this effect must then be closely related to the molecular structure of the body struck by radiation; it comes as no surprise that the study of this phenomenon has led to the resolution of many problems in chemistry and physics. Ultimately, photonic theory explains many phenomena, predicts many others, but leaves numerous classical physics phenomena unexplained; thus, we cannot answer the question that Euclid posed more than twothousand years ago: what is light? We shall see how, in the years of Raman’s discovery, wave mechanics answered the age-old question.

7. THE STRUCTURE OF MATTER

RADIOACTIVE DECAY 7.1 Radioactive transformation Ernest Rutherford (Fig. 7.1) was born to English parents in Spring Grove (later called Brightwater), New Zealand on 30 August 1871; after having received his degree, he went to Cambridge to begin scientific research under the guidance of Thomson, who immediately noticed his unusual intelligence. After his initial experimental work on electromagnetic waves and gas ionization, in 1898 he was appointed professor of physics at the University of McGill in Montreal (Canada), where he continued his studies on radioactivity-induced ionization. His work, among the most advanced experimental research of the first decades of the 20th century, will be briefly presented in the following pages. Here we simply say that, in addition to his scientific merits, Rutherford was also known for his pedagogical excellence, first acquired in Montreal and then honed at the physics institutes of Manchester and Cambridge, which he directed as an enlightened monarch, encouraging the youth, enhancing the equipment, and giving off a vibrant atmosphere of optimism. He died in Cambridge on 19 October 1937. In Montreal, in 1899, his colleague Robert Bowie Owens (1870-1940), who became famous in the United States as an electrical engineer, informed him that thorium radiation was susceptible to air currents. The observation appeared a curious one; Rutherford, following this indication, experimentally demonstrated that radioactivity from thorium compounds, which was noticeably of constant intensity as long as the observation was made in a closed container, rapidly diminishes if the experiments are done in a room. Furthermore, he noticed that the radiation is lifted by weak air currents. Lastly, bodies near the thorium compounds, after some time, also emit radiation themselves as if they are radioactive: Rutherford called this property “excited activity”.

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Fig. 7.1 – Ernest Rutherford.

Rutherford quickly understood that these phenomena would be immediately explained if one assumed that thorium compounds, besides radiating Į particles, also produced other particles which were themselves radioactive. To the substance made up of these last particles he gave the name emanation, likening it to a radioactive gas that, when deposited in extremely thin, invisible layers on the bodies near the thorium (which produces it), imparts apparent radioactivity to the bodies themselves. Guided by this hypothesis, Rutherford, using a simple waft of air that licked the thorium sample, was able to separate radioactive gas from the matter that produced it, driving it into an ionization chamber to study its behaviour and physical properties. In particular, he demonstrated that the intensity of the emanated radiation (later christened thoron, much like radon and actinon were the names given to the radioactive gases of radium and actinium, respectively) falls off rapidly and regularly, halving every minute following a geometrical progression with time: after ten minutes it becomes negligible.

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Meanwhile, the Curies had also discovered that radium possessed the property of exciting neighbouring bodies. They made use of the theory advanced by Becquerel to explain the radioactivity present in precipitates of radioactive solutions, calling the new phenomenon induced radioactivity. More precisely, the Curies held that induced radiation arose from an excitement of the neighboring bodies when they struck by the rays emitted by radium: something akin to phosphorescence, to which the Curies explicitly likened the phenomenon. Besides, even Rutherford, when speaking of “excited” activity in the early days, must have been considering an induction phenomenon, something which 19th century physics was quite prepared to handle. Yet Rutherford knew something that the Curies did not: he knew that the excitement or induction was not a direct effect of thorium, but rather of the emanation, while the Curies still had not discovered the emanation of radium, obtained by Friedrich Ernst Dorn (1848-1916) in 1900, who repeated on radium the work that Rutherford had carried out on thorium. In the same year, Crookes, with his typical youthful vigour (at 68 years of age nonetheless), conducted a noteworthy experiment in the course of which, using fractional crystallization, he obtained an inactive uranium nitrate. This result led him to think that the radioactivity of uranium and its salts “was not a property inherent to the element, but rather due to the presence of an external body” of yet unknown chemical properties and therefore “provisionally” known as uranium X. Independently of Crookes and almost at the same time, Becquerel also obtained a uranium salt of very weak activity, which could even lead one to believe that pure uranium was inactive. But how then did a given commercial uranium salt, whatever its origins and whatever treatments it had undergone, always exhibit the same radioactivity? Is it not better to believe that a uranium salt whose activity has been reduced would regain its original intensity with time? To clear this doubt, in 1901 Becquerel reexamined the weakly radioactive products that he had prepared 18 months prior and found that they had spontaneously regained their lost activity: the hypothesis of uranium’s own activity regained its value, although it could not rule out that uranium was amalgamated with another, very radioactive body, which could not be separated by chemical operations. Yet what was the mechanism through which the body regained the temporarily lost radioactivity? Becquerel answered this embarrassing In the spring of 1900, having published his discovery of emanation, Rutherford suspended his research and returned to New Zealand to marry. Upon returning to Montreal later that year, he met Frederick Soddy (18771956), who had completed a chemistry degree at Oxford in 1898 and had

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come to Montreal in search of an academic position, which he did not obtain. The meeting between the two young scientists was a fortunate event for the history of physics. Rutherford explained to Soddy that he had discovered and separated thoron, showing him a glimpse of the vast and budding field of research, and inviting him to join him in the physical and chemical study of thorium compounds. Soddy accepted. The research occupied the young scientists for two years. Soddy in particular studied the chemical nature of thorium’s emanation, demonstrating that the new gas was completely impervious to any known chemical reaction: one had to consider it, then, part of the family of inert (noble) gases; precisely argon, as he conclusively demonstrated at the beginning of 1901 (today we know it to be an isotope), which Lord Rayleigh and Ramsay had discovered in the air in 1894. A new, critical discovery rewarded the tenacity of the young scientists: along with thorium, there was another element present in their samples which had distinct chemical properties from thorium and a radioactivity almost a thousand times greater. This element was chemically separated from thorium through ammonium precipitation and called thorium X, in analogy with uranium X. The activity of the new element halved in four days, a sufficient time for it be comfortably studied and to recognize that, undoubtedly, the emanation of thorium was not produced by thorium, as it had initially appeared, but by thorium X. If in a sample one separated the thorium X and the thorium, the radioactive intensity of the latter was much smaller than the former, but increased exponentially with time, causing a continuous production of new radioactive matter. In a 1902 paper published by the Transactions of the Chemical Society of London, the two scientists announced the revolutionary theory of the transformation of elements. In a following paper that same year, they reasserted that “radioactivity is at once an atomic phenomenon and accompanied by chemical changes in which new types of matter are produced, these changes must be occurring within the atom, and the radioactive elements must be undergoing spontaneous transformation… question with a barely sketched theory of atomic transformations. Some months earlier, Perrin had likened each atom to a planetary system, from which certain negatively charged particles could escape. These were the first vague indications of a process that would be clarified in a few years. In the spring of 1900, having published his discovery of emanation, Rutherford suspended his research and returned to New Zealand to marry. Upon returning to Montreal later that year, he met Frederick Soddy (18771956), who had completed a chemistry degree at Oxford in 1898 and had come to Montreal in search of an academic position, which he did not

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obtain. The meeting between the two young scientists was a fortunate event for the history of physics. Rutherford explained to Soddy that he had discovered and separated thoron, showing him a glimpse of the vast and budding field of research, and inviting him to join him in the physical and chemical study of thorium compounds. Soddy accepted. The research occupied the young scientists for two years. Soddy in particular studied the chemical nature of thorium’s emanation, demonstrating that the new gas was completely impervious to any known chemical reaction: one had to consider it, then, part of the family of inert (noble) gases; precisely argon, as he conclusively demonstrated at the beginning of 1901(today we know it to be an isotope), which Lord Rayleigh and Ramsay had discovered in the air in 1894. A new, critical discovery rewarded the tenacity of the young scientists: along with thorium, there was another element present in their samples which had distinct chemical properties from thorium and a radioactivity almost a thousand times greater. This element was chemically separated from thorium through ammonium precipitation and called thorium X, in analogy with uranium X. The activity of the new element halved in four days, a sufficient time for it be comfortably studied and to recognize that, undoubtedly, the emanation of thorium was not produced by thorium, as it had initially appeared, but by thorium X. If in a sample one separated the thorium X and the thorium, the radioactive intensity of the latter was much smaller than the former, but increased exponentially with time, causing a continuous production of new radioactive matter. In a 1902 paper published by the Transactions of the Chemical Society of London, the two scientists announced the revolutionary theory of the transformation of elements. In a following paper that same year, they reasserted that “radioactivity is at once an atomic phenomenon and accompanied by chemical changes in which new types of matter are produced, these changes must be occurring within the atom, and the radioactive elements must be undergoing spontaneous transformation… Radioactivity may therefore be considered as a manifestation of subatomic chemical change.”250 Moreover, the following year they wrote: “The radio-elements possess of all elements the heaviest atomic weight. This is indeed their sole common chemical characteristic. The disintegration of the atom and the expulsion of heavy charged particles of the same order of mass as the hydrogen atom leaves behind a new system lighter than before, and possessing chemical and physical properties quite different from those of the original element. The disintegration process, once 250

“Philosophical Magazine”, 6th series, 4, 1902, p. 395.

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started, proceeds from stage to stage with definite measurable velocities in each case. At each stage one or more Į ‘rays’ are projected, until the last stages are reached, when the ȕ ‘ray’ or electron is expelled. It seems advisable to possess a special name for these now numerous atomfragments, or new atoms, which result from the original atom after the ray has been expelled, and which remain in existence only a limited time, continually undergoing further change. Their instability is their chief characteristic. On the one hand, it prevents the quantity accumulating, and in consequence it is hardly likely that they can ever be investigated by the ordinary methods. On the other, the instability and consequent rayexpulsion furnishes the means whereby they can be investigated. We would therefore suggest the term metabolon for this purpose.”251 The term did not catch on, because this first cautious theory was quickly corrected by the two young scientists, who clarified certain ambiguous points that the reader may have noticed. The corrected theory had no need for the new term and ten years later was described by one of the two, who had by then become a world-famous scientist and Nobel prize winner, as follows: “The atoms of a radioactive substance undergo a spontaneous transformation. A minute fraction of the total number of atoms is supposed each second to become unstable, breaking up with explosive violence. In most cases, a fragment of the atom – an Į-particle – is ejected at a high speed, while in some other cases the explosion is accompanied by the expulsion of a high speed electron or the appearance of Röntgen rays of very penetrating nature, known as Ȗ-rays. Radiation accompanies the transformation of atoms and serves as a direct measure of the degree of their disintegration. It has been found that the transformation of an atom leads to the production of an entirely new type of matter totally different, in physical and chemical properties, from the original substance. This new substance is, in turn, unstable and is transformed by the emission of radiation of a characteristic type… It has thus been clearly established that atoms of certain elements undergo spontaneous disintegration, accompanied by an enormous emission of energy, relative to that released in ordinary molecular transformations.”252 In the referenced 1903 paper, Rutherford and Soddy compiled the following table of “metabolons” which, admitting their theory, could be regarded as products of disintegration, based on their own and others’ experiments:

251 252

Ibid, 5, 1903, p. 586 E. Rutherford, The Structure of the Atom, in “Scientia”, 16, 1914, p. 339.

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uranium uranium X ?

thorium thorium X thorium emanation thorium I thorium II ?

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radium radium emanation radium I radium II radium III ?

These were the first “family trees” of radioactive substances. Gradually, other members joined these families of naturally radioactive elements, and three distinct families were defined, of which two had uranium as a “progenitor” while the third had thorium. The first counted 14 “descendants”, namely 14 elements obtained one from the other in successive disintegrations, the seconds counted 10 and the third 11: now, many physics texts reproduce these family trees in full. At this point, we indulge in a brief digression: nowadays the conclusion that Rutherford and Soddy drew from their experiments may seem entirely natural, even obvious in hindsight. What did it basically entail? In essence, after some time a new element appeared mixed in with the original fresh thorium, and this new element in turn produced a gas that was also radioactive. The genesis of new elements was truly before one’s eyes. It was before one’s eyes, but not quite literally. We must keep in mind that the quantities of the new elements were still far from reaching the minimum amounts necessary for the most delicate chemical analyses of the time; scientists were working with the most minute traces, perceptible only using radioactive methods of detection, photography, and ionization. These effects could be explained in other ways (like induction, or their presence in the prepared samples from the start, as had occurred for the discovery of radium). That disintegration was not the first conclusion drawn from the experiments is evident if one looks to Crookes, who from analogous experiments deduced that uranium was not radioactive, or the Curies, who did not even stop to discuss the idea of atomic transformations, which was suggested almost as a stutter by Perrin and Becquerel. Much courage was required in 1903, with atomic theory reigning utterly triumphant, to speak of transformation of elements: it was a hypothesis certainly not shielded from criticism, and perhaps would have fallen if Rutherford and Soddy had not, with admirable consistency, continued to support it for decades, backing it up with numerous pieces of evidence that in part we will see shortly. It is opportune to add that the theory of radioactive induction also rendered great services to science, avoiding a waste of resources searching

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for new radioactive elements each time that radioactive effects in inactive elements were discovered. Let us momentarily return, however, to the 1903 paper by Rutherford and Soddy, whose last section is dedicated to the calculation of the energy emitted during radioactive transformations. The computation is carried out supposing that during a transformation, each atom emits at least one “beam” of mass m, velocity v, and charge e, and therefore of kinetic energy

1݉ ଶ 1 ݉‫ ݒ‬ଶ = ‫ = ݁ ݒ‬2 ή 10ିଶ଴ 2݁ 2 in electromagnetic units, having assumed for m/e and e the values then known for ȕ-rays. Taking into account of the number of atoms in a gram of radium, the five successive transformations then known, and the number of ions produced by the radioactive process per second, the two scientists arrived at the conclusion that a gram of radium produces 15,000 calories a year. Some time earlier, Pierre Curie and Laborde had measured (§ 4.8) 100 calories per hour (= 880,000 calories per year); but since they were still drafting their paper, Rutherford and Soddy could not have known of the French scientists’ result. In 1907, when Rutherford left Montreal to accept a professorship at the University of Manchester, the theory of disintegration had been generally accepted by physicists, with the exception of the usual skeptics who, faithfully guarding old ideas, neither die nor surrender.

7.2 The Nature of Į-rays A critical point of the theory of disintegration, or decay, which we have thus far overlooked in the interest of linearity, is the nature of the Įrays emitted by radioactive substances. Being slow and easily absorbed by matter, after their discovery Į-rays had not attracted the attention of physicists, which was concentrated on the faster, preferred ȕ-rays, one hundred times more penetrating than Į-rays. The minor importance given to Į-rays by physicists was in part due to observational difficulties, as also follows from the nomenclature then in use: scientists spoke of “deviable rays” and “non-deviable rays” in a magnetic field; by “deviable” they curiously meant ȕ-rays and J-rays (§ 4.7). Furthermore, Becquerel, Villard, Dorn, and others, by analogy with the production of X-rays, were inclined to believe that the radiation of

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radioactive substances was mainly a non-deviable one that upon striking a body, in some way provokes the emission of deviable radiation (namely a ȕ particle), which in turn leaves behind an Į particle. In summary, Į-rays were thought of as secondary radiation produced by ȕ-rays. The chemical change therefore precedes the radioactivity. Rutherford and Soddy, on the other hand, took another point of view: radioactivity accompanies the chemical change. This was the key part of Rutherford and Soddy’s work, leading to the great importance attached by Rutherford to the study of Įrays, one of the main factors for his success. In 1900, Robert Rayleigh (Robert Strutt, son of John William) and, independently, Crookes had advanced the hypothesis that Į-rays were positively charged, without any experimental confirmation. For two years Rutherford attempted to observe the deflection of the rays in a magnetic field, invariably reaching uncertain results. Only at the end of 1902, having been able to procure a respectable quantity of radium through the good offices of Pierre Curie, could he observe their deflection, both in a magnetic field and an electric field. The observed deflection was such to report a positive charge for the Į particle; based on the extent of the deflection, Rutherford deduced that the velocity of the particle is about half that of light (a value that was subsequently reduced to roughly a tenth); the ratio e/m resulted equal to around 6000 electromagnetic units. It followed that if the charge carried by the Į particle was the elementary charge, the mass of the particle would be roughly twice that of a hydrogen atom. Rutherford realized that these results were only a rough indication, but a qualitative conclusion could still be drawn: Į-rays have a mass of the order of atomic masses and are therefore comparable to the anode rays observed by Goldstein, but of considerably higher velocity. This result, said Rutherford, “projected a torrent of light onto the radioactive process,” and we have seen its reverberation in the quoted excepts from his papers written in collaboration with Soddy. In 1903, Marie Curie confirmed Rutherford’s discovery with an apparatus that today is described in all physics texts, in which, utilizing the scintillation produced by all the rays emitted by radium, one can simultaneously observe the opposite deviations of Į and ȕ-rays and the lack of deviation of Ȗ-rays in electric and magnetic fields. The theory of radioactive decay led Rutherford and Soddy to the view that each stable substance produced during the transformation of radioactive elements has to be present in radioactive minerals, where the transformations have been occurring for millennia. Could the helium that Ramsay and Morris Travers had found in uranium minerals then be a

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product of radioactive transformations? At the beginning of 1903, radioactivity studies received a sudden and vigorous impulse from the fact that Giesel from the Chininfabrik in Braunschweig was able to put pure radium compounds, like RaBr2, which contained 50 percent of the pure element by weight, on the market at very cheap prices (10 marks and later 20 marks a milligram). Up until then scientists had had to work with compounds that, at best, contained 0.1 percent radium! Soddy, who meanwhile had returned to London to continue the study of the properties of emanation in Ramsay’s chemical laboratory, the only one in the world equipped for such research, bought 20 milligrams of this commercial compound, a quantity that was sufficient for him to spectroscopically show in 1903, together with Ramsay, that helium is present in several month old radium (that is, prepared several months prior), and that emanation produces helium. Where did helium then fit in the scheme of radioactive transformations? Was it the final product of a transformation of radium or was it a product of some intermediate stage of its evolution? Rutherford guessed that helium is supplied by Į-rays; to be precise, that every Į is a helium atom with two positive charges. Yet it took years of experimental work to demonstrate this property. Only when he and Geiger devised a counter for Į particles (§ 6.8) was a demonstration found: measuring the charge carried by each Į particle and finding the ratio e/m, it immediately resulted that m was equal to the mass of a helium atom. Nevertheless, all of these experiments and calculations still did not provide definitive proof that the identity of Į-rays was that of helium ions. Indeed, if a helium atom was released simultaneously with the emission of an Į particle, for example, the experiments and calculations would give the same results, but the Į particle could be a hydrogen atom or another unknown substance. Rutherford understood this possible criticism and, to dismiss it, gave a definitive demonstration in 1908, along with Thomas Royds. Using an apparatus that collected the Į particles radiated by radon and channeled them through a tube for spectral analysis, he observed the characteristic spectrum of helium. From 1908 then, there was no longer any doubt that Į-rays are helium ions and that helium is a constituent of naturally radioactive bodies. Before leaving this matter, we add that some years after the identification of helium in uranium minerals, the American chemist Boltwood, from an examination of minerals containing uranium and thorium, deduced that the last non-radioactive product of the successive transformations of uranium is lead, and in addition that radium and actinium were products of

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uranium’s decay. The table of Rutherford and Soddy’s “metabolons” therefore had to be profoundly modified, as we have already noted. A new and interesting study arose from the theory of atomic disintegration: because radioactive transformations occur in accordance with rigid laws, unchanged by any physical agent known before 1930, from the ratio between the quantities of uranium and lead or helium present in a uranium mineral one can deduce the age of the mineral, and by extension, the age of the Earth. An initial calculation gave a value of 1.8 billion years, but Jean Joly (1857-1933) and Robert Rayleigh (18751947), who conducted important studies on the subject, believed this value to be an overestimate. Today, the age of uranium minerals is calculated to be of the order of 1.5 billion years, not very far from the initial estimate.

7.3 The fundamental law of radioactivity We have already mentioned that Rutherford had experimentally discovered that the radioactivity of thorium’s emanation exponentially decreased with time and was halved after about a minute. All of the radioactive substances examined by Rutherford and others qualitatively followed the same law, but each with their own half-life. This experimental fact is expressed with a simple formula253 that relates N0, the number of existing radioactive atoms at time 0, and N, the number of radioactive atoms still intact at time t. The law can also be expressed by saying that the percentage of disintegrated atoms in each time interval is a constant specific to the element in question, called the decay constant, and its reciprocal the mean lifetime. 253

The formula was first introduced by Rutherford (A Radioactive Substance Emitted from Thorium Compounds, in “Philosophical Magazine”, 5th series, 49, 1900, pp. 1-14) in a rather different context, and was studied by Rutherford and Soddy in the field of decay theory in the referenced 1903 paper (ibid, 6th series, 5, 1903, pp. 576-91). It is the following: ܰ = ܰ଴ ݁ ିO௧ where O is the decay constant and its inverse is the mean lifetime of the element; the time required for the number of atoms to be reduced by half is called the period or half-life. After a mean lifetime, the initial quantity of radioactive substance, that is the original number of atoms, is reduced to 1/e = 0.368. One can see that the mean lifetime and the half-life are proportional; to be precise, the mean lifetime is 1.445 times the half-life. As we have said, Ȝ varies much from one element to another, and therefore all of the other properties that we have just defined also vary. For example, the mean lifetime of uranium I is 6.6 million years; for actinium A it is three thousandths of a second.

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Until 1930, no known agent was capable of affecting the natural rate of the phenomenon at all. Since 1902, Rutherford, Soddy, and later several other physicists placed radioactive bodies in a wide range of physical conditions, never succeeding in changing the decay constant. “Radioactivity, according to present knowledge,” wrote Rutherford and Soddy, “must be regarded as the result of a process which lies wholly outside the sphere of known controllable forces, and cannot be created, altered, or destroyed.”254 The mean lifetime is a well-defined, immutable constant for each element, but the individual lifetime of an atom of that element is entirely indeterminate; furthermore, the mean lifetime does not diminish with time: it remains the same for both a group of recently formed atoms and another formed in a different geological epoch. In short, employing an anthropomorphic metaphor, the atoms of radioactive elements die, but do not age. At its core, the fundamental law of radioactivity proved to be entirely incomprehensible. From the preceding characteristics one can see, as scientists did at the time, that the law of radioactivity is a probabilistic law; precisely, it said that the probability of decay in a given instant is the same for all the radioactive atoms present. It is then a statistical law, increasing in accuracy with an increasing number of atoms considered. If external causes affected the phenomenon of radioactivity, the interpretation of the law would be fairly simple, as in this case the atoms transformed in a given instant would be those which happened to be in a particularly favourable condition for the action of the external cause. Thermal agitation could then provide an explanation for the particular conditions of the atoms that decay. In short, the statistical law of radioactivity would assume the same meaning as the statistical laws of classical physics, considered to be aggregates of the individual laws of dynamics, for which a statistical consideration is necessary only due to the multitude of entities involved. However, experiments yielded no possibilities of retracing the statistical law to individual laws determined by external causes. Once external causes were ruled out, scientists tried to find the reasons for the transformations inside the very atoms that were transformed. “Since among a great number of atoms present, some are immediately destroyed, while others can exist for a very long time,” wrote Marie Curie, “it is no longer possible to consider all the atoms of a single simple substance as entirely similar, but we must admit that the difference in their destinies is 254

“Philosophical Magazine”, 6th series, 5, 1903, p. 583.

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determined by individual differences. A new difficulty then presents itself: the differences that we introduce must be such that they cannot determine that which can be called the ageing of the substance; they must be such that the probability that an atom lasts another given amount of time is independent from the time for which it has already lived. Every theory of atomic structure must satisfy this condition, if we take the preceding point of view.”255 Following Marie Curie’s line of thought was her student Debierne, who supposed that every radioactive atom passes rapidly and continuously through many different states, such that the average state remains constant and independent of external conditions. It follows that, on average, all the atoms of the same species have the same properties and equal probabilities of decay, caused, from time to time, by an unstable state assumed by an atom. Yet the constant probability of decay entailed an atom of great complexity, made up of many components in disordered motion. This intra-atomic excitation, limited to the central part of the atom, could lead to the definition of an internal temperature of the atom, much greater than the external one. The speculations of Marie Curie and her disciple, not backed by any experimental results and consequently sterile, did not lead to anything. We have recalled and highlighted them because this fruitless attempt to apply a classical interpretation to the law of radioactive decay was the first, or at least the most convincing example of a statistical law that did not come from laws of an individual nature. A new form of statistical law arose, one which applied directly to the collective, irrespective of the evolution of its individual components: this concept was to become clear a little more than a decade after the vain efforts of Curie and Debierne.

7.4 Radioactive isotopes In the first half of the 19th century, chemists, in particular JeanBaptiste Dumas, had indicated certain relations between the atomic weights of elements and their chemical and physical properties. This research was furthered by the Russian chemist Mendeleev, who in 1896 published his revolutionary theory on the periodic table of the elements, one of the most powerful syntheses in chemistry. Mendeleev organized the known elements in order of increasing atomic weight. Here are the first ones, along with the atomic weights known at the time: 255

Rapport et discussion du Conseil Solvay tenu a Bruxelles du 27 au 31 avril 1913, Gauthiers-Villars, Paris 1921, pp. 68-69.

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7 Li; 9.4 Be; 11B; 12C; 14 N; 16 O; 19 F 23 Na; 24 Mg; 27.4 Al; 28 Si; 31 P; 32 S; 35.5 Cl Mendeleev noticed that the physical and chemical properties of the elements are periodic functions of their atomic weights. For example, in the first row of the elements above the density grows regularly with atomic weight, reaches a maximum near the halfway point of the series, and then decreases; the same periodicity, although not equally marked, is found with other physical and chemical properties (fusibility, dilatability, conductivity, oxidizability) of elements contained in both the first and second rows. These variations occur with the same pattern in the elements for the two rows, such that elements in the same column (Li and Na, Be and Mg, etc.) display analogous chemical character. The two series written above are called two periods: all the elements can be thus ordered in periods having the characteristics to which we have referred. The law of Mendeleev follows: the properties of elements are periodically related to their atomic weights. Mendeleev’s table was missing the noble gases (helium, neon, argon, krypton, xenon, radon) which had not yet been discovered in his time. However, the existence of other new elements that had not yet been isolated was predicted, and later confirmed by their discovery (scandium, gallium, germanium, polonium, rhenium, technetium). We need not continue to trace the ulterior complications of the periodic table, the lively criticism it generated, and its progressive confirmation by the valuable services that it provided to science. It is enough to remember that by the end of the 19th century, it had become almost universally accepted by chemists, who took it as an empirical fact since all attempts at a theoretical interpretation had been fruitless. In the first years of the 20th century, the manufacturers of Ceylon gems discovered a new mineral, thorianite, which today is known to be a mineral of thorium and uranium. Several tons of thorianite were shipped to England for examination; but in the initial analyses, due to an error that Soddy attributed to a well-known German treatise on analytical chemistry, thorium was confused with zirconium, and so the material, thought to be solely a uranium mineral, was treated with the method used by Marie Curies to separate radium from pitchblende. In 1905, having used this procedure, the young German chemist Otto Hahn (1879-1968) traveled to Ramsay’s laboratory in London, more to learn English than radioactivity, and obtained a substance that in chemical analyses seemed thorium, but differed due to a much stronger radioactivity; like thorium, it produced thorium X, thoron, and other radioactive elements.

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The intense radioactivity revealed the presence of a new radioactive element, yet to be chemically defined, in the resulting substance. He called it radiothorium and it was immediately clear that it was an element of thorium’s disintegration series, which, although it had escaped the previous analyses of Rutherford and Soddy, belonged between thorium and thorium X. The mean lifetime of radiothorium was found to be around two years, long enough that radiothorium could be used as a substitute for very expensive radium in laboratories. Besides scientific interests, this economic reason also pushed many chemists to attempt (in vain) its separation. No chemical process was able to separate it from thorium. Quite the opposite, in 1907 the problem seemed to grow more complicated, because Hahn discovered mesothorium, the generator of radiothorium, which itself was also inseparable from thorium. The American chemists Herbert McCoy and W.H. Ross had the audacity to interpret the lack of their own and others’ success as an inherent impossibility, but to contemporaries this seemed a convenient excuse. Meanwhile, in the period from 1907 to 1910, other cases of radio-elements that could not be separated from other radioactive elements were observed; the most typical examples being thorium and iodine, mesothorium I and radium, and radium D and lead. The already muddled picture was further complicated by the proliferation of alleged discoveries of new elements: a veritable obsession with the discovery of “a new element” had possessed researchers. Hahn recounts that while working on radiothorium, he happened to routinely find, in certain circumstances, a small quantity of a new precipitate. He informed Ramsay, the director of the institute and far from an expert on radioactivity, who immediately believed it to be a new element, for which he wanted to give a brief announcement to the Royal Society. But Hahn, having some qualms, pleaded him to hold off a bit longer on the communication. In this way, he was able to discover that the new element suspected by Ramsay originated from the dust that fell from the ceiling of the basement laboratory, which was made of glass reinforced by rusted iron bars.

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Fig. 7.2 - Diagram drawn by Soddy in 1913 to illustrated the law of displacements in radioactive transformations. The ordinate axis shows atomic masses and the abscissa shows the order of the element in the periodic system (atomic number). Source: F. Soddy, The Origins and the Conceptions of Isotopes, in Les prix Nobel en 1921-1922, Santesson, Stockholm 1923.

Certain chemists likened the inseparability of the new radio-elements to the case of rare earth metals, which had arisen in the course of the 19th century. Initially, the similar chemical properties of rare earth metals led all of the elements to be considered homogeneous, and only later, gradually, were scientists able to separate them with more refined chemical methods. Yet Soddy objected that the analogy was forced: in the case of rare earth metals the difficulty did not lie in separating the elements, but in noticing the separation; on the other hand, in the case of radio-elements the difference between the two was evident from the start, but they could not be separated. In 1911, Soddy systematically examined a commercial sample of mesothorium that also contained radium and found that it was impossible to increase the content of one element or the other, even resorting to innumerable fractional crystallizations. He concluded that two elements can have different radioactive properties and, nonetheless, other physical and chemical properties so identical that the elements prove inseparable

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through ordinary chemical procedures. If two such elements have identical chemical properties, they must be placed in the same position within the periodic classification of elements: therefore he called them isotopes. Having realized this fundamental concept, Soddy attempted a theoretical interpretation of it, formulating the “law of radioactive displacements”: the emission of an Į particle leads to a shift of the element two spots to the left in the periodic table; but the transformed element can later return, by emitting two ȕ particles, to the same square of its original generating element, thus the two elements will have identical chemical properties despite the different atomic masses (Fig. 7.2). However, in 1911 the chemical properties of the radioactive elements that emitted ȕ-rays, almost all having short lifetimes, were still poorly understood, so before admitting this interpretation, the character of ȕ-rays emitters first had to be clarified. Soddy entrusted this task to his assistant Alexander Fleck. The work required much time, and the two assistants to Rutherford, Smith Russell and Joseph George von Hevesey also participated; later Casimir Fajans joined as well. The work was completed in 1913 and Soddy’s law was demonstrated with no exceptions. It could be stated in a very simple manner: the emission of an Į particle reduces the atomic mass of an element by four units and shifts the element two spots to the left in the periodic table; the emission of a ȕ particle does not noticeably change the mass of an element, but shifts it one spot to the right in the periodic table. Therefore, if a transformation generated by an emission of Į-rays is followed by two consecutive transformations with the emission of ȕ-rays, the element returns after the three transformations to its initial position and assumes identical chemical properties to the generating element, while having a lower atomic mass by four units. Through this mechanism, it is also clear that isotopes of two different elements can have the same atomic mass and different chemical properties. Steward called such elements isobars.

NON-QUANTUM ATOMIC MODELS 7.5 First ideas regarding the complexity of atoms The second part of the paper by J.J. Thomson in which the method of paragraph 11.3 is explained is dedicated to considerations about the structure of matter. It resulted from the first experiments, as we have said, that “corpuscles” had a mass that was certainly smaller than that of the simplest atom, which however for Thomson was not lower in order of magnitude than one tenth or one hundredth the mass of the hydrogen atom.

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To him the hypothesis advanced in 1815 by William Prout appeared therefore acceptable, according to which the different chemical elements were different collections of atoms of the same nature; substituting the hydrogen atoms hypothesized by Prout with “corpuscles”, the theory of the primordial element was in agreement with experimental results and with the hypothesis recently proposed by Norman Lockyer to explain the characteristics of stellar spectra. Yet how did the “corpuscles” that formed the atoms stay together? According to Thomson, one could adopt the Boscovich model or even simply the model “of a number of mutually repulsive particles, held together by a central force”. Unfortunately, the theoretical study of such a collection of particles increases so rapidly in complexity with the number of particles to become practically impossible; thus it is preferable to resort to models to visualize the possible structure of these atoms composed of corpuscles. In 1878-79, the American physicist Alfred Marshall Mayer (18361897) published several short papers, immediately republished by the English journals “Nature” and “Philosophical Magazine”, that sparked much interest: William Thomson praised them in another “Nature” article that same year and used their results for his vortex atom theory. Mayer placed small, floating cork cylinders in a bowl full of water, each vertically skewered by a magnetic needle whose tip barely protruded from the top of the cork; all needle tips had the same polarity. Over these floating magnets, a vertical magnetic cylinder was placed with its opposite pole at a distance of 60 cm from the tips of the needles. Subjected to both mutual repulsion among themselves and attraction to the large magnet, the floating magnets spontaneously arranged themselves in different equilibrium configurations. Mayer distinguished the configurations he obtained by numbers of needles in groups or classes (primary, secondary, tertiary…), and observed that the stable configuration of one class formed the “nucleus” of successive ones. The same number of needles can produce many configurations, one of which is more stable than the others: vibrations of the external magnet, arising as a result of its vertical motion, cause the transitions from one configuration to another. For example, with 29 total floating magnets, one remains at the centre,surrounded by a first ring of 6 magnets, a second wider ring of 10 magnets, and an external third ring of 12; or one can have a more stable configuration that consists of four rings which, from interior to exterior, contain 1, 6, 9, and 13 magnets. The behavior of the floating needles could be taken as a model, according to Mayer, for molecular structures that explained the real behavior of bodies: for example, increases in volume during certain solidifying

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processes, allotropy, and isomerism.256 This floating magnet model was revived by Thomson not as a model for the arrangement of atoms in a molecule, as Mayer had proposed, but as a model for the atom, with the number of magnets proportional to its atomic weight. It immediately followed that if some property depended, for example, on the number of magnets in the innermost ring, this type of ring could be repeated in other models. In this way, this model provided the beginnings of a rational explanation for the periodic classification of the elements:257 quite a seducing prospect for a physicist at the end of the 19th century! There is no doubt that Mayer’s experiments, reintroduced by Thomson and his school, had considerable influence on Rutherford’s thinking. Nevertheless, the reliability of this atomic model necessitated that the number of “corpuscles” making up the atom be of the order of ten; however, if for the composition of the simplest atom one supposed a cloud of thousands of corpuscles, the model would no longer offer anything to experimental or theoretical study. As Thomson and others established that “corpuscles” (which from now on we will refer to as electrons) had a much smaller mass than that of the hydrogen atom, the English scientist was forced (1899) to modify his model in the following way: he postulated that neutral atoms contain a great number of electrons, whose negative charge is compensated by “something which causes the space through which the corpuscles are spread to act as if it had a charge of positive electricity equal in amount to the sum of the negative charges on the corpuscles.”258 Thomson’s second atomic model was still conceptually muddled, but compared to the first represented great progress because the atom was no longer composed only of electrons, but of electrons and “something else” (still unspecified) which acted as a coagulant. 256

A.M. Mayer, A Note on Experiments with Floating Magnets; Showing the Motions and Arrangements in a Plane of Freely Moving Bodies, Acted on by Forces of Attraction and Repulsion; and Serving in the Study of the Directions and Motions of the Lines of Magnetic Force, in “The American Journal of Science and Arts”, 3rd series, 15, 1878, pp. 276-77; On the Morphological Laws of the Configuration Formed by Magnets Floating Vertically and Subjected to the Attraction of a Superposed Magnet; with Notes on Some of Phenomena in Molecular Structure Which These Experiments May Serve to Explain and Illustrate, ibid., pp. 477-48; 3rd series, 16, 1879, pp. 247-56. 257 J.J. Thomson, Cathode Rays, in “Philosophical Magazine”, 5th series, 44, 1897, p. 293. 258 J.J. Thomson, On the Masses of the Ions in Gases at Low Pressures, in “Philosophical Magazine”, 5th series, 48, 1899, p. 547

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More explicit, although still following the trail of Thomson, was Perrin, who at a conference on 16 February 1901, which unfortunately was not publicized, hypothesized that the atom was composed of “one or more masses very strongly charged with positive electricity” and “a multitude of corpuscles, small negative planets of a sort […] having a total negative charge equivalent to the total positive charge.”259 Perrin, however, did not specify the position of the “corpuscles” relative to the mass or masses of positive charge.

7.6 Thomson’s atom-fragments Thomson’s second model was adopted by Lord Kelvin, whose great authority certainly impacted the future fortunes of the model. His study, which appeared in 1901,260 supposed that the negative electric fluid was composed of electrons (which Kelvin, perhaps on firmer etymological grounds, called electrions, in analogy with the term ion), permeating freely not only though empty atomless space, but also through atoms themselves. Kelvin further postulated an attraction between an atom of ordinary matter and an electron governed by the following law: the attraction of an atom on an external electron is inversely proportional to the square of the distance of their respective centres; on an electron inside the atom, the attraction is proportional to the distance between the two centres. This law, arising from a famous theorem of Newton, is equivalent to assuming a uniform distribution of positive electricity in the space occupied by the ordinary matter atom. It followed that there were two types of electricity: the negative kind, which was granular; and the positive kind, which was vaguely thought to be of continuous constitution, as fluids and in particular ether were imagined. Ordinary uncharged matter is then a collection of atoms inside which enough electrons are mingled to cancel any external electric forces. Fundamentally, Lord Kelvin’s model consisted of a uniform spherical distribution of positive electric charge in which a certain number of electrons were intermingled, like raisins in plum pudding. From Lord Kelvin’s paper, it is not clear whether he thought that the so-called matter acted as a background for the electric charge, or if the two were one and the same: this ambiguity would propagate to later atomic models. 259

J. Perrin, Œuvres scientifiques, Centre national de la recherche scientifique, Paris 1950, pp. 165-67. 260 Lord Kelvin, Aepinus Atomized, published for the first time in the celebratory volume in honour of Johannis Bosscha (1831-1911), and republished in “Philosophical Magazine”, 6th series, 4, 1902, pp. 257-83.

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In any case, the simplest “plum pudding” model of an atom was made up of a uniform spherical distribution of positive charge with a single electron at the centre. Stability problems arose for atoms with two or more electrons, which Lord Kelvin mentioned without fully discussing, concluding that additional electrons were probably arranged in concentric spherical surfaces inside the atom. In the model, an electron shifted from its equilibrium position by any cause, called “excited”, can oscillate and even rotate around the centre. The electron oscillations produce electromagnetic waves with characteristic frequencies, explaining the presence of spectral lines according to the hypothesis proposed by Stoney in 1891. Thomson began his theoretical studies looking at these spectra, developing and maintaining his loyalty to the Kelvin model for almost fifteen years, a model which later came to be called the “Thomson atom”, or less frequently the “Kelvin-Thomson atom” by physicists in the first two decades of the century. Thomson immediately recognized261 that electrons can also rotate around the atom’s centre, producing elliptically polarized light waves. As far as the magnetic field produced by rotating charges, it results from classical theory that electrons rotating due to a force proportional to distance cannot explain the magnetic properties of objects, as this phenomenon occurs without dissipation of energy. However, assuming the electron motion to be quenched by an unknown force, one could perhaps explain paramagnetism. Yet was energy dissipated? Thomson did not answer the question, knowing well that an answer could have led to grave consequences. The year before, Voigt too had attempted to explain paraand diamagnetism with an equally complex hypothesis involving rotating electrons obstructed in their motion by continuous collisions. Contrary to the conclusions of Voigt and Thomson, Langevin, in a very important 1905 paper that is still drawn from by those who study magnetism, believed it possible to give Ampère’s hypothesis a precise interpretation if one supposed that electrons moved in closed trajectories. In addition, Langevin demonstrated that the Zeeman effect could be explained by the hypothesis of rotating electrons, even ignoring the attractive force that keeps them in orbit. The mechanics of atomic structures that was established was studied by Thomson in a following paper in 1904. He concluded that electrons have to be in rapid rotational motion with velocities no lower than a 261

J.J. Thomson, The Magnetic Properties of Systems of Corpuscles Describing Circular Orbits, in “Philosophical Magazine”, 6th series, 6, 1903, p. 673.

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certain limiting value; that if there are numerous (more than eight) electrons, they are arranged in multiple rings (connecting the structure of atoms to the periodic table of elements); and lastly, that the number of electrons in each ring diminishes with the ring’s radius. In radioactive atoms, as a consequence of their radiation, the velocity of electrons slowly decreases, and when it reaches its limiting value, stability is lost and a genuine explosion follows in which particles are emitted and a new atomic configuration is attained. While Thomson’s atom did not withstand criticism and experimental tests, it would be a mistake to believe its study entirely sterile. Instead, this model acted as a valuable guide for experimental and theoretical inquiries of the time, and highlighted the fundamental problems that had to be resolved for any atomic model that had electrons as its constituents. These problems could be reduced to three: the number and distribution of electrons in relation to the mass of the atom; the nature and distribution of positive charge that compensates the total negative charge of the electrons; and the nature and distribution of the mass of the atom. Thomson understood that the essential problem for any model was finding an approach that allowed one to uncover clues about the number of electrons in an atom from experimental results. He found this approach by taking each electron to be a scattering centre for the radiation that struck the atom. Based on this hypothesis, four phenomena could provide the clues he sought: x-ray scattering, the absorption of cathode rays, the dispersion of light, and the deviation of charged particles as they rapidly pass through matter. Following these tracks, Thomson and several other experimenters, the most notable of which were James Arnold Crowther (1883-1950) and Charles Glover Barkla (1877-1944), concluded that the number of electrons in an atom had to be proportional to its atomic weight. Crowther deduced from x-ray diffraction experiments (1910) that the number of electrons was 2-3 times the atomic weight of the element; on the other hand, Barkla, also studying x-ray diffraction, inferred that the number of electrons in light atoms was around one-half of their atomic weight. These were contradictory results, but they had one thing in common: the number of electrons was in some way tied to the mass of the atom in consideration. If only for this qualitative result, which led others to more accurate research, Thomson’s atom is worth remembering. It was more difficult to tackle the second problem: the positive charge of the atom. Here, the evidence was not nearly as obvious as it had been for the negative charge. Nevertheless, according to Thomson, there were two signs of the existence of a positive charge inside the atom: the emission of Į particles by radioactive substances could indicate that they

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originated from the inside of atoms; did this also apply to stable elements? “As I see it,” wrote Thomson in 1913, “it is very likely that certain atoms besides those of radioactive elements can be decomposed, and that helium can be obtained as a product of this decomposition.” Furthermore, the second sign was his belief to have observed the emission of helium by certain metals bombarded with high-velocity cathode rays; but this observation was disputed by Marie Curie (as Galvani acutely noted, often one sees what one wants to see).

7.7 The Nagaoka-Rutherford Atom Lord Kelvin had faintly hinted at the possibility of a central nucleus of positive charge inside the atom. This hint, ignored by Thomson, was embraced, perhaps unwittingly, by the Japanese physicist Hantaro Nagaoka (1865-1950) in a communication to the Physical and Mathematical Society of Tokyo in December 1903, which was later published in the English journal “Nature”. While Thomson proposed his model with electric phenomena in mind, Nagaoka set out to study a Saturn-like system in an attempt to explain emission spectra. “The system,” explained the scientist, “consists of a large number of particles of equal mass arranged in a circle at equal angular intervals, and repelling each other with forces inversely proportional to the square of the distance between the particles; at the centre of the circle is placed a large particle attracting the other particles forming the ring according to the same law of force,” Observing that if the particles moved around the centre, the system remained stable for small transversal or longitudinal oscillations, Nagaoka added: “Evidently the system here considered will be approximately realised if we place negative electrons in the ring and a positive charge at the centre. Such an ideal atom will not be contradictory to the results of recent experiments on cathode rays, radio-activity, and other allied phenomena.”262 The Nagaoka model did not have good press. It was immediately pointed out that the model was unstable; furthermore, applying Nagaoka’s theory to spectral lines gave results that contradicted experiment. In those years, Thomson’s model was consistently given more credence, despite the poor agreement between theoretical predictions and experimental results. 262

H. Nagaoka, On a Dynamical System Illustrating the Spectrum Lines and the Phenomena of Radio-activity, in “Nature”, 69, 1904, p. 392. The theory was also exposed, with more detailed calculations, in “Philosophical Magazine”, 6th series, 7, 1904, p. 437.

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In 1908, the German physicist Hans Geiger (1882-1945) and the Australian teenager Ernest Marsden (1889-1970) began, in Rutherford’s laboratory, the experimental study of particle scattering through thin foils made of gold and other metals. They observed that the majority of particles crossed the foil almost in a straight line and continued their trajectory as if they had met no obstacle in the matter: the same observation made by William Bragg in 1904. However, Geiger and Marsden also noticed that, alongside this behaviour of the majority, in exceptional cases a particle about 1 every 8000- was significantly deviated at an angle greater than a right angle, that is, reflected. What had transpired was undoubtedly a collision between an Į particle and the atoms of the traversed matter. Thomson and Nagaoka’s atomic models, however, required a radical alteration to the concept of a collision between atoms, beyond the kinetic theory of gases. This likened atoms to elastic spheres and treated their collisions exactly as if they were billiard balls. The new atomic models no longer allowed this idealization, because when two atoms and their corresponding electric charges approach each other, non-negligible repulsive forces arise between the respective electric charges, modifying the original trajectories of the atoms: a “collision” in the mechanical sense does not occur. The theory of the motion of two atoms approaching each other is considerably more complicated from a mathematical point of view. In a first approximation, however, the change in the motion of the two supposed atoms with their respective charges is equivalent to that which would arise from a mechanical collision, thus we can continue to speak of collisions with the convention that we are dealing with collisions of a fictional nature. Having assumed this approximation, Thomson interpreted the significant deflection observed by Geiger and Marsden not as a single collision of an Į particle against an atom, but as the sum of many small deflections experienced by the particle in its successive collisions against the atoms of the traversed matter. Responding to criticisms of this theory, Thomson attempted in vain to explain why the consecutive small deviations were all in the same direction, such that they added up to a large total deflection, as observed. Thomson’s model did not allow for another interpretation of the experimental results obtained by Geiger and Marsden, as intuitively, even without following Thomson’s calculations, a diffuse distribution of positive charge – the Į particle – crossing another diffuse distribution of amalgamated electrons – the atom of the metallic foil – can only experience slight deviations.

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In a historic paper263 published in May 1911, Rutherford examined Thomson’s theory and showed, with a straightforward computation, that the probability of obtaining deflections greater than 90o from the effect of numerous consecutive collisions of Į particles against atoms was practically zero, confirming what was already intuitively understood by many physicists. He therefore advanced the hypothesis that, at least for some particles, the large deflection had to be attributed to a brusque deviation experienced by the Į particle while passing through a strong electric field inside an atom, or in other words, a single atomic collision. But if the deflection was due to a single collision, it was necessary to assume that the atom contained an extremely small, positively-charged central core that enclosed the bulk of the atomic mass. In short, it was necessary to adopt an atomic model with a nucleus and rotating electrons, whose stability, which had been disputed, was not a pressing concern for Rutherford, as the problem of stability could once again arise when even smaller atomic structure was discovered. This model immediately explained the strong deflection. The Į particle crosses the electronic atmosphere (whose effects can be ignored), approaches the nucleus, and due to the large Coulomb force that arises between the two positive charges, is significantly deviated into a cometlike orbit. Rutherford detailed the quantitative theory of Į particle scattering: in 1913, Geiger and Marsden, with a meticulous experiment that has since made history, confirmed it. Langevin observed that Rutherford’s atomic model did not seem in the least bit supported by radioactive phenomena, since radioactive bodies also emit ȕ particles, which seemed to originate from the deepest parts of the atom. Therefore, electrons also had to be present in the atomic nucleus, and for this reason Thomson’s model appeared more reliable. Marie Curie insisted on the necessity of assuming that electrons existed inside the nucleus: these essential electrons, as she called them, or nuclear electrons, as they were later called, could not escape without the destruction of the atom, while the others, called peripheral by Curie, could be detached from the atom without changing its chemical nature. For twenty years, nuclear electrons were a part of the presumed atomic structure (perhaps a residual homage to Thomson’s model); on Rutherford’s suggestion, physicists ended up attributing to them a coagulating role, uniting the positive charges of the nucleus that the mutually repulsive Coulomb forces tended to separate. 263

“Philosophical Magazine”, 6th series, 21, 1911, p. 669.

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The experiments that had led to the conclusion that nuclear charge is one-half the atomic weight were obviously also applicable to this new model. Yet Rutherford noted that, at least for light nuclei, this law was not exact. Indeed, having identified the Į particle emitted by radioactive substances as a helium ion with two positive charges, it is difficult to suppose that this particle maintains any of its peripheral electrons: thus the neutral helium atom should have two electrons and, by analogy, the hydrogen atom should have one. The nuclear charge of hydrogen is then one, and the law is erroneous. 1913 was the critical year, or as Soddy called it, the “dramatic” year for Rutherford’s model. Four fundamental results converged almost simultaneously to make the nuclear model of the atom appear highly likely to be correct. These four results intertwined with and influenced one another, not solely due to the narrow geographical region in which they were obtained (with the exception of one) -between Cambridge, Manchester, and Glasgow- but also because of their common inspiration, Ernest Rutherford. We have already discussed the first result, the experimental law of radioactive displacements explicitly articulated by Soddy in the spring of 1913. We now lay out the remaining three: the concept of “atomic number” proposed by the Dutch physicist Antoon Johannis Van den Broek (1870-1926) a bit earlier, the quantization of electronic orbits asserted by Bohr in the summer, and the experimental law that Moseley formulated that winter. Van den Broek noticed that the observations regarding Į particle scattering were better interpreted by Rutherford’s model if one supposed that the nuclear charge to be equal to the product of the numerical position, which he called “atomic number”, of the element in Mendeleev’s periodic table and the electron charge. Therefore, if Z is the element number and e is the unit charge, the electric charge of the nucleus is Ze. The element number is approximately half of the atomic weight for lighter atoms, and slowly decreases in ratio until reaching a value of about 0.4 for uranium, the last element classified at the time. As a result, while Van den Broek’s idea was not too different from the preceding experimental law, it was of alluringly simplicity. Its justification, however, was fairly tenuous. Giving it a solid experimental foundation was the last important result of that year (we will talk extensively of the third in § 7.13): Moseley’s law. Henry Moseley (1887-1915), who two years later would die in combat on the peninsula of Gallipoli, began his research into x-ray spectra as a volunteer assistant to Rutherford, employing the crystal reflection method that Bragg had recently introduced. The specific aim of his research, as

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Rutherford attested, was to decide if the atomic number of an element was more important than its atomic weight in the production of high frequency spectra. Barkla had already demonstrated that if fairly energetic x- rays struck an ordinary substance, they generated others, called secondary xrays, which were homogeneous, that is of the same frequency. Moreover, secondary x-rays were characteristic to the substance that was struck, but independent of the frequency of the primary x-rays. Moseley measured the frequency of the primary spectral lines discovered by Barkla for successive elements in the periodic table, finding that they are proportional to the square of a number that varies by one unit for each passage from one element to the next in the table. This was an experimental fact that did not involve any theoretical considerations regarding composition of the atom and the origin of its radiation. However, it acquired a deeper meaning when Moseley showed that his observations could be interpreted as proof, as he wrote, “that in the atom there exists a fundamental quantity that increases regularly in passing from one atom to the next. This quantity can only be the charge of the central nucleus.” Moseley later showed that only three elements were missing between aluminium and gold in the periodic table and predicted their spectra. In addition to these experimental results by Moseley, Soddy’s law also confirmed Van den Broek’s idea. Because any emission of an Į particle reduces the atomic weight by four and the atomic number by two, while the emission of a ȕ particle raises the charge by one and leaves the atomic weight invariant, one only needs to know a radioactive family’s type of emission and the atomic number of its progenitor to calculate the atomic number and mass of all its elements, which can then be compared to experimental results. Such comparisons invariably showed excellent agreement with the theory. Rutherford’s fervent school sought a direct measurement of nuclear charge, but the first world war slowed and impeded research, because unlike what happened during the second world war, nobody remotely thought that these studies could be of military interest; they belonged entirely to the realm of “safe peaceful philosophy” which had to be sacrificed to wartime demands. Consequently, only in 1920 did James Chadwick (1891-1974) make the first accurate determination of the nuclear charges of certain elements, deducing them from the percentage of Į particles deviated at a given angle by atoms of the element in question. Using this method, he found for copper, silver, and platinum the charges of 29.3, 46.3, and 77.4, respectively, and atomic numbers 29, 47, and 78: an acceptable accord.

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7.8 Artificial decay of elements Around 1913, the evidence supporting Rutherford’s atomic model came mainly from radioactive phenomena. These were so unique, however, that it was possible that the structure of radioactive atoms was completely different from that of stable atoms. Extending the composition of a radioactive atom to a stable atom was pure extrapolation. “I think that there is more than one advantage in establishing a theory of atomic structure on a broader base than that of radioactivity,” warned Thomson in 1913, “the chemical properties and a great number of the physical properties depend on the distribution of electrons in the vicinity of atomic surfaces; these electrons do not intervene in radioactive transformations, so they cannot give us any information about them.”264 It would have therefore been of utmost importance to have some experimental evidence that the structure of stable atoms was comparable to that of radioactive ones. In 1902, Rutherford attempted to demonstrate this, guided by a very simple argument that is best conveyed in his own words: “the Į particle of radium is the most concentrated source of energy at our disposal and we have seen that there is good reason to believe that the fast Į particle of radium C is able to penetrate the nuclei of light atoms and perhaps also heavy atoms. Unless atomic nuclei are excessively stable structures, it is to be presumed that they experience a rupture under the influence of the strong forces that act during the impact with an Į particle… In view of the small dimensions of nuclei, the chances of a central collision would be quite small and, even in the most favourable cases, no more than one particle in 10000 effectively produces a decay.”265 Rutherford thus bombarded nitrogen with Į particles produced by a radioactive substance. He observed the production of hydrogen ions with a single charge, which Marsden had already observed and called protons in 1914. The number of protons produced was extremely small: a million Į particles bombarding the nitrogen gave rise to barely 20 protons, which Rutherford detected with the scintillation method. The observed effect, commented Rutherford, “is weak and difficult to measure, but by and large gives the impression that H atoms are born out of the disintegration of the nitrogen nucleus.”266 Protons were also obtained by bombarding other elements (boron, fluorine, sodium, aluminium, phosphorus) with Į particles from radium C. In 1921 and 1922, Rutherford and Chadwick 264

Rapports et discussions du Conseil de physique Solvay, Paris 1913, p. 51. Atomes et electrons. Rapports et discussions du Conseil de physique tenu à Bruxelles du 1 au 6 avril 1921, Paris 1923, pp. 50-51. 266 Ibid., pp.51. 265

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obtained other nuclear reactions, and in 1925 Rutherford interpreted all of these results as atomic decays, indicating that the Į particle could be captured by a nucleus. Later, when artificial decays became common, physicists realized that Rutherford had really produced decays as early as 1920, but at first the announcement of his results had been met with reservation, if not skepticism. Rutherford’s first experiment, the bombardment of nitrogen, for example, should be interpreted as follows: an Į particle is captured by a nitrogen nucleus that then emits a proton, forming a nucleus of mass 17, which is an isotope of oxygen.

7.9 Non-radioactive isotopes The isotopy (or isotopism) of radioactive elements arose as an experimental fact, independent of any hypothesis regarding atomic structure. To highlight this historical process, we presented this idea before discussing atomic models at all, unlike modern writers, who invert the chronological order for didactic convenience (Soddy believed, perhaps exaggerating, that this historical inversion was deliberately planned to spite him). Yet after Soddy’s law of radioactive displacement was invoked as one of the most convincing pieces of evidence backing Rutherford’s model, extrapolating to stable atoms, the isotopy of non- radioactive elements seemed to be a direct consequence of the theory and shed new light on a discovery that Thomson had made in 1912. Previously, in 1910, Herbert Edmeston Watson (1886-1980) had measured the atom weight of neon with extreme care and found it equal to 20.200 (having set the atomic weight of oxygen equal to 16). In 1912, Thomson, subjecting neon anode rays to the action of simultaneous electric and magnetic fields according to his method (§ 4.3), observed that the beams did not seem homogeneous, splitting into two parabolas, one corresponding to a particle of atomic weight 20 and the other, weaker but still clear, a particle of atomic weight 22. Thomson suspected that atmospheric neon was a mixture of two different gases. Soddy immediately commented: “the discovery is a most dramatic extension of what has been found for the elements at one extreme of the periodic table, to the case of an element at the other extreme, and strengthens the view that the complexity of matter in general is greater than the periodic law alone reveals.”267 267

F. Soddy, Annual Report in Radioactivity, in “Chemical Society’s Annual Reports on the Progress of Chemistry”, 1913, p. 266.

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In 1913 and 1914, Francis William Aston (1877-1945), an assistant to Thomson, attempted to separate the two supposed constituents of atmospheric neon through fractionalized distillations, failing completely; a later laborious attempt using a semipermeable membrane resulted in highly uncertain results. The first world war broke out; Aston was called to render his services to the Royal Air Force and suspended his research. He resumed in 1919, when the existence of radioactive isotopes had been confirmed beyond any reasonable doubt, and approached the problem theoretically in collaboration with the German physicist Frederick Alexander Lindemann (1886-1957), who later became Lord Cherwell: together they showed that out of all the physical methods of separating isotopes (diffusion, distillation, centrifugation), the one with most promising results was Thomson’s electromagnetic method. That same year, Aston began his own experiments and perfected them the following year, developing an instrument that he called, as it is still called today, a mass spectrometer. In the mass spectrometer, Thomson’s method is applied to deflect charged particles using an electric field and a magnetic field. Soon after, Aston drastically increased its precision by introducing photographic detection and, in particular, arranging for the electric and magnetic deflections to be in the same plane but different directions. The physical principles at play in the mass spectrometer were accordingly well known. Ions of the element being examined cross an electric field and then a magnetic field, strike a photographic plate, and leave a mark. Now, the deviations undergone by the ions depend on the ratio e/m, (or better ne/m, because an ion can carry more than a single elementary charge) between their charge, which is the same for all the ions, and their mass. Therefore, all ions of the same mass are concentrated at the same point on the photographic plate, and the ions of different masses at different points: the mass of an ion can then be deduced from the point struck on the plate. The new instrument proved to be incredibly productive right away: Aston immediately obtained confirmation that neon was composed of two isotopes of respective atomic masses 20 and 22; the next element to be examined was sodium, and two isotopes of masses 35 and 37 were found. Aston began a systematic study, and the squares in the periodic table filled up with isotopes. In 1921, 11 out of 21 elements that had been examined had isotopes; ten years later, 42 out of 64 elements examined had isotopes (tin had the most, with 20 isotopes ranging in mass from 112 to 124). In 1945, across 83 elements only 10 were “pure”, while the others had 283 isotopes: a veritable profusion! Now, the multitude of different atoms brought about serious difficulties in the study of atomic energies.

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As time went on, spectrometers were improved and perfected: Aston built a second spectrometer in 1925 that had a precision of one part in 10000; in 1937, Aston’s third spectrometer reached a precision of 1 part in 100000, provoking Soddy’s assessment: “these instruments can be considered among the most marvellous and precise apparatuses that the human mind could have ever invented.”268 Aside from utilizing the mass spectrometer, a precision instrument oftused in research, non- radioactive isotopes, or as they were more commonly called, stable isotopes, could also be separated through other approaches (diffusion, chemical exchange, electrolysis, centrifugation) whose descriptions we leave for more technical treatments. Aston’s discovery had theoretical consequences of exceptional importance. As an immediate consequence, it once again posed the age-old problem of the definition of an element. The dilemma consisted in knowing if each isotope should be considered a different element, and thus every substance containing multiple isotopes as an amalgam of elements and not as a single chemically defined element. For now, the large majority of chemists remains anchored to the classical conception, holding that each element is chemically defined by its chemical properties and emission spectrum under given conditions; for elements with identical chemical properties but different nuclear mass, the concept of isotopes is introduced: atoms whose nucleus contains the same number of protons. In other words, isotopes are considered chemically equivalent but physically distinct elements. If we do not split hairs, however, this definition of isotopy includes the definition of an element. Here we see the first inadequacy of the periodic table, which is valid for classifying the majority of chemical and physical proprieties of elements, like volume, valence, electric and thermal conductivity, that is those properties, called electronic, that depend on the number and arrangement of the outermost electrons from the nucleus. On the other hand, for others, like mass, radioactivity, and x- ray spectrum, which depend on the composition of the nucleus and the closest electrons to it, a classification based only on atomic number is insufficient.

7.10 Matter and energy However, since the first measurements in 1920, the most surprising fact for Aston and his collaborators was that the atomic weights of all of the light elements turned out to be whole numbers, at least within the 268

F. Soddy, The Story of Atomic Energy.

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limits of the experimental errors at the time, which were estimated to be of the order of 1 part in 1000. This “whole number rule” resulted slightly flawed for heavy atoms, from Z = 30 onwards, for which the deviation from the law is still minuscule and grows regularly with atomic number. The whole number rule, Aston argued, allows for the most alluring simplification of our concept of atomic mass. It reinstates Prout’s hypothesis: protyle was not hydrogen though, as Prout had believed, but rather the proton, which was combined with a corpuscle that was around two thousand times smaller, the electron. It followed that the decimals found by chemists for the atomic masses of many elements turned out to be the weighted average of the mixture of isotopes constituting the chemical element. These were therefore merely statistical effects due to the relative quantities of the constituent isotopes. The whole number rule had an exception, however, which proved to be much more important than the rule itself: the atomic mass of hydrogen, even using the mass spectrometer, was not 1 (having set the mass of the oxygen atom to 16), but 1.008, as chemists had established. The difference was minute, but nevertheless large enough that it could not be attributed to experimental errors, which were smaller than the discrepancy. Aston thought to explain this apparent anomaly by first invoking the non-additivity of masses predicted by Lorentz’s electronic theory, and subsequently relativity, which was spreading in 1920 despite the bitter criticisms that it faced. According to Aston, when protons unite to form the nucleus of an element, a part of their mass is transformed into binding energy of the nucleus. This phenomenon, called the packing effect by Aston, is known as the mass defect today, and explains the apparent loss of mass of a hydrogen nucleus when it joins others to form a new nucleus. We add that this interpretation of the mass defect is now a landmark in nuclear theory. “Theory indicates,” Aston wrote, “that when such close packing takes place the effective mass will be reduced, so that when four protons are packed together with two electrons to form the helium nucleus this will have a weight rather less than four times that of the hydrogen nucleus, which is actually the case. It has long been known that the chemical atomic weight of hydrogen was greater than one-quarter of that of helium, but so long as fractional weights were general there was no particular need to explain this fact, nor could any definite conclusions be drawn from it. The results obtained by means of the mass spectrograph remove all doubt on this point, and no matter whether the explanation is to be ascribed to packing or not, we may consider it absolutely certain that if hydrogen is

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transformed into helium a certain quantity of mass must be annihilated in the process. The cosmical importance of this conclusion is profound and the possibilities it opens for the future very remarkable, greater in fact than any suggested before by science in the whole history of the human race. We know from Einstein’s Theory of Relativity that mass and energy are interchangeable and that in c.g.s. units a mass m at rest may be expressed as a quantity of energy mc2, where c is the velocity of light. Even in the case of the smallest mass this energy is enormous. The loss of mass when a single helium nucleus is formed from free protons and electrons amounts in energy to that acquired by a charge e falling through a potential of nearly thirty million volts. If instead of considering single atoms we deal with quantities of matter in ordinary experience, the figures for the energy become prodigious. Take the case of one gram-atom of hydrogen, that is to say the quantity of hydrogen in 9 grams of water. If this is entirely transformed into helium the energy liberated will be … 6.93(1018) ergs. Expressed in terms of heat, this is 1.66(1011) calories, or in terms of work 200,000 kilowatt hours. We have here at last a source of energy sufficient to account for the heat of the sun. In this connection Eddington remarks that if only 10 percent of the total hydrogen on the sun were transformed into helium, enough energy would be liberated to maintain its present radiation for a thousand million years. Should the research worker of the future discover some means of releasing this energy in a form which could be employed, the human race will have at its command powers beyond the dreams of scientific fiction; but the remote possibility must always be considered that the energy once liberated will be completely uncontrollable and by its intense violence detonate all neighbouring substances. In this event the whole of the hydrogen on the earth might be transformed at once and the success of the experiment published at large to the universe as a new star.”269 In a two-year period (1919-20), the problem of the atomic mass of neon had led to the reconsideration of one of the most important cosmological questions. This is indicative of the importance of the discovery of isotopes not only for physics, but for all of science, as often great discoveries relate apparently very distant phenomena with the same underlying principle.

269 F.W. Aston, Mass-Spectra and Isotopes, Nobel Lecture, in Les prix Nobel en 1921-1922, pp. 13-14.

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7.11 Atomic weights and radioactive measures While William Prout’s hypothesis on the primordial element, protyle, was still accepted, the atomic weights of all elements were expressed as multiples of the atomic weight of hydrogen, assumed to be one. Soon however, the measurement of atomic weights demonstrated that Prout’s hypothesis was no longer acceptable, because the atomic weights of elements, having taken H = 1, did not turn out to be whole numbers. Nevertheless, the original system lived on, spurred by Dumas, who with his authority reinstated Prout’s hypothesis in 1878. The knowledge of the atomic weights of elements was of such importance for chemistry that in Germany a special commission for atomic weights was created with the task of critically evaluating and tabulating the most reliable measurements. The discovery of the electron and radioactive phenomena once again challenged Prout’s hypothesis, causing the commission for atomic weight, led by Ostwald, to propose (1900) to the chemical societies of other nations the formation of an international commission for atomic weight, and the choice of oxygen’s atomic weight of 16 as the reference weight. This choice had already been recommended by the Gesellschaft Deutscher Chemiker in 1897, harking back to Berzelius, who in 1826 had assumed the atomic weight of oxygen to be 100. In 1902, the International Commission for Atomic Weights was formed, counting among its first members the English scientist Thomas Eduard Thorpe (1845-1925), the American Frank Clarke (1847-1931), and the German Karl Friedrich Otto Seubert (1851-1942): starting in 1905, all scientists based atomic weights on oxygen and the commission regularly published its findings until 1916. After the First World War, German scientists were excluded from the commission, one of the many instances of postwar foolishness. For this reason, they created their own, which in 1921 published its first report. In the years between 1916 and 1921, however, an important development occurred that the German commission had to take into account: the discovery of stable isotopes. The commission factored in this breakthrough by dividing the table of atomic weights into two parts. In the first part, the “practical atomic weights” used by chemists were reported; the second part included the “chemical elements and atomic species in order of atomic number” necessary for physicists: in both tables the reference element was oxygen 16. The records of the German commission continued until 1931, when the commission was dissolved, because by 1930 Germans had been readmitted into the international commission, which however had not dealt with isotopes.

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Tasked by the Gesellschaft Deutscher Chemiker, Hahn continued to publish results relating to isotopes until 1938; then, due to the increasing role of physical methods (mass spectroscopy, nuclear physics), the task was entrusted to Siegfried Flügge (1912-1997) and Joseph Mattacuh (1895-1976), who in 1942 collected the enormous material they had accumulated in a volume (Kernphysikalische Tabellen). Incidentally, Mattauch had given the laws governing isobars in 1937, which bear his name, saying: if the atomic weight is odd, there are no stable isobars; if the atomic weight is even, there are several stable isobars. These laws are accompanied by Aston’s rule: elements with an even atomic weight have at most two stable isotopes. After the discovery of the isotopes O17 and O18 in 1935, it became clear that the atomic weight of the isotopic mixture of “natural” oxygen and other simple elements could not be constant. For years scientists discussed the introduction of a common mass scale for physics and chemistry. In 1953, Mattauch proposed that the isotope carbon-12 serve as the standard for atomic weights: in 1961 the first “international table of chemical atomic weights” appeared based on C12. The accelerated progress of radioactive research and especially the medical applications of radium soon made accurate measurements of radioactivity imperative. In 1910, an international congress on radium gathered in Brussels, deciding to create an international radio-standard commission. Marie Curie, who was part of the commission, and Otto Honigschmid (1878-1945) of the Vienna Institut für Radiumforschung were entrusted with the task of independently preparing samples of radium from anhydrous radium chloride. In March 1912, the Commission internationale des étalons de radium met in Paris and compared the sample prepared by Curie to the three samples prepared by Honigschmid: they were found to be in perfect accord. Curie’s sample (21.99 mg of pure radium chloride) was delivered to the international office of weights and measures, while one of Honigschmid’s samples (21.17 mg of radium chloride) was brought to Vienna as a secondary sample. The sale of radioactive substances, whose cost had become exorbitant due to their widespread therapeutic use, was then regulated based on their activity levels, which was determined from comparison with previous samples. The curie (c) was established as a unit of measurement, defined as the quantity of emanation (radon) in equilibrium from one gram of elemental radium, which emitted 3.7(1010) Į particles per second. The lower denominations of millicurie (mc = 10-3 c), microcurie (ȝc = 10-6 c), and nanocurie (nc = 10-9 c) are more frequently used. In 1960, the general conference of weights and measures bestowed

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upon the international office of weights and measures the authority to extend its jurisdiction to the metrology of ionizing radiation, inviting it to establish a division of measurement standards to work towards their unification and the precise definition of units.

THE BOHR ATOM 7.12 Spectral series Since the first studies of light spectra (§ 1.8), physicists had realized that despite the apparent disorder, there is a certain regularity in the distribution of the spectral lines of elements. Perhaps the first to write down a relation, however, was Stoney, who in 1870 observed that the frequencies of the C, F, and H lines of the solar spectrum, which correspond to the Į, ȕ, and Ȗ lines of the hydrogen spectrum, were in a ratio of 20:27:32. This observation led him to think that the three lines were caused by a periodic event inside the hydrogen molecule. His belief was confirmed the following year, when, in collaboration with the chemist James Emerson Reynolds (1844-1920), he verified that the frequency of the lines in the spectrum of CrO2Cl2 were closely related to the frequency of harmonics emitted by a violin string under certain conditions. In 1885, Johann Jacob Balmer (1825-1898) showed how Stoney’s observations fit in a more comprehensive law. He found that the wavelengths (or their inverses, called “wave numbers”) of the lines in the visible spectrum of hydrogen could be expressed through a simple formula, in which successive spectral line wave numbers were obtained by letting a variable take integer values (starting from 3).270 Encouraged by Balmer’s success, spectral research was begun by Johannes Robert Rydberg (1854-1919) in 1889, and Johann Heinrich Kayser (1852-1940) and Carl David Runge (1856-1927) in 1890. The former gave the series for thallium and mercury, while the latter physicists 270

If v0 is the wave number of a hydrogen line, the Balmer series is expressed by the formula 1 1 ‫ݒ‬଴ = ܴ ൬ ଶ െ ଶ ൰ 2 ݊

where R is a constant, called the Rydberg constant, equal to 10967758.3 m-1; n must be set to integer values (3, 4, 5, …). In theory, the number of spectral lines is infinite, but it follows from the formula that the successive lines become closer and closer and, after a certain point, can no longer be identified because their intensity falls as one moves toward violet.

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made history with their work, which lasted many years and employed the photographic technique. They also studied the intensity of various lines, the differences between spark and arc spectra, and gave the series for several elements, in particular for alkali and alkaline earth metals. In the 20th century, study focused on the hydrogen series. In 1904, the American Theodore Lyman (1874-1954) found another hydrogen series in the ultraviolet; in 1909, Friedrich Paschen (1865-1947) found a series in the infrared and in 1922, Patrick Maynard Blackett (1897-1974) found a second infrared series.271 The previous series are given by formulas that resemble the formula for Balmer’s series, the common constant involved, called Rydberg’s constant by spectroscopists, was known to a high degree of accuracy. All of these equations were experimentally deduced and all had the following characteristic: the wave numbers of the spectral lines were always obtained by adjusting a quantity in the formula in whole number steps. Only two cases were known in physics in which, instead of changing continuously, physical values varied in integer steps: interference phenomena and harmonics of vibrating bodies (such as strings). Finally, in 1908, the Swiss physicist Walter Ritz (1878-1909) outlined the principle of combination, which was fundamental to modern spectroscopy and can be stated in its simplest form as follows: the frequency of a spectral line is often equal to the difference (or sum) of the frequencies of two other lines in the same spectrum. Until 1913 nobody knew how to theoretically interpret spectral lines, much less the principle of combination: the two remained exact but mysterious empirical formulas. One thing was clear in the first decade of the century, that this wide collection of irrefutable experimental results had to result from the structure of the smallest particles of matter. Classical electromagnetism attributed emission to the existence of electric charges that are normally at rest in radiating matter, but with external excitations can vibrate around their equilibrium position at fixed frequencies, with a simultaneous emission of radiation; with time these oscillators, as the vibrating charges were called, lose their energy in the form of radiation and become still. On this basis, classical electromagnetism 271

If the formula from the previous footnote is written 1 1 Q଴ = ܴ ൬ ଶ െ ଶ ൰ ݉ ݊ and one sets m = 1 and successively n = 2, 3, 4, … one obtains Lyman’s ultraviolet series; if m = 2 and n = 3, 4, 5, … one had Balmer’s series; if m = 3 and n = 4, 5, 6, … one has Paschen’s infrared series; and lastly, if m = 4 and n = 5, 6, 7, … one has Blackett’s series.

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explained spectral lines, which however resulted in very different positions and conditions from those found in experiments. Poincaré, noticing this failure of classical electromagnetism, reflected: “A first study of the distribution of the lines reminds one of the harmonics that are found in acoustics, but the difference is vast; not only are the vibrational numbers not the successive multiples of the same number, but one cannot even find something analogous to the roots of the transcendental equations which arise from many problems of mathematical physics: that of the elastic vibrations of a generic body; that of the Hertztian waves in a generic oscillator; Fourier’s problem of cooling a solid body. The laws are simple, but of a totally different nature… We have not accounted for this fact yet, and I believe that therein lies one of the most important secrets of nature.”272

7.13 Bohr’s theory To Rutherford’s great merit, he intuited that radioactive phenomena took place in the nucleus and dedicated himself to its study, not allowing himself to be misled by problems relating to the electrons or the major criticisms to which he was subjected. In fact, Rutherford’s model was irreconcilably at odds with classical electromagnetism, according to which a rotating electron had to continually emit electromagnetic waves, and therefore continually lose energy and move closer to the nucleus until it fell in, neutralizing or even destroying it. Therefore, one either had to renounce his model to remain faithful to classical physics, or abandon classical physics and find new principles to account for the model. The second option was chosen by a young Danish physicist, Niels Bohr (Fig. 7.3), born in Copenhagen to Christian Bohr, a professor of philosophy, on 7 October 1885, and died in the same city on 18 November 1962. He graduated in 1911 in his home town and moved to Cambridge later that year, joining Thomson’s laboratory. From there, he moved to Rutherford’s Manchester laboratory in 1912 with the aim of conducting experimental research. Yet Rutherford, noticing the extraordinary theoretical abilities of the youth, persuaded him to dedicate himself to theory: a detail worthy of mention because Rutherford, who did not have a soft spot for theoretical physics, has often been reproached in England for having negatively influenced the country’s theoretical research.

272

Poincaré, La valeur de la science, p. 205.

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Fig. 7.3 – Niels Bohr

In Rutherford’s laboratory, Bohr was exposed to the most advanced physical concepts of the time: the structure of the atom, relativity, and quantum theory. Having returned to Copenhagen in 1913, in 1914 he went back to Manchester, and in 1916 he was appointed professor of theoretical physics at the University of Copenhagen. In 1920, the Danish academy of science, funded by a famous beer factory, gave Bohr the financial means to open an institute for physics and give fellowships to young scientists from all over the world. The institute soon became a centre of scientific activity to which young (and not so young) scientists from all over flocked (Paul Adrien Maurice Dirac, Wolfgang Pauli, Werner Heisenberg, Léon Rosenfeld, George Gamow, J. Robert Oppenheimer, and Edward Teller, just to name a few), attracted by Bohr’s fame and his polite, welcoming, and entirely informal manners. In 1922, Bohr was given the Nobel prize in physics, and in 1957 he was the first physicist to receive the “atoms for peace” award. His research on atomic structure and new paradigms in the philosophy of science revitalized physics research, giving it a new face.

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In the famous paper On the constitution of atoms and molecules,273 Bohr began by observing that the electron system of Rutherford’s atom is unstable. Nevertheless, one can attempt to treat the problem with the novel quantum ideas. “The result of the discussion of these questions seems to be a general acknowledgement of the inadequacy of the classical electrodynamics in describing the behavior of systems of atomic size, Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i.e. Planck’s constant, or as it is often called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed.” Bohr then studied the equilibrium conditions of an atom composed of a nucleus and an electron rotating around it. Here he took the electron to have variable mass compared to the mass of the nucleus, and velocities much slower than the speed of light. If there is no radiation, the electron follows stationary elliptic orbits; but if the atom radiates, the electron approaches the nucleus until the size of its orbit becomes of the order of the nuclear dimensions: clearly this could not be the mechanism of radiation. A different approach had to be taken. Bohr based the new approach on two postulates; that classical mechanics can be applied to the dynamical equilibrium in stationary states, but not to the transitions between two stationary states; and that the transition between a stationary state and another is accompanied by the emission of homogeneous radiation for which the relation between frequency and energy is given by Planck’s theory. Eight years later, Bohr more clearly summarized the essence of his theory: “We will base our considerations on the following fundamental postulate, here presented in the form which we will use in what follows on quantum theory: An atomic system, which emits a spectrum formed by distinct lines, can assume a certain number of distinct states, which we call stationary states; the system can exist in such a state for some time without emitting radiation, which only occurs through a process of complete transition between two stationary states, and the radiation emitted is always composed of a series of simple harmonic waves. In the theory, the frequency of radiation emitted during a process of this type is not directly determined by the motion of particles in the atom in a manner corresponding to the ideas of the classical theory of electromagnetism; it is simply tied to the total quantity of energy emitted during the transition, the 273

“Philosophical Magazine”, 6th series, 216, 1913.

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product of the frequency Q and Planck’s constant h being equal to the difference of the values E’ and E’’ of the atom’s energy in the two states involved in the process, such that one has hQ = E’ – E” ”.274 Essentially, therefore, the electron rotating around the nucleus obeys all the laws of classical mechanics, but not those of electromagnetism, in the sense that during rotation it does not emit radiation. Jeans gave an imaginative analogy for Bohr’s atom: suppose there is a racetrack with many parallel lanes, all separated by hedges so tall that a spectator can see neither horses nor jockeys while they race in their own track. If a horse jumps over the hedge from one lane to another, however, it becomes visible to the observer during the transition. Using classical mechanics one can treat the dynamical equilibrium of a system in a stationary state and, knowing the orbital radius, calculate the velocity, frequency, potential energy, and total energy of the rotating electron. The electron, however, cannot simply be found in any orbit, but rather only those for which the energy difference is a whole multiple of the quantum of action h. These are the quantized orbits, which, from Bohr onward, were taken to begin from the one closest to the nucleus. All atoms can thus exist in a series of stationary states, each corresponding to the particular orbit in which the electrons are effectively found; at any given time, the atom is in a stationary state corresponding to some energy value, so for each atom there is a series of energy values that correspond to the different stationary states it can assume. Bohr rightfully observed that his quantized planetary system could only partly be likened to the planetary systems of astronomy. Indeed, the laws of gravitation allow us to study and explain the motions of the planets with a high degree of accuracy, but they do not allow us to predict their orbits, which are not completely determined by the planetary masses and the Sun. Planetary orbits essentially depend on the conditions in which the solar system was formed, that is, its history. As a result, there is a fundamental difference between a planetary system and an electronic one: in the first the orbits of planets remain constant; in the second they change. What are the fundamental properties of the stationary states of electronic systems? How must the problem of their stability be dealt with? Bohr was initially very conciliatory in this regard: “Even if we are obligated,” he writes, “to introduce modifications in the classical theory of electrons that imply a radical change in our ideas concerning the mechanism of radiation, we are not at all forced by this to admit that motion always differs fundamentally from that which would result from 274

Atomes et electrons (cit.), p. 230.

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classical electronic theory. On the contrary, we are very naturally led to assume – a hypothesis which currently is the basis for all applications of quantum theory to atomic problems – that it is possible, to a high degree of accuracy, to describe the motion of the particles in the stationary states of an atomic system like that of point masses moving under the influence of their mutual repulsions and attractions, due to their electric charges. Secondly, if we envisage the problem of the “stability” of stationary states, we immediately recognize that a necessary condition for stability is that, in general, the action of external agents on the motion of the particles in an atom is not exerted following the ordinary laws of mechanics. Actually, we will see that the properties expressed by the state equations, which distinguish stationary states from the possible mechanical motions of an atomic system, are not characterized, in accordance with their nature, simply by the velocities and the distribution of particles at a given instant, but critically depend on the periodic properties of the orbits to which the velocities and instantaneous distributions correspond. If we thus consider an atom subject to variable external conditions, it is not enough – to find the changes in motion produced by a change in these conditions – to simply imagine, as in ordinary mechanics, the effect of the forces acting on the particles in a given moment, but rather the resulting motion must fundamentally depend on the change in the character of the possible orbits corresponding to the variation in external conditions.”275 Having established the theory on these semiclassical foundations, Bohr supposed, for a reason that we will soon see, the electron orbits to be perfectly circular, and immediately computed that the radius of the smallest orbit (orbit 1) was 0.556(10-8) centimeters, equal in order of magnitude to the atomic radius given by kinetic gas theory. Bohr then applied the theory to the simplest atomic model, that of hydrogen, composed of a central nucleus of mass 1 having a single positive charge and an electron that orbits around it. With a rapid calculation, Bohr was able to obtain the general formula for the spectral series of hydrogen (substantially the same formula given in the second footnote of § 7.12), that is, the Balmer and Paschen series that were already known, and the Lyman and Blackett series that at the time (1913) had not been discovered. Furthermore, he computed Rydberg’s constant, obtaining a value coinciding with the experimental one. This was a great success because it was the first rational explanation for the mysterious series; one could now calculate atomic radii and Rydberg’s constant a priori. Bohr’s theory thus on one hand validated the 275

Ibid., p. 232.

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hypothesis of the atom with a nucleus, and on the other successfully extended the quantum hypothesis to atomic theory. From the hydrogen atom, Bohr moved on to simply ionized helium, a system which, in Rutherford’s model, was made up of a nucleus of mass 4 and charge 2, and a single orbiting electron: the problem was therefore identical to that of hydrogen, with a few quantitative corrections. Bohr also obtained an undiscovered series analogous to Balmer’s for the spectral lines of simply ionized helium. He identified it as the series discovered in 1909 by the American astronomer Edward Charles Pickering (1846-1919) in the spectra of certain stars, which had been misattributed to hydrogen. Yet when Bohr tried to extend his theory to atoms with more than one orbiting electron, the difficulties became serious. With an ordinary helium atom, which only has two orbiting electrons, the mathematical problem already becomes complicated. Indeed, one must take into account that each electron is acted on by both the nucleus and the other electron: this is a version of the “three body problem”, made more difficult that the astronomical problem (first studied by Newton) by the fact that the interaction between the two electrons is of the same order of magnitude as that between the nucleus and one electron, while in the astronomical problem the attraction between the two planets is much smaller than that between one planet and the Sun. Aside from the mathematical difficulties that became insurmountable with more complicated atoms, there was a conceptual difficulty, in the sense that with multiple electrons, quantization became unclear. Awaiting further progresses in quantum theory that would allow this conceptual difficulty to be surmounted, Bohr limited himself, in 1913, to an attempt to at least extend the theory to hydrogen-like atoms, namely atoms composed in the following way: if the atom contains N electrons, N-1 belong to a central region, and the outermost Nth electron orbits around the shell made up of the nucleus and the other N-1 electrons; the transitions of the Nth electron (called luminous) from one orbit to another determine the emission spectrum of the element. This category of atoms includes lithium, sodium, potassium, etc. Applied to this class of atoms, however, Bohr’s theory proved to disagree with experimental results.

7.14 Sommerfeld’s theory When Bohr formulated his theory in 1912, the quantization procedure was known only for motion that depended on a single variable. For this reason, Bohr was forced to assume that the orbits of peripheral electrons

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were perfectly circular, while the laws of mechanics indicated that electronic orbits had to be Keplerian ellipses. However, two variables are needed to specify the position of a point on an ellipse, and thus to quantize an elliptical orbit one must know how to quantize motion in two variables. In 1916, almost at the same time, William Wilson (1875-1965) and Arnold Sommerfeld (1868-1951) proposed a method to quantize mechanical systems defined by multiple variables. Sommerfeld immediately had the idea of applying it to Bohr’s atomic model, consequently introducing elliptical orbits; but this introduction did not lead to significant modifications to Bohr’s conclusions. Sommerfeld was only able to explain a few other empirical formulas known to spectroscopists, in particular Ritz’s principle of combination. Yet Sommerfeld observed that Bohr, strictly adhering to classical orthodoxy, had taken the mass of the rotating electron to be constant, while his own formulas indicated that the electron velocity was too high to ignore relativistic corrections. It was necessary, therefore, to modify Bohr’s theory by introducing the relativistic mass of the electron, which varied with velocity, in formulas. With this substitution the electron orbit was no longer a fixed ellipse, but an ellipse that rotated in-plane around one of its to foci, which was occupied by the nucleus. With this relativistic basis and his own method of quantization, in 1916 Sommerfeld was not only able to explain the fine structure of the hydrogen spectrum, but also that of x-ray spectra. Sommerfeld’s theory made a forceful impression at the time, acting as a fundamental confirmation of both quantum methods and relativity. Yet it was soon followed by more thorough critical reflection and more accurate experimental tests that quenched the initial enthusiasm. The number of spectral lines in Sommerfeld’s theory was less than the number observed with a spectroscope; furthermore, the theory was not applicable to atoms with multiple orbiting electrons. Even its extension to helium was unsatisfactory due to the mathematical difficulties we discussed above.

7.15 The correspondence principle The failures of the theories of Bohr and Sommerfeld were not disastrous. Besides the shorfalls, there was a markedly positive outcome of the Bohr doctrine; for the first time the immense field of spectroscopy had been unified, baring the nature of its laws to the world. The failures could only mean that some details of the theory were not quite sorted out, but did not invalidate the fundamental concepts.

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More serious, on the other hand, were the general criticisms levelled at the theory. While electromagnetic theory described radiation exactly, giving not only the frequency of a monochromatic wave, but also its intensity and polarization, Bohr’s theory could not give any indication on these last two properties, which are critical for completely defining monochromatic radiation. Aside from these inadequacies, Bohr’s theory had a striking contradiction: the interaction between nucleus and electron was treated in a classical manner, but in this classical framework Bohr had coarsely introduced a quantum caesura. The electron’s jumps between one orbit and another could not be described by classical mechanics, and yet the electron, while in a stationary state, obeyed classical laws, though it did not radiate energy, in contradiction with the laws of electromagnetism. The electron seemed to exist outside of the space and time that framed the classical laws of mechanics and electromagnetism. In short, Bohr started from classical concepts and arrived conclusions that were incompatible with classical physics: the deep incongruity manifested itself in the theory. Physicists, including Bohr, realized this. Waiting for a breakthrough, they attempted to limit the largest discrepancies between the consequences of the theory and physical reality, introducing preliminary guiding criteria which had an important impact on the ensuing scientific development. Bohr gave the canonical example, elaborating the correspondence principle in 1918, the origins of which can be traced back to his first works in 1913. A heuristic principle, it was stated by Bohr as follows: in developing the theory, one must be guided by the idea that as quantum numbers become large, the radiation emitted asymptotically tends to that which would be emitted if the system obeyed classical laws. In other words, the laws of the new physics have to reduce to the laws of classical physics when quantum discontinuities become very small, that is when the quantum of action goes to zero. In this way, classical physics, while generally recognized to be inexact, became a sort of guide in the discovery of quantum laws. Across several works, Bohr explained how this principle, which was anything but simple to understand, should be applied. Among the concrete results obtained by applying the principle of correspondence were Bohr’s approximate calculation of the intensity of spectral lines in quantum theory and the quantum dispersion formula given by his students Hendrik Kramers and Werner Heisenberg in 1923, which, while not in agreement with the classical version, was in accord with experiment. But as the applications of the correspondence principle grew, difficulties abounded. Physicists were forced to introduce more and more rules of

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selection or prohibition, which appeared unintuitive and mysterious. Among these, for example, was the exclusion principle formulated by the Austrian physicist Wolfgang Pauli (1900-1958) in 1925: it affirms that there cannot exist two or more electrons in the same physical conditions in a quantum system (Pauli exclusion principle). Physicists puzzled over the deeper meaning of this strange rule, whose application nevertheless led to results in agreement with experiment. There was much discussion: some tried to justify it with the indistinguishability of electrons. Many were more inclined to believe it simply a corrective rule that eliminated the negative consequences that arose from thinking of electrons as corpuscles, perhaps at odds with reality.

7.16 The composition of atoms By the end of the first world war, it was clear that, based on the hypotheses of Rutherford and Bohr, it was not possible to arrive at an anatomical theory of the composition of atoms through a mere synthesis of previous knowledge. On Bohr’s suggestion, physicists then attempted to confront the problem through an inductive approach: that is, making use of the abundant experimental results on chemical behavior, magnetism, and spectral lines of various atoms that could be presumed to depend on their constitution to deduce their structure and, in particular, the distribution of electrons. This was an impressive endeavor that led to great results, naturally more of an empirical nature than those which Bohr’s theory had been hoped to bring about. Atomic number was made the starting point for the study of the periodic classification of elements. Because there were 92 elements, there existed 92 atoms in which the number of peripheral electrons grew steadily from 1 to 92. Yet how were these electrons arranged? Were they all in the same orbit or were they in different ones? We have already seen that starting from the first models proposed by Thomson, electrons were assumed to be arranged in different layers, and only in this way the periodicity of Mendeleev’s table could be explained. Even Bohr’s theory – for the same interpretative requirement – supposed that on each quantized level there could be no more than a certain maximal number of electrons. This principle of level saturation and the overarching physical principle that states that the stable state of a system is always that of minimum energy were fundamental in the atomical theory of the atom. We will not go into detail, choosing instead to sketch out the methods followed. The fundamental observation was the following: atoms in the first column of Mendeleev’s table (hydrogen, lithium, rubidium, caesium,

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copper, silver, cold) have an analogous spectrum to that of hydrogen, are easily positively ionized, significantly unstable, and chemically reactive. From these experimental observations, one can reasonably infer that all of these atoms have an analogous structure to that of hydrogen, thus belonging to the class of hydrogenoids having the properties described above. The second column in the periodic classification can be thought of as arising from the previous one with the simultaneous addition of a new orbital electron to each atom and an elementary charge to the nucleus. The successive columns are analogously related. The distribution of electrons in the frame of hydrogenoid atoms is found using a unique sequential process in which the prohibition rules come into play. The hydrogen atom is composed, as we have described many times, of a nuclear proton and an orbital electron. In its stable, nonexcited state, the system has minimal energy, and thus the electron rotates in the orbit closest to the nucleus, the energy level that is labeled by K. If a second electron is added to this K level, giving it certain characteristics, and a positive charge is simultaneously added to the nucleus, a helium atom is obtained. With the addition of the second electron, the K level becomes saturated, or closed. If a third electron is added to the atom, it cannot go in the K level, but rather is placed in the immediately successive level, labeled by L: in this way (while still simultaneously adding a positive charge to the nucleus) lithium is obtained. Thus, one by one, Bohr, Pauli, and their other collaborators from the Copenhagen school constructed the 92 different atoms of the periodic table, for each one indicating the distribution of the peripheral electrons in the various levels: by 1927 this impressive inductive structure was basically complete. The existence of quantized levels and the preceding structural schemes of atoms were fully confirmed by ionizing radiation experiments and the structure of x-ray spectra.

8. WAVE MECHANICS

THE NEW MECHANICS 8.1 A statistical extension of the radiation law The new mechanism introduced by Bohr for the emission and absorption of radiation freed quantum theory from the restrictions that had tied it to linear oscillators or equivalent special cases. The question of deriving the blackbody radiation formula from these new hypotheses naturally arose. Albert Einstein dedicated himself to solving this problem, having carefully followed Bohr’s works and holding the correspondence principle in high regard, which appeared to many a prelude of quantum theory’s embedding into classical mechanics. In 1917, Einstein’s greatest contribution to quantum theory appeared, his famous paper in which the probabilistic notions of the radioactive decay law are applied to Bohr’s atom. Much like every radioactive atom decays at an unpredictable time through an apparently causeless process, the transition of an electron in an atom is entirely unpredictable and should be studied with statistical laws. Einstein formulated these laws, stating: 1) in the presence of a radiation field, the electron’s probability of transition per unit time, both in the sense of emission and absorption, is proportional to the intensity of the radiation; 2) even in the absence of external perturbations, the spontaneous transition of electrons from higher energy states to lower energy states will occur, with a probability proportional (per unit time) to the number of atoms initially in the excited state. Thus, notions from the law of radioactive emission, to which Einstein evocatively referred, were brought to the phenomenon of radiation. With these foundations, tracing and extending his theory of Brownian motion, Einstein was not only able to obtain Planck’s formula for blackbody radiation, but also discussed the problem of impulse exchange between atomic systems and radiation in a general manner; concluding that for each elementary radiative process there had to be an impulse of hȞ/c emitted in an entirely random direction. This conclusion exacerbated the wave-particle conflict, since such a description of the emission process forbade the possibility of spherical waves. Einstein regretfully noted this consequence at the end of his paper. “These characteristics of elementary

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processes,” he wrote, “seem to render almost inevitable the development of a suitable quantum treatment of radiation. The weakness of the theory lies in the fact that on one hand it does not permit steps forward towards the wave conclusion, and on the other it resorts to ‘chance’ when time and the direction of elementary processes are concerned; yet despite this I have full confidence in the validity of the path we have taken.”276 The influential starting point for indeterministic physics advanced by Einstein is worth underlining, as he remained a convinced determinist for the rest of his life. For Einstein, his appeal to statistics remained firmly within the realm of the causal laws of classical physics. According to him, the exact instant of an electron’s transition in an atom is determined by causal laws that depend on the structural properties of the excited atom; only our ignorance and the complexity of these laws forces us to resort to statistical considerations, which are consequently only of practical value. Yet history does not lie: it was Einstein himself who first extended the statistical approach from radioactivity to other fields of physics.

8.2 The wave-particle debate Around 1923, studies of the photoelectric effect, research on Bohr’s atom, Einstein’s theory mentioned above, and the discovery of the Compton effect all forced physicists to take the possibility of discontinuous radiation seriously. The number of phenomena that could no longer be interpreted using the framework of classical optics was increasing almost daily. Yet again experiment had shattered the confines of a theory. By this point, in the course of its history, physics had often found itself in the presence of theoretical insufficiencies, which it had always been able to surpass by proposing novel and more comprehensive theories that explained both old and new phenomena. However, this new crisis gave rise to a rather different situation from the ordinary pattern. The new theory, that of light quanta, did not seem more comprehensive than the wave theory, in the sense that however hard one tried, certain phenomena described by classical optics simply could not be explained by the new photonic theory. The clash between wave and particle, two paradigms that seemed irreconcilable in the first years after World War I, was often compared to the 18th century disagreement between Newton’s emissive theory and 276

A.Einstein, Quantentheorie der Strahlung, in “Physikalische Zeitschrift”, 18, 1917, p. 128.

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Huygens’ wave theory. However, we do not think this a fitting comparison at all. In the 18th century, each of the two theories more or less explained all of the optical phenomena that were then known; the choice of one or the other reduced to applying a criterion of simplicity or personal preference, becoming almost an aesthetic judgement on the part of the scientist deciding. The intuitive type was undoubtedly inclined towards Newton’s theory, which could explain the most common optical phenomenon, rectilinear propagation of light, in an immediately intuitive manner; while the scientist prepared to sacrifice some intuition for greater logical coherence chose Huygens’ theory. It is then indicative of this predisposition that the wave tradition was carried on into the 18th century almost exclusively by mathematicians: Jean Bernoulli, fils (1710-1790) and Euler. In the 20th century, on the contrary, the disagreement was not between physicists, but in physics. Each physicist was forced to attribute a wavelike nature to light to explain some phenomena (diffraction) and a corpuscular nature to light to explain others (the photoelectric effect). In short, each physicist, to repeat a joke told by William Bragg, was forced to consider light as composed of particles on Monday, Wednesday, and Friday, and suppose it wavelike on Tuesday, Thursday, and Saturday. Only for certain phenomena (the rectilinear propagation of light and the Doppler effect) could both theories be equally applied. A last-ditch effort to save classical optics that has been completely forgotten today, but was much discussed in the years between 1924 and 1927, was attempted by Bohr, who had hoped to frame the theory of light quanta in terms of wave theory through the correspondence principle, or at least bridge the gap between the two theories. Together with Kramers and John Slater in a 1925 paper, Bohr advanced the curious hypothesis that atoms constantly emit all the radiation corresponding to the possible transitions from their current state to any other stationary state. Yet such radiation would be “virtual” – devoid of physical action. It becomes real only in the cases governed by Einstein’s probabilistic law. Among other things, the theory also claimed that the principles of energy and momentum conservation were not valid for individual processes, but held only in a statistical sense for the aggregate of many elementary processes. And so, in essence, classical optics was saved by interpreting all laws in a statistical manner. Despite their best efforts to remain within the classical framework, physicists could not avoid statistical laws! Sommerfeld wrote the following response to Bohr’s attempt: “It is currently too early to make judgements on the theory. Yet it seems that the compromise that is sought through a statistical approach between the wave

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theory and the theory of quanta has the characteristics of an experiment. Nor does it seem to us any less delicate to introduce certain entities in physics like the field of virtual radiation which, by assumption, are not observable. This is analogous to what we saw with luminous ether which, as a consequence of the relativity of motion, lost its relevance and finally ceded to the progress of science. We consider the opposition between the wave theory and the theory of quanta as a provisional enigma, and we presume that to resolve it we will need deep modifications of the fundamental concepts of the electromagnetic field or, as in the theory of relativity, of the fundamental principles of physical knowledge.”277

Fig. 8.1 – Louis De Broglie

277 A.Sommerfeld, Über die letzten Fortschrifte der Atomphysik, in “Scientia”, 39, 1926, p. 18.

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8.3 The wave associated to a particle When Sommerfeld wrote these words, new and profoundly revolutionary ideas had already been advanced three years earlier, which for the moment did not resolve the wave-particle debate, but, on the contrary, extended it from the photon to the electron. The young French physicist Louis De Broglie (1892-1987; Fig. 8.1) was the architect of this breakthrough. In 1919, as soon as he was discharged from the military, he began to work in the private laboratory of his brother Maurice (1875-1960), which was open to young researchers who dedicated themselves to the study of x-ray spectra and the photoelectric effect, the most advanced fields in physics research at the time, for which Maurice De Broglie had already gained international recognition. X-ray studies had naturally led Louis to reflect on the nature of radiation and in particular Einstein’s works on light quanta. We can trace these reflections in two notes from 1922 concerning blackbody radiation, which pointed the young physicist “to the idea that it was necessary to find a general synthetic scheme that allowed for the unification of the corpuscular point of view.” With admirable simplicity, De Broglie later described the analogies between the corpuscular, or particle, and wave theories that had captured his imagination in his Nobel lecture, when they had already been accepted by the physics mainstream: “Firstly the lightquantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation W = hȞ, which contains a frequency Ȟ. Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. On the other hand, the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations.”278 All of a sudden, through a psychological process that he acutely analysed in a semi-autobiographical writing, at the end of the summer of 1923 De Broglie experienced a sort of crystallization of thought: “the spirit of the researcher instantly beholds, with great clarity… the grand outlines of new ideas that had been obscurely formed inside him, and he gains in an instant the absolute certitude that the employment of these ideas will allow for the 278 L. De Broglie, Conférence Nobel prononcée à Stockholm, le 12 décembre 1929, p. 4, in Les prix Nobel en 1929, Santesson, Stockholm 1929.

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resolution of the majority of the problems posed and clear up the whole question, shedding light on analogies and harmonies that had hitherto been ignored.”279 The prospects of the synthetic theory that had so suddenly dawned upon De Broglie were the subject of three famous papers presented to the Académie des sciences of Paris, marking the genesis of wave mechanics. These three papers would surprise a modern reader not only for the audacity of their ideas, but also for the great simplicity of the mathematical methods employed, especially when compared to the complex and at times excessively flamboyant mathematical theories of theoretical physicists. De Broglie begins his first paper, which like the second was presented to the academy by Perrin, as follows: “Consider a moving material of proper mass m0 which is in motion with respect to a fixed observer with a speed v = ȕc (ȕ < 1). According to the principle of the inertia of energy, it must have an internal energy equal to m0c2. On the other hand, the quanta principle leads us to attribute this internal energy to a simple periodic phenomenon of frequency Ȟ0, such that hȞ0 = m0c2, with c being the limiting velocity in the theory of relativity and h the Planck constant.”280 In short, this fundamental postulate says that each matter particle is the source of an intrinsic vibration with frequency determined by the relation above. If the particle is in motion, relativistic reasoning leads to the conclusion that the vibration appears to a fixed observer as a wave moving with greater velocity than the particle. Having obtained this result, De Broglie studied the uniform motion of an electron in a closed trajectory, demonstrating that the quantized trajectories of Bohr-Sommerfeld theory could be interpreted as a resonance effect of the wave’s phase along the length of the closed paths. In other words, if an electron travels in a closed loop, the wave must follow and form a continuous loop of undulations, and thus the length of the loop must be equal to to a whole number of wavelengths. This is akin to the laying of wallpaper, which fits along the walls of a room with no visible juncture only if the perimeter of the walls contains a whole number of motifs of the ornamental pattern. The stationary orbits are then those that contain exactly 1, 2, 3, … wavelengths associated with the circling electrons. A calculation shows that these orbits turn out to be precisely those postulated by Bohr.

279

L. De Broglie, Continu et discontinu en physique moderne, Michel, Paris 1941, pp. 82-83. 280 L. De Broglie, Ondes et quanta, in “Comptes-rendus de l’Académie des sciences”, 177, 1923, p. 507.

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In his second paper, De Broglie applied his ideas to the case of photons and outlined a theory of interference phenomena and light diffraction supposing that the distribution of photons is determined by waves, and allowing for regions of darkness where waves destructively interfere with one another. In the third paper281, the new theories led De Broglie to Planck’s blackbody radiation law. Furthermore, he established a now classic correspondence between Maupertuis’ principle of least action applied to the motion of a particle and Fermat’s principle applied to the propagation of its associated wave. With this correspondence, the new mechanics explained the link, which had been noted by Hamilton and later Jacobi (§ 1.4), between classical dynamics and geometrical optics. While this connection had captivated De Broglie since he was young, it does not appear that it was, as some now claim, the inspiration for his research. The three brief papers of 1923 were fused, integrated, and historically framed in his 1924 doctoral thesis, where the new ideas had even broader applications. In it, for example, De Broglie showed that the new principles could quantitatively explain the Doppler effect, reflection from a moving mirror, and radiation pressure, arriving at formulas identical to those obtained by wave theory. The reception of De Broglie’s ideas in the scientific environment is best summed up by the following anecdote told by Born. In 1925, Einstein invited him to read De Broglie’s theory, saying: “Read it! While it may seem the writings of a madman, it is a solid construction.”282

8.4 Quantum Mechanics The principles of wave theory were still mostly unknown when another rather different attempt was made to overcome the conceptual difficulties obstructing theoretical physics. This effort was undertaken by a very young student of Sommerfeld, Werner Heisenberg (1901-1976), and rapidly developed thanks to the contributions of Born and Pascual Jordan. Heisenberg was imbued with the spirit of the “Copenhagen school” formed around Bohr, and had dedicated his first works to the application of the correspondence principle. His theory arose from the union of this principle and his phenomenological approach to studying physical problems. 281

De Broglie, Les quanta, la théorie cinétique des gaz et le principe de Fermat, ibid., pp. 630-32. 282 M. Born, La grand synthèse, A. George (editor), Louis De Broglie physicien et penseur, Michel, Paris 1953, p. 185.

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According to Heisenberg, all quantities that are not accessible to experiment should be excluded from the construction of a physical theory, which can then only include quantities that are observable. In the preceding theories of the atom, for example, scientists had introduced the trajectories, positions, and velocities of electrons, yet who had ever observed the path of an electron? Who had ever experimentally measured the position or velocity of an electron? What was known was the stationary states of an atom, the transitions between these states, and the energy emitted or absorbed in these transitions. Any atomic theory could only take into account these quantities. Whether Heisenberg actually succeeded in realizing his philosophical programme of excluding all unobservable quantities from theory is not quite as clear. Nevertheless, this new direction given to the theoretical approach marked an important stage in the evolution of modern physics. Notions connected to experiments could not be expressed through the usual continuous functions because of quantum discontinuities; a new formalism was needed, for which Heisenberg perhaps drew inspiration from the correspondence principle. Classical theory expressed each quantity associated with a quantum system using Fourier series expansions, quantum theory instead decomposed the same quantity into elements corresponding to different atomic transitions: according to the correspondence principle, these two procedures had to asymptotically converge for a very large number of quanta. Based on these considerations, Heisenberg had the radical idea to decompose every quantum quantity and represent it with a table of numbers, analogous to what mathematicians call a (infinite) matrix. In reality it was Born, then his assistant at Göttingen, who applied the matrix interpretation to Heisenberg’s quantization conditions and found himself in front of the strange formula pq – qp = (h/2ʌ)i, where q indicates the position coordinate and p the momentum (more precisely the respective matrices of position and momentum). Following the discovery of this formula, known today as the commutation relation, there was a period of feverish work by Born and his student Jordan, who were soon joined by Heisenberg, that culminated in a 1926 paper enumerating the fundamental principles of quantum mechanics (an expression introduced by Born) and its extension to electrodynamics. The mathematical rules governing the objects of this new theory turned out to be the same ones discovered by Charles Hermite for matrices, which had been studied in mathematics but had up until then lacked significant physical applications. Now, these rules are not always the same as those of ordinary algebra; in particular, the product of two matrices does not obey the commutative law, as the commutation relation found by Born shows.

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By the end of 1925, a few months before the aforementioned work by Born, Heisenberg, and Jordan appeared, Paul Adrien Maurice Dirac (19021984), a very young English physicist, published a paper inspired by one of Heisenberg’s conferences at Cambridge which contained the same results obtained by the Göttingen researchers. The only difference was that he did not refer to matrices, but instead reinvented and elaborated the theory of such non-commutative objects. Indeed, non-commutativity appeared to him a concept so fundamental for quantum mechanics that he based his entire analysis on it. According to Dirac, the transition from classical mechanics to quantum mechanics is brought about by substituting the quantities represented by ordinary numbers in classical mechanics with quantities represented by “quantum numbers”, whose product was not commutative. Heisenberg, guided by the correspondence principle, introduced Planck’s constant into his scheme such that for macroscopic phenomena, where h is small, the product of mechanical quantities is always commutative, thus recovering classical mechanics. We cannot further expound this theory without the help of mathematical formalism, which is practically essential for the discussion quantum mechanics. The rigour and precision of the formalism and the results it obtained (the demonstration of the existence of stationary states and quantized energies, a calculation of the energy levels of a linear oscillator and the hydrogen atom, etc.) filled young physicists with enthusiasm. However, the exceedingly abstract formulation of the theory, which substituted physical concepts with mathematical symbols, made some believe that physics had gone astray and perplexed those physicists who, like Einstein, held that “every physical theory should, without any calculation, be able to be illustrated by images so simple that even a child could understand it.”283

8.5 Wave equations The logical soundness of De Broglie’s construction, momentarily ignoring the question of observability of the associated wave, attracted the attention of the Austrian physicist Erwin Schrödinger (Fig. 8.2). Schrödinger was born in Vienna on 12 August 1887, and started as a professor of theoretical physics at Zurich in 1921, after which he succeeded Planck in Berlin from 1927 to 1933, then moved to Oxford (1933-36), and later Graz (1936-38). In 1938, Nazi persecution forced him to take refuge in Dublin, where he was granted a professorship that he 283

Referred by De Broglie, Nouvelles perspectives en microphysique, cit., p. 236.

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maintained until 1956, when he returned to Vienna, where he died a few years later on 4 January 1961. A prolific scientist and philosopher, he is mainly known for having found the explicit equation describing the waves of wave mechanics in 1926, and subsequently deducing from it a rigorous method to study quantization problems. For these works he received the Nobel prize in physics in 1933 along with Dirac.

Fig. 8.2 – Erwin Schrödinger

Schrödinger’s equation, obtained by transforming classical equations following the approach indicated by Hamilton, has the property that its coefficients are not all real numbers, but also include imaginary numbers. In classical physics, on the other hand, the equations describing the propagation of waves are always real, and while real functions are sometimes substituted with imaginary (or complex) ones, this is simply a computational tool. However, in the Schrödinger wavefunction, traditionally denoted by the Greek letter ȥ, the imaginary factors are unremovable and thus appear intrinsic to the very phenomenon that they describe. In other words, while waves in classical physics correspond to vibrations of a real medium (air, for example, for sound waves) or a hypothetical one (the

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ether in the case of light propagation), the waves of wave mechanics cannot be considered vibrations of a physical medium. This conclusion only served to confirm De Broglie’s intuition: it is not possible to attribute a physical existence to a wave that does not carry energy, like his “associated wave”; this is a fictitious wave, as De Broglie had called it, or a phantom wave, as Einstein had baptized it with his habitual imaginative humour. To construct his equation, Schrödinger started from the classical equations of wave propagation and thus from Newtonian mechanics. It followed that the equation did not take into account relativistic corrections for particles at high velocities; it was essentially only valid for slowmoving particles. Because of this, Schrödinger’s equation had a few shortcomings, which physicists tried to compensate by applying the necessary relativistic corrections. It was only in 1928, however, that Dirac, after having perceptively critiqued the previous attempts at a relativistic wave equation, demonstrated the necessity of a broader generalization. In consequence, he formulated a theory of considerable physical importance that however was even more unusual and abstract than quantum theory and wave mechanics. One of the greatest strengths of Dirac’s theory was that the hypothesis of the rotating electron followed naturally from it. This hypothesis proposed by George Uhlenbeck and Samuel Goudsmit in 1925 had attracted much attention: the two physicists likened the electron to a charged sphere that rotates about one of its own axes. The electron thus comes to possess an angular momentum and magnetic moment of its own. To denote the intrinsic rotation and moment of the electron, Uhlenbeck and Goudsmit used the word spin, now adopted by all physicists. The hypothesis of spin has been since been fully confirmed, and in retrospect seems to have been the missing ingredient in the previous theories of the atom. Dirac’s theory harmoniously reconciled relativity, quanta, and spin, which on the surface had appeared independent concepts. Another surprising consequence of the theory was its prediction that electrons could also exist in states of negative energy having properties that appear rather unusual to our ordinary physical intuition. To accelerate negative energy electrons, their energy would need to decrease, and therefore they would seemingly slow down; on the other hand, when brought to rest, their energy would grow. These conclusions brought about heated discussions between advocates and opponents. Dirac attempted to interpret the strange consequences with a clever hypothesis that, however, seemed too artificial: predicting the (uncommon and ephemeral) existence of positive electrons. Physicists

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were generally skeptical regarding this prediction when, in 1932, first Anderson and then Blackett and Giuseppe Occhialini demonstrated that in atomic decays induced by cosmic rays, there appeared particles that behaved exactly like the positive electrons predicted by Dirac’s theory. Later, positive electrons, called positrons, were artificially produced by bombarding certain heavy elements with ܵ-rays and their charge to mass ratio was measured, which resulted equal to that of negative electrons. Today there is no doubt regarding the existence of positrons, which number among the basic constituents of matter.

8.6 The equivalence of wave and quantum mechanics With his equation, Schrödinger could treat the problem of the stationary states of a quantum system and calculate its energy values. In doing this, he calculated energy levels and thus spectral terms, in many cases obtaining the same values that had been obtained with the older theory of quanta. In many other cases, however, the values found by Schrödinger were different from those obtained by the older theory, but more in line with experimental results. The simplest example is that of the linear oscillator. Planck, at the dawn of the theory, had already quantized the linear oscillator by supposing that its energies were integral multiples of the energy quantum hv. Yet certain physical phenomena (for example the band spectra of diatomic molecules) were better interpreted by supposing that the linear oscillator was quantized in half-integer multiples, that is as a product of the quantum h and a number from the sequence 1/2, 3/2, 5/2, … (2n-1)/2. The quantization method in Schrödinger’s theory predicted precisely these half-integer multiples, setting it apart from the quantum theory of the past. Heisenberg’s matrix mechanics also led to the same results. Schrödinger sensed that this was not simply a coincidence, but rather that behind this convergence of results there lay a deeper reason, the fundamental equivalence of wave mechanics and matrix mechanics. This result, which Schrödinger rigorously demonstrated in a famous 1926 paper, astounded the physicists of the time because of the stark difference in concepts and mathematical formalism employed by the two theories. Nevertheless, Schrödinger’s paper showed without a hint of doubt that matrix mechanics is simply a mathematical reformulation of wave mechanics. The latter is slightly more intuitive and requires fewer mathematical methods, making it the more frequently utilized; matrix mechanics, on the other hand, often leads more quickly to results.

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8.7 Experimental verifications Ideas as innovative as those of wave mechanics require experimental evidence to become solidly established. An experimental test of wave mechanics was proposed by De Broglie: “A flux of electrons,” he wrote, “that passes through a small enough aperture would give rise to diffraction phenomena. It is in this way that experimental confirmation of our ideas must perhaps be sought.”284 Of course, this is an idealization. One can easily calculate that the wavelength associated with a slow-moving electron, ignoring relativistic corrections, is at most a few angstroms,285 of the order of X-ray wavelengths. We have already seen that even the finest artificial gratings have lines that remain too distant from one another to obtain x-ray diffraction without recourse to the Compton effect. Consequently, electron diffraction could not be produced by the best gratings available, much less the simple fissure described by De Broglie. A few years before, De Broglie published his papers on wave mechanics, Clinton Joseph Davisson (1881-1958) and his collaborators from the Bell telephone company in New York conducted experimental research into the emission of secondary electrons (that is, electrons emitted from the collision of other electrons, called primary, with an object), obtaining results that were interesting, but difficult to interpret theoretically. In the summer of 1926, Davisson discussed his experiments in London with Owen Richardson, Born, and James Franck, reaching the conclusion that his research could lend itself to the experimental confirmation of De Broglie’s theory. Returning to his experimental work with this aim, Davisson and his collaborator Lester Alberto Germer (18961971) were able to announce the following spring that they had experimentally observed electron diffraction, obtained by firing electrons perpendicularly towards the face of a nickel crystal and collecting the diffracted electrons in a Perrin cylinder, which could move in a circle around the crystal. A few months later George Paget Thomson (18921975) and his student Alan Reid (who died in a car accident not much later at the age of 22) were able to obtain electron diffraction independently of Davisson by shooting electrons at metallic foils and crystalline powders 284

L. De Broglie, Quanta de lumière, diffraction et interférences, ibid., p. 549. To be precise, the wavelength associated with the motion of a nonrelativistic electron is found to be: ଵଶ,ଶସ O = ௏ 10ି଼ cm (V in volt) and since electrons required must be accelerated by at least ten volts, the calculation above gives the result indicated in the text.

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and then conducting observations through photographic means, which aside from simplifying Davisson’s cumbersome setup, showed electron diffraction ad oculus and allowed for its immediate comparison to images of x-ray diffraction. On the near simultaneity and independence of the two research efforts undertaken in New York and the small city of Aberdeen, Davisson remarked: “That streams of electrons possess the properties of beams of waves was discovered early in 1927 in a large industrial laboratory in the midst of a great city, and in a small university laboratory overlooking a cold and desolate sea. The coincidence seems the more striking when one remembers that facilities for making this discovery had been in constant use in laboratories throughout the world for more than a quarter of a century. And yet the coincidence was not, in fact, in any way remarkable. Discoveries in physics are made when the time for making them is ripe, and not before; the stage is set, the time is ripe, and the event occurs – more often than not at widely separated places at almost the same moment.”286 Put more simply, laboratories had been equipped for the discovery of electron diffraction for 25 years, but had lacked wave mechanics to propose and guide the research. The tests continued, with varying conditions and apparatuses, until in 1929 Arthur Rupp was able to observe electron diffraction using a simple optical grating and an oblique angle of incidence, such that the electrons were nearly parallel to the grating, in accordance with the setup previously proposed by Compton. As often happens, the experimental confirmation of electron diffraction, which required much skill on the part of the first experimenters, today is so simple due the perfecting of technical instruments that it can be repeated as a lesson demonstration in school. Diffraction was also observed with atoms (as Otto Stern and Immanuel Esterman first showed in 1929 working with hydrogen atoms), molecules, and any sort of corpuscular beam in general. The tests, which were conducted in variety of velocity conditions, confirmed the accuracy of the fundamental relations that connected waves and particles; in the case of high velocities, experimental results were exact after taking into account relativistic corrections, and thus these experiments also provided an indirect confirmation of relativity. Electronic diffraction was immediately applied to the study of surface phenomena and the composition of small crystals, proving itself much more suited than x-rays for the investigation of thin membranes due to the reduced penetration of electrons. In particular, this method is now 286

C.J. Davisson, The Discovery of Electron Waves, Nobel Lecture, p. 1 in Les prix Nobel en 1937, Santesson, Stockholm 1938.

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industrially employed in the study of the lubricating power of oils and the nature of the surface layers of metals. It is not easy to list all the applications of electronic diffraction, which has become a science in its own right with its methods, techniques, and specialists. Yet, because of its fame and remarkably widespread use (especially in biological fields), we must at least mention the electron microscope, which was pioneered by Max Knoll, Ernst Ruska, and Ladislaus Marton, and constructed for the first time in 1931 by Bodo von Borries and Ruska, and later in 1933 in the physics laboratory of the science department of Besançon, by René Fritz and Jean-Jacques Trillat. Electron microscopes typically attain magnifications of 20,000 times ordinary size, in some cases reaching 100,000, allowing for the ability to distinguish between two points a few tens of angstroms apart (1 Å = 10-8 cm). It is then easy to see how much this instrument furthered the fields of medicine and biology in the study of bacterial and viral morphology and cancer research. We note that despite this incredible usefulness, many physicists initially believed that the electron microscope could not be applied to biological research because it was said that organic matter would be destroyed by electronic bombardment. The use of wave mechanics in nuclear theory obtained its first great success in 1928, when the Russian-American physicist George Gamow explained the passage of particles through the potential barrier of heavy nuclei; a few years later, in 1934, Fermi, applying the ideas of wave mechanics, successfully explained a paradoxical phenomenon that he had discovered: slow neutrons (whose velocity is of the order of the velocity of thermal excitations) are particularly effective in producing artificial radiation if they are utilized as projectiles. Yet we will return to these and other applications of wave mechanics in the next chapter. Since wave mechanics had such wide-ranging applications in nuclear physics, its success in theoretical chemistry is no surprise, where it described and predicted a great number of chemical reactions with its fundamental interpretation of chemical valences, which had remained a mystery until then. In conclusion, wave mechanics renewed all theories regarding the structure of matter; today it is also indispensable for the understanding of macroscopic phenomena. It is not only a science for physicists, chemists, and biologists, but also for engineers. If the value of a theory is measured by the number and importance of its applications, wave mechanics, contrary to popular misconceptions, is one of the most fruitful ideas of modern physics.

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8.8 Quantum statistics Having conceived of the photon as a particle, physicists attempted to derive the radiations laws by treating radiation as a photon gas and applying the same statistical criteria that classical physics had so successfully applied to ordinary gases, which determine the partition of velocities and thus energies of n molecules in a gas of energy E and volume V. For the photon gas, the problem differs from that of a molecular gas essentially due to the fact that a molecular gas has a fixed number of molecules, while a photon gas contains a variable number of photons because the walls of the container can absorb and emit radiation, and thus photons. Another difference arises from the quantum hypothesis that reduces the total number of available states of the photonic system in consideration. Taking these into consideration, the problem was formulated according to the ideas of classical physics, and for a photon gas in a blackbody cavity one obtained Wien’s radiation law, which, as we have seen (§ 6.3), did not agree with experiment. In 1924, the Indian physicists Satyendranath Bose overcame this issue by postulating that photons cannot be described by the statistical criteria of ordinary matter particles, which had until then seemed to physicists the only possible, or even imaginable ones. Bose’s reasoning essentially amounted to the following: all statistical questions in essence reduce to determining how to distribute a certain amount of objects in a certain amount of cells. To give an example, suppose we have two objects, indicated by the symbols + and -, to distribute in two cells, where each cell is represented by a set of parentheses (). According to classical statistics, there are the following four ways to distribute them: (+) (-); (-) (+); (+-) ( ); ( ) (+-). However, Bose, observing that photons are indistinguishable from one another, further deprived them of individuality, which classically is quite different from indistinguishability, and concluded that the first two distributions were the same, leaving the following three possible distributions of two photons (now both indicated by the same sign: +) in two cells: (+) (+); (++) ( ); ( ) (++). Whether or not Bose’s justification was acceptable, the fact of the matter is that by applying the corrections he suggested to the classical

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statistical procedure, one arrived at Planck’s law, which as we know had been fully confirmed by experiment. Einstein edited the German translation of Bose’s paper and simultaneously received the manuscript of De Broglie’s thesis from Langevin, who was the supervisor at the young French physicist’s doctoral examination, in which he had deduced Planck’s law from his wave hypothesis. De Broglie observed that if the motion of a particle was associated with the propagation of a wave, only the standing waves that were resonant in the particle’s container could be considered, thus modifying the statistical calculation of classical mechanics. Einstein was greatly impressed by De Broglie’s considerations and brought them closer to those of Bose in two papers written in 1924 and 1925. In these papers, Einstein also applied the new statistical criteria to ordinary gases, deriving an interesting theory that explained the classical behaviour of gases in ordinary conditions and at very low temperatures. The success of this approach, called Bose-Einstein statistics, encouraged physicists to look for other cases in which it might be convenient to modify the classical statistical criteria. In 1926, Fermi observed that if Nernst’s principle was to remain valid even for an ideal gas, then the specific heat at constant volume of a monoatomic gas given by classical thermodynamics could only be an approximation valid for high temperatures; in reality this specific heat tends to zero as the absolute temperature tends to zero. Yet to notice the variation in question, the motions of ideal gases must also be quantized. Fermi dedicated an important paper to this quantization problem that appeared that year in the “Rendiconti dell’Accademia dei Lincei”. The quantization rules given by Sommerfeld appeared insufficient to him, as they led to an expression for specific heat that went to zero as absolute temperature did, but also depended on the total quantity of gas, in disagreement with all experimental results. Sommerfeld’s rules therefore had to be corrected by keeping in mind that, according to Fermi, ideal gases contain indistinguishable components. Drawing inspiration from Pauli’s principle, Fermi stated his fundamental hypothesis: “We will admit that in our gas there can be contained at most one molecule, whose motion is characterized by certain quantum numbers, and we will show that this hypothesis leads to a theory perfectly consistent with the quantization of the ideal gas, and that in particular explains the predicted decrease of specific heat for low temperatures, and leads to the exact value for the

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entropy constant of the ideal gas.”287 This hypothesis is equivalent to assuming that this gas, which obeys the Pauli exclusion principle, also obeys another statistical criterion, which was independently and almost simultaneously adopted by Dirac. Returning to the simple example that we presented earlier, this criterion consists in supposing that the two particles can only be distributed across the two cells in a single way: (+) (+). The two new statistics and their corresponding thermodynamics, which were slightly different from one another, asymptotically approach classical mechanics when quantum discontinuities tend to disappear. When applied to real gases, the differences between these new thermodynamic theories and classical thermodynamics are so small as to be undetectable. Therefore, the new statistics could not be experimentally confirmed through the study of real gases. Bose-Einstein statistics were experimentally confirmed by blackbody radiation and Fermi-Dirac statistics were verified by electronic theories of metals, as Sommerfeld first demonstrated. Physicists now accept that particles on atomic scales are divided in two categories: those that obey the Pauli exclusion principle, like electrons, protons, neutrons, and some atomic nuclei follow Fermi-Dirac statistics, and Dirac proposed to call them fermions; those that do not obey the exclusion principle, like Į-particles, photons, and other atomic nuclei follow Bose-Einstein statistics, and Dirac proposed to call them bosons.

THE INTERPRETATION OF WAVE MECHANICS 8.9 The position of the particle in the wave As we have observed, De Broglie demonstrated that wave phenomena were also connected to the corpuscles that were traditionally considered matter particles completely distinct from radiation. In substance, De Broglie extended the wave-particle dichotomy by showing that matter also exhibits wavelike behavior. His scope was to use this interpretation to overcome the conflict, through a synthetic theory that conserved the traditional aspects of the notions of particle and wave. De Broglie, whose scientific views had been formed around the classical ideas of a strictly deterministic physical reality firmly rooted in the framework of space and time, immediately thought to obtain the synthesis by considering particles 287

E. Fermi, Sulla quantizzazione del gas perfetto monoatomico, in “Rendiconti dell’Accademia dei Lincei. Classe di scienze fisiche, matematiche, e naturali”, 6th series, 3, 1926, p. 147.

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as a kind of wave singularity. In 1924 he wrote, “The whole theory will not become truly clear until we succeed in defining the structure of the wave and the nature of the singularity constituted by the corpuscle, whose movement should be predictable only by taking the undulatory point of view.”288 Schrödinger’s gave a different and much bolder interpretation, while still remaining faithful to classical rules. Schrödinger had resolved the antithesis between particle and waves by denying the existence of one of the two. According to this interpretation, only waves exist physically; particles do not have an objective reality, but rather are purely manifestations of wave propagation. To be precise, particles are small packets of waves. De Broglie, Einstein, and others rejected this interpretation, especially because wave packets have a tendency to disperse in space and therefore cannot represent a particle, which is endowed with prolonged stability. Yet Schrödinger objected “that the thing which has always been called a particle and, on the strength of habit, is still called by some such name is, whatever it may be, certainly not an individually identifiable entity.”289 He held this view for a long time, maintaining that a subtler and less “naïve” version of this interpretation, based on second quantization, was preferable to the “transcendental, almost psychic” interpretation now accepted by most theoretical physicists, which we will discuss shortly. In any case, Schrödinger’s interpretation did not have much following. De Broglie continued to contemplate the interpretation of a particle as a wave singularity. In this scheme, the motion of the singularity, tied to the time evolution of the wave, no longer obeys the laws of classical mechanics, which is a mechanics that describes particles governed only by the forces applied to them, but rather feels the effects of all obstacles encountered by the moving wave. This explains interference and diffraction phenomena, and all of wave mechanics then fits in the framework of classical physics. However, the development of this intuitive idea faced the significant difficulty that wave mechanics, like classical optics, described continuous waves without a singularity. After an intense period of preliminary work, De Broglie arrived at a fairly subtle theory that he called the double solution, whose second reconceptualization (occurring more than two decades later) we will briefly mention. Yet when it came time to analytically develop his 288

L. De Broglie, Sur la dynamique du quantum de lumière et les interférences, in “Comptes-rendus de l’Académie des sciences de Paris”, 179, 1924, p. 1029. 289 E. Schrödinger, The Meaning of Wave Mechanics, in Louis De Broglie physicien et penseur, edited by George, p. 24.

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conceptual theory, De Broglie encountered mathematical difficulties that he could not surpass. Therefore, when Lorentz asked him in the summer of 1927 to write a report for the 5th Solvay congress held in Bruxelles in October of that same year, De Broglie, fearing criticism stemming from the lack of mathematical rigour in his theory, decided to present a reduced version of his theory at the conference, at which the foremost physicists of the time were present. In this new version he admitted the existence of particles and placed them within the wave, in some sense acting as their pilot, hence the name pilot wave theory. This was not a successful attempt. By now everyone including De Broglie had accepted that ȥ was not a physical wave, but a phantom wave of statistical nature: to govern the motion of a particle with an entity devoid of physical meaning was neither a logical nor acceptable idea for physicists. In the discussion that followed De Broglie’s report, the young Pauli brought up serious objections, which De Broglie was unable to brush off; Schrödinger, convinced that particles did not exist, could not agree with De Broglie’s ideas; Einstein advised him to persevere in his attempt, which nevertheless he did not entirely agree with; and the young physicists present at the conference, like the majority of those absent, accepted Heisenberg’s interpretation.

8.10 The uncertainty principle Max Born, born in Wroclaw on 11 December 1882, played a leading role in the interpretation of quantum mechanics. Having been a professor of theoretical physics in Berlin since 1916, in 1922 he moved to Göttingen; in 1933, to escape Nazism, he moved to Britain, first to Cambridge and later to Edinburgh, where he held the chair of theoretical physics until 1953. In 1939 he became a British citizen, and in 1954 he returned to Göttingen, where he died on 5 January 1970. Born made important contributions to the theory of specific heats, optics, combustion, and piezoelectricity. His most influential contributions, however, were in the field of quantum mechanics, which was approached from the perspective of matrix mechanics at the Göttingen institute that he directed (§ 8.4). In 1928, he interpreted the square of the modulus of the wavefunction as a measure, at each point in space and instant in time, of the probability that the associated particle is observed in that point at that instant. For his research in quantum mechanics and especially, as the Nobel commission wrote, “for his statistical interpretation of the wavefunction”, he shared the Nobel prize in physics with Walter Bothe in 1954.

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Heisenberg agreed with this interpretation and further developed it. According to him, physics had to dispense with models and explicatory analyses. From a philosophical point of view this idea was not new; this was essentially the belief of pragmatist philosophy and had been placed at the basis of the theory of relativity. But, taken as the foundation of the new mechanics, and pushing its logical consequences as far as possible, it contributed to a revival of the pragmatic theories of Mach, Ostwald, and Vailati, quickly winning over many scientists and philosophers. As the pragmatists of the 20th century had opposed atomic theory, considering it a crude and naïve concept, so the new school declared that at the root of the crisis in physics lay the naïve image of the electron as a corpuscle – the “point particle” of classical mechanics. Heisenberg thought to overcome the particle-wave conflict by attributing to the two concepts only the value of analogy and limiting himself to the declaration that “the collection of atomic phenomena is not immediately describable in our language.” We must then abandon the concept of a point particle exactly localized in space and time. Physics, devoid of any metaphysical tendencies, can either give the exact position of the particle and a complete indeterminacy in time or an exact instant in time and a complete indeterminacy in space. More precisely, our physics can only tell us (in the best of cases) what probability we have of finding, in a certain instant, the entity we call a particle in a certain volume of representative space (not necessarily the same as physical space): this is the inevitable consequence of the introduction of quantization in physics. In 1927 Heisenberg proposed, with the support of Bohr and his school, to take this ineliminable uncertainty, whose magnitude can be calculated in many ways, as the characteristic principle of the new physics. For instance, Heisenberg calculated the limitations of the accuracy one could attain in determining the position and velocity of an electron, basing himself on the non-commutative relations of quantum mechanics; the language of wave mechanics, however, offers more intuitive approach, as Bohr demonstrated in 1928. Seemingly even more intuitive is the argument based on the following popular thought experiment proposed by Bohr that same year. Suppose, he said, that we want to determine the position and momentum of an electron at a certain instant. To this end, the most precise physical technique that we can use is that of illuminating the electron with a beam of photons; a collision between the electron and a photon will occur and the position of the electron will be determined to less than one wavelength of the photon employed, as is known even in classical optics. Consequently, we should use photons of small wavelength and thus high

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frequency, meaning that they will also have high energy hv and momentum hv/c. It follows that the photon will further alter the electron’s momentum. To obtain the exact position of the electron, the photon’s frequency would have to be infinite, but then its momentum would likewise be infinite and thus the electron’s momentum would be entirely indeterminate; if, on the other hand, we would like to measure the momentum, an analogous argument shows that the position of the electron would remain indeterminate. Concretely, if this argument is described quantitatively, one finds that if 'q is the uncertainty in the measurement of position and 'p is the corresponding uncertainty in the measurement of momentum, the following relationship always holds:

ο‫݌‬ο‫ ݍ‬൒ ݄/4ߨ If instead of p and q, two other conjugate quantities were chosen, like energy and time, one would find an analogous inequality. These inequalities are called the uncertainty relations of quantum mechanics.

8.11 Indeterminacy More intuitively, one can say that the fundamental principle of quantum mechanics is based on the observation that every instrument and measurement technique alter the quantity that is being measured in a manner that is not perfectly predictable. That instruments alter the quantities they measure was a well-known and almost trivial fact for classical theory. Yet it was also known that by refining the instruments the error could be reduced, so that at least theoretically, an error-free measurement could be attained in the limit of fine-tuning. It was this last limiting step that modern indeterminists rejected. We cannot maintain that the uncertainty could be zero unless we simultaneously give a hypothetical experimental procedure to obtain this error-free measurement. And since this experimental procedure does not exist, we must, in adherence to facts and not biases, hold that no physical quantity is exactly measurable, save for at the expense of the absolute indeterminacy of another quantity conjugate to the first. Nowhere in his writings was Heisenberg more explicit in this belief than in the following passage from a 1941 book: “When only a relatively low precision is needed, it is certainly possible to speak of the location and velocity of an electron to a level of detail, adopting the criteria of objects in our daily life, that is already extraordinarily minute. If instead we take into account the size of atoms, this precision is poor, and a natural law

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particular to this microscopic world impedes us from knowing the location and velocity of a particle with the precision we desire. Indeed, experiments can be performed that allow us to determine the position of a particle with great precision, but to carry out this measurement we must subject the particle to a strong external action, from which follows a great uncertainty in its velocity. Nature thus avoids a precise fixing of our intuitive concepts due to the inevitable perturbation that is tied to each observation. While originally the aim of all scientific inquiries was that of possibly describing all of nature as it would have been in and of itself, that is without our influence and our observation, now we understand that this very aim is unobtainable. In atomic physics it is completely impossible to disregard the modification that each observation produces in the observed object.”290 It was this basic question regarding the scientific knowledge of reality itself that caused the irreconcilable schism between the philosophy of modern physics and that of classical physics. Heisenberg, Bohr, Born, and perhaps the majority of modern physicists fully welcomed the neopositivist thesis, rejecting that the concept of physical reality had any meaning independent of an observer. Admittedly, “positivism” is not a univocal term. Nevertheless, what most physicists meant for positivism was the philosophical doctrine that ties all physical reality to our senses. Any object, for example a table, is only the collection of sensations that we receive from it. Asking what the table is “in and of itself” is a meaningless question. Thus, we can no longer speak, as the physics that had dominated the previous century claimed, of a real world separate and independent from us, sending us signals that activate sensory perceptions organized by our brain in coherent forms, giving us an experiential world distinct from the real world. The arrogant and unattainable ideal of 19th century physics had been to discover this real world hidden behind the experiential world. Such a purpose was meaningless for neopositivists. To them, the aim of science was not the discovery of fragments of the absolute reality of an external world, but the rational coordination of the collection of human experiences. It follows that physical laws are not “natural laws” in the classical sense, but convenient rules that economically describe the succession of our perceptions; rather than “discoveries”, they are purely “inventions”. The law of universal gravitation did not exist in nature before Newton invented it, in exactly the same way as the Ninth Symphony did not exist before Beethoven composed it. 290 W. Heisenberg, Mutamenti nelle base della scienza, trans. A. Verson, Einaudi, Torino 1944, pp. 81-82.

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This interpretation, which fundamentally unsettled the traditional ideas of classical physics, was firmly opposed by the proponents of classical mechanics. We will later quote a significant passage on this problem written by Einstein a few years before his death; for now we refer to what Planck wrote in 1923 and often repeated in his role as one of the most respected leaders of the classical current of thought: “The foundation and preliminary condition of all truly fruitful science is the metaphysical hypothesis, certainly not substantiated by purely logical means, but also such that logic will never disprove it, of the existence of an external world in and of itself, completely independent from us, from which we cannot obtain direct knowledge save using our senses. It is as if we could only observe an object through a pair of glasses whose color was slightly different from person to person. We certainly would not think to attribute all of the properties of the image we perceive to the glasses, as far as evaluating the object, we concern ourselves with determining to what extent the color in which it appears is due to our glasses. Analogously, scientific thought requires that the distinction between exterior and interior world be recognized and imposed. The branches of science have never preoccupied themselves with justifying this leap into the transcendental, and they have done well. If they had done otherwise, they would not have made such rapid progresses; moreover, what matters more is that a refutal was not and will never be something to fear, because questions of this type cannot be answered with reason.”291 Precisely because the problem could not be solved through reasoning, indeterminist physicists continued to believe that any physical process was inseparable from the instrument with which it was measured and the sensory organ with which it was perceived: the observed, the instruments of observation, and the observer thus constitute a physical totality. In this neopositivist epistemology a contradiction has yet to be found. This is the reason for their primacy, and not, as Planck acutely observed, that they introduced quantities that were observable in principle or questions that had physical meaning, because a certain quantity is observable or has physical meaning according to the theory with which it is interpreted; even classical physics, from its point of view, considers observable quantities and problems which make physical sense. In the philosophical view espoused by Heisenberg and Bohr, another serious consequence for classical physics was implicit: the uncertainty principle was immediately interpreted by them as a repudiation of the 291

M. Planck, La conoscenza del mondo fisico, trans. Enrico Persico, Einaudi, Torino 1954, p. 117.

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principle of causality. At the time, Bertrand Russel observed that it was a sophism to hold that if a phenomenon was not determined in the sense of not being measured – the sense given by Heisenberg – it must also be indeterminate in the entirely different sense of having no cause.292 To accept Russel’s point of view, we would need an exact definition of causality, an infallible criterion according to which one say with certainty, ‘this phenomenon is the cause of that one’. Such a criterion, however, was missing. “We cannot compel anyone with purely logical reasons,” wrote Planck, one of the most tenacious proponents of causality, “to admit a causal relationship even where there is total correlation. One need only think of Kant’s example of night and day. The causal link is not logical in nature, but transcendental.”293 Physicists had always connected the concept of a cause with our ability to predict future events. The possibility of making an exact prediction regarding the future had been an indication of the existence of a causal relationship, without pointing out exactly what this relationship was. Now, if no measurement can be called exact, then no previously-calculated prediction can coincide exactly with the result of the measurement; in other words, it is impossible to exactly predict a physical event. Born gave a simple example: suppose we have a small sphere that elastically bounces back and forth, at constant velocity, between two walls a distance l apart. If its velocity is known with an uncertainty e, the uncertainty in the prediction of the sphere’s position at time t will be et and hence grows with time, therefore if one obtains etc = l, the sphere can be at any point between the two walls; for times greater than tc, determinism gives way to complete indeterminacy. All indeterminist physicists adopted this point of view. In this sense, they maintained that rigorously exact laws, subject to a causal principle, did not exist in nature. Those that physics had called laws of nature were instead simply rules that had a very high degree of accuracy, but never reached absolute certainty. The new physics was thus forced to look for a statistical root in all ‘physical laws’ and reformulate them in terms of probability. Much like relativity had dethroned the Kantian categories of a priori space and time, quantum mechanics dethroned the category of a priori causality. New principles substituted the a priori characterizing these categories; for causality, for instance, this principle was probability. From this change in paradigm followed a shift in the direction of research. For example, confronted with a radioactive process, we saw that Marie 292

B. Russel, The Scientific Outlook, 1934, p. 94 Planck, La conoscenza del mondo fisico, trans. Enrico Persico, Einaudi, Torino 1954, p. 124.

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Curie had attempted to find out why one radioactive atom decays today while another nearby one decays in a thousand years. In the new conception, the problem is not posed in these terms: the question that is truly of interest is that of knowing, for a given radioactive element, how many atoms decay in a second, without seeking if and why this or that atom decays – in short, ignoring individual destinies. Classical physicists firmly opposed this consideration, as can be seen by Planck’s words: “It is true that the law of causality cannot be demonstrated any more than it can be logically refuted: it is neither correct nor incorrect; it is a heuristic principle; it points the way, and in my opinion it is the most valuable pointer that we possess in order to find a path through the confusion of events, and in order to know in what direction scientific investigation must proceed so that it shall reach useful results. The law of causality lays hold of the awakening soul of the child and compels it to continually ask why; it accompanies the scientist through the whole course of his life and continually places new problems before him. Science does not mean idle rest upon a body of certain knowledge; it means unresting endeavour and continually progressing development towards an aim which the poetic institution may apprehend, but which the intellect can never fully grasp.”294 If Planck’s attitude may appear unscientific, we add that certain neopositivist physicists also went too far and, focusing on the subject of consciousness, ventured to make idealistic affirmations, at times also tinged by a mathematical mysticism reminiscent of the Pythagoraeans. Sommerfeld, for example, considered the wave-particle duality analogous to the matter-spirit duality; for Eddington, the physical-mathematical universe was a shadow world created by the selective activity of our minds. According to him, one of the most significant recent progresses was the recognition that “physics operates in a world of shadows,” positions very similar to those taken by Jordan, Jeans, and Federigo Enriques.

8.12 The principle of complementarity Faced with the uncertainty relations, Bohr adopted a unique philosophical position, which he first explained at the international congress of physicists held in Como in September 1927, in the occasion of the centennial of Alessandro Volta’s death.

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M. Planck, Causality in nature, trans. W.H. Johnston, 1932.

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Bohr began by asking himself why an entity like the electron could be represented by two models so different from one another, the particle model and the wave model. He then demonstrated that, because of the uncertainty relations, the two models could never contradict one another, as the more one model was made precise, the more the other became indistinct. The two manifestations of the electron, wave and particle, are not in conflict because they are never simultaneously present: the sharper the particle-like properties of an electron are in a phenomenon, the more faded and evanescent are its wavelike properties. In short, the electron has two appearances, sometimes presenting one and sometimes the other, but never both at the same time; these two aspects are mutually exclusive and yet complete each other, like the faces of a medallion. What holds for the electron also holds for photons and any other elementary entity in physics. To express both the ideas of exclusion and completion, Bohr called the two aspects complementary. According to Bohr, complementarity was an underlying feature of physics, and he made it into a philosophical doctrine, transposing it to the relationship between brain and consciousness, the question of free will, and other philosophical questions. For some, the concept of complementarity was not entirely clear. Einstein claimed to have never been able to formulate it exactly and De Broglie found it “un peu troublé”; others, on the other hand, interpreted it as an expression of the fact that the electron is neither a particle, nor a wave, at least as conceived by our classical mentality: it is something else that only a new mentality will allow us to understand. This interpretation is based on the concept, often expressed by Bohr, that our choice of a physical scheme, or as he called it, our “idealizations” (perfectly localized particle, perfectly monochromatic wave, etc.) are overly simplified products of our minds that do not describe reality. In any case, Bohr and his school widely applied the principle of complementarity not only to physics but also biology, in an attempt to understand the dual chemical-physical and vital aspects of biological phenomena. Attempts to transpose physical laws to biology arose at almost the same time as physics itself, Borelli’s work295 provides the best account. Yet the new physical ideas led to a disconcerting observation: if the same devices that physics used to study atoms were employed to study the atoms of an organism, then that organism would cease to be alive; if, instead, one wanted to maintain the integrity of biological tissues, the physical and chemical analyses would remain incomplete. “The incessant 295 § 5.9 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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exchange of matter, which is inseparably connected to life, would also mean the impossibility of regarding an organism as a well-defined system of matter particles, like those considered in the study of ordinary physical and chemical properties of matter.”296 In other words, physics faced the dilemma: either describe phenomena outside of ordinary space-time and maintain the principle of causality, or describe phenomena in the habitual space-time and accept the uncertainty relations. These alternatives posed Heisenberg and Bohr seemed to some physicists a concession to the principle of causality. In the end, they argued, the type of description matters little when one arrives at the same results, and no mental sacrifice is necessary to admit that the uncertainty relations are now useful, and even necessary for physics. They need not, however, be a consequence of the lack of causality in nature, but simply a correction for our mistake in representing particles, which we still do not understand, classically. However, this interpretation is perhaps a bit of a contortion of Heisenberg and Bohr’s thought.

8.13 Probability waves Even classical physics had been forced to introduce probabilistic laws: the statistical descriptions in the study of gas evolution (§ 2.15). Yet these statistical laws were introduced as syntheses of individual dynamical, and thus causal, laws, which the human mind could not keep track of due to their great number and complexity. Statistical laws were therefore a corrective for our ignorance. The view of quantum physics is completely different. We have already mentioned it with regard to the law of radioactive decay, which was not deduced from the study of the evolution of a single radioactive atom, since quantum physics is not interested in the evolution of individual entities. The statistical law was given immediately, with no previous enumeration of individual cases: this was the fundamental difference between the statistical laws of classical mechanics and quantum mechanics. Furthermore, quantum physics does not describe positions and velocities of individual particles to predict their path, as classical physics had. It only yields statistical laws, which are then applicable to aggregates and collections, not individual particles. When a statistical law is applied to an individual entity, the predictions it makes can only be of a probabilistic nature. For example, if the license plate numbers of the 296

46

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automobiles that circulate in a city are one-half even and one-half odd, we cannot predict with certainty if the license plate number of the next automobile that passes will be odd or even; we can only say that the probability is 1/2. In this way, if we consider light to be composed of photons and say that a beam of light incident on a mirror is ¾ reflected and ¼ refracted, we are stating a statistical law and thus saying that out of many photons that strike the mirror, ¾ will return and ¼ will pass through it. Yet this law does not allow one to predict with certainty what will happened to a single photon that is incident on the mirror; we can only say that there is a 75% chance that it will be reflected. This was the framework in which wave mechanics was given a physical interpretation. That the associated wave to a particle was a phantom wave, lacking physical reality, appeared immediately clear from the beginning, but if it did not represent a concrete wave, what physical meaning could it be endowed with? In the study of interference phenomena and diffraction, classical optics assumes that light energy is distributed in space in a manner proportional to the intensity of the wave (the principle of interference). With the introduction of the photon, this becomes the requirement that the intensity of the wave at each point is proportional to the number of photons passing through it. However, interference patterns were observed in experiments with lights so dim that photons had to have struck the apparatus not in bursts, but one at a time. One is thus led to think that the intensity of the wave associated with a photon represents the probability that the photon is found at each point. Analogous considerations can be made for the electron, which obeys the same laws of diffraction. Therefore, we are led to the fundamental principle stated by Born in 1926 (§ 8.10): the square of the modulus of the wavefunction denotes, at each point and instant, the probability that the associated particle is observed at that point and that instant. This was the interpretation accepted by Heisenberg and Bohr in 1927, hence the wave function \ was called a probability wave: this is simply a mathematical expression to which a name was given to evoke concrete classical physics concepts. These waves are purely abstractions, however, introduced to predict the variation of probability with time. More specifically, Born, Heisenberg, and Bohr acknowledged the existence of the particle and the wave, but for them, the particle has neither position, velocity, nor determining trajectory; only when a measurement or observation is made can it shown to be in a certain position or have a certain velocity; the wave represents the probability of its presence or velocity at a certain point in space and instant in time. In this interpretation, the concept of a particle localized in space and time

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then disappears, becoming a collection of possibilities associated with various probabilities; the wave is a mathematical expression, a representation of probability, and thus a ‘subjective’ entity that is modified with the knowledge of the experimenter. With this understanding, the particle-wave dichotomy is overcome by the principle of complementarity.

8.14 A movement to return to determinism The 5 year period between 1923 and 1928 can be considered one of the most fertile periods in the history of physics. The need to probe a world that was hidden from our direct perception pushed physicists to forge novel theories, develop new conceptions of the world, and sometimes substitute physical entities with abstract symbols, eventually forcing them to abandon the traditional ideas and instead find inspiration in a new philosophy of science. We also note the following detail, which regardless of any partisan judgements of the past, is purely a historical fact. The discipline of physics, born in the 17th century at odds with the philosophy of the time, had opposed it by posing as authentic knowledge and the only truly meaningful intellectual activity. This polemical attitude continued for centuries, softening slightly in the first two decades of the 20th century. In the years between 1923 and 1928, this enmity almost disappeared, as the fervent physical discussions turned to the laws of thought and each physicist became a philosopher armed with mathematics, the instrument considered most suitable to express a peculiar concept. Today, the reciprocal antipathy between philosophers and physicists is much attenuated compared to the 19th century, precisely due to the effects of those five years. The role of theoretical physics after 1928 was comparable to the role of physics the generation after Newton: ordering, arranging, further elucidating concepts, extending the theory to new applications, creating a broad new mentality through academic teaching, and popularizing scientific ideas; in short, the problem-solving of the earlier years became axiomatization. This process of consolidation, however, was not carried out with universal consensus, nor was it always free from erroneous returns to past ideas, as is natural and certainly not always harmful in the history of physics. The new ideas were opposed by the physicists of the older school: Lorentz, Planck, and Einstein. During the 1927 iteration of the Solvay conference, Lorentz, who presided over it, clearly reaffirmed his faith in classical determinism. He said, “The image that I want to form of phenomena must be absolutely clear and defined, and it seems to me that

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we cannot form such an image without the system of space and time. For me an electron is a corpuscle which, at a given instant, is found at a welldefined point in space. And if this electron comes upon an atom and enters it and, after many adventures, leaves it, I develop a theory in which the electron conserves its individuality – that is, the electron passes through the atom.”297 Lorentz died the following year, and the battle for determinism was continued by Planck and Einstein (among others). At the following Solvay conference in 1930 (Einstein participated religiously in all of these scientific gatherings) the discussion, according to Bohr’s account, “took a truly dramatic turn.” Einstein proposed a thought experiment that, while taking relativistic considerations into account, purportedly contradicted the uncertainty principle. The experiment is performed using a device consisting of a box with an aperture on one side. The aperture is covered by an obstruction, which is controlled by a clock contained inside the box. If the box contains radiation, the conditions can be regulated such that, at a given instant signalled by the clock, the obstructing plug opens, allows a single photon to escape, and then closes again. By weighing the box before and after the emission, one can deduce the mass of the escaped photon and therefore its energy: in this way one would obtain, without the reciprocal uncertainties postulated by quantum mechanics, exact measurements of energy and time. This argument greatly preoccupied scientists, who nevertheless eventually became convinced that it was not valid. A later recollection by Bohr, whose discussion we postpone,298 very clearly outlines the careful arguments necessary to refute Einstein’s example. 297

N. Lorentz, in Electrons et photons. Rapport et discussions au Vème Conseil de physique Solvay, Gauthier-Villars, Paris 1928, pp. 248-49 298 N. Bohr, Discussion with Einstein on Epistemological Problems in Atomic Physics, in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schlipp, 1949, pp. 199-241. This volume is composed of a wide collection of essays written in honour of Albert Einstein; a collection that is important not only for the authority of its contributors (Sommerfeld, De Broglie, Pauli, Heitler, Bohr, Born, Reichenbach, Mine, Infeld, Laue, Gödel, to name a few) and the incisiveness of the essays, but also as textual evidence for the uninhibited critical approach of modern physicists. Despite the fact that the essays were written in honour of Einstein, whom all of the contributors regarded as their great teacher, his work was subjected to serious and sometimes even biting criticism, as they continually reproached his classical mentality and his “rigid adherence to classical theory”. Einstein recounted to those close to him: “This work is not a celebratory book, but rather an indictment against me.” The elderly scientist felt obligated to close the volume with a wonderful Reply to Criticism, in which he reaffirmed his scientific

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Yet Einstein did not give in: throughout his writings, conferences, and friendly conversations he continued to support the fundamental tenets of 19th century epistemology. Einstein highlighted the real point of discord between the now-orthodox conception of the new physics and his own views in a page written in 1949: “Above all, however, the reader should be convinced that I fully recognize the very important progress which the statistical quantum theory has brought to theoretical physics.” “In the field of mechanical problems – i.e., wherever it is possible to consider the interaction of structures and of their parts with sufficient accuracy by postulating a potential energy between material points – this theory even now presents a system which, in its closed character, correctly describes the empirical relations between statable phenomena as they were theoretically to be expected. This theory is until now the only one which unites the corpuscular and undulatory dual character of matter in a logically satisfactory fashion; and the (testable) relations which are contained in it, are, within the natural limits fixed by the indeterminacyrelation, complete. The formal relations which are given in this theory – i.e., its entire mathematical formalism – will probably have to be contained, in the form of logical inferences, in every useful future theory.” “What does not satisfy me in that theory, from the standpoint of principle, is its attitude towards that which appears to me to be the programmatic aim of all physics: the complete description of any (individual) real situation (as it supposedly exists irrespective of any act of observation or substantiation). Whenever the positivistically inclined modern physicist hears such a formulation his reaction is that of a pitying beliefs, justified his point of view with numerous examples, and revealed his apprehension for the new direction taken by physics, which he indirectly summed up in the following passage: “I close these expositions, which have grown rather lengthy, concerning the interpretation of quantum theory with the reproduction of a brief conversation which I had with an important theoretical physicist. He: ‘I am inclined to believe in telepathy.’ I: ‘This has probably more to do with physics than with psychology.’ He: ‘Yes’ ”. How distant do the years from 1918 to 1928 seem, when awareness of relativity was spreading to the public and relativistic ideas seemed so shocking to people’s orthodox mindsets that debates on their validity transformed into rallies. Einsteins ideas were so disconcerting that they even alarmed religious authorities. In 1928, the cardinal of Boston exhorted the youth to stay away from such atheist theories; accordingly, the Rabbi of New York sent a telegraph to Einstein, peremptorily asking: “Do you believe in God?” Einstein responded by telegram: “I believe in Spinoza’s God, Who reveals Himself in the lawful harmony of the world, not in a God who concerns himself with the fate and doings of mankind.” The Rabbi concluded that Einstein believed in God and reassured the faithful.

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smile. He says to himself: ‘there we have the naked formulation of a metaphysical prejudice, empty of content, a prejudice, moreover, the conquest of which constitutes the major epistemological achievement of physicists within the last quarter-century. Has any man ever perceived a 'real physical situation'? How is it possible that a reasonable person could today still believe that he can refute our essential knowledge and understanding by drawing up such a bloodless ghost?’ Patience! The above laconic characterization was not meant to convince anyone; it was merely to indicate the point of view around which the following elementary considerations freely group themselves. In doing this I shall proceed as follows: I shall first of all show in simple special cases what seems essential to me, and then I shall make a few remarks about some more general ideas which are involved.”299 Some time after Einstein had written this passage and independently from him, a revisionist movement championed by De Broglie emerged. After the 1927 Solvay conference, De Broglie -partially swayed by the clever arguments of Bohr and Heisenberg, and partially overcome by the enthusiasm with which the new ideas were met, especially among young physicists- endorsed the probabilistic interpretation of wave mechanics and made it the subject of the first official course in this field, held in Sorbonne in 1928. Nevertheless, it is not difficult to notice a certain moderation regarding the probabilistic doctrine in the many accounts written by De Broglie. For example, De Broglie, perhaps influenced by the unconscious yet powerful appeal of tradition, attempted in 1941 to rescue at least a part of the pillar which, for millennia, had supported every attempt to understand the world, introducing a principle of “weak causality”, outlined as follows: If the event A is succeeded by any one of the phenomena B1, B2, …, Bn, and if none of the B phenomena occur without the occurrence of A, we can define A to be the cause of the B phenomena; there is thus a causal relationship between A and the B phenomena, but no longer determinism, in the sense that we cannot predict which B phenomenon will take place once A has occurred. It follows that determinism and causality are different concepts and a weak principle of causality can continue to hold in an indeterministic science. Yet De Broglie’s deep, perhaps unconscious, intellectual unease clearly came to light in his later years, brought about by external

299

Reply to Criticisms, in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schlipp, 1949, pp. 665-88.

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influences, but having effects would have been too disproportionate had they not been primed by a slow internal escalation. In the summer of 1951, the young American physicist David Bohm sent a paper to De Broglie that he intended to publish in “Physical Review”, which indeed later appeared in the 15 January 1952 edition. In this paper, Bohm returned to a deterministic interpretation of wave mechanics in the form of pilot wave theory; but the young physicist, perhaps not fully familiar with De Broglie’s analogous attempt, added certain subtle considerations to the measurement processes described by the theory that were sufficient to dismiss the objections that Pauli had raised in 1927. Bohm’s paper caught the attention of De Broglie, who that same year had centred his course at the Institut Poincaré on a systematic and critical examination of the probabilistic interpretation of quantum mechanics, remaining struck by the strength of the objections of its opponents and the abstruse arguments employed by the advocates of probabilism. De Broglie also became acquainted with the work conducted at his institute by another young physicist, Jean-Pierre Vigier, who had renewed the theory of the double solution, integrating it with the new ideas of Bohm and general relativity. Here, the attempted incorporation of general relativity is of note because wave mechanics had up until that point only taken special relativity into account, ignoring Einstein’s generalization. It is evident that the hope to one day unify the two great physical theories of our time could not help but excite De Broglie, who consequently examined the problem in two preliminary papers, and on 31 October 1952 held a conference at the Centre de synthèse de Paris, later published in instalments along with a paper by Vigier and several pertinent documents. At the conference, he declared that the probabilistic interpretation of wave mechanics, which he had believed and taught for almost twenty years, had to be subjected to a fresh critical examination. To those who accused him of inconsistency, De Broglie responded quoting Voltaire, “L’homme stupide est celui qui ne change pas.” In various scientific notes organically synthesized into a survey article, and other, more intuitive papers and conferences collected in the second part of Nouvelle perspectives en microphysique (1956), De Broglie laid out a new interpretation of his old double solution theory. According to this interpretation, the propagation equation for the wave u associated with the particle is not linear, but rather can be thought of as the superposition of something akin to a very fine needle u0 (of dimensions inferior to 10-13 cm) embedded in a monochromatic plane wave v having the same form as a classical light wave, which becomes indistinguishable from u at large distances from the singularity ‫ݑ‬଴ . Due to the nonlinearity of the (yet

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unknown) wave equation for u, the motion of this singularity (which represents the particle) is then determined by the current lines of the wave v. The wave u, both in the v portion and at ‫ݑ‬଴ , is an objective wave that exists independently of observation; but an observer can construct a wave \ that is proportional to v everywhere, where the arbitrarily chosen coefficient of proportionality depends on the information available to the observer. Therefore, \ is still a ‘subjective’ wave, but it is tied to v, explaining its ability to yield exact statistical predictions. This new interpretation allowed the theory to overcome the obstacle it had encountered in 1927: the explanation of interference phenomena. For example, in Young’s double slit experiment, when a photon crosses the screen the theory implies that the singular point ‫ݑ‬଴ only passes through one of the two slits, but the dimensions of ‫ݑ‬଴ are so small compared to the macroscopic diameter of the slit that one can suppose that the entire surface of the slits coincides with the classical wave v, allowing for the traditional optics that explains the interference pattern to hold. This new interpretation of wave mechanics is deterministic and fits within the classical framework. Heisenberg’s uncertainty relations are still valid, but they are interpreted by the theory as predictive uncertainties and not measurement uncertainties of the position and velocity of the particle. According to De Broglie, the new theory could open up vast prospects for physics. For example, an exact knowledge of the function u would give rise to a complete description of the structure and properties of the microscopic particles (mesons and hyperons) that were being continually discovered in those years. In addition, the new theory, as we previously mentioned, allowed for the integration of quantum physics and general relativity in a vast unified relativistic theory of all kinds of fields. Only the future will pass a judgement on De Broglie’s ideas and the efforts of physicists towards this alternate path. For now, an objective assessment must note the relatively lukewarm reception of this new theoretical approach by physicists, demonstrated also by the scant literature on the topic, especially when one compares to the veritable blossoming that has arisen from the indeterminist philosophy of Born, Heisenberg, and Bohr.

9. ARTIFICIAL RADIOACTIVITY

PARTICLE ACCELERATORS 9.1 The proton We have already mentioned Rutherford’s attempts to split stable atoms by bombarding them with Į particles. While the first results obtained in 1920 were uncertain and met with a healthy dose of thinly-veiled skepticism that lurked just beneath the respect warranted by his intrepid endeavour, the mere possibility of success was so alluring that it was worth pushing onwards with the experimental effort. The first definitive result of Rutherford’s efforts was the experimental evidence for the existence of the proton, that is the hydrogen ion, as a constituent of the atomic nucleus. The theoretical existence of the proton, or as it was initially called, the nucleon, had been postulated by Rutherford and Nuttall in 1913, and explicitly or implicitly assumed, as we have seen, by almost all atomic physicists. Yet it was one thing to assume, even given the indirect evidence, and another to demonstrate through concrete experiments that protons truly exist inside atomic nuclei. The experiments of Rutherford and Chadwick, which we have already mentioned (§ 7.8), were repeated and modified by other physicists who found analogous results that, while themselves modest, together were of great theoretical and psychological importance, because they convinced scientists that elements were in fact transmutable. The definitive proof, however, of the existence of the proton, and with it the legitimacy of interpreting the experimental phenomena as transformations, was obtained in 1925 by Blackett, a past student of Rutherford in Manchester who, having returned from internment in Germany, rejoined him at Cambridge. This turn of events explains, and in part justifies, the use of two different expressions – hydrogen ion and proton – to refer to the same thing. Blackett, following Rutherford’s advice and with his assistance, was able to photograph the emission of a proton in the collision between an Į particle and a nitrogen nucleus using a Wilson cloud chamber. This was an extremely rare event: Blackett examined 23,000 photographs containing 460,000 trajectories of Į

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particles and found only 8 cases of proton emission. In each of these cases, the particle seemed engulfed by the nitrogen atom, as no trace of it remained after the collision. It was this critical detail that allowed Rutherford to interpret the collision as we discussed (§ 7.8): the nitrogen nucleus captures the Į particle and immediately emits a proton, transforming into an isotope of oxygen. Once the experiment had been repeated by many physicists (Heinz Pose, Lise Meitner, Bothe, De Broglie, Louis Leprince-Ringuet, Constable), it became certain that the proton is emitted by the struck nucleus through a “breaking” process of the atom, the first confirmed example of the artificial transmutation of elements. This man-made decomposition may have been confirmed, but it was extremely rare; so rare that millions of Į particles were needed to obtain a few dozens of protons and thus transformed atoms, an absolutely insignificant quantity compared the scales detectable by even the most precise chemical analysis. The meagre production of protons was not only due to the rarity of collisions, but especially because not all collisions were followed by the emission of a proton. The photographs taken with the Wilson chamber showed many sudden stops in the trajectories of Į particles that were not followed by the expulsion of a proton. In short, the particles were not effective in breaking up the atoms they struck. Further experimental efforts were guided by Gamow’s theory (§ 8.7), which was based on wave mechanics. According to the theory, the positive charge of an atomic nucleus creates a strong potential around it: the nucleus is thus surrounded by a potential barrier. Clearly, to cross the barrier from interior to exterior or vice versa, a particle must have an energy greater than a certain lower limit. Physicists were surprised to calculate that the radioactive substances could emit Į particles that had too little energy to justify, using classical corpuscular mechanics, the penetration of the potential barrier. However, if a particle was associated with a wave, Gamow showed that the potential barrier behaved like a refractive medium for a light wave. Much like a light wave incident upon an absorptive medium, which slightly penetrates the medium at depths related to the various wavelengths contained in the beam (albeit with extremely low intensities in the case of perfect reflection), and for thin media even passes through it; the wave associated with the particle can be transmitted through the potential barrier, although significantly weakened on the other side, even if the particle does not have sufficient energy. Taking the matter wave to mean a probability, one can express this result by saying that even a particle of insufficient energy has a small probability to cross the potential barrier: this is the “Gamow effect”, or as it is called

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more frequently, “quantum tunnelling”. More precisely, Gamow’s theory predicted that for constant incoming particle energy, the probability of crossing the barrier is higher with lower particle mass. An immediate consequence of this result is that protons are much more effective in breaking up atoms than D particles at the same energy. For historical fairness, we add that at almost the same time as Gamow the English physicist Ronald Gursey and the American Edward Condon arrived at the same conclusions, also basing their argument on wave mechanics considerations. Until the development of Gamow’s theory, only Į particles had been used in the bombardment of elements, as being the most energetic particles known at the time, they seemed the most effective for attaining atomic disintegration. The energies of emitted D particles had been measured during the first years of the century. It is now useful to refer to results using the units of energy that, around 1930, rapidly became popular among atomic physicists: the electron-volt (eV), the product of the electron charge e and the potential difference of one volt, equal (as a simple calculation shows) to 1.59(10-12) erg. In nuclear physics, the unit most often used is actually the mega-electronvolt (MeV). The fastest Į particles were emitted from radioactive substances with an energy of 8,000,000 eV = 8 MeV. Now, around 1925, the most powerful induction coils used for x-ray production reached a potential difference of 100,000 volts; meaning that a proton or an electron accelerated by the field produced inside one of these coils could, at most, reach energies of 100,000 eV = 0.1 MeV: nowhere near the energy of an Į particle. Gamow’s theory, however, renewed the hopes of scientists, predicting that protons of energy 1 MeV had the same effectiveness as Į particles of energy 32 MeV. Because the dramatic improvement in projectile efficacy, it was therefore no longer necessary to produce electrostatic fields of millions of volts, which at the time appeared a mere fantasy and discouraged even the most audacious entrepreneurs. The current intensities attained would only have to be multiplied by a factor of 5 or 6, a target not outside the realm of possibilities for the laboratory techniques of the time. Because of these considerations, Gamow’s theory jolted experimental physicists, who immediately understood that one could produce artificially accelerated particles that rivalled and even surpassed the efficacy of the natural projectiles produced by radioactive substances. As a result, scientists also freed themselves from their dependency on costly and rare radioactive materials for atomic research: a condition which had limited

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the study of radioactivity to five laboratories in the world: Manchester, Cambridge, Paris, Berlin, and Vienna.

9.2 High voltage devices It was therefore necessary to create devices that allowed for very high voltages to be obtained. Many experimenters dedicated their efforts to the achievement of this goal (William Coolidge, Charles Lauritsen, Merle Tuve, Brasch, and others), but the best results were obtained almost simultaneously by Van de Graaff, John Douglas Cockroft and Ernest Thomas Watson, and Ernest Orlando Lawrence. Robert Jemison Van de Graaff (1901-1967) was inspired by classical electrostatic machines which, after having provided precious services to physics the previous century, seemed relegated to the dustbin of scientific relics. In 1931, Van de Graaff began the construction of an electrostatic induction device that Righi had already designed in 1872. The Van de Graaff generator, which he finished in 1933, is built as follows: electric charges are produced by the metallic tip of a small generator (of the order of 10,000 volts); the electric charges are shot at an insulated support that is in continuous motion; the support, in the course of its motion, enters a large hollow electrode (a Faraday cage, which was in fact first developed by Beccaria), and through another point connected to the cavity, transfers the charges onto the outer surface of the electrode, whose potential can in theory then increase without limit. Because of this, this device has allowed physicists to create potential differences of over 5 million volts with an effective output of around 6 kW. To put this quantity into perspective, the most powerful induction machine of the 1800s, the many-disked Wommelsdorf machine, could obtain a potential difference of 300,000 volts with an effective output of 1.2 kW. The biggest inconvenience of this machine was that the surface on which it was installed had to be of enormous dimensions to avoid an electric discharge of genuine artificial lighting between the electrode and the walls. To circumvent this, modern Van de Graaff generators are enclosed inside sturdy iron containers filled with an high pressure gas: this modification allows the machine to achieve a potential difference of millions of volts while maintaining modest dimensions. The idea to produce high voltages to accelerate protons and use them in atomic collisions was also proposed in Rutherford’s Cambridge laboratory. Cockroft and Walton began work towards this in 1930, and two years later developed a device in which the voltage of a transformer was rectified and then amplified through a system of vacuum tubes and

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capacitors. Using this apparatus, the two young scientists obtained a nearly constant potential difference of 700,000 volts, which they then applied to an experimental tube built to produce hydrogen ions. The positive ions were accelerated and focused by special electrodes to obtain proton currents of the order of 10 micro-amperes. The apparatus became very influential, and we shall soon see the historic experiments performed by its two creators in 1932.

9.3 The cyclotron The acceleration of elementary particles was obtained in a rather different and ingenious way by the American physicist Ernest Orlando Lawrence (1901-1958). It appears that it was the reading of the German physicist Rolf Wideröe’s paper that pushed him to build the new device, which produced high-velocity atomic particles not by accelerating them with strong fields but by imparting regular impulses to them. The first device of this type was built in 1930 by Lawrence with the help of Edlersen, his collaborator at the University of California: the prototype was 10 cm in diameter and made of glass and red wax sealant. The first metallic model of the same dimensions was built by Lawrence and Stanley Livingstone, and was capable of supplying hydrogen ions with an energy of 80,000 eV, needing only a potential difference of 2000 volts in the machine. Encouraged by his success, Lawrence later build a device that was 28 cm in diameter through which hydrogen ions could receive 1.25 MeV of energy. He described his machine in a famous 1932 article, and thus the birth of the cyclotron, as it was called, is attributed to that year. While the most powerful cyclotrons can assume gargantuan proportions, the fundamental principle of the device is fairly simple: an oscillating field is applied in resonance to the spiral trajectory of an ion such that its rhythmic impulses increase the particle’s velocity. A shallow cylindrical container cut in two “D”-shaped halves is placed in a strong magnetic field perpendicular to its base, which is produced by a powerful electromagnet with flat poles of circular cross section. The two D’s are placed in a vacuum and a high frequency alternating voltage is applied across them, producing an oscillating electric field in the space between the two D’s, while inside them the electric field is zero due to the well-known shell theorem of electrostatics. The ions produced in the centre of the container enter one of the D’s and are subjected only to the magnetic field, which bends their trajectories so that they turn around and once again enter the region between the D’s. Now, the electric field is

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regulated such that the time taken by the ion to traverse the semicircular path inside the D region is equal to one-half of the oscillating field’s period. Consequently, each time that the ions enter the middle region, the electric field has changed direction and they receive a new impulse incrementing their velocity as they enter the other D. Because the radii of the circular trajectories are proportional to the velocities of the ions, the time needed to traverse each spiral is independent of velocity. Thus, if the oscillation of the electric field is synchronized once with the time taken by the ions to travel a half-circumference, it will remain synchronized for the entire time that the ions are inside the device. The trajectory of an ion is therefore a sort of spiral composed of semicircles of increasing radius until it reaches the outer edge of the apparatus, where it strikes a charged deflecting plate and is deviated outwards through a thin mica window. One can see that the final energy of the ion is greater the more numerous are the impulses it receives inside the cyclotron, and thus the more semicircles traversed (typically thousands), which as radius increases are limited by the size of the region containing the magnetic field. The diameter of the magnet, along with the strength of the magnetic field, are therefore indicative of the energies that can be attained. In 1932, Lawrence followed the construction of his first device with a new one whose magnet was 94 cm in diameter and weighed 15 tons; and in 1937, after years of hard work, he built a more powerful cyclotron with a 150 cm diameter magnet of that weighed 220 tons in total and could produce currents of 100 micro-amperes and energies of 8 MeV. The remarkable progress made in the acceleration of particles by the cyclotron were vividly demonstrated by experiments in which one could see matter particles, after having been artificially accelerated with potential differences of millions of volts, cross a few metres of air at ordinary pressures and become visible, as had been previously only observed in very dilute gases. This was a stark contrast from the turn of the century, when it had seemed incredible that Į particles were emitted by radioactive substances with so much energy that they could cross a few centimetres of standard-pressure air without deviation. The powerful cyclotrons highlighted a fact that the theory had already predicted: the process of particle acceleration, which we have briefly described above, is only valid when variations of the particle’s relativistic mass can be ignored. However, to impart the same kinetic energy of a proton to an electron, it must be accelerated to much higher velocities to compensate for its lower mass, and thus it is no longer possible to neglect relativistic effects, making the cyclotron process no longer applicable. In short, cyclotrons cannot be used to accelerate electrons.

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Despite this hurdle, physicists were still able to attain electron acceleration by making one of two modifications to the cyclotron: in the synchrocyclotron, the frequency of the alternating voltage applied to the two D’s is reduced so that the relativistic mass of the electron increases; in the synchrotron, on the other hand, constructed by Edwin McMillan and Vladimir Veksler, the strength of the magnetic field is increased to increase the electron’s relativistic mass. Another electron accelerating machine was the betatron, built by Donald Kerst in 1945, which obtains accelerated electrons through magnetic induction. We leave the description of other types of accelerators, built to be ever more powerful, to more specialized treatments, but underline that they all employed the same physical principles in essence.

1932, THE MARVELLOUS YEAR FOR RADIOACTIVITY The year 1932 was officially called the “annus mirabilis for radioactivity”, perhaps without the grandiloquence that the name suggests. In that year, physicists made four momentous discoveries, all of which had been expected, a fact that does not diminish the importance of the events, but only serves to highlight the skill of the theoreticians and experimentalists of the time (in particular Rutherford) who had predicted them. These discoveries were the positive electron, or positron; the artificial transmutation of elements, along with the subsequent experimental demonstration of the reciprocal transformation of matter into energy; the discovery of the neutron, a particle of mass 1 and charge 0; and the discovery of deuterium, or heavy hydrogen, the hydrogen isotope of mass 2. We now discuss each of the last three discoveries, deferring discussion of the positron to a later paragraph.

9.4 Deuterium Rutherford, in a 1920 conference that we will address later (§ 9.6), had predicted the possibility of a hydrogen isotope of mass 2 and charge 1. Yet neither chemical analyses nor mass spectrography, which had not yet reached its current refinement, had ever found it. On the contrary, it seemed like there was perfect agreement between physical and chemical methods in the measurement of the atomic mass ratio of hydrogen and oxygen. However, studies of oxygen’s band spectra showed that ordinary oxygen contained traces of two isotopes, of respective masses 17 and 18. From calculations based on these spectroscopic observations, in 1931

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Birge and Menzel deduced that, in order for these new measurements to agree with the aforementioned ratio, one had to admit the existence of a hydrogen isotope of mass 2, present in ordinary hydrogen with an incidence of one part per 4500 parts of “light” hydrogen. Birge and Menzel’s calculation aroused interest, much like Rutherford’s prediction had ten years earlier. The enthusiasm went well beyond what normally would be generated by the discovery of an isotope. Indeed, “heavy” hydrogen was a unique isotope in that its mass was predicted to be twice that of the known element, while in general the masses of any two isotopes differ only by a small percentage. It was therefore reasonable to wonder if even with double the mass, the atom would still be chemically indistinguishable from its isotope. Harold Clayton Urey (1893-1981), then a professor of chemistry at Columbia University in New York, attempted to experimentally demonstrate the existence of the heavy hydrogen theoretically confirmed by Birge and Menzel. The theory predicted that heavy hydrogen in liquid form evaporated more slowly than light hydrogen, so Urey thought it possible to obtain a certain degree of separation between the two isotopes through the distillation of liquid hydrogen. Indeed, through this very laborious approach, he was able to obtain a type of hydrogen that, after undergoing chemical analyses, revealed the presence of mass 2 hydrogen atoms beyond a reasonable doubt. In January of 1932, Urey published his discovery. Of even more interest was the fact that, for the first time, a difference was observed in the chemical properties of two isotopes, such that they could be separated with relative ease into their respective pure states. Urey called the hydrogen isotope of mass 2 deuterium, and its ion deuteron. The terms diplogen and diplon, later proposed to indicate the atom and its ion, did not catch on. When deuterium is combined with oxygen it forms “heavy water”, a molecule with different properties than its ordinary counterpart: it freezes at 3.8 °C, boils at 101.2 °C, and has a greater maximum density (measured at 11.5 °C) than “light” water. Small quantities of heavy water are present in regular water. Immediately after Urey’s discovery, Edward Washburn (1881-1934) proposed to separate heavy water and light water through electrolysis, because during the process, heavy water is concentrated in the part of the liquid that has not yet split. Many facilities were created for the large-scale production of heavy water based on this principle, as it was frequently used in scientific

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research. By 1934, there was already a functioning facility in Rjukan, Norway that could produce half a kilogram of heavy water a day. The discovery of deuterium opened up interesting areas of research for various branches of physics. Perhaps its most interesting property is its efficacy in producing elemental transformations when accelerated by a cyclotron and used as a projectile. Bombarding deuterons with other deuterons releases an enormous amount of protons and forms nuclei of tritium, the hydrogen isotope of mass 3. Up until its discovery in collisions, tritium, which is radioactive, had been searched for to no avail in heavy water.

9.5 Artificial transformations with accelerated particles Cockroft and Walton immediately demonstrated the great services that accelerating machines could provide to science. Even Rutherford’s faith in these devices must have been solid if, according to what is said, after Cockroft and Walton had toiled for a few years to create the device, he, more impatient to see results than his collaborators, told the scientists: “Good; you have worked enough: now try.” This episode may have been true, since Rutherford, in a 1932 note, remembered the first experiments done by the two scientists as employing potentials of 125.000 volts, but the original paper by the two scientists describes “experiments which show that protons having energies above 150.000 volts are capable of disintegrating a considerable number of elements.”300 The method used is very simple in principle. Hydrogen ions are first accelerated by a potential difference that can reach up to 600,000 volts and then fired at the metallic sheet under examination. The eventual products of the atomic splitting strike a fluorescent zinc sulphide screen on which scintillations can then be observed with a microscope. The first element to be bombarded was lithium; the screen displayed bright scintillations, whose number was proportional to the intensity of the proton current. The scintillations appeared similar to the ones obtained when Į particles were fired at fluorescent screens: to confirm this impression, the particles were examined in a Wilson chamber and an ionizing chamber, allowing the scientists to conclude that there could be no doubt regarding the particles’ nature. More precisely, the explanation proposed by Cockroft and Walton was the following: a lithium nucleus of mass 7 captures the proton and immediately splits into two D particles. If 300

J.D. Cockroft and E. E. S. Walton, in “Proceedings of the Royal Society of London,” A, 137 (1932), p. 229.

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this process corresponded to what was really occurring, the two Į particles produced had to be emitted in opposite directions, in accordance with the principle of action and reaction. This conclusion was verified as well, through an experiment in which a second fluorescent screen-microscope device was placed on the opposite side of the lithium sheet, which was created by the thin layer left over after evaporation on a sheet of mica: the scintillations were simultaneously observed on the two screens at symmetric points, confirming the hypothesis that D particles were emitted in pairs. It was soon determined that the Į particles were emitted with an energy of 8.76 MeV, almost equivalent to that of Į particles emitted by thorium. Where did this energy come from? Certainly not from the incident proton, which was produced with an energy less than one-sixth that of a single Į particle. Now, if one summed the mass of the lithium atom with the mass of the incident proton, one obtained a mass slightly greater than that of two Į particles. In other words, a mass defect was experimentally detected: if 7 grams of lithium were transformed, one would observe the disappearance of around 18 milligrams of mass: this was the quantity of matter that in the process had become the energy of the Į particles. The experiment then, besides producing the artificial transformation of elements, experimentally demonstrated the transformation of matter into energy for the first time. This conclusion was reached quite naturally by the two physicists, as especially after Aston’s works, physicists no longer had any doubts regarding the transformation of matter into energy. It might surprise some that after this experimental confirmation had been achieved, the scientists categorically dismissed the possibility of using the phenomenon for the practical production of energy. The reason was quite simple: the freeing of atomic energy was obtained only with the use of a greater quantity of energy than that produced; applying this process to yield atomic energy would have been, to use a rough but fitting analogy, like trying to produce hydroelectric energy by pumping water up mountains to then utilize the energy of its downhill flow. However, here we make a more general observation. Until the discovery of fission, no scientist had ever thought to utilize atomic energy for good or nefarious purposes. This was not an anomaly in the historical progression of science, and must be kept in mind throughout the now-widespread debates regarding its responsibility. One cannot predict where a scientific discovery will lead and thus it is impossible to establish a specific point where the chain of progress should be halted, assuming an end to be advisable, which has yet to be proven.

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After lithium the two scientists subjected other elements to experimental testing: beryllium, carbon, oxygen, fluorine, sodium, aluminium, potassium, calcium, iron, cobalt, nickel, copper, silver, lead, and uranium. Using the same experimental apparatus all of these elements gave rise to scintillations on the screen, indicating a nuclear transformation, and thus a transformation of elements. The new alchemy, as Rutherford called it in a popular book, had arisen. Using the notation introduced by Aston, which amounts to adding the mass of an element to its chemical symbol as a subscript, the first reaction of the new chemistry that the two scientists wrote was F19 + H1 = O16 + He4 which meant that a fluorine nucleus struck by a proton captures it, immediately splitting into an oxygen nucleus and a helium nucleus.

9.6 The neutron To explain the results obtained from the collision of Į particles with light atoms (§ 7.8), Rutherford, in addition to predicting the isotope of hydrogen (§ 9.4), had also hypothesized the existence of a particle with the same mass as a hydrogen nucleus and zero charge. Actually, according to Rutherford, it was less a new particle and more a new type of hydrogen atom, with the electron situated very close to the nucleus and tightly bound to it. During a Bakerian Lecture on 3 June 1920, a conference that was part of a larger series organized by the Royal Society using an endowment from the naturalist Henry Baker (1690-1774), Rutherford presented a discussion on the constitution of nuclei and isotopes. It seemed natural to him to assume that the fundamental constituents of elements were hydrogen nuclei and electrons. He continued: “If we are correct in this assumption, it seems very likely that one electron can also bind two H nuclei and possibly also one H nucleus. In the one case, this entails the possible existence of an atom of mass nearly 2 carrying one charge, which is to be regarded as an isotope of hydrogen. In the other case, it involves the idea of the possible existence of an atom of mass 1 which has zero nucleus charge. Such an atomic structure seems by no means impossible. On present views, the neutral hydrogen atom is regarded as a nucleus of unit charge with an electron attached at a distance, and the spectrum of hydrogen is ascribed to the movements of this distant electron. Under some conditions, however, it may be possible for an electron to combine much more closely with the H nucleus, forming a kind of neutral doublet.

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Such an atom would have very novel properties. Its external field would be practically zero, except very close to the nucleus, and in consequence it should be able to move freely through matter.”301 In 1921, J.L. Glasson attempted in vain to obtain the neutral particle in a high-voltage vacuum tube containing hydrogen. Salomon Rosenblum, in 1928, outlined the experimental difficulties that were faced in trying to observed these hypothetical neutral particles (if they existed). The obstacles were essentially two: neutrons, when passing through a Wilson chamber, do not produce a trail of water droplets and thus cannot be detected using this approach; and neutrons cannot be deviated by electric or magnetic fields, and thus are not detectable using traditional electromagnetic techniques. Nevertheless, the combination of a general reverence for Rutherford and the allure and practical applications of his idea made the demonstration of the neutron’s existence, neutron being the name proposed by Nernst, an objective for English physicists in the decade between 1920 and 1930. Rutherford, along with Chadwick, who had assisted him in the experiments described in the cited Bakerian Lecture, once again attempted an experimental demonstration in 1929, finding negative results. In 1930, Walter Bothe and Herbert Becher bombarded the atoms of light elements like beryllium and boron with Į particles from a radioactive sample of polonium, obtaining a very penetrating form of radiation that they interpreted to be of electromagnetic nature, originating from the disintegration of the bombarded nucleus. At the time, L’Institut Curie in Paris enjoyed the largest supply of polonium in the world and was thus capable of preparing an Į particle source almost ten times stronger than that of any other laboratory in the world. Irène Curie (1897-1956), daughter of Pierre and Marie, and her husband Frédéric Joliot (1900-1958) worked at the institute and were the best equipped to continue Bothe and Becher’s experiments. The couple thus began bombarding beryllium and lithium with a high-energy Į-ray source in 1931. They observed the highly penetrating radiation, but realized that when the beryllium radiation passed through paraffin or another hydrogen-containing substance, the ionization produced by the radiation increased, and they demonstrated that this increase was due to the emission of high-velocity protons from the paraffin. In addition, they discovered that the radiation produced by beryllium could cause the emission of atomic nuclei that it encountered along its path; with the 301

L. Rutherford, Nuclear Constitution of Atoms, in “Proceedings of the Royal Society of London,” A, 97 (1920), p. 396.

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emission occurring, as was demonstrated using a Wilson chamber, less frequently with increasing mass of the atoms struck. A systematic study of the absorption of the new radiation revealed its corpuscular nature to the two scientists, challenging the previous supposition of an electromagnetic nature. Further experiments showed that the radiation could easily pass through matter; it penetrated even 10 to 20 cm of lead without difficulty. On the other hand, protons of equivalent velocities were stopped by a quarter of a millimetre of lead: the new radiation, concluded the JoliotCuries, could not be made up of protons. The Joliot-Curies therefore came close to discovering the neutron (as Bothe had), but did not quite achieve the feat, because they held that “elementary” particles could only include those which were admitted by the science of the time: electrons, protons, and Į particles. It is true that Rutherford had advanced the hypothesis of the possible existence of an atom of mass 1 and nuclear charge 0, but while the Joliot-Curies had attentively read all of his publications, they had overlooked the Bakerian Lecture of 1920 in the belief that it would be difficult to find novel, unpublished ideas in such conferences. At the time, personal contacts between scientists were rare and conferences infrequent, and in consequence there was less exchange of ideas. Rutherford’s neutron hypothesis thus remained primarily limited to British circles. On the other hand, the situation at Cambridge was different: Rutherford and Chadwick had continued to think about the neutron and Chadwick had attempted its experimental observation several times throughout the decade. Under Chadwick’s supervision, Harvey Webb also worked at Cambridge, for two years intermittently researching the excited radiation in beryllium arising from Į particle bombardment. He noticed that the radiation emitted by beryllium in the direction of the incident Į particles was much more penetrating than that emitted in other directions. It thus occurred to him and Chadwick that this radiation could be corpuscular, but for the moment there was still no definitive proof. After Webster’s paper was sent to a publisher, Chadwick received the aforementioned work of the Joliot-Curies and he once again wondered if the radiation studied by the French scientists was made up of neutrons. A careful critical examination of the experiments performed by Webster and the Curie-Joliots, and an intense few days of experimentation convinced him that the radiation could not be of electromagnetic nature, unless the principle of momentum conservation was to be abandoned. At the end of February 1932, a brief communication appeared in the English journal “Nature”, cautiously titled Possible Existence of a Neutron.

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A few years later, the discovery of the neutron, a cornerstone of atomic physics, earned him the Nobel prize in physics. In his Nobel lecture, he declared: “The penetrating power of particles of the same mass and speed depends only on the charge carried by the particle, it was clear that the particle of the beryllium radiation must have a very small charge compared with that of the proton. It was simplest to assume that it has no charge at all. All the properties of the beryllium radiation could be readily explained on this assumption, that the radiation consists of particles of mass 1 and charge 0, or neutrons.”302 For example, the neutron hypothesis immediately explained why it was more difficult for the radiation to cause the emission of heavier nuclei from the bombarded sample. Yet where and how did neutrons originate? Chadwick proposed an analogous mechanism to the one indicated by Rutherford for artificial decays through Į particle bombardment. In this conception, the neutron is then a constituent of the nucleus and is emitted after a collision with a particle. More precisely, an Į particle, striking a beryllium nucleus, is captured and the resulting nucleus immediately emits a neutron, transforming into a carbon atom. Using the notation we have already introduced and indicating a neutron with n1, the following reaction would occur: Be9 + He4 = C12 + n1 and in the case of Boron, bombardment the decay process would be the following: B11 + He4 = N14 + n1 . Chadwick’s interpretation was universally accepted, and in the following years could boast many experimental confirmations, a large part of which were indirect. The next step was to study the properties of the neutron. Its mass was found to be very close to that of the hydrogen atom, but concentrated in a volume thousands of times smaller. The absorption of neutrons by matter only occurs if the particles strike the matter nuclei, so this effect is therefore small and equal in all directions. Norman Feather, also in 1932, discovered another critically important property of neutrons: bombarding nitrogen with Po + Be radiation, he observed pairs of trails that originated from the same point in a Wilson chamber. Feather attributed these to the decay of a nitrogen nucleus caused by a collision with a neutron, and with 302

J. Chadwick, The Neutron and Its Properties. Nobel Lecture, p. 4 in Les prix Nobel en 1935, Santesson, Stockholm 1937.

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much difficulty was able to distinguish two different decay processes, one characterized by the capture of the neutron and the other without capture. Still in 1932, Lise Meitner (1879-1968) and Kurt Philipp also attained the decays of oxygen atoms bombarded by neutrons: following this observation, many other decays of this type were produced by experimenters. We will soon discuss these decay processes more fully, for now we simply note that neutrons have proven themselves extraordinarily useful in breaking up atoms for a fairly simple reason given by Fermi, as we will shortly see. The discovery of the neutron had three main immediate effects: the formulation of a new nuclear model, to which quantum mechanics could be applied; the impetus for many new research efforts that led to important discoveries (artificial radioactivity, nuclear fission, etc.); and the genesis of practical applications of nuclear energy. We will discuss all of these effects in the following pages, beginning from the first. We have already mentioned that the nuclear model of protons intermingled with electrons was more or less considered unsatisfactory by all physicists. This widespread dissatisfaction explains the unanimous agreement on the need to change the model, but this agreement evaporated when it came to describing the new nucleus: some believed that electrons in the nucleus were bound to protons to form neutrons, such that light nuclei were composed of Į particles, protons, and neutrons, while heavy nuclei could also contain a few free electrons; others, like Perrin, held that the nucleus contained special groupings made up of a proton and a neutron, called half-helium. The Russian physicist Dmitri Ivanenko, in a brief communication that appeared in “Nature” in 1932, supposed that the nucleus was composed only of protons and neutrons; immediately after, Heisenberg based the theory that gave the stability conditions of a nucleus and the laws of radioactive decay on this hypothesis. As time went by, Ivanenko’s alluringly simple hypothesis was confirmed by subsequent studies of nuclear transformations and quickly spread, establishing itself as the prevalent theory in very little time. In his scheme, a nucleus of mass number A (the whole number closest to the nuclear mass) is made up of Z protons and N = A – Z neutrons. Clearly Z is also equal to the number of the atom’s planetary electrons, the atomic number. The constituents of the nucleus, protons and neutrons, were generically called nucleons, reusing a term that had already been applied in other contexts: we will explain the reasons for this identification later. The new theory led to the nowuniversal practice of modifying the notation introduced by Aston to describe nuclei: in the adapted notation, the atomic number of the element

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(the number of protons) is also added as an index. After various versions were used initially, the notation of the Joliot-Curies prevailed, in which the two indices were both placed to the left of the chemical symbol such that the mass number was at the top and the atomic charge (atomic number) on ଶ଻ the bottom: for example, ଵଷ Al . Once the theory of the nucleus’ composition was accepted, the number of neutrons and protons that compose it could be determined: they are almost equal in number, except in the nuclei of heavy atoms, where neutrons are more abundant. Yet what forces ensure the stability of the nucleus? This question was studied by Heisenberg and Ettore Majorana (1906-1938), the young Italian physicist who mysteriously disappeared in 1938.303 But we will return to this problem later.

9.7 Beta decay If the nucleus only contains protons and neutrons, how is it that experiments showed that many radioactive nuclei could emit beta rays, or electrons? Heisenberg’s theory did not answer this question, nor was an answer provided by the previous theory of the nucleus, which had theorized the existence of electrons inside the nucleus (§ 7.7): an embarrassing presence and a thorn in the side of nuclear theory, clashing even with the principle of conservation of energy. Indeed, if a radioactive atom A transforms into another atom B through the emission of an electron, the nucleus of A and the nucleus of B have fixed energies, and therefore the emitted electron must always have the same energy, and thus the same velocity, related to the difference in energy of the two nuclei. Instead, in 1914 Chadwick had shown that electrons emitted from radioactive substances have variable energies, ranging from zero to 0.7 MeV. Was the principle of conservation of energy then violated in this case? By 1931, Bohr was almost willing to make this concession. Yet the majority of physicists did not go this far: the balance was restored by assuming that the emitted electrons came from the deepest layers of the radioactive substance and lost part of their energy while traversing the material; but during emission they all had the same velocity. 303

His scientific writings are collected in a volume published by the Accademia nazionale dei Lincei, La vita e l’opera di Ettore Majorana, Rome 1966. The importance of his work, rediscovered thirty years after his death, is elucidated well by a fellow scholar, E. Amaldi, Ricordo di Ettore Majorana, in “Giornale di fisica”, 9, 1966, pp. 300-18.

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This patched theory held until 1927, when careful experiments demonstrated that the electrons emitted by nuclei can have very different energies at the moment of emission. In 1931, Pauli reexamined the problem and augmented it with a new hypothesis, which then seemed more of a ready-made expedient to salvage the model without breaking the laws of physics. Pauli supposed that along with the electron, another very small, neutrally-charged particle is emitted, randomly dividing with the electron the energy difference between the initial and final nuclei in the radioactive transformation. Because of its size and lack of charge, the hypothetical particle would not be observable and the electron could thus have all the possible energies in the observed range. It was a curious particle, proposed to allow atomic physicists to save their theory without having to dispense with the principle of conservation of energy, despite experimental evidence to the contrary - and in the heyday of neopositivism too! Pauli named this new particle the neutron, the same name given the following year by Chadwick to the particle he discovered (§ 9.6). One day, at the University of Rome, Fermi gave a lecture on Pauli’s theory during which a student asked him if Pauli’s neutron was the same one as Chadwick’s. “No,” Fermi answered, “Pauli’s neutron is much smaller and much lighter.” He then added, with his typical sense of humour: “It is a neutrino,”304 and it is with this tongue-in-cheek name that the particle hypothesized by Pauli is known today. The nature of the neutrino, which physicists could not experimentally detect, sparked discussion for many years: initially, some disputed its existence; then it was thought that the neutrino was the quantum of some field, like the photon is for the electromagnetic field. Only in 1956, with the use of the powerful nuclear reactor at the Savannah River Site, Clyde Cowan and Frederick Reines were able to experimentally demonstrate its existence, bombarding protons with the supposed neutrinos and obtaining neutrons and electrons. Moreover, the number of neutron-electron pairs resulted exactly as Pauli’s theory had predicted. This theory described an atom whose nucleus was assumed to contain electrons; but what can be said regarding Ivanenko and Heisenberg’s nucleus, which was free of electrons? In 1932, Fermi proposed a theory of beta decay, which maintained Pauli’s hypothesis of the existence of the neutrino and supposed that neither the neutrino nor the electron are found in the nucleus. However, a neutron in the nucleus can transform into a 304

Translator’s note: In Italian, the word for neutron, neutrone, has the -one ending, indicating an augmentative, while neutrino is the diminutive form of the same word, having the suffix -ino.

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proton, with the simultaneous emission of an electron-neutrino pair and energy. In Fermi’s theory, the proton and neutron were not two distinct particles, but two quantum states of the same particle, generically called a nucleon: the process of beta decay thus appears quite similar to the emission of a light quantum by charged particle that jumps from a higher quantum state to a lower one. According to Fermi, the emission of positrons by radioactive substances is due to the reverse process, that is the transformation of a proton into a neutron. A free neutron can also spontaneously transform into a proton and an electron, with a half-life that Fermi experimentally measured to be 14 minutes following the construction of the nuclear battery. The device he used, also known as a “Fermi bottle”, consisted of an empty spherical container placed inside a functioning atomic battery for a long time. Most of the neutrons generated by fission pass through the bottle without encountering any obstacles. However, some neutrons decay into a proton and an electron in the bottle, remaining trapped inside it because the walls are impermeable to charged particles. The quantity of hydrogen formed in the bottle in a certain time then allowed Fermi to calculate the mean lifetime of the neutron.

NUCLEAR ENERGY 9.8 Induced radioactivity At a meeting of the Académie des sciences de Paris on 15 January 1934, Perrin presented a communication by Irène and Frédéric Joliot-Curie announcing the discovery of artificially radioactive elements. The two physicists, continuing their research on the effects of Į particle bombardment, had discovered the previous year that some light elements (beryllium, boron, aluminium) emit positrons when bombarded with Į particles. They wanted to uncover the mechanism of this emission, which appeared different from the other transformations that were known at the time. To this end, they placed an Į-ray source at a distance of one millimetre from a sheet of aluminium, irradiating the sheet for around ten minutes and then placing it on top of a Geiger counter. The scientists observed that the sheet emitted radiation with an intensity that exponentially decreased with time, having a characteristic decay time of 3 minutes and 15 seconds. Analogous results were found with boron and magnesium, but with different decay times: 14 minutes for boron and 2.5 for magnesium.

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No such effect was observed with hydrogen, lithium, carbon, beryllium, nitrogen, oxygen, fluorine, sodium, calcium, nickel, and silver. These negative results, however, still provided an important indication, revealing that the radiation that had been observed in aluminium, magnesium, and boron could not be ascribed to a contamination of the Įray source. The boron and aluminium radiation, observed in a Wilson chamber, appeared to be made up of positrons. This was undoubtedly a new phenomenon, differing from other decays in that all nuclear reactions that had previously been observed were instantaneous phenomena akin to explosions, while the positrons produced by the irradiated aluminium continued to be emitted even after the Į particle source was removed. The two scientists concluded that what they observed was an authentic radiative phenomenon, made evident by the emission of positrons. This interpretation overturned the scientific consensus of the time, which had held that atoms formed by bombardment with heavy particles are always common stable isotopes. Energy considerations led the Curie-Joliots to interpret the phenomenon as follows: first, an Į particle is captured by an aluminium nucleus, causing the emission of a neutron and the ensuing formation of the radioactive isotope phosphorus-30 (stable phosphorus has a mass of 31); this unstable atom, named radiophosphorus by scientists, then decays, emitting a positron and transforming into a stable atom of silicon, obeying the same law as naturally radioactive elements. Despite the minute yield of these transformations and the meagre mass of the transformed elements (only a few million atoms), the two scientists, through high-precision experiments, were able to determine the chemical nature of the elements obtained. The discovery of artificial, or induced, radioactivity seemed (and remains) one of the greatest discoveries of the century: physicists foresaw its great theoretical importance and other scientists were galvanized by the endless range of applications to biology and medicine. The Curie-Joliots were deservedly recognized with a Nobel prize in chemistry the following year, and experimenters all over the world imitated their work, also bombarding elements with other projectiles. Especially in Britain and the United States, where physicists had access to powerful accelerators, many new radioactive elements were found through bombardment by accelerated protons and deuterons. In particular, one of the first major successes of the cyclotron was the production of radioactive sodium, radium E (bismuth210), and radium F (polonium), obtained by shooting high-energy deuterons at table salt for sodium, and at bismuth for the other two elements.

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9.9 Neutron bombardment A soon as news of the Curie-Joliots’ experiments reached Rome, Fermi, “a theoretical physicist of the finest type,” as the old Rutherford had fondly called him, decided to repeat them using neutrons as projectiles.

Fig. 9.1 - Enrico Fermi in the year of the Nobel Prize

Enrico Fermi (Fig. 9.1) was born in Rome on 29 September 1901, and after having completed his high school studies there, he enrolled in the Scuola normale di Pisa, where received a degree in physics in 1922. His inclination towards theoretical physics was entirely an independent choice, as the subject was then unheard of at Italian universities. Although there were many renowned mathematical physicists in Italy (Tullio Levi-Civita and Vito Volterra, for example), they focused only on the problems of classical physics, which they addressed from a strictly mathematical point of view. Despite being in an unfavourable, if not hostile, environment,Fermi

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quickly leapt to the highest ranks of theoretical physics in Europe because of his important contributions, some of which we have mentioned in the previous pages. In Italy, it was an old-school experimenter who immediately noticed his genius. Orso Mario Corbino (1876-1937), then professor and director at the physics institute of Rome, leveraged his scientific prestige and political position as a senator and former minister of education and the economy to establish the first Italian professorship of theoretical physics in Rome: Fermi was granted this title on November 1926, and in 1929 he became a member of the Accademia d’Italia at its formation. With the beneficial backing of Corbino, he created an active school of young physicists around him in Rome, who first mainly dedicated themselves to atomic and molecular spectroscopy, and then after 1932 to nuclear physics. In 1938, he received the Nobel prize in physics for his work on induced radiation, which we will soon discuss. From Stockholm, where he had traveled to accept his prize, he left for the United States, where he sought shelter from the Italian government because of his staunch antifascist views and to protect his family, in particular his Jewish wife, from anti-semitic persecution. He became a professor of theoretical physics at Columbia University, and collaborated, as we will discuss, with other scientists in the construction of the atomic bomb. After the war, he dedicated himself to the study of mesons and cosmic rays. He died in Chicago on 28 November, 1954. Let us return to neutrons. Right after their discovery, nobody considered them suitable to act as disintegration agents, so much so that in their first article on induced radiation, the Curie-Joliots, while inviting their colleagues to repeat the experiments with other atomic projectiles, did not mention their possible use at all. Fermi explained the reasons for the general lack of confidence and, in contrast, his own insight to employ them: “The use of neutrons as bombarding corpuscles has the disadvantage that the number of neutrons that can be practically prepared is far inferior to the number of alpha particles that can be obtained from radioactive sources, or the number of protons or deuterons that can be produced using high voltage vacuum tubes; however this inconvenience is partly compensated by the greater efficacy of neutrons in producing artificial decays. Neutrons also have the advantage that the number of elements that can be activated by their bombardment is much greater than the number of radioactive elements that can be obtained using any other projectile”.305

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E. Fermi, Radioattività prodotta con neutroni, in “Il nuovo Cimento”, n.s., 11, 1934, p. 430.

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The new technique, experimentally applied at the old institute on via Panisperna in Rome, was very simple. The neutron source, a thin glass tube containing beryllium powder and radium emanation, was placed inside cylinders made up of the substance to be examined. This substance was subjected to a heavy bombardment of neutrons for some time and then quickly taken to a Geiger counter in another room, where its radioactive impulses were measured. Fermi, after a few unsuccessful attempts due to the scant intensity of the source, was able to make many elements radioactive with neutron bombardment: fluorine, aluminium, silicon, phosphorus, chlorine, iron, cobalt, silver, and iodine. In many cases he was also able to describe the chemical nature of the radioactive elements obtained. Having attained these results, Fermi enlisted the collaboration of a group of young researchers from his school (Edoardo Amaldi, Oscar D’Agostino, Franco Rasetti, Emilio Segrè, Bruno Pontecorvo), with them subjecting 65 elements to neutron radiation, out of which 57 unmistakeably exhibited induced radiation. Among the activated elements, the effect appeared to have no dependence on the atomic weight of the bombarded element. In the course of these investigations, Fermi made another discovery in 1934. The experimenters had realized that the intensity of the induced radiation depended on the conditions of the bombardment. Fermi conducted a systematic study, filtering the neutrons directed at the samples using various screens; he began with a lead screen and later used a paraffin one. To his great surprise, instead of absorbing the neutrons, the paraffin screen heightened their inductive power. This result was entirely unexpected, but Fermi quickly realized how to interpret it. Hydrogenated substances, or better, the hydrogen atoms they contain, slow down neutrons more than other substances because protons and neutrons have about the same mass, and so each elastic collision of a neutron and proton gives rise to an almost equal apportionment of kinetic energy. Fermi demonstrated that after twenty collisions with a hydrogen atom, the energy of a 1 MeV neutron is reduced to that of thermal excitations. It follows that high energy neutrons crossing a hydrogen-rich layer rapidly lose energy, transforming into “slow,” or “thermal,” neutrons, which thus have velocities of the order of thermal excitations. However, one still had to explain the apparent paradox that slow neutrons are more effective in producing decays than fast ones, or more precisely, as Fermi and others showed, that the greatest effect occurs at a given neutron energy, which is different for every substance. Wave mechanics gave Fermi the explanation for this effect, analogous to one given by Bohr, who modelled it as a resonance phenomenon. These slow

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neutrons became Fermi’s primary research interest, remaining his focus for many years. Experiments using slow neutron bombardment were fruitful beyond all expectations: almost all elements that were bombarded gave rise to radioactive isotopes. Right before the second world war there were already more than 400 new radioactive substances, some of which even exhibited a radioactive intensity greater than that of radium.

9.10 Transuranium elements Chemical analyses and theoretical considerations based on the distribution of isotopes allowed Fermi to identify three distinct processes in the creation of induced radiation. All three begin with the capture of the incident neutron by the nucleus it collides with; this nucleus simultaneously emits either an Į particle, a proton, or nothing, in each case transforming into a radioactive element. The first two processes occur more frequently when light nuclei are bombarded, while the third occurs often for heavy nuclei. In the spring of 1934, Fermi, bombarding uranium and thorium with neutrons, observed the production of a complex amalgam of radioactive elements that underwent a series of transformations in conjunction with ȕ particle emission. Fermi and his collaborators attempted to chemically isolate the elements involved. Their experimental studies brought them to conclude that some of these elements were neither isotopes of uranium nor lighter elements, but elements with an atomic number greater than 92: in short, new and entirely artificial elements. Fermi, Rasetti, and D’Agostino, who were the first to think they had produced and identified elements of respective atomic numbers 93 and 94, called them ausonium and esperium. The discovery, however, was viewed with skepticism by many physicists (Aristid von Grosse, Ida Noddack, and others). Hahn and Meitner, on the other hand, ostensibly corroborated it, and even believed to have created another new element of atomic number 96. Later though, after the discovery of fission (§ 9.11), it became clear that Fermi and the German physicists had not created transuranium elements in their experiments, but rather products of the fission of uranium. The first transuranium element, of atomic number 93, was discovered by McMillan and Philipp Hauge Abelson in 1939-40; the second element, of atomic number 94, was identified in 1941 by Joseph Kennedy, Glenn Seaborg, Segrè, and Arthur C. Wall: the two elements were called neptunium and plutonium, respectively, to extend the planetary analogy.

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During and after the second world war, another ten or so transuranium elements were artificially created. Besides neptunium and plutonium, the properties of the elements of atomic numbers 95 and 96, called americium and curium are fairly well-known. All transuranium elements are radioactive, though with very different half-lives: neptunium has a half-life of around 2 days, plutonium of 24,000 years, americium of 500 years, and curium of 5 months. Of particular importance due to its wartime use was plutonium, produced in appreciable quantities in the United States through the use of cyclotrons, which first bombarded uranium with deuterons and then with neutrons. By the end of 1942, half a kilogram of plutonium had been produced, a sufficient quantity to study its principal chemical characteristics. Later, plutonium was obtained from nuclear reactors (§ 9.12) in quantities that were kept secret, overcoming enormous technical challenges and requiring massive monetary investments.

9.11 Nuclear fission In 1938, Irène Curie and Pavel Savic observed that in uranium samples that had been radioactively induced with Fermi’s method, an element similar to lanthanum appeared. Yet the two French physicists, following the common practice in those years, concluded that the product had to be “a transuranium element, which has rather different properties from those of other known transuranium elements, a hypothesis which raises the question of interpretation.”306 The same experiments were repeated later that year by Hahn and Fritz Strassmann, who confirmed the experimental results of their French colleagues, but established that the new element was in fact lanthanum. Yet the question of interpretation left many physicists perplexed. Hahn and Strassmann privately communicated the results of their experiments to Meitner, who had been their collaborator for thirty years at the KaiserWilhelm-Institut in Berlin and, as an Austrian Jew, had been forced to leave Germany after the annexation of Austria; she found shelter in Stockholm, where Bohr’s support helped her obtain a position at the physics institute. Meitner and Otto Frisch, another German-Jewish refugee in Sweden, realized the true nature of the phenomenon and communicated their interpretation to both their Berlin colleagues and Bohr, who announced their discovery on 26 January 1939 at a meeting of the 306 I. Curie and P. Savic, in “Comptes-rendus de l’Académie des sciences,” 208, 1938, p. 1645.

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American Physical Society. That same day, scientists present at the conference experimentally confirmed the phenomenon, immediately sending news of the result to the press, and thus anticipating the publication of Meitner and Frisch’s original paper, which although it was sent to the periodical “Nature” on 16 January, was only published in March of that year. The phenomenon, called “burst” by Hahn and Strassmann, and “fission” by Meitner and Frisch, consisted in the fact that when a neutron strikes an atom of uranium-92, the atom splits into two more or less equal parts: Hahn and Strassmann for example, had obtained barium-56 and krypton-36 in their experiments. Joliot, who had also independently interpreted the effect in the same way, immediately understood the importance of the new form of decay. Fundamentally, the phenomenon of fission is a consequence of arithmetic. As we have already mentioned, the number of protons is almost equal to the number of neutrons in light nuclei; with increasing atomic weight, the number of neutrons grows faster than the number of protons. For uranium nuclei, for example, the ratio between the number of neutrons and protons is 1.59, while it oscillates between 1.2 and 1.4 for elements that are at about half its atomic number. Therefore, if a uranium nucleus splits into two parts, the total number of neutrons contained in the fragments must be less than the neutrons contained in the original nucleus to ensure the stability of the two products. The net effect of the fission of a uranium atom is thus a release of neutrons. These neutrons could then cause the splitting, or fission, of other atoms, which in turn again produce neutrons, and so on. In short, physicists realized that this process could give rise to a chain reaction, analogous to certain chemical chain reactions occurring in explosions. In 1939 Francis Perrin, son of Jean-Baptiste, published the first calculation of the “critical mass” needed to trigger the chain reaction: as might be expected, it was a preliminary estimate, now only of historical value. Today it is known that no quantity of ordinary uranium could trigger a chain reaction, because the neutrons produced by the fission of the few atoms of uranium-235 would be absorbed through “resonant capture”, to use the technical term, by atoms of uranium-238. This would then produce uranium-239, which through two subsequent decays would transform into neptunium and plutonium. A critical mass exists only for fissionable materials like uranium-235 and plutonium. Furthermore, the calculation of the mass lost in the fission of a uranium atom showed that the fission process would be accompanied by an energy production of 165 MeV: an enormous quantity, equivalent to four times the total energy released in the complete sequence of natural radioactive transformations from uranium to lead.

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Joliot’s predictions were soon experimentally confirmed, and it was also established that the fission of uranium nuclei follows their capture of slow neutrons. Applying theoretical considerations, Bohr attributed the fission not to common uranium of mass 238, but to its isotope of mass 235; soon after, in 1940, Nier experimentally confirmed Bohr’s prediction, discovering that plutonium was another easily fissionable atom. These, more or less, were the limits reached by nuclear physics on the eve of the second world war. Nuclear research was then covered by a thick veil of mystery, shrouded from the view of ordinary people. Even much of the scientific community only came to know of the technological and scientific progress of nuclear physics though a terse war communiqué: on 6 August 1945, an atomic bomb was dropped on the Japanese city of Hiroshima, producing death and destruction on apocalyptic scales. Once the war was over, news of the colossal amounts of work done by scientists residing in the United States to harness atomic energy began to trickle out to the world. In 1945, Henry Smyth of Princeton published on official United States report, Atomic Energy for Military Purposes, which contained all the information available to the general public. The report did not discuss certain critical scientific and technological details, and remained vague on others. Today the scientific processes are well-known, and only certain technical details of military interest are secret, but it is obvious that “non-governmental” science will never be able to experimentally test such military applications on the same scale. Smyth’s report tells us that the idea of harnessing atomic energy for military purposes came from a group of foreign physicists who had escaped to the United States during the war, among them the Hungarians Leo Szilard, Eugene Paul Wigner, and Teller, the Austrian Victor Frederick Weisskopf, and the Italian Fermi. This group attracted the attention of the U.S. President, Franklin Delano Roosevelt, through Einstein, who on 2 August 1939 wrote him a famous letter, saying “Some recent work by E. Fermi and L. Szilard, which has been communicated to me in manuscript, leads me to expect that the element uranium may be turned into a new and important source of energy in the immediate future… and it is conceivable […] that extremely powerful bombs of a new type may thus be constructed.”307 The effort to produce atomic energy on large scales had two different objectives: the gradual and controlled release of energy for ordinary industrial use, and the construction of a super-explosive device. The second objective was seen as the more immediately urgent of the two in 307

Albert Einstein, Letter to Franklin D. Roosevelt, 2 August 1939.

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that tragic period of world history. Yet soon the scientists realized that the fastest way to obtaining the second objective was achieving the first. Indeed, as we have said, fission occurs in atoms of plutonium and uranium-235, which only accounts for 0.7% of ordinary uranium. The atomic bomb required large quantities of uranium-235, which was very difficult to separate from uranium-238, while the gradual production of energy did not necessitate a preliminary separation of the isotopes, only a large initial quantity of uranium, and generated plutonium as a byproduct. Accordingly, the “atomic battery” was born, a name perhaps given due to its simple construction: this name has now fallen into disuse, replaced by the more accurate expression nuclear reactor. The key problem was reducing the number of neutrons resonantly captured by uranium-238 and thus rendered unavailable in the chain reaction process, although they do retain some use as fertilizers, producing uranium-239 that can then transform into either neptunium or plutonium. It was therefore necessary to quickly remove the fast neutrons from the uranium sample, reduce their kinetic energy, and make them once again penetrate the uranium sample, now as thermal neutrons that could cause the fission of uranium-235 atoms. The mediating role in this process had to be played by lighter atoms, such that the neutrons would lose most of their energy in collisions with the mediators without causing atomic decays. At the time, only two substances appeared suitable for this function: heavy hydrogen in the form of heavy water, and carbon. As the former was extremely expensive, carbon in graphite form became the preferred solution. The first nuclear reactor, made up of alternating layers of uranium and graphite, was built by Fermi in collaboration with Anderson, Walter Zinn, Leona Woods, and George Weil, and began to operate on 2 December 1942 on a University of Chicago tennis field. This reactor generated a power of 0.5 watts, and after 10 days of successful operation was scaled up to produce 200 watts. This was the first machine of one of the most advanced branches of modern engineering. It is difficult to overstate the importance of this first step for humanity’s progress in the 20th century. This pilot trial allowed a careful experimental study of the production of plutonium, which led to the conclusion that this approach gave rise to a real possibility of producing the plutonium necessary for an atomic bomb. By the end of 1943, the project to construct an atomic bomb had entered its final phase: the first test was successfully performed at 17:30 on 16 July 1945 at the aerial base of Alamogordo, around 200 km from Albuquerque in the New Mexico desert.

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9.12 Cosmic rays The gradual discharge of a charged electroscope that is left alone308 was the subject of a detailed experimental study by Coulomb. However, with the discovery of radioactivity and the ionizing action of the radiation given off by radium, this century-old explanation began to seem a bit too simplistic. Around 1903, the discharge phenomenon was once again studied by many physicists (John McLennan, E.F. Burton, Rutherford, Lester Cook), who, by experimenting on electroscopes that had been shut inside containers with and without lead screens, established that the discharge could be slowed down by filtering the air inside the container using wet cotton balls or screening it with lead. These experiments showed that Coulomb’s explanation was not exhaustive, as the discharge also arose from causes that he had not predicted. Since it had been recently discovered that radioactive substances emit gamma rays, and Jean Holy and Poole had shown that radioactive substances are prevalent inside the Earth’s crust, the spontaneous discharge effect was initially attributed to the gamma radiation emitted by radioactive substances present inside the Earth. For many years, this interpretation was accepted without debate by physicists. Perhaps the first to draw attention to its shortcomings was Domenico Pacini (1878-1934), who between 1908 and 1911 conducted systematic experiments that led him to suggest that the radiation was not of terrestrial origin but instead came from above. In 1909, the Swiss scientist Albert Gockel launched a balloon containing an electroscope and observed that it discharged more rapidly at an altitude of 4000 metres than it did at sea level. From 1911 to 1913, the Austrian physicist Victor Hess (1883-1964), who received the Nobel prize for his pioneering contributions, repeated Gockel’s experiments, confirming and extending his results. Because electroscopes discharged more rapidly at high altitudes than at sea level, it was difficult to maintain the assumption that the agents responsible for the effect originated in the Earth’s crust. Hess very prudently proposed the hypothesis that the radiation had extraterrestrial origins, although not excluding the possibility of atmospheric origins. Other physicists, especially Werner Kolhoster (18871946), who launched electroscopes to altitudes of around 9,000 meters, confirmed that at high altitudes the discharge could be even 8 to 10 times faster than on the Earth’s surface. Kolhoster noted that if the radiation was 308

§ 7.33 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambrdge Scholars Publ. 2022.

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of extraterrestrial origin, it would need incredible penetrating power, at least 5 to 10 times as much as the strongest radioactive gamma rays, considering that it would have to traverse the entire atmosphere to act at the Earth’s surface. Having been suspended because of the war, experiments resumed in 1922 with Ira Bowen and Millikan, who in San Antonio, Texas launched automatically data-collecting electroscopes to heights of 16,000 metres. These experiments found puzzling results: the speed of electroscope discharge, or equivalently the increase in ionization, did not constantly increase with height. This result seemed to rule out extraterrestrial origins and support the hypothesis of a radioactive distribution in the upper atmosphere. Several years of uncertainty followed, which experimental approaches could not clear up because atmospheric balloon launches were risky and expensive ventures that often gave dubious results. To bypass this difficulty, in 1923, Kolhoster in Germany, and two years later Millikan and George Cameron in America, had the idea of studying the radiation not by going upwards, but by going down, as Pacini had done in Livorno in 1911, conducting experiments 3 metres underwater in the ocean. Kolhoster measured the ionization in a deep crevasse in the Alpine ice; Millikan and Cameron plunged their electroscopes into the waters of Lake California, reaching 20 metres of depth. These experiments were repeated by Reneger in Lake Constance, who reached depths of 220 metres: in this way scientists confirmed that the ionizing effect of the yet-unknown radiation diminished with the depth of submersion. This result immediately led to the conclusion that the radiation was not of terrestrial origin. Furthermore, the experiments also showed that the radiation was exceptionally penetrating, because in water it could reach depths more than three times thicker than the atmosphere. If penetrating power can be taken as a measure of energy, the new radiation had to be much more energetic than any known earthly radiation. These results and other experimental evidence convinced Millikan that the radiation was of extraterrestrial origin (in 1926, he further showed that it was of extragalactic origin). Accordingly, in a paper published in 1925, Milikan proposed the term “cosmic rays” to refer to the extragalactic agents that continually bombarded the Earth. Nevertheless, the belief that the radiation originated from the upper levels of the atmosphere survived for a few years, and was not entirely ungrounded, as we will soon see, so the ionizing agent that caused discharge in electroscopes continued to be called penetrating radiation.

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Once it had been confirmed that cosmic rays were of extraterrestrial origin, the problem became determining their nature. In 1927, Dmitri Skobelzyn was able to take the first photographs of cosmic ray trajectories in a Wilson chamber. Given that the time period in which this problem was studied, the years around 1929, was the heyday of wave mechanics, distinguishing whether cosmic rays were photons or electrons mattered less than directly measuring their energy. To this aim, Millikan and Anderson conducted experiments in Pasadena using the classic method of deviating charged particles with a magnetic field. They therefore built a vertical Wilson chamber and placed it inside a very powerful field. In the summer of 1937 the scientists obtained their first results, which were surprising because of the exceptional amount of energy measured in the rays, of the order of billions of electronvolts, while the previously recorded energies for radioactive bodies did not surpass 15 MeV. The study of the trajectories and their deviation allowed Anderson, Millikan, and many other physicists who began their own research after the initial successes (distinguished among the pioneers of this period were Auguste Picard for his famous stratospheric ascent and Bruno Rossi for his 1933 expedition to Asmara) to establish the rather conglomerate nature of the ionizing radiation, which is composed of Į-rays, high-velocity electrons, protons, neutrons, positrons, and gamma rays. To clearly distinguish positive and negative particles in the Wilson chamber, Anderson thought that the simplest approach was to determine the direction in which their trajectory curves in a magnetic field. However, the particles were so energetic that their curvature was almost imperceptible. Anderson thus had the idea to place a 6mm thick lead sheet transversally inside the chamber: a particle that crosses it is slowed down and is therefore more deviated by the magnetic field upon re-emerging. The photographs taken in August 1932 with this simple modification of the Wilson chamber (Fig. 16.2) display a new feature, by and large the most important discovery in these series of investigations: the existence of a particle having the same mass as an electron but the opposite charge. The photographs could not be interpreted otherwise; the curvature indicated a positive charge, which, because of the magnitude of the curvature and the amount of energy remaining after passing through the lead, could not have been carried by a proton. In September 1932, Anderson announced the discovery of the positive electron, or positron, predicted, as discussed earlier, by Dirac’s theory.

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Fig. 9.2 - The first photograph of the trail produced by a positron obtained by Anderson. As the positron crosses the lead sheet (horizontal white stripe), its energy is reduced from 63 to 23 MeV, and therefore the radius of curvature of its trajectory decreases. The magnetic field is normal to the plane of the photograph. Source: C. D. Anderson, The Production and Properties of Positrons, in Les prix Nobel en 1936, Santesson, Stockholm, 1937.

In the spring of 1933, Blackett and Occhialini confirmed the discovery, using a similar apparatus that however had been attached to a Geiger counter automatically signalling the passage of cosmic rays. The discovery of the positron was the first fundamental contribution made by the study of cosmic rays to nuclear research. As often happens in such cases, this immediately led to widespread interest among atomic physicists in research on cosmic rays, which had up until then seemed rather marginal compared to the general direction of physics research at the time. From 1925 to 1932, as we have said, it was believed that the penetrating radiation observed on the Earth’s surface came directly from extragalactic regions, after having crossed the entire terrestrial atmosphere.

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However, having established its heterogeneous composition and its similarity to the products of laboratory-produced atomic disintegrations, physicists began to suspect that penetrating radiation was a secondary product created by collisions between a type of primary radiation with atmospheric atoms: from 1933 to 1937, more evidence was collected supporting this view, and today there is no doubt that penetrating radiation originates in Earth’s atmosphere from the collision of its atoms with primary cosmic rays. In 1934, Carl David Anderson (1905-1991) and Seth Neddermeyer, studying certain trails left in a Wilson chamber, concluded that they were not produced by electrons, as it had seemed, but rather by new particles that had a mass between that of electrons and that of protons. They provisionally called them W particles, and these W’s were later observed by other experimenters as well. From this point on, the study of cosmic rays became intertwined with the theory of nuclear forces, that is the mutual forces that are exerted by the constituents of the nucleus. In the research and discussion that followed the introduction of the idea of nuclear forces (§ 9.13), which joined the already established gravitational and electromagnetic forces in the pantheon of interactions, Heisenberg supposed in 1932 that their effects could be interpreted as arising from the exchange of an electron, or more generally, an electric charge between a proton and a neutron. Soon after, Fermi developed the theory of beta decay (§ 9.7) based on the neutrino hypothesis of Pauli. This theoretical direction suggested that nuclear forces could be reduced to the exchange of electrons and neutrinos or positrons and neutrinos between two nucleons, much like electromagnetic forces were thought to arise from the exchange of photons between charged particles. Yet the forces obtained from these processes turned out too small, as Igor Tamm and Ivanenko observed in 1934, because beta decay was too slow of a process compared to the supposed rapid exchange of electric charges responsible for nuclear forces. In 1935, the Japanese physicist Hideki Yukawa (1907-1981) from the University of Osaka realized that the calculation could be corrected if one hypothesized the existence of a new particle that has a mass between that of a proton and an electron and a positive or negative charge. Such a particle would be emitted or absorbed when a nucleon transitions from a proton state to a neutron state or vice versa. In short, a neutron and proton could interact with each other by exchanging this particle, like two charged particles interact through photon exchange – Yukawa’s particle would then be the quantum of the nuclear field. The mass of the hypothesized particle

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was calculated by Yukawa to be about 300 times greater than the mass of the electron. When he first formulated the theory, Yukawa did not know of the works of Anderson and Neddermeyer, so he concluded that “as such a particle has never been observed in experiment, the previous theory seems to be off the mark.” It was not off the mark, but rather “incomplete for various reasons,” as Yukawa himself said at the conference on the development of the theory of the meson, which he held in 1949 in the occasion of his Nobel prize in physics. For the particle hypothesized by Yukawa, whose existence was definitively demonstrated in 1937, Anderson proposed the name mesoton and Bohr the name meson: out of the two terms, which both referred to the particle’s intermediate mass between the electron and proton, Bohr’s prevailed. Starting in 1947, the study of mesons experienced considerable growth, especially due to the work of Cesare Lattes, Occhialini, Cecil Powell, and Lawrence. After having established that the mass of the meson discovered by Anderson was around 290 times that of the electron and that its charge could be both positive and negative, another meson weighing 210 electron masses was discovered; the first meson was called the ʌ meson, or pion, while the second was the ȝ meson, or muon. With the introduction of the new photographic sheet technique (§ 4.7), a considerable number of new particles of greater mass than the pion were discovered. These are generically called heavy mesons, and their mass can in some cases be larger than that of nucleons. Heavy mesons can have a positive, negative, or neutral charge, and they have a shorter half-life than pions, whose half-life is already quite brief (a few hundred thousandths of a second). The large number of new particles that were being discovered, each named and labelled differently by the various scientists involved in this enterprise, brought about general confusion, leading some physicists to propose a new, standardized classification and notation for elementary particles in 1954, which we will discuss. There is still another mystery regarding the origin of cosmic rays, despite the many theories that have been proposed to explain it. The shortcomings of these theories lie in explaining the enormous energies carried by cosmic rays, which have been calculated to be 6 GeV on average (with peaks of more than 20 GeV), 100 times greater than the energies involved in radioactive phenomena, 150 times greater than the maximum energy that the most powerful cyclotrons could impart to particles in 1950, and at least 30 times greater than the energy released by

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the fission of uranium. One of the most extensive theories was formulated in 1949 by Fermi. According to him, cosmic rays originate and are accelerated mainly by collisions with moving magnetic fields in interstellar regions. The main weakness of this theory, as Fermi himself recognized, was that it could not explain the presence of the heavy nuclei observed in the radiation in a simple manner. A later theory, advanced by Sterling Colgate and Montgomery Johnson, supposed that cosmic rays are produced by a collision mechanism during stellar supernovae. We hope that the observations made by the instruments on artificial satellites, which were first launched from a Russian base in the autumn of 1957 and later spread throughout the world, will open a new chapter in the physics of cosmic rays and the structure of matter.

9.13 The nuclear field The fundamental problem of nuclear physics is describing the nature of the forces that act between the constituents of the nucleus. Already in 1910, from Rutherford’s study on the deviation of Į particles in the proximity of nuclei, it appeared, though it was not emphasized at the time, that the forces that act near the nucleus are no longer those predicted by classical mechanics. Around 1932, once it had been accepted that nuclei were formed by protons and neutrons, it became clear that their cohesion had to result from a mutual attraction independent of electric charges (an attraction between protons and neutrons and neutrons with themselves) that overcame Coulomb repulsion (between protons). Consequently, physicists embraced the idea of a nuclear field that ensured the stability of nuclei and whose influence was only felt at nuclear distance scales. The nature of this field nevertheless remained uncertain, as did the question of whether it was associated with particles like the electromagnetic field. It is obvious that the forces arising from this field cannot be of electromagnetic nature, since the neutron is neutrally charged, and that they cannot be of gravitational origins either, as a calculation assuming this hypothesis gives forces 1038 times smaller than the real values. We must therefore conclude that we are dealing with a new type of field. After the existence of the meson was confirmed, Yukawa’s theory gained a good deal of credibility and physicists modelled many other nuclear field theories on it, treating the interactions as arising from meson fields. Efforts were even made towards the unification of the electromagnetic and meson fields. In any case, physicists today consider the nuclear field as real as the electromagnetic or gravitational field.

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A feature of the nuclear field is that it gives rise to forces with an extremely small range of action. If one considers any nucleus and a charged particle, for example an Į particle, the interaction between them at relatively long distances reduces to Coulomb repulsion, which is inversely proportional to the square of the distance between the two (assumed) point charges. With decreasing distance, a critical value R is reached where the repulsion deviates from Coulomb’s law; R is called the nuclear radius and can be measured experimentally. In this way, it was found that R is proportional to the cube root of the nuclear mass number A, with the proportionality constant called the nucleon radius and given by 1.42(10-13) cm. It follows that the volume of the nucleus is proportional to A, and thus the density of all nuclei is almost equal and extremely high. This detail led Bohr to propose his liquid drop model of the nucleus, which was among the most successful of the proposed nuclear models. Bohr likened the nucleus to a liquid droplet, since both in a nucleus and in a drop of liquid the density is independent of the number of constituent particles. In Bohr’s model, nuclear reactions occur in two sequential phases. In the first phase, every particle incident on the nucleus is captured and its kinetic energy is rapidly divided amongst the nuclear constituents. Then, in the second phase, the excess energy is either re-emitted by the nucleus in the form of gamma radiation, or it accumulates on a single nuclear particle, which then gains sufficient energy to escape the nucleus: the emission of the particle becomes then essentially like the evaporation of a droplet. In conclusion, we still do not know much about the nature of nuclear forces: they appear to be attractive, around a hundred times stronger than electrostatic forces, of very short range, independent of the charge of particles, but perhaps dependent, in a manner still unknown, on their spin.309

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Translator’s note: Since the 1960s, much has been discovered regarding the nature of the nuclear interaction, as is detailed in Chapter 11.

10. CONTEMPORARY DEVELOPMENTS IN PHYSICS

SOLID STATE PHYSICS 10.1 Characteristics of solid state physics It may seem strange that almost at the end of our story, after having discussed solids at length, we only now adopt the expression “solid state physics”, as if it were a new topic that was only developed in the second half of the 20th century. In reality, solid state physics was born at the same time as physics itself, but up through the entire 19th century its study was purely macroscopic and phenomenological – without any consideration, that is, for the microscopic structure of solids, whose behaviour was experimentally studied under various mechanical, thermal, and electrical conditions. In this sense, the history of solid state physics up to the 20th century is contained in the previous pages. Although the atomic hypothesis was more than two thousand years old, only in the 19th century did scientists apply it to the study of solids, advancing fields like crystallography (§ 4.5). In the 20th century, after the discovery of x-ray diffraction, the atomic hypothesis was systematically applied to the study of solids, producing a wealth of results. Today, the term solid state physics refers to the study of general properties of matter in the solid state, or better, in the condensed state310 (solid and liquid), deduced from a few general hypotheses about the crystalline state, the modes of vibration of crystal lattices, and electronic band structures (§ 10.3). Therefore, the study of crystal structures, the theory of specific heats (§ 6.12), and electrical conductivity (§ 3.12) are also chapters belonging to solid state physics. Other chapters of this extensive field of modern physics are the cohesion of solids; their mechanical, thermal, electrical, magnetic, and optical properties; and the interactions between solids and gases. Solid state physics developed as 310

Note of the translator: In modern usage, the term solid state physics has given way to the broader Condensed Matter Physics.

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much from technical research on elasticity and plasticity of materials, as from the theoretical research of many physicists, including Born, Nevill Francis Mott (1905-1996), and Frederick Seitz (1911-2008). The guiding ideas and calculational tools were provided by quantum mechanics, but the complexity inherent in any collection of atoms led to a frequent use of models, especially in the early years. After the second world war, solid state physics underwent dramatic development, having overcome a sort of impotence complex that had convinced it in the forties that it was impossible to describe many properties of solids on a fundamental level, leading it to only employ phenomenological approaches. The complexity of matter and the lack of historical perspective make it difficult to present a cohesive historical picture. We will thus only note a few characteristic concepts of these new developments and some important applications, leaving further detail to more specialized accounts. We also add that the frenetic modern scientific production, the great mobility of scientists, their frequent conferences and symposia, and the extreme (and not always equitable) competition between researchers often make it difficult to assign credit for discoveries, inventions, and research methods.

10.2 Crystal structure It was known since the 19th century that the traditional classification of states of mater -solid, liquid, and gas- was only valid as a first approximation applicable to everyday life. Before the discovery of x-rays, the term crystal was used to refer to solids that seemed symmetric and anisotropic from their outward appearance. It was later discovered, however, that even solids that outwardly seem amorphous can be a collection of small crystals joined together. Crystals should then be classified based on their atomic structure and not their external appearance: on the atomic scale, limestone, which is called amorphous, is a crystal like calcite. There are, however, amorphous solids even on the atomic scale: the quintessential example is glass. A new classification of the states of matter was necessary; one that would keep track of the degree of order in the crystal structure. The limiting cases were the ordered state and the disordered state: the former is characteristic of crystals, the second is typical of amorphous solids and liquids if the density is high, and gases if the density is low.

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Research on crystal structures, pioneered by the Bragg father-son duo, has continued with ever-growing success to this day, benefitting from continual chemical and metallographical advances. Crystal lattices, discussed in paragraph 4.5, are characteristic of ideal crystals. Such perfect lattices do not exist in nature: real crystals contain defects that are not simply occasional but structural, that is inherent in their structure and therefore ubiquitous and characteristic. For example, some vertices of a lattice can be missing atoms or ions, creating empty spaces, or lacunae, called Schottky defects, named after the German physicist Walter Schottky (1886-1976), who first observed them in 1935. Extra atoms or ions can also be found in interstitial spaces, that is between the vertices of a lattice. Such an imbalance occurs when a hole is formed by an atom leaving its place on the lattice: this is called a Frankel defect, named after the Russian physicist Yakov Il’ich Frenkel (1894-1952), who identified it in 1926.311 It is also possible for a lattice to be missing parts of certain planes, in which case the contiguous planes adjust in the appropriate manner: this is a dislocation. Lastly, a crystal can contain atoms or ions foreign to the crystal structure called impurities. The defects listed above can be created artificially and have considerable influence on physical properties, like strength, plasticity, electrical conductivity, and ferromagnetism. Yet how are the atoms and ions of crystal lattices attached to each other? This problem was addressed from a quantum mechanical point of view by the German Walter Heitler (1904-1981) and Fritz London (19001954) in 1927. The two physicists mainly studied the bonds between hydrogen molecules. This research was further developed by Linus Pauling (1901-1994) and John Clarke Slater (1900-1968) and, especially for bonds in solids, was continued until 1937 by many other scientists (Victor Goldschmidt, John Bernal, Seitz, etc.). Various types of bonds were recognized, providing a set of criteria for classifying solids. More precisely, the classes are ionic solids, like sodium chloride, in which the vertices of the lattice are occupied by ions instead of atoms; molecular solids, like iodine, in which the lattice points are chemically stable atoms with a full valence shell; covalent solids, like diamond, germanium, and silicon, in which atoms form covalent bonds between one another, each with two valence electrons shared between adjacent atoms – quantum mechanics and the Pauli exclusion principle provide an adequate 311 J.I. Frenkel, Nouveaux développements de la théorie électrique des métaux, in Atti del Congresso internazionale di fisica, Como 1927.

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theoretical scheme to describe this type of bond. There are also metallic solids, like copper, in which the lattice is occupied by elements that do not have a full valence shell: the valence electrons, also called conduction electrons, move freely in the interior of metallic conductors.

10.3 The electronic structure of atoms in a crystal Related to the previous research were studies of how electronic structure changes in crystal atoms. In 1900, Drude supposed that the electrons in a metal are free and in disordered motion, leading to reciprocal collisions between electrons and ions: in short, that they behave like gas molecules. His theory interpreted the electrical conductivity of metals like the theory of ionized gases interpreted the current that arises from an electric field in a gas. The fundamental assumption of Drude’s theory, the presence of free electrons in metals, was experimentally confirmed in 1917 by the Americans Richard Tolman and Thomas Dale Stewart (1890-1958). The experiment they conducted involved suddenly halting a copper coil that was rotating about its axis at a high velocity, such that the charge carriers inside it continued to move due to inertia and thus produced an electric current, which was then read by a galvanometer connected to the ends of the coil. One could then deduce the ratio e/m, which resulted identical to the ratio for cathode rays. While Drude’s theory explained some phenomena very well, it did not account for other experimental results, such as the increase in resistivity with temperature for most conductors (linear according to experiment; proportional to the square root of absolute temperature according to Drude’s theory); the exception to this rule observed in a few conductors, in which resistivity decreases with increasing temperature; the increase in conductivity due to light; the inverted Hall effect observed in certain conductors (iron, zinc, cadmium, etc.), in which the free charges that are deviated by a magnetic field appear positive instead of negative, while for those same metals the photoelectric and thermoionic effects are identical to those found in other metals. The theoretical study of electrons in metals was taken up by Sommerfeld in 1928, who applied the equations of wave mechanics and Fermi-Dirac statistics to the collection of valence electrons in a metallic crystal. He deduced that valence electrons have to be distributed in a series of discrete energy levels, where each level cannot contain more than two electrons with opposite spin: this is the extension of the Pauli exclusion principle to the atoms of a crystal. The theory was later improved by

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taking into account the electrostatic field produced by atomic nuclei as well. Assuming that there is only one free electron per atom, Sommerfeld found that the electron gas is “degenerate”, meaning either highly concentrated or at low temperatures. At absolute zero, electrons fill all of the lowest energy levels. The highest filled level is called the Fermi level. The main fault of Sommerfeld’s theory was its excessive simplicity. It assumed that the electrons in a metal are completely free to roam inside the space occupied by the metal, but unable to escape it. Yet, precisely because the electrons cannot escape the metal save for exceptional cases (if the metal is very hot, part of an electric circuit, or struck by electromagnetic radiation), as has been experimentally demonstrated, they must be held by the attractive force of the positive ions that were once the atoms containing the free electrons. The presence of ions then influences the motion of electrons and modifies their associated De Broglie wavelength. If a single free electron moves along a lattice segment, its half-wavelength has to be submultiple of the segment length, and so its energy changes in discrete quanta. In a crystal, however, where there are a multitude of ions and electrons that interact with each other, the problem becomes extremely complicated. In 1928, right after the appearance of Sommerfeld’s theory, Félix Bloch (1905-1983), a student of Heisenberg, began to study the problem with approximate analytical methods. It is more useful pedagogically, though, to refer to the approximation method described by Heitler and London. Suppose there are n identical atoms, far enough from each other such that they do not interact. This system will have n energy levels that are all the same. If we now bring the n atoms together to form a crystal lattice, the waves associated with the electrons change and the energies are no longer the same: they separate and form groups of energies, called bands, at different levels. This effect is akin to coupling n oscillating circuits with the same frequency, which then become a system with n distinct frequencies. The bands of allowed energies are separated by intervals of prohibited energy. The width of the bands depends on the interatomic spacing, and thus on the type of crystal. Due to the Pauli exclusion principle, each level can contain at most two electrons, which have to have opposite spins. Band theory has been fundamental for solid state physics, and its development was furthered by many other physicists besides those already mentioned, including Léon Brillouin, Rudolf Peierls, and Morse. The mechanism of conduction in solids had puzzled physicists for some time, as we saw earlier. In particular, electronic theory had not been able to explain why, unlike metallic conductors, the conductivity of some

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substances increases with temperature. These substances were called semiconductors and this property was taken as their defining phenomenological characteristic. Band theory explains the different behaviour of metals and insulators when an electric field is applied between two points in a material. If the valence band is completely filled by electrons and above it there is an unoccupied band separated by a large enough gap of prohibited energy, the substance behaves like an insulator. In other words, this means that the electrons in an insulator are bound tightly enough that they can only gain energy, and thus velocity, in large jumps, requiring elevated electrostatic fields. If instead the valence band is only partially occupied by electrons and the forbidden energy gap above it is not too large, the crystal behaves like a conductor. In other words, some electrons in a conductor can change their energy (and velocity) by small quantities, such that even a weak electrostatic field can give rise to a current. According to band theory, there is no substantial difference between insulators and semiconductors. Both insulators and semiconductors have filled valence bands underneath a higher energy band which, at sufficiently low temperatures, is not completely filled. The two bands are separated by a band of prohibited energies, called a band gap, that is relatively large (a few eV) in insulators and considerably smaller in semiconductors. For instance, in diamond, an insulator with a crystal lattice structure, the band gap is around 7 eV; in pure silicon and germanium, typical intrinsic semiconductors, the band gaps are 1.2 and 0.78 eV, respectively. The conductivity of semiconductors can be altered by thermal excitement, electromagnetic radiation, magnetic fields, and the Seebeck and Peltier effects. The addition of impurities in the crystal changes the number of conduction electrons and the number of holes, and therefore also the conductivity of the material. This field of research is extremely vast; we thus leave it to more specialized texts. By 1932, the theory of conduction in solids, or at least its main points, was almost complete. Despite this, most practical consequences, especially relating to semiconductors, were only found after 1950, although semiconductors had been used since the beginning of the century (for example as rectifiers for receiving electromagnetic waves). We will now discuss one of the most widespread applications of semiconductors. In certain types of semiconductors (for example, pure germanium and silicon), the Seebeck effect is much stronger than that in any other metal, nearing one hundred times ordinary magnitudes. Thermoelectric couples made up of semiconductors are then used to

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convert thermal energy into electrical energy, producing a sizeable yield. Scientists also attempted to use the inverse Peltier effect; the best results were attained by Abraham Ioffé (1886-1960) from the Semiconductor Institute of the U.S.S.R, who a little after 1950 was able to build small experimental refrigerators with an efficiency almost on par with that of absorption-based household appliances. An application of semiconductor physics that has revolutionised electronics is the transistor, created in 1948 by William Shockley (19101989), John Bardeen (1908-1991), and Walter Brattain (1902-1987), researchers at Bell Telephone Laboratories who in 1956 received the Nobel prize in physics. Transistors, which now come in many dozen varieties, can substitute electronic tubes in many applications (portable radio-receivers, electronic calculators, measurement instruments, telephone switchboards, etc.) with added technical advantages: highly reduced space, no filament necessary for supplying electricity, high performance, and numerous benefits. In 1954 at the Bell Telephone Laboratories, semiconductors, especially silicon and selenium, were used for another important application: the solar battery, the first practical device that transformed light energy directly into electrical energy. Solar batteries, now used in many technologies (telephones, radiotransmission, etc.), have become irreplaceable components of artificial satellites, in which the use of ordinary batteries or capacitors is almost impossible because of their excessive weight. The introduction of transistors and other new devices like superconductors gave rise to a curious linguistic effect. While until the end of the second world war “electronics” referred to the science and technology related to the conduction of electrons in gases and vacuum, today the term has a much broader, and strictly speaking, more correct meaning. It is difficult to define exactly, but according to many, electronics is related to the production, use, and transmission of information. In commercial jargon, moreover, the word is considered so all-encompassing that it is used to indicate any electric device.

10.4 Magnetic properties The most complete classical theory of magnetism was developed by Langevin (§ 7.6) in 1905. Langevin’s theory, based on the Amperian hypothesis of molecular currents (inadmissible in classical physics) ignored molecular interactions, which can disturb the natural orientation of the elementary magnetic units. This interaction was addressed by the French physicist Pierre Weiss (1865-1940), who from 1896 to 1905

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conducted careful experiments on highly anisotropic ferromagnetic crystals (magnetite, pyrrhotite). In 1907, he extended Langevin’s theory, introducing the idea of a molecular field, a mutually orienting effect on elementary magnetic units. According to Weiss, the molecular field is proportional to magnetization. This theory led Weiss to show that at medium and low temperatures, a ferromagnetic object experiences a spontaneous magnetization even in the absence of external fields. In most experiments, spontaneous magnetization does not appear because of the presence of small domains, magnetized regions that are differently oriented such that together they compensate any individual fields. According to this theory, a piece of iron is made up of a collection of domains that have different magnetizations, resulting in zero net magnetization. If the iron is placed in a magnetic field, the domains align and the whole material becomes magnetized. For some time, it was thought that magnetization was due to a sudden collapse of the domain directions. But later, still from the work of Weiss and his school, it became clear that magnetization did not arise from the effects of a violent catastrophe on atomic ordering, but rather from the gradual growth of a domain of preferred orientation at the expense of others. Essentially, the surface separating two contiguous domains shifts more or less rapidly, giving rise to a larger region of favourable magnetization. With this image of the separating surface, or domain wall, one can explain the hysterisis cycle, of fundamental interest in magnetic applications. Weiss’ theory initially appeared rather strange, but little by little, as the new phenomena it predicted were discovered, it gained the confidence of physicists. In 1931, Francis Bitter (1902-1967) was able to observe domains with a microscope and study their behaviour in an external magnetic field, using an experimental apparatus similar to the old method of iron filings. Well before this experimental confirmation, Weiss and his school had conducted numerous measurements of paramagnetic magnetization coefficients with the aim of deducing, through Langevin’s formulas, the value of atomic moments. By 1922, Weiss had verified that all of the values obtained were whole multiples of an elementary moment, which he called the magneton (as opposed to electron). With the introduction of Bohr’s atomic theory, which marked the beginning of the quantum theory of magnetism, many physicists noticed that the quantization of electronic orbits leads to the existence of an elementary magnetic moment whose value depends on the electron charge, its mass, and Planck’s constant, and therefore is itself a constant. The

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value of the Bohr magneton, as it was called, was calculated to be 9.17(1020 ) in cgs units: five times greater than the Weiss magneton, which was later shown to have been an erroneous measurement. The first experimental confirmation of the existence of the Bohr magneton was obtained in 1921 by Stern and Gerlach, who observed the deviation experienced by a beam of metallic atoms passing through an inhomogeneous magnetic field in vacuum (§ 2.14). For the atoms of certain metals (silver, copper, gold), they found a magnetic moment very close to the Bohr magneton; for others (lead, tin, zinc, mercury) they found no magnetic moment; still others gave rise to more complicated results. From 1860, Maxwell had attempted to demonstrate the gyromagnetic effect, namely the fact that the rapid rotation of a magnetic bar about its own axis produces an internal magnetization (direct gyromagnetic effect) and, vice versa, that the instantaneous magnetization of a bar induces a rotational impulse opposite to the internal angular momentum gained (inverse gyromagnetic effect). The gyromagnetic effect, theoretically described by Owen Richardson in 1914; was experimentally verified in 1915-16 by Einstein and Wander de Haas, and in 1917 John Stewart demonstrated the inverse gyromagnetic effect. The hypothesis of the rotating electron (§ 8.5) brought about a radical change in how physicists thought of magnetism. To begin with, the spin of the electron produces another magnetic moment distinct from the orbital magnetic moment due to the electrons revolution around the nucleus. Quantum mechanics shows that spin can assume only two orientations: one parallel and the other antiparallel to the field. In 1932, Bethe gave a theory of ferromagnetism based on the quantum interpretation of Weiss’ molecular field formulated by Heisenberg in 1928. According to this theory, ferromagnetism is due to electron spins and not orbital magnetic moments, while the existence of Weiss’ domains is demonstrated by the fact that magnetization occurs in discrete jumps and not continuously. To the traditional division of magnetic materials into diamagnetic, paramagnetic, and ferromagnetic, quantum mechanics and in particular Heisenberg’s theory allow for the addition of antiferromagnetism and ferrimagnetism. Antiferromagnetism, predicted independently by LouisEugène Néel in 1932 and Lev Landau in 1933 and theoretically formulated by Jan Van Vleck in 1941, consists in the fact that in certain substances (for example manganese dioxide) below a characteristic temperature, half of the magnetic moments are aligned in one direction and the other half are aligned in the opposite direction, giving zero net magnetization. The characteristic temperature is marked by anomalies in physical properties

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(specific heat, thermal expansion, susceptibility) analogous to those occurring near the Curie point. Ferrimagnetism is characteristic of certain ferrites (iron oxides of the form MFe2O3, where M is a divalent metal with a particular crystal structure) that in 1946 Jacob Snoek (1902-1984) found to possess special magnetic properties: like antiferromagnets, ferrimagnets also have magnetic moments in two opposite directions, but they are neither equal in magnitude nor in number, giving rise to a nonzero net magnetization. Ferrimagnetic materials, besides exhibiting spontaneous magnetization below a critical temperature, have a high permeability and a very high resistivity, which can reach 106 Ohm/cm (the resistivity of iron is of the order of 10-5 Ohm/cm). Experimental tests of antiferromagnetism were performed in 1949 by Clifford Shull, who used a method based on the diffraction of thermal neutrons; the same method was employed in 1951 by W. A. Strauser and Ernest Wollan (1902-1984) to test ferrimagnetism. New experimental techniques have engendered the discovery of ever more complex magnetic structures, which are today subjects of active study. Furthermore, practical applications of ferromagnetic and antiferromagnetic materials have abounded (amplifiers, magnetic memory, voltage stabilizers, etc.).

EXTRATERRESTRIAL EXPERIMENTATION 10.5 Atmospheric exploration Up through almost the entire 19th century, human daily life – commerce, communication, war – took place on the surface of the Earth. As a result, an understanding of the space above us seemed merely a secondary and mostly speculative aim. Once the Aristotelian theory that held that the atmosphere was divided into three regions – the lowest adjoining the Earth, the highest connected to the celestial sphere and in motion with it, and an extremely cold middle separating the two – was discredited, few advances were made in this field, often relating to meteors and astronomical events. The invention of the barometer (1644) provided a vigorous impulse for the study of the atmosphere, which initially concentrated on physical characteristics (pressure and its variation with time and altitude) and was later extended, in the second half of the 17th century, to its chemical composition. However, it was only in 1856 that Bunsen established that atmospheric composition was not constant regardless of Earth’s surface region below it, but instead varied slightly with latitude and the presence

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of vegetation or bodies of water, as well as with altitude, but not in an appreciable manner in the zone where breathing remains possible. In the course of the 19th century and the first years of the 20th, scientists attempted to find the height of the atmosphere using three different methods: the observation of meteors (Livingstone), the duration of sunsets (Alfred Lothar Wegener [1880-1930] in 1910), from which Arab astronomers had deduced that it was 92 km thick, and the observation of auroras (Lars Vegard [1880-1963], starting in 1913). The first method gave a thickness of 150-300 km, the second of 64 km, and the third of 500 km. These values were so different that they not only generated controversy, but also a lack of confidence in the methods employed. Balloon ascents, which began in the 18th century (§ 2.2), continued into the first years of the next century with almost no scientific results, leading to their abandonment for several decades. In 1850 they recommenced with two ascents that made history, those of Jacques-August Barral (18191884) and Giacomo Bixio (1808-1885), brother of Nino Bixio, the famous captain in Garibaldi’s army. In their second ascent, they reached an altitude of 7000 m and measured a temperature of -39 °C, which at the time was thought too low by scientists. The English meteorologist James Glaiser (1809-1903) conducted systematic ascents in an air balloon from 1862 to 1866, from which it seemed that temperature decreased at a decreasing rate with altitude. In 1875, the Russian chemist Mendeleev gave a formula for this variation, predicting that air at the outer limit of the atmosphere had to assume a temperature of -28 °C. This theoretical result was disproved by the ascents (1891-1901) of the Polish scientist Arthur Berson (1859-1942), who, having reached a height of 9000 m, measured a temperature of -48 °C. Scientists thus returned to the original hypothesis that temperature regularly decreases with height at a rate of around 10 °C per kilometre. Furthermore, extrapolating from experimental results, it was held that the decrease continued until the outer limits of the atmosphere. In 1893 Gustave Hermite (1863-1914), a nephew of the mathematician Charles, and Georges Léon Besançon (1866-1934) were among the first to launch weather balloons (also known as a sounding balloons), closed rubber balls partially filled with hydrogen that carried recording instruments, which were suspended in a cylinder with reflective walls and no ends. The ball inflated as it rose, eventually exploding and beginning its descent, which was slowed by a parachute. This new technique valuable contributions to scientists’ knowledge of the atmosphere. However, weather balloons immediately highlighted a serious source of error: the thermometers took too long to come into equilibrium with the surrounding air, leading to a sizeable difference (sometimes greater than

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10 °C) between the temperatures measured during the ascent and the descent of the balloon. This issue was resolved, though, by Léon Teisserenc De Bort (1855-1913), who in 1898 began systematic investigations using weather balloons whose ascent was slowed to reduce, in conjunction with other correctives, the error. In 1902, following observations made with 236 weather balloons, of which 74 had attained altitudes of 14 km, he announced an unexpected discovery: the temperature always decreases with height, “but the decrease, instead of remaining the same with altitude, as it had been supposed, reaches a maximum, and then rapidly falls off to become almost zero at an altitude that in our regions is 11 km, on average. Starting from a height that varies with the atmospheric conditions (from 8 to 12 km), there begins a zone of very gradual temperature decrease… We cannot specify the thickness of this zone, but from the current observations it appears to extend for at least several kilometres.”312 Teisserenc called this region the “isothermal zone”. His discovery was initially met with skepticism, especially because he was unable to provide a theoretical interpretation of his results. However, still in the first years of the century, the mechanism came to be understood, mostly due to Lord Rayleigh’s detailed works on the diffusion of sunlight (§ 1.1). The theory explained that violet and ultraviolet radiation is absorbed by the upper layers of the atmosphere, while red and infrared radiation is absorbed by the water vapour and carbon dioxide present in the lower layers of the atmosphere. Other radiation reaches the surface of the Earth, heats it, and is subsequently re-emitted as infrared radiation that is then newly absorbed by water vapour and carbon dioxide. Ultimately, a large part of the solar energy remains trapped in the atmosphere, causing continual mixing of the lower atmospheric layers. This is why this region is called the troposphere, while the “isothermal zone” of Teisserenc is called the stratosphere because it does not host any vertical motion, but only motion in horizontal “strata”. Initially, it was thought that the stratosphere extended all the way to the outer reaches of the atmosphere. A study on meteor trajectories published in 1923 by Frederik Alexander Lindemann (1886-1957) and Gordon Miller Dobson (1889-1976) jeopardized even the new theory: the two scientists showed that the ignition and extinguishing of meteors can be explained if one assumes that the temperature of the upper atmosphere increases from -55 °C in the stratosphere to 30 °C at a height of 50 km. 312

L. Tesserenc De Bort, Variation de la température de l’air libre dans la zone comprise entre 8 et 13 km d’altitude, in “Comptes-rendus de l’Académie des sciences de Paris”, 134, 1902, p. 987.

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The conclusions of Lindemann and Dobson were qualitatively confirmed multiple times by markedly different research approaches: the study of “silent zones” in the propagation of sounds (that is, the fact that the sound of an explosion can be heard at a great distance yet not at lesser distances in the same direction), the calculation of temperature from the thermodynamic equilibrium between matter and radiation, the spectrum of the aurora borealis, and ionized atmospheric layers (§ 10.6). The fervour of theoretical study made it ever more pressing to conduct direct measurements not with a weather balloon, but with aerostats that could bring both scientists and apparatuses to high altitudes. As a result, scientific ascents were once again undertaken, and in the 19th and early 20th centuries increasingly organized for competitive races or military uses, to the extent that almost all armies had a special division to manage the use of aerostats in war by the middle of the 19th century. Even these non-scientific uses, however, provided precious critical and technical data. After the first world war, this method was utilized by the French-Swiss physicist Auguste Piccard (1884-1962), then professor of physics at the free university of Brussels, who up until 1925 studied a special aerostat design in which a spherical aluminium basket is hermetically closed. The basket could host a scientist, an assistant, and various scientific apparatuses. In his first ascent in 1931, Piccard reached an altitude of 15 km but brought back few scientific observations because of the various incidents he experienced. More successful was his second attempt in 1932, during which he reached a height of 16 km and collected ample scientific measurements. Piccard’s ascents caused widespread global stir, popularizing the term “stratosphere”, which in common parlance came to mean an almost unreachable metaphoric height (of intelligence, speculation, spirit, etc.), while scientists used it and continue to use it in its specific sense. Piccard’s exploits were imitated by many Soviet and American scientists and soldiers: in 1935, Americans reached the altitude of 22,066 km. Direct experimentation in the atmosphere led to natural modifications and corrections to what had been deduced from theory and indirect measurement. These advances clarified, for instance, the composition of the atmosphere, which had been long been a topic of tortuous and confused study (Gouy; H.B. Morris; Raul Sophus Epstein). Once the belief that the stratosphere was characterized by an absence of wind was disproven, it followed that the atmosphere beyond the troposphere did not contain aerial gases in layers of decreasing density, ending with a layer containing only hydrogen. Instead, it seemed that the atmosphere was made up of mainly nitrogen and oxygen throughout, which at a certain

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altitude in the hundreds of kilometres are in atomic and not molecular form. The discontinuous surface between the troposphere and stratosphere, called tropopause by Charles Féry (1865-1935), is found at temperate latitudes near the altitude of 11 km. A second interface separates the stratosphere from the upper atmosphere and signals the end of the constant temperature zone and the beginning of temperature increase: this is found at heights of around 40 km and Féry proposed to call it the stratopause. Perhaps there are other discontinuous boundaries above 40 km: D. H. Marty and O. O. Pulley hypothesized two more, at respective heights of 60 and 80 km. What is certain is that there is no limiting surface of the Earth’s atmosphere as a whole, which Laplace placed at six Earth radii above the equator, reasoning that at that point the centrifugal force “exactly balances with gravity”313 After the second world war, atmospheric science saw tremendous advances thanks to the use of new techniques with carrier rockets and artificial satellites. The new direct measurements improved and at times corrected certain details, but the general picture remained not too different from that developed during the interwar period. Out of these developments, we limit ourselves to recounting that in 1951 the International Geodetic Union proposed the following nomenclature for the division of the atmosphere into temperature-based regions: the troposphere is the region between the Earth’s surface and the tropopause, the level at which temperature ceases to decrease, and has a thickness that varies from 7 km at high latitudes to 18 km at the equator; the stratosphere is the region between the tropopause and the stratopause, at around 35-40 km in altitude, where the temperature begins to increase; the mesosphere is the region between the stratopause and the mesopause, the height at which the minimum temperature is reached (about -100 °C); the thermosphere is the region that follows the mesosphere, in which temperature increases rapidly until reaching the thermopause, after which it remains essentially constant; the exosphere, finally, is the outermost part of the atmosphere. A theoretical study by Marcel Nicolet on the chemical composition of the atmosphere in the different regions leads to a distinction between the conditions in the stratosphere, the mesosphere, and the lower thermosphere. This distinction is fundamentally due to the behaviour of the oxygen atom. It begins in photochemical equilibrium with ozone in the stratosphere, becomes more abundant than it in the mesosphere, and breaks equilibrium

313

P.S. Laplace, Exposition du système du monde, Paris 1824, p. 270.

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in the lower thermosphere.314 As far as the dynamical conditions in the different atmospheric regions, R. J. Murgantroyd held that “in general, however, the upper mesosphere can be dealt with very similarly to the troposphere, the mesopause region to the tropopause region and the lower thermosphere to the lower stratosphere in the study of general circulation.”315

10.6 The ionosphere In 1896, when Guglielmo Marconi began his first official experiments with radiotelegraphy, science held that electromagnetic waves propagated exactly like visible light waves, and consequently that natural obstacles and the curvature of the Earth placed strong limitations on the transmission distance of signals. Yet on 12 December 1901, Marconi was able to radiotelegraphically connect Poldhu in England with St. John’s in Newfoundland across the 320 km stretch of the Atlantic ocean that separated them. Contrary to the predictions of the theory, Marconi thus demonstrated that the curvature of the Earth did not impede the transmission of electromagnetic waves. Scientists initially tried to explain this phenomenon by attributing it to the diffraction of radio waves. In a popular article, Poincaré maintained this view, adding that the fundamental role of the antenna was to increase wavelength. Antenna height is increased because “there is an increase in wavelength and consequently in the diffraction effect through which the obstacle due to Earth’s curvature can be bypassed.”316 The book from which this sentence is cited is the second expanded edition of another one of the same title published in 1899, in which there was no mention of cableless telegraphy, which Marconi had already accomplished three years earlier: this is indicative of how radiography was viewed by official science at the end of the century. Sommerfeld, in a series of publications from 1909 to 1926, gave a rigorous theory of propagation through diffraction in the presence of a 314

M. Nicolet, Le problème physique de la stratosphère à la thermosphère inferieure, in “Proceedings of the Royal Society of London”, A (288), 1965, p. 491. 315 R.J. Murgantroyd, Winds in the Mesosphere and Lower Thermosphere, ibid., p. 588. The articles cited in this paper, and the previous one, are part of a larger discussion between specialists held on 3 December 1964. All of these articles contain an ample bibliography, to which we direct the reader in the case of further historical interest in the topic. 316 H. Poincaré, La théorie de Maxwell et les oscillations hertziennes. La telegraphie sans fil, Naud, Paris 1904, p. 98.

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finitely conducting and flat surface. The curvature of the Earth, however, complicated the problem enormously. One of classical electromagnetism’s most loyal devotees, Lord Rayleigh, also returned to the problem of electromagnetic wave propagation: he concluded that Marconi’s result was inexplicable within the current framework, and thus new conceptions of the propagation of electromagnetic waves and the constitution of the Earth’s atmosphere were necessary. Poincaré, towards the end of the volume cited above, wondered how Marconi’s successes were to be interpreted, in particular the communication channel that had been established between Poldhu and the Carlo Alberto, a ship anchored at La Spezia, which was even more impressive than the earlier signal sent to Newfoundland because this signal had surmounted the obstacle posed by the Massif Central of France. Poincaré did not rule out that the successes were due to the ample power transmitted and the sensitivity of the detection instruments, but also wondered “whether the theory [was] defective and insufficient to account for these facts.”317 In 1902, the American engineer Arthur Kennelly (1862-1939) and, independently, the British physicist Oliver Heaviside (1850-1925) and the Japanese physicist Hantaro Nagaoka hypothesized the existence of a layer of highly ionized gas in the upper atmosphere functioning as a mirror for radiation, reflecting electromagnetic waves that come from emitting stations on the ground back towards the Earth. This was purely a hypothesis at this point, and one advanced very cautiously at that. Heaviside, for example, after writing that the Earth’s crust perhaps takes part in the propagation of the wave, added in passing: “There may possibly be a sufficiently conducting layer in the upper air. If so, the waves will, so to speak, catch on to it more or less.”318 It is worth noting, though, that the possibility of conducting layers in the upper atmosphere had really been first proposed by 1882 by Balfour Stewart to interpret the small but regular daily changes in the Earth’s magnetic field. This layer was called the Kennelly-Heaviside layer, or more commonly the Heaviside layer, perhaps because the British scientist had published his proposal in a more widely read work. However, the existence of the Kennelly-Heaviside layer remained a hypothesis without experimental evidence, so it came to be believed among radiotelegraphers that the propagation of Hertzian waves on the 317 318

Ibid., p. 109. O. Heaviside, in Encyclopaedia Brittanica, 1902, Vol. 33, p. 215.

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surface of the Earth occurred through diffraction, giving rise to the practice of using long wavelengths that could better bypass obstacles. Meanwhile, however, amateur radio operators surprised engineers and scientists by communicating amongst themselves with wavelengths shorter than 200 m, which were held to be inadequate for radiocommunications: in 1921 American and British operators succeeded in communicating across the Atlantic, and in 1923 Americans and French operators did the same. At this point, the work of a World War I veteran entered the picture: Edward Victor Appleton (1892-1965), an ex-officer of the radio section of the British Engineer Corps. It resulted from experiments that began in Cambridge in 1922 that the radio signals emitted by the BBC in London, which were practically constant during the day, weakened at night. Appleton thought that a possible explanation of the phenomenon could be provided by the interference between radio waves that propagate directly from the emitter to the receiver and waves that reach the receiver after having been reflected by Heaviside’s layer. If the difference between the two paths is a whole number multiple of the wavelength, then the receiver measures the maximum possible intensity of the radio signal, while the minimum intensity is obtained if the path length difference is an odd number of half-wavelengths. Appleton tested this hypothesis using an ingenious method called frequency modulation, which also allowed him to find the height of the reflecting layer. Suppose that the emitter produces waves of slowly varying frequency (and thus slowly varying wavelength). The receiver will observe a succession of n maxima and minima in the radio signal. If the initial and final wavelengths are ߣଵ and ߣଶ , one has

݊=

‫ܦ ܦ‬ െ ߣଵ ߣଶ

where D is the path length difference between the two trajectories. Once D is known, simple triangle geometry permits one to calculate the height of the layer. Applying this method, Appleton, with the help of his assistant M. A. F. Barnett, demonstrated the existence of the Kennelly-Heaviside layer and determined that it was at an altitude of 80 km.319 Still with the same type of method, in the winter of 1926-27 Appleton discovered the existence of a second, even more ionized layer that he called the F layer, whose lower

319 E. A. Appleton and M. A. F. Barnett, Local Reflection of Wireless Waves from the Upper Atmosphere, in “Nature”, 115, 1925, pp. 332-34.

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limiting surface was around 230 km above the Earth.320 A bit earlier, in 1925, the Americans Gregory Breit and Merle Tuve (1901-1982) measured the height of the layer based on the time it took for electromagnetic signals to return to Earth after being emitted by radio stations and subsequently reflected by the layer. In this way, they demonstrated that the region of the atmosphere between 50 and 500 km above the Earth can be ionized to an appreciable extent by incoming external radiation: it was therefore called ionosphere, a term proposed by Robert Watson in 1926. The Heaviside layer, characterized by a maximum in ionization density (1.0-1.3(109) ions per m3) is found at around 100 km; a second layer (Appleton’s layer or the F layer) is found between 200 and 400 km, above which there are other layers denoted by other letters in the alphabet. The F layer plays an essential role in the reflection of waves with wavelength below 40 m, meaning a frequency greater than 7.5 MHz (1 MegaHertz = 106 Hertz). Ionospheric research using electromagnetic waves has had important applications in radar techniques (§ 10.10) and radiocommunications with extraterrestrial receivers, which became critical in the era of cosmic exploration. The relationship between ionospheric perturbations, the variation in the Earth’s magnetic field, and auroras are areas of active scientific study. Perturbations in the ionosphere also influence radio-propagation, and in general the propagation of electromagnetic waves is tied to the presence and constitution of the ionosphere in a complex way, far from the simple relationship that Kennelly and Heaviside had imagined at the beginning of the century. In the middle of the 20th century, the study of the ionosphere benefitted greatly from the direct measurements of ionic densities taken by instruments installed on missiles and artificial satellites. History was made in 1958 with the launch of the first American artificial satellite, Explorer I. A group of researchers headed by James Van Allen had installed a Geiger counter and a transmitter on board the satellite. The sides of the Geiger counter were thick enough such that only protons with energies greater than 30 MeV and electrons above 3 MeV could pass through them. It was found that near the equator, starting from a height that varied between 400 and 1200 km depending on the longitude, the number of particles detected by the Geiger counter rapidly increased, until over 1,700 particles per square centimetre were detected every second, 1,000 times more than what had been expected from cosmic radiation. It followed that the Earth is 320

E. A. Appleton, The Existence of More than One Ionised Layer in the Upper Atmosphere, ibid., 120, 1927, p. 330.

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surrounded by highly radioactive zones. The thickness of these zones, together called the Van Allen belt, was measured in March 1958 when the satellite Pioneer III, launched with the aim of reaching the Moon, failed and instead followed a lengthened elliptical trajectory. During flight, the Geiger counter that Van Allen’s group had placed on board allowed scientists to establish that the radiation belt stretched as far as 40,000 km from the Earth, with two distinct peaks at 3000 km and 15,000 km. The same year, Van Allen was able to draw a map of his eponymous belt, which is a consequence of the Earth’s magnetic field and is populated by protons and electrons.

10.7 Rockets and artificial satellites We have frequently recognized the services rendered to pure scientific research by rockets and satellites. While their construction and use are described by some of the most advanced techniques of modern applied physics and engineering, we nevertheless think it opportune to briefly mention them as well in this history of physics, if only to mark a few important dates that unveiled a new world of scientific inquiry. The possibility of launching artificial satellites or space vehicles was already pointed out by Newton in the first pages of his Principia.321 Missile launches, namely launches of bodies with a propulsion mechanism of their own and non-gravitational acceleration for all or part of their trajectory, dates back to the Chinese in the first years of the 13th century, who used gunpowder in bamboo or cardboard tubes. This military technique spread throughout Asia and Europe and was used up through the entire first half of the 19th century. Significant improvements in the cannon (rifling, breechloading, hydraulic brake) led to the abandonment of missiles as weapons of war, leaving them almost exclusively in domain of amateur pyrotechnics. At the start of the 20th century, however, thanks to a handful of pioneers who at the time appeared merely dreamers, the scientific study of stratospheric and interplanetary flight began. Among them, the designation of forefather is reserved for the Russian Konstantin Eduardovich Tsiolkovsky (1857-1935), who in 1903 published a paper on spatial exploration with reaction-based apparatuses, followed by numerous other writings on the same topic. Later research was influenced by three critical ideas of Tsiolkovsky: the rocket is the only possible means of propulsion outside 321 § 6.4 of M.Gliozzi: A History of Physics from Antiquity to the Enlightenment, Cambridge Scholars Publ. 2022.

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the atmosphere, multistage rockets are necessary to reach high velocities, and liquids are the most convenient type of propellent. Also among the pioneers was the Romanian (later naturalized German) Hermann Oberth (1894-1984), who starting in 1910 experimented on rockets and studied astronautics; during the second world war he worked in Werner von Braun’s (1912-1977) group on the study and use of longrange missiles. Lastly, we mention the American Robert Goddard (18821945). In 1919, he published a pivotal and widely-circulated article that, aside from discussing the theory of the rocket, which essentially coincided with Tsiolkovsky’s, also reported a series of experimental tests performed in 1918 with powder rockets, the precursors to the bazooka, the American antitank weapon employed during the second world war. Goddard then dedicated himself to experimentation on rockets with liquid combustibles and in 1926 launched the first rocket of this type, naturally with modest results by today’s standards, but of great importance at the time, as they demonstrated the feasibility of this new form of propulsion. German experiments followed in 1928-29, as did Russian experiments in 1933. Research efforts in this field, increasingly concealed as military secrets, continued in all highly industrialized nations. When in the course of the second world war the Germans bombarded London with the V1 and V2 rockets designed by von Braun, it became apparent that their progress in the study of rockets had been faster than that of the Americans and Russians. Further advances were made evident to the scientific community outside the USSR by a Russian communiqué on 20 August 1957 announcing the launch of an intercontinental missile; another communiqué released the following 4 October informed the world of the launch of the first artificial satellite, which was followed by a second launch on 3 November, and the first successful American launch on 31 January 1958. Space flights continued and continue, despite their enormous cost, the great risks involved, and the sacrifice of human lives: this is the price that humanity has always paid for its great conquests in the name of progress. In 1961, satellites began to carry human crews; in 1968 three Americans landed on the Moon: the expedition was repeated another six times, until 1972. Today, satellites are sent into orbit for various scientific aims relating to research in the upper atmosphere: studies and measurements of the ionosphere, radiation belts, cosmic rays, and magnetic fields, as well as to make astronomical observations. Among the many practical applications to daily life that have followed, the most popular are the uses of artificial satellites in radiocommunications and intercontinental television.

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10.8 Microwaves For thirty years, the field of radiocommunications employed long and medium wavelength waves, leading scientists and engineers to focus on this part of the electromagnetic spectrum. Around 1925, however, their attention was drawn to shorter wavelengths, both for their aforementioned use by amateur radio operators and for the objective of ionosphere exploration, leading to the birth of a new science, radioastronomy. The dawn of this discipline can be traced to 1932, when Karl Jansky, while studying the disturbances in short wavelength transmission caused by electric discharges, picked up galactic electromagnetic waves; following observations were shrouded in military secrecy, and consequently the new field only blossomed after the second world war.322 As we have previously discussed, in the laboratory, the study of short waves dates back to their namesake Hertz, who had employed waves of wavelength around 66 cm, while Righi had used waves of about ten centimetres (§ 3.29), but also produced ones of 2.5 cm. In 1897, the Indian physicist Jagadis Chandra Bose (1858-1937), using a two-spark oscillator based on Righi’s, succeeded in producing waves of 6 mm wavelength. Yet these small wavelengths were obtained with instruments that yielded small quantities of energy. To radiate more energy, one had to resort to instruments of sizeable dimensions and therefore to longer wavelengths. Electromagnetic waves between 10 cm and 0.2 mm in wavelength, at the upper limit of the infrared region, are called microwaves, and have become particularly important in the last few decades, not only due to their technological applications for radiocommunications and radar, but also in other fields of scientific research. The techniques that employ microwaves are markedly distinct from those used in the applications of longer wavelengths: electronic tubes, oscillating circuits, and transmission lines for microwaves are different from the analogous long wavelength devices. We restrict our attention to the transmission lines for microwaves, which are waveguides, rectangular or cylindrical tubes with conducting walls, filled with a dielectric in which waves can propagate. The transversal dimensions of the waveguide must be of the same order of magnitude as the wavelengths that it transmits, making it impossible to use for long wavelengths.

322

Poincaré notes that the coherer had been used to “research whether the Sun emits Hertzian radiation; the result was negative. Perhaps such radiation is absorbed by the solar atmosphere.” (Poincaré, La théorie de Maxwell et les oscillations hertziennes, cit. p. 42).

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The discovery of the properties of waveguides has its roots in the end of the 19th century, when experimenters placed oscillators in a metallic tube to shield them from external perturbations. Lodge first observed some unusual properties of wave propagation, and in 1879 Lord Rayleigh conducted a cursory study of propagation inside a waveguide. Other illustrious physicists like Thomson, Larmor, and Drude also became involved, yet did not manage to develop a satisfactory theory. Only in 1936 was an acceptable mathematical theory of waveguides published by the Americans John Renshaw Carson (1857-1940), Sallie Mead, and Sergei Schelkunoff (1897-1992) of Bell Telephone Laboratories. That same year, the Frenchman Léon Nicol Brillouin (1889-1969) interpreted the properties of electromagnetic wave propagation in a tube as arising from the interference of plane waves reflected by the tube walls. From a physical point of view, the mechanism is the following: the metallic walls of the waveguide act as a screen and prevent the electromagnetic wave generated at the tube opening from dispersing into the surrounding space, forcing the electromagnetic energy to propagate along the tube. It follows from Brillouin’s theory that there is always a component of the electric field or magnetic field in the direction of propagation, whereas in typical waves travelling inside conductors the fields are perpendicular to the direction of propagation: these waves are called “transverse electromagnetic waves”, or TEM, and propagate at the speed of light in the dielectrics that surround conductors. The waves in waveguides, on the other hand, are called “transverse magnetic” (TM) or “transverse electric” (TE), depending on their character. At the beginning of the century only TEM waves were known, and thus it had been impossible to give a theoretical interpretation to waveguide propagation. Out of the many devices built using waveguides we mention the cavity resonator, formed by part of a waveguide that is closed off by conducting walls. If inside the cavity one produces an electromagnetic wave whose wavelength is a submultiple of the distance between the two walls that seal the cavity, stationary waves arise from the reflection of the original waves on the two ends. The resonant cavity is therefore the analogue of the oscillating circuit, but with the added advantage of exhibiting a threefold infinity of possible resonant frequencies. The production of short wavelength electromagnetic waves requires special instruments; 43 cm waves had been obtained as early as 1919 using triodes. However, the triode is no longer adequate for shorter wavelengths, because when the time required for an electron go from cathode to anode is of the same order as the period of the desired wave, phase shifts arise that render the triode dysfunctional. Scientists attempted to overcome this

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hurdle by using other devices: “lighthouse triodes” were developed by the Russians Deviatkon, Gurevch, and Khokhov in 1940; klystron was created in 1939 by the Americans Russel and Sigurd Varian; R. Kompfuer developed the travelling wave tube in 1942, for which John Pierce gave the theory in 1947; and finally the magnetron, which had its origins in Hull’s diodes in 1921, in which a current between a cylindrical cathode and a coaxial anode is regulated by a magnetic field parallel to its axis. After undergoing many modifications, the magnetron has become the most powerful source for high power emission and is used especially in radar.

10.9 Microwave spectroscopy Traditional optical spectroscopy analysed radiation in the infrared, visible, and ultraviolet ranges, reaching further towards the lower limit of wavelengths. The spectroscopy of electromagnetic waves began with Hertz, who studied the dielectric constant of different substances (resin, amethyst, potassium permanganate). Banly later studied the properties of Hertzian waves (a frequency range that encompasses what today are called microwaves and radio waves) in certain liquids, like alcohol and petroleum, while Righi experimented on their absorption in certain nonmetallic materials like wood and mirror glass. Then, electromagnetic wave spectroscopy turned towards the study of long wavelength radiation for reasons we discussed in the previous section. Hertzian spectroscopy, or more precisely microwave spectroscopy, began in 1934. That year, C. E. Cleeton and N. Williams observed that at ordinary pressures, ammonia absorbs electromagnetic waves of 1.25 cm wavelength. After the second world war studies of this sort multiplied, benefitting from the progress that had been made in microwave research. Microwave spectroscopy is often used to study the absorption of microwaves in matter and fill the gaps in the incomplete information that visible spectra provide on electronic energy levels. Through microwave spectroscopy experiments, for example, Willis Eugene Lamb, Jr. and Robert Retherford studied the fine structure of the hydrogen spectrum in 1950. The instruments used in microwave spectroscopy are naturally different from those used in optical microscopy: dispersive devices like prisms and gratings are replaced by electronic sources of essentially monochromatic radiation, which consequently have much higher sensitivity and resolving power that optical spectrographs. We will describe a few important applications of microwaves and microwave spectroscopy below. Here we mention in passing that microwave

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spectroscopy was also applied to atomic clocks, first appearing in 1949 due to the work of Harold Lyons, who built an ammonia clock and later a caesium clock.

10.10 Some applications: radar, maser, laser Hertz had showed that electromagnetic waves were reflected in 1886, a finding corroborated by many other experimenters (§ 3.29). Starting from the beginning of the 20th century, scientists proposed to use this effect to detect obstacles for marine navigation; it appears that the first patent request for such a device was submitted in 1904 in Germany. Yet practical applications remained distant due to the insufficient technical means of producing and receiving waves. The possibility of using electromagnetic waves to detect distant objects was suggested in 1925, after Breit and Tuve had used a similar impulse technique to explore the ionosphere (§ 10.6). Research commenced in all scientifically advanced countries, but the best results were obtained by British and American scientists. In 1930, a group of British scientists led by Watson Watt began working on such applications in military secrecy enforced by the British government. By 1935, the group began to build its first instruments, which they called radar, an acronym for Radio Detecting and Ranging. In 1938, again through the work of Watson Watt, the British coastal defence radar network was completed, and was one of the determining factors in the British victory in the Battle of Britain (August 1940). During and after the second world war, radar technology made rapid progress closely tied to the techniques of microwave production and reception. The use of microwaves (falling between 1.5 m and 3 cm in wavelength, depending on their use) was especially convenient because highly focused antennae could be used, which thus radiated very narrow bands. Besides its military uses, radar has been used in many civilian applications: naval and aerial navigation, the detection of cyclones in meteorology, and several others. Radars have also been employed in astronomy, especially in the study of the solar system: the detection of radio echoes from the Moon (1948), Venus (1956), and the Sun (1959) have allowed for incredibly precise measurements of their distance from the Earth, with uncertainties of a few parts per thousand, unimaginable a few decades earlier. The operation of the radar is based on an elementary physics principle analogous to the one used by Langevin during the first world war and in

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submarine ultrasonic depth finders. An electromagnetic wave strikes the object that one wants to measure and becomes reflected or re-emitted; the returning wave is captured by instruments on the radar that originally emitted it. Based on the time taken by the signal to reach the target and return to its source, the distance of the object is deduced. Yet while the physical principle is simple, the devices necessary for its operation are complicated and require the use of advanced techniques, as is evident from the lengthy incubation period of the invention and its numerous improvements throughout the years. To discuss the next application, we must first take a step back and recall some concepts we have already discussed. Einstein’s theory of radiation (§ 7.13), in its later quantum version, holds that spontaneous emission occurs when an electron passes from a higher energy level to a lower energy level, emitting a photon of energy H = hQ, where Q is the frequency; absorption, on the other hand, occurs if an electron jumps from a lower energy level to a higher one. When an excited atom, namely one with higher energy than its ground state, is struck by an electromagnetic wave, i.e., a photon, it is stimulated into emitting a second photon of the same frequency, and subsequently falls to a lower energy level. This is called stimulated emission and corresponds to what Einstein had called “induced emission”. Consequently, the incident wave gains the energy lost by the atom and is amplified. However, for the stimulated energy to be greater than the absorbed energy, the atoms at a higher energy level must outnumber those at a lower level. In 1954 Charles Townes, James Gordon, and Herbert Zeiger succeeded in building a high-frequency amplifier that used this effect, which they named maser, an acronym for Microwave Amplification by Stimulated Emission of Radiation. In this device, stimulated ammonia atoms are separated by an inhomogeneous electric field; they then pass through a resonant cavity where they amplify a high-frequency wave. In the appropriate conditions, this system is also a very stable automatic oscillator, and can therefore be used as an atomic clock. From the maser originated the laser (Light Amplification by Stimulated Emission of Radiation) or optical maser, as it was initially called. The essential characteristic of the laser is the emission of coherent light, whose properties were studied by Lord Rayleigh in 1895 in papers that had appeared purely theoretical exercises because nobody knew of any method to obtain it. In ordinary conditions, multiple perfectly monochromatic point sources emit waves whose phases randomly change very rapidly, because the components that emit the radiation are individual atoms or molecules that radiate independently of each other. Lord Rayleigh

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demonstrated, through a lengthy calculation, that n identical point sources of incoherent light produce light n times as intense as that produced by a single source. However, if the n sources all emit light in phase with each other (and the radiation remains in phase), then the intensity of the light is n2 times that of a single source – much greater than what n incoherent sources can produce. This result may appear to contradict the principle of energy conservation, but Lord Rayleigh noted that incoherent light (from a flame, lightbulb, the Sun, etc.) radiates outwards in all directions, while coherent light propagates almost entirely in one direction. The most valuable property of coherent light is precisely that it is focused, allowing for radiation to be emitted in specific directions. In the first solid state lasers, the active material, in which the excited atoms outnumber the lower-energy ones, was a pink ruby rod composed of synthetic Al2O3 crystals with a small percentage of chromium impurities (around 0.5 parts per thousand), which Theodore Maiman used in 1960 to construct the first laser. Through the optical pumping technique developed by Alfred Kastler in 1950, the chromium impurities are excited to obtain population inversion, that is more atoms in the higher energy level than in the lower one. A visible light wave of the appropriate frequency strikes the ruby rod and stimulates the chromium atoms to emit its same frequency: the emission of coherent red light follows. In the first ruby lasers this emission was intermittent, with impulses that lasted around a thousandth of a second. Today there are solid state lasers that can emit at different frequencies and even continuously. The most well-known continuous emission laser, however, is made up of a gas, a mixture of helium and neon which was first built in 1960 by Ali Javan, William Bennet, and Donald Harriot. If the coherence of laser radiation is used, very high amounts of energy can be concentrated in minute spaces: lasers are therefore employed in nano-surgery, photoincision, and welding. The fact that the light is monochromatic, while certainly not perfectly, but to a degree unattainable by other light sources, finds useful applications in spectroscopy. The stability in frequency is used as a measure of time and in experimental studies on relativity, while the minimal dispersion of the beam has applications in telecommunications and measuring large distances (for example, the distance between the Earth and the Moon).

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MATTER AND ANTIMATTER 10.11 A note on the nature of scientific research after the second world war The second world war accentuated the ties between large administrative entities and scientific research (§ 4.11), which became increasingly more collective and organized by governments. Some scientists expressed disdain for this change, as can be seen from Joliot’s nostalgia for the “artisanal” character of nuclear research in its earlier years, when the researcher felt a proximity with the phenomenon being studied and could give free reign to his creativity, or Hahn’s wistful memory of scientific work in Montreal under Rutherford’s guidance, when electroscopes were made from cigarette boxes balanced on top of preserve cans. Both were futile aspirations for an individual freedom of the past that war and technological progress had crushed, perhaps definitively. The financial support of the state, today an indispensable part of nearly all serious scientific research, inevitably pushes research towards utilitarian aims. On the other hand, in the first 40 years of the 20th century, physics had been overrun with new ideas, new mathematical techniques, and new philosophies. What followed, as had transpired the previous century after the works of Ampère and Faraday, was a period of consolidation, critical reflection, and especially application of the new scientific ideas through the ingenuity of engineers. After the second world war, research became more applied than speculative, as is evidenced by the previous pages of this chapter. There were no new revolutionary ideas, but rather a revolution in technology, which became so tightly interwoven with physics through a continual exchange of knowledge and ideas that it could be almost completely identified with it.

10.12 Thermonuclear reactions Since 1920, Aston had thought that the mass defect in nuclear reactions was transformed into nuclear binding energy. An impressive series of experiments, which we have already mentioned, confirmed this interpretation and allowed for the calculation of the binding energies. For example, the binding energy of a helium nucleus, an Į particle, is around 28 MeV; the binding energy of lithium is about 30.5 MeV, and so on. Now, the binding energy of a nucleus that results from the union of two light nuclei is generally greater than the sum of their individual

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binding energies, while the binding energy of a heavy nucleus that is split into smaller parts is less than the binding energy of the parts, as we have discussed (§ 9.11). In both processes, energy is released because the total mass decreases: the decrease in mass multiplied by c2 gives the amount of energy released through Einstein’s relation. The binding energy can thus be used for two opposite aims: either to join lighter elements, i.e. fusion, or to break up heavy elements, i.e. fission (§ 9.11). To obtain nuclear energy from fusion, one must overcome serious obstacles. In order for two nuclei to fuse they must be in extremely close contact, but nuclei have a positive charge and therefore repel each other; an amount of energy equal to their electrostatic repulsion is required bring them together. Calculations show that this energy is of the order of 20,000 V. This quantity of energy could be found, by thermal excitation, in a gas that has a temperature of 200 million degrees: at the beginning of the century scientists did not know if such a temperature even had physical meaning. Despite not having direct observations of stellar interiors, there are good reasons to believe that the matter they contain is in a rather different state that what we are used to. In particular, the inside of stars hosts the conditions necessary for thermonuclear fusion reactions to occur, leading to the production of enormous quantities of energy, first estimated by Eddington (§ 7.10). The thermonuclear reactions (nuclear reactions caused by elevated temperatures) that are thought to occur in stars were fervently studied in 1939-40 by several physicists. One of the most well-known ideas to come from this research was the cycle proposed in 1940 by Hans Albrecht Bethe (1906-2005), an American physicist of German origins, which today is called the Bethe cycle or carbon-nitrogen-oxygen cycle. In this process, through six nuclear reactions, four protons are fused into a helium nucleus without consuming carbon or nitrogen. Another well-known process is the proton-proton cycle, studied by Fowler in 1951, through which four protons are also fused into a helium nucleus. The study of these fusion processes has for now remained within the realm of astrophysics. For a fusion reaction to be triggered and continue for a certain amount of time, two fundamental conditions must be met: very elevated temperature of the order of millions of degrees and a very dense medium, such that there are enough collisions between particles. It is thus clear why prewar physicists, despite having observed some exothermic fusion reactions, could not harness energy from nuclear fusion. The construction of the atomic bomb created the conditions required by a fusion process. The possibility of constructing a fusion-based bomb

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was hypothesized as early as 1943, as soon as the realization of a fission bomb began to appear likely. A fission bomb is to a fusion bomb like a match is to a gas oven. The actual development, however, began in the United States after 1949, following the development of the first Soviet atomic bomb. Research was conducted by a group of physicists under the direction of the Hungarian-born physicist Edward Teller (1908-2003). In 1951, the first experiments were conducted. The fusion bomb, hydrogen bomb, or H-bomb showed the world its terrifying potential, which is thousands of times greater than the atomic bomb that had destroyed Hiroshima.

10.13 The physics of plasma After the appearance of the H-bomb, scientists immediately thought to contain the energy of nuclear fusion and use it for peaceful purposes. The efforts and money dedicated to this goal by industrialized nations was enormous. However, the problems faced in trying to produce energy from nuclear fusion were equally imposing, mainly because temperatures beyond hundreds of millions of degrees are required to obtain thermonuclear reactions. The first problem is that of the container in which nuclear fusion would occur. All known Earth materials evaporate long before having reached the necessary temperatures. Various devices were designed, the most ingenious of which seems to be the “magnetic trap” (also known as magnetic mirror), proposed in the Soviet Union in 1953 and in the United States the following year. The magnetic trap is a space in which ionized gas would be confined by suitable magnetic fields that are either constant or pulsating: prototypes giving rise to both types of fields have been built. Equally intractable problems arise when one tries to heat the gas to the temperatures necessary to keep the reactor running for a sufficiently long time. In a gas at a temperature of millions of degrees, all orbiting electrons are stripped away from their atoms through ionizing collisions. What formerly was a gas is then made up of rapidly moving nuclei and electrons. The ensuing substance is neither a solid, liquid, nor gas, but a new state of matter, which the American physicist and chemist Irving Langmuir (1881-1957) called plasma in 1928. Langmuir, who received the 1932 Nobel prize in chemistry for his research and discoveries in the field of surface chemistry, introduced an auxiliary electrode, later called a Langmuir probe, in a vacuum tube containing ionized gas, and from the current-potential relation in the probe he deduced the density of electrons and ions in the tube, as well as their velocity distributions. Through these

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experimental considerations, Langmuir distinguished two regions in the tube: one, which he called plasma, was electrically neutral; the other, in which the electric field was considerably strong, formed a sheath around the electrode. In 1926, Frans Penning discovered that plasma can oscillate: if a collection of electrons shift in position, the region with an electron deficit pulls them back in with a restoring force dependent on their shift. Oscillations therefore arise, with a frequency that depends only on the average density of the electrons in the plasma, according to the formula given in 1929 by Langmuir and Lewi Tonks: ݂ = 9000ξ݊ where n is the number of electrons per cubic centimetre and f is measured in Hertz. It follows that plasma can host longitudinal electromagnetic waves, transverse electromagnetic waves, and hydromagnetic waves. Of great interest in current physics research is the study of plasma shock waves, which are analogous to shock waves in gases and could be used to heat plasma. The production of very high temperatures in the laboratory is one of the most striking consequences of plasma studies. After the second world war, Pyotr Kapitza and Igor Kourtchatov in the Soviet Union and Winston Bostick in the United States obtained temperatures of the order of several millions of degrees for very short time periods. A new technique using plasma channels, first devised by H. Macker in 1951, even allowed for temperatures like 50,000 °C to be stably obtained, while before the war the maximum temperature produced been through the electric arc, which at best attained 6000 °C. It is evident that the new instruments rendered a precious service to the study of chemical reactions at high temperatures in addition to plasma studies. The definition of plasma given by Langmuir has been extended today to include any mixture of oppositely charged particles of almost equal density with high mobility of at least one of the two species (generally electrons). With this wider definition, metals also become plasmas, as conduction electrons can freely move within them. Plasma is also found in the ionized gas in neon signs, lightning, and stars. The problem of magnetic confinement, the discovery of strong stellar magnetic fields (Horace Babcock, 1949), and other astronomical phenomena (cosmic radiation, the ionosphere, auroras, magnetic storms) engendered a new field of mathematical physics, magnetohydrodynamics (or magnetofluid dynamics, or hydromagnetics), which began to systematically

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develop in 1942, when Hannes Alfvén theoretically studied a particular wave phenomenon, called magnetohydrodynamic, that was observed in experiments conducted on sample of mercury (Stig Lundquist and Bo Lehnert) or liquid sodium (Bostick and Morton Levine) subject to a strong magnetic field. This field induces electric currents in the conducting fluid that modify the original magnetic field; the mechanical forces present in the interaction between field and current had not yet been studied by electromagnetism or fluid dynamics. In the new discipline, they are theoretically described by associating, with the appropriate modifications, the equations of electromagnetism with those of hydrodynamics, typically neglecting displacement currents. Magnetohydrodynamics considers plasma to be a continuous medium characterized by fluid quantities, like viscosity and conductivity. A more thorough study relates the properties of plasma to mechanisms between microscopic components. On this basis, scientists have constructed more refined theories, which nevertheless are not only of speculative interest. For instance, it was deduced that Langmuir waves cannot occur in the absence of thermal excitation, and that because of the Brownian motion of electrons, they can propagate longitudinally as electronic waves. Lev Landau in 1946, and David Bohm and Eugene Gross in 1949 formulated a theory describing the movement of electrons in plasma, showing that plasmas can also host ionic waves that at low frequencies resemble sound waves. Not only do plasmas host longitudinal, transverse, and more complex waves, they can also radiate energy through different mechanisms, like the Cherenkov effect, the gyromagnetic effect, collisions, and deceleration. The first effect was discovered by the Russian physicist Pavel Cherenkov (1904-1990) when he observed the emission of light in liquids placed near a radioactive source. In 1937, his compatriots Ilya Frank (1908-1990) and Igor Yevgenyevich Tamm (1895-1971) interpreted the phenomenon, whose causes are not necessarily limited to radioactivity. To be precise, if an atomic or subatomic charged particle travels through a medium with a velocity greater than the phase velocity of light in the medium, then light of very specific characteristics is emitted: classically, this was interpreted as an electromagnetic shock wave. The theory, which can also be formulated quantum mechanically, had many extensions, including to highly ionised media.

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10.14 Counters and detectors Although increasingly powerful accelerators were constructed (§ 9.13), producing beams of particles emitted in very short time intervals, they risked being unusable because of the excessively slow detection provided by Geiger counters (§ 6.8) and spinthariscopes (§ 4.1). The Geiger counter, used since 1908, is essentially made up of a metallic tube, closed by insulating caps on its ends, containing gas and a central metallic wire on its axis. An elevated potential difference is established between the tube and the wire, which is what eventually produces signals through discharges. When a charged particle enters the cylinder, it sets off the process of gas ionization inside the tube, producing a brief discharge, which then is amplified several times so that it can be detected. Bothe and Rossi observed that two Geiger counters placed near each other detected a considerable amount of simultaneous events, called coincidences, that indicated the movement of a particle in both of the devices. Rossi devised an ingenious valve circuit with which the coincidences could be easily counted. Today, coincidence counters are mainly used to determine the direction from which particles originate. Spinthariscopes were also greatly improved by 1947 by J. W. Coltman and F. H. Marshall, who added special photoelectric cells, known as photomultipliers, which allowed them to transform the light impulse into a current impulse. What resulted was an instrument called a scintillation counter that could be used for ȕ and Ȗ-ray spectroscopy, as well as the detection of Į particles. Scintillation counters allow for the study of phenomena that occur in time intervals of the order of a billionth of a second. The Cherenkov counter, on the other hand, is based on the eponymous effect (§ 10.13); it works by transforming the light emitted by a moving particle into a current impulse through a photomultiplier, which then indicates the particle’s presence. A counter of this type lends itself to sorting of particles with different velocities and determining the directions of their trajectories. Alongside these counters, trajectory-based detectors are now indispensable to physics. We have already mentioned the Wilson, or cloud, chamber (§ 4.3), which, however, has a fundamental drawback: the droplet trail that marks a particle’s trajectory is only produced when the gas has the right degree of saturation. After a detection, several minutes go by before the chamber can once again be used. This intermittent functioning not only places time limits on its performance, but also greatly reduces the probability of capturing a new phenomenon, which, to be observed, must

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occur exactly in the brief time in which the chamber is in a condition to detect it. Detection then becomes the time-based analogue of finding a needle in a haystack. Beckett and Occhialini partly compensated for this deficiency (§ 9.12) with a method inspired by the coincidence counter of Bother and Rossi. In 1932, Beckett and Occhialini stacked two Geiger counters on top of each other and placed them below a vertically oriented Wilson chamber, such that each particle that passed through the two counters had to first cross the chamber. A relay mechanism triggered the expansion of the Wilson chamber for each coincidence detected, and the expansion was so rapid that the tracks left by the particle did not have time to disappear before it was complete. In short, the device made it so that each cosmic ray that entered the chamber was “photographed”. This chamber was placed in a strong magnetic field produced by a solenoid. In 1938, a method that had been used in the discovery of radioactivity was re-studied by Cecil Powell (1903-1969), who received the Nobel prize in physics in 1950. This technique consisted in directly recording particle paths with photographic emulsions. Powell realized that the method could be used to determine the energy of neutrons by observing the trails left by recoil protons. Contrary to the current opinion of the time, he found that the length of the trail left by a charged particle in the emulsion provided an acceptable measurement of the order of magnitude of its energy. This method became commonly used following the fruitful collaboration between Powell and Occhialini that began in 1945, and especially after Lattes, Occhialini, and Powell discovered meson decay in 1947 with an improved photographic plate technique. An ionizing particle crossing a photographic emulsion sensitizes granules of silver bromide along its trajectory. If the number of sensitized granules is sufficient, after the plate is developed, microscopic observation shows a trail of black granules. The preparation of the plate, which is relatively thick (around 300 microns) but very finely grained, is an extremely delicate operation; special attention is required for development and fixing, and the microscope observations are long and demanding. In exchange, its use is very simple and does not require magnetic fields, electric currents, or moving mechanical parts. The photographic plate method, which has the additional advantage of allowing straightforward collaboration between distant researchers, has provided a significant impetus for research, leading to the discovery of new particles and effects. The photographic plate method, however, due to the excessive atomic collisions arising from the elevated emulsion density, does not allow for measurements of magnetic field effects. In 1952, Donald Arthur Glaser (1926-2013), recipient of the Nobel prize in physics in 1960, succeeded in

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building a detector that eliminated the flaws and combined the advantages of Wilson chambers and photographic plates. He based his instrument on a physical principle that had been known since the 18th century: the introduction of an external object in a heated liquid causes bubbling. What happens if the heated liquid is traversed by an elementary particle? His experiments confirmed what the theory had predicted: an ionized particle that passes through a heated liquid in a particular temperature interval leaves a trail of microscopic bubbles along its path, marking the beginning of the aforementioned boiling effect. Trails of these bubbles can then be photographed. The bubble chamber was thus created, an apparatus that can also be submerged in a magnetic field to measure the curvature of particle trajectories and their momentum. Using a bubble chamber, one can observe many more events than through a Wilson chamber in a set time interval. Various liquids are used (propane, freon, xenon, etc.) depending on the type of experiment; liquid hydrogen is a frequent candidate. Glaser’s first chamber was only a few centimetres long, but today such detectors have reached enormous dimensions. It is said that Glaser was inspired to create his bubble chamber when he observed the formation of bubbles around irregularities inside a bottle of beer: even if this story is not true, it underlines the simplicity of the physical principle on which the bubble chamber is based.

10.15 Particles and antiparticles The detectors and counters discussed in the previous section allowed for the observation of many new elementary particles. Several dozens of species are known today, making “elementary” particles somewhat of a misnomer. The growing number of particles discovered after the second world war, together with the lack of a general theory of elementary particles and their interactions, brought about a general confusion in the scientific community. In 1954, there was an attempt to restore order with the creation of a unified taxonomy and nomenclature for elementary particles. To be precise, particles were classified in one of four “families”: particles affected by the forces responsible for nuclear stability, called hadrons, which are further subdivided into heavier baryons (including neutrons, protons, and heavier hyperons) and lighter mesons (including pions and kaons, or k mesons); leptons (muons, electrons, neutrinos); and finally massless photons, which are in a family of their own. We have already described the processes through which various particles are produced (electrons, protons, neutrons, muons, photons);

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mesons and hyperons arise from the collision of two particles, like two nucleons, or a nucleon and a photon. For such collision processes to produce new particles, however, a great deal of energy is required, which can only be found in nature in cosmic rays, explaining why many elementary particles have been discovered through the study of cosmic radiation. Artificially, elementary particles have been observed using powerful accelerators. Because of this, the vast field of elementary particle physics is often called high energy physics. In the processes that produce elementary particles, all physical conservation principles are obeyed: energy, momentum, electric charge, and other particle properties are the same before and after particle production. Dirac’s theory of the electron, which had led to the prediction of the positron’s existence (experimentally confirmed by Anderson [§ 9.12]), was extended to the nucleon, that is to protons and neutrons. “In any case,” said Dirac in his 1933 Nobel lecture, “I think it is probable that negative protons can exist, since as far as the theory is yet definite, there is a complete and perfect symmetry between positive and negative electric charge, and if this symmetry is really fundamental in nature, it must be possible to reverse the charge on any kind of particle.”323 Later, leaping from elementary particles to the universe, Dirac concluded: “If we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we must regard it rather as an accident that the Earth (and presumably the whole solar system), contains a preponderance of negative electrons and positive protons. It is quite possible that for some of the stars it is the other way about, these stars being built up mainly of positrons and negative protons. In fact, there may be half the stars of each kind. The two kinds of stars would both show exactly the same spectra, and there would be no way of distinguishing them by present astronomical methods.”324 A particle with the same mass as the proton but the opposite charge, predicted by Dirac and later required by many other theoretical results, was indeed observed in 1955 by Owen Chamberlain, Segrè, Clyde Wiegand, and Tom Ypsilantis, and was given the name antiproton. The following year, at Lawrence Berkeley National Radiation Laboratory, where the antiproton had been observed, another group of physicists (Bruce Cork, Glen Lambertson, Oreste Piccioni, and William Wenzel) bombarded protons with antiprotons and obtained antineutrons, which are 323

P. A. M. Dirac, Theory of Electrons and Positrons, pp. 5-6, in Les prix Nobel 1933, Santesson, Stockholm 1935. 324 Ibid., p. 6.

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distinct from neutrons in that they have an equal and opposite baryon number. Today, every particle is believed to have a corresponding antiparticle, and antiparticles have been experimentally observed for all known particles. Particles and antiparticles have the same mass and spin; if they are charged, then they have equal and opposite electric charges. If a particle is neutral, it still has an antiparticle, which in certain cases is identical to the particle itself and in others is distinct. When particles and antiparticles interact, they can annihilate, transforming into other lighter particles. Given that protons, neutrons, and electrons comprise atoms; antiprotons, antineutrons, and positrons could be combined to create anti-atoms. For example, the element anti-hydrogen, made up of a positron orbiting an antiproton, has been created in the laboratory. Combining anti-atoms, one can imagine antimatter, which however cannot exist naturally on Earth because it would come into contact with matter and annihilate itself. Yet does antimatter exist in the universe? The future will perhaps answer this question.

10.16 Strange particles Elementary particles interact with each other through forces that depend on the particles involved. Between 1948 and 1949 there were long discussions among physicists regarding these interactions, coming to the conclusion that they should be classified in four types. An electrically charged particle acts on another charged particle with an effect described by Maxwell’s equations. Such interactions are called electromagnetic, with the condition that the classical wave aspect of Maxwell’s equations is replaced, in processes of emission and absorption, by the particle-wave duality of quantum mechanics. The forces that hold the nucleus together, however, are much stronger than electromagnetic interactions. To be precise, they are about 100 times stronger than the Coulomb force: this type of interaction is called the strong interaction (a comparative term based on the electromagnetic standard). Based on the hypotheses of his theory of beta decay (§ 4.12), Fermi deduced that the forces in play were weaker than the forces of the electromagnetic field. This type of interaction, which occurs in many other decay processes, is thus called the weak interaction. Many orders of magnitude weaker are the gravitational interactions between elementary particles – of the order of 10-38 times them magnitude of the strong interactions.

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It is evident that the stronger an interaction, that is the stronger the forces it induces, the faster its effects occur. Strong interactions are, therefore, the fastest, occurring in an almost unimaginable time interval of around 10-23 seconds. Electromagnetic interactions occur in a time interval that is 100 times longer. Almost all weak interactions occur in the same time interval of 10-9 seconds, or a billionth of a second, and thus have a much longer duration than strong interactions. In 1947 in Manchester, George Rochester and Clifford Butler, while studying a shower of particles formed by the passage of high energy cosmic rays through a layer of matter (like a sheet of lead in a Wilson chamber), observed bifurcated trails that were formed by a yet unknown process. They interpreted the phenomenon as arising from the decay of a new neutral particle that had decayed into two charged particles. Powell, using the photographic emulsion technique in 1949, obtained a photograph of a cosmic ray that he interpreted as a new particle that had decayed into three mesons. Research continued, intensifying in 1952, when scientists in Long Island (New York) built the cosmotron, the first machine that could produce energies of tens of GeV, quantities comparable to those found in cosmic rays. It was established that the new neutral particles were of at least two types, a baryon (called lambda, /) and a meson (called kappa zero, K0). Yet the study of the decay of these particles led to a strange discovery: while they are formed through the strong interaction in around 10-23 seconds, they decay through the weak interaction after 10-8 to 10-10 seconds. In short, these particles have much longer lifetimes than what theory predicts. They were thus called strange particles. The explanation for these strange phenomena was provided in 1952 by Abraham Pais (1918-2000) and Yoichiro Nambu (1921-2015), who hypothesized the law of associated production: in the collision of two nucleons or of a pion and a nucleon there are always two particles produced, and never only one. In 1953, Murray Gell-Mann (1929-2019) and Kazuhiko Nishijima (1926-2009) also independently came to this conclusion; Gell-Mann introduced a new and physically non-intuitive quantity, strangeness, postulating its conservation. Another conserved quantum number that is useful in describing the phenomenology of strong interactions is isospin, which has the same formal properties as spin. Protons and neutrons arise as two different isospin states (of isospin +1/2 and -1/2, respectively) of the same particle, the nucleon. Analogously, the three pions of electric charge +1, 0, and -1 form an isospin triplet.

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10.17 Conservation and violation of parity The idea of symmetry, which brings to mind the harmony and unity of all parts, is not only important in the arts but in all of science as well. Symmetry criteria have been invoked by physics since its inception to interpret various phenomena. We have already mentioned (§ 4.5) the usefulness of crystallography for solid state physics, in which symmetry plays a central role. Symmetry in physics was holistically studied by Pierre Curie from 1884 to 1894, who summarized its practical value in the maxim “symmetry elements present in causes must also be found in effects,” capturing the deterministic ideas of the time. In quantum physics, the idea of symmetry gained more and more importance as time went on. Dirac’s theory, from which the concept of antiparticles arose, is based on the principle of (special) relativistic symmetry. A particular example of symmetry that has been studied since ancient times by mathematicians and philosophers is mirror symmetry, which amounts to the symmetry between left and right. The corresponding physical question that naturally arises is the following: is there some intrinsic method, independent of our conventions, to distinguish a phenomenon from its mirror image? For example, if a person moves in front of a mirror, can an observer who sees both distinguish which one is real? She cannot, although she will realize that one of the two is left-handed while the other is right-handed. Yet is it the real person who is right-handed or is it the mirror image? The observer cannot decide, as either one could very well exist. The idea of mirror symmetry can be expressed mathematically. If one takes an analytic function and changes the sign of one variable, the resulting function is the mirrored version of the original, having been reflected across a plane. If the function does not change sign when its coordinates are reversed, it is called even; if it changes sign, it is called odd. In 1932, Eugene Paul Wigner (1902-1995) called this inversion of axes a parity operation. This amounts to a change of sign (equivalent to a reflection and a successive rotation of 180 °) in the three coordinates of the wave equation associated to a particle: it follows that it too can have odd or even parity depending on whether the wavefunction changes sign. The indistinguishability of a phenomenon from its mirror image is thus translated to the principle of conservation of parity, which states that the parity of the final state is always equal to the parity of the initial state. For several decades, this principle was a pillar of quantum physics.

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In 1949, Powell had given the name IJ to a meson that decayed into three ʌ mesons; but after some time it was discovered that there was a new meson, called ș, that decayed into two ʌ mesons. Later, more accurate measurements showed that IJ and ș had the same mass and behaved in the same manner in many processes, thus appearing to be the same particle, which then had two decay mechanisms. On the other hand, they could not be the same particle because they did not have the same parity. In conclusion, physicists faced a new enigma, which was the subject of ample discussion in the years between 1954 and 1956. To overcome this apparent contradiction, in the summer of 1956 two young Chinese-American physicists from Columbia University, ChengNing Yang (b. 1922) and Tsung-Dao Lee (b. 1926), advanced the theory that parity conservation is not valid for weak interactions, and consequently for the decay mechanisms of IJ and ș mesons, allowing for them to be the same particle, as is known today (K meson). The two physicists also proposed an experiment that could show whether weak interactions were distinguishable from their mirror images. According to them, radioactive cobalt-60 was well-suited to this purpose, as its nuclei have spin and decay by emitting an electron and a neutrino. A strong magnetic field can cause the alignment of the nuclei, meaning that their spin axes all point in the same direction, but to avoid problems caused by thermal excitation this alignment must be done at low temperatures, of the order of one hundredth of a degree Kelvin. If the nuclei emit equally in all directions and, in particular, in both opposing directions along their spin axis, their decay is indistinguishable from its mirror image. However, if electrons are emitted more in one direction than the other, then the effect and its mirror image can be distinguished. An observer on the privileged side of the axis would then see the current produced by the magnetic field move in a counter-clockwise direction, while the mirrored observed would see it move clockwise. In conclusion, if cobalt-60 emits more electrons in the direction of the magnetic field than in the opposite direction, parity conservation is violated. The experiment proposed by Yang and Lee was particularly delicate. The onerous task of carrying it out fell to the Chinese physicist ChienShiung Wu (1912-1997, Fig. 10.1)) and her collaborators at the Bureau of Standards in Washington, D.C., who spent six months preparing for it. In the meantime, physicists placed bets on its outcome: many of the leading theorists of the time were certain of the experiments’ impending failure because, as Dirac said, it cannot be the case that God is a “weak lefthander”, and much less, as Pauli added, that he appear ambidextrous only “when He expresses Himself strongly.” Yet God, imprudently invoked,

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resoundingly proved the theorists wrong and showed himself to be lefthanded in weak interactions. The experiment, conducted in 1957 in the span of only 15 minutes, demonstrated without any room for doubt that Colbalt-60 emits more electrons in the opposite direction of an applied magnetic field. Numerous other experiments that followed confirmed this result. The experiment, for which Yang and Lee received the Nobel prize in 1957, had immediate ramifications for physics. If the principle of parity conservation was not valid for weak interactions, what was to say that all the other conservation principles are always valid?

Fig. 10.1 - Chien- Shiung Wu

The fact that parity is not conserved in weak interactions led Yang and Lee to postulate that the spin of the neutrino, which is only relevant to weak phenomena, can only be in one direction, bringing back an idea that Weyl had expressed in 1929 based on symmetry considerations, though he had later abandoned it because it violated parity conservation. Several neutrino experiments performed in 1957 confirmed this prediction. Yang and Lee also observed that parity symmetry was conserved if, during

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reflection, matter is also replaced with antimatter. In other words, anticobalt-60 in a magnetic field would give rise to the same emission as oppositely-oriented cobalt-60 in the same field.

11. FUNDAMENTAL INTERACTIONS

11.1 Quantum electrodynamics Out of the four fundamental forces in nature, the one that describes the interaction between electromagnetic radiation and matter is without a doubt the force that is most relevant to our daily lives, being responsible for the stability of atoms and all chemical and biological processes. The theory that describes this type of interaction is known today as quantum electrodynamics. The other interactions at play are the force of gravity, strong interactions, which are responsible for the forces that hold nucleons together in the nucleus, and weak interactions, which are involved in radioactive decays. The birth of quantum electrodynamics, often referred to using the acronym QED, can be traced back to a work by Dirac in 1927325 in which, for the first time, quantum mechanics was applied not only to the atom, through Schrödinger’s equation, but also to radiation, giving rise to the first quantum field theory. This was the beginning of a series of interesting conceptual developments. With the advent of quantum field theory, empty space, which after Einstein’s theory of relativity could no longer be considered a continuous medium filled with an intangible ether, became populated by other entities. In Dirac’s formulation, the electromagnetic field in a vacuum is described by an infinite family of harmonic oscillators. According to quantum mechanics, these oscillators cannot have zero energy without violating Heisenberg’s uncertainty principle (§ 8.10). Consequently, atoms interact with the electromagnetic field even in the vacuum, giving rise to spontaneous emission and absorption of photons, which act as the mediators of electromagnetic interactions. This theory, combined with the relativistic wave equation of the electron also discovered by Dirac (§ 8.5), provided a rather precise description of the behaviour of electrons inside atoms. Soon, however, it 325

P.A.M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation, in “Proceedings of the Royal Society of London”, A(114), 1927, pp. 243-265.

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was realized that the theory had some embarrassing flaws. The predictions of quantum field theory are obtained by successive approximations, using a perturbative series whose expansion parameter is the electron charge e, or more precisely the fine-structure constant,

ܽ=

2ߨ݁ ଶ 1 ؄ 137 ݄ܿ

where h is Planck’s constant and c is the speed of light. Keeping only the first term in the expansion, which was enough to study the Dirac equation, physicists obtained reasonable results. If, instead, further approximations were taken into account, which in principle should have given a small correction to the first approximation, the results obtained were meaningless or infinite. The first to realize that these difficulties arose was probably Oppenheimer326, but the first complete calculation that revealed the embarrassing presence of infinities was performed by Weisskopf,327 to whom Pauli had proposed to calculate the contribution to the electron mass m0 that comes from its interaction with the electromagnetic field. This calculation is the quantum version of a well-known problem in classical electrodynamics: the energy due to the electric field of a sphere with charge e and radius r is

݁ଶ 2‫ݎ‬ which therefore diverges when r goes to zero. In the quantum version the result is significantly more complicated, but continues to diverge and is therefore infinite when applied to the electron, which is considered a point particle. Another worrying result was obtained studying the properties of the vacuum. From a quantum point of view, the vacuum behaves as a dielectric medium due to the effects of the constant production and annihilation of virtual electron-positron pairs. An electric charge e0 in vacuum thus causes a polarization of these virtual pairs (vacuum polarization), leading to an effectively observed charge of 326

J.R. Oppenheimer, Note on the Interaction of Fields and Matter, in “Physical review”, 35, 1930, pp. 461-77. 327 V. Weisskopf, Über die selbstenergie des Elektrons, in “Zeitschrift für Physik”, 89, 1934, pp. 27-39; Erratum Z, ibid., 90, 1934, pp. 53-54.

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e=

471

௘బ ఢ

where ߳ is the dielectric constant of the vacuum. Quantum electrodynamics gave an infinite value for ߳, so to observe a nonzero effective charge of the electron, its “true” charge also had to be infinite. These apparently contradictory results were a great impediment to the further development of the theory, which remained largely unchanged for almost fifteen years. In this landscape of general frustration, a few brilliant works by Ernst Stückelberg stood out,328 in which he developed a manifestly relativistic formulation of electrodynamics that suggested a systematic technique for the elimination of the infinites. Despite the extremely innovative content of these papers, they went almost entirely unnoticed. A vigorous push for the recommencement of research in this field was instead given by new experimental results. In April 1947, Willis Lamb and his student Robert Retherford of Stanford University succeeded in measuring the energy difference between the 2s and 2p levels of the hydrogen atom by using microwave sources that had been developed for radar applications during the war329 (§ 7.12). Based on the Dirac equation, these two energy levels should have been exactly degenerate (that is, of the same energy). The difference observed, called the Lamb shift, then had to be attributed to successive terms in the perturbative expansion of quantum electrodynamics – the very terms that gave infinite contributions. In the same period, Henry Foley and Polycarp Kusch observed that the electron, which behaves like a small magnet because of its spin, has a slightly different magnetic moment from that predicted by the Dirac equation.330 A few months after these important discoveries, for which Lamb and Kusch shared the 1955 Nobel prize, the Japanese physicist Sin-Itiro Tomonaga (1906-1979) and the Americans Julian Schwinger (1918-1994), Richard Feynman (1918-1988), and Freeman Dyson (1928-2020) developed, using different methods, a general formulation of quantum 328

E.C.G. Stückelberg, Relativitisch invariante Störingstheorie des Diracschen Theorie, in “Annalen der Physik”, 21, 1934, pp. 367-89; Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte (Teil I), in “Helvetica Physica Acta”, 1938, pp. 225-44. 329 W.E. Lamb and R.C. Retherford, Fine Structure of the Hydrogen Atom by a Microwave Method, in “Physical Review”, 72, 1947, pp. 241-243. 330 H.M. Foley and P. Kusch, On the Intrinsic Moment of the Electron, in “Physical Review”, 73, 1948, p. 412.

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electrodynamics that allowed one to systematically eliminate the infinities from the theory. The procedure applied, now known as renormalization, consists in the elimination of the initial parameters of the theory, namely the mass m0 and charge e0 of the electron, in favour of the ones that are actually measured, m and e. For each calculation in quantum electrodynamics, the mass and the charge must be suitably redefined such that the infinites are relegated to the physically unobservable initial parameters. This process allows for the unambiguous and extremely precise calculation of both the Lamb shift and the correction to the magnetic moment of the electron. More than sixty years have now passed since the publication of these works (for which Tomonaga, Schwinger, and Feynman shared the 1965 Nobel prize in physics), and although the calculational techniques have been greatly refined, the conceptual basis has remained unchanged. Quantum electrodynamics has been thoroughly tested, and there are currently no known significant discrepancies between theory and experiment. To appreciate the degree of accuracy of its theoretical predictions one only has to look at a few examples. The frequency associated with the Lamb shift is experimentally measured to be 1057.862 MHz, with an uncertainty of about 20 in the last two digits, while the theoretical prediction is 1057.864 MHz, with an uncertainty of about 14 in the last two digits. Similarly, the anomalous magnetic moment of the electron turns out to be, in appropriate units, 1.00115965221 (with an uncertainty of 4 in the last digit), while the theory predicts a value of 1.00115965246 (with an uncertainty that is five times greater). There are not many other examples in physics of such close agreement.

11.2 Renormalization Despite its undeniable success in the extremely accurate description of the interaction between electromagnetic radiation and matter, the method used to eliminate infinities provoked a good deal of perplexity and suspicion. In 1980, Dirac concluded his speech at a symposium on the birth of particle physics with the following words: “I am fairly certain that the correct basic equations have not yet been discovered. Some new relativistic equations are needed; new kinds of interactions must be brought into play. When these new equations and new interactions are thought out, the problems that are now bewildering to us will get automatically explained, and we should no longer have to make use of such illogical processes as infinite renormalization. This is quite nonsense physically, and I have always been opposed to it. It is just a rule of thumb

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that gives results. In spite of its successes, one should be prepared to abandon it completely and look on all the successes that have been obtained by using the usual forms of quantum electrodynamics with the infinities removed by artificial processes as just accidents when they give the right answers, in the same way as the successes of the Bohr theory are considered merely as accidents when they turn out to be correct.”331 Nevertheless, by that time the process of renormalization rested on much more solid foundations. Already in 1952, Ernst Stückelberg and André Petermann, and Murray Gell-Mann (1929-2019) and Francis Low the following year, had given a general and mathematically rigorous formulation for the infinity removal process.332 From this formulation, it results that the value of the effective electric charge depends on the distance at which it is measured. For long distances, it is constant, but it grows as one approaches the electron, giving rise to well-documented experimental effects. The infinite value of the “true”, or “undressed”, initial charge is attained when a probing particle reaches the exact point where the electron is found. To get rid of this undesired possibility, the field theory is modified at short distances in such a way that it becomes impossible to get closer to the electron than a certain predetermined distance scale, called the cut-off. This scale has no physical meaning, but serves to avoid contradictions in the mathematical formulation of the field theory, which in this new form is called a regularized field theory. Clearly the physical properties of the theory cannot depend on this artificial scale. The renormalization process thus establishes a series of rules to apply so that one can extract physical information from the theory independently of the cut-off scale. While renormalization certainly elicited a slew of criticisms, perplexities, and doubts from the physics community, today the theory is universally accepted. Furthermore, it has turned out to be a very fruitful principle in predicting and explaining new phenomena. Along these lines, it was discovered that all quantum field theories suffer from infinities in the initial values of their coupling constants (which have an analogous role to the electric charge in electromagnetism), but that only for a particular class of them, called renormalizable theories, does the infinity cancellation 331 P. A.M. Dirac, The Origin of Quantum Field Theory, in L. M. Brown and L. Hoddeson (editors), The Birth of Particle Physics, Cambridge University Press, Cambridge 1983, p. 55. 332 E.C.ZG. Stückelberg and A. Petermann, La normalisation des constantes dans la théorie des quanta, in “Helvetica Physica Acta”, 26, 1953, p. 499. M. GellMann and F. Low, Quantum Electrodynamics at Small Distances, in “Physical Review”, 95, 1954, p. 1300.

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method work. It is then natural to look for models to describe the other fundamental forces in this more restricted space of theories. The most surprising discovery, however, was that of the American Kenneth Wilson (1936-2013). He began by noticing a deep analogy between the quantum fluctuations that can be ascribed to the Heisenberg uncertainty principle and the thermal fluctuations of a classical thermodynamic system. In systems described by thermodynamics there is always a microscopic length scale involved, like the interatomic distance or the lattice spacing of a crystal. However, the physical behaviour of this system near a second order phase transition does not depend on this microscopic scale. Consequently, the scale is analogous to the cut-off of regularized quantum field theories. Elaborating on this analogy, in 1971 Wilson was able to give new and much more general framework for renormalization, providing a unified description of classical critical systems and quantum field theories. One of the most conceptually important consequences of Wilson’s approach (which earned him a Nobel prize in 1982) was the discovery of universality classes. It had been known for some time that very different physical systems could have the same behaviour near their critical points. The idea, which was mainly due to the Russian physicist Leonard Kadanoff, was that in the proximity of these points the correlation length becomes much larger than any other length scales involved and small-scale details become irrelevant. Wilson demonstrated that the properties of such systems only depend on a few general characteristics, in particular the symmetries of the laws that describe their behaviour. Almost all systems that have the same symmetry also have the same critical behaviour, and are thus said to belong to the same universality class. For instance, the liquid-vapour interface, liquid mixtures, certain alloys, and some magnetic systems are all part of the same universality class. Similarly, in systems described by quantum field theories that have the same symmetry, the coupling constants depend on the relevant physical parameters in the same way.

11.3 Gauge symmetries The first systematic studies of the application of symmetry notions to atomic and nuclear physics were conducted in the forties, mainly by the Hungarian-born physicist Eugene Paul Wigner (1902-1995), who was given the 1963 Nobel prize in physics for these important contributions to quantum mechanics.

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The mathematical tool of choice to study symmetries is group theory. A group is a set of transformations that obeys the following axiom (among others): the product of two transformations is once again a transformation in the same set. For example, rotations about an axis form a group because the composition of two rotations is also a rotation. It is easy to see that the set of transformations that leave the laws describing physical system invariant (unchanged) form a group, called the symmetry group of the system. There is a very deep connection between symmetry groups and conservation laws. For example, if a system is invariant under spatial translations, linear momentum is conserved; if it is invariant under rotations, angular momentum is conserved. Similarly, energy conservation is related to time translation symmetry. These examples naturally bring up a question: what is the symmetry associated with the conservation of electric charge? To answer this, we must first explain a few more details regarding the matter field of a charged particle. Supposing, for simplicity, that the particle is spinless, a field configuration amounts to assigning a complex number a + ib to each point in spacetime P(x;t), that is two real numbers (a, b) that can be considered the coordinates of a point Q(a,b) in the complex plane (Fig. 11.1). The laws of quantum electrodynamics are invariant under the group of rotations about the centre of this plane (a rotation of angle Į from point Q to point Q’ in the complex plane is shown in Fig. 11.1). The conservation law associated with this symmetry is indeed electric charge conservation.

Fig. 11.1 – Complex plane

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However, QED also has a larger symmetry: its laws are invariant even in the case that the rotational angle Į depends on the point P, that is when Į is an arbitrary function of spacetime. A transformation of this type is called a gauge transformation. Crucially, this group of transformations is only symmetry of the system if the matter fields are coupled with the electromagnetic field. In other words, gauge symmetry not only requires the conservation of electric charge, but also the existence of the photon, the mediator of interactions between charged particles. The symmetry group of quantum electrodynamics is a one-parameter group (the angle of rotation Į) that is typically denoted by U(1). It is abelian, meaning that it is a group in which the effect of two consecutive transformations does not depend on the order in which they are applied. Starting towards the end of the 1940s, the discovery of a wealth of new particles suggested the introduction of new conserved charges besides electric charge, namely isospin and strangeness (§ 10.17). From the close relationship between symmetries and conservation laws, it followed that one could expect the laws describing these new particles to be invariant under a group of transformations larger than U(1), meaning a group that depends on more parameters. In a famous 1954 work, the very young Chinese-American physicist Chen-Ning Yang (who two years later would propose the hypothesis of parity violation in weak interactions with Tsung-Dao Lee) and the American Robert Mills extended the notion of gauge transformations to these new symmetries as well, supposing that all transformation group parameters were arbitrary functions of spacetime.333 The consequences were simple to interpret but highly impactful. Gauge invariance implies the existence of new spin-1 particles, one for each parameter, that act as mediators in interactions between particles carrying this new form of charge. If, additionally, the group is not abelian (meaning that the composition of two transformations depends on the order in which they are applied), the mediators themselves are charged and thus subject to self-interactions. While the significance of this theoretical development was not lost on a few farsighted physicists, for many years the theory of Yang and Mills was considered merely an elegant mathematical exercise and did not find concrete applications, as it suffered from two main complications.

333

C. N. Yang and R. L. Mills, Conservation of the Isotopic Spin and Isotopic Gauge Invariance, in “Physical Review”, 96, 1954, p. 191.

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The first was that the gauge fields responsible for the interactions described apparently massless bosons. This precluded the possibility that they were the mediators of non-electromagnetic interactions for two reasons. On one hand, the only massless spin-1 particle known to date is the photon. On the other hand, the strong interactions are short-range and it was for this reason that Yukawa had been able to predict the existence of the ʌ meson in 1935, a mediator of the forces ensuring the stability of the nucleus (§ 9.12). These problems were resolved in 1964 with three independent publications, respectively by Peter Higgs of the University of Edinburgh, Robert Brout and François Englert of the Université Libre de Bruxelles, and Gerald Guralnik, Karl Hagen, and Thomas Kibble of Imperial College, who showed that by introducing an opportune multiplet of scalar fields in Yang-Mills theory, which, unlike the other fields in the theory, could have a nonzero vacuum expectation value, the gauge bosons would become massive. The Higgs mechanism, or Brout-Englert-Higgs mechanism, as we will soon see, was immediately adopted as the starting point for a unified theory of the electromagnetic and weak interactions. The other serious issue facing Yang-Mills theory was its lack of compatibility with the principles of quantum mechanics. The non-abelian nature of its symmetry group gives the theory a much more complicated structure than that of QED, and many studies initially seemed to show that it was not renormalizable. It was a brilliant doctoral student at Utrecht, Gerard ‘t Hooft (b. 1946), who surmounted these difficulties, demonstrating in two monumental 1971 papers334 the renormalizability of the gauge theories, both in the original version with massless bosons and in the presence of the Higgs mechanism. For these advances and other contributions to gauge theories, ‘t Hooft and his supervisor Martinus Veltman (1831-2021) shared the 1999 Nobel prize in physics. Starting in the 1970s, gauge symmetry, which twenty years earlier had seemed nothing more than an elegant mathematical exercise, proved to be one of the most fruitful ideas in elementary particle physics, replacing numerous ad hoc theories that had been designed to explain phenomena appearing quite different from one another. Furthermore, gauge theory also plays an important role in other fields of physics. We will see in the following pages that some of these applications have become an important source of inspiration for the resolution of one of the most difficult problems in elementary particle physics: quark confinement (§ 11.5).

334

G. ‘t Hooft, Renormalization of Massless Yang-Mills Fields, in “Nuclear Physics”, B(33), 1971, p. 173; Renormalizable Lagrangian for Massive Gauge Field Theories, ibid., B(35), 1971, p. 167.

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11.4 Quantum fluids In solid state physics there exist systems that manifest quantum effects on a macroscopic scale. A prototype of such a system was predicted by Einstein as early as 1926: because of the indistinguishability of identical particles (§ 8.8), an ideal boson gas (a gas composed of non-interacting bosons) undergoes a phase transition at a critical temperature, below which a macroscopic fraction of its molecules occupies the energy ground state. These molecules make up Bose-Einstein condensate, a fluid that exhibits quantum properties like the typical interference phenomena of wave propagation. A quantum fluid is characterized by the fact that the De Broglie wavelength (§ 8.3) associated with its constituent particles (molecules, atoms, electrons, etc.) is greater than their average separation distance. Because Ȝ is inversely proportional to velocity, and the root mean square velocity is in turn related to the temperature of a fluid (§ 2.14), one finds that the temperature at which Bose-Einstein condensation occurs is lower at lower densities. It is therefore very difficult to observe this phenomenon, since a real gas only approaches its ideal gas approximation when it is very dilute, which implies a very low critical temperature. The first such condensate was created only in 1995 at a temperature of the order of one millionth of a degree Kelvin. The experimenters responsible, Americans Eric Cornell and Carl Wieman of the National Institute of Standards and Technology and the German Wolfgang Ketterle of MIT, received the 2001 Nobel prize in physics for their work. A much easier quantum fluid to produce is helium-4 at temperatures below 2.17 degrees Kelvin, which was first obtained by Kamerlingh Onnes in 1908 in his Leiden laboratory (§ 6.14). This liquid exhibits unusual behaviour that cannot be explained by classical physics: it has an infinite thermal conductivity, causing heat to propagate in its interior through constant velocity waves (second sound), contrary to the usual laws of diffusion. Another characteristic of cold helium-4 is its superfluidity, namely the ability to flow through tubes with apparently zero viscosity. At times, during the initial experiments, helium in its superfluid phase would mysteriously disappear from the containers in which it was held, seeping out of microfractures so thin that even gaseous and ordinary liquid helium could not pass through them. It was the Russian physicist Lev Landau (1908-1968) who first proposed the hypothesis that superfluid helium is a quantum liquid characterized by the presence of a macroscopic fraction of its molecules moving coherently in the lowest energy state. With this assumption, he

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was able to provide an accurate description of the main effects that characterize superfluidity, receiving the Nobel prize in physics in 1962. Other quantum fluids are also known. One of these originates from the Hall effect: if a conducting bar traversed by a current I is placed inside a perpendicular magnetic field, a potential difference VH arises between the two edges of the bar. The Hall conductivity associated with this potential is defined by the ratio

ߪு =

‫ܫ‬ ܸு

In 1980, the German physicist Klaus von Klitzing (Nobel prize recipient in 1985) discovered that at very low temperatures (a few degrees Kelvin) and in the presence of rather strong magnetic fields, the Hall conductivity dramatically departs from classical behaviour: when the magnetic field is varied the conductivity does not also continuously vary like Hall described a century earlier, but rather takes on integer multiple values of the quantity

݁ଶ ݄ where e is the electric charge and h is Planck’s constant. This behaviour has been confirmed in a wide variety of materials with extremely high accuracy (nearly one part per 100 million!). A quantization law so simple and so accurate suggested an explanation based some general physical principle. In a brilliant 1981 work, the American physicist Robert Laughlin demonstrated that the origin of this quantum Hall effect was simply QED gauge invariance applied to conduction electrons in a magnetic field. Laughlin was also responsible for the interpretation of an analogous phenomenon discovered in 1982 by Daniel Tsui, Horst Störmer, and Robert Gossard in a high quality sample of gallium arsenide, in which the Hall conductivity was quantized in fractional multiples of von Klitzing’s constant (fractional quantum Hall effect, or FQHE). The first fraction observed was 1/3, but different materials with different FQHE constants are known today. According to Laughlin’s explanation, at very low temperatures and elevated magnetic fields, the conduction electrons of these materials condense to form a quantum fluid quite different from the others that we have considered. Among the many unexpected properties of the FQHE, the most emblematic is the following: if an electron is added to

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the system, the whole fluid is affected, and collective modes arise that behave like particles with fractional electric charge (for example 1/3 the electron charge). The discovery of this effect and its interpretation earned Störmer, Tsui, and Laughlin a share of the 1998 Nobel prize in physics. Another quantum fluid of considerable interest because of its numerous important applications is the one formed inside superconductors, metals which below a characteristic temperature host a collection of effects encapsulated by the term superconductivity. The most surprising property of these phenomena is an electric resistance that appears to be exactly zero, as if conduction electrons flow through the metal without friction, exactly like superfluid helium flows across surfaces. Yet electrons are fermions (§ 8.8) and cannot then undergo BoseEinstein condensation. The key to this mystery was found in 1957 by the Americans John Bardeen, Leon Cooper, and John Schrieffer, who shared the 1972 Nobel prize in physics. According to their theory, now known as BCS, the vibrational effects of phonons, the quanta of thermal vibrations in crystal lattices, produce attractive forces between electrons that overcome Coulomb repulsion and give rise to bound states of two electrons called Cooper pairs. These two-electron “molecules” have spin zero and are therefore bosons: at sufficiently low temperatures they too can become Bose-Einstein condensate. It was Landau again, in collaboration with the Russian Vitaly Ginzburg, who proposed a model field theory that gave a comprehensive description of superconductivity. In this model, Cooper pairs are described by a charged scalar field coupled to the electromagnetic field. The formation of the condensate corresponds to the fact that the field has a nonzero vacuum expectation value. This is nothing other than the Higgs mechanism that generates the mass of the coupled gauge field, in this case the photon. Consequently, a photon inside a superconductor acquires nonzero mass. This implies that magnetic fields cannot penetrate the interior of the superconductor. Thus, the Higgs mechanism provides a simple explanation for a phenomenon that accompanies the superconducting phase transition of a metal: the expulsion of magnetic fields inside the conductor (Meissner effect). Another scientist of the Russian school, Alexei Abrikosov, demonstrated that the Landau-Ginzburg model predicted the formation of vortices in the condensate of Cooper pairs. A magnetic field can penetrate along the axis of the Abrikosov vortex, but its flux must be an integral multiple of

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݄ܿ ݁ where h is Planck’s constant, c is the speed of light, and e is the electron charge. These predictions were promptly confirmed by experiments, and many years later, in 2003, Abrikosov shared the Nobel prize in physics with Ginzburg and the Englishman Anthony Leggett. We will see that Abrikosov vortices play an important role in the solution to the problem of quark confinement.

11.5 Quarks and quantum chromodynamics Throughout the history of scientific thought there have been moments in which scientists thought to have found the fundamental constituents of matter, which, subject to simple laws, can explain the entirety of physical phenomena. Examples of these “basic units” include atoms in the 1800s and the electron, proton, and neutron in the 1930s. Initially these building blocks are relatively few, leading to the belief that ultimate order and simplicity have been attained; the picture is then invariably complicated as the “elementary” constituents are probed further and new constituents are discovered. This process occurred again at the beginning of the sixties, when the number of new hadrons (mesons and bosons) discovered had already surpassed a few dozen. Many scientists began to think that these “elementary” particles were not the fundamental constituents of matter. Convincing evidence for the composite nature of hadrons came from the fact that some of them are simply excited states that decay into nucleons by emitting mesons after very short time intervals, much like excited atoms decay to their ground state by emitting photons. In 1964, Murray Gell-Mann335 (and, a few weeks later, George Zweig336) proposed a systematic classification in which the elementary constituents, called quarks, were particles of spin 1/2 and electric charge 2/3 or 1/3, with the proton charge set to 1. Baryons are made up of three quarks, while mesons are made up of a quark and an antiquark. Protons and neutrons are then simply two of the many baryons that can be obtained 335

M. Gell-Mann, A Schematic Model of Mesons and Baryons, in “Physics Letters”, 8, 1964, p. 210. The manuscript was received by the journal on 4 January 1964. 336 G. Zweig, An SU(3) Model for Strong Interaction Symmetry and Its Breaking, CERN TH-412; February 1964.

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by combining three quarks in an allowed permutation. Many of these baryons had already been observed, yet some of the slots in the table of allowed possibilities remained empty, much like the undiscovered elements in Mendeleev’s periodic table a century earlier (§ 7.4). In this way, physicists predicted the existence of new particles with properties that corresponded to the missing quark combinations, which were then experimentally observed in the course of a couple years. Despite these unmistakeable successes, the quark model was met with great skepticism in the physics community: even in 1971, the elderly Heisenberg curtly interrupted a conference speaker, exclaiming “But the quark model is not physics!”337 Nevertheless, by that time experimental techniques had already been developed that allowed physicists to “see” quarks. A composite object can be resolved into its constituents by bombarding it with radiation of smaller wavelength than the object’s dimensions. Between 1968 and 1972, at CERN and the Stanford Linear Accelerator (SLAC), high energy electron beams used to strike protons and neutrons sometimes experienced violent collisions with deviations of up to 180 degrees in their trajectories, suggesting the presence of apparently point-like constituents inside nucleons, which were identified as quarks. In many ways, these experiments are reminiscent of the ones Rutherford had performed a half-century earlier to reveal the presence of the apparently point-like nucleus inside the atom (§ 7.7). However, there is a crucial difference. While it is always possible to extract electrons from an atom or nucleons from a nucleus by using sufficiently energetic radiation, an analogous result has not yet been obtained for quarks. Free, isolated quarks have never been observed, and today’s physicists believe that the laws governing their interactions prohibit their existence as free particles. This fact is expressed by saying that quarks are permanently confined. At the end of the sixties, quark confinement appeared to be in stark contrast with the results of the electron-beam experiments, which clearly showed that quarks behaved like free particles inside nucleons. Most physicists thought that no quantum field theory could explain such strange behaviour. Dozens of ingenious theories attempted to interpret this phenomenon, but none of them were able to withstand criticism and experimental tests; among these attempts were the beginnings of string

337

Episode recounted by David I. Olive at the Dirac Centenary Conference, Baylor University (Texas), October 2002.

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theory, which would later be resurrected as a candidate for a theory of quantum gravity, as we shall see. After the monumental works by ‘t Hooft in 1971, many began to suspect that Yang-Mills theories were somehow involved in quark interactions. Finally, in 1973 David Politzer of Harvard and David Gross and Frank Wilczek of the Institute for Advanced Study at Princeton discovered a general property of non-Abelian gauge theories that turned out to be the key to solving the mystery: the vacuum surrounding a YangMills field source is polarized in an opposite manner to that of QED (§ 11.2), such that the closer a test particle gets to the source, the more the strength of the source appears to decrease, until it behaves like a free particle decoupled from the field in the limit of zero distance (which could only be attained with an infinite test particle energy). This property of Yang-Mills sources is called asymptotic freedom338; Gross, Politzer, and Wilczek were awarded the 2004 Nobel prize for their discovery. The effect they described was the following: if quarks are the sources of a gauge field, when they are close to each other or when they interact with a beam of energetic electrons (which can therefore get very close to the sources) they behave like weakly interacting particles, but when they move away from each other the interaction force increases. There are, as we will see, various types (or “flavours”) of quarks, but protons, neutrons, and pions are made up of only two, up quarks (denoted by u), and down quarks (denoted by d), which have charges of 2/3 and -1/3, respectively. The proton is formed by a uud triplet, and therefore its charge is 2/3 + 2/3 – 1/3 = 1. The neutron is formed by the triplet udd and is therefore neutrally charged. If quarks are the sources of the Yang-Mills field, besides electric charge they must carry another type of charge that couples to the gauge fields, called gluons (because they glue quarks together), like electric charge couples to photons. The new type of charge was called colour for no particular reason, and the corresponding theory that describes the interactions between quarks quantum chromodynamics, or QCD for short. Whereas electric charges can only be positive or negative, there are three colours, red, green, and blue, each of which can be positive or negative. Colour changes sign when going from a particle to an antiparticle. If, for example, a quark has a positive red colour, then its 338

In reality, this property had been known to ‘t Hooft in 1971, and he had already communicated it at a congress in Marseille in 1972, but did not publish anything regarding the subject because, as he later said, it was difficult to publish something without the approval of one’s supervisor at the time, who in his case was not interested in the subject.

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antiparticle will have negative red colour. Like electrodynamics, equal charges repel and opposite charges attract. For this reason, a quark and an antiquark pair form a meson. The existence of three colours makes the rules governing quarks more complicated than their electrodynamic counterparts. The gauge symmetry group that determines these laws is SU(3). The number 3 refers to the number of colours that are transposed by group transformations. The U indicates that from a mathematical point of view, these transformations are unitary matrices. Lastly, the S stands for “special” and refers to the fact that these matrices have determinant one. This group depends on eight parameters and consequently, based on the argument of § 11.3, implies the existence of eight different types of gauge bosons or gluons that act as the mediators of quark interactions. Gluons also cannot exist as free particles, but, unlike photons, they can interact with each other, giving rise to conglomerates called glueballs. From an experimetal point of view, it is difficult to distinguish glueballs from ordinary mesons; nevertheless, several candidate particles have already been observed. Asymptotic freedom provides an accurate description of the phenomena that depend on short-range quark interactions, in which they behave like weakly interacting particles. On the other hand, when quarks move away from each other the coupling constant of the theory becomes too large: the perturbative methods of field theory are no longer applicable and QCD becomes mathematically intractable. In the years between the seventies and eighties, correctly describing the mechanism responsible for quark confinement became one of the central themes of particle physics research. The first hint came in 1974 from Kenneth Wilson (§ 11.2). He reformulated Yang-Mills theory on a four dimensional crystal lattice instead of the usual continuous spacetime background. In this way, the confining nature of the quark attraction was confirmed near an infinite value of the coupling constant, but this value was too large to draw comparisons to the real world. At the beginning of the 1980s, ‘t Hooft and the American Stanley Mandelstam advanced the hypothesis, backed by highly plausible heuristic arguments, that the vacuum of non-abelian Yang-Mills theories contains a condensate of magnetic monopoles. Much like a condensate of Cooper pairs is responsible for the expulsion of magnetic fields in superconductors (§ 11.4), the monopole condensate in QCD expels the chromoelectric field. The field flux generated by a quark is then channeled through an Abrikosov vortex. The energy of the vortex is proportional to its length, so if a quark were isolated then its energy would be infinite. Conversely, an antiquark can reabsorb this vortex and the energy of the quark-antiquark

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pair is then proportional to their distance. Confinement was thus explained, but how could one demonstrate ‘t Hooft and Mandelstam’s conjecture? A short paper by the Americans Michael Creutz and Laurence Jacobs and the Italian Claudio Rebbi that appeared in “Physical Review Letters” in the autumn of 1979 caught the physics community by surprise. In it, the authors demonstrated that it was possible to numerically simulate Wilson’s model with a computer and obtain valuable information regarding the physical values of the coupling constants, a feat that no analytic method had been able to achieve. Numerical simulations are a sort of experiment in which a computer is programmed to evolve a virtual physical system according to a set of predetermined rules, for example those thought to describe quark interactions, and then the ensuing properties of the virtual system are compared to the real world. In this way, one can check the validity of a hypothesis or predict the behaviour of a system even when the laws describing it are too complex to be studied through analytic methods. With the relentless development of ever more powerful calculators and computers, numerical simulations and computational physics in general gave become a valuable and potent tool in all branches of physics. Through these methods, lattice gauge theories have now reached a level of precision such that they can not only corroborate ‘t Hooft and Mandelstam’s conjecture, but also accurately reproduce the spectrum of baryons and mesons. However, as accurate as numerical simulations can be, they always contain a margin of error. Consequently, while they can confirm the plausibility of a conjecture, they can never replace a mathematical proof. Thus, the problem of quark confinement that had beset the physics community in the eighties, while not entirely resolved in a rigorous sense, has received enough extensive numerical confirmations to be considered more than surpassed.

11.6. Electroweak unification Fermi’s theory of beta decay (§ 9.7) was the first theoretical formulation of the physics of weak interactions. It is based on the point interaction that transforms a neutron into a proton, an electron, and an antineutrino pictured in the diagram of Fig. 11.2:

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Fig. 11.2

The same diagram could be used, applying the diagrammatic techniques developed by Feynman, to describe the scattering of neutrinos given by the reaction

n + Ȟ ĺ p + e -, but while the theory worked well at low energies, at high energies it gave rise to the untameable infinities typical of non-renormalizable theories (§ 11.2). With the advent of Yang-Mills theory, it was natural to try to find a more satisfactory description using gauge bosons that mediated weak interactions, in analogy to electrodynamics. The current theory of weak interactions was formulated in the sixties by the Americans Sheldon Glashow (b.1932) of Harvard and Steven Weinberg (1933-2021) of MIT and the Pakistani physicist Abdus Salam (1926-1996), who was then at the Imperial College of London. Glashow first developed the theory in 1961, when it was still unclear how to give mass to gauge bosons to make interactions weak at short-range. After the discovery of the Higgs mechanism (§ 4.3), Weinberg and Salam formulated a consistent theory that, as ‘t Hooft later demonstrated in 1971, was renormalizable. Several years later, in 1979, once it appeared clear that this was the correct theory even if the gauge bosons had not yet been experimentally detected, they received the Nobel prize in physics in recognition of their work. In the Weinberg-Salam model, the diagram in the previous figure is replaced by the following one:

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Fig. 11.3

In this process, the neutron emits a W- (a gauge boson of spin 1) and transforms into a proton; in turn, the W- decays by emitting an electron and an electron antineutrino. The same process can be described on a quark level by replacing the neutron and proton with down and up quarks, respectively. The mass of the W boson predicted by the model is of the order of 85 proton masses, so the W emitted by the neutron cannot be real, as it would violate the principles of energy and momentum conservation. However, a quantum system can violate this principle for a very short time interval įt as long as the energy conservation violation įE satisfies Heisenberg's uncertainty relation:

ߜ‫ ܧߜ ݐ‬൒

݄ 4ߨ

A particle which violates energy or momentum conservation is called virtual339. All fundamental interactions in nature are caused by exchanges of virtual particles. Because the virtual W boson emitted by the neutron is very heavy, it has a short lifetime and quickly decays into an electron and antineutrino. The new diagram is then not very different from the previous one, explaining why Fermi’s theory works so well for describing neutron decay. On the other hand, in a scattering process involving very energetic neutrinos the energy difference įE between the virtual W boson and a real 339

In the relativistic formulation of the theory, the emission of a virtual particle conserves both energy E and momentum p, but violates the conservation of rest mass, which is given by the relation: ‫ ܧ‬ଶ + ܿ ଶ ‫݌‬Ԧଶ = m2c4 where c is the speed of light.

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one is smaller, and the intermediary boson therefore has a longer lifetime. This detail allows for the elimination of the divergences found in Fermi’s model. The gauge group that describes this type of interaction is U(1) × SU(2), where U(1) is the single parameter group that we have previously encountered and SU(2) can be thought of as the group of rotations in three dimensions. The overall group then depends on four parameters and thus there are four gauge bosons. The application of the Higgs mechanism gave rise to a surprise, however. Only three of these gauge bosons acquire mass; one of the four remains massless. As the only boson satisfying this property is the photon, the theory also contained quantum electrodynamics. Another surprise was that the trio of massive bosons, besides the W- and its antiparticle, the W+, also includes a neutral boson Z of mass around 100 times the proton mass. This theory predicted that this boson was responsible for weak processes that had not yet been observed. Indeed, the Z boson, unlike the photon, can be absorbed and emitted by neutrinos. An exchange of a virtual Z boson generates a weak interaction between neutrinos and other particles, called the weak current interaction, pictured in the following diagram:

Fig. 11.4

Despite widespread skepticism, these new phenomena were in fact photographed in a bubble chamber at CERN in 1973. Ten years later, two different experiments at CERN were successful in producing W and Z bosons from a high-energy collision between a beam of protons and a beam of antiprotons, conclusively confirming Weinberg and Salam’s theory. In recognition of this experimental feat, the Italian

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Carlo Rubbia and the Dutchman Simon Van der Meer received the 1984 Nobel prize in physics. Up until now we have only dealt with two leptons, the electron and the electron neutrino Ȟe, and two types of quarks: u and d. For a currently unknown reason, nature has repeated this scheme two more times. There are thus three families, or “generations”, of fundamental particles, each formed by two leptons and two quarks. The second generation is formed by the leptons ȝ and Ȟȝ (the muon and the muon neutrino) and the quarks c and s (charm and strange).340 The third generation includes the leptons IJ and ȞIJ (the tau and the tau neutrino) and the quarks t and b (top and bottom, or, if one wishes, truth and beauty). All interactions between these particles are mediated by the same gauge bosons, namely photons, gluons, and W and Z bosons, meaning that quarks and leptons from different generations have the exact same properties aside from their masses. For instance, the muon has a mass of about 206 electron masses and the tau of 2491 electron masses. In the original Weinberg-Salam model, neutrinos were described as massless particles, but it was recently discovered that they must have a small, but nonzero, mass. Neutrinos interact very weakly with matter, making it very difficult to detect them. The Sun is a powerful source of electron neutrinos, which are constantly produced by nuclear fusion reactions in its interior. Most of the energy produced at the centre of the Sun takes millions of years to reach the solar surface and is emitted as electromagnetic radiation. Neutrinos, on the contrary, leave the sun after a few seconds and zip through space. Each second, five million neutrinos strike every square centimetre of the earth’s surface. Neutrino detectors, made up of enormous containers filled with hundreds of tons of liquid (trichloroethylene, for instance), which are placed in mines several thousands metres below sea level to shield them from other radiation, detect less than ten neutrinos a day. The American Raymond Davis (Nobel prize in physics in 2002) was the first, in 1967, to measure the flux of solar neutrinos with a groundbreaking experiment that he conducted in a South Dakota gold mine. He found that the number of observed neutrinos was significantly smaller than what was predicted, as if a portion of the neutrinos had been lost on the way to the detector. Every experiment performed in the successive decades confirmed this observation independently of the techniques used. 340

The strange quark was the third quark predicted in Gell-Mann’s original scheme in 1964. This is the quark that characterizes strange particles (§ 10.16). The theory of weak decays of strange quarks had already been developed by the Italian physicist Nicola Cabibbo in 1963.

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Fig 11.5 - The combination of the electroweak theory and QCD is called the “Standard Model”. Although the details of the model are complicated, the figure is intended to summarize its fundamental ingredients.

In reality, Pontecorvo had shown in 1958 that if the neutrino has a small nonzero mass, it can oscillate between different generational types, such that a detector that is only sensitive to the presence of electron neutrinos would detect fewer neutrinos than theoretically predicted. The first piece of evidence for the existence of oscillations among the muon neutrinos emitted by upper-atmosphere cosmic rays was obtained only in 1998 by the Super-Kamiokande experiment in the Kamioka mine of Japan. In 2002, an experiment at the Sudbury Neutrino Observatory of Ontario confirmed that solar neutrinos can oscillate by transforming into neutrinos of another generation, thus solving the mystery of their disappearance. Quarks and leptons are, as far as we know, free of internal structure: together with the gauge bosons that mediate the electroweak and strong interactions they make up the elementary constituents of all known matter. The U(1) × SU(2) × SU(3) gauge theory that describes their interactions, now known as the Standard Model (Fig. 11.5), is one of the broadest syntheses of scientific thought and has withstood ample tests; all of its elementary components have been observed, including the elusive Higgs boson that gives mass to the other gauge bosons. The experimental discovery of the Higgs boson occurred only after the construction of the largest particle accelerator in the world at CERN. Composed of a ring 27 kilometres in circumference, the Large Hadron

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Collider (LHC) was built, among other things, to detect the Higgs boson. After a disastrous accident in 2008, only nine days into testing, in which the fusion of an electrical junction led to the escape of tons of liquid helium and the destruction of dozens of extremely expensive superconducting magnets, on 4 July 2012, scientists at CERN were finally able to announce the discovery of this particle, also referred to as the God particle by the press. The following year, Peter Higgs and François Englert received the Nobel prize in physics for their theoretical prediction.

11.7 Gravity Despite being the dominant force in the universe, gravity is by far the weakest of the fundamental interactions. Its current formulation is not very different from the one developed by Einstein in his critical 1916 paper, Grundlage der allgemainen Relativitätstheorie. We have already discussed the ample tests to which relativity was subjected (§ 5.10), including the gravitational deflection of light, the precession of the perihelion of Mercury, and the redshift of spectral lines caused by the slowing of clocks in a gravitational field. The most precise measurement of this effect was performed in 1976 by researchers at Harvard in collaboration with NASA: a hydrogen maser clock (§ 10.10) was placed inside a rocket that was sent into orbit at an altitude of 10,000 km, and its frequency as a function of height was compared to that of an identical clock on the ground. The results of this experiment confirmed the predictions of general relativity to an accuracy of seven parts per 100,000. Some of the most interesting objects studied in general relativity are black holes, that is solutions to Einstein’s equations with singularities. They are surrounded by an event horizon, a spacetime surface that divides events for an observer A into two (nonempty) sets: those that were, are, or will be observable by A, and those that can never be observed by A. The event horizon acts as a membrane that is permeable in only one direction, as it can only be crossed by a physical system moving into it. Conceptually, one can say that the gravitational field at the event horizon is so strong that the escape velocity is equal to the speed of light, meaning that nothing, not even photons (at least classically), can escape from beyond it. Every black hole is characterized by a mass, an angular momentum, and possibly an electric charge. Black holes of any mass can exist. Inside almost every galaxy there is a supermassive black hole whose mass can reach several billion times the mass of the Sun. In 2019, at the centre of the elliptical galaxy Messier 87, a photograph of the event

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horizon of a black hole was obtained for the first time, a culmination of the collaborative efforts of dozens of radio-telescopes across the world. Einstein had shown as early as 1916 that that his field equations could admit wavelike solutions similar to those of electromagnetism. Every massive accelerated body emits gravitational waves that travel at the speed of light and interact in a negligible manner with the matter that they strike, transporting important information that can originate from very distant sources without disruption. Yet how could one “read” this information? The effects produced by these waves seemed so small as to preclude any experimental relevance. For a long time, the existence of gravitational waves was far from accepted in the scientific community. Einstein himself wrote to Born in a 1937 letter: “Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist, though they had been assumed a certainty to the first approximation. This shows that the non-linear general relativistic field equations can tell us more or, rather, limit us more than we have believed up to now.”341 Yet starting in 1960, new developments rekindled interest in gravitational waves. Hermann Bondi and his collaborators were able to show in a manifestly covariant way, that is independently of the choice of coordinates or reference frames, that gravitational radiation is without a doubt physically observable, and that it carries energy and angular momentum from the systems that emit it. In the same time period, Joseph Weber of the University of Maryland began to build the first gravitational antennas in an attempt to detect the radiation. Since then, gravitational wave research has become one of the most active branches of general relativity. Starting at the end of the 1990s, enormous laser interferometers were built for their detection. In particular, in the Cascina countryside near Pisa, the arms of the Virgo interferometer are 3 kilometres long. In the United States, there are a couple interferometers of the same type under the umbrella of the Laser Interferometer Gravitational-Wave Observatory (LIGO) with 4 km arms, one in Livingston, Louisiana and the other in Hanford, Washington, and other interferometers have been constructed or are under construction around the world. To an outside observer, dedicating such extensive and costly resources to the detection of gravitational waves may have seemed a risky decision. In reality, the great drive of these gravitational wave hunters stemmed from the fact that there was already observational evidence of their 341

A. Einstein and M. Born, Scienza e vita. Lettere 1916-1955, Einaudi, Torino 1973, p. 149.

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existence. In the summer of 1974, Russel Hulse and Joseph Taylor, researchers from Princeton University using the Arecibo radio telescope in Puerto Rico, observed a tightly orbiting binary system in which one of the two constituents was a pulsar, that is a neutron star342 with a strong magnetic field that rotates while emitting light in pulses, much like a lighthouse. In this instance, as was later discovered, the pulsar and its companion weighed around 1.4 solar masses and their orbit was so tight, having an average distance about one solar radius, that orbital velocities reached 300 km/s and the orbital period was a mere eight hours. Under these conditions, one can expect notable relativistic effects. Moreover, the pulsar proved to be (and still is) one of the most accurate clocks in the universe, given that its rotational period of 59 milliseconds varies less than a fourth of a nanosecond each year, and can thus be used as a highprecision instrument to measure the properties of the binary system. The agreement between these observations and Einstein’s theory was spectacular, but an even greater result pertained to gravitational waves: a binary system like the one observed is a powerful source of gravitational waves. The loss of energy due to this emission had to be manifested as a contraction of the orbital radius accompanied by a very small reduction in the orbital period. Hulse and Taylor gathered data on the binary system until 1983, finding a striking accord with the predictions of general relativity so as to consider their result definitive evidence for the existence of gravitational waves. In recognition of their work, they received the Nobel prize in physics in 1993. The first direct observation of gravitational waves occurred with a pronounced 0.2 second signal measured by both of the LIGO interferometers on 14 September 2015. This event was later interpreted as arising from the merger of a 29 solar mass black hole with another 36 solar mass black hole, giving rise to a black hole of 62 solar masses, an event that occurred 1.3 billion years ago in the direction of the Great Magellanic Cloud. The 3 solar masses of missing energy in this violent process were thus emitted in the form of gravitational waves. After this event, thanks to an upgrade in the LIGO and Virgo instruments that increased their sensitivity by a factor of ten, many other events were observed simultaneously by all three interferometers, allowing for a more accurate localization of the sources of these gravitational waves. On 17 August 2017, the three instruments observed a new kind of 342

Translator note: A neutron star is formed at the end of a giant star’s lifetime if its mass is too small to form a black hole. After an enormous explosion called a supernova, the core of the star collapses to form an extremely dense ball of mostly neutrons, hence the name neutron star.

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phenomenon that produced a signal of one minute duration, which was interpreted as the merger of two neutron stars. 1.7 seconds after the end of the gravitational signal, the Fermi Gamma-Ray Telescope, a satellite telescope, recorded a gamma ray burst. A few hours later, optical telescopes on Earth, having been alerted by LIGO and Virgo, observed in the same position of the gravitational source what can be considered an astronomical cataclysm – a kilonova, namely a stellar explosion one thousand times brighter than the typical nova. This series of coordinated observations marked the beginning of a new astronomy based on the integrated study of gravitational and electromagnetic signals originating from the cosmos, providing much more precise information on the universe surrounding us. The 2017 Nobel Prize in physics was awarded to the Americans Barry Barish, Kip Thorne, and Rainer Weiss for their role in the discovery of gravitational waves as founders and directors of LIGO. By 1917, Einstein had realized that general relativity had cosmological implications. Applying the theory to the universe on a large scale and hypothesizing that it could be described, in a first approximation, by a homogeneous matter distribution, he was disappointed to discover that his equations did not admit static solutions. The only possible solutions described an expanding or contracting universe. He therefore modified them by adding a new ad hoc term, called a cosmological constant, that allowed them to also describe a static universe. He later referred to this modification as the “biggest blunder” of his life. Indeed, in 1929 Edwin Hubble had already realized by studying the redshift in the spectra of distant galaxies that the universe was in fact expanding. The first calculations of the age of the universe, however, gave results that were even inferior to the age of the Earth, which is calculated based on the percentage of radioactive elements in rocks (§ 7.1). The first realistic calculations of the time since the hypothesized Big Bang birth of the universe were only performed towards the end of the fifties. Strong evidence for the cosmological model suggested by relativity was provided by the accidental discovery made in 1965 by two researchers from Bell Telephone Laboratories, Arno Penzias and Robert Wilson, who received the 1978 Nobel prize. While attempting to find the source of noise that disrupted telecommunications in the microwave bands, they discovered that this background “noise” radiation was of cosmic origin and that it uniformly permeated all of space. This radiation constitutes the remnant of the electromagnetic blackbody radiation that abounded in the primordial universe, which was filled with a very hot plasma of protons and electrons. When neutral atoms formed, the electromagnetic radiation decoupled from matter and the universe became transparent. The background radiation that

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we now observe is a fossil image of this moment, called recombination. With the expansion of the universe, the temperature associated with this radiation has decreased to its present value of 2.7°K. Many other further pieces of evidence were added to support this cosmological model, among which the calculation of the relative abundance of helium and hydrogen observed in interstellar space stands out. Today the relativistic model of the Big Bang has become the commonly accepted description of the evolution of the physical world, and physicists are already attempting to study the properties of the universe before the Big Bang.

11.8 String theory A review of nature’s fundamental interactions would not be complete without at least a mention of string theory. Despite not having any experimental evidence thus far, this theory, sometimes denoted TOE, an acronym for “Theory of Everything” (not without a hint of irony), has enjoyed the attention of many theorists for the past several decades. When it was born at the end of the 1960s, the aims of string theory were much less ambitious. At that time, it had not been confirmed yet that the strong interactions were described by a gauge theory, and the guiding principle for much of hadronic physics research was “nuclear democracy”, an idea developed mainly by Geoffrey Chew (1924-2019) of the University of Berkeley. This principle affirmed that all hadrons are fundamentally the same: each hadron can be considered to be composed of the other hadrons, and the forces that arise from hadron exchanges are the same ones that are responsible for their stability. Unlike QED, where the force (photon exchange) is clearly distinct from the particles (sources of the electromagnetic field), in hadron physics there appeared to be a perfect symmetry, or duality, between forces and particles, with every hadron acting as both a mediator of interactions and a source. In 1968, the young Italian physicist Gabriele Veneziano (b. 1942) realized that a mathematical function studied by Euler two centuries earlier could explicitly yield nuclear democracy and force/particle duality, which until then had remained at the level of philosophical speculation. He found that this mathematical tool – the Euler Beta function – successfully described the collision of two mesons. In the course of a few months, the model was extended to other processes involving an arbitrary number of particles, giving rise to a systematic theory of hadronic processes called the dual resonance model. Thanks to the work of Veneziano and the Italian physicist Sergio Fubini (1928-2005), it was discovered that this

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model predicted the existence of an extremely rich spectrum of physical states. The spectrum contained an infinite number of resonances with masses that increased with increasing intrinsic angular momentum, following a scheme known as the linear Regge trajectory (named after the Italian physicist Tullio Regge [1931-2014], who had introduced this notion ten years earlier in the theory of quantum diffusion). In 1970, Yoichiro Nambu, Holger Nielsen (b. 1941), and Leonard Susskind (b. 1940) independently realized that the quantum numbers that classified the resonances of the theory exactly coincided with the normal modes of vibration of a relativistic open string. This provided a very simple intuitive model for the structure of hadrons: mesons are vibrating strings with quarks attached on their ends, and the elastic force of the string can then explain quark confinement. However, as Veneziano’s model and its generalizations were formulated on more solid mathematical bases, it became clear that there were discrepancies with hadron phenomena. At that time – the beginning of the 1970s – quantum chromodynamics (§ 11.5) was being honed, and its extraordinary success quickly led to the abandonment of the vibrating string description of hadrons. The dual resonance model, however, became string theory, and continued to attract attention because of its coherence and mathematical beauty. Within the rich spectrum of the theory, certain vibrational modes appeared to behave exactly like gauge bosons (§ 11.3), and could therefore be gluons, the mediators of inter-quark forces. On the other hand, the internal consistency of the theory also required the existence of closed strings, like rubber bands, which invariably included a massless spin two boson that had nothing to do with hadrons but behaved exactly like the graviton. In 1974, Joël Scherk (1946-1980), from the École Normale Supérieure of Paris, and John Schwartz (b.1941) of the California Institute for Technology, were the first to make the leap to approaching string theory not as a hadronic theory, but rather as a theory of fundamental interactions that included gravity. At that time, however, the theory still had a physically unacceptable property: it predicted the presence of particles with an imaginary mass, called tachyons, and was consequently abandoned by most. A small group of researchers nevertheless continued to advance it, eliminating the troublesome presence of tachyons. In 1984, Schwartz and Michael Green (b.1946), then at Queen Mary College, showed that string theory could be regarded as a unified theory of all the interactions present in nature, thus realizing Einstein’s dream. For the first time, gravity had been made fully compatible with the laws of quantum mechanics and unified with the other interactions.

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The enthusiasm generated by this work was immense: between 1984 and 1986 there were more than a thousand papers written on the subject. In this period, later baptized the “first superstring revolution”, it was established that there were five possible string theories. In a sort of collective credulity, it was thought that the only thing left was to discover was which of these had been chosen by nature to describe our world. In reality, string theory admits a wealth of different solutions, and physicists soon realized that it can describe an infinite number of possible universes. The theory encompasses many other possibilities besides the standard model; in particular it does not explain why there are three generations of particles and gives no new predictions on quark and lepton properties. In 1995 there was another great burst (“the second superstring revolution”) that gave rise, among other things, to a unification of the five types of string theory in a single theory whose nature remained mysterious. By then, however, it had been somewhat stripped of its guise as a “theory ofeverything”, and has since come to be considered an important generalization of relativistic quantum field theories that is particularly well-suited to describe the effects of quantum gravity. Since its birth, string theory has always been an inexhaustible source of inspiration even in other fields of physics. At the beginning of the seventies, it provided the first example of supersymmetry, a symmetry between bosons and fermions that predicts the existence of a fermionic partner for each boson and vice versa. Supersymmetry was later formulated in four-dimensional spacetime by Julius Wess (1934-2007) of the University of Karlsruhe and Bruno Zumino (1923-2014) of CERN in 1974. Today, many physicists believe t his to be a hidden symmetry of nature, and the first experimental signatures may soon be found by LHC (§ 11.6). Another important contribution dating back to the “first revolution” was the development of conformal field theories in two dimensions, which are an invaluable tool in the study of phase transitions in two dimensional systems. Lastly, the influence of string theory was felt in many branches of mathematics, in particular algebraic geometry. It is no coincidence that Edward Witten of the Institute for Advanced Study at Princeton, one of the most influential architects of the two superstring revolutions, was awarded the 1990 Fields Medal, the greatest honour in mathematics.

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INDEX OF NAMES

Abbe E., 37 Abelson P.H., 415 Abraham M., 224-225, 243, 259260 Abria J., 128 Abrikosov A.A., 480-481, 484 Adams W., 266 Aepinus F.U.T., 146, 149, 330 Alfvén H., 458 Amagat E., 50 Amaldi E., 408, 414 Amici G.B., 20, 30, 36 Ampère A.M., 12, 59, 85, 94, 96106, 111-112, 120-122, 127128, 160, 172, 176, 191, 195, 331, 454 Anderson C.D., 422-425 Anderson W.C., 17-18, 419 Andrade J.F., 240 Andrews T., 49-50 Ångström A.J. , 30-32, 58 Antinori V., 126, 238 Appleton E.V., 444-445 Arago J.F.D., 4, 6-12, 17-19, 25, 33, 35, 46, 94-97, 100-102, 111, 121-123, 126, 237-239 Archimedes, 156 Arrhenius S., 137, 139 Arzberger J., 46 Aston F.W., 340-343, 345, 402-403, 407, 454 Aury G., 1 Avenarius R., 116 Avogadro A., 76, 97-98, 138, 147, 278, 284, 286, 288 Babbage C., 121 Babcock H., 457 Baccelli L., 122

Baisch, 272 Baker H., 403 Baliani G.B., 59, 261 Balmer J.J., 346-347, 352-353 Bancalari M.A., 155 Banly, 450 Banks J., 85 Bardeen J., 434, 480 Barkla C.G., 283, 332, 337 Barlow P., 97, 106-107, 143, 161, 217 Barnett M.A.F., 444 Barral J.A., 438 Barré de Saint-Venant A.J.C., 239, 268 Battelli A., 214 Bauer, 285 Baumé A., 47 Becher H., 404 Beckett, 460 Beckmann E., 272 Becquerel A.C., 88, 98, 107, 111. 115-116, 140, 155 Becquerel A.E., 35, 151, 155-156, 218 Becquerel H., 218-224, 229, 313, 317-318 Behrens T., 85, 89 Bell A.G., 164 Bellani A., 39 Belopolsky A., 20 Bémont G., 222 Bennet W., 453 Benzenberg J.F., 239 Bérard J.E., 24, 50-51, 53 Bergson H., 247 Berkeley, 248 Bernal J., 430 Bernoulli, 75, 79, 360

A History of Physics over the Last Two Centuries Berson A., 458 Berthelot M., 49 Berthollet C.L., 41, 43, 90 Berzelius J.J., 91, 101, 135, 344 Besançon G.L., 438 Bessel F.W., 40 Bétancourt A. de, 45-46 Bethe H.A., 436, 455 Betti E., 178 Bingham E., 268 Biot, 6-7, 11, 15, 44, 60, 88, 89, 9596, 102, 104, 106 Birge R.T., 400 Bitter F., 435 Bixio G., 438 Black J., 63 Blackett P.M., 347, 352, 369, 393, 423 Bloch F., 432 Bogoljubov N., 267 tzBohm D., 391, 458 Bohnenberger J.G.F. Von, 85, 238 Bohr N., 190, 336, 346, 348-360, 363-364, 378, 380-392, 408, 414, 416, 418, 425, 427, 435436, 473 Boltwood B., 287, 320 Bolzmann L., 58, 69, 80-82, 187, 242, 259, 272-275, 289, 299, 304 Bond W., 36 Bondi H., 492 Borda, 190 Borelli, 239, 384 Born M., 301, 305-306 Borries B. von, 372 Boscovich R., 153, 328 Bose S., 373-375 Bose J.Ch., 448, 478, 480 Bostick W., 457-458 Bothe W., 308, 368, 394, 404-405, 459-460 Botto D., 120 Bowen I., 421 Boyle, 45, 49, 64, 76, 284 Brace D., 22

505

Bragg W.H., 217, 334, 336, 360, 430 Bragg W.L., 217, 430 Branly E., 182 Brasch, 396 Brattain W., 434 Bravais A., 215-216 Breguet L., 18 Breit G., 445, 451 Brewster D., 6, 9, 14-15, 29 Bright C., 192 Brillouin M., 278, 280, 282, 285, Brillouin L.N., 432, 449 Brodhun E., 33 Brown R., 289-292, 294 Brugmans A., 155 Brugnatelli L., 43, 85-86, 88 Buisson H., 20 Buffon, 14 Bunsen R., 31-33, 72, 140, 225, 437 Burattini T.L., xii, 191 Burton E.F., 420 Butler C., 464 Cagniard de la Tour C., 48-49, 55 Cailletet L., 50 Calinon A., 249 Callendar H., 63 Calzecchi Onesti T., 182 Cameron G., 421 Campbell L., 166 Cantoni G., 289 Carbonelle I., 289-290 Cardano G., 92 Carlisle A., 85- 86 Carnot L.N., 61, 239 Carnot S.N., 61-65, 67-68, 70-71, 73, 75, 157, 232 Carré F., 48 Carson J.R., 449 Cartesio, 167, 170, 246 Casboi N., 40 Cassina U., 291 Cauchy A.L., 13 Cavallo T., 47

506 Cavendish H., 43, 58, 86, 142, 144, 146, 167, 191, 261 Cazin A.A., 185 ýerenkov P., 458-459 Chadwick J., 337, 338, 394, 404406, 408-409 Chamberlain O., 462 Chappuis P., 45 Charles J.A, 43-44 Chazy J.F., 255 Chew G., 495 Children J., 91-92 Christie H., 112, 121, 149-150 Cigna G.F., 47 Clapeyron B., 54, 64-65, 70-71 Clark L., 113, 140-141 Clarke E.M., 161 Clarke F., 344 Clarke Slater J., 430 Clausius R., 58, 72-78, 120, 136139, 145, 149, 157, 173, 181, 193, 275, 290 Cleeton C.E., 450 Clément N., 52-54, 60 Clouet J.F., 48 Cockroft J.D., 396, 401 Cohen R., 297 Colding L.A., 65, 75 Colgate S., 426 Colladon J.D., 101 Coltman J.W., 459 Compton A., 304, 307-310, 371 Condon E., 395 Condorcet N., 14, 190 Configliachi P., 93 Constable, 394 Cook L., 420 Coolidge W.D., 212, 396 Cooper L., 480, 484 Copernicus, 29 Corbino O.M., 413 Coriolis G.G. De, 235, 239 Cork B., 462 Cornell E., 478 Cornu M.A., 17 Cotes, 178

Index of Names Cotton, 152 Coulomb, 93-95, 104, 127, 142-144, 146, 149, 155, 172, 191, 196, 208, 261, 286, 335, 420, 426427, 463, 480 Cournot A., 239 Cowan C., 409 Crawford A., 58 Creutz M., 485 Crookes W., 31, 183, 193, 195-199, 205, 214-215, 286-287, 313, 317, 319 Crowther J.A., 332 Cruikshank W., 85-86 Cumming J., 115 Curie I., 404, 405, 408, 410-413, 416 Curie M. (Sklodowska M.), 220226, 278, 280, 287, 313, 317, 319, 322-324, 333, 335, 345, 383, 404 Curie P., 220-227, 280, 313, 317319 Curtet, 91 Czapski S., 141 D’Agostino O., 414-415 Daguerre J.M., 34-35 D’Alembert, 237 Dal Negro S., 126, 161 Dalton J., 39, 44, 46, 51, 61 Daniell J.F., 48, 114, 136, 140, 185 D’Arsonval J.A., 99, 164 Darwin C., 51 Darwin E., 51 Davis R., 489 Davisson C.J., 370-371 Davy H., 34, 59, 87, 91-92, 95, 97, 101, 107, 133, 150, 159 Debierne A., 220, 222, 287, 323 De Broglie L., 16, 249, 267, 361364, 366, 368, 370, 374 De Broglie M., 279, 362-377, 384, 388, 390-392, 394, 432, 478 Debye P., 189, 301, 308 De Candolle A., 58

A History of Physics over the Last Two Centuries Del Buono P., 23 Delafosse G., 215 Delambre J.B.J., 56, 100, 101 De la Hire, 41 De la Rive A., 58, 92, 104, 106, 116, 151, 185 Delaroche F., 50-51, 53 De La Rue W., 36 Dellmann J.F., 94 Delsaulx J., 289-290 De Luc J.A., 38, 41, 52, 85, 89 De Mairan J.J., 47 Demarçay E.A., 222 De Saussure H.B., 24, 41 Deschales C., 29 Desormes C., 52-54, 60, 89 Despretz C., 58, 193 Deviatkon, 450 Dewar J., 287, 300 Dirac P.A.M., 349, 366-368, 375, 422, 431, 462, 465, 466, 469473 Dobson G.M., 439-440 Dollond J., 27, 30 Donati G.B., 30 Doppler C., 19-20, 190, 360, 361 Dorn F.E., 313, 318 Draper W.J., 35 Drude P., 120, 208, 275, 298, 308, 431, 449 Duhamel J.M.C., 122 Duhem P., 157, 172 Dulong P.L., 39-40, 45-46, 54-55, 66, 297-300 Dumas J.B., 121, 323, 344 DuMond J., 297 Dyson F., 471 Ebert H., 227 Eddington A., 264, 267, 343, 383, 455 Edison T., 164, 229 Edlersen, 397 Edlund E., 130-132 Ehrenhaft F., 207 Einstein A., 153-154, 241-244, 247-

507

254, 256-266, 278-282, 285, 291-297, 299-301, 304-306, 309, 343, 358-362, 364, 366, 368. 374-377, 381, 384, 387391, 418, 436, 452, 455, 469, 478, 480, 491-494, 496 Elster J., 220, 228 Enriques F., 383 Eötvös L. von, 261, 262 Epstein P.S., 190 Epstein R.S., 440 Erman G., 101 Esterman I., 371 Euclid, 63, 263-264, 310 Euler L., 2, 141, 237, 240, 248, 360, 495 Evrard B., 35 Exner F., 289 Fabbroni G., 87 Fabry C., 20 Fajans C., 327 Faraday M., 48-49, 92, 94-95, 97, 102, 118-121, 123-130, 132136, 139-140, 144-160, 168169, 172-175, 179, 184-189, 195, 204, 229, 259, 287, 396, 454 Faure C., 88 Favy, 45 Feather N., 406 Fechner G.T., 107, 110 Feddersen W., 127 Fedorov E., 216 Felici R., 103, 128, 131, 172 Fermat, 4, 364 Fermi E., 213, 372, 374-375, 407, 409-416, 418-419, 424, 426, 431-434, 485, 487-488, 494 Ferraris G., 162-163, 193 Féry C., 441 Feynman R.P., 471-472, 486 Fievez C., 185-188 Fitzgerald G.F., 21-22, 187, 189 Fizeau A.H.L., 5, 17-22, 35, 40-41, 161, 171, 186, 245, 256

508 Fleck A., 327 Flügge S., 345 Foley H.M., 471 Fontana F., 46 Forbes G., 17 Forbes J.D., 26, 165 Foucault L., 18-19, 30, 35, 126, 158, 162, 235-239 Fourcroy A.F., 85, 90 Fourier J.B.J., 38-39, 56-57, 60-61, 108-109, 116, 125, 193, 348, 365 Fowler, 455 Fox Talbot W., 29 Franck J., 370 Frank I.M., 458 Franklin, 56, 58, 93 Franz R., 58, 120 Fraunhofer J., 20, 27-29, 31, 35 Frenkel Y.I., 430 Fresnel A.J., 1, 5, 8-15, 18-21, 26, 40, 151, 184, 245, 285 Friedrich W., 217 Fritz R., 372, 416 Fubini S., 495 Galileo, 59, 239, 240, 244, 245, 247, 255, 261-263, 266 Galvani L., 43, 85 Gamow G., 349, 372, 394-395 Garbasso A., 214 Garnett W., 166 Gassiot J.P., 140 Gaugain T.M., 99, 110 Gaulard L., 162 Gauss J.F.C., 99, 111, 114, 127, 145, 178, 191, 238 Gautherot N., 87 Gay-Lussac J.L., 11, 24, 39, 43-47, 51-54, 64, 96, 101, 138, 284 Geiger H., 287, 320, 334, 335, 410, 414, 423, 445-446, 459, 460 Geissler H., 30, 197 Geitel H., 220, 228 Gell-Mann M., 464, 473, 481, 489 Gerlach W., 77, 436

Index of Names Germer L.A., 370 Gibbs J.W., 291 Giese W., 229 Giesel F., 220, 223, 320 Gilbert, 92 Ginzburg V., 480-481 Glaiser J., 438 Glaser D.A., 460-461 Glashow S., 486 Glasson J.L., 404 Glitscher K., 258 Gockel A., 420 Goddard R., 447 Goldschmidt V., 430 Golstein E., 197, 199, 319 Gordon J., 452 Gossard R., 479 Goudsmit S., 368 Gouy L.G., 289-291, 440 Graetz L., 271 Graham T., 137 Gramme Z., 162 Grassmann, 104 Green G., 13, 141, 143-145 Green M., 496 Grimaldi, 3, 9, 29 Gross D.J., 483 Gross E., 458 Grosse A. von, 415 Grotthus C., 90-91, 134, 136, 147 Grove W.R., 140 Grüneisen E., 301 Guericke, 51, 227 Guglielmini, 239 Gurevch, 450 Hachette J.N.P., 89 Hagen K., 275, 477 Hahn O., 324-325, 345, 415-417, 454 Haldat Ch.N. De, 155, Hall E.H., 187, 431, 479 Halley, 2 Hallwachs W., 227-228 Hamilton William (1788-1856), 165, 282, 364, 367

A History of Physics over the Last Two Centuries Hamilton William Rowan (18051865), 14-16 Hankel W., 85 Hansteen Ch., 94 Harriot D., 453 Harris W., 118 Hasenöhrl F., 259-260, 278 Haukance, 41 Hauksbee, 1, 197, 227 Haüy R., 215 Heaviside O., 443-445 Heisenberg W., 349, 355, 364-366, 369, 377-382, 385-386, 390, 392, 407-409, 424, 432, 436, 469, 474, 482, 487 Heitler W., 388, 430, 432 Helmholtz H. von, 1, 32, 67-69, 75, 78, 99, 105, 127-128, 169, 172, 180-182, 193, 195, 241, 275, 286, 288 Henri V., 292 Henry J., 96, 110, 126-128, 164 Henry W., 86 Hermite C., 365 Hermite G., 438 Herschel J., 7, 10, 13, 24, 29, 30, 121, 150, Herschel W., 23-24 Hertz H., 176, 179-183, 199, 201, 203, 227, 228, 240-241, 244245, 259, 260, 278, 448, 450451 Hess V., 420 Hevesey J.G. Von, 327 Higgs P., 477, 480, 486, 488, 490, 491 Hirn G.A., 67 Hittorf J.W., 136, 197, 227 Holborn L., 116 Honigschmid O., 345 ‘t Hooft G., 477, 483-486 Hooke R., 41 Hope T., 41 Hoppe E., 110, 132 Hubble E., 494 Hull, 450

509

Hulse R., 493 Hume D., 241 Huttel A., 17 Huygens C., 4, 5, 10, 190, 360 Icilius Q., 117 Infeld L., 256, 304, 388 Ingenhousz J., 56, 58 Ioffé A., 434 Ivanenko D., 407, 409, 424 Jacobi C.G., 14, 16, 112 Jacobi M., 161, 364 Jacobs L., 485 Jamin J., 9-10 Jansky K., 448 Janssen P., 30 Javan A., 453 Jeans J., 275, 280, 351, 383 Jenkin H., 192 Jenkin W., 126 Johnson M., 426 Joliot F., 404, 417-418, 454 Joliot-Curies, 405, 408, 410-413 Jolly Ph., 137 Joly J., 321 Jordan P., 364-366, 383 Joule J.P., 51, 65, 67-69, 72, 75, 114, 117-120, 158 Kadanoff L., 474 Kamerlingh Onnes H., 188, 278, 280, 296, 301-302, 478 Kapitza P., 302, 457 Karolus A., 17 Kastler A., 453 Kauffmann W., 220, 224, 258 Kayser J.H., 346 Keesom W., 296, 302 Kelland Ph., 165 Kelvin Lord (Thomson W.), 22, 6869, 71, 113-114, 164, 166, 187, 188, 192, 194, 285, 330-331, 333 Kennedy J., 415 Kepler, 23, 354

510 Kerr J., 17, 151-152, 186-187 Kerst D., 399 Ketteler E., 20 Ketterle W., 478 Khokhov, 450 Kinnersley, 92, 117 Kirchhoff G.R., 31-32, 108, 111, 176, 180, 193, 240, 270, 275 Klitzing K. von, 479 Knipping P., 217 Knoll M., 372 Kohlrausch F.W., 99, 113, 136-137, 139, 164 Kohlrausch R., 107, 112, 137, 171, 176, 192 Kolhoster W., 420-421 Kompfuer R., 450 Konig K.R., 10 Kourtchatov I., 457 Kramers H., 355, 360 Krönig A., 76-77 Kundt A., 55, 151-152, 187, 189 Kurlbaum F., 271 Kusch P., 471 Laborde A., 225, 318 Lagrange J. L., 56, 141, 190, 235, 237, 282 Lalande, 100, 122 Lamb W.E., 450, 471-472 Lambert, 23, 41 Lambertson G., 462 Lamé G., 13 Landau L., 436, 458, 478, 480 Landriani M., 45 Lang E., 238 Langevin P., 158, 231, 255, 267, 279-280, 293, 331, 335, 374, 434-435, 451 Langley S.P., 27 Langmuir I., 456-458 Laplace P. S. de, 2, 4, 11, 44, 51-54, 75, 82-83, 95-97, 100-101, 104, 122, 141-142, 144, 172, 295, 441 Larmor J., 189, 260, 449

Index of Names Lattes C., 425, 460 Laue M. von, 215-217, 267, 388 Laughlin R., 479-480 Lauritsen C., 396 Lavoisier A.L., 50, 58, 75, 85, 125, 190 Lawrence E.O., 396-398, 425 Lawrence W., 217 Lebedev P., 272 Le Bel J.A., 7 Lecat M., 266 Lecher E., 132, 182 Leclanché G., 140 Leduc S.A., 187 Lee T.D., 466-467, 476 Legendre A.M., 14 Leggett A., 481 Lehnert B., 478 Leibniz G.W. von, 248, 277, 283 Lenard Ph., 199, 205, 207, 209, 210, 213, 229, 231, 259, 260, 303 Lenz E.Kh., 110, 120, 127, 193 Leprince-Ringuet L., 394 Levi-Civita T., 263, 412 Levine M., 458 Libri G., 93 Liebig J. Von, 47, 65 Linde K., 48, 50 Lindemann F.A., 301, 340, 439-440 Lippmann G., 35, 199 Lister J., 36 Livingstone L.&S., 397, 438 Lloyd H., 9, 15 Lo Surdo A., 152, 189 Lockyer N., 328 Lodge O., 21, 182, 449 Lommel E., 27 London F., 430 Lorentz H.A., 21-22, 187-189, 207, 244-247, 249, 253-255, 257, 263-264, 269, 275, 278-280, 283, 302, 307, 342, 377, 387388 Loschmidt J., 79, 196, 284 Low F., 473 Lumière L.J. & A., 35

A History of Physics over the Last Two Centuries Lummer O., 33, 271-272, 277, 305 Lundquist S., 458 Lyman Th., 347, 352 Lyons H., 451 Mach E., 20, 78, 193, 239, 241-243, 248, 291, 378 Macker H., 457 Macquer P., 59 Magnus H.G., 27, 44-46, 58, 137 Maiman Th.H., 453 Majocchi A., 35 Majorana E., 408 Malus E.L., 5-6, 33 Mandelstam S., 484-485 Marconi G., 183-184, 213, 442-443 Marianini S., 88, 106-108, 111-112, 127, 139 Mariotte E., 23 Marsden E., 334-335, 338 Marshall F.H., 459 Marton L., 372 Marty D.H., 441 Mascart E., 193 Mascheroni L., 45 Masson A., 126, 161 Mattauch J., 345 Matteucci C., 19, 26, 110, 126, 155 Maupertuis P.L. Moreau de, 364 Maxwell J.C., 20, 77, 79-82, 105, 122, 128, 144, 147, 149, 154, 165-181, 184-185, 187-188, 192, 194-195, 244-247, 250, 252, 254, 257, 259-260, 264, 269. 272, 274-275, 278-281, 290-292, 299, 436, 442, 448, 463 Mayer A.M., 206, 297, 328-329 Mayer J., 58 Mayer R., 65-68, 75, 79 McMillan E.M., 399, 415 McCoy H.N., 325 McLennan J., 420 Mead S., 449 Meissner W., 480 Meitner L., 394, 407, 415-417

511

Melloni M., 25-27, 98, 116 Mendeleev D.I., 49, 323-324, 336, 356, 438, 482 Menzel D.H., 400 Merchaux P., 89 Meucci A., 164 Meyer S., 223 Michelotti V., 97-98 Michelson A.A., 5, 18, 20-22, 244, 250-251, 254-255 Miller D., 22 Miller W.A., 30 Miller Dobson G., 439 Millikan R.A., 207-208, 297, 306, 421-422 Mills R., 476-477, 483-484, 486 Minkowski H., 257 Mitscherlich E., 40 Mittelstaedt O., 17 Mohor K.F., 65 Mojon B., 84 Moll G., 96, 161 Mollet, 161 Monge G., 41, 48, 56, 90 Montgolfier J.E. e J.M., 43 Morichini D., 149-150 Morley E.W., 5, 20-22 Morosi G., 65 Morozzo C.L., 84 Morse S., 111-112, 432 Moseley H., 336-337 Mossotti O.F., 29, 148-149, 173174 Mott N.F., 429 Moulin, 285 Mouton G., 152, 190 Murgantroyd R.J., 442 Nacken, 258 Nagaoka H., 333-334, 443 Nambu J., 464, 496 Napoleon, 8, 43, 56, 61, 84, 142, 191 Natterer J., 49-50 Neddermeyer S., 424-425 Néel L.E., 436

512 Nernst W., 139, 187, 278, 280, 299301, 374, 404 Neumann F.E., 13, 127-128, 172, 178 Newcomb S., 18 Newton I., 2, 4, 6, 9-11, 15, 19, 24, 40, 51-53, 84, 93-94, 144-146, 153, 175, 178, 235, 239-243, 246, 248, 253, 260-262, 264266, 277, 305-306, 330, 353, 359-360, 368, 380, 387, 446 Nicholson W., 85-86, 89 Nicol W., 7 Nicolet M., 441, 442 Nielsen H., 496 Niepce J.N., 34 Nier A.O., 418 Nishijima K., 464 Nobili L., 25, 98, 106, 116, 121122, 126-127 Noble H.R., 22 Noddack I., 415 Noll W., 268 Nollet J.A., 57, 137 Northmore Th., 48 Nuttall J.M., 393 Ohm G.S., 106, 108-113, 117, 119, 127, 192, 230 Oberth H., 447 Occhialini G., 369, 423, 425, 460 Oersted H.C., 88, 92-95, 97-98, 100-101, 105, 115-116, 128, 177 Oppenheimer J.R., 349, 470 Ostwald W., 78, 87, 137, 139, 266, 291, 344, 378 Owens R.B., 311 Pacini D., 420-421 Pacinotti A., 162 Pacinotti L., 117 Page C., 161, 164 Painlevé P., 257 Pais A., 464 Palmieri L., 126, 161

Index of Names Paoli D., 65 Papin D., 48 Parisi G., 213 Parker J., 156-157 Paschen F., 258, 272, 277, 347, 352 Pasteur L., 7 Pauli W., 349, 356-357, 374-375, 377, 388, 391, 409, 424, 430432, 466, 470 Pauling L., 430 Peano G., xii, 291 Pearson F., 22 Pease F., 22 Péclet J.C.E., 112 Peierls R., 432 Peltier J.C.A., 116-117, 433-434 Penning F., 457 Penzias A., 494 Perkin W., 232 Perrin F., 417 Perrin J.B., 199-201, 213, 278, 290, 292-295, 314, 317, 330, 363, 370, 407, 410 Petit A.T., 39-40, 45, 297-300 Petermann A., 473 Pfaff C., 85 Pfeffer W., 137 Philipp K., 407 Phillips R., 153 Picard J., 190-191, 422 Piccard A., 440 Piccioni O., 462 Pickering E.C., 353 Pictet M.A., 24 Pictet R., 50, 101 Pierce J., 450 Pixii H., 101, 160-161 Planck M., 69, 72, 75, 82, 139, 207, 215, 233, 267, 275-283, 288, 298, 302, 304-306, 350-351, 358, 363-364, 366, 369, 374, 381-383, 387-388, 435, 470, 479, 481 Planté G., 88 Plücker J., 30, 130, 156, 197 Poggendorff J., 65, 67, 99, 108, 110, 113, 117, 162

A History of Physics over the Last Two Centuries Poincaré J.H., 82, 105, 168, 176, 218, 240, 243-245, 247-249, 255, 278, 280-282, 348, 391, 442-443, 448 Poisson S.D., 11, 13, 15, 52, 105, 121-122, 141-147, 149, 172, 237 Politzer D., 483 Pontecorvo B., 414, 490 Poole, 420 Popov A.S., 184 Porta G.B., 23, 47 Pose H., 394 Pouillet C., 98, 111, 155 Powell C., 425, 460, 464, 466 Poynting J.H., 259 Preece W., 183 Prevost P., 121 Priestley, 41, 86 Pringsheim E., 272, 277, 305 Prout W., 205, 328, 342, 344 Pulley O.O., 441 Quincke G.H., 10, 137 Raman C.V., 307, 309-310 Ramanathan K., 309 Ramsay W., 220, 289, 314-315, 319-320, 324-325 Rankine W., 69, 74, 78, 169, Raoult F.M., 138 Rasetti F., 414-415 Rayleigh Lord (J.W.Strutt), 19, 22, 36, 45, 141, 166, 193, 273-275, 277-278, 285, 308-309, 314315, 439, 443, 449, 452-453 Rayleigh R., 319, 321 Réaumur, 41, 43 Rebbi C., 485 Reech F., 240 Regener E., 286 Regge T., 496 Regnault H.V., 45-47, 49, 67, 70, 72, 77, 114, 232 Reich F., 31, 239 Reid A., 370

513

Reines F., 409 Reiss Ph., 164 Retherford R., 450, 471 Reynolds J.E., 346 Ricci Curbastro G., 263 Riccioli, 4 Richardson O., 370, 436 Richmann, 55, 57-58 Richter H., 31 Riemann G.F.B., 104, 146 Righi A., 164, 183, 187, 212, 228, 396, 448, 450 Ritchie E.S., 161 Ritter J., 24, 34, 85, 88, 114, 195 Ritz W., 347, 354 Robertson S., 85 Rochester G., 464 Roiti A., 212 Romagnosi D., 93 Röntgen W.C., 152, 209-214, 218219, 316 Rood A.N., 33 Roosevelt F.D., 418 Roscoe H., 31 Rosenblum S., 404 Rosenfeld L., 349 Ross W.H., 325 Rossetti F., 41 Rossi B., 422, 459-460 Rowland H., 29, 105-106, 131, 187188, 193 Royds T., 320 Rubbia C., 213, 489 Rubens H., 275, 278 Rudberg F., 44-45 Ruhmkorff H.D., 151, 161, 186, 188 Rumford, count of, 41, 58-60, 65 Runge C.D., 346 Rupp A., 371 Ruska E., 372 Russell B., 382 Russell S., 327, 450 Rutherford E., 213, 220, 222-224, 278, 286-287, 311-322, 325, 327, 329, 333-339, 348-350, 353, 356, 393-394, 306, 399-

514 401, 403-406, 412, 420, 426. 454, 482 Rydberg R., 346-347, 352 Sacerdote G., 45 Salam A., 486, 488-489 Savart F., 95-96, 104 Savary F., 96 Savic P., 416 Scheel C., 41 Schelkunoff S.A., 449 Scherk J., 496 Schmidt G.K., 220 Schoenflies A., 216 Schottky W., 430 Schrieffer J.R., 480 Schrödinger E., 16, 366-369, 376377, 479 Schulze J.H., 24 Schuster A., 185, 203, 227, 229 Schwarz J., 496 Schwarzschild K., 190, 266 Schweidler E. von, 223 Schweigger J., 97, 109 Schwinger J., 471-472 Seaborg G., 415 Seebeck T.J., 24, 109, 115-116, 433 Seeber L.A., 215 Segrè E., 213, 414-415, 462 Séguin M., 65 Seitz F., 429-430 Secchi A., 185 Sella Q., 285 Sénarmont H.H. de, 58 Senefelder A., 34 Seubert K.F.O., 344 Shockley W., 434 Shull C., 437 Siemens C.W., 193 Siemens E.W., 112, 163, 192-193 Sklodowska M. see Curie M. Skobelzyn D., 422 Slater J.C., 360, 430 Smoluchowski M.R., 296 Smyth H., 418 Snoek J., 437

Index of Names Soddy F., 220, 313-322, 324-327, 336-337, 339, 341 Sohncke L., 216 Soldner J. von, 264 Sommerfeld A., 189, 258, 278, 280, 282, 353-354, 360-361, 363-364, 374-375, 383, 388, 431-432, 442 Somov P., 145 Spallanzani L., 289 Stancari V.F., 41-42 Stark J., 152, 189-190 Stefan J., 271-274, 288 Stern O., 77, 80, 371, 436 Stevin, 61, 190 Stewart B., 443 Stewart J., 436 Stewart Th.D., 431 Stokes G.G., 20, 24-25, 31, 105, 166, 206, 244 Stoney G.J., 196, 205, 208, 286, 331, 346 Störmer H., 479-480 Strassman F., 416-417 Strauser W.A., 437 Strutt J.W. see Rayleigh Lord Strückelberg E., 471-473 Sturgeon W., 96-97, 121, 158 Sturm J.Ch., 24 Susskind L., 496 Szilard L., 418 Tait P., 116, 166, 185, 187 Talleyrand-Pérygord C.M., 190 Tamm I.Y., 424, 458 Targioni Tozzetti, 239 Taylor J., 493 Teisserenc de Bort L., 439 Teller E., 349, 418, 456 Tesla N., 162, 183 Thénard L.J., 24, 90 Thicion, 289 Thompson B., 59 Thomson J.J., 195, 201-207, 213214, 224, 228-231, 259, 308, 311, 327, 329-335, 338-340, 348, 356, 449

A History of Physics over the Last Two Centuries Thomson G.P., 370 Thomson W. see Kelvin Lord Thorpe T.E., 344 Tillet M., 190 Tohplygione S.A., 268 Tolman R., 431 Tomonaga S., 471-472 Tonks L., 457 Townes Ch., 452 Townsend J., 224, 231 Traube M., 137 Travers M., 319 Tremery J., 98 Trillat J.J., 372 Trouton F., 22 Tsui D., 479-480 Tuve M., 396, 445, 451 Tyndall J., 25-27, 59, 69, 285 Uhlenbeck G., 368 Urey H.C., 400 Vailati, 378 Van Allen J., 445-446 Van de Graaff R.J., 396 Van den Broek A.J., 336-337 Van der Meer S., 489 Van der Pol B., 267 Van der Waals J., 78, 278, 284, 288 Van Marum, 48, 86 Van’t Hoff J.H., 7, 137-139, 294 Van Trostwijk P., 86 Van Vleck J., 436 Varian S., 450 Varley, 197 Vassalli Eandi, 45, 98, 114 Vegard L., 438 Veksler V., 399 Veltman M., 477 Vendermonde, 41 Veneziano G., 495-496 Verdet E., 13, 151 Vigier J.P., 391 Villard P., 224, 318 Villari E., 152 Violle J., 33

515

Viviani, 238 Voigt W., 247, 331 Volta A., 41-46, 58, 64, 84-90, 93, 111, 114-115, 191, 284, 383 Voltaire, 391 Volterra V., 267, 412 von Braun W., 447 von Katman, 301 von Klitzing K., 479 Walton, 396-401 Wartmann E.F., 184 Washburn E., 400 Watson E.T., 396 Watson H.E., 339 Watson R., 445 Watt J., 46, 61, 232 Watt W., 451 Weber H.F., 298 Weber J., 492 Weber W., 98-99, 104-105, 111, 114, 127-128, 130, 156, 158, 171-172, 176-177, 185, 191, 195 Webster, 405 Wedgwood T., 34 Wegener A.L., 438 Weil G., 419 Weinberg S., 486, 488-489 Weiss P., 434-439 Weiss R., 494 Weisskopf V.F., 418, 470 Welter J.J., 54, 101 Wennelt, 162 Wenzel W., 462 Wess J., 497 Weston E., 141 Wheatstone C., 17, 29, 107, 110, 112-113, 155 Whewell W., 134 Whittaker E., 63 Wiederöe R., 397 Wiedemann E.E.G., 227 Wiedemann G.H., 58, 107, 120, 151, 187, 199 Wiegand C., 462

516 Wieman C., 478 Wien W., 116, 272-273, 277-278, 280, 373 Wiener O., 289 Wigner E.P., 418, 465, 474 Wilcke J., 147, 483 Wilczeck F., 483 Williams N., 450 Williamson A., 137 Wilson Ch.Th.R., 206-207, 258, 308, 393-394, 401, 404-406, 411, 422, 424, 459-461, 464 Wilson K.G., 474, 484-485 Wilson R., 494 Wilson W., 354 Winckler C.L., 40 Witten E., 497 Wollan E., 437 Wollaston W., 24, 28, 48, 84-45, 87, 89, 92, 132 Wommelsdorf H., 396 Woods L., 419

Index of Names Wren Ch., 190 Wu Ch.Sh., 466-467 Yang C.N., 466-467, 476-477, 483484, 486 Young Th., 1-5, 9, 12, 24, 59, 68, 278, 307, 392 Young J., 17 Ypsilantis T., 462 Yukawa H., 424-426, 477 Zamboni G., 89 Zeeman P., 152, 186-189, 195, 207, 244, 331 Zeiger H., 452 Zeiss C., 37, 232 Zinn W., 419 Zöllner J.C.F., 20 Zhukovskij N.E., 268 Zumino B., 497 Zweig G., 481