The Concept of Matter: A Journey from Antiquity to Quantum Physics 3031365577, 9783031365577

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History of Physics

Florestano Evangelisti

The Concept of Matter A Journey from Antiquity to Quantum Physics

History of Physics Series Editors Arianna Borrelli, Institute of History and Philosophy of Science, Technology, and Literature, Technical University of Berlin, Berlin, Germany Olival Freire Junior, Instituto de Fisica, Federal University of Bahia, Campus de O, Salvador, Bahia, Brazil Bretislav Friedrich, Fritz Haber Institute of the Max Planck, Berlin, Berlin, Germany Dieter Hoffmann, Max Planck Institute for History of Science, Berlin, Germany Mary Jo Nye, College of Liberal Arts, Oregon State University, Corvallis, OR, USA Horst Schmidt-Böcking, Institut für Kernphysik, Goethe-Universität, Frankfurt am Main, Germany Alessandro De Angelis , Physics and Astronomy Department, University of Padua, Padova, Italy

The Springer book series History of Physics publishes scholarly yet widely accessible books on all aspects of the history of physics. These cover the history and evolution of ideas and techniques, pioneers and their contributions, institutional history, as well as the interactions between physics research and society. Also included in the scope of the series are key historical works that are published or translated for the first time, or republished with annotation and analysis. As a whole, the series helps to demonstrate the key role of physics in shaping the modern world, as well as revealing the often meandering path that led to our current understanding of physics and the cosmos. It upholds the notion expressed by Gerald Holton that “science should treasure its history, that historical scholarship should treasure science, and that the full understanding of each is deficient without the other.” The series welcomes equally works by historians of science and contributions from practicing physicists. These books are aimed primarily at researchers and students in the sciences, history of science, and science studies; but they also provide stimulating reading for philosophers, sociologists and a broader public eager to discover how physics research – and the laws of physics themselves – came to be what they are today. All publications in the series are peer reviewed. Titles are published as both printand eBooks. Proposals for publication should be submitted to Dr. Angela Lahee ([email protected]) or one of the series editors.

Florestano Evangelisti

The Concept of Matter A Journey from Antiquity to Quantum Physics

Florestano Evangelisti Università degli Studi Roma Tre Rome, Italy

ISSN 2730-7549 ISSN 2730-7557 (electronic) History of Physics ISBN 978-3-031-36557-7 ISBN 978-3-031-36558-4 (eBook) https://doi.org/10.1007/978-3-031-36558-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Fulvia and Fabrizia

Acknowledgements

This book had originally been written in Italian; this English version was possible thanks to the translation made by my friend Martin Bennett, to whom I am very grateful.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1

Matter and Classical Science

2

The Legacy of Antiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Knowledge from Crafts and Techniques . . . . . . . . . . . . . . . . . . . . . 2.2 Matter in Ancient Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Pre-socratics’ Concept of Matter . . . . . . . . . . . . . . . . 2.2.2 Atomism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Nature in the Aristotelian System . . . . . . . . . . . . . . . . . . . 2.3 Alchemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 7 9 11 14 16

3

Matter at the Beginning of the Scientific Revolution . . . . . . . . . . . . . . 3.1 Empirical Knowledge of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Matter and Substances in Arab, Medieval and Renaissance Alchemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Matter and Substances in the Philosophical and Scholarly Tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18

Matter in the Seventeenth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Alchemical and Chemical Quests in the First Half of Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Matter and Physical Properties: Air and Vacuum . . . . . . . . . . . . . . 4.2.1 Experiments with Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Combustion, Respiration and Calcination. The Complex Nature of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Mechanical Philosophy and Atomism in the Seventeenth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Atomism of Gassendi, Boyle and Newton . . . . . . . . . . . .

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19 24 26

30 32 34 37 39 40

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4.4 Boyle’s Criticism of Aristotelian and Alchemical Concepts . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 43

Matter in the Eighteenth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Phlogiston Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Matter in the Gaseous State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Black and Fixed Air (Carbon Dioxide) . . . . . . . . . . . . . . . 5.2.2 Cavendish and Inflammable Air (Hydrogen) . . . . . . . . . . 5.2.3 Oxygen, Nitrogen and Air . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Combustion and Calcination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Composition of Water and Confutation of the Phlogiston Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Chemistry and Matter at the End of Eighteenth Century . . . . . . . . 5.5.1 Conservation of Matter and Chemical Elements . . . . . . . 5.5.2 Compounds and Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Combination and Equivalents . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 49 50 51 53

6

Matter and Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Voltaic Pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Electrolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 69 71 73

7

Chemistry and Matter in the Nineteenth Century . . . . . . . . . . . . . . . . 7.1 Revival of Atomism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Avogadro’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Dualistic Theory of Affinity . . . . . . . . . . . . . . . . . . . . 7.1.3 Faraday and the Electrolysis . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Chemical Bonds and Valency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Elements and Periodic Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 79 80 81 82 84 87

8

Physics and Matter in the Nineteenth Century . . . . . . . . . . . . . . . . . . . 8.1 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Nature of Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Loschmidt’s Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Nature of Electric Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Cathode Rays and the Discovery of Electrons . . . . . . . . . 8.2.2 Canal Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 91 93 94 94 96 98 99

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Part II 9

56 57 58 59 60 61

Matter and Old Quantum Theory

The Appearance of Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.1 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Contents

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9.3 Quantum Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10 Bohr’s Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Early Atomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Light and Atoms: Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Bohr’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Moseley’s Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Experiment of Franck and Hertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Elliptical Orbits and Space Quantization . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 114 118 120 122 122 124

11 Electrons and Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Electronic Structure and Periodicity of the Elements . . . . . . . . . . . 11.1.1 Bohr’s Aufbauprinzip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fine Structure and Magnetic Field Effects . . . . . . . . . . . . . . . . . . . 11.3 Sommerfeld’s Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . . 11.4 A New Quantum Number: Electron Spin . . . . . . . . . . . . . . . . . . . . 11.5 Wavelike Nature of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 128 130 131 133 136 137

Part III Quantum Mechanical Representation of Matter 12 Quantum Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Helium and Quantum Properties of Multielectron Atoms . . . . . . . 12.4 Molecules: Covalent Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 143 146 149 151

13 Solid Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Atomic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Electrons in Solid Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 The Free Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Bloch Waves and Band Theory . . . . . . . . . . . . . . . . . . . . . 13.2.3 Metals and Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 156 156 158 161 162

14 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Doping and Microelectronic Devices . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 165 169 171

Appendix A: On Gases and Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Appendix B: On Nineteenth Century Physics . . . . . . . . . . . . . . . . . . . . . . . . 177

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Appendix C: On Old Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Appendix D: On Matter and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 193 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter 1

Introduction

This book outlines the long journey leading to the present concept of matter. Thanks to quantum mechanics and advances in physics and chemistry, we now have a fairly accurate knowledge of matter’s structure and properties. As is well known, it can be summarized as follows: Matter is composed of atoms and molecules. Atoms are complex structures made up of a central nucleus and encircling electrons. The large number of different substances known to date are aggregates of relatively few atomic species. On the atomic scale matter obeys the laws of quantum mechanics, which indeed stems from the attempts to explain matter’s properties. This knowledge, whose assessment represents one of scientific research’s greatest successes, was developed relatively recently, despite being the answer to age-old questions, some of which arose contextually with the birth of philosophy in ancient Greece. The said journey can be divided into three phases: A first phase, which saw the development of concepts based on classical physics and chemistry and closed at the end of the nineteenth century; a second phase, which saw the construction of the old quantum theory in the attempt to explain the mysterious properties of matter and closed with the formulation of the new quantum theory; then a third phase when the modern concept of matter, based on quantum mechanics, was elaborated. This periodization is reflected in the three parts of this book. The first part deals with the following topics: The legacy of antiquity, rediscovered and appropriated by Western culture in the late Middle Ages and the Renaissance. The revision process that began with the scientific revolution of the sixteenth and seventeenth centuries, and revealed the inadequacy of the dominant theories of Aristotelian origin. The new experiments and the new theories developed in the eighteenth century, in line with the ideas brought about by the new science, representing fundamental steps towards the modern concept of matter. The final assertion, in nineteenth century, of matter as consisting of a definite number of elementary substances, the elements, endowed with peculiar physical and chemical properties and whose combination generates chemical substances. The demonstration that the elements are made of atoms, that inside atoms there are positive and negative electric charges, and that the negative charges are electrons. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_1

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The second part depicts the attempts to understand matter’s mysterious properties that could not be explained by classical physics. Attempts that proved the effectiveness of quantum concepts and led to the construction of the old quantum theory. In particular, the following topics are covered: The appearance in the physics world of the first quantization hypotheses thanks to Planck and Einstein; Niels Bohr’s theory of the hydrogen atom and its generalization by Arnold Sommerfeld; the discovery of new purely quantum–mechanical properties of matter (electron spin, exclusion principle, and wave-particle duality). The third part outlines the new quantum theory and exemplifies its use to assert the final properties of material substances. In particular, the following topics are covered: The ultimate clarification of atomic structure, the nature of chemical bonds, the electrical properties of metals and insulators, the nature of semiconductors.

Part I

Matter and Classical Science

Chapter 2

The Legacy of Antiquity

In sixteenth and seventeenth centuries, at the beginnings of the scientific revolution, when the journey that has led to the modern concept of matter began, the first problems to solve dealt with the soundness of the cultural heritage inherited from the past. This chapter aims to illustrate this legacy, which was based on ideas, empirical knowledge and belief regarding matter stemming for the most part from the ancient world, and which can grouped into three strands. The first and oldest strand was that of the technical skill and empirical knowledge of artisans, which started with the civilisation process and defined historical epochs. The second, which we could define as learned tradition, includes the theories concerning matter found inside antiquity’s philosophic systems, and was born together with the philosophical reflection. The third and most recent strand includes the description of laboratory practices as well as the handling of substances, which made up the most convincing and positive part of alchemy books, free from the mishmash of fantasies, abstruseness and superstitions contained therein.

2.1 Knowledge from Crafts and Techniques An important legacy concerning knowledge of matter inherited from the ancient world is represented by the set of notions and empirical practices accumulated by those who worked metals, manufactured ceramic objects, dyed fabrics, worked leathers, made jewellery. All these activities required an appropriate skill and a degree of experimentation which, little by little, would give rise to the awareness of the existence of well-defined substances, each endowed with characteristic properties, a knowledge eventually evolving into the modern concept of chemical substance. The accumulation of such expertise began on prehistorical times and featured a category of individuals possessing specific technical knowledge handed down principally by word of mouth. Written accounts of these activities were rarely preserved in remote antiquity. Such knowledge (or sets thereof) evolved alongside the tradition of learned © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_2

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knowledge, remaining for the most part distinct from it, albeit over time some of its concepts came to be included in the philosophical systems from which ancient science would evolve. Knowledge about metals had a prominent role. This was largely due to the combination of particular physical and chemical characteristics that distinguish them from other substances, enabling their ready identification from the dawn of history. Equally significant was their impact on social life in the form of weapons, tools and jewellery. Not by chance the civilisations following the Stone Age are recognisable by the metals most widely in use: the Copper and Bronze Ages, then the Iron Age. How exactly technical knowledge regarding metals originated is lost in the night of time and is difficult to trace. The first metals known to man were almost certainly gold and copper, both being found pure in their native state [1]. Evidence of their being worked appears as far back as 3400 BC in Egypt and Mesopotamia. The Sumerians were expert at working gold, silver and copper [2]. Within a thousand years from the discovery of the processes of extraction and purification toward the end of the fourth millennium BC, metallurgy, here meaning the range of processes making it possible to impart metal with new and controlled forms, had been developed in both Egypt and the Middle East. At the start of the third millennium mankind had already mastered simpler metallurgical techniques for some of the most common metals, gold, copper, lead, antimony, tin, and bronze [3]. The iron age began later, since this metal was more difficult to shape than other metals in previous use. Only after a long period of experimentation new technical procedures and instruments more suited to working this metal were introduced. From the second half of the second millennium BC, Middle East and Egypt witnessed an extensive use of iron tools and weapons. It is probable that the first iron came from the land of the Hittites around the Black Sea. Greco-Roman metallurgy was nothing other than a continuation of that of the Middle East. The Greeks and Romans brought only two major contributions to knowledge of metals: the production of mercury, mainly used to extract gold, and the production of brass, an alloy of copper and zinc [4]. From Greek antiquity onwards the seven most common metals (gold, silver, iron, mercury, tin, copper and lead) and some of their alloys, most of all bronze (alloy of copper and tin), had become an important financial and political factor. As a result, reflection upon their properties would also influence philosophical thought. In the classical period, as well as metals, a certain number of chemical substances were known and widely used, their being found pure in nature or easily separated from the minerals that contained them. The most common among these were sulphur, charcoal, pigments, salt, and alum [5]. These substances, along with some simple chemical operations, were described in the writings of Dioscorides (around 40–90 DC), Pliny the Elder (23–79 AD) (author of the famous Naturalis Historia in 37 volumes) and Galen (129–201 AD). Although a few written documents, such as those just mentioned, survived from antiquity, this legacy of knowledge about matter and the handling of substances was largely passed down orally and through the institution of apprenticeship, so assuring its survival. However, it is also important to bear in mind that many of

2.2 Matter in Ancient Philosophy

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these recipes, such as those regarding the preparation of precious stones and above all metal working, were included in the alchemy texts that, however controversial, would become common from the Alexandrine period onward.

2.2 Matter in Ancient Philosophy The Greeks were the first to reflect extensively on the nature of matter and have left a vast literature containing their speculations. One can affirm that the reflection on the being of the world, of which matter is a relevant part, and the becoming of the world, of which the transformation of bodies and substances is an essential manifestation, was intrinsically linked to the first schools of philosophy in ancient Greece.

2.2.1 The Pre-socratics’ Concept of Matter Traditionally, the birth of philosophic thought is traced back to Thales (around 640– 546 BC), precisely because his theory can be taken as the first to focus clearly on the problem of the world’s nature and to have identified its essence as a physical entity rather than a divine one. Thales was the first to ask whether the things surrounding us could be manifestations of a single reality which presents itself in various forms, a question that became central to Greek philosophical speculation for generations. For Thales the answer was in the affirmative: He believed in the existence of a primary substance, the arche, the cause and origin of everything, and which remained itself while underlying whatever apparent transformations. Thales identified the arche with water. Aristotle in the first book of Metaphysics declares that Thales had chosen water as the arche based on the observation of moisture’s role in vital phenomena. Water for Thales was a mobile, changeable, fluctuating essence, subject to a cycle of existence whereby it passed from the sky and the air to the earth, and thence to animal and vegetal bodies then back to the air and the sky [6]. Water is normally liquid but becomes vapour when it is heated and changes to ice when cooled, whereby the same substance manifests itself in all three states of matter (solid, liquid and gas). The most important aspect of Thales’ reflection is that he tried to answer the question of the reality concealed behind the phenomena that we observe, a question also featuring the beginning of Greek philosophy [7]. Thales was the first of the three philosophers of Miletus who formed the Ionic School and are considered the first philosophers of nature. Aristotle refers to them with term physikoi, physicists, from the Greek term physis, meaning nature in its most general sense, as opposed to the earlier theologoi (theologians), in that they, the physikoi, had been the first to try to explain nature’s phenomena in purely physical terms instead of supernatural ones. The second philosopher of the Ionic School was Anaximander (around 610–545 BC), a younger contemporary of Thales, who further generalized the concept of

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primary substance. Anaximander believed that Thales’ water could not be the arche, since it already possessed form and specific qualities, while the primary substance must be undifferentiated and indeterminate. This substance he called the apeiron. The term has been translated as the indefinite and infinite, this to mean that the substance was neither limited nor defined by specific properties. For Anaximander this undifferentiated substance had given rise to many worlds, these existing in every epoch, appearing and disappearing in time like bubbles in the apeiron. Such worlds, emerging from the apeiron, resulted from the action of two opposing qualities, heat and cold. With such a concept Anaximander introduced the idea of opposites as the driving mechanism for explaining the world cosmologically, a concept characteristic of the primitive science and typical of the Greek physics to come [8]. The Ionian school’s third philosopher was Anaximenes (around 560–500 BC), a pupil of Anaximander. Anaximenes also held that the only conceivable explanation for physical reality was that all things derive from one and return to one. This primary substance he identified as the pneuma (air, breath), which in its eternal movement assumes various forms. According to the sixth century AD philosopher Simplicius, Anaximenes had claimed that [7]: the pneuma differs in rarefaction and density according to the various substances. Rarefied it becomes fire, condensed it becomes first wind, then cloud, and when condensed still further water, then earth and stones. Everything else is made of these substances..

Under the thrust of the opposite processes of condensation and rarefaction, to which the pneuma is subject, these transformations occur continually in both directions. Translated into modern language, Anaximenes invoked the mutual conversions between the states of matter (gas, liquid and solid), which are apparent in everyday life, to explain the evolution of the nature in its entirety. Therefore, besides identifying the arche, he also sought to describe the natural phenomena through which primary substance assumes one form or another, thus completing a further step toward the development of scientific thought. Simplicius notes that Anaximenes also postulated eternal motion, which is indeed the cause of the change. Another Greek philosopher, whose thought was centred on nature and matter, was Empedocles of Agrigento (around 490–432 BC). According to Aristotle, he was the first to establish that there are four primary substances or primitive elements (earth, air, water and fire). These he defined as the roots of everything, affirming that from these things sprang all things that were, are and will be. The four primitive elements are eternal and always identical to each other, although they also mix to form other substances, which are necessarily composed. Clearly the three elements earth, water and air correspond to the solid, liquid and gaseous states of matter, while fire, defined by Empedocles an ethereal substance, occupies a special place in the world, since, due to its lightness, can be the substance composing the celestial bodies. Empedocles thus enunciated clearly for the first time the theory destined to dominate the concept of matter in antiquity and in the Middle Ages, up to the coming of the modern world. To explain the becoming of the world and things Empedocles also assumed the existence of two further actors in nature: love and strife. Under the influence of these two forces the four elements combine and separate, with love pushing them toward

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union and strife trying to separate them, producing a situation of continuous alternate change. Here appears for the first time the concept of the existence of forces, distinct from matter, as the physical cause of natural phenomena. The cosmos is conceived as a state of dynamic equilibrium between opposite forces, transformations taking place when one of these becomes dominant. Empedocles’s theory, taken up and developed by Aristotle, has been one of science history’s most durable, destined to last more than two thousand years. As well as the philosophers mentioned above, two other philosophical systems from classical Greece clearly need to be considered for the impact they have had on the evolution of the concept of matter, namely Leucippus and Democritus atomistic theory and Aristotle’s conception of the world. The former contained in embryonic form some of the concepts used and developed in the modern notion of matter. The latter dominated the philosophical and scientific domain until the beginning of the modern era.

2.2.2 Atomism The atomistic theory, formulated by Leucippus of Miletus (probably living between the early and late fifth century BC) and his pupil Democritus of Abdera (around 460– 370 BC), was taken up and modified by Epicurus (342–270 BC). Later it became known to the Roman world thanks to the poet-philosopher Lucretius (94–55 BC). This theory was in strong contrast to the world vision upheld by the other Greek schools of philosophy. Leucippus and Democritus believed that the universe is an empty space extending into infinity, occupied by indivisible particles which they called atoms, from the Greek term atomos, meaning just that, indivisible. The empty space, the vacuum (or void), is defined as that which is not and is considered a primary reality as are atoms. The acceptance of vacuum is something exceptional in ancient philosophical thought. In fact, for most Greek philosophers the universe was one of common sense, that is to say a universe that was full [9]. According to Democritus, atoms are indestructible; their number is infinite; they all are made of the same matter and differ from each other only in shape, size and position. Atoms move in the void, their movement being due to reciprocal collisions. Just as atoms are eternal and uncreated, so is their movement, which is atoms’ natural state, originating by a previous movement and needing no explanation, as is evident by its nature. According to the atomists, the qualities which we distinguish in things result from collisions giving rise to different orderings of atoms. Two aggregates of atoms, two objects, can differ through the size, form and number of atoms that compose them but also through the ordering of atoms within the aggregate. The collisions between atoms thus account for the origin of the world, or rather, the infinite number of worlds that populate empty space. The earth was born from vortex-like movements which dragged the heavier atoms toward the centre, where they bound together and could not move except by making small oscillations. The finer atoms of water, air, and

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fire, were driven out into space, where they are in a vortex-like motion round the earth. Some atoms were hurled far into space and, triggered by friction between vortices, formed the celestial bodies. Since atoms are infinite as space is also infinite, there are an infinite number of worlds similar to earth, some coming into being, others in the process of dying [10]. Given that everything is made up of eternal and immutable atoms, it follows that every birth and extinction is nothing else than a different combination of atoms. We ourselves are nothing but temporary aggregates of atoms, which will very soon separate to become part of other beings and entities. In the course of time we possibly might be regenerated, should our own atoms become reconnected. In such dissolution and aggregation of atoms, history is perpetually repeated. In Democritus’ theory there is also an attempt to explain the mechanisms whereby we perceive the world. On the basis of hypotheses already expressed by Empedocles, he posited that every sensory perception is due to flows of atoms. Visual perception of a body, for example, is due to a flow of atoms emanating from the body and striking the eye. That which exists is nothing other than this collision of particles, while colour is a secondary effect of the sensory organ. The same can be said for taste, touch and smell. As for the perception of sounds, it could be triggered by vibrations of air interposed between a sound’s source and the ear. The atomistic theory was a sophisticated construct, endowed with an evident modernity, which had a strong influence on the development of scientific thought. It expounds a theory that is mechanic and materialist, describing natural phenomena in terms of purely mechanical laws and clearly opposed to the concept of a world with a preordained purpose or harmonious order. It postulates a universe from which is excluded any divine intervention or interference. For these reasons it has been contested by both the more traditional schools of philosophy and the various religions dominant in antiquity and the middle ages. According to Geymonat [11]: Democritus’s atomism, later taken up and partly modified by Epicurus, is the most precious legacy, in the field of general interpretations of nature, that the Greeks have passed down to successive generations; it played a fundamental role in shaping modern science in the sixteenth and seventeenth centuries. It was the deaf struggle against the philosopher of Abdera, conducted without let-up by all the branches of Idealism (from Plato, who never even mentions Democritus’s name, to the Hegelians), which buried in silence many results of his marvellous research.

Condemned by the Church, the atomistic theory would be revived with the emergence of the scientific revolution. We need to bear in mind, however, that Democritus’s atoms, like their movement and the void where this took place, were pure hypotheses, not founded on any direct evidence. On the other hand, testing them would have required an investigation capability only attained tens of centuries later.

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2.2.3 Nature in the Aristotelian System Aristotle (384–322 BC) had a more practical and scientific mind than Plato and constructed a vast body of theories by which he sought to explain nature in a more detailed way than any of his predecessors. The description of nature which he developed became over time the tenet accepted by science and philosophy and remained so until the Renaissance. Greek philosophers, Arab and Christian scholars studied Aristotle so deeply that his ideas on natural philosophy became the unconscious inheritance of all thinkers for almost two thousand years. In this span of time every observed fact and every new speculation were interpreted within the Aristotelian context. The general scheme and the cosmology of the Aristotelian system derived from the Pythagorean-Platonic tradition and its distinction between the imperfect and transitory world, situated within the lunar sphere, and the outer celestial region which was considered perfect and eternal. The earth was an immobile sphere at the centre of a cosmos consisting of a series of concentric crystalline spheres. The earthly region was bordered by the revolving sphere in which the moon was embedded, the first of antiquity’s seven planets. Then followed the rotating spheres, of ever wider radius, in which the other planets were embedded (in order: Mercury, Venus, Sun, Mars, Jupiter and Saturn) and the sphere of the fixed stars. Each sphere was subject to the influence of those external to it. All the celestial spheres were crystalline and perfect. This general conception, of Pythagorean origin, was borrowed by Eudoxus of Cnidus (408–355 BC) [6]. As for the substance of which the celestial region was made up, from the sphere of the moon to the universe’s edge, Aristotle had adopted from Anaxagoras (499–428 BC) the concept of ether. The ether or quintessence was a perfect matter, immutable, eternal, whose movement was circular and therefore also perfect. Regarding matter inside the lunar sphere, Aristotle adopted the belief held by many civilisations in the ancient world, that nature’s complexity could result from the transformation and combination of a few simple substances in a state of continuous becoming. This belief permeated, with different shades, the principal Greek philosophical schools. More specifically, Aristotle adopted Empedocles’ conception whereby all things result from a combination of four primitive elements: earth, air, water and fire. These elements, if left free to follow their natural tendencies, would arrange themselves in order of density, with the immobile earth at the centre, surrounded by concentric layers of water (the ocean), of air (the atmosphere) and fire. However, the movement of the celestial spheres, acting through the lunar sphere, creates a continual disturbance; consequently, the elements mix and can never be found in their pure state. Therefore, in the earthly region the four primitive elements are only found in combination and do not coincide with the substances to which we give the names of earth, water, air and fire. For example, the substance which we know as water is mainly made up of the primitive element water but also contains small proportions of the other three primitive elements. The primitive element water

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constitutes water’s essence, which humans have no way of experiencing. The same goes for the other primitive elements. According to Aristotle the becoming of the physical world is explained by the elements’ constant attempt to return to their own natural position. The natural movement of earthly objects is therefore toward their natural place, such that, if earth is displaced upwardly in air and then released, it falls in a straight line, while air, freed into water, rises and similarly fire rises in air. The straight movement is temporary, since it ceases when objects reach their natural position. Aristotle was the first to explicitly ask which is the process that leads to existence (actualisation) of individual beings, objects and substances, all characterised by a specific combination of the four primitive elements, and what renders specific a determined combination. Solving this problem is the primary motivation behind Aristotle’s physics, and the focus of the first two books of its Physics [12]. The answer resides in a philosophical conception in which each thing (being, substance, object) is conceived as the indissoluble union of form and matter. The concept of form takes up Plato’s theory of ideas. Platonic ideas, just like Aristotelian forms, are the ideal essences of things. Acting on matter, they render things actual and make a thing what it is. For example, the form of the horse acting upon matter does so in such a way that a certain animal turns out to be a horse, or by analogy the form of gold ensures that such a mineral is gold, etc. The process of actualization, or the way by which form acts on matter, is rather convoluted and would subsequently become one of the most debated concepts of Aristotelian doctrine. Aristotle, in line with the theories of previous thinkers, assumes that all substances originate from a primary matter, defined as the common substrate to all bodies, on which whatsoever form can be brought to bear. This primary matter is clearly a development of the arche of the first Ionian philosophers. Primary matter is pure potentiality, upon which the different forms can be impressed, much as a sculptor can make different statues from the same block of marble, although Aristotle preferred to think of form as something evolving from within, as in the growth of organic substances. There is, however, the need to introduce a preliminary step, a future source of endless controversy. Since all things are formed by a combination of the four primitive elements, it follows that the undifferentiated primary matter must, first of all, be actualized in the primitive elements that will compose the object to be actualized. Only as a last step, can the process of actualization be completed by the introduction of the form of the object to be actualized. It follows that at the heart of the actualization process comes to be the assumption (and the associated theory) that the four elements can transform into each other and merge to form more complex substances. This process comes about through the action of the active qualities (hot and cold, moist and dry), assigned by Aristotle to the primitive elements. Each element is, in fact, characterized by two of the four qualities as illustrated in the diagram of Fig. 2.1. Fire is hot and dry, air is hot and moist, water is cold and moist, earth is cold and dry. The Aristotelian elements contained in substances are not therefore immutable; each can be transformed into any of the others through the change of one or both qualities. Fire can become air by means of heat, air manages to turn to water thanks

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Fig. 2.1 Relationship between primitive elements and qualities according to Aristotle

to humidity and so on. Finally, each form can be removed and substituted with a new form: from this concept was born the idea of the transmutation of elements, which would find its way to the heart of alchemy. Aristotelian ideas that properties depend on the balance of the four elements and the four qualities appear in all explanations of natural phenomena in successive Greek and Roman philosophers They were applied to medicine by Galen of Pergamon (129– 199 AD.) in the form of theory of the four humours (blood, phlegm, black bile and yellow bile), which the body must maintain in a state of balance if one wants to stay healthy. Illnesses are caused by the predominance of one humour and is cured by the opposite humour. This theory was equally influential in medicine as the four elements were in the conception of matter. Aristotle’s matter is a continuum which does not allow for the existence of the vacuum. The polemic against the vacuum is a cornerstone of his physics. The main argument Aristotle posits to support his thesis stems from his theory of movement. According to Aristotle, when an object is thrown, it advances in space, maintaining in part its initial velocity only because the air around it, stirred by the object launch, continues to push it forward albeit with an ever-decreasing force. If the object had been thrown in vacuum and thus not surrounded by air, it would not be able to continue to move. A second argument against the existence of vacuum is drawn from the analysis of the speed of objects thrown into different media (for example, in air and in water). Observation tells us that an object’s speed is inversely proportional to the density of the medium in which the object moves: the lower the density, the higher the velocity. It follows, according to Aristotle, that in vacuum, whose density is zero, velocity would be infinite, something which he considers manifestly absurd. The continuity of matter, however, opened up a problem, set to have a great influence in the future debate on matter’s structure, that is to say, the question of its divisibility into ever smaller parts. Aristotle solved it by introducing the concept of minima naturalia (natural minima). By this he meant that in each substance there is a minimum dimension in which substance can be divided without losing its essential characteristics. In principle a natural minimum can be further divided, but when divided it loses the characteristics of the substance in question. Thus, the minima naturalia are not indivisible like the atoms of atomistic theories; rather, they are the limit of substances’ identity. Having survived and been passed down to the western world thanks to the efforts of conservation and interpretation of ancient science on the part of the Arabs, then re-discovered in its original form by Renaissance humanists, the Aristotelian system

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dominated ‘learned’ thought from the Late Middle Ages to the sixteenth and seventeenth centuries’ scientific revolution. Regarding its concept of matter, the main reason for such longevity was due to the theory’ ability to furnish a qualitative explanation to a great many daily observations and its closeness to ongoing sensory experience. Such explanations were difficult to confute without the introduction of the quantitative experimental analysis in the new spirit brought about by the scientific revolution.

2.3 Alchemy The third strand of knowledge regarding matter, handed down from antiquity, is that coming from alchemy. In modern times alchemy has become synonymous with obscurantism, smoky symbolism and fraud. However, alchemy was not only a conglomeration of fantastical beliefs; it was also marked out by multiple contents and aspects of value. In general, it has always had a double identity: On the one hand there was a set of more or less smoky theories, often mystical in nature; on the other there were the collections of practical recipes for the preparation of chemicals and the illustration of laboratory instruments and procedures. It is now recognised that this second aspect contributed to the birth of experimental chemistry and materials technology. As such over the centuries it played an important role in the evolution of the concept of matter [13] There is general consensus that alchemy, understood as a more or less organic body of doctrines, originated in the Hellenistic period when Greek and Egyptian culture met in Egypt’s Alexandria, where the great library and museum attracted scholars and philosophers from all the known world [14]. In this great cultural crucible alchemy was born as a fusion of three distinct traditions of thought: the speculations on matter deriving from Greek philosophy, the oriental mysticism, and, thirdly, the ancient Egyptians’ empirical knowledge of the arts and technologies [8]. In Alexandria, for the first time, artisans came in contact with philosophical ideas and religious and mystical practices. Philosophical theories and religious beliefs were applied to the explanation of technical processes, while the more abstruse speculations of philosophers and priests were reined in by the comparison with the actual matter behaviour highlighted by the craftsmen. Two of the main aspects that have characterized alchemy are due to the influence of Greek philosophy. As we have seen above, the Greek philosophers, primarily Aristotle, had developed concepts of matter that could be used to clarify technical practices. From the notion that the objects originated from the mixing of a few primitive elements, the basic idea of the alchemical practice was born, according to which the substances could be, at least in principle, analysed to trace their composition. Furthermore, the belief that the four primitive elements were convertible with each other by changing the imprinted qualities had as a corollary the idea that the transmutation of substances was possible, an idea that nurtured the spread of alchemy.

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The influences of Aristotelian doctrine are clearly visible in the relationship that alchemists had with metals. Alchemistic tradition appropriated the belief that common metals are made up of impure or intermediate (immature) forms of a metallic substance which, in its pure and definitive form (mature), is gold. This belief was based on Aristotle’s treatise Meteorologica, where the ancient notion was adopted that the less perfect metals evolved slowly toward the more noble and mature forms. Such transformations took place deep inside the earth through the effect of two effluvia, one dry and inflammable, and the other, damp and vaporous, which were generated thanks to the heating by the sun. Metals, being malleable and able to melt, were the result of damp exhalation, while dry exhalation determined the formation of the non-metallic minerals [15]. The belief that the different metals were the result of different degrees of imposed qualities within the earth endowed alchemy with its central utopia: the possibility of substances’ transmutation. In the Alexandrine period alchemists arrived at the conviction that it was possible to replicate in laboratory the natural processes, or even accelerate them [16]. If a common metal could be appropriately purified and ripened, it would be transformed into gold. Consequently, alchemy’s main aim became to strip ordinary metals of their properties, returning them to primitive matter, and, subsequently, to impose the qualities of gold on this primitive matter. As alchemy progressed, the belief arose that the power to impose such qualities resided in one substance alone, the philosophers’ stone or quintessence, its discovery becoming the alchemists’ dream. Although alchemy was generally considered as mostly interested in metallic transmutations, it should be stressed that its real concern was chemical changes as a whole. A few remaining treatises afford us a clear enough picture of alchemy in the Alexandrine period. Despite a setting in which philosophical influences dominate, mixed with religious ideas and animism, the texts also report recipes and processes of eminently practical value. They bear witness to direct knowledge of techniques of the time, whether of metallurgy, dying or glass-making, and can hardly be defined as alchemical in the pejorative meaning commonly given to this word today. The texts clearly show a mastery of distillation, sublimation and filtration processes, as well as containing descriptions of apparatus such as alembics, condensers, ovens, etc. Among the manipulation processes described, in addition to those that deal with substances of more ancient tradition such as the seven metals, the texts also deal with new substances, such as compounds of zinc, antimony and arsenic [17]. It appears that sulphur was well known and widely used, even if other flammable materials were often referred to by the same name. A certain number of salts such as sodium carbonate, alum, iron sulphate and common salt were also used in quite pure form. To conclude, critical study of alchemical texts from the Alexandrine period highlighted a genuine knowledge of metallurgy and chemical manipulations supporting the thesis of those who see in ancient alchemy the embryo of chemistry and materials science. However, such re-evaluation is not meant to downplay the influence (eminently negative for modern sensibilities) exercised by the other side of alchemy, namely the mishmash of mystical beliefs, irrational practices and superstition, of which it was also made up right from its beginning. It was the inclusion and

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later the predominance of this side which would reduce alchemy to archetype of pseudo-science [5].

References 1. Bromehead C N (1954) Mining and quarrying. In: Singer C, Holmyard E J, Hall A R (ed) From Early Times to Fall of Ancient Empires. A History of Technology, vol. 1. Oxford University Press, London, pp. 567–580 2. Partington J R (1989) A Short History of Chemistry. Dover Publications, New York 3. Forbes R J (1954) Extracting, smelting, and alloying. In: Singer C, Holmyard E J, Hall A R (ed) From Early Times to Fall of Ancient Empires. A History of Technology, vol. 1. Oxford University Press, London, pp. 581–609 4. Forbes R J (1956) Metallurgy. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Mediterranean Civilizations and the Middle Ages. A History of Technology, vol. 2. Oxford University Press, London, pp. 41–82 5. Brock W H (1993) The Chemical Tree. W. W. Norton & Company, New York 6. Singer C (1959) A Short History of Scientific Ideas to 1900. Oxford University Press, London 7. Freely J (2010) Aladdin’s Lamp. Vintage Books, New York 8. Leicester H M (1971) The Historical Background of Chemistry. Dover Publications, New York 9. Sassi M M (2015) Dai Presocratici ad Aristotele. In: Eco U, Fedriga R (ed) Storia della Filosofia, vol. 1. La Biblioteca di Repubblica-L’Espresso, p. 91 10. Mason S F (1962) A History of the Sciences. Collier Books, New York 11. Geymonat L (1970) Storia del Pensiero Filosofico e Scientifico, vol. I. Garzanti, Milano 12. Lewis E (2018) Aristotle’s Physical Theory. In: Jones A, Taub L (ed) Ancient Science. The Cambridge History of Science, vol. 1. Cambridge University Press, pp. 196–214 13. Holmyard E J (1956) Alchemical equipment. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Mediterranean Civilizations and the Middle Ages. A History of Technology, vol. 2. Oxford University Press, London, pp. 743–764 14. Viano C (2018) Greco-Egyptian Alchemy. In: Jones A, Taub L (ed) Ancient Science. The Cambridge History of Science, vol. 1. Cambridge University Press, pp. 468–482 15. Clericuzio A (2013) Grecia: Scienze e Tecniche. In: Eco U. (ed) L’Antichità, vol. 8. La Biblioteca di Repubblica-L’Espresso, p. 190 16. Levere T H (2001) Transforming Matter. The Johns Hopkins University Press, Baltimore 17. Ihde A J (1984) The development of modern chemistry. Dover Publications, New York

Chapter 3

Matter at the Beginning of the Scientific Revolution

Briefly outlined in the previous chapter was the set of ideas and notions which were at the basis of the concept of matter in antiquity and which, acquired by the West through a complex transmission process, underlay the vision of nature until the Renaissance and the beginning of the scientific revolution. The Aristotelian system, adopted by the church after an initial rejection, represented the vision of the world and nature of educated people starting from the late Middle Ages. This knowledge had been rediscovered in its organic form in a long recovery process, which began in the late Middle Ages through the translations of the ancient texts previously translated and commented on in Arabic. Their conquest of Egypt in 640 AD had brought the Arabs into contact with what remained of the Greco-Hellenistic civilization. An immediate result was a far-reaching commitment, encouraged by Arab rulers, to translating Greek books into Arabic. The Christian West’s recovery of classical culture had continued through direct translations of the ancient Greek texts, made available in the period of Humanism. However, in the long process of transmission and acquisition, the body of notions on materials and substances, while remaining substantially unchanged in its general theoretical structure, had evolved and increased considerably, both from the viewpoint of critical analysis and empirical knowledge. This evolution had occurred through, roughly, the three strands that had characterized its development in classical antiquity: the strand represented by theories and debates within philosophy and science; the strand of the empirical knowledge of craftsmen who worked materials and substances; the strand represented by that branch of alchemy interested in the manipulation of chemical substances and which would slowly transform into modern chemistry.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_3

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3.1 Empirical Knowledge of Matter In the long period from antiquity to the end of the Middle Ages, artisans had continued to contribute significantly to knowledge of materials, working processes, and the transformation of substances. This contribution had become ever more relevant as gradual economic growth and the development of commerce increased demand and the spread of consumer and luxury goods, such as metal utensils, drugs, perfumes, textiles, and dyed skins, etc. It should also be remembered that many alchemical recipes derived from artisans’ practice. On the threshold of the modern era, along with purification and working of metals, craft industry was normally able to producing a variety of chemical substances such as alkalis, soaps and other detergents, acids, ceramic products, glass, pigments, dyes, combustibles, incendiary materials, explosives, drugs, liquors [1]. The number of substances known and in use had grown greatly in respect to the Ancient World. Nevertheless, except for the information scattered inside alchemical texts, written documentation had been scarce. Most of the oldest technical procedures had been handed down by oral tradition and the practice of apprenticeship. This situation started to change in the Middle Ages with the compilation of the first manuals for craftsmen. The oldest is the treatise known as Compositiones ad tingenda (Compounds for dying), its most important sections compiled in Alexandria around 600 AD and translated into Latin around 200 years later. Until the thirteenth century literary material on the practise of artisans had remained scarce. From this date on, however, the quantity of technical documents increased, leading to improved instruction and training of artisans, whose number also grew significantly. In the Renaissance, thanks to the introduction of the printing press, works on practical chemistry became widely available, in particular, treatises on distillation, preparation of perfumes, solvents, enamels, ceramics and glass, as well as descriptions of instruments and apparatus. One of the innovative aspects of the Renaissance, highly relevant for successive scientific and technological developments, was the progressive breaking down of the barrier between the wisdom and erudition to be gleaned from books and the more hands-on knowledge of artisans, a gap that in the ancient and medieval society had separated the liberal arts from the mechanical arts. On the one hand, exponents of the learned world began to cultivate an interest in the techniques and methods of the artisans, searching for explanations [2]. On the other hand, artisans and those directing manufacturing activities began to show more interest in the evolution of culture and knowledge. The secrecy of the guilds started to dissolve as artisans wrote texts that gathered and divulged their trade secrets. An important example of the new cultural attitude comes from metallurgy, whose empirical development accompanied the progress of civilisation from its beginnings. In the sixteenth and seventeenth centuries, the expansion of artillery and firearms greatly increased interest in mechanical and metallurgical processes, especially regarding the production of bronze and iron, whose consumption rose constantly due to the frequency of wars and the mechanization of armies. The spread of mining and

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metallurgical activity resulted in the discovery of new minerals and metals, such as zinc, bismuth and cobalt. Thanks to such activity scholars began to pay attention, albeit in a still embryonic way, to the special proprieties of metals and their transformations. The handling of metals became the school from which chemistry would be born. It began to develop a general comprehension of the processes of oxidization and reduction, of distillation and amalgam. The assay, essentially consisting of a small-scale melting and accurate weighing, became the most advanced field of applied science in the sixteenth century [3]. It had two main purposes: evaluating the quality of minerals, to determine the possibility of working them profitably, and evaluating coinage and jewellery to determine their purity and detect possible fraud. Having progressed via centuries of purely empirical practice, the assay became a precise method of quantitative testing and the model for the development of chemical experimentation and analysis. This new cultural approach encouraged the codification of metallurgical notions in treatises precisely describing metals, alloys and their handling techniques. A notable example is the treatise De la Pirotechnia, written in Italian by Vannoccio Biringuccio from Siena (1488–1537). Published in 1540 it was the first printed book on metallurgy. As well as laying out general rules on mineral research and exploration, on mining construction and the instruments necessary for it, Biringuccio illustrated in detail the contemporary knowledge regarding the main metals and the methods for producing steel and brass. In the same period, another important contribution came from the German scientist and mineralogist Georg Bauer (1494–1555), better known by the Latinized name Agricola and held to be the father of mineralogy. Agricola wrote numerous books on mineralogy and metallurgy, the most famous being De re metallica. In his books he described characteristics of minerals, processes transforming minerals in metals, preparation of chemical agents used in metallurgy, and analytical methods. Another writer on metallurgy, whose book was to have a more lasting impact than Agricola’s, was Bohemian metallurgist Lazarus Ercker (1530–1594), author of Treatise on minerals and assay [4].

3.2 Matter and Substances in Arab, Medieval and Renaissance Alchemy Europe had no direct knowledge of the alchemical texts of the Greco-Hellenistic period. Alchemy was transmitted to the West through the Arab world. Alchemy spread very rapidly among Muslims, who also significantly developed the empirical knowledge of substances and processes. The first Arab treatises on the subject appeared as early as around 700 A.D. The basic ideas on which Arab alchemy was founded are those already contained in the Greco-Hellenistic alchemy and can be summarised thus: The Aristotelian theory of the four elements was generally accepted. One metal’s transmutation into another was believed possible through a modification in the ratio of the primitive elements

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contained there. Gold was considered the noblest and purest of the metals. In Arab alchemy one meets for the first time in explicit form the notion of the existence of a precious substance’, variously called elixir, fifth element, quintessence or the philosopher’s stone, capable of transmuting a base metal into a noble one. It should be highlighted, moreover, that the Arab treatises clearly show how the authors had direct knowledge of the techniques of substances manipulation they described. Arab alchemists made many contributions to laboratory practices, introducing, among other things, refined distillation techniques. Furthermore, we owe to the Arabs the discovery and production of many new substances. The successes attained by Arab science in the field of chemical substances were due to the emancipation from class prejudices which had made Greek scientists tend to ignore the body of craftsmen’s knowledge. Arab philosophers and physicians were interested in the rich collection of new techniques and processes then developing, the knowledge of which was no longer confined to the artisan community, as it had been in the classical age. In this cultural context new theories regarding the composition and transformation of substances were developed and new rational methods to tackle the inherent problems were introduced. A direct and reasonably accurate picture of the knowledge of matter in the Arab world as it would be handed down to the medieval west can be obtained by briefly reviewing the contributions of three leading Arab and Persian scholars. The first is Jabir ibn Hayyan, who possibly was alchemist at Caliph Haroun alRashid’s court in the eighth century. In the west he has come to be known as Geber, and thus he was “venerated” during the long period of European alchemy [5]. He was author of a number of works containing genuine chemical insights, although many writings attributed to him deal with esoteric and occult alchemy and are probably the work of successive compilers. These works demonstrate that Jabir possessed a vast knowledge of chemical and metallurgical operations. In this regard he concerned himself, among other things, with the methods of refining metals, with preparation of steel, with distillation of acids, with glass manufacture, and with the dying of silk. The Jabirian corpus also placed important emphasis on the use of fractional distillation of substances in order to isolate the four natures (i.e. four qualities, heat, cold, dryness, and moisture) before recombining them. This subject had a huge impact in the medieval West [6]. Jabir was also one of the first to attempt a classification of inorganic substances, showing Arab alchemists’ insight. Substances are classified into three categories, based on their physical properties [7]. There are substances which, once subjected to heat or combustion, evaporate completely and which he calls spirits.1 Then there are the metals, that is, shiny substances that could melt and were malleable, and produced noise when hammered.2 Lastly come the minerals, or substances which are not malleable but shatter and are pulverized when hammered and may melt or 1

For Jabir spirits include sulphur, arsenical compounds, mercury, camphor, ammoniacal salt (ammonium chloride). 2 The metals of Jabir are gold, silver, lead, tin, copper, iron. In some of his books mercury is classified as a metal.

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not melt when heated. Minerals are numerous and, in their turn, are divided into three groups: those containing an appreciable quantity of spirit 3 ; those with a small quantity of spirit 4 ; those without hardly any spirit at all.5 This tendency to classify substances based on essentially physical proprieties is highly characteristic of the best Arab alchemists. Regarding theories of matter, Jabir’s basic ideas are those of Alexandria’s alchemists and therefore of Aristotle. He holds that every substance is the result of a precise numerical balance between the primitive elements that compose it. Furthermore, as for Aristotle, matter is characterized by the four qualities (heat, cold, dryness, and moisture), these being real natures or principles, which can become separated from material substances and can recombine in definite proportions to form new substances. The alchemist’s task is, therefore, to, determine the proportions whereby they enter into substances, to extract them in pure form and then to recombine them in the appropriate quantities to generate new desired substances. Finally, Jabir introduced the sulphur/mercury theory of metals, this later having a considerable impact on Renaissance alchemy [8]. We have seen how in Meteorologica Aristotle explained that metals are generated and evolved within the earth through the influence of two opposed exhalations or principles, one dry and inflammable, the other moist and vaporous. Jabir gave these two principles a material identity, associating the dry exhalation with sulphur and the moist exhalation with mercury. Consequently, according to his theory, for long adopted in alchemistic circles, the various metals are a combination of mercury and sulphur in different proportions. The second great name in Muslim alchemy, Abu Bakr Muhammad ibn Zakariya alRazi (860–925), Rhazes in Latin, was a Persian physician and alchemist practising in Baghdad. He was also the main Arab writer of alchemical treatises. He wrote Secretum secretorum (Secret of secrets), which despite its arcane title, is basically a practical chemistry manual. It offers a clear classification of many substances, as well as describing laboratory instruments necessary for their handling. The treatise is divided into sections on substances, apparatus and recipes. Al-Razi shows a direct knowledge of a vast number of chemical products and of many highly technical procedures. His classification of substances is more sophisticated than Jabir’s [9]. He distinguishes substances as being in four classes: mineral, vegetal, animal, and derivative. The minerals are in their turn grouped into six dominant subclasses: spirits, bodies, stones, vitriols, boraces, and salts. A glimpse at these subclasses provides a clue as to how far chemical knowledge contained in the works of alchemy had advanced. Al-Razi made alchemy more systematic and orderly. He introduced into his study of substances a practical and scientific approach rarely seen previously. The third scientist whom we must mention is the Persian Ali ibn-Sina (980– 1037), called Avicenna in Latin. A man of great culture and versatility, he exerted a considerable influence on philosophy and medicine both in the East and in the 3

Examples are malachite, lapis lazuli, turquoise and mica. Examples are shells, pearls, vitriols. 5 Examples are onyx, dust and aged vitriols. 4

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West. In the section of his Canon of medicine dedicated to pharmacology he lists as many as 760 substances, among them various narcotics such as mandragora, opium, hemlock and Indian hemp. Although his influence on alchemy has been less than that of the previous two authors, he is to be remembered in alchemical tradition as being the first to deny transmutation. Notwithstanding his theoretical scheme, which essentially reproduces the Aristotelian theory of metals, taken up and developed by Jabir, Avicenna held that the process for the formation of metals cannot be reproduced artificially. For him alchemists can at most change the apparent state of substances without, however, altering their essence [10]: I do not exclude - he says- that it is possible to reach such a level of perfection in the imitation so as to deceive the shrewdest expert, but the possibility of transmutation has never been clear to me. Rather I believe it to be impossible since there is no way to distinguish one metallic combination from another.

The first information concerning alchemy arrived in Medieval Europe with the translations of Arab treatises made in Spain, whose beginning can be dated to the fall of Toledo in 1085, during the crusades for the reconquest of Muslim Spain. The most active period for such translations was between 1125 and 1280 [2]. Spain was the main centre of contact between the Muslim and Christian worlds, for here were found Christians assimilated by the Muslim community, and Muslims assimilated by Christian communities, both bilingual, plus a large number of Jews, many of them trilingual. Although in the West alchemy was distrusted and opposed by the Church, in the medieval period the body of knowledge was extended, also thanks to respected scholars such as Albertus Magnus (1200–1280), Roger Bacon (1219–1292), Thomas Aquinas (1225–1274), Ramon Lull (1232–1316) and Arnold of Villanova (1240– 1312) [9]. In the fifteenth century, despite the aura of secrecy around its writings, alchemy consolidated its position, especially in the courts of kings and princes, and became more and more closely linked with medicine. Alchemical texts, even if unoriginal from a theoretical viewpoint, were wide-spread in various cultural contexts of Europe. Medieval alchemy, like its Arab counterpart, concerned itself with all aspects of the technology and the theory of substances. It also had close links with pharmacology, albeit the treatment and transmutation of metals continued to be its main interest. Both the alchemists and the natural philosophers of the late middle ages saw alchemy as a means to increase knowledge of the mineral world, something that the age of antiquity had left incomplete. The manuscripts of medieval alchemy show a growth in the number of processes and techniques described, in line with the increased number of chemical products on the market and the improvement of purification processes through techniques such as sublimation, distillation, solution, and crystallization. Better equipment made it possible to prepare alcoholic distillates to a high level of concentration and to produce three important acids: oil of vitriol (sulphuric acid (H 2 SO4 )), aqua fortis (nitric acid (HNO3 )) and aqua regia (a mixture of nitric acid and hydrochloric acid (HNO3 + HCl)). From the early fourteenth century alchemists were the first to conduct experiments on then recently discovered mineral acids.

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However, it is also to be noted that, despite major advances, alchemical practices and recipes remained eminently qualitative [5]: Whoever examines the equipment in an alchemist’s laboratory cannot fail to note the absence or scant importance of balance. The fact that alchemists attached little value to quantitative determination is no surprise if one considers that the idea of the existence of the chemical elements in the pure state still lay in the future.

In the Middle Ages alchemy thus covered a vast range of activities, taking in every branch of chemical and mining technology and substantially improving the empirical knowledge of matter. Furthermore, it is important to stress how observations were frequently made to fit into the context of a theoretical scheme. In fact, the characteristic structure (although not exclusively) of a medieval alchemical treatise consisted not only of descriptions of practical recipes but also of a theoretical section usually containing a recitation of Aristotle’s theory of elements. While in the Medieval period alchemy was rooted in the Aristotelian system, Renaissance alchemy introduced new notions derived from the revival of doctrines linked to philosophical traditions significantly different from Aristotelianism, Neoplatonism in particular. These doctrines prepared the way for an animistic conception of alchemy, with strong cosmological and religious elements, to be adopted in the sixteenth century by many Renaissance alchemists and physicians. An important novelty of the period was the rise of a new school of alchemy, iatrochemistry, a medical doctrine, which places chemical substances, natural or synthetic, at the basis of therapy aimed at restoring the body balance altered by disease. The iatrochemists held that alchemy should be mainly applied to the preparation of medicines and to better understanding the processes of living organisms. Iatrochemistry’s founder was Theophrastus Bombastes von Hohenheim (1493– 1541), better known as Paracelsus, a revolutionary and iconoclast. He unleashed a revolution in alchemy and medicine, which for him are closely related, with the aim of overthrowing both traditional medicine and alchemy. Paracelsus saw the processes of nature as essentially alchemical [11]: The growth of animals and plants, the ripening of fruit and vegetables, the processes of fermentation in making wine and beer, the digestion of foods, indeed all natural processes involving transmutation, growth and development are alchemical. Nature is an alchemist and God, who rule nature, is the supreme alchemist.

Paracelsus believed in the philosopher’s stone and in the elixir of life. He also believed in Aristotle’s four elements, but he thought that they manifested themselves in bodies as the tria prima (three principles): salt, sulphur and mercury. Salt is the fixity and non-combustibility principle, mercury is the fusibility and volatility principle, sulphur the inflammability principle.6 To justify the tria prima theory, Paracelsus used the example of the destructive distillation of wood, already used by Aristotle [8]: The smoke produced by combustion is the manifestation of the volatile part, mercury; the light and flash demonstrate the presence of sulphur; ash, 6

It is not entirely clear what the relationship between the tria prima and the Aristotelian four elements was, nor what form and matter became in the Paracelsian scheme.

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incombustible and non-volatile, is the residue due to the presence of salt. According to Paracelsus the alchemical transformations of all bodies can be explained on the basis of tria prima, this constituting a material trinity corresponding to the three constitutive principles of man (vital spirit, soul and body) and to the divine trinity (Father, Son and Holy Ghost). Paracelsus’s figure was much discussed by both his contemporaries and those who came after him. It is necessary to emphasize, however, that his theories had an important role in the development of the concept of the “purity” of substances at the basis of chemistry and material science. According to Paracelsus, alchemy’s task was to remove impurities, converting impure substances into pure ones, separating beneficent substances from harmful ones, converting useless or maleficent substances into healthy and medicinal ones. Paracelsus and his followers believed that [5]: the more the extracts for medicine were free of contaminating substances, the more effective they would be. This led to the development of chemical assays and specifications of purity, clearing the way for the concept of the individuality of chemical substances, a concept to be slowly but surely consolidated later by the works of Boyle, van Helmont, Black, Lavoisier and Dalton.

3.3 Matter and Substances in the Philosophical and Scholarly Tradition As far as philosophical reflection on matter and substances is concerned, it must be borne in mind that, in the long period from the beginning of Christianity to the late Middle Ages, the study of nature had a marginal interest in the speculations of the philosophical schools. Official philosophical knowledge, oriented by the dominant religions, focused principally on problems such as the nature and knowledge of God, the purpose of the universe, the nature of man, the meaning of human knowledge as compared to the revealed truth. However, with the revival of Aristotelianism in the late Middle Ages, the nature of philosophical speculation began to change. Aristotelianism was a complete encyclopaedia of learning, also including the sciences of nature, which, consequently, assumed a recognised place in the philosophic research. The science of nature taught and practiced in medieval universities coincided with what Aristotle had defined as natural philosophy.7 As philosophy’s attention to nature grew, the commentaries on Aristotle’s natural philosophy multiplied. This is particularly evident in the effort to assimilate the Aristotelian system on the part of scholastic philosophy. The renewed interest in nature of medieval philosophers led to a critical analysis of the Aristotelian concept of matter and, at the same time, to its use in an attempt to bring order to the growing number of discovered substances and observed phenomena. It was a question of explaining the mutations induced in matter by heating and combustion and of identifying the true transmutations of substances, a 7

In Aristotle’s definition, natural philosophy encompassed the study of all aspects of nature and included the fields that are today the subject of physics, chemistry, natural sciences, and biology.

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problem of paramount significance for alchemical beliefs. Starting from the thirteenth and fourteenth centuries, at the center of debate was the Aristotelian doctrine of the actualization of substances as an indissoluble union of form and matter. The actualization mechanism had been the subject of controversy since its inception. Among the aspects most debated it was the thorny problem of the mixts (mixed substances), namely whether a chemical compound (for example, a metal alloy produced by combining two molten metals) still contained the forms of the component substances and the forms of the primitive four elements and what was the relationship among all these substantial forms. From a philosophical point of view, the problem of the mixts was a particular aspect of a broader problem, deemed of extreme importance since the inception of the Greek philosophy: the continuity and divisibility of matter. As we have seen, Aristotle had tried to solve the problem of divisibility into ever smaller parts by introducing the concept of natural minimum, meaning that in every substance there is a minimum dimension in which the substance itself can be divided without losing its essential characteristics. The natural minimum did not have a physical existence, but was a limit to the uniqueness of a substance and the subsistence of its form. In the Middle Ages, solving this type of problems (and also more philosophically complex versions) was considered to be crucial as proof of the validity of Aristotelian theories of matter. As a consequence, these issues were tackled by the most famous philosophers and scholars of the time,8 who not only gave different and often conflicting answers, but also highlighted how, within the Aristotelian principles, it was impossible to find criteria for establishing which was the correct solution [12]. In the Renaissance, the Aristotelian philosophers attempted to resolve the state of crisis and great uncertainty regarding the entire theoretical scheme, taking a path which would pave the way for the introduction of corpuscular conceptions, something traditionally foreign to Aristotelianism. First came the hypothesis that natural minima have a real existence. According to these theories,9 when two substances react chemically, it is their respective natural minima which react and thus the natural minima of the compound are formed [13]. It was further hypothesized that the natural minima of the different substances have different dimensions and that their mutual collisions and subsequent aggregations give rise to the chemical composition. With the introduction these further hypotheses the theory of natural minima became in practice indistinguishable from the atomistic and corpuscular theories, which in the meantime were being rediscovered. The final step of interpreting natural minima in decidedly atomistic terms would be made by Daniel Sennert (1572–1637) in the early seventeenth century. In the Renaissance, when the awareness was growing as to the inadequacy of the dominant Aristotelian theories, some philosophers and scholars began to use 8

Among them are Albertus Magnus (1200–1280), Avicenna (980–1037), Averroes (1126–1198), Roger Bacon (1219–1292), Thomas Aquinas (1225–1274), John Duns Scotus (1266–1308), Albert of Saxony (1320–1390). 9 These theories were held by Agostino Nifo (1469–1538), Giulio Cesare Scaligero (1484–1558), Franciscus Toletus (1532–1596).

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hypotheses explicitly atomistic or corpuscular for interpreting complex experimental data. These were alternative conceptions of matter based on the existence, at a microscopic scale, of atoms or corpuscles, from whose characteristics and aggregations originate observed macroscopic phenomena and the properties of bodies. Such theories would be strengthened in the following century, as result of the diffusion of the mechanism ideas in the wake of the success of mechanics in explaining motion, both on earth and in the heavens. Already in the Middle Ages there had been philosophers and scholars who had proposed explanations of material phenomena in atomistic and corpuscular terms, both in the Arab world10 and in the Christian West.11 Such voices, however, had remained marginal. The veritable rebirth of atomistic theories occurred during Humanism. The process started with the translation into Latin of the Lives of the Philosophers of Diogenes Laertius (the principal source of knowledge of Epicurus’s philosophy) by Ambrogio Traversari (1386–1439) and the rediscovery of Lucretius’s De rerum natura (a poem highly influential during the Renaissance) by Poggio Bracciolini (1380–1459). Through these works Renaissance philosophers acquired the awareness of a possible explanation for natural phenomena antithetical to the Aristotelian one. As a consequence, atomistic theories were introduced into many cultural circles of the time. In the philosophical field atomism was supported explicitly by Giordano Bruno (1548–1600), for whom all bodies are composed of atoms, indivisible and indestructible. In the medical field, corpuscular theories emerged in the discussions on the causes of infection. For examples, Girolamo Fracastoro (1478–1550) claimed that infection was spread by seeds, which he viewed as microscopic corpuscles. In the field of metallurgy, the metallurgist Vannoccio Biringuccio (1488–1537) interpreted metals’ chemical-physical proprieties in corpuscular terms. His theories were a compromise between the Aristotelian doctrine of elements and the atomistic conception. He conceived of the four primitive elements as ensembles of special particles, qualitatively determined, whose aggregation in various proportions give rise to the proprieties of the different metals.

References 1. Taylor E S, Singer C (1956) Pre-Scientific industrial chemistry. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Mediterranean Civilizations and the Middle Ages. A History of Technology, vol. 2. Oxford University Press, London, pp. 352–388 2. Mason S F (1962) A History of the Sciences. Collier Books, New York 3. Forbes R J, Smith C S (1957) (1956) Metallurgy and assaying. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) From Renaissance to Industrial Revolution. A History of Technology, vol. 3. Oxford University Press, London, p 63 4. Singer C (1959) A Short History of Scientific Ideas to 1900. Oxford University Press, London 10

Mu’tazili (eighth century) and al-Razi (860-925). ] Nicolas d’Autrecourt (about 1300 -1350), a follower of William of Ockham and a critic of Aristotle’s philosophy.

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5. Holmyard E J (1956) Alchemical equipment. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Mediterranean Civilizations and the Middle Ages. A History of Technology, vol. 2. Oxford University Press, London, pp. 743–764 6. Newman W R (2013) Medieval Alchemy. In: Lindberg D C, Shank M H (ed) Medieval Science. The Cambridge History of Science, vol. 2. Cambridge University Press, pp. 385–403 7. Leicester H M (1971) The Historical Background of Chemistry. Dover Publications, New York 8. Brock W H (1993) The Chemical Tree. W. W. Norton & Company, New York 9. Ihde A J (1984) The development of modern chemistry. Dover Publications, New York 10. Bernardoni A (2014) Medioevo Centrale. In: Eco U. (ed) Il Medioevo, vol. 5. La Biblioteca di Repubblica-L’Espresso, p. 221 11. Levere T H (2001) Transforming Matter. The Johns Hopkins University Press, Baltimore 12. Joy L S (2006) Scientific Explanation from Formal Causes to Laws of Nature. In: Park K, Daston L (ed) Early Modern Science. The Cambridge History of Science, vol. 3. Cambridge University Press, pp. 82–105 13. Clericuzio A (2012) Il Cinquecento. In: Eco U. (ed) L’Età Moderna e Contemporanea, vol. 3. La Biblioteca di Repubblica-L’Espresso, p. 261

Chapter 4

Matter in the Seventeenth Century

In the seventeenth century the cultural change, associated with the scientific revolution begun in the second half of the sixteenth century, was fully under way. It took the form of a new approach to scientific problems, based on the revaluation of empirical research and experimentation and on growing predominance of practice over theory. Attempts were made to introduce a mathematical formulation of scientific theories. Past theories were subjected to critical examination, which would lead to the collapse of the Aristotelian edifice. The period saw a considerable increase in the number of those dedicating themselves to scientific activities as well as of scientific and technical publications. The practice of publishing detailed descriptions of experiments and images of the equipment became widespread, so allowing the reader to repeat or check them. Science and technology assumed a growing influence in the economy, politics and intellectual life of most European countries. A complex phase of transformation began, in which technical progress was put to use in scientific experiments. This in turn resulted in increased technical progress, favouring, as seldom before, the expansion of scientific research linked to practical applications, such as metallurgy, cartography, ship building, artillery, the science of water and practical chemistry. The period includes the first great breakthroughs in scientific observation and experimentation. Among these are: Copernicus’s new conception of the solar system and its definitive affirmation by Galileo, despite the Church’s condemnation; the development and applications of the telescope and microscope, vastly extending the field of observation; Gilbert’s description of the terrestrial magnetic field; Harvey’s discovery of circulation of the blood. Regarding the properties of matter, two parallel strands began to appear, albeit often intersecting, and which, using modern classification, we could define as chemical and physical investigations. The first tied in with the slow but continuous evolution of the alchemical practices towards more rigorous procedures of analysis and manipulation of substances. A greater attention was paid to reactions’ quantitative aspects, to the reproducibility of results and to the explanation of experiments. Such an evolution also initiated dismantling the legacies of ancient and medieval

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philosophy and old-style alchemy, so giving birth to chemistry as a scientific discipline. The second strand occurred within the context of seventeenth century natural philosophy1 and was closely linked to new problems and techniques, born from advances in mechanics and growing interest in new phenomena, such as electricity and magnetism, and novel instrumentation.

4.1 Alchemical and Chemical Quests in the First Half of Century In the early seventeenth century, in contrast with breakthroughs arising from applying the scientific method in mechanics and astronomy, the application of the new methodology into matter’s chemical properties was decidedly slower. Thanks to the practical activities and chemical processes reported in technical literature and alchemical texts, a mass of data was available as well as a sound empirical knowledge regarding the properties of metals, minerals and substances in general. However, there remained the problem of classifying this vast empirical knowledge in a satisfactory way and explaining it convincingly. The complexity of most substances and chemical processes made any attempt at theorisation extremely difficult. In this period there was still no perception of the difference between inorganic and organic compounds, substances of animal and vegetal origin and minerals being put on the same level. Nor was there any clear awareness of difference between simple and compound substances, except for the vague and qualitative way surmised from theories linked to the Aristotle’s four elements or Paracelsus’s three principles. Then there was the problem of understanding what happened when substances were treated using the most familiar processes, such as combustion, heating, chemical reaction, solution in liquids. Very often the changes observed were still interpreted as proof of the transmutation of substances. Finally, a universally agreed terminology was lacking [1]: in alchemy’s long tradition different substances were often confused one with the other; conversely identical substances went by diverse names and were considered different. As the century went on this situation changed. As old theories were being criticised, focus was brought to bear on more specific problems, leading to a definite identification of substances, resulting in improved understanding of their characteristics and classification. With the growing influence of mechanical philosophy, one sees a more radical revision of research themes and problems to be tackled. More attention was paid to the nature of reactions and the definition of substances. Meanwhile there was a decline of experiments looking for transmutation, a topic that was losing its credibility albeit most philosophical chemists of the time did not exclude its possibility a priori. In short, alchemy was gradually becoming chemistry.

1

In the Renaissance and in the seventeenth century natural philosophy mainly dealt with the problems of physics.

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Among the most representative authors of this renewal we find German physician Daniel Sennert (1572–1637), physician Sebastien Basson from Lorraine (1573died after 1625), French physician Jean Rey (1583–1645), and, above all, Flemish physician Joan Baptista van Helmont (1579–1644). It is necessary to note, however, that these like many other contemporary scientists, were still influenced by alchemical tradition, especially when they sought to draw up theoretical explanations for observed phenomena. A brief overview of van Helmont’s activities and theories pictures precisely the conceptions and problems of the period. Van Helmont was an iatrochemist, i.e. a physician practitioner of alchemy in the Paracelsian sense. He rejected the Aristotelian theory of the four elements and that of Paracelsus’s three principles. Nevertheless, reasoning along the old lines, he still held that there must be a primary substance and that this was water, since the waters had been mentioned in the Bible as the primordial chaos preceding the rest of the Creation [2]. This said, van Helmont believed that this opinion of his was also supported by a series of experiments he had made. In the most famous one, he monitored the growth of a weeping willow over five years, weighing the soil in its container before and after. His conclusion was that the willow’s growth and increase in weight were simply due to water, in that water was the only substance added and there were no observable changes in the soil in the container. Later on, faced with the practical impossibility switching back air to water, he also accepted air as a primary substance, concluding that all substances originated from water and air. Despite these old-fashioned ideas, basically in line with the traditional approach whereby the composition of different substance was believed to be a mixture of a few primitive elements, van Helmont made two important contributions to the transition from alchemy to chemistry. The first is linked to the eminently quantitative nature of his activity, in contrast with the prevalent qualitative approach of the time. According to Partington [3]: He made extensive use of the balance and expressed clearly the law of indestructibility of matter.

The second important contribution involves his pioneering research into gaseous substances. Van Helmont was the first to clearly recognize that many chemical processes produce aeriform substances different from air, namely gases. Thus, he was the precursor of a new research sector, that of gaseous substances, later to prove crucial in promoting the eighteenth century’s chemical revolution. His interest had been aroused by the observation that often, during chemical reactions, substances behaving like air were released. He named them effluvia or spirits. For example, describing a typical experiment, he recounts heating 62 lb. of charcoal in air until he was left with 1 lb. of ash, all the rest having disappeared as spiritus sylvester (literally wild spirit, today’s carbon dioxide). When, however, charcoal was heated in a sealed vessel, combustion would either not occur, or would occur with violence as the spirit escaped from the exploding vessel. These disruptive experiments led van Helmont to coin the word gas, meaning chaos in Flemish, and to his definition of gas [1]:

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It is clear from his writings that van Helmont also obtained various other gases, including an inflammable gas (today’s methane) evolved in putrefaction, which he called gas pingue ( fatty gas) [4]. Van Helmont was therefore the first to perceive clearly the existence of aeriform substances other than air. Note here that natural philosophers and alchemists of the time, like their predecessors, believed there existed only one type of aeriform substance, air,2 and that all gaseous fluids evidenced were only air that had been contaminated by differing levels of impurities. It was the impurities that gave the air the characteristic of being poisonous or odorous or flammable or coloured. Furthermore, following the alchemical tradition which posited that bodies were composed of matter and spirit (Aristotle’s form), it was believed that the effluvium, emitted at times upon heating the body, was nothing other than the spirit which escaped and could therefore be isolated through condensation. It was on the basis of similar beliefs that van Helmont had undertaken to isolate these spirits, not only using traditional pyrotechnic methods, but also by exploiting the action of acids on substances. Those he called gases were spirits that escaped condensation. Since the spirits of substances were specific, he believed that the gases differed one from another according to their origin, and that they differed from air and condensable vapours. This was a highly innovative way of thinking, to which, however, his contemporaries paid little attention, since at the time it was really impossible to draw a distinction between these aeriform emissions from a chemical viewpoint, due to a lack of instrumentation to gather and analyse them.

4.2 Matter and Physical Properties: Air and Vacuum In the seventeenth century the resurgence of atomistic and corpuscular theories and the increasing doubts on the validity of the Aristotelian system prompted a heated debate on one of the crucial problems underlying both: the existence of empty space, the vacuum (or void). This debate was strictly linked to that concerning the physical proprieties of air, these at the time being the object of considerable experimental investigation.3 A vision antithetical on the existence of an empty space had characterized philosophical theories right from their original formulation in ancient Greece. For Democritus the vacuum was essential for allowing atoms’ movement and combination. Vice-versa, for Aristotle the polemic against the void was one of the cardinal points of his philosophical system. This dichotomy reflected the clash between the concept of matter composed of discrete entities and that of matter as a continuum. 2

A fortiori, it was not suspected that air was a complex mixture of different gases. Chemical properties and composition of the air would only become the subject of study in the eighteenth century.

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Centuries of debate on the subject had shown how theoretical discussion was unable to resolve the problem. The answer could only come from experiments, one of the achievements of the seventeenth century scientific revolution being to have strongly affirmed such a concept. The first convincing direct experimental evidence for the vacuum, understood as absence of air in a vessel, was the result of experimentation on air’s weight, in the context of fluid mechanics’ development then taking place. Air’s nature had been discussed from ancient times along with whether it had weight or not. In the Middle Ages alchemists and philosophers generally believed that it had none. They often went as far as to attribute to it a negative weight, since, like fire, it tended to rise upward [5]. A further problem was its relation with the primitive element air present in the Aristotelian theory of the four elements, still widely shared in the early seventeenth century. The first famous experiment to show unequivocally how air in effect does have a weight and how its removal gives rise to a void was conducted in 1644 by Italian mathematician and physicist Evangelista Torricelli (1608–1647). Torricelli had been Galileo’s assistant during the latter’s final months of life. Galileo was intrigued by the observation that, when water was sucked from a well via a pipe, its height did not exceed nine metres. He had mentioned this problem to Torricelli4 [6]. Analysing the evidence available, Torricelli concluded that the phenomenon was attributable to the weight of the atmospheric air. It was the weight of the air on the water’s surface which produced the pressure capable of pushing the water in the pipe upwards. This was only possible as long as the pressure of the water column in the pipe remained less than the pressure exerted by the atmospheric air. Then the maximum height of water in the tube matched the atmospheric pressure. To verify this hypothesis Torricelli conceived of his famous experiment. In it he used mercury, a liquid much heavier than water, so as to equal the atmospheric pressure with a column much shorter and more manageable than the one needed with water. As described in all elementary physics texts, Torricelli, after filling with mercury a one-meter-long tube sealed at one end, immersed the other end in a basin also filled with mercury. After opening the immersed end, he observed that the mercury in the tube dropped to a height of around 760 mm, above the surface of mercury in the basin (Fig. 4.1). From such experimental evidence Torricelli drew two important conclusions. The first confirmed the hypothesis that the mercury of the tube could not drop because of the pressure of air exerted on the free surface of the mercury in the basin. Therefore, this pressure must be equal to that exerted by the weight of mercury in the tube. Consequently, not only did air have a weight, but he had hit upon a simple way to measure it, a discovery of incalculable practical importance. The second conclusion was that the space left free of mercury at the tube’s closed top was deprived of air and

4

It is said that, following the failure of his technicians in lifting water from a depth of 15 m by means of a suction pump (experience showed a lifting limit of about 9 m), Cosimo II dei Medici, Grand Duke of Tuscany, turned to Galileo for explanations.

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Fig. 4.1 Sketch of Torricelli’s apparatus

thus void. Torricelli’s results were confirmed soon afterwards by Blaise Pascal (1623– 1677), who in 1647 published the results of his own first barometric experiments in Expériences nouvelles touchant le vide (New experiments on the void). Here he stated explicitly that the tube’s upper part is empty, that the vacuum is quite possible in nature and that so-called horror vacui (abhorrence of the vacuum) is nothing but the tendency of fluids which surround empty spaces to fill them.

4.2.1 Experiments with Vacuum In the mid-seventeenth century, in the triumphant spirit of the scientific revolution, the ever-increasing list of new instruments included the vacuum pump, an apparatus able to reduce the quantity of air in a sealed vessel. This invention opened the way to a new experimental methodology, contributing greatly to knowledge regarding the properties of matter, and, more generally, to science and technology with effects felt down to this day. In its time the vacuum pump enabled researchers to study the air’s properties and conduct sophisticated experiments on combustion and respiration. The first vacuum pump was invented by German physicist Otto von Guericke (1602–1686), burgomaster of Magdeburg. It was essentially a cylinder fitted with

4.2 Matter and Physical Properties: Air and Vacuum

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a piston, attached to a container (Fig. 4.2). By extracting the piston it was possible to remove part of the air contained in the vessel. The seal was far from perfect and the extraction of air very laborious. The maximum void obtainable corresponded to pressures of the order of tens of millimetres of mercury (mm-Hg),5 namely to the reduction of air in the container to a value about 80 times smaller than that corresponding to atmospheric pressure. In 1654 von Guericke demonstrated how his pump functioned to the dignitaries of the Holy Roman empire at the Diet of Ratisbon, conducting the famous Magdeburg hemispheres experiment, dramatically proving the force of atmospheric pressure and winning himself fame as a result. In the experiment he joined together two metal hemispheres, about 50 cm in diameter, equipped with leather gaskets, and extracted the internal air thanks to the pump. As the air was sucked out, the pressure of the external air, no longer counterbalanced, made the two hemispheres stick together. The force exerted by the pressure was such that not even 16 horses (8 per part) succeeded in detaching them. The two hemispheres separated on their own as soon as air was let in again through a valve. English chemist and physicist Robert Boyle (1627–1691) had the merit of being the first scientist to use the vacuum pump for a systematic study of air’s properties. Boyle made important contributions to the knowledge of matter and science’s evolution from traditional alchemy to modern chemistry. Between 1658 and 1659, after learning of von Guericke’s invention, Boyle collaborated with Robert Hooke (1636–1703), to construct his own much improved vacuum pump, which allowed to remove air with relative ease. The apparatus enabled him to carry out sophisticated experiments, initiating a new phase of experimental studies which would become a model of new scientific practice. Studying the behaviour of air pressure as a function of volume occupied, Boyle showed that air is an elastic fluid and established the law that bears his name (Boyle’s law or also Boyle-Mariotte’s law). This states that the pressure p of a determined quantity of air is inversely proportional to the volume V it occupies: the smaller the volume the larger the pressure and vice-versa (i.e.: pV = constant). The law was first spelled out in 1662 in the publication of A Defence of the Doctrine Touching the Spring and Weight of the Air. In 1676 the law was reformulated more precisely by French physicist Edme Mariotte (1620–1684), who, confirming Boyle’s data, specified a very important aspect, how this is valid only if the temperature of the air remains constant. The law is the first quantitative formulation of a physical propriety valid for all gases at low density and high temperature. These would be later known as perfect gases.6 As we shall see, the capacity to explain Boyle’s law quantitatively would constitute a key element in the kinetic theory of gases and the corpuscular interpretation of matter.

5

As shown by Torricelli, atmospheric pressure corresponds to about 760 mm-Hg. It should be borne in mind that in Boyle’s time, despite the observations made previously by van Helmont, there was actually no awareness of the existence of gases other than air, and the term gas was not yet in use. 6

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Fig. 4.2 Drawings of von Guericke’s vacuum pumps. Above, his first pump with a wooden barrel as a container. Below, his second pump with a copper sphere as a container. Reproduced from Ottonis de Guericke, Experimenta nova, Courtesy of The Linda Hall Library of Science, Engineering and Technology

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4.2.2 Combustion, Respiration and Calcination. The Complex Nature of Air The availability of the first vacuum pumps in the second half of the 1600 s, made it possible to study air ‘s role in essential phenomena like combustion and respiration, and to demonstrate, for the first time, how air was not a simple substance but a complex mixture. We need to remember, as stated previously, that the natural philosophers and the alchemists of the time believed that there was only one aeriform substance, air. Gaseous fluids, for which there was ample evidence, were nothing other than air more or less contaminated by impurities. These first experiments can be seen as a prelude for the development of physics and chemistry of gases, leading in the next century (eighteenth century) to the discovery of various elementary gases such as hydrogen, oxygen, nitrogen, and carbon dioxide and to understanding oxygen’s role in combustion and respiration, each a successful and fundamental step for understanding the nature of matter. Pioneers of such research in the seventeenth century were Boyle, Hooke and the English physician John Mayow (1641–1679), the first to conduct quantitative studies into air’s role in combustion and respiration. Typical experiments consisted of studying the combustion of candles or inflammable substances placed in containers where a void has been produced and in analysing the conditions under which flames were extinguished. From observation that, without air, no flame developed, only for it to reappear as soon as new air was introduced into the evacuated chamber, it was proved that air is needed to produce and maintain combustion. By analogy, by observing what happened to small animals inside evacuated vessels, it was also demonstrated how the presence of air was essential for respiration. Successive more precise experiments showed that, in reality, combustion was extinguished when a large quantity of air still remained inside the container. It was observed that this residual air was no longer able to support combustion. Other experiments, where a mouse was inserted into the closed recipient together with a lighted candle, showed that, as with combustion, respiration also used up only a fraction of air. Accounts of these experiments and their inherent theories were published in Boyle’s 1672 work New experiment touching the relation betwixt flame and air, in Hooke’s great work Micrographia e in Mayow’s Tractatus quinque medico-physici, published in Oxford in 1674. The observation that only part of the air was used in combustion and respiration processes was highly relevant: It provided the first evidence that air is not a simple substance, but a mixture of different fluids. The phenomenon was well described by Hooke in his Micrographia (1665) [5], where he stated that air is the universal solvent of all sulphurous bodies (i.e. bodies that can burn), their dissolution determined by a substance inherent in air and mixed with it. It is this action of dissolution that causes fire. Mayow, like Boyle and Hooke, concluded that air must contain two types of component substances: One type, which was removed during the combustion and respiration processes, that he called spiritus nitro-aereus; a second type, which was unable to sustain either combustion or respiration and remained after the process had

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ended, occupying a reduced volume. According to Mayow all combustible bodies contain sulphurous particles and the heat produced in combustion is due to the violent collision between sulphurous particles and those contained in the spiritus nitro-aereus [3]. Hooke’s substance inherent in aria and mixed with it and Mayow’s spiritus nitro-aereus were nothing but oxygen. In this sense Boyle, Hooke and Mayow got very close to discovering it. However, they were unable to identify what this new substance was, as there were still no convenient methods for preparing and analysing gases nor an awareness of their existence. Awareness of the existence of oxygen would only come more than a century later, thanks to Priestley and Lavoisier. Alongside combustion and respiration, another specific research field attracting seventeenth century scholars’ attention, and which would prove of great importance for the understanding of matter, was the calcination of metals. From the times of ancient alchemy, the term calcination indicated a process of heating solid substances at high temperature, protracted for the time necessary to eliminate all the volatile substances. Regarding metals, it was known since ancient times that they, gold and silver excepted, underwent notable changes when heated in an open furnace. The process determines the formation of a residue, called calx (from Latin calx, meaning lime).7 In the sixteenth century, when more attention was paid to understanding the phenomenon and what distinguished a metal from its calx, it was noted that the calx was heavier than the metal whence it came, a fact difficult to explain. The explanations usually given were very vague. For example, an aereus spirit escaped from the metal or the metallic matter grew denser or some kind of acid was absorbed from the fire or the fire itself had a weight which it transferred to the metal in forming the calx. In the seventeenth century a highly accurate study of calcination was made by French doctor Jean Rey (1583–1645) and illustrated in his Essays de Jean Rey sur la Recerche de la cause pour laquelle l’Estain et le Plomb augmentent de poids quand on les calcine, published in 1630. In his essay Rey advanced a theory for calcination incredibly precise for his times. Having observed that the weight of lead and tin increased when calcinated, he attributed the increase to the weight of air incorporated in metals during heating. Rey’s explanation contained two important novelties. The first was that air had a weight. It is claimed that Rey’s hypothesis, which father Mersenne brought to the attention of scientific community of the day (including Torricelli, Galileo, Decartes, Pascal), helped stimulate Torricelli and Pascal’s research into the subject [7]. The second novelty was that calcination implied air’s incorporation in metals. This hypothesis, largely ignored by contemporaries, would be confirmed by Lavoisier over a century later. In the second half of the seventeenth century, both Boyle and Mayow conducted precise studies into the calcination of metals. Boyle presented his results, acquired by experimenting mainly with tin and mercury, in the text New experiments to make fire and flame stable and ponderable of 1673. He accurately described the weight increase of the residue of calcinated metals. However, he misjudged the role of air, believing that the effect was due to fire’s igneous corpuscles, which, passing through 7

Today we know that this is the oxide of the calcined metal.

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the glass, were absorbed by the metal. From this he concluded that fire had a weight. On this point Boyle’s opinions are clearly behind those of Rey. Mayow’s view (1641–1679) regarding the phenomenon was, instead, essentially correct. In his Tractatus quinque medico-physici he noted that when some metallic antimony in dust was calcinated on a marble slab by means of slow burning, the calx was heavier than the original metal, despite the observable escape of abundant fumes. Mayow stated that the particles of spiritus nitro-aereus of atmospheric air fixed themselves in the metal.8

4.3 Mechanical Philosophy and Atomism in the Seventeenth Century The atomistic-corpuscular concept of matter, which had begun to spread in the late sixteenth century, was strengthened by the triumph of seventeenth century mechanical philosophy. In interpreting the properties of matter the idea took hold that they resulted from the dynamics of the particles which it was made from. Thus, the first half of the century saw various interpretations of phenomena based on atomistic and corpuscular concepts. German doctor Daniel Sennert (1572–1637) drew up a theory that can be defined as a compromise between atomism and Aristotelianism. He reformulated in atomistic terms the theory of natural minima, asserting that the Aristotelian four elements cannot be infinitely divided; hence they are reducible into minimal parts which he identified with atoms. Sennert postulated the existence of four types of atoms corresponding to the four Aristotelian elements [8]. His atomism can be rightly defined as a transition from Aristotelian conceptions to atomism. A radical confutation of the Aristotelian theory of forms and elements was contained in Philosophia naturalis adversus Aristotelem published in 1621 by the Lorraine physician Sebastien Basson (1573-died after 1625). Basson adopted an atomistic conception whereby the local movement of atoms, their position, aggregation or disaggregation, all accounted for the phenomena that the Aristotelians believed resulted from the actions of forms. Italian physician and chemist Angelo Sala (1575–1637) explained fermentation as a reaggregation of elementary particles triggering the formation of new substances. More attentive and detailed examination of the single properties of matter, and the efforts to construct corpuscular models to explain them, spawned the problem of identifying which property was intrinsic to matter and which, instead, resulted from the reaction of man’s senses. In response to such problems seventeenth century natural philosophers, in particular Galileo, developed the doctrine of subjectivity of sensory qualities, thus dividing qualities of material bodies and substances into two categories, primary qualities, belonging strictly to matter, and secondary qualities, resulting from the reaction of our senses. This distinction was strongly supported by Galileo, who affirmed, more clearly than anyone thereto, that the objective and 8

Remember that Mayow’s spiritus nitro-aereus was none other than oxygen.

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real qualities of bodies, the only ones susceptible to mathematical treatment and thus truly knowable, are those of quantitative character, namely shape, size, position and movement (primary qualities). All the others, colour, odour, taste, etc. (secondary qualities) are not attributes intrinsic to matter, but result from interaction with our senses. It follows that for Galileo and the new science’s supporters matter is reduced to an aggregate of corpuscles in movement, its properties being strictly geometricmechanical, and that experiments on matter should be limited to studying primary qualities of shape, size and movement [9]. In the second half of the seventeenth century mechanical and corpuscular philosophy acquired a growing influence and was taken up by the major scientists of the time. There was no longer any conceptual difficulty in accepting that substances and bodies could be an aggregate of minute corpuscles, albeit for now destined to remain invisible. Corpuscular theories replaced the explanations of the proprieties of substances in Aristotelian terms of forms e qualities. Without doubt the development of the microscope by Robert Hooke and others helped stimulate the imagination into envisaging a world of infinitely small things. Francis Bacon (1561–1626) had said that [1]: If only Democritus had a microscope, he would perhaps have leaped for joy, thinking a way was now discovered for discerning the atom.

4.3.1 Atomism of Gassendi, Boyle and Newton In the mid-seventeenth century a notable contribution in diffusing the atomism was made by the French Abbot Pierre Gassendi (1592–1655). He showed how this can provide an innovative coherent mechanical model for depicting matter’s macroscopic properties at microscopic level. Gassendi’s atomistic hypotheses matched perfectly the mathematical-corpuscular tendency of the age. According to Geymonat [10]: Gassendi, restoring atomistic philosophy, wanted to show how this could meet the demands of the new scientific spirit, offering a working hypothesis able to explain recent experiments.

In his work rehabilitating atomism, Gassendi strove to Christianize Epicurean philosophy, eliminating aspects traditionally believed incompatible with Christian principles. In particular he denied that atoms were eternal and infinite in number, affirming that they were created by God and that their motion through the vacuum was not random but the result of divinely-imparted initial conditions [11]. In line with the new mechanics, Gassendi’s atoms were material corpuscles, characterized by size, shape and weight. To explain the vast variety of natural bodies and substances, he hypothesised a vast number of different types of atoms. Gassendi took up the thinking of Democritus and Epicurus according to whom without the void between the parts of bodies one can explain neither the structure and transformations of bodies and substances nor their movement. To prove the vacuum’s existence he cited the barometrical experiments of his time, some performed by himself. Finally, drawing on Epicurean tradition, he held that the movement of atoms alone is enough

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to explain many of the physical phenomena observed, without having to introduce hypotheses ad hoc for each of these phenomena. This highly important concept would later underlie the success of the kinetic theory of gases, as we shall see. Gassendi’s atomistic-corpuscular model enjoyed great success among his contemporaries. It showed how, on the basis of Galileian and Cartesian dynamics, it became easier to explain matter’s structure and transformations by envisaging aggregates of small particles rather than by imagining the evolution of a continuous constituent matter, as demanded by the vacuum’s opponents. In addition to the substantial agreement between the atomistic conception and the most admired scientific achievements of the time, the success was also due to the modifications that Gassendi had made to the old Democritan model to align it with Christian dogma. At the time of the foundation of the Royal Society (1660), when mechanical philosophy had become new orthodoxy, atoms and corpuscles of some sort were inserted into the theories that dominated the scientific debate across Europe. In full harmony with this new climate, Boyle, a scientist of Galilean inspiration who combined experimental investigations and mechanical ideas, also made important contributions to the corpuscular conception of matter. He posited the existence of one universal matter, common to all bodies, extended, divisible and penetrable, formed of solid and imperceptible corpuscles, whose unique attributes were shape and size. These elementary corpuscles Boyle named prima naturalia [5]. Prima naturalia combined to form primitive clusters (aggregates), which constituted the fundamental units of substances, and were also characterized by shape, size and movement. These primitive clusters, made up of a few elementary corpuscles (prima naturalia), were in principle separable, albeit this occurred rarely, their structure being extremely compact and resistant to separation. Therefore, primitive clusters remained intact in a great number of chemical reactions and could be recovered. As a demonstration, one could cite, for example, the possibility of recovering gold, silver and other metals subjected to the action of solvents. Boyle’s primitive clusters were endowed with chemical properties, not simple mechanical ones as was the case for prima naturalia. Primitive clusters and prima naturalia could also further combine, leading to more complex regroupings endowed with characteristic properties. Boyle called the dispositions whereby these aggregated textures. The diverse textures determined substances’ sensory qualities. A substance’s texture was not something stable and immutable, but could be modified by adding or subtracting other corpuscles (primitive clusters and prima naturalia), or by changing the constituent corpuscles’ layout. According to Boyle every substance was constantly exposed to the action of corpuscles; these, being in constant movement, modified its physico-chemical properties. In Boyle’s opinion it was thus possible to explain the proprieties of different substances as being simply a result of the corpuscles’ four proprieties: size, texture, shape and movement. As we can see, primitive clusters and prima naturalia had a notable analogy with modern molecules and atoms. Isaac Newton (1642–1727), the model of the modern man of science, was also an expert in alchemy, with a notable interest in problems linked with the structure of matter. Newton, like Boyle, was an adept of corpuscular philosophy. Nevertheless,

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though both invoked mechanical explanations for matter, the details of their models differed. While Boyle and his followers held that different chemical phenomena were linked with shape, size, disposition and movement of corpuscles and groups of corpuscles, Newton’s explanations were based on the idea that bodies acted and interacted via active principles (i.e. forces of interaction). Starting from the observation that macroscopic bodies act upon each other through forces such as gravity, electricity and magnetism, he pondered whether these same forces or other unknown ones might not also be at work on the small particles of bodies [12]. In the case of chemical phenomena, these forces acted at minute distances and thus it was also necessary to understand their behaviour at short range.

4.4 Boyle’s Criticism of Aristotelian and Alchemical Concepts With Boyle, criticism of Aristotelian conceptions of matter, still widely shared by chemists and alchemists of the time, took on, from a scientific viewpoint, a more definite slant, in line with the mechanical philosophy that had come to dominate in the second half of the century. In harmony with the new scientific methodology, he sought to demonstrate the incongruity of trying to explain experimental results by using Aristotle’s, or Paracelsus’s or, more generally, alchemical theories. His ideas are reported in The sceptical chymist of 1661, where he attacks both what he deems wrong ways of reasoning and incompetence in conducting experiments. In particular, he attacks the theories of primitive elements (Aristotle’s four elements and Paracelsus’s three principles). Both, says Boyle, were conceived by someone more seduced by abstract theorising than experimental evidence [11]. In his critique Boyle takes as a starting point the analysis of experiments cited in support of Aristotelian theories and lists the inconsistencies in their explanation [1]. A prominent example is the burning of wood, from Aristotle’s time commonly quoted to prove the theory of the four elements. When one burns a green branch, one first observes the production of smoke. This, according to the Aristotelians, demonstrates how wood contains a part of the element air. Next, if the branch is green, some boiling liquids are seen to escape, this, according the Aristotelians, coming from the component of the element water. The flame released by combustion shows the presence of the component of the element fire dissolving. Finally, the remaining ash is the manifestation of the presence of the element earth. Boyle raises various objections to this interpretation. First, he notes that the substances obtained during and after combustion, whether wood or other substances, are not elemental substances, but complex products, as can be shown by further chemical treatment. He points out that combustion’s residues vary according to modality and intensity of the heating. Likewise, in general the number of substances released and combustion’s residues depends on the object burnt and can also be more than four. As an example, he cites gold and silver, which remain unaltered, and blood, which produces five

References

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residual substances. Boyle concludes that such analyses are completely inappropriate for demonstrating that all substance are composed of a few primitive elements. He notes how corpuscular theories are more suited for explaining observed phenomena, although he also specifies that, for the time being, it is not possible to perform any experiment to show this in a scientifically unequivocable way. In his book The sceptical chymist Boyle did not limit himself to attacking the old Aristotelian ideas. He proposed a new concept of elements, highly similar to the modern one. In his view, chemists should not have preconceptions concerning the number of elements existing, but should consider as element any substance that they cannot decompose into other substances. Boyle writes [4]: As for elements, as do chemists who use the simplest language based on their principles, I mean certain primitive and simple bodies, or not mixed perfectly; these, being constituted by no other body, are the ingredients of which all the so-called bodies perfectly mixed are intimately composed and in which they are definitively decomposed... I should not consider any body as a fundamental substance or element, if it is not perfectly homogeneous, and is decomposable into whatever number of diverse substances.

Nevertheless, this empirical concept of the chemical element, which, when reproposed by Lavoisier at the close of the eighteenth century was rapidly accepted, had, in Boyle’s time, few followers and was long ignored. It is easy to understand why. This replaced previous schemes, elegantly deceptive in their simplicity, with a multiplicity of possible new elements, without supplying or suggesting practical methods for their research. Consequently, chemists of the period had the impression that it had no practical use.

References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12.

Brock W H (1993) The Chemical Tree. W. W. Norton & Company, New York Mason S F (1962) A History of the Sciences. Collier Books, New York Partington J R (1989) A Short History of Chemistry. Dover Publications, New York Holmyard E J (1958) The chemical industry. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Industrial Revolution. A History of Technology, vol. 4. Oxford University Press, p. 222 Singer C (1959) A Short History of Scientific Ideas to 1900. Oxford University Press, London Gribbin J (2003) Science: a History. Penguin Books W. A. D. (1908) The discovery of the weight of the air. Nature 78: 219 Ihde A J (1984) The development of modern chemistry. Dover Publications, New York Bernal J D (1954) Science in History. Watts & Co., London Geymonat L (1970) Storia del Pensiero Filosofico e Scientifico, vol. II. Garzanti, Milano Levere T H (2001) Transforming Matter. The Johns Hopkins University Press, Baltimore Baggott J (2017) Mass. Oxford University Press, London

Chapter 5

Matter in the Eighteenth Century

Regarding concepts of matter, the eighteenth century saw the definitive transition from conceptions still linked with the old classic heritage to theories and viewpoints in line with changes brought about, in physics and chemistry, by the previous century’s scientific revolution. Early in the century, despite advances in experimental methods and the spread of mechanics and corpuscular models, the conceptions of matter of alchemists, iatro-chemists and, in general, those occupied with substances’ reactions and manipulations, were still largely conceived according to philosophical ideas of Aristotelian, alchemical and Paracelsian tradition. Conversely, by the final years of the century such theories’ unsustainability was confirmed and a vision of matter decidedly nearer today’s asserted itself. In the eighteenth century, empirical knowledge of materials and substances made notable progress. New theories were advanced, methodological practices were affirmed and discoveries were made, all being fundamental steps towards the development of the modern concept of matter. Among these were the formulation of the principle of the conservation of matter in chemical transformations and reactions, the discovery of gases and the gaseous state, and the formulation of the modern theory of combustion. The distinction between physical1 and chemical properties of matter increased. These were investigated with different experimental techniques and were characterized by often divergent and sometimes even opposite explanatory theories.

5.1 Phlogiston Theory From the early eighteenth century, in order to explain the phenomena of combustion and calcination, problems which had received considerable attention in the previous centuries, a new theory took root. This was the phlogiston theory. Substantially in line with Aristotelian-Paracelsian tradition, it constituted for a long period the most 1

The term physics began to be used in place of natural philosophy after Newton’s death.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_5

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accepted explanation for combustion and calcination and, however erroneous, greatly influenced the period’s scientific research. The phlogiston theory’s birth goes back to the ideas of German chemist Joachim Becher (1635–1682). In line with the then widespread iatro-chemical opinions, in his 1669 Physicae subterraneae he held that all solid bodies were made up of a mixture of air, water and three earths: terra lapidea (vitreous earth), which was the principle of fusibility; terra fluida (mercurial earth), which contributed fluidity, subtility, volatility and metallicity to substances; terra pinguis ( fatty earth), which produced oily, sulphurous and combustible properties [1]. These three earths corresponded to Paracelsus’s and the iatro-chemists’ salt, mercury and sulphur. Becher believed that combustible substances were rich in terra pinguis and that combustion could be nothing other than the manifestation of the release of terra pinguis from the body on fire. Metals also contained a certain quantity of terra pinguis which escaped during calcination. In the first decades of the eighteenth century this conception gained momentum in many scientific circles thanks above all to the work of German physician and chemist Ernst Stahl (1660–1734), its foremost supporter who popularised the idea in the highly successful 1723 book Fundamenta chymiae. Following Becher, Stahl also accepted three different types of earths, but describes them in a rather different way [2]. Mercurial earth accounted for metals’ brightness and malleability, their capacity of being moulded and worked by goldsmiths or blacksmiths. Vitreous earth made substances able to melt and accounted for minerals’ weighty and massive nature. Finally, sulphurous earth enabled bodies to burn and produce flame. Stahl rebaptized sulphurous earth (Becher’s terra pinguis) with the name phlogiston (from the Greek word for combustible) and considered it to be the flammability principle. For Stahl, seeking to explain the great variety of chemical substances and properties, these three earths were chemical principles, or the cause of the particular properties of the bodies that contained them. They maintained a crucial property of the old Aristotelian elements: Although material substances, they cannot be isolated. Phlogiston confers on bodies the capacity to burn. According to Stahl, all substances contain a certain quantity, great or small, of phlogiston. When they burn the substances transfer the phlogiston contained into the surrounding air. When all the phlogiston has been given up, the substance stops burning. The amount of heat given off by the burning substance is a measure of the amount of phlogiston it contained. Air does not participate directly in the reaction of combustion; rather, it takes up the phlogiston from the burning substance and finally transfers it to other substances. This action of air is essential for combustion and explains why substances do not burn in the absence of air. This theory of combustion was in line with the beliefs of the first alchemists and iatro-chemists, according to whom substances are generally composed of matter and spirit, the spirit freeing itself from matter when the substance is heated [3]. Phlogiston is not only released in every combustion process, but also during the calcination of metals. When a metal is heated, phlogiston escapes, leaving calx behind it. When the calx in turn is heated, phlogiston recombines once again with the calx (or, rather, re-enters in the calx) reconstituting the metal. Thus,

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according to the phlogiston theory, metals are not simple substances, but compound ones, a mixture of calx and phlogiston. Although phlogiston theory was supported by most eighteenth century’s chemists and applied to a variety of transformations, phlogiston’s nature remained rather vague. Its attributed properties were multiform. Sometimes it was the matter of fire, sometimes a dry earthy substance (soot), sometimes a fatty principle (in sulphur, oils, fats and resins), and sometimes a mixture of invisible particles emitted by a burning candle [4]. It was contained in animal, vegetable and mineral bodies and was the same everywhere and capable of being transferred from one body to another. Furthermore, it was the cause underlying metals’ properties. Finally, since in calcination calx was heavier than the original metals, many of the theory’s followers believed that phlogiston had a negative weight and could render the substances it penetrated lighter. That the phlogiston theory found so much favour in the eighteenth century was due to it offering a unitary interpretation to the various processes of combustion and calcination. It brought together a vast amount of experimental data into one coherent, if finally mistaken, theoretical scheme. Stahl also supported the existence of atoms. However, he did not believe that atomistic or corpuscular theories could explain the large amount of changes observed in matter nor that they could describe chemical processes. According to Stahl the qualities of bodies are chemical ones and cannot be reduced to the mechanistic behaviour implicit in atomistic theories. Concerning alchemy, in which he had believed in his youth, Stahl highlighted its illusory and bogus aspects. To confute the possibility of transmutation and the belief that metals ripen in the earth to become gold, he cited the case of British tin which, he said, had remained the same as when it was exploited by the Phoenicians.

5.2 Matter in the Gaseous State In the eighteenth century a significant contribution to the knowledge of matter was made by the study of aeriform substances and the discovery of gases. Scientists finally became aware that air is not a single substance, but is made up of several components, initially given the generic name of airs by the discoverers, each air having its own characteristic properties. More than a century after the observations of van Helmont came the final recognition of the existence of gases as substances endowed with their own chemical identities. Such studies led to the definitive abandonment of beliefs based on previous traditions, whether Aristotelian, alchemical, or Paracelsian. In the past, the study of chemical properties and the manipulation of substances essentially involved matter in the solid and liquid state. The existence of substances in a gaseous state, with a chemical identity, was virtually unknown to the alchemists and first chemists. Even when smell and combustion manifested the presence of aeriform substances with characteristics markedly different from those of ordinary air, this was imputed to effect of impurities or other alterations of the air itself.

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This situation had a historical reason. From time immemorial it had been possible to treat solid and liquid substances, heating them, burning them, exposing them to tests and examining how they changed in various reactions. However, neither the old alchemists, nor the new chemists, nor the artisans using chemical substances thought it necessary to develop techniques for analysing the only aeriform system then known, atmospheric air, or for investigating the possible impurities it contained. The same Van Helmont, who had invented the name gas and described at least two new ones, spiritus sylvester (carbon dioxide) and gas pingue (methane), thought that gases could be neither captured nor analysed. Consequently, for the alchemists and first chemists the study of matter was limited to solely the analysis of solids and liquids. The seventeenth century, as we have seen, brought the first evidence of the complex nature of air, even though this was generally ignored by a still nascent chemistry community. While there was a growing awareness that atmospheric air played a role in combustion, respiration and other reactions, this was not attributed to the properties of air itself but to the existence of substances that air could absorb or release according to circumstances. According to both the corpuscular conception of matter and that propounded by Stahl and his followers, air was not a chemically active substance, but one with solely physical–mechanical properties. For Stahl, as for Becher, air was a physical environment where chemical phenomena could take place. For example, as we saw in the theory of combustion, it operated like a “sponge” able to absorb a limited amount of phlogiston. This situation began to change in the mid-eighteenth century, partly owing to the English pastor Stephen Hales (1677–1761). Hales engaged in botany, chemistry and theology. He is considered one of the founders of vegetal physiology. He was the first to consider the problem of determining the amount of air escaping from various substances when heated. To this end he invented the pneumatic trough (Fig. 5.1), an instrument which would prove essential for chemical experimentation and for gathering matter in the gaseous state when studying substances’ chemical and physical properties. The pneumatic trough consists of an ampoule filled with water and turned upside down inside a vessel in turn full of water. Thanks to a suitably modelled tube leading inside the ampoule, the gases produced during heating of the substances could pass into the trough. In such a way it became possible to gather the gases released by the reactions in a reproducible manner and to measure their volume. Using his pneumatic trough Hales conducted a vast range of experiments on gas emissions due to heating and combustion, their descriptions contained in his book Vegetable Staticks published in 1727. However, being interested principally in determining the volume of airs emitted by various substances, he neglected to study their properties, thus missing the chance to discover the existence of different gases. For Hales the emissions were all of air. He concluded vaguely that air abounds in animal, vegetal and mineral substances [4]. For this reason, his studies and conclusions made little impact on his contemporaries. It was decades before his experiments were appreciated for their novelty and consequences at a theoretical level.

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Fig. 5.1 Pneumatic trough, reproduced from a old copy of Stephen Hales Vegetable Staticks. Wikipedia Public Domain

5.2.1 Black and Fixed Air (Carbon Dioxide) The scientific community first became aware of the multiplicity of gaseous substances and air’s complex nature thanks to the research of Scottish chemist and physician Joseph Black (1728–1799). His research would lead to the discovery of carbon dioxide, which he named fixed air. It all began when Black, still a medical student, chose to study for his doctoral thesis the chemical properties of magnesia alba, given its possible use in treating bile and kidney stones. Magnesia alba [which we now know to be basic magnesium carbonate (MgCO3 )] was then introduced into medicine as a remedy for acid stomach. In his experiments, Black observed that when magnesia alba was heated, as well as the residues of calcination (i.e. calcinated magnesia and a small amount of water), there was a notable quantity of air released. Black also found that the magnesia alba was heavier than the residues of calcination. He attributed this difference to the weight of air liberated. Finally, he discovered that this air rendered opalescent the limewater (solution of calcium hydroxide (Ca(OH)2 ) in water) when passed through it, something that did not happen with ordinary air. From such evidence Black deduced that he must be dealing with a substance different from ordinary air, and which he called fixed air, since, before heating, it was fixed in the solid magnesia alba.

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Through successive experiments with limestone, alkalis and acids, Black demonstrated that fixed air was also present in limestone and mild alkalis and that its release determined the passage to these substances’ caustic form. It was an important discovery in that, hitherto, substances’ caustic properties had been held to result from the uptake of some form of igneous matter coming from the furnace. Moreover, by showing that fixed air could be newly re-introduced into calcination residues, Black proved that the new air could combine with liquids and solids to form new substances. Black described his research in his doctoral thesis On the acid humour arising from food, and magnesia alba, published in extended form in the 1756 essay Experiments upon Magnesia alba, Quicklime, and some other Alcaline Substances. The precision of his experiments, their methodological rigour and the incontrovertible proof that fixed air was a chemical substance distinct from atmospheric air earned him immediate fame. As we have seen, the existence of airs (or gases, as he had called them) differing from ordinary air had been implied by van Helmont, but without being clearly shown and remaining practically unobserved. In a further series of experiments Black proved that this new air was denser than atmospheric air and unable to support either combustion or respiration. Furthermore, from the observation that quicklime, exposed to atmospheric air, was slowly transformed into slaked lime, he deduced that fixed air must also be contained in small quantities in atmospheric air. Here we find the first explicit evidence that atmospheric air is a mixture of gases. Black finally showed that fixed air is produced in the respiration of animals, in fermentation processes and in the combustion of coal. Black’s work’s great importance derives from its being the first detailed study to focus exclusively on chemical reactions [5]. He not only identified the new air qualitatively on the basis of its chemical behaviour; he also performed quantitative analysis, accurately weighing the substances before and after reactions. Black may not have been the first to hypothesize that a substance can be characterized by identifying its constituent parts and their proportions by weight or to posit that quantitative analysis can reinforce its qualitative equivalent. He was, though, the first to apply such methods to the study of gases. In the mid-eighteenth century, after Black’s seminal experiments, research into airs was actively pursued by a number of important English scientists. These socalled pneumatic chemists within a few years had isolated the principal gases, albeit they continued to be called airs until the reformed chemical nomenclature introduced by Lavoisier at the century’s end.

5.2.2 Cavendish and Inflammable Air (Hydrogen) We owe to Henry Cavendish (1731–1810), one of England’s famous pneumatic chemists, the discovery of hydrogen, which he named inflammable air after the ease with which it burned. In the past various experimenters, in particular Paracelsus as far back as in the sixteenth century, and later Boyle, had noted how inflammable air

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evolved when some metals, e.g. iron, were treated with acids. In this regard there was considerable confusion, one reason being that other experiments had found inflammable airs which differed from that produced by treating metals with acids (they were hydrocarbons, as we know today). Cavendish was the first to gather this air, by using a sophisticated version of Hales’s pneumatic trough, and subjecting it to systematic studies demonstrating its chemical identity. In his experiments he obtained inflammable air through the action of diluted oil of vitriol (sulphuric acid (H 2 SO4 )) and spirit of salt (solution of hydrochloric acid (HCl) in water) on zinc, iron and tin. Proving inflammable air to be soluble in neither water nor alkalis, contrary to what occurred with Black’s fixed air, he showed that the two were different airs. He then found that inflammable air, mixed with atmospheric air, formed an explosive mixture. Finally, Cavendish, a precise and expert experimenter, managed to determine the weight of the new air and demonstrated that it was very light, eleven times so than atmospheric air [4]. Cavendish believed in the phlogiston theory. Seeing how this aeriform substance was released in highly diverse reactions, he concluded that inflammable air had the same properties as phlogiston and that, even, it could be the phlogiston itself. According to Stahl, metals were a compound of calx and phlogiston, therefore Cavendish now speculated that acids had the property of releasing phlogiston from metals.2 Cavendish described his discoveries in a series of three articles published in I766 in Philosophical Transactions entitled On factitious airs, to denote [6] any kind of air which is contained in other bodies in an anelastic state, and is produced from thence by art, in other words, gases produced artificially in the laboratory and distinct from atmospheric air.

5.2.3 Oxygen, Nitrogen and Air Oxygen, the other gas which, together with hydrogen, plays a vital role in substances’ composition, was discovered independently by Joseph Priestley (1733–1804), another famous English pneumatic chemist, and by Swedish chemist and pharmacist Carl Wilhelm Scheele (1742–1786). Priestley was a highly successful chemistry experimenter, preparing and identifying between 1770 and 1780 around twenty new airs, including, to apply their modern names, ammonia, hydrochloric acid, sulphur dioxide, nitrogen monoxide and carbon monoxide. Evidence for each of these gases had been found previously, but it was Priestley who collected them systematically, analysed their chemical properties and showed that they were diverse airs. His experimental success was mainly due to his capacity to conceive new apparatus. For example, by using an improved version of Hales’s pneumatic trough, where the gases bubbled through mercury instead of 2 Cavendish represented the action of acids on metals by the following reaction: metal (calx + phlogiston) + acid ⇒ salt (calx + acid) + inflammable air (phlogiston).

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water, he was able to isolate and analyse, for the first time, gases soluble in water. His experiments on airs were described in his Experiments and observations on different kinds of air (1774–1777), in his Experiments and observations relating to various branches of natural philosophy (1779–1786) and in a dozen articles in the Royal Society’s Philosophical Transactions [4]. The discovery for which Priestley is most famous, is that of oxygen, or dephlogisticated air as he named it. It was made in 1774, during a series of experiments in which he tried to extract airs from a variety of chemical substances. One of these was the red precipitate, or calx, of mercury (i.e. mercuric oxide, prepared through heating mercury in air). Priestley found that this calx, when heated, was transformed into metallic mercury, developing a colourless air insoluble in water. This result was rather surprising for a convinced adherent of phlogiston theory like Priestley. As we have seen, according to phlogiston theory, the calx of a metal was the metal which had lost its phlogiston during heating. Therefore, further heating would in no way affect it. Priestley subjected this new air to the series of experiments he normally performed to characterize the airs he was discovering. Among these there were experiments on combustion and respiration. In the former he placed candles in a receptacle filled with the new air and observed the brilliance and duration of the flame. In the latter experiments in the receptacle were placed mice, of which he measured the survival time. He found that the new air, though showing the same characteristics as atmospheric air, was much better at supporting both combustion and respiration. Interpreted according to phlogiston theory, the experimental results suggested that the new air was totally free of phlogiston, since, during combustion, it seemed to absorb significant quantities of it much quicker and better than could ordinary air.3 Consequently, Priestley named it dephlogisticated air. In the same period, this same air (oxygen) was also isolated and characterized independently by Swedish pharmacist Carl Wilhelm Scheele (1742–1786), who called it fire air. Scheele arrived at the discovery of fire air starting from investigation of atmospheric air’s behaviour in chemical reactions and in combustion. These experiments were motivated by his observation that the amount of air contained in a closed receptacle decreased when put in contact with phlogiston-rich substances. In each case he had also observed that the decrease of air stopped when around a quarter of the original air had disappeared and that the remaining air had lost its capacity to support combustion, as shown by the extinction of the candles’ flame.4 Having accurately characterized residual air and shown that it weighed less than ordinary air, Scheele concluded that [4]: 3

The explanation of the observations according to the phlogiston theory is as follows. A burning candle emits phlogiston and ordinary air allows combustion by absorbing it. In a closed container, the flame goes out after a certain time because the air becomes saturated with phlogiston and cannot absorb any more. The total amount of phlogiston that ordinary air can absorb depends on how much it already contains. 4 Today we know that the substances used by Scheele react by absorbing oxygen and that the reaction continues until the oxygen contained in the air is exhausted.

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air is composed of two fluids, differing from each other, the one of which does not manifest in the least the property of attracting phlogiston, whilst the other, which composes between the third and fourth part of the whole mass of the air, is peculiarly disposed to such attraction.

He named the first fluid (today’s nitrogen) foul air, since it was unable to support respiration or the burning of the combustible material; the second (today’s oxygen) he named fire air. Scheele was thus among the first to show unequivocally that ordinary air was not a single substance (or element, according to Aristotelian tradition), but a mixture composed principally of two gases, oxygen and nitrogen. Meanwhile Priestley and Cavendish also had perfected their studies on the composition of atmospheric air, obtaining similar results. Thus, by the late eighteenth century it was considered an established fact that atmospheric air was a mixture of different gases. In this regard, one needs to remember an experiment bearing witness to the level of accuracy achieved by researchers of the time, in particular Cavendish, in determining airs quantitively. In one of numerous experiments Cavendish analysed hundreds of atmospheric air samples, making them react with various substances capable of fixing the different airs and monitoring the diminution in volume. In one of pertaining essays he writes: having condensed all the phlogisticated air possible, I then absorbed dephlogisticated air. There remained only a small bubble of air which constitutes 1/120 of the volume of the phlogisticated air.

Without knowing it or having the means to do so, Cavendish had made the first ever observation of a noble gas, argon, now known to be a component of atmospheric air.5 Mention of this mysterious residue was not followed up and soon forgotten. The experiment was repeated a century later by Ramsay and Rayleigh, who found a residue in a 1/107 ratio with nitrogen, matching Cavendish’s result. These authors showed that they were dealing with a new unknown gas which did not react with other elements, earning it the name argon (lazy).

5.3 Combustion and Calcination When dephlogisticated or fire air was discovered, neither Priestley nor Scheele, adherents of the phlogiston theory, were able to realise fully this new air’s importance. We owe the great French chemist Antoine Lavoisier (1743–1794) the full recognition of the oxygen’s relevant role in matter’s chemical properties. In the 1770s Lavoisier also had undertaken the study of airs. His prime aim, though, was to understand the mechanism of combustion, his being highly sceptical of the phlogiston theory from the outset. In fact, although the theory was generally 5 Dehumidified air on the ground is composed of 78.09% nitrogen (N ), 20.9% oxygen (O ), 0.93% 2 2 argon (Ar) and 0.031% carbon dioxide (CO2 ), plus other components in smaller quantities, including solid particles in suspension, which make up the so-called atmospheric dust.

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accepted by chemists, there were those who remained unenthusiastic or frankly critical. Despite the apparent logic of the system built around this hypothetical substance, some important facts lacked any simple explanation. One was air’s role in combustion. It had been observed repeatedly that air was crucial to combustion and that in its absence combustion failed to occur. The explanation of the phlogiston theory’s supporters was that air attracted phlogiston and that combustion could not happen without air or if the air was already saturated with phlogiston, since in such cases phlogiston could not leave the burning substance. Many found this explanation inadequate. Furthermore, a greater difficulty, of a quantitative nature, arose when trying to explain the weight increase observed during the calcination of metals. As we have seen, according to the theory, when a metal was heated, phlogiston escaped, leaving the calx of metal as a residue of the calcination process. This explanation implied that, according to the phlogiston theory, metals were not simple but compound substances composed of calx and phlogiston. To explain why the calx should have a weight greater than that of the original metal, despite the loss of phlogiston, many of the theory’s adherents believed that one characteristic of phlogiston was negative weight, a lightness which pushed upward the substances which contained it.6 Such was the situation that Lavoisier proposed to clarify early in his scientific career. Based on some of his previous speculations on the role of air, he thought that a probable explanation was that air was somehow fixed in the metal during combustion and this incorporation of air caused the weight increase. Starting in 1772, Lavoisier carried out a series of experiments to study quantitatively combustion and calcination processes, using various materials and measuring accurately the weight of all the substances before and after heating. He studied the combustion of non-metallic substances, such as sulphur and phosphorus, the calcination of metals, such as tin and lead, and the reduction of lead oxides, such as litharge and minium (for more details see Lavoisier’s experiments in Appendix A). At the end of the first cycle of experiments, Lavoisier showed that the weight increase observed in combustion and calcination was effectively due to the incorporation of some air, the nature of which was still uncertain. At first, he thought he could be dealing with Black’s fixed air and later with ordinary air or a part of it. The situation was clarified when Priestley disclosed that he had achieved the release of dephlogisticated air (oxygen) by heating the red calx of mercury. With this new information, Lavoisier immediately embarked on a second cycle of studies, at the end of which he was able to confirm Priestley’s and Scheele’s discover that two different airs are actually present in atmospheric air. These he named, respectively, pure air (Priestley’s dephlogisticated air and Scheele’s fire air) and mofette atmosphérique (Priestley’s phlogisticated air and Scheele’s foul air). Furthermore, and more importantly, he developed a new theory according to which pure air (oxygen) is the agent of combustion and calcination.

6

We recall that the upward motion was a property of the Aristotelian fire element and that in the eighteenth century theories of matter still existed which referred to Aristotle’s philosophical framework.

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Lavoisier presented the new combustion theory in 1777 in an essay entitled On combustion in general. The theory can be summarised in the following salient points. A body can burn only in the presence of pure air (oxygen). In combustion there is destruction or decomposition of pure air and an increased weight of the burnt body exactly equal to the weight of the pure air destroyed or decomposed. Because of the addition of pure air the body burnt is changed into an acid.7 To explain the flame and heat developed in combustion, which he named, respectively, light and fire’s matter, Lavoisier postulated the existence of an imponderable fluid, heat’s matter, later called caloric. This gave rise to a theory of heat which would stay in vogue for all the first half of the nineteenth century.8 Caloric was the union of light and fire’s matter and combined chemically with substances. In particular, pure air was a substance composed of caloric and a base. In the combustion pure air was split: the base was fixed in the body that burnt and the caloric was released as flame and heat. Regarding calcination, Lavoisier affirmed that this was none other than slow combustion which produced not an acid but a calx, a substance composed of metal and the base”of the pure air. In Lavoisier’s theory caloric took the place of phlogiston as the principle of combustibility. At first sight it would seem that the only change was that phlogiston, with a due change of name, was transformed from a property of combustibles, as laid down by the old theory, to a property of pure air. In reality, there were major differences between the two concepts. Caloric was absorbed or emitted during most reactions and present in all substances, while phlogiston was present only in combustible substances. Furthermore, when added to a substance, caloric caused its expansion or change of state, from solid to liquid or from liquid to gas. Finally, caloric could be measured with thermometric techniques while phlogiston could not. The term oxygen to denote pure air was used by Lavoisier for the first time in an essay dated 5 September 1777, concerning the nature of acids [6]. The term, from Greek, means generator of acid. This essay developed the theory that the acid character of substances results from the presence of oxygen. Lavoisier had shown previously that all acids formed from sulphur, carbon, nitrogen and phosphorus contained pure air, his conclusion being that such air was essential for their formation. Lavoisier had also conducted a series of experiments on respiration, whose results, using modern nomenclature, can be summarised thus. Respiration is a form of slow combustion. In respiration, oxygen burns the carbon of food to form carbon dioxide, which is expelled. Meanwhile, the heat released in the process is the source of animal’s internal warmth. The non-respirable part of air, the mofette atmosphérique (that he later called nitrogen i.e. lifeless), is exhaled unaltered. In conclusion, though Lavoisier did not discover oxygen, he has been the first to understand it is a chemical element of utmost importance and to establish the true

7

Lavoisier calls “acids” the anhydrides. The notion of “imponderable fluid” was very common in eighteenth-century science. Electricity was generally thought to consist of “imponderable fluids,” and it was believed that magnetism could be explained in a similar way.

8

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chemistry of combustion and calcination. In this way he revolutionised chemistry completely.

5.4 Composition of Water and Confutation of the Phlogiston Theory Notwithstanding his combustion theory’s pertinence and novelty, at the time of its presentation Lavoisier was still unable to proclaim its superiority to phlogiston theory. The main reason for his caution was due to his inability to explain some important observations from recent experiments on airs and metals. It was found that inflammable air (hydrogen) was emitted when some metals, e.g. tin and zinc, were dissolved in acids forming their respective salts. When instead it was the calx of a metal to be dissolved in acids, it formed a salt, but without emitting inflammable air. With his theory Lavoisier was unable to explain either the emission of inflammable air or the difference between the two reactions. Quite to the contrary, using the phlogiston theory, the explanation was almost immediate! According to the theory, inflammable air was nothing other than phlogiston, metal was calx + phlogiston, salt was calx + acid. In the first experiment one obtained the reaction: Metal(calx + phlogiston) + acid => salt(calx + acid) + in f lammableair ( phlogiston) In the second: calx + acid => salt(calx + acid) Lavoisier also had trouble explaining the combustion of inflammable air (hydrogen). After Cavendish’s discovery, various investigators had conducted experiments on this subject. None, however, had noted the presence of clearly identifiable combustion products, while, according to Lavoisier ‘s theory, combustion of inflammable air should produce an acid.9 The presence of moisture following combustion was until then ignored or undervalued. There were two reasons for this. The first reason was that moisture was widespread in chemical reactions conducted in experimental conditions of the time; the second that, traditionally, water was considered an element and therefore was scarcely expected to appear when inflammable air was burnt. Lavoisier managed to solve these problems only years later, when he got news of Cavendish’s experiments showing that inflammable air (hydrogen) and dephlogisticated air (oxygen) combine to form water (for more details see Composition of water in Appendix A). Lavoisier became aware of Cavendish’s experiments 9

Remember that the combustion of hydrogen in air results in the formation of water.

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in 1783, when Charles Blagden, Cavendish’s assistant and future secretary to the Royal Society, visited Paris. Lavoisier would soon repeat and expand them. First, he confirmed water’s composition through the process of synthesis, i.e. producing water by exploding together inflammable air and dephlogisticated air. Later he demonstrated its nature of compound substance by using the process of analysis, i.e. by passing aqueous vapor over red-hot iron and liberating the two composing airs. Following these experiments, in 1784, Lavoisier came to the conclusion that water is not an elementary substance as believed in Aristotelian tradition and more recently by the followers of the phlogiston theory; it was, rather a complex substance composed of two airs. In line with this vision he rebaptized inflammable air with the name hydrogen, meaning producer of water in Greek. His prediction that combustion of inflammable air would produce an acid once confirmed, Lavoisier immediately inserted the new information into the general theory he was developing. He could now offer an alternative explanation to the phlogiston one for what was observed in reactions of metals and calxes with acids. A metal dissolved in a diluted acid took up the oxygen of the water present in the solution, forming its calx (an oxide). This then united with the acid to give salt, while the hydrogen of the water was released. When, instead, it was the calx that was dissolved in acid, salt could form directly without the intervention of water and without the escape of hydrogen. At this point Lavoisier was convinced that his theory could explain all chemical phenomena far more satisfactorily than phlogiston theory could. In 1786 he published an essay titled Reflections on phlogiston, attacking the old theory of Stahl and showing how all of chemistry’s main processes could be explained without resorting to that hypothetical substance.

5.5 Chemistry and Matter at the End of Eighteenth Century Thanks principally to the work of English pneumatic chemists (Black, Cavendish and Priestley) and of Lavoisier and his assistants, in the eighteenth century a series of concepts and methodological practices took root. These were milestones on the road to the modern concept of matter and related chemical knowledge. Among them are: Formulating the principle of conservation of matter in chemical transformations and reactions; the working definition of the concept of chemical elements; the concept of equivalents in chemical reactions; the awareness of the existence of substances of well-defined composition.

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5.5.1 Conservation of Matter and Chemical Elements The belief that in chemical transformations and reactions, matter was neither destroyed nor created, was present, more or less explicitly, in many measuring and analytic procedures, also outside the scientific field, this even prior to the eighteenth century. In the seventeenth century this belief had sometimes surfaced directly. For example, Francis Bacon wrote [5]: It is sufficiently clear that everything becomes and nothing really perishes, and the quantity of matter remains absolutely the same.

Equivalent passages can be found in Boyle’s writings. The idea of the conservation of matter was confirmed definitively in eighteenth century chemistry experiments, when the determination of weight, before and after chemical transformations, became a rigorous practice and the balance an essential instrument. Systematic application of quantitative methods in the study of reactions had begun with van Helmont, Boyle and Black. However, the champion of quantitative chemistry was Lavoisier. He gave clarity and coherence to concepts of matter’s indestructibility and mass’ conservation at the basis of quantitative methods. Lavoisier says [4]: …it can be taken as an axiom that in every operation an equal quantity of matter exists both before and after the operation…and that only changes and modifications occur. The whole art of making experiments in chemistry is founded on this principle: we must always suppose an exact equality between the principles of the body examined and those of the products of its analysis.

Thanks to Lavoisier the concept of mass’ conservation of in chemical reactions and transformations has become an explicit postulate. Also of great methodological importance is the clear distinction established in the eighteenth century between elementary and compound substances and the final framing in a modern sense of the concept of chemical element. This also is to be credited to Lavoisier. Resuming Boyle’s ideas, he defined element as a substance that cannot be decomposed further by any known method of chemical analysis. Thus, the elements no longer constituted metaphysical entities or principles established a priori, as in Aristotelian-type doctrine. Rather they were the end result of experimental analysis in the laboratory. Based on this definition, Lavoisier identified 32 substances, which he classified as elements in that they could not be decomposed. His list included, as well as the true elements, also light and caloric, Lavoisier believing these to be imponderable material substances. With the scientific community’s acceptance of the concept of a plurality of elements and Lavoisier’s theory of the formation of chemical compounds starting from these elements, the late eighteenth century sees a breakthrough toward the modern concept of matter and the final ditching of theories of Aristotelian and medieval origin.

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5.5.2 Compounds and Affinity In the early eighteenth century, Georg Stahl (the phlogiston theory’s proponent) strongly claimed that, to understand matter’s structure, it was necessary to study compound substances and characterize them with chemical methods. He highlighted the need to distinguish between chemical compounds and mixtures, i.e. simple aggregates of different substances [2]. A chemical compound was a substance whose properties were not the simple sum of the properties of substances which united to form it. A chemical compound’s formation involved changes in constituents, which were modified in the combination process, as when, for example, an acid and a base reacted to form a salt. Aggregates or mixtures, instead, were the result of the simple physical mixing of invariable constituents, in processes resembling the mixing of different types of sand. As the eighteenth century proceeded, attention to the nature of chemical compounds stimulated the scrutiny of chemical reactions. There arose the problem of their selectivity, i.e. why a certain substance combined with other determined substances and with these only. In reality this problem was an old one, which had interested scholars since the Middle Ages and to which they had tried to respond with the affinity concept. Alchemists had hypothesized a relationship of friendship between substances that combine. In the thirteenth century Albertus Magnus, trying to solve the problem of the forces holding the parts together in compound substances, had introduced the concept of an affinitas between bodies that combine [6]. In the seventeenth century Glauber, Boyle and Mayow had all sought to explain substitution reactions10 using the concept of relative affinity. Newton talked of sociability and attraction between components. He suggested that there were very strong attractive powers between the particles of bodies that combine. However, these attractions extended only for minute distances and their force varied from one substance to another. For example, according to Newton aqua regia acts on gold since it has an attractive force towards this metal, so allowing it to penetrate inside, while lack of such a force in respect to silver prevents it from acting likewise. In the eighteenth century the study of elective affinities (as chemists of the time called them) became one of the main problems in investigating matter’s structure. Tables of affinity were drawn up, acids being ordered according to their propensity to bind with a given base, and the metals according to their reactivity with sulphur. Relative grades of affinity were assessed ascertaining whether one base displaced another base from a given compound and one metal displaced another metal. The first tables were compiled in 1718 by the French physician Etienne-Francois Geoffroy (1672–1731). More elaborate tables were drawn up by Swedish chemist and mineralogist Torbern Bergman (1735–1784) and by others from 1750 onwards. A highly influential text, Elements of theoretical chemistry, published in 1749 by French 10 Substitution reactions are defined as: Reactions in which the functional group of one chemical compound is substituted by another group; reactions which involve the replacement of atoms or molecules of a compound with other atoms or molecules; reactions such as those where, in salts, one base substitutes another base or an acid another acid.

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chemist Pierre Joseph Macquer (1718–1784), had a whole chapter dedicated to the subject, helping to promote interest in tables of affinity. It concluded [2]: All the experiments which have been hitherto carried out, and those which are still being daily performed, concur in proving that between different bodies, whether principles or compounds, there is an agreement, relation, affinity or attraction. This disposes certain bodies to unite with one another, while with others they are unable to contract any union. It is this effect, whatever be its cause, that will help us to give a reason for all the phenomena furnished by chemistry, and to tie them together.

Actually, although affinity tables constituted a first step toward a classification of substances, the reality of chemical reactions was far more complex. In the early nineteenth century the studies of French chemist Claude-Louis Berthollet (1748– 1822) showed how a compound’s formation was linked to a variety of factors other than affinity. His Essai sur les etats chimiques, published in 1803, demonstrated how factors such as concentration, temperature, pressure, and the quantity of reagents, determine whether a particular reaction happens or not [1]. Berthollet concluded that these factors make it impossible to establish the relative affinities of two substances towards a third one, striking a heavy blow to the practical use of tables.

5.5.3 Combination and Equivalents Strictly linked with the affinities problem and thus much debated at the time, were the concepts of equivalent weight and equivalent ratio in substitution reactions. The concept of equivalent weight derived from the following observations. If one considered, in reactions forming salts, the amounts of different bases necessary to neutralise a given quantity of acid, it was found that the ratio of their weight did not vary when the reactions were repeated using different quantities of acid. An analogous constancy was found in the ratios of the weights of different acids needed to neutralize a given quantity of a base and in the weight ratio of two metal displacing each other in acid solutions. In the first half of the eighteenth century various chemists, including Johann Kunckel (1630–1703), Nicolas Lémery (1645–1715), Georg Stahl (1659–1734), Wilhelm Homberg (1652–1715) and Carl Friedrich Wenzel(1740– 1793), studied the problem with the aim of determining equivalent weights. In the second half of the century, thanks above all to Cavendish, Torbern Bergman (1735– 1784) and Irish geologist and chemist Richard Kirwan (1733–1812), knowledge grew, leading to the concept’s generalization into the so-called law of reciprocal proportions or law of equivalent proportions. Such a law was elaborated in 1791 by German chemist Jeremiah Richter (1762–1807) and formulated thus [3]: The weight of a substance A, that combined with a known amount of a substance B, would also combine exactly with that weight of a substance C, which entered into combination with the same known amount of the substance B.

After such development scientists began to prepare tables of equivalent weights, reporting the relative quantities of chemical elements which combine.

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The usefulness of the concept of equivalent weight was sanctioned in the last years of the century with the work of French chemist Joseph Louis Proust (1755–1826). Proust, using in his analysis substances of the best purity available,11 found that a given compound substance, however obtained, in nature or the laboratory, always contained the same simple bodies (i.e. the same components) combined in identical proportions of weight. In his first experiments he had noted that pyrites (a mineral containing iron and sulphur) always had the same composition irrespective of its place of origin. In particular, for each gram of iron there were always present 0,57 g of sulphur and such a composition was also respected in the pyrites he had obtained in the laboratory. Performing a long series of investigations into the composition of many minerals and artificial compounds of metals, he ascertained that such regular behaviour was quite common in minerals and metallic substances. These observations led him to formulate the law of definite proportions (known also as Proust’s law), which states that: When two or more elements react to form a determined compound, they always combine according to definite and constant proportions of weights. Proust drew up the law in 1797, commenting on it thus [4]: We must recognize an invisible hand which holds the balance in the formation of compounds. A compound is a substance to which Nature assigns fixed ratios, it is, in short, a being which Nature never creates other than balance in hand, pondere et mensura.

References 1. 2. 3. 4. 5. 6.

11

Brock W H (1993) The Chemical Tree. W. W. Norton & Company, New York Levere T H (2001) Transforming Matter. The Johns Hopkins University Press, Baltimore Mason S F (1962) A History of the Sciences. Collier Books, New York Partington J R (1989) A Short History of Chemistry. Dover Publications, New York Singer C (1959) A Short History of Scientific Ideas to 1900. Oxford University Press, London Ihde A J (1984) The development of modern chemistry. Dover Publications, New York

We can see Proust’s results as a practical consequence of the widespread awareness of the importance of the purity of materials in chemical experimentation, which had established itself in the eighteenth century, thanks mainly to the works of Black, Cavendish and Lavoisier.

Chapter 6

Matter and Electricity

A decisive contribution to our knowledge of matter derives from the study of electricity. The discovery of electrical properties leads naturally to questioning their relationship to the matter in which they are observed. The history of electricity begins with the first observations concerning the particular qualities of amber (elektron in Greek), a translucent yellowish resin. When rubbed with a piece of leather, amber develops the capacity to attract small pieces of light material such as feathers. Greek tradition reports that Thales of Miletus (640– 546 B.C.) was among the first to mention this. It is unclear, however, if he personally discovered this property or if he learned it from Egyptian priests and others during his long journeys. In the ancient world there had been other mentions of phenomena linked to electricity. For example, Aristotle wrote about a fish called torpedo, which gave shocks, paralyzing the muscles when touched [1]. The Etruscans were said to attract lightning by firing metal arrows toward the clouds. Over the centuries mention of such properties was repeated in the writings of scholars right up to the seventeenth century when interest in electric phenomena and a more systematic study of them became a feature of the nascent scientific revolution. The modern history of electricity begins with the English physician William Gilbert (1544–1603). In 1600, after almost twenty years of experiments, Gilbert completed his work De magnete, magneticisque corporibus et de magno magnete tellure (On the Magnet and Magnetic Bodies, and on the Great Magnet the Earth). This was dedicated principally to the study of magnetic properties, but included studies concerning electrical phenomena. In his book Gilbert analysed static electricity generated by amber and showed how the same property is possessed by other substances, such as glass, resin, sulphur, etc. From the Greek name for amber, “elektron”, he coined the Latin neologism electricus meaning similar to amber in its attractive properties. The work included the first classification of substances into electrical ones, which, on rubbing, attracted objects, and non-electric ones, which after rubbing had no attractive properties. It is the first example of systematic study

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Fig. 6.1 Old drawing of Gilbert’s versorium. Reproduced from S. P. Thompson, Elementary Lessons in Electricity and Magnetism, 1902. Wikipedia Public Domain

of matter’s electrical properties, made possible by the use of the first electroscope,1 invented by Gilbert and which he named the versorium. Gilbert’s versorium is a metal needle that can turn freely on a pivot, as shown in Fig. 6.1. It is similar to a compass needle but unmagnetized. The needle is attracted by bodies charged through rubbing then placed nearby, and rotates toward the charged object. The response of the versorium’s needle is independent of the polarity of the electric charge; thus, it cannot distinguish between a positive charge and a negative one, unlike the compass needle, which has a north and south extremity and can distinguish between a north and a south pole. The primitive state of knowledge at the time prevented Gilbert understanding the reason behind such attraction (electrostatic induction); however, his versorium was useful for revealing if the rubbed object acquired electrical properties. Gilbert hypothesised that electrical substances possessed a special matter that he called electric effluvium [2]. As a result of rubbing, this effluvium is removed from substances and spreads in the surrounding air, so displacing it. When the effluvium returns to its original position, the displaced air pushes the light bodies, making them move toward the attracting body. In his experiments Gilbert observed and analysed only the attractive effects generated by rubbed objects. The property whereby bodies can also repulse each other, electrostatic repulsion, seems to have been first noted accidentally by the Jesuit philosopher and mathematician Niccolò Cabeo (1586–1650). In his Philosophia Magnetica (1629), he described how small objects, attracted by charged amber, sometimes recoil from each other at a distance of even several centimetres after contact with the amber itself. Following Gilbert, Cabeo also described electrostatic attraction in terms of an electrical effluvium released by rubbing materials together and spreading into the surrounding air which, becoming displaced, causes light bodies to move. The necessity of air for observing electric phenomena was later disproved by experiments carried out by Robert Boyle (1627–1691) around 1675, using the vacuum pump. These experiments showed how electrical phenomena persist in the absence of air. By the mid seventeenth century the realisation that electric phenomena are important properties of matter, worthy of investigation, was taken as a given by the scientific community, the second half of the century witnessing notable progress in their study. The first machines to generate electric effluvia in a more controlled and reproducible way, electrostatic machines (or electrostatic generators) were invented. The first of these machines was invented around 1660 by Otto von Guericke (1602–1686), also 1

The electroscope is an instrument that measures the electric charge.

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Fig. 6.2 Drawings of von Guericke’s friction machine. Reproduced from Ottonis de Guericke, Experimenta nova. Courtesy of The Linda Hall Library of Science, Engineering and Technology

famed, as we have seen, for inventing the first vacuum pump and for the famous hemispheres of Magdeburg experiment. The electrostatic generator, a friction machine, invented by von Guericke consisted of a sphere of sulphur on an iron axle turned by a handle (Fig. 6.2). Electrification was produced by keeping a cloth in contact with the sphere while in rapid rotation. An electrostatic generator, significantly improving on the more primitive version invented by Otto von Guericke, was constructed in 1706 by British scientist Francis Hauksbee (1666–1713). Hauksbee’s generator consisted of a glass sphere made to rotate rapidly by a wheel with a pulley. The machine generated enough electricity to demonstrate another phenomenon linked to static electricity: electrical discharges and the concomitant emission of light. Hauksbee’s first generators produced enough light to permit reading and were the true precursors of today discharge lamps. In 1709 Hauksbee published PhysicoMechanical Experiments on Various Subjects. Containing an Account of Several Surprizing Phenomena Touching Light and Electricity, a book representing an important addition to the knowledge of electrical phenomena. In it Hauksbee reported his experiments, in which he observed not only electrostatic attraction, but also repulsion, something first described by Niccolò Cabeo, but which had gone unnoticed. The first half of the eighteenth century saw a notable advance in the knowledge of electrical properties thanks to the experiments of English scientist Stephen Gray (1666–1736). Gray was the first to discover and investigate the phenomenon of electrical conduction. Prior to his experiments attention had been only concentrated on the generation of static electricity and related phenomena. Despite the difficulties stemming from the use of such unreliable sources of electricity as the friction machines, Gray was able to show that electrical virtue (as he called electrical phenomena) was

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not just static, but resembled, rather, a fluid able to propagate itself. As often occurs in scientific research, the first observation of the phenomenon happened by chance. One night, Gray noted that the stopper of the tube he used to generate electricity through rubbing, attracted small pieces of paper and straw. As a rule, in experiments the stopper was not electrified. That night, however, climactic conditions and the effects of humidity were such that the stopper had accumulated electrical virtue. In a successive cycle of tests Gray discovered that metal wires and thick damp strings were able to transport the electrical virtue, while silk wires could not. These were the first observations of conducting materials opposed to insulating ones. Gray also provided the first unambiguous description of the phenomenon of electrostatic induction, showing that it was possible to transmit electric virtue to metal objects. At the time these were considered non-electric substances, since they could not be charged through rubbing. Important studies on electricity were conducted in the same period in France by Charles-François de Cisternay du Fay (1698–1739) and abbot Jean-Antoine Nollet (1700–1770), both in close contact with Gray. Inspired by the discoveries of the latter, du Fay, after methodical research, verified that all materials, except metals, can be electrified by rubbing. His conclusion was that electricity is a general property of matter. Furthermore, starting out from the observation that rubbed objects do not always attract small bodies but in certain cases repelled them, du Fay studied this phenomenon systematically. He found that two pieces of electrified amber repelled each other when brought at close distance and that the same happened to two electrified glass pieces. However, if one placed a piece of electrified amber next to an electrified piece of glass, the two attracted one other. Du Fay thus deduced that there must be two types of electricity, his names for them being resinous electricity and vitreous electricity (corresponding to today’s negative and positive electricity). In line with this, he distinguished substances into resinous and vitreous, according to how they repelled, respectively, electrified amber or glass. In 1733 du Fay formulated the first theory of electricity, called the two fluids theory. It held that bodies contained two distinct electric fluids. These are present in equal measure in nonelectrified bodies and become inequal following rubbing, with the one or the other predominating in electrified bodies. An important progress in the field came in 1745 thanks to a significant advance of the electric instrumentation: The invention of the Leyden jar (the oldest form of electric condenser, Fig. 6.3) by, independently of one another, German Canon Ewald von Kleist (1700–1748) and Dutch physicist Pieter van Musschenbroek (1692–1761). The name derives from the city, Leyden, location of the university where van Musschenbroek, the first who presented it to the international scientific community in 1746, held the chair of professor. The first form of Leyden jar was a glass container filled with water; the water forming the inner plate of the capacitor. A metal nail driven through the cork stopper made contact with the water, allowing the water to be charged with electricity and discharged. The jar was held in one hand, which formed the other plate of the capacitor. The later types consisted of a glass container which had coatings of metal foils

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Fig. 6.3 First form of Leyden jar, consisting of a bottle with a metal spike through its stopper making contact with the water. Right, more common type, which had coatings of metal foil on the inside and outside. Reproduced from W. J. Harrison, C. A. White, Magnetism and Electricity, 1898 and R. A. Houstoun Elements of Physics, 1919. Wikipedia Public Domain

on the inside and outside walls. The two metal foils formed the plates of the capacitor, while the glass wall served as the dielectric. The electrode for the internal was a conducting rod or a metal chain. Leyden jars possessed a rather high electric capacitance, which combined with the glass’s high dielectric strength and thickness, made them ideal as high voltage capacitors. Thanks to this invention investigators had at their disposal the first practical equipment to store electricity (more precisely, the electrostatic energy that is generated in presence of opposite electrical charges). The Leyden jar rapidly became a basic instrument in eighteenth century electrical laboratories, making possible a whole series of new experiments. In this regard it is amusing to recall some jaw-dropping experiments conducted by abbot Nollet to support the theory of electricity as a fluid running continuously among objects in contact with one another. In a demonstration organized for king Louis XV in 1746, he discharged a battery of Leyden jars in such a way that the current passed through a line of 180 royal guards linked by iron wires. Soon afterwards he repeated the experiment, lengthening the chain to try and measure the propagation speed of electric phenomena. Using iron rods, he joined 200 Carthusian monks to form a circumference of around 1.6 km. Discharged through this human chain a battery of Leyden jars, he observed that the electric shock felt by the monks was almost simultaneous and concluded that electrical propagation was instantaneous. Similar attempts, if less spectacular, to determine the propagation velocity of electric phenomena were also made in England of the same period. On August 1747 William Watson (1715–1787) drew up, along with Henry Cavendish and others, a famous experiment to measure the velocity of propagation through a wire almost 2 km long. Shortly afterwards, in a further experiment, the wire’s length stretched to around 3.7 km. Failing to measure any time difference between the departure and arrival of the discharge, Watson also concluded the propagation was instantaneous. In the same period, inspired by the then prevailing belief, originated by Newton, of the existence of imponderable fluids, Watson and American scientist and statesman

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Benjamin Franklin (1706–1790) hypothesised that the cause of these electrical phenomena resided in an atmosphere or fluid diffused in space, inside and outside material bodies [3]. According to this new theory the two types of electricity, viteous and resinous (for the first time named by Franklin as positive and negative electricity) were the manifestation of an excess or defect of one single electric fluid. When the density of the fluid within matched that outside, bodies were electrically neutral (non-electrified). Bodies were electrified when fluid was transferred from one object to another. A body that acquired an excess of fluid in respect to its outside, became positive, while a body defective in fluid became negative. The electric fluid was conserved, neither created nor destroyed, but only transferred from one object to the other [4]. In 1759 German physicist Franz Aepinus (1724–1802) published the book Tentamen Theoriae Electricitatis et Magnetismi (An attempt at a theory of electricity and magnetism) describing the electrical and magnetic effects then known. It set out to explain them as due to an interaction similar to Newton’s law of gravity. That is to say, as due to attractive and repulsive forces between electrified bodies, which act at a distance and diminish in proportion to the inverse of the square of the distance. Following and improving on Franklin’s theory, Aepinus hypothesised that one electrical fluid alone is normally present in all material bodies and that the relative abundance or lack of fluid manifests itself as positive or negative electricity. One theory of electricity matching Aepinus’s was drawn up in the same period, although independently, by English physicist and chemist Henry Cavendish (1731–1810) but published only in 1771. In the same period the English chemist Joseph Priestley (1733–1804) also conducted important researches on electricity, published in 1767 in the book History and Present State of Electricity, where he examined all the data available at the time. Carrying out many studies on the electrical conduction of a variety of substances, Priestley showed that there is no clear demarcation between conductive and nonconductive materials, but that a capacity, more or less marked, to conduct electricity exists in many substances. With such experimental evidence he overturned one of the hitherto most commonly held beliefs, namely that only water and metals could conduct electricity. Furthermore, he believed that there must exist a relationship between electricity and chemical reactions. Finally, based on experiments performed with charged spheres, Priestley, like Aepinus, was among the first to propose that electrical forces vary in proportion to the inverse of the square of the distance, as with Newton’s law of gravity. Definitive and quantitative proof that the interaction between electrified bodies is similar to gravitation came in the late 1780s by French physicist Charles-Augustin de Coulomb (1736–1806). His classic experiments were made possible by his invention of torsion balance. Coulomb invented the torsion balance (Coulomb’s balance), using the results of his studies on the elastic torsion of wires, published in 1784 in Recherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de metal. The torsion balance allows the measurement of even very weak forces. The attraction or repulsion exerted between the electric charges causes a twist of the wire, which rotates until the elastic torsion force balances the electrostatic force.

6.1 The Voltaic Pile

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The force is therefore proportional to the torsion angle. In this way, by varying the magnitude of charges and the distance between the spheres, Coulomb was able to demonstrate the famous law bearing his name. It establishes that between two electric charges q1 and q2 placed at a distance r a force F is exerted proportional to the charges themselves and inversely proportional to the square of the distance, i.e.: F =C

q1 q2 r2

with C a constant of proportionality. The force is directed along the line joining the charges and is repulsive or attractive according to whether the charges have the same or opposite sign. From a conceptual viewpoint the demonstration that electrical interactions are due to Newtonian forces proved of great importance for the understanding of electric properties and the development of this field of study. It made it possible to extend to electric phenomena all physical properties that mechanics had calculated for gravitational interactions, from Newton on. From this moment on, electrostatics as a theoretical discipline made very rapid progress, thanks, precisely, to the established link with mechanics.

6.1 The Voltaic Pile In the early months of the nineteenth century the scientific community received news of an invention destined to have a great impact on science and technology, and which, in the immediate aftermath, would allow a series of experiments fundamental for the knowledge of matter: the invention of the voltaic pile (the forerunner of the modern batteries) by Italian scientist Alessandro Volta (1745–1827). For the first time the scientific world had at its disposal an apparatus able to produce an intense and continuous flow of electricity (electric current), so paving the way for the revolutionary discoveries in electro-magnetism and electrolysis. Volta announced his invention to the scientific community with a letter dated March, 1800, addressed to the President of the Royal Society, Sir Joseph Banks. The letter, containing the first description of the voltaic pile, was published in Philosophical Transactions with the title On the Electricity excited by the mere Contact of conducting Substances of different Kinds. Initially named artificial electric organ or electromotor apparatus, the voltaic pile was later given this name after its characteristic structure. The physical effects on which the Voltaic pile is based had been observed and discussed long before Volta’s invention. In 1762 Swiss philosopher and scientist Johann Sulzer (1720–1779) found that, when he touched simultaneously with his tongue the tips of two different metals in contact with each other, he felt a strange tingling. This, essentially, was the first observation of the Volta effect. Years later, in 1791 Luigi Galvani (1737–1798), anatomy professor at Bologna University, observed by chance another surprising manifestation of electricity, described in detail in his

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essay De Viribus Electricitatis in Motu Musculari Commentarius (Comment on the effects of electricity on the movement of the muscles). Galvani observed that frogs’ legs put out to dry contracted when the brass hook whence the frogs were hung came into contact with an iron mesh. After verifying that the same effects occurred also in closed spaces and far from possible sources of electricity, he concluded that the contractions were caused by a new form of electricity, which he named animal electricity, stored or produced by the frogs’ muscles. In 1796, when he began to conduct experiments similar to Johann Sulzer’s with metals in the mouth, Volta believed he was experiencing effects of the animal electricity discovered by Galvani. He found, however, that it was possible to produce a flow of electricity (an electrical current) also in the absence of animal tissues, using, in place of the tongue, a piece of cardboard soaked in brine. He deduced that the effect was due to the contact of two different metals with a damp object. To amplify the effect, Volta thought of using many individual cells, each made up of two metals in contact with an acidic liquid. In the first version of the device each cell was a wine glass filled with brine in which were immersed two electrodes of different metals (Fig. 6.4). Connecting the two electrodes by means of a conductor, he obtained the flow of electric current. He also found that the metal pair most effective for producing electricity was composed of zinc and copper. In the device’s next configuration, the wine-glasses were substituted by cells (voltaic elements) consisting of a zinc disk on top of a copper one, with an intermediary layer of felt or cardboard soaked with water and sulphuric acid. The voltaic pile consisted of a column of these cells, one on top of the other (Fig. 6.4). The functioning of the voltaic pile was so complex that it would only be fully understood much later, and Volta’s theory could only be approximate. Nonetheless, the difference between its characteristics and those of electrostatic generators previously available as sources of electricity was grasped immediately. As early as 1801 English chemist and physicist William Hyde Wollaston (1766–1828) demonstrated how electricity obtained from the voltaic pile was less intense but produced in much greater quantities compared with rapid and explosive discharges generated by electrostatic machines [5]. The theoretical problem was posed of establishing if the galvanic fluid, as electricity generated by a voltaic pile was then called, was identifiable with the electric fluid produced by electrostatic generators. The problem remained a subject for debate for some years, although from 1802 onward Volta maintained that they coincided. Despite these conceptual difficulties, the voltaic pile’s potential was immediately recognised and the new device became hugely popular. In the years after its invention, the voltaic pile was substantially improved so that scientists had at their disposal electric generators (batteries) able to produce intense and stable electric currents. Applied for investigating electromagnetic phenomena, they allowed a rapid succession of new discoveries and the development of new technologies. In the immediate aftermath, the use of voltaic piles rendered possible various experiments, which contributed significantly to the understanding of matter.

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Fig. 6.4 Volta’s original illustration of the different piles, from the letter sent to sir Joseph Banks. Wikipedia Public Domain

6.2 Electrolysis The first important experiment was conducted in May, 1800, just weeks after Volta’s announcement. The chemist William Nicholson (1753–1815) and the surgeon Anthony Carlisle (1768–1840) constructed the first English version of the battery. They found that if they passed electric current through water, water was decomposed into its constituent gases, hydrogen and oxygen, forming gaseous bubbles above the electrodes. They also observed the still more surprising fact that each gas escaped at a distinct electrode, hydrogen at what Volta had called the negative pole and oxygen at the positive one. These experiments were the first evidence of electrolysis, i.e. the decomposition of substances through the passage of an electric current, one of the

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most important chemical effects produced by electricity in liquids.2 They demonstrated how the battery could be a new and powerful instrument for investigating matter. In September of the same year, German scientist Johann Wilhelm Ritter (1776– 1810) managed to deposit copper by way of electrolysis. A little later, in an essay published in 1803, Swedish chemists Jöns Jacob Berzelius (1779–1848) and Wilhelm Hisinger (1766–1852) showed that, in general, chemical compounds are decomposed by electric current, their components collected at the electrodes, with hydrogen, alkalis, earths and metals going to the negative electrode and oxygen, acids and oxidised compounds going to the positive one [6]. Such researches had a vast impact, immediately capturing scientists’ interest, and leading to the emergence of a new discipline, electrochemistry. With electrochemistry research pursued with chemical methods crossed paths for the first time with the problems connected to the relationship between the nature of electricity and that of matter. In the years that followed, a series of ground-breaking experiments were performed by English scientist Humphrey Davy (1778–1829). In 1806 he formulated the hypothesis that attraction between chemical substances was electric in nature. Consequently, he believed that many substances, which had hitherto resisted every attempt at scission, could be decomposed by the passage of a strong electric current [7]. Two of these substances were caustic soda (NaOH) and caustic potash (KOH), which Lavoisier, with his special intuition but no tangible evidence, had assumed were metallic compounds containing oxygen. In 1807 Davy passed an intense electric current coming from a large battery built by himself, through potash and soda in a molten state; he showed that both could be decomposed: the caustic soda into oxygen, hydrogen and a new soft white substance, sodium, and the caustic potash into the same two gases and a new substance of similar aspect, potassium. The following year, still using electrolysis, Davy isolated calcium, boron, barium, strontium and magnesium, much enriching the list of known elements. In 1810 he went on to demonstrate that the gas chlorine, isolated by the Swede Scheele in 1774 and believed to contain oxygen, was instead a single element. In his 1813 book Elements of agricultural chemistry Davy, adopting Lavoisier’s definition, introduced for the first time into an English text the term element in its modern chemical sense. He said [8]: All the varieties of material substances can be resolved into a relatively small number of bodies which, being not further decomposable, are in the present state of chemical science to be considered as elements.

Davy listed 47 elements.

2

The term electrolysis was introduced by Faraday in 1832 and derives from the Greek words elektron (amber) and lysis (dissolution).

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References 1. Sarkar T K, Salazar-Palma M, Sengupta D L (2006) Introduction. In: Sarkar T K, Mailloux R J, Oliner A A, Salazar-Palma M, Sengupta D L (ed) History of wireless. John Wiley & Sons, Hoboken, New Jersey, p. 1–52 2. Heathcote N (1967) The early meaning of electricity. Annals of Science 23: 261–265 3. Mason S F (1962) A History of the Sciences. Collier Books, New York 4. Gribbin J (2003) Science: a History. Penguin Books 5. Lindell I V (2006) Evolution of electromagnetics in the nineteenth century. In: Sarkar T K, Mailloux R J, Oliner A A, Salazar-Palma M, Sengupta D L (ed) History of wireless. John Wiley & Sons, Hoboken, New Jersey, p. 165–188 6. Partington J R (1989) A Short History of Chemistry. Dover Publications, New York 7. Holmyard E J (1958) The chemical industry. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Industrial Revolution. A History of Technology, vol. 4. Oxford University Press, p. 231 8. Singer C (1959) A Short History of Scientific Ideas to 1900. Oxford University Press, London

Chapter 7

Chemistry and Matter in the Nineteenth Century

The nineteenth century sees the definitive assertion of our modern vision of matter composed of a precise number of elementary substances, the elements, each endowed with its own peculiar physical and chemical properties, the combination of which gives rise to chemical substances. In the period a large number of new elements and compounds are discovered. The number of known elements (23 when Lavoisier provided the first empirical definition of element) becomes 63 in 1871 [1]. Together with the definition and the discovery of elements comes the relaunch of modern atomism. The epitome of scientific research’s success in the period are the Periodic law of elements and the Periodic table, drawn up almost simultaneously and independently by Dmitrij Ivanovich Mendeleev and Julius Lothar Meyer. This great scientific breakthrough was helped by the formulation of the concept of valency and a reliable determination of atomic weights.

7.1 Revival of Atomism In the early nineteenth century, thanks to English chemist John Dalton (1766–1844), atomism was relaunched. The theory introduced, for the first time, quantitative parameters subject to experimental verification in explaining the chemical phenomenology of substances. The emergence of such a conception evolved naturally from research into chemical reactions and the analysis of substances conducted in the previous century. At the end of the eighteenth century most chemists used, more or less explicitly, the concept of microscopic entities to explain chemical phenomena. It was a legacy of the seventeenth century, when corpuscular ideas had become popular and the concept that all matter was in the last analysis composed of solid, hard, impenetrable and mobile microscopic particles had become a self-evident truth [2]. However, these corpuscles or particles were essentially objects characterized by physical properties, such as size, shape and velocity, and were of little help in explaining chemical phenomena, such as reactions and combinations of substances. As a result, no real © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_7

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attention was paid to them by chemists when it came to the theorization of chemical phenomena. The situation changed thanks to Dalton’s work. Atomistic theory was formulated by Dalton in such a way that it responded to the then most topical problems regarding the definition of chemical element, the formation of compounds, and the meaning of equivalent weights. He presented a preliminary sketch of the theory in an article read to Manchester’s Literary and Philosophical society in 1803 and a full account was made in his 1808 New System of Chemical Philosophy [3]. Starting out from the concept of the multiplicity of elements asserted in the late eighteenth century principally by Lavoisier, and from the idea that matter is composed of atoms, Dalton proposed a theory whereby each element was characterized by specific atoms, equal and indistinguishable, which conserved their individuality in all chemical transformations. These atoms were indestructible and indivisible microscopic particles, but so small as to be undetectable by the most powerful microscope. Atoms of different elements were characterized by different weights; this was the basic property distinguishing one element from another and accounting for the different densities of the elements themselves. In line with the concept of matter’s conservation in chemical transformations, the atoms of elements could be neither created nor destroyed, but combined to form compound atoms (nowadays molecules). In forming a compound, the atoms of a given element combined only with whole and small numbers of atoms of other elements. Dalton’s theory matched perfectly the concept of equivalent weights developed in the eighteenth century and the law of definite proportions, formulated by Proust in 1797 which states: when two or more elements react to form a determined compound, they always combine according to definite and constant proportions of weights. Dalton extended the study of equivalent weights to cases in which the same elements combine to form different compounds, as happens, for example, when the same quantity of carbon reacts with two different quantities of oxygen to form carbon monoxide and carbon dioxide. The results of this research led him to formulate, in 1804, the law of multiple proportions, which states: If two elements combine, forming different compounds, the quantities of one of them combining with a fixed quantity of the other are in rational ratios, expressed by whole and small numbers. In the case of the carbon oxides (carbon monoxide and carbon dioxide) the quantities of oxygen combining with the same quantity of carbon are in the ratio of 1 to 2. In the case of nitrogen oxides, which Dalton himself had investigated, the quantities of oxygen combining with the same quantity of nitrogen are in the ratio of 1:2:4. Praise is due to Dalton for intuiting that, using his theoretical scheme, it was possible, starting out from equivalent weights, to determine the relative weight of atoms, even if their dimensions or weights were too small to be measured. To this end it was sufficient to know the elements forming a compound and their relative weights. For example, if m grams of an element A joined with n grams of an element B to form a substance AB, whose compound atoms (molecules) were formed of one atom A and one atom B, the ratio of the atomic weights of the two elements was exactly m/n. Based on this concept, Dalton constructed the first table of relative weights (atomic weights) of elements. He took as a reference the weight of the atom

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of the lightest substance, hydrogen, and expressed the atomic weights of the other element in relation to hydrogen taken as equal to 1. Proceeding thus he managed to compile a table with the relative weights of 37 substances. This he published in his 1808 work A New System of Chemical Philosophy. The importance of Dalton’s cannot be overstated. Even if atomic particles could never be weighed individually, he struck upon a method which joined, for the first time in history, atomic theory with a quantity measurable in the laboratory (relative weight), so bridging the gap between experimental data and hypothetical atoms. Finally, some concreteness was given to entities hitherto solely hypothetical. However, in Dalton’s time the scheme had an evident difficulty, which was to remain a source of major controversy for many years. To determine the relative weight of atoms starting from proportions of combination one needed to know the number of atoms forming the constituent compound atom (molecule) of the substance. At the time, however, there was little experimental information on this point. Consequently, Dalton tried to solve the problem by hypothesising some general rules, which, unfortunately, in many cases proved incorrect, giving rise to disputes that much delayed the acceptance of his atomism. His proposed rules of combination stipulated that: (a) If two elements, A and B, formed a single compound, it was assumed to be binary (i.e. the molecule is formed of one atom A and one atom B), unless there was explicit cause to believe the contrary; (b) If the two elements formed two distinct compounds it was assumed that one was binary and the other ternary (i.e. molecules formed of one atom A and two atoms B or two atoms A and one atom B); (c) If the compounds were three, one was binary, the other two ternary, and so on for compounds of increasing complexity. Excepting the criteria of simplicity, there were no real reasons why this scheme should reflect reality. In this scheme’s actual implementation, there was a further potential element of confusion stemming from another of Dalton’s assumptions regarding compound atoms (molecules). Years earlier he had asked himself why the atmospheric air, a mixture of gases of different weight (mainly nitrogen and oxygen), remained homogenous and the heaviest component did not pile up at the bottom and the lightest rise to the top. He had concluded that this happened because the atoms of the same element (equal atoms) repelled each other, while the atoms of different elements (different atoms) did not. Consequently, for Dalton, the ultimate constituents of single elements could not be compound atoms (molecules), but must be single atoms. Thus, for Dalton the elements most common in the gaseous state, such as nitrogen, oxygen and hydrogen must consist of single atoms. Since (as we now know) these consist of bi-atomic molecules, this further assumption would become another source of great confusion and uncertainty. Given the relevant consequences for the evolving concept of matter, it is worth illustrating it with some examples. At that time, water was the only known compound of oxygen and hydrogen. In line with his rules, Dalton assumed that the compound atom (molecule) of water was composed of one atom of each of the two elements. Weight analysis showed that, to form water, eight parts by weight of oxygen combined with one part by weight of hydrogen. Thus, oxygen’s atomic weight (referred to hydrogen) for Dalton was 8, instead of the correct value of 16. Similarly, the only compound of nitrogen and

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hydrogen known to Dalton was ammonia, whose compound atom he therefore took as consisting of one atom of each of the two elements [4]. According to weight analysis the relative weights of hydrogen and nitrogen were in a ratio of one to five. Therefore, for Dalton the atomic weight of nitrogen was 5, instead of 14, the correct value. In other cases, however, his rules proved correct as in the case of the two carbon oxides (carbon monoxide and carbon dioxide). Dalton considered the first to be a binary compound constituted of one atom of carbon and one of oxygen, and the other to be ternary compound formed by one atom of carbon and two of oxygen. Through weight analysis he therefore attained the correct ratio of the relative weights of oxygen and carbon atoms. Despite its modernity and innovativeness, Dalton’s atomistic theory was received with caution by the scientific community. Though adopting his general ideas, many still considered atoms only as a heuristic instrument, a tool for understanding better the behaviour of elements, without necessarily accepting the real existence of atoms themselves. It would take more than half a century for Dalton’s atoms to become a reality in the chemical concept of matter. Contradictions in Dalton’s theoretical framework began to emerge when the amount of elementary gases contained in gaseous compounds were measured more precisely. In 1805 French physicist and chemist Joseph Louis Gay-Lussac (1778– 1850), along with German naturalist and geographer Alexander von Humboldt (1769–1859), published an essay demonstrating that, when hydrogen and oxygen combine to form water, two volumes of hydrogen and one oxygen give rise to two volumes of water vapour. In his atomic theory Dalton had instead assumed that that one volume of hydrogen combined with one volume of oxygen to produce one volume of water vapour. In the years following, Gay-Lussac continued his experiments using other gases, so expanding the knowledge of combination ratios. He found, for example, that one volume of chlorine and one of hydrogen gave two volumes of hydrochloric acid, that two volumes of nitrogen and one of oxygen gave two volumes of nitrous oxide, that one volume of nitrogen and three of hydrogen gave two volumes of ammonia, and so on. He published his results in the 1808 article Sur les combinaisons des substances gazeuses les unes avec les autres, where he formulated his law on combination ratios of gaseous substances, stating: When two gaseous substances react to form new substances, also gaseous, the volumes of gases reacting and those produced are in ratios expressed by whole and simple numbers. The results of Gay-Lussac, although objectively bolstering atomism, were not well received by Dalton (who believed them to be mistaken) since they did not agree with his combination schemes. Apparently, Dalton never realized the importance of Gay-Lussac’s law for the progress of atomism and continued to call its validity into question.

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7.1.1 Avogadro’s Law In 1811 Italian chemist Amedeo Avogadro (1776–1856) showed that the results of Gay-Lussac on the volumes of combination could be explained assuming what would become known as Avogadro’ s law (or principle), namely that equal volumes of different gases, at the same pressure and temperature, contain the same number of molecules.1 Referring to Gay-Lussac’s results, Avogadro affirms that [5]: It must be admitted that very simple relations also exist between the volumes of gaseous substances and the numbers of simple or compound molecules which form them. The first hypothesis to present itself in this connection, and apparently even the only admissible one, is the supposition that the number of integral molecules in all gases is always the same for equal volumes …; the ratios of the masses of the molecules are then the same as those of the densities of different gases at equal temperature and pressure.

However, Avogadro’s hypothesis explained the combination ratios observed by GayLussac only if one assumed that the particles forming gases could also be molecules containing equal atoms (i.e. atoms of the same element). For example, the observation that two volumes of hydrogen and one volume of oxygen gave two volumes of water vapour, was explained if the particles constituting oxygen and hydrogen were molecules containing two atoms each. Upon combining, each molecule of oxygen gives an atom to a molecule of hydrogen, so forming a number of molecules of water equal to those originally present in the two volumes of hydrogen (in modern notation: 2H2 + O2 = 2 H2 O). Similarly, the observation that one volume of hydrogen combines with one volume of chlorine to give two volumes of hydrochloric acid, is explained naturally if the particles constituting hydrogen and chlorine are molecules containing two equal atoms and if, in the chemical combination, the initial molecules split, forming new molecules composed of one hydrogen atom and one of chlorine (H2 + Cl2 = 2 HCl). Avogadro’s understanding that many gaseous elements exist in the form of a bi-atomic molecule was a crucial intuition. As a result of his considerations Avogadro formulated the first clear definition of molecule: By molecule is meant the smallest aggregate of atoms, equal or different, able to exist independently and possessing all the chemical and physical properties of the resulting substance, constituted of a collection of molecules.

With Avogadro’s hypotheses, the atomistic theory of matter started to take a definitive form. These hypotheses supplied a general method for determining molecular and atomic masses of substances in the gaseous state. In fact, if equal volumes of different gases, at the same pressure and temperature, contained the same number of molecules, the ratio between their weights became the ratio of the masses of the single particles contained in them. However, in the first decades of the nineteenth century Avogadro’s hypotheses were not generally accepted by the major scientists of the time. In particular, existing proofs seemed insufficient for the idea of the existence of molecules formed of 1

It should be remembered that, in 1814, also the French scientist André-Marie Ampere (1775–1836) suggested this hypothesis.

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equal atoms to be accepted. The widespread opinion, as held by Dalton, was that atoms of the same element repelled each other and so could not combine to form molecules. Meanwhile, strong support for this belief had also arrived from electrolysis experiments. According to the field’s most authoritative researchers (Humphry Davy and Jöns Jacob Berzelius), these experiments seemed to demonstrate that the forces linking atoms were electrical in nature, and, therefore, provided a straightforward reason (i.e. having the same charge) for the hypothesis that equal atoms repelled each other. Hence, Avogadro’s theory was practically ignored in the first half of the nineteenth century. Not until 1860, when Avogadro was already dead, was the scientific world’s attention drawn again to his work by another Italian chemist, Stanislao Cannizzaro (1826–1910). Neglect of such an important law delayed the progress toward understanding the structure of matter for nearly half a century.

7.1.2 The Dualistic Theory of Affinity Davy’s idea that attraction between elements, responsible for forming chemical compounds, is essentially electrical in nature, was taken up and developed by Swedish chemist Jöns Jacob Berzelius (1779–1848). From the observation that in electrolytical decomposition elements and compounds are attracted either to the positive pole or the negative one, Berzelius, in 1812, proposed the electric theory of chemical affinity whereby atoms and/or groups of atoms were electrically charged and characterized by their charge’s sign: defined electropositive (i.e. with a positive charge) when attracted toward the negative electrode, or electronegative (i.e. with a negative charge) when attracted toward the positive electrode. Berzelius claimed that the formation of compound substances was due to the electrical attraction between electropositive and electronegative atoms or groups of atoms, their union leading to the partial neutralization of opposite charges. Then, electrolysis was the exact opposite of chemical combination: the electric charges lost in combination were restored in the two fragments of the compound, which separated in the electrolytic process and appeared as ions (so called from the Greek term ion, meaning traveller). Furthermore, according to Berzelius, in chemical combination neutralization was seldom total, and the charge remaining allowed a group of atoms to form a more complex compound, though linked more weakly, with another group of atoms of opposite charge. The more complex compound was closer to electrical neutrality than were its constituent groups. This theory, which posited the existence of two fundamentally different types of elements and groups of elements, electropositive ones and electronegative ones, was named dualistic theory. It is worth emphasizing that, like Dalton’s theory, the dualistic theory presupposed that equal atoms, having identical electric charge, repelled each other and, therefore, denied the existence of binary molecules formed by equal atoms. As we have seen, this had negative consequences for the ready acceptance of Avogadro’s theory. Berzelius was also an enthusiastic supporter of Dalton’s atomic theory and contributed greatly to the fixing of atomic weights. From 1810 he conducted a series

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of experiments to measure ratios of combination and equivalent weights (by 1816 he had already studied 2000 different compounds) [1]. This allowed him to compile, in the 1830s, a reasonably precise table of atomic weights of all the forty or so elements then known, determined in relation to oxygen. Berzelius also developed a theory of gases based on the volumes’ law of GayLussac. Through it he was able to assign the correct formulas to a certain number of important compounds, such as water vapour, hydrochloric acid and ammonia, which Dalton, using the criteria of maximum simplicity, had determined incorrectly. In his discussion of the theory of volumes Berzelius stated [5]: Experience shows that just as the elements combine in fixed and multiple proportions by weight, they also combine in fixed and multiple proportions by volume, so that one volume of an element combines with an equal volume, or 2, 3, 4, or more volumes of another element in the gaseous state …The degrees of combination are absolutely the same in the two theories, and what is named atom in the one (Dalton’s) is called volume in the other.

By means of his theory of volumes Berzelius deduced that the correct formula of water, hydrochloric acid and ammonia were H2 O, HCl e NH3 , since the combinations between volumes came in the ratios 2:1, 1:1 e 1:3, respectively. Notably he did not use Avogadro’s law to get this result. In particular he did not assume that the ultimate particles of hydrogen, oxygen, chlorine and nitrogen were bi-atomic molecules (H2 , O2 , Cl2 , and N2 ), believing the formation of molecules composed of equal atoms to be impossible. To Berzelius we also owe the concept of radical, i.e. the idea that a group of atoms can enter, unchanged, in a whole series of compounds, behaving as if it were a simple element. It is also worth remembering that Berzelius invented the modern system of symbols for the elements, which uses the first letters of the name to identify them.

7.1.3 Faraday and the Electrolysis Significant contributions to the progress of electrolysis and of that branch of chemistry spawned from it, electrochemistry, came from research conducted by the great English physicist Michael Faraday (1791–1867), who rendered the investigation of these processes quantitative. In collaboration with philosopher and scholar William Whewell (1794–1866), he contextually developed electrochemistry’s terminology, coining terms like electrode, anode, cathode, anion, cation, electrolyte and electrolysis [6]. Faraday used the term ion to indicate the particles, of still unknown nature, which, inside the electrolyte, migrated toward the electrodes transporting the electric current. From the observation that often metals dissolved and entered in solution at one electrode and other metals escaped from the solution at the other electrode, he deduced that the ions could transport matter from one place to another. Therefore, ions must be of two types, those attracted by the cathode and therefore positive, which he called cations, and those attracted by the anode and thus negative, and which he named anions.

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His research into electrolysis led Faraday to believe in a quantitative relationship between the amount of decomposed substance and the amount of current passed through the solution. He was able to show how the only important factor in electrolytic processes was the amount of current, while other parameters, such as dimension and number of electrodes, had little influence. The discovery of the relationship between current and amount of substance was not easy, since secondary reactions often complicated the results, making their analysis uncertain. In the years 1832–33 F formulated the two laws of electrolysis bearing his name and summarizing the gist of his research: The quantity of substances released or deposited at an electrode during electrolysis is directly proportional to the quantity of electricity passed through the electrolyte. When the same quantity of electricity is passed through solutions of various substances, the quantity of substances released are proportional to their equivalent weights.

Faraday’s research had the great merit of establishing a precise quantitative relationship between electricity and chemistry. The gist of the second law was that the amount of current liberating 1 g of hydrogen liberated other substances in quantities equal to the chemical equivalents of these substances. Faraday commented thus [6]: I have proposed to call the numbers representing the proportions in which they are evolved electro-chemical equivalents. Thus, hydrogen, oxygen, chlorine, iodine, lead, tin are ions; the four former are anions, the two metals are cations, and 1, 8, 36, 125, 104, 58 are their electro-chemical equivalents nearly.

7.2 Chemical Bonds and Valency The second half of the nineteenth century saw the need to explain a great quantity of data about chemical reactions and to classify a multitude of newly synthesized compounds. Consequently, it became a matter of urgency to understand why some elements could combine to form compounds and others not, as well as why the formation of compounds occurs according to proportions which depend on the type of atoms involved. Toward the mid-century issues such as affinity, equivalents and atomic weights became once again central. One also had a revival of the atomism, which had slipped into the background of chemical theorizing. English chemist Edward Frankland (1825–1899) was the first to make, in 1852, a real step toward solving the combination numbers problem. Frankland, investigating various organic compounds, both metallic and not, discovered that each element only combined with a well-defined number of organic groups [3]. This number he called atomicity or valency of the element. The valency (or valence) thus came to define a measure of one element’s ability to combine with other elements (or, as soon became clear, the ability of an atom to combine with other atoms). The name valency derived from the term equivalency used previously. Frankland also noted (making the first step toward the periodic system of elements) that groups of elements displayed the same valency: for example, antimony, arsenic, nitrogen, and

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phosphorus always combined with three or five organic radicals, while mercury, zinc and oxygen combined with two. The concept of valency was clarified and deepened a few years later by Scottish chemist Archibald Couper (1831–1892) and German chemist Friedrich August Kekule (1829–1896). They introduced into chemistry the idea of bonds between atoms, initiating the modern representation of valency and the ways atoms combine. For example, if water is examined in light of these new concepts, we would say that hydrogen (H) has valency one since it can form a single bond, while oxygen (O) has valency two, and we could represent the elementary units of water as H–O–H, the dash indicating the bond between two atoms. Similarly, examining ammonia, we deduce that nitrogen has valency three and we could represent the elementary unit of ammonia as

These representations of elementary units (today molecules) of substances were called structural formulae and helped greatly in understanding matter’s structure. Furthermore, according to these researchers, the experimental data showed that bonds could also be made between atoms of the same element, even if the nature of such bonds was far from clear. Further important clarification of the concept of valency came in 1857 thanks to Kekule’s suggestion that an atom of carbon, the most important element inside organic compounds, can combine with a number of atoms or groups of atoms varying from two to four. On this basis Kekule designed structural models of several organic compounds, and used the models to interpret these compounds’ reactions. It should be pointed out, however, that in the years of their formulation such ideas of valency were still ill-defined through lack of any unanimously accepted criterion to establish an atom’s combination number. Most chemists concerned with combinations of elements preferred to use equivalent weights which were determined directly, rather than atomic weights which involved the uncertain estimate of the number of atoms that combined. The rejection of Avogadro’s hypothesis left chemists without a general method for determining the number of atoms that combined. Despite the developments due to Dalton and Berzelius, from 1820 until 1860, the atomic theory played only a small part in the chemical scrutiny of matter. The stalemate was broken in 1860 thanks to the work of Italian chemist Stanislao Cannizzaro (1826–1910). In 1858 Cannizzaro wrote an essay in which he showed how, starting out from Avogadro’s hypothesis, a gaseous substance’s molecular weight could be found from determining the density of its vapour. In fact, accepting that equal volumes contain the same number of particles, the ratio of the density of any given substance to the density of a standard substance supplies the ratio of molecular weights. As the standard the lightest element, hydrogen, should be chosen. Cannizzaro also noted that, as already clarified by Avogadro in1811, the hydrogen

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molecule contains two atoms. Thus, the density relative to hydrogen must be doubled to give the correct molecular weight. Cannizzaro distributed copies of a booklet laying out his ideas in a more complete form during the First International Chemical Congress held in Karlsruhe in 1860. Furthermore, during the congress, he promoted approval of an important proposal, clarifying the distinction between atom and molecule, so eliminating a source of confusion going back to the times of Dalton and Avogadro [1]. The proposal stated: It is proposed to adopt different concepts for molecule and atom, considering as a molecule the smallest quantity of a substance entering into a reaction and conserving its physical and chemical character, and understanding an atom to be the smallest quantity of an element entering the molecule of its compounds.

Cannizzaro’s booklet and the ensuing research activity rapidly convinced most chemists that Avogadro’s hypothesis was generally valid.

7.3 Elements and Periodic Law In the first half of the nineteenth century, with the number of discovered elements increasing,2 it became evident that they could be grouped into families exhibiting similar chemical properties. In 1817 German chemist Johan Dobereiner (1780– 1849) showed that the atomic weights of calcium, strontium and barium formed an approximate arithmetic series. In 1826 French chemist Antoine Jérôme Balard (1802–1876), after discovering bromine and analysing its properties, predicted that chlorine, bromine and iodine would form another arithmetic series. Berzelius showed that this was approximately true and called the three elements, collectively, halogens (i.e. formers of salts). In the following years French chemist Jean Baptiste André Dumas (1800–1884) used the similarity between reactive properties in an attempt to regroup the elements into natural families, and placed boron, carbon, and silicon in one group and nitrogen, phosphorus, and arsenic in another. In the 1860s, thanks to the acceptance of Avogadro’s law and his hypotheses, finally atomic weights and valences became self-consistent, generally approved and based on precise analysis. New attempts were made at classification. English chemist John Newlands (1837–1898) and French mineralogist Alexander Béguyer de Chancourtois (1820–1886) independently observed that if chemical elements were arranged in order of atomic weight, at regular intervals there appeared elements with very similar chemical properties. In particular, Newlands noted that [5]: the eighth element, starting from a given one, is a kind of repetition of the first, like the eighth note in an octave of music.

He also underlined that the regularity was observed only when one used the new atomic weights proposed by Cannizzaro. In 1865 English chemist William Odling 2

Between 1790 and 1830, 31 new elements were discovered.

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(1829–1921) ordered the elements into a table very similar to the one proposed by Mendeleev a few years later. It is well known that the periodic law and the modern classification of elements in the form of periodic table, were proposed in practically definitive form almost simultaneously if independently by Dmitri Ivanovich Mendeleev (1834–1907) in Russia and by Julius Lothar Meyer (1830–1895) in Germany. Although he had formulated it previously, Meyer only published his version in 1870, a year after Mendeleev, who had the honour of giving the Periodic table its name. The law of the periodicity of the elements stated that the chemical properties of elements varied in a periodic manner as a function of their atomic weight 3 and the periodic table arranged the elements so as to display such periodicity. In the first version drawn up by Mendeleev (shown in Fig. 7.1) groups of elements with similar properties were arranged into horizontal rows while the columns, containing the elements whose properties vary continuously, constituted periods. In a later version Mendeleev changed the role of the rows and the columns, the table assuming a form more similar to today’s. In Mendeleev’s day, many elements, including the entire group of noble gases, had still not been discovered. Therefore, in formulating his periodic table, Mendeleev introduced some important modifications to the simple sequence of elements according to atomic weight. These were later confirmed by experiment, greatly bolstering his construction’s credibility. To maintain the correct alignment of properties of the elements into the different groups, he introduced three empty boxes. According to him, these corresponded to elements as yet undiscovered. Based on the group they belonged to, he named them eka-boron, eka-aluminum, eka-silicon, the properties of which he predicted with surprising precision. Mendeleev also noted that, sorting strictly by atomic weight, there were a few discrepancies and was daring enough to change the order of some elements, re-inserting them in groups of belonging based on their properties. For example, according to its atomic weight, tellurium found itself below bromine, which has completely different properties. However, the atomic weight of tellurium 127.6 is only slightly greater than iodine’s 126.9, and Mendeleev thought fit to exchange the two elements’ order so that under bromine could be found iodine, which exhibits highly similar properties. The periodic system of elements was initially greeted with a certain scepticism. This turned to real appreciation when Mendeleev’s predictions about the missing elements were confirmed by their discovery. eka-alluminum, renamed gallium, was discovered in 1874 by French chemist Paul-Émile Lecoq de Boisbaudran (1838– 1912). In 1879 Scandinavian chemist Lars Fredrik Nilson (1840–1899) discovered eka-boron, which he renamed scandium. Finally, German chemist Clemens Alexander Winkler (1838–1904) isolated eka-silicon, renaming it germanium, in 1885. These discoveries gave periodic table and the hypotheses on which it was based wide-ranging credibility.

3

In 1913, alongside his investigations into the structure of atoms, English physicist Henry Moseley (1887–1915) showed how the correct arrangement of elements was according to atomic number and not to atomic weight.

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Fig. 7.1 Mendeleev’s 1869 periodic table of elements. Wikipedia Public Domain

In the second half of the nineteenth century some very important facts therefore were definitively confirmed: material substances were underpinned by a considerable number of elementary substances, the elements, whose combination gave rise to compound substances; these elements were distributed into seven groups (which became eight after the discovery of the noble gases) characterized by similar properties regarding valency and chemical activity. Henceforth the central problem became what exactly these elements’ microscopic structure was. However, surprising as it may seem in looking back, the insights acquired were still not universally seen as proof of atoms’ real existence, and the division into two types of atomism professed by nineteeth century scientists still held its ground. Indeed, there was a chemical atomism, accepted by chemists, but in general only implicitly, and which formed the conceptual basis for assigning relative atomic weights and molecular formulas. Then there was a much more controversial physical atomism, with its claims regarding the ultimately mechanical nature of all substances. One difficulty still to be overcome when applying atomic theories to chemistry was due to the fact that the atoms of

References

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the different elements were believed to be basically equal to one another. This belief hindered any explanation of the highly specific character of chemical processes, as Boyle had already noted in the seventeenth century. Although from the 1870s the peculiarities of these two atomic views started to become clearer, their unification would come only in the first years of the twentieth century. Meanwhile physics had taken a different path which would finally lead to incontrovertible proof of atoms’ existence and the comprehension of atomic structure.

References 1. 2. 3. 4.

Gribbin J (2003) Science: a History. Penguin Books Brock W H (1993) The Chemical Tree. W. W. Norton & Company, New York Mason S F (1962) A History of the Sciences. Collier Books, New York Holmyard E J (1958) The chemical industry. In: Singer C, Holmyard E J, Hall A R, Williams T I (ed) The Industrial Revolution. A History of Technology, vol. 4. Oxford University Press, p. 231 5. Partington J R (1989) A Short History of Chemistry. Dover Publications, New York 6. Ihde A J (1984) The development of modern chemistry. Dover Publications, New York

Chapter 8

Physics and Matter in the Nineteenth Century

In the nineteenth century atomism and the particle conception of matter, revitalised in chemical circles by Dalton, received a boost in physical circles by the completion of that imposing theoretical edifice known as the kinetic theory of gases. It was a theoretical construction, based on the concept of ensembles of microscopic particles (atoms or molecules) which had taken more than a century for its definitive formalization and whose extraordinary success greatly contributed to the final acceptance of atomism. A decisive support for the idea of a matter composed of atoms and molecules came at the century’s end, thanks to studies and discoveries in a highly specialized branch of physics, electric conduction in rarefied gases. Such studies would culminate with the discovery of electrons, thus demonstrating unequivocally the existence of microscopic corpuscles, endowed with definite mass and charge, within matter.

8.1 Kinetic Theory of Gases In the context of atomism and corpuscular theories it had become natural to hypothesise a gas as a collection of a great number of microscopic corpuscles (also referred to as atoms or molecules or particles by proponents of the various theories), distributed uniformly in empty space and in constant random movement. It had been also hypothesised that these corpuscles collided frequently with themselves and the walls of the vessel containing them and that their movement was subject to the laws of Newtonian mechanics. What goes under the name of Kinetic theory of gases is the result of advances stretching two centuries in the attempt to calculate the various physical macroscopic properties of this ideal system by using classical mechanics. Its acceptance as a true model of microscopic reality kept pace with the success obtained in explaining a growing number of experimental observations and fundamental concepts, such as the conservation of energy and the second law of thermodynamics.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_8

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The first bricks of this edifice had been laid by Newton, a convinced atomist, with his demonstration of Boyle’s law. Indeed, applying the laws of mechanics to a model gas where the corpuscles repelled each other with a force inversely proportional to the square of the distance, Newton had shown that gas’s pressure and volume were inversely proportional, that is they behaved according to the law demonstrated experimentally by Boyle. A more significant step was due to Swiss mathematician and physicist Daniel Bernoulli (1700–1782). His Hydrodynamica, published in 1738, introduced explicitly the basic tenets upon which kinetic theory was built [1]. Namely: (a) gas consists of particles in rapid and chaotic motion, which collide into each other; (b) in the time interval between collisions, the particles move with rectilinear and uniform motion in a random direction; (c) the collisions of the particles on the walls of the container give rise to the macroscopic pressure that we experience. From these hypotheses Bernoulli was able to calculate the dependence of gas pressure on the volume occupied, which agreed with Boyle’s law. Moreover, Bernoulli established that an increase in temperature involved an increase of the kinetic energy of the gas, supporting the hypothesis that heat is the macroscopic manifestation of the movement of particles. Bernouilli’s theory was the first example of how, starting from a model of corpuscles assimilated to the material points of mechanics, one could deduce a mathematical relationship between macroscopic physics quantities, such as pressure, volume and temperature. However, despite its intrinsic value, Bernoulli’s theory was practically ignored till the early nineteenth century. This was for various reasons, the most important being that most scientists of the time still held to phlogiston theory to explain thermal phenomena and looked suspiciously on atomistic concept of matter. Kinetic theory returned to the attention of researchers in the first decades of nineteenth century when new concepts of heat and energy were being developed and there was renewed interest in atomistic theories of matter. In 1845 the Scottish civil engineer and amateur physicist John James Waterston (1811–1883) drew up a version of the kinetic theory, able to explain in an exhaustive way the physics findings. His article, submitted to the Royal Society, was not published, however, and his contributions went completely ignored. Waterston’s work was only re-evaluated in 1892, when Lord Rayleigh recovered the manuscript. Recognising that it contained an almost complete development of the kinetic theory, he had it published for its historical interest [2]. It was in the following decade that the theory’ status changed drastically thanks to the works of German physicist and chemist August Krönig (1822–1879) and German physicist and mathematician Rudolf Clausius (1822–1888). In 1856 Krönig published an article in which, using kinetic theory, he derived the relationship linking a gas’s pressure and volume to its kinetic energy. The following year Clausius, independently of Krönig, obtained the same relationship, now known as KrönigClausius’ equation (or law). These works contributed to the definitive systemization of kinetic theory and its growing acceptance by the scientific community. In their articles Krönig and Clausius, like previous authors, defined gas as the ensemble of a great many identical particles, characterized by mass and velocity, distributed uniformly in space and in constant random motion. The characteristics assumed for

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the gas made the problem a well-defined one, solvable by elementary mathematical methods (for more details see Krönig-Clausius Equation in Appendix B). The authors demonstrated that, in a gas, the product pressure times volume equals two thirds of the average kinetic energy of the particles which constitute it (Krönig-Clausius’s equation). A further decisive contribution to the construction of kinetic theory was made in the same period by the great English physicist James Clerk Maxwell (1831– 1879). Indeed, among his many extraordinary accomplishments, one of the most significant was to calculate the distribution function of gas particle velocities, now referred to as Maxwell’s distribution law of velocities. This contribution, expounded in the 1860 article Illustrations of the dynamical theory of gases, is a noteworthy advance in the theoretical construction of ideal gas. In previous theories information on the velocity of single particles was missing; as such, only physical quantities depending on average values could be calculated. Maxwell’s theory supplied the velocity information in the form of probability that a particle has a given speed. For a system composed of a great number of particles, this equates to specifying the number of particles with a given speed. This information allowed kinetic theory to calculate, a priori, a series of important physical quantities and to predict the results of many experiments, thus verifying a posteriori the justness of the theory itself. Thanks to Krönig, Clausius and Maxwell, the kinetic theory of gases received a widespread consensus in the scientific community. As well as its undoubted capacity in explaining the physics of gases, its success was due to the fact that it bolstered the concept of heat as kinetic energy and supplied a microscopic interpretation for the then recently formulated principle of energy conservation.

8.1.1 Nature of Heat In the first decades of the nineteenth century there were two competing concepts of heat and thermal phenomena. One considered heat to be a material substance; the other believed that heat and thermal phenomena resulted from the motion of the microscopic particles composing matter. The first notion, the dominant one at the time, was provided by the caloric theory, proposed by Lavoisier as replacement for phlogiston theory. It was the final and most sophisticated version of the idea that heat is a fluid substance, an idea that was born in the Greek world and had led many philosophers, primarily Empedocles and Aristotle, to designate fire as one of the primitive elements. Caloric was envisaged by Lavoisier as a thin and imponderable fluid, similar to the fluids postulated previously to explain electrical and luminous phenomena. The quantity of caloric contained by a body determined its being hot or cold, in such a way that, if two bodies having different temperatures were put in contact, caloric flowed from the hotter one to the colder. The second notion was based on the idea that heat and thermal phenomena are the sensory manifestation of the movement of microscopic particles of which substances

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are composed. This concept took hold in the seventeenth century, with the emergence of mechanical philosophy and the revival of corpuscular theories of matter. The concept was also consistent with the assumptions underlying the kinetic theory of gases. Indeed, it was already shown by Bernoulli, in his formulation of the theory, that an increase in temperature leads to an increase of vis viva (live force), as kinetic energy was then called, suggesting that heat was the macroscopic manifestation of the latter. The problem of the nature of heat was solved in mid-century in parallel with the development of thermodynamics.1 Following the experiments of English physicist James Prescott Joule (1818–1889), who had demonstrated the conversion of mechanical work into heat, Clausius in Germany and Lord Kelvin (William Thomson) in England developed the concept of energy as quantity that is conserved, so formulating the first principle of Thermodynamics. Clausius, in his formulation of the kinetic theory of gases, also showed that the internal energy of a gaseous system corresponded to its vis viva (kinetic energy). Finally, using the equation of state of the gases, he demonstrated that the kinetic energy of the particles is proportional to the temperature of the gas and that, in a transformation at constant volume, it undergoes a change equal to the amount of heat received or ceded (for more details see Kinetic Energy and Temperature in Appendix B). Kinetic theory attained its final form thanks to Austrian physicist Ludwig Eduard Boltzmann (1844–1906), to whom we owe the introduction of new statistical methods into the theory itself. In 1871 Boltzmann generalized Maxwell’s formulation, drawing up what goes under the name of Maxwell–Boltzmann’s distribution. His theoretical contributions are germane to all branches of physics, linked to fundamental problems such as the temporal evolution of complex systems and the significance of the direction of time. Following in Maxwell’s tracks, Boltzmann showed that, whatever a gas’s initial configuration (i.e. the distribution of the velocities of molecules to begin with) such velocities evolve toward distributions with a higher probability of occurring. The distribution of highest probability corresponds to the configuration of system’s final equilibrium. Boltzmann stated that this evolution is a universal probability law and goes beyond gaseous systems. He also highlighted how this behaviour fits in with that of the new physics quantity, entropy, then recently introduced into thermodynamics. Thus, the concept of entropy growth can be identified with the tendency of systems composed of large numbers of particles to evolve from a state of smaller probability to one of higher probability. Entropy, a function of state hitherto somewhat mysterious, became, in Boltzmann’s formulation, the measure of disorder in the motion of the microscopic constituents of matter. The relationship between the entropy of a system and the probability of its macroscopic state is given by the famous formula S = k log (W ), where S is the entropy, W is the probability of the distribution and k a universal constant, called Boltzmann’s constant in his honour (This formula is inscribed on Boltzmann’s tomb in Vienna). The work of Clausius, Maxwell, and Boltzmann clearly showed how a mechanical model of gas (i.e. a gas thought of as a combination of microscopic particles 1

The term thermodynamics was introduced by William Thomson in 1851.

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obeying the laws of Newtonian mechanics) make it possible to express all its macroscopic physical properties (pressure, density, temperature, and new properties, such as energy and entropy), as a function of microscopic parameters, such as particle velocities, collision rates, etc. This represented a strong, albeit indirect, support to the atomistic and corpuscular nature of matter. We conclude the exposition of kinetic theory by briefly illustrating two subjects where kinetic theory played a decisive role, specific heats and Loschmidt’s experiment, both important in the quest of the matter’s structure.

8.1.2 Specific Heats Among kinetic theory’s successes was its capacity to explain specific heat (i.e. the amount of heat required to raise the temperature of one gram of a substance by one degree). The study of specific heats had an important role both in understanding the nature of heat and the structure of matter, and in the development of quantum mechanics in the early twentieth century. The concept of specific heat was first introduced around 1760 by Scottish scientist Joseph Black, who had found that the quantity of heat needed to obtain the same increase in temperature varied from substance to substance. These observations implied that thermal behaviour was a specific property of substances. Evidence that the study of specific heats could play an important role in understanding matter’s structure came in 1819, when French scientists Pierre Louis Dulong (1785–1838) and Alexis Thérèse Petit (1791–1820) published the article Sur quelques points important della théorie de la chaleur, containing the data they had collected on a great number of solid elements. They found that the specific heat decreased as the atomic weight of elements increased in such a way that the product of the two quantities remained almost equal for all the elements. They named this new quantity atomic heat and enunciated the law whereby the atoms of all simple bodies have exactly the same capacity for heat (law of Dulong-Petit). The same year (1819) saw the publication of the work of two French chemists Nicolas Clément (1779–1841) and Charles Bernard Desormes (1777–1862) on the specific heats of gases, in which the authors were able to determine precisely the ratio between specific heats measured, respectively, at constant pressure and constant volume. Kinetic theory well accounted for these properties of specific heats. Clausius calculated the specific heat at constant volume and that at constant pressure for a gas in which the energy of the molecules was only kinetic, showing how their ratio was in line with experimental data. In 1871 Boltzmann showed that it was possible to explain the law of Dulong and Petit, assuming that, for the purposes of calculating thermal properties, a substance in the solid state can be considered as an ensemble of microscopic harmonic oscillators (i.e. particles (atoms or molecules) that vibrate around their position of equilibrium) (for more details see Harmonic oscillator in Appendix B). According to Boltzmann’s results, the specific heats of substances in

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the solid state have a universal value equalling 3R, where R is the universal constant of gases.

8.1.3 Loschmidt’s Experiment In 1865, when atoms and molecules, despite their increased presence in the theories of chemistry and physics, continued to be considered hypothetical ill-defined entities, Austrian scientist Joseph Loschmidt (1821–1895) conceived an ingenious procedure which, using the concepts of kinetic theory, made it possible to assess for the first time their dimensions as well as to estimate the Avogadro number. To gauge the size of the hypothetical microscopic particles, Loschmidt reasonably assumed the volume of a determined quantity of substance in the liquid state to be about equal to the product of the number of particles (atoms or molecules) present multiplied by the volume of each particle. Considering, instead, the same quantity of substance in the gaseous state, Loschmidt was able to obtain its volume by exploiting the kinetic theory. Using an elementary mathematical development, Loschmidt obtained, for particles assumed to be spherical, the remarkable result that the diameter of the particles was equal to eight times the product of the mean free path times the condensation ratio.2 Finally, Loschmidt applied this relationship to the air, the only gaseous substance whose condensation ratio he knew, obtaining a value for the diameter of the particles equal to, approximately, one billionths of a meter, somewhat larger than the true one, but of the right order of magnitude. Knowledge of the molecules’ diameter also allowed Loschmidt to assess the Avogrado number (also called Loschmidt-Avogadro number), i.e. the number of molecules contained in a mole of gas under standard conditions of pressure and temperature, obtaining a value notably near the current accepted value of 6.022 × 1023 .

8.2 Nature of Electric Charges In the second half of the nineteenth century, in scientific circles, the acceptance of Avogrado’s hypotheses had made it quite natural viewing chemical reactions and combination processes in terms of particles which combine and separate. There was a growing belief, above all in the English scientific world, that this microscopic vision of things must correspond with reality. However, especially where the influence of chemistry was stronger, many scientists continued to believe that atoms were nothing but a heuristic concept, its only value being that they helped to describe the behaviour of chemical elements. A fundamental difficulty for the final acceptance of the real 2

The mean free path of a gas particle is the mean value of the distance it travels between two successive collisions. The condensation ratio is the ratio between the volume of the gas and that of the liquid.

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existence of atoms and molecules was linked to the fact that the structure of these hypothetical components of matter was still completely unknown. Many, invoking the scientific method, viewed with suspicion the introduction into science of objects that could be neither revealed nor measured directly. Then, towards the century’s end, a decisive support for the idea of a matter composed of atoms and molecules came, quite unexpectedly, thanks to studies and discoveries in a highly specialized branch of physics, electric conduction in rarefied gases. These researches aimed to unravel another great problem of the time, the nature of electrical charges. In the second half of the nineteenth century, some of the most pressing scientific problems were linked with electricity and magnetism. Maxwell’s great synthesis had ruled that electric charges and electric currents are sources of electromagnetic fields, but it had left open the problematic nature of the charges themselves. For his part, Maxwell envisaged electricity as a continuous immaterial fluid permeating substances. However, research into electrolytic phenomena in the early nineteenth century, and the formulation of the concept of ions, as particles responsible for the flow of electric current in electrolytes, had begun to spread the idea of the electric charges being corpuscular and discontinuous in nature. In the second half of the century, such hypotheses became explicit and open to scientific debate. The first to suggest explicitly the existence of charged elementary corpuscles was Irish physicist George Johnstone Stoney (1826–1911). At the 1847 meeting of the Britannic Association held in Belfast, he pointed out that the ultimate significance of Faraday’s laws on electrolysis is that electricity, analogously to matter, is made up of corpuscles. Stoney noted that these corpuscles were all equal and indivisible, like atoms of electricity, and suggested calling them electrons. Furthermore, taking the density of atoms calculated by using the kinetic theory of gases as a basis, he estimated for the first time the value of this hypothetical elementary electric charge. The estimated value differed little from today’s 1.6 × 10–19 C. However, Stoney’s thesis received little attention at the time of its presentation. Indeed, in the literature, the great German scientist Hermann von Helmholtz (1821–1894), who in 1881 came up with the same explanation as Stoney, is often credited as the first to propose elementary electric charges’ existence. In this context is worth remembering that American physicist Henry Augustus Rowland (1848–1901) performed some experiments, inspired by Helmholtz, which demonstrated that an electric charge in movement is equivalent to an electric current and, reciprocally, a current is the result of the movement of electric charges. The idea of the existence of particles of electricity, was taken up by Dutch physicist Hendrik Antoon Lorentz (1853–1928). Prompted by Stoney, Lorentz also called these hypothetical particles electrons and in 1878 proposed a Theory of the electron able to describe with notable success the interaction of light with matter. The problem of the nature of electric charges was finally solved in the 1890s, thanks to studies and discoveries in a specific field of physics, electric conduction in rarefied gases.

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8.2.1 Cathode Rays and the Discovery of Electrons The study of electric phenomena in gases had begun following the invention of the vacuum pump in 1654 by German physicist Otto von Guericke (1602–1686), which enabled natural philosophers of the time to investigate the properties of rarefied airs. Among the many new phenomena to attract experimenters, were sparks and flashes of light generated when electricity was produced by rubbing the surfaces of the evacuated glass containers. It was noted that the sparks’ length increased upon decreasing the air pressure. However, due to the primitive nature of instruments available and the limited knowledge of the time, in the seventeenth and eighteenth centuries in-depth understanding of these phenomena was impossible. Such experiments remained essentially a matter of simple curiosity. Systematic experimentation into electric discharge in rarefied gases began in the mid-nineteenth century. It was facilitated by a general improvement in experimental techniques and measuring instruments, in particular by the invention of new vacuum pumps and new sources of high-voltage electric sources. Around 1856 German physicist and instrument-maker Heinrich Geissler (1814–1879) developed the mercury pump, a new kind of vacuum pump able to empty containers at a pressure of around 0.1 mm-Hg, one order of magnitude lower than that obtainable in the previous centuries. In the same context, Geissler invented a new type of tube for the electrical discharge in gases evacuated at these lower pressures. These tubes, commonly known as Geissler tubes, forerunners of current gas discharge lamps, were sealed glass cylinders of various forms with a metallic electrode at each end. Between the two electrodes was applied a very high voltage, varying from a few thousand to hundreds of thousands of volts, thanks to a new kind of high-voltage generators, induction coils, invented some years before. In these conditions inside the tube an electric discharged was unleashed and the gas emitted a brilliant glow, its characteristics depending on the pressure and type of gas used. Studies on electrical conduction in rarefied gases took a decisive step toward solving the problem of electric charges a decade later thanks to improved vacuum pumps, which allowed a more powerful evacuation of the tubes down to pressures of a thousandth (10–3 ) of mm-Hg. British physicist William Crookes (1832–1919), in the late 1860s, was among the first to study the discharge in these tubes, named later as Crookes’ tubes. It was found that, when the pressure dropped toward such low values, the space between the two electrodes became completely dark. In these conditions Crookes noted that, at the anode, the glass of the tube began to shine, as if there were something which, departing from the cathode, proceeded to strike the walls around the anode. These mysterious rays, also observed independently in the same period by German physicist Julius Pluecker (1801–1868), were characterized by propagating themselves in a straight line and stopping in the presence of an obstacle. The nature of these mysterious rays, baptized cathode rays by the German physicist Eugen Goldstein (1850–1930) in 1876, soon became a problem of great interest for a scientific community then actively engaged in trying to understand the nature of electromagnetic phenomena, the principles of thermodynamics and, if they existed,

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the nature of atoms and molecules. The last quarter of the century saw a lively debate on how these rays could be fitted into current theories of electromagnetic phenomena. Two currents of thought took shape. One, supported mainly by English physicists, held that cathode rays were particles of radiant matter, electrically charged. The other, supported mainly by German physicists, held that they were ethereal waves, meaning, a new form of electromagnetic radiation, an entity different from what transports the electric current in the tubes. To resolve the controversy numerous experiments were conducted. In particular, the effect of magnetic fields and electric fields on the trajectory of cathode rays was studied, these fields being already known to deflect the trajectories of electrically charged objects, but not electromagnetic radiation. However, these experiments continued to give contradictory and inconclusive results. The reason, as it would be discovered, was that the pressure under which the experiments were conducted was still too high and the rays’ trajectories were affected by the residual gas in the tube. The problem of the nature of cathode rays was fully resolved in the two years of 1896–97. Thanks to the experiments of English physicist Joseph John Thomson (1856–1940) and the German Philipp Lenard (1862–1947), the rays were identified as flows of new particles, hitherto unknown and hence given the name electrons. For his studies into the nature of cathode rays Lenard received the Nobel prize in 1905. A year later Thomson received the same award acknowledging him as discoverer of electrons. Let us summarise the main aspects of this ground-breaking discovery by citing Thompson’s exposition on being awarded the Nobel prize, his speech being intended for an audience of non- specialists [3]. The arguments in favour of the rays being negatively charged particles are primarily that they are deflected by a magnet in just the same way as moving, negatively electrified particles. We know that such particles, when a magnet is placed near them, are acted upon by a force whose direction is at right angles to the magnetic force, and also at right angles to the direction in which the particles are moving. ... I was able to obtain the electric deflection of the cathode rays. This deflection had a direction which indicated a negative charge on the rays. Thus, cathode rays are deflected by both magnetic and electric forces, just as negatively electrified particles would be.

Upon analyzing the effects of magnetic and electric fields on the trajectory of cathode rays, Thomson was able to determine the velocity of the particles and the ratio e/m between their charge e and their mass m (for more details Measurement of e/m in Appendix B). In his Nobel prize speech, Thompson commented thus on the results of the measurements [3]: The results of the determinations of the values of e/m made by this method are very interesting, for it is found that, however the cathode rays are produced, we always get the same value of e/m for all the particles in the rays. ...The value of e/m is not merely independent of the velocity. What is even more remarkable is that it is independent of the kind of electrodes we use and also of the kind of gas in the tube. This constant value, when we measure e/m in the c.g.s. system of magnetic units, is equal to about 1.7 x 107 ... hence for the corpuscle in the cathode rays the value of e/m is 1,700 times the value of the corresponding quantity for the charged hydrogen atom. ... Now it has been

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8 Physics and Matter in the Nineteenth Century shown by a method which I shall shortly describe,3 that the electric charge is practically the same in the two cases; hence we are driven to the conclusion that the mass of the corpuscle is only about 1/1700 of that of the hydrogen atom.4 Thus the atom is not the ultimate limit to the subdivision of matter; we may go further and get to the corpuscle, and at this stage the corpuscle is the same from whatever source it may be derived. The corpuscle appears to form a part of all kinds of matter under the most diverse conditions; it seems natural therefore to regard it as one of the bricks of which atoms are built up.

8.2.2 Canal Rays In 1886 German physicist Eugen Goldstein (1850–1930) discovered that, if one used a tube with a perforated cathode and applied an electric voltage of thousands of volts, weak luminous rays were observed to start from the holes and extended into the region behind the cathode. These rays’ luminosity depended on the gas contained in the tube, yellowish for air, pink for hydrogen, etc. Goldstein named these rays, Kanalstrahlen (canal rays) since they were produced by holes (canals) in the cathode. The first studies showed that, in presence of an electric field, these rays behaved as if they had a positive electric charge and came from the anode (thus their alternative name, anode rays). In the decade after their discovery, canal rays received much less attention than cathode rays. Only in 1898 was an in-depth study undertaken by German physicist Wilhelm Wien (1864–1928), the results being published in 1902. Using a very strong magnetic field he was able to deflect canal rays and establish definitively that they had positive charge. He also found that, at low pressure, it was possible to deflect them with an electric field and so was able to determine the ratio between their charge and their mass. The ratio was found to depend on the species of the gas left in the tube. The amount of positive charge was equal to or a whole multiple of the electron charge, while the mass matched that of the gaseous substances making up the residual atmosphere. Such evidence clearly indicated that canal rays were made of corpuscles (atoms or molecules) of gases that had lost one or more electric charges, in strong analogy with the positive ions of electrolytes. The insights into cathode rays and canal rays showed unequivocally that matter is made of atoms, that inside atoms exist positive and negative charges which balance each other, and that negative charges are electrons. Consequently, at the end of the nineteenth century the reality of atoms was widely accepted by the scientific community; finding out their structure became the new dominant problem. It seems ironic that the existence of atoms (indivisible, in Greek) was confirmed definitively by experiments showing their divisibility. 3

The method used by Thomson was the use of the Wilson’s cloud chamber. Around 1899 C. T. R. Wilson (1869–1959) had introduced Wilson’s cloud chamber, the first device that allowed to visualize the trajectory of charged particles. 4 In reality the mass is 1840 times smaller than that of hydrogen.

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References 1. Jeans J H (1904) The Dynamical Theory of Gases. Cambridge at the University Press 2. Waterston J J (1892) On the Physics of Media that are Composed of Free and Perfectly Elastic Molecules in a State of Motion. Philosophical Transactions MDCCCXCII:1–79 3. Thomson J J (1906) Carriers of negative electricity. Nobel Lecture, December 11, 1906

Part II

Matter and Old Quantum Theory

Chapter 9

The Appearance of Quanta

In the decade bridging the nineteenth and twentieth centuries, research into the structure of matter saw completely unpredicted developments, which proved to be of great relevance for science in general: To understand the properties of matter, the structure of atoms and molecules, and phenomena such as the interaction between matter and radiation, physics needed to be refounded; it became necessary to conceive a new mechanics valid for the microscopic world, namely quantum mechanics. From the beginning of the twentieth century, the story of the concept of matter identified with the story of quantum mechanics! Such a revolution originated from developments in two branches of research of the time: One (subject of the present chapter) involved the attempts to explain various experimental observations impossible to understand using concepts and principles of then-known mechanics and electromagnetism; the other (subject of the chapter to follow) concerned research to determine the structure of atoms.

9.1 Black Body Radiation Among the effects and experimental observations that the late nineteenth century had difficulty in explaining and whose solution would require the introduction of new physics concepts, there was the radiation emitted by the black body. The black body is an abstract construction emerging from the study of thermal radiation and its interaction with matter. It is well-known that it was precisely to explain black body radiation that German physicist Max Planck (1858–1947) put forward the quantization hypothesis, the idea underlying the conceptual revolution leading to the building of quantum mechanics. Although the fact that a hot body emits heat is a common experience man has been aware of since the discovery of fire, the scientific study of heat radiated from hot objects, thermal radiation, is relatively recent. The first of these investigations began in the late sixteenth century with the experiments of Italian philosopher, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_9

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alchemist, playwright and scientist Giovambattista Della Porta (1535–1615). Subsequent research by Marc-Auguste Pictet (1752–1825), Horace Bénédict de Saussure (1740 -1799) and Pierre Prevost (1751–1839), led to the first relevant result in 1790, with the definition of Prevost’s exchange theory. This affirms that all bodies continually emit and absorb heat, that the amount of emitted radiation increases with increasing temperature, and that, when two bodies at different temperatures are in sight of one another, the hotter supplies heat to the colder until they attain the same temperature. This result marks the starting point of the modern theory of thermal radiation developed in the nineteenth century and culminating in Planck’s formula for black body radiation [1]. The demonstration that thermal radiation occurs by emission of rays similar to those of light we owe to German and naturalised British astronomer, William Herschel (1738–1822), who, in 1800, discovered the existence of invisible rays. He noted by chance that glasses of different colours, when used in a telescope, had diverse heating effects. This observation induced him to study the heating effect of the various components of visible light. Using a prism and thermometer, he also discovered heating effects in the after red region (infrared), in which there was no visible radiation. He found, furthermore, that these new rays could be reflected and refracted like ordinary light and that they could also be absorbed by glass transparent to visible rays. Herschel, a follower of the corpuscular theory of light and caloric theory, called these non-visible rays caloric rays and believed there were two types of radiation, luminous radiation and caloric radiation. From 1820, thanks to other experiments, above all by Latvian physicist Thomas Johann Seebeck (1770–1831), the existence of caloric radiation (today’s thermal or infrared radiation) became commonly accepted among physicists. Around the mid nineteenth century, after the final demonstration of the wavelike nature of light, scientists were agreed concerning the notion that an object at fixed temperature (a hot body in the jargon of physics) emits energy in the form of waves, commonly called thermal or infrared radiation. It was also agreed that this phenomenon has the following characteristics: a) all bodies emit such radiation toward the surrounding environment and absorb it from the environment; b) the amount of radiation emitted increases with increasing temperature; c) at thermal equilibrium, emission and absorption rates are equal. It thus follows, that a body initially hotter than the environment is bound to cool, since the amount of energy emitted surpasses the energy absorbed. These concepts were set on a quantitative basis thanks to the laws of absorption and emission of radiation, formulated by German physicist Gustav Kirchhoff (1824– 1887). In 1859, in papers read before the Berlin Academy of Sciences, he proved what would be called Kirchhoff’s Law of thermal radiation [2]: The ratio between the powers of emission and the powers of absorption for rays of the same wavelength is constant for all bodies at the same temperature. To this law the following corollaries are to be added: 1) The rays emitted by a substance, excited in whichever way, depend upon the substance and the temperature; 2) every substance has a power of absorption which is a maximum for the rays it tends to emit. Finally, Kirchhoff drew up three laws of spectroscopy which summarise the modes of interaction between

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radiation and matter (which we will illustrate later). The first of these concerns thermal radiation and states that: a body, solid or liquid, or a dense gas, brought to incandescence, emit radiation of all wave-lengths, giving rise to a continuous spectrum. A consequence of the Kirchhoff’s Law of thermal radiation is the concept of black body, whose study entailed the first step toward quantum mechanics. Indeed, it follows from Kirchhoff’s law that the emissive power of a perfectly absorbing surface (i.e. a surface that absorbs all the incident radiation at all wavelengths) is a universal function. Accordingly, it was defined as black-body, a body whose surface is able to absorb all the radiation falling on it. Consequently, the spectrum of radiation it emits, named black body radiation, is a universal function of frequency and temperature. Black body radiation, as well as its great conceptual importance for physics, has also a significant practical value in that it supplies us, with fair approximation, the spectrum of thermal radiation emitted by a radiant object at a fixed temperature. For these reasons its experimental characterization and theoretical study were actively pursued in the second half of the nineteenth century. A conceptually important realization of a perfectly absorbing body is represented by a cavity, provided with a small hole (Fig. 9.1). A cavity is an enclosed region of space, kept at a given temperature, in thermal equilibrium with the walls that enclose it, the interior of which is occupied by thermal radiation, constantly emitted and absorbed by the walls. Consider now a small hole in the wall of the cavity. If the size of the hole is negligible compared to the cavity’s total surface, the internal equilibrium of the system is not appreciably disturbed by the presence of the hole itself. Furthermore, the external radiation, which enters through the hole, cannot get out because, to exit, it would have to bounce off the walls a large number of times and would end up being completely absorbed (as schematized in Fig. 9.1). However, the hole allows the exit of the thermal radiation in equilibrium within. Therefore, a small hole in the cavity constitutes a perfectlyabsorbing emissive-zone, a black body. Fig. 9.1 Outline of a black body

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The behaviour of the radiation emitted by the black body as a function of the wavelength at different temperatures is shown in Fig. 9.2. The explanation of this behaviour was a challenge for the physicists of the second half of the nineteenth century, which would be fully met only at the century’s end thanks to a series of contributions distributed over thirty years. The first important step was taken in 1874, when the Czech physicist Josef Stefan (1835–1893), analysing experimental data, found that the total energy S, radiated per unit of area and per unit time, depends only on the absolute temperature (raised to the fourth power) multiplied by a universal constant σ (i.e. S = σ T 4 ). The constant σ, subsequently named Stefan’s constant, does depend neither on the nature of the radiating surface nor on the construction details of the black body. The exactness of this relationship, subsequently renamed Stefan-Boltzmann’s law, was demonstrated theoretically by Austrian scientist Ludwig Boltzmann (1844–1906) in 1884. Boltzmann evaluated the entropy variation due to the cavity expansion resulting from the pressure of thermal radiation and demonstrated that the Stefan’s relationship could be obtained using one of the mathematical properties of the entropy function. In his memorial address devoted to Boltzmann in 1907, the Nobel laureate Hendrik Antoon Lorentz (1853–1928) called this demonstration a veritable pearl of theoretical physics [3]. In the 1890s further advances in understanding the black body radiation were made by German physicist Wilhelm Wien (1864–1928). In 1893 he drew up an empirical law, later called Wien’s displacement law, which correlated the wavelength λmax of the maximum emission of radiation to the temperature T of the black body. The law affirms that λmax varies with varying temperature in such a way to maintain constant the product λmax T. In 1896, Wien surmised, also empirically, a relationship able to

Fig. 9.2 Behaviour of the radiation intensity emitted by the black-body as a function of the wavelength at different temperatures. The intensity increases and has a maximum that moves towards short wavelengths as the temperature increases. The emission goes to zero, with different trends, both at long and short wavelengths Wikipedia Creative Commons

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reproduce in a satisfactory way the experimental dependence of the radiation intensity as a function of wavelength. He assumed that the radiation within the cavity had the same energy of the particles on the walls that emitted it, and thus its dependence was given by Maxwell’s distribution. Although Wien’s formula reproduced the experimental spectrum well enough at short wavelengths, it differed notably at larger ones. Another noteworthy contribution to solving the problem was made by English physicists Lord Rayleigh (1842–1919) and James Jeans (1877–1946) in 1900, few months before the presentation of Planck’s theory. They calculated the energy density of electromagnetic waves within a cubic cavity with metallic walls. Under these conditions, standing waves are formed inside the cavity subject to the condition that the electric field is zero on the wall surfaces. It was therefore possible to calculate the wavelength of the allowed waves (modes of the cavity) and their distribution. They obtained an expression for black body radiation, later named Rayleigh-Jeans’ formula, that faithfully reproduced the experimental trend at large wavelengths. However, at short wavelengths the formula was not only completely mistaken, but even predicted an intensity that goes to infinity. The results of Wien and of Rayleigh and Jeans demonstrated dramatically how it was impossible to explain black body radiation using current physical theories. The problem was resolved at the end of the year 1900, when German physicist Max Planck (1858–1947) presented his famous mathematical law which reproduced perfectly the spectrum of black-body radiation and its dependence on temperature. To obtain his formula Planck first used an empirical approach. He started out from the evidence that at short wavelengths black-body spectrum follows Wien’s law, while at long wavelengths the spectrum is well reproduced by the Rayleigh-Jeans formula. Consequently, he searched for a mathematical expression able to reproduce these two limiting cases. To this end, Planck, as he clearly explained in his Nobel prize lecture, exploited Kirchhoff’s laws, according to which the absorption and emission of radiation at thermal equilibrium does not depend on the characteristics of cavity walls. This property allowed him to hypothesize that the molecules present on the cavity surfaces behaved like microscopic harmonic oscillators, whose absorption and emission properties were already known. He then showed that, as a function of the radiation wavelength, the energy of a harmonic oscillator in thermal equilibrium with black-body radiation was proportional to the energy density of the black body itself. Finally, he showed how the energy of a harmonic oscillator could be expressed as a combination of the two limiting cases represented by Wien’s law and RayleighJeans’s one. Thus, he obtained his famous formula which perfectly reproduced the experimental data and contained a new physical constant he indicated with the letter h and which would become the famous Planck’s constant, the quantum of action. Later Planck sought a theoretical justification for his empirical formula. He found that this could be obtained if, in calculating the average value of energy of the oscillators, one imposed the condition that the energy of an oscillator cannot have just any value, but must be quantized in whole multiples of a minimum value ε given by the equation:

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ε = hν where h is the Planck constant, the quantum of action, and ν the oscillation frequency. The hypothesis that energy of a microscopic harmonic oscillator cannot vary continuously but has quantized values, linked to its oscillation frequency, and varies with discontinuity in whole multiples of its lowest value, was difficult to accept by Planck’s contemporaries, for whom physical properties were continuous variables. Furthermore, macroscopic harmonic oscillators were among the models most studied by classic mechanics and the behaviour of their energy was well known. Finally, it should be noted how microscopic harmonic oscillators were only models introduced in the attempt to simulate macroscopic properties of matter, without there being direct experimental observation. Consequently, for some years Planck’s hypothesis of quantization remained confined to the level of a heuristic expedient, used to solve the intricate problem represented by the black body radiation, but not paid much attention regarding possible conceptual and methodological implications for the edifice of physics as a whole. This situation changed rapidly from 1905 onward when Albert Einstein (1879–1955) extended the hypothesis of quantization to the electromagnetic radiation and introduced the concept of photon, showing how these new hypotheses supplied the key for explaining other mysterious phenomena, in particular the photoelectric effect and the dependence on the temperature of specific heat.

9.2 Photoelectric Effect Among the numerous experimental observations difficult to explain in the decade spanning the turn of the twentieth century was the strange behaviour of metals when struck by short-wavelength light, what became known as photoelectric effect. In 1887 German physicist Heinrich Hertz (1857–1894), in a series of experiments he conducted to verify the accuracy of Maxwell’s theory of electromagnetic waves, discovered that, when he illuminated metallic electrodes with ultraviolet light, the electric discharge signalling the waves’ arrival at the detector was facilitated. Studies of such an effect were taken up the following year by German physicist Wilhelm Hallwachs (1859–1922), who found that a disc of zinc, if negatively charged, rapidly lost its charge when illuminated by ultraviolet light, while no such effect was observed with radiation of greater wavelengths or if the disc was positively charged. In 1890 German physicists Julius Elster (1854–1920) and Hans Geitel (1855–1923) showed that, among all the metals, the alkali ones were the most sensitive to light and discovered that sodium and potassium also responded to visible light [4]. However, the situation remained confused until 1899, when J. J. Thomson, experimenting with vacuum tubes, clarified the effect’s experimental features. Flashing light on the cathode, he showed that ultraviolet light caused the emission of charged particles from the cathode metal, giving rise to an electric current (photoelectric current). He also demonstrated that the particles emitted were identical to those that make up

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cathode rays, the electrons. However, the peculiarities which rendered the photoelectric effect problematic were discovered later in 1902, when German physicist Philipp Lenard (1862–1947) investigated photoelectric current ‘s dependence on the intensity and frequency of incident light and found some disconcerting behaviour inexplicable on the basis of current physical theories. Among these the most relevant was that the threshold of photoemission and the kinetic energy of photoemitted electrons depended on the frequency of light, something that could be in no way explained by any classical interaction mechanism between electromagnetic radiation and matter (for more details see More on Photoelectric Effect in Appendix C). In 1905, in one of four articles published in his annus mirabilis, titled On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Einstein showed how all the characteristics of photoelectric effect could be explained if one accepted Planck’s quantization concept and extended it to the electromagnetic radiations. According to Einstein’s revolutionary hypothesis the energy of an electromagnetic wave is quantized and linked to its frequency according to the scheme proposed by Planck in his explanation of the black body radiation, i.e.: ε = hν The quantization hypothesis introduced a particle-like behaviour of the electromagnetic radiation seemingly at odds with the accepted wave nature of light. As result it was initially received with caution by the scientific community. The final proof that Einstein’s theory of photoelectric effect is exact was supplied a decade later, in 1916, by American physicist Robert Millikan (1868–1953).

9.3 Quantum Specific Heat We have seen how the ability to explain specific heats of gases and solid matter boosted the credibility of kinetic theory and the corpuscular concept of matter. The measurements made on a great number of elements and summarised in Dulong-Petit’s law seemed to show that the specific heat of matter in a solid state has a universal value equal to 3R, where R is the universal constant of gases. The demonstration of the law’s exactness, performed by Boltzmann using the methods of statistical mechanics, confirmed this opinion. However, at the end of the century DulongPetit’s law began to run up against some notable exceptions. The first evidence had come from measurements of specific heat of the diamond, providing a value much less than expected (0.73 R instead of 3R). Subsequently, with the development of cryogenic technologies, it became possible to measure specific heats at temperatures lower than the ambient one. These revealed a totally unexpected behaviour: instead of being constant, specific heats showed a marked dependence on measurement temperature and diminished along with it. Such behaviour was not comprehensible in the light of kinetic theory and statistical mechanics hitherto developed.

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Again, it was Einstein who showed how the introduction of the concept of energy quantization made it possible to explain specific heats’ anomalous behaviour. Following his 1905 article on the photoelectric effect, where he had proposed the quantization of the energy of electromagnetic radiation, Einstein, in his 1907 article Planck’s Theory of Radiation and the Theory of Specific Heat, suggested that the microscopic vibrations of solid matter could be described in terms of harmonic oscillators, quantized according to Planck’s rule. Upon calculating the specific heat, he showed how the energy quantization led to a dependence on temperature which reproduced the two principal characteristics of the experimental data: At high temperatures specific heat assumed the value given by Dulong-Petit’s law, while, at low temperatures, its value decreased with decreasing temperature.

References 1. Putley E H (1984) The development of thermal imaging systems. In: Ring E F, Phillips B (ed) Recent Advances in Medical Thermology. Plenum Press, New York, p. 151 2. White H E (1934) Introduction to atomic spectra. McGraw-Hill, New York 3. Sommerfeld A (1955) Thermodynamics and statistical mechanics. Academic Press, New York 4. Salazar-Palma M, Sarkar T K, , Sengupta D L (2006) A Chronology of Developments of Wireless Communication and Supporting Electronics. In: Sarkar T K, Mailloux R J, Oliner A A, SalazarPalma M, Sengupta D L (ed) History of wireless. John Wiley & Sons, Hoboken, New Jersey, pp. 53-164

Chapter 10

Bohr’s Atom

Physics research in the early twentieth century was deeply influenced by the recent discovery of the existence of electrons. The evidence that atoms are not simple and indivisible particles but have a complex structure focused the scientific community’s efforts on the understanding of their structure and the distribution of the charges and masses inside them. Within few years a number of major advances culminated in the Bohr’s innovative theory for the hydrogen atom, which marked the beginning of quantum atomic physics.

10.1 Early Atomic Models Right from the mid-nineteenth century, with the progress of the mathematization of physics and the development of electromagnetism, there had been attempts to construct atomic models more sophisticated that simple corpuscles given mass, volume and shape, as conceived by the atomistic theories of previous centuries. The first mathematically refined model was proposed in 1867 by William Thomson (Lord Kelvin) (1824–1907) [1]. In line with current debates on the existence of ether as the medium of electromagnetic phenomena, Thomson suggested that atoms were none other than vortices in the ether. The idea that material’s properties could be the result of swirling motion of a medium goes back to Descartes and had been taken up in more recent times by Hermann von Helmholtz (1821–1894). Helmholtz had also shown that vortices exercised forces of mutual interaction and that these forces were similar to the magnetic forces between electric wires. According to William Thomson vortex atoms were perfectly elastic bodies, the various types of atoms corresponded to different types of nodes present in the vortices, and their properties could be evaluated mathematically. Between 1870 and 1890 the theories whereby atoms are vortices in the ether became very popular among British physicists and mathematicians. The same J. J. Thomson, who would discover electrons in 1897, in

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his degree thesis tried to explain the chemical combination of atoms that form biatomic molecules, such as hydrogen and hydrochloric acid, as a mutual attraction of vortex atoms [2]. However, between 1895 and 1904, spurred by knowledge brought by the studies on cathode rays and canal rays and by the discovery of electrons, models based on vortices in the ether were replaced by models in which ions and electrons were simulated in states of equilibrium, static or dynamic. French physicist Jean-Baptiste Perrin (1870–1942) was the first to advance, in 1901, the hypothesis that atoms were formed of nuclei around which electrons rotated like planets of the solar system. Starting out from the experimental observation that microscopic particles ionize dividing themselves into a smaller portion, negatively charged, and a larger portion positively charged, Perrin claimed that each atom was constituted by [2] one or several masses strongly charged with positive electricity, in the manner of positive suns whose charge will be very superior to that of a corpuscle, and on the other hand, by a multitude of corpuscles, in the manner of small negative planets, the ensemble of their masses gravitating under the action of electrical forces.

In 1902 William Thomson (Lord Kelvin) proposed a model consisting of electrons disposed in a position of mutual equilibrium, in presence of the positive electric charge necessary to obtain the atom’s neutrality. In his view, the positive charge could be thought of, interchangeably, either as a uniform charge density, distributed within the volume occupied by each atom, or as a charge situated within a smaller concentric globe. A further model was proposed in 1903 by German physicist Philipp Lenard (1862– 1947), who with his experiments on cathode rays of 1894 had obtained the first evidence for the prevalence of empty space inside atoms. Lennard envisaged the atom as a mostly empty shell, inside which electric dipoles (i.e. couples of positive and negative charges), were distributed. He called dynamids these dipoles, whose number was proportional to the atomic weight of the element considered [3]. Yet another model was proposed by Japanese physicist Hantaro Nagaoka (1865– 1950) in 1904. His is a central positive sphere with a halo of electrons, resembling the planet Saturn. J. J. Thomson, discoverer of the electron, also proposed, in the report On the structure of the atom of 1904 his own atomic model, which for a time enjoyed wide popularity. His atom consisted of a sphere of positive charge evenly distributed, inside which were disseminated negative point-like charges, the electrons. The configuration resembled that of seeds inside a water-melon or, for the English, plums inside a pudding. The electrons were located in position of equilibrium where the attraction due to the positive charge balanced the repulsion exerted by the negative charges of the other electrons. Thomson assumed that the state of lowest energy was that where the electrons were stationary in the position of equilibrium. The interaction with the electromagnetic radiation excited the atom, making the electrons vibrate or rotate around their position of equilibrium. Overall, however, the model failed to match experimental data then available on atoms.

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The problem of the distribution of electric charges within the atoms was finally resolved by experiments carried out between 1906 and 1913 at Manchester University by New Zealand physicist Ernest Rutherford (1871–1937) and his assistants. To obtain information on charge configuration, they bombarded a series of thin metallic foils of different thicknesses with subatomic particles and analysed the scattering which these underwent as a result of interaction with the atoms. The particles used were α (alfa) particles, which Rutherford had previously shown to be doubly ionized helium atoms, and thus having a positive charge equal to 2e.1 It was expected that the scattering of α particles, due to Coulomb interaction with the electric charges inside the atoms, characteristically depended on the distribution of the charges themselves. In particular, if the distribution of the charges was that hypothesized by J. J. Thomson, the scattering angles should be small, since the scattering centres, electrons and uniformly distributed positive charges, had a much smaller mass than that of the incident α particles. The experiments produced surprising results. Though most of the α particles were scattered at small angles (less than one degree) as expected, it was found that an appreciable number were diffused a wide angle and that about one particle every 8000 was backscattered at an angle greater than 90°. The backscattering could happen only if the impact occurred with objects whose mass was sensibly greater than that of the incident α particles. Therefore, the experimental evidence seemed to show unequivocally that inside atoms a restricted zone existed where both charge and mass were concentrated. Based on such evidence, in 1911 Rutherford proposed his famous atomic model (Fig. 10.1) in which it was assumed that the atomic mass and the positive charge were concentrated in a region much smaller than the total atomic volume [4]. Rutherford’s atom resembled a tiny solar system in which electrons (planets) orbited around the nucleus (sun), attracted by the forces due to their electric charges. When an α particle passes through this type of atom, it is highly probable that it impacts only with one of the orbiting electrons, so explaining the high number of small-angle scattering events observed. Occasionally, however, it happens that the α particle passes close to the nucleus, resulting in a diffusion at a large angle. In his article Rutherford also showed that, based on this model, the distribution of measured scattering events could be reproduced theoretically in a quantitative way. Despite the strong support of experimental evidence, Rutherford’s model at first received little recognition in that it presented insurmountable difficulties at theoretical level. In fact, an orbiting electron, moving on a curved trajectory, is subject to acceleration, but an accelerated electric charge radiates electromagnetic waves and loses energy. The laws of classic physics (equations of motion and equations of electromagnetism), if applied to Rutherford’s atom, imply that all the energy of the orbiting electrons would be lost through radiation in a time less than a billionth of a second and that the electrons would collapse onto the nucleus. α Particles are helium nuclei emitted by many radioactive substances and were a subject of study at that time. Rutherford had determined their mass/charge ratio by studying their trajectory subject to electric and magnetic fields. Since the concept of nucleus was not yet clear, he had only been able to define them as doubly ionized helium atoms.

1

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Fig. 10.1 Ernest Rutherford’s atomic model

In the early 1910s, the scientific community interested in the microscopic structure of matter found itself in considerable difficulty. Experimental evidence of phenomena measured at microscopic scale seemed to contradict the predictions obtained by applying known physical laws. To this one had to add the numerous and detailed observations on the emission and absorption of electromagnetic radiation by atoms, which there was no way of explaining. These properties of radiation/matter interaction were part of a branch of physics, spectroscopy, which had undergone significant development in the second half of the nineteenth century and was shortly to provide Niels Bohr with the key for the construction of his triumphant atomic theory. Consequently, before continuing the review of atomic models, it is appropriate to briefly outline the knowledge of spectroscopy of the period.

10.2 Light and Atoms: Spectroscopy Since ancient times, optical phenomena, in particular the absorption and emission of luminous radiation, have provided one of the main sources of information on matter’s structure. The relevance of this field of investigation has grown, from the seventeenth century on, in parallel with advances in understanding the nature of light. From these studies, in particular the break-throughs made in the second half of the nineteenth century, was born a powerful investigative tool, spectroscopy, based on the analysis of the radiation emitted or absorbed by substances. We can trace spectroscopy’s origin to Isaac Newton’s discovery that if one passes sunlight through a glass prism, one obtains a set of coloured bands, from violet to red, similar to the rainbow, which he called spectrum (using the Latin term for image, vision). The phenomenon was already known, but Newton was the first to grasp that

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white light from the sun is none other than the sum of various chromatic components, as they are perceived by our eye. The first significant step toward the use of light as a tool for investigating matter was made by Scottish physicist Thomas Melvill (1726–1753), who studied the differences in colour observed when burning different gases. Presenting his results in a lecture entitled Observations on light and colours held at the Medical Society of Edinburgh in 1752, he showed how the spectra, obtained when a thin beam of light from the gas flame was passed through a prism and projected onto a screen, were composed of a large number of bright lines of different colours, later called emission lines [5]. Years later, in 1802, English scientist William Hyde Wollaston (1766–1828), analysing in greater detail the spectrum of the sun light, noticed the presence of dark lines. It was the first observation of the so-called absorption lines, i.e. dark lines due to the absorption of specific frequencies of the radiation passing through a medium. The process is the reverse of that giving rise to emission lines. However, Wollaston attached little importance to his discovery since he thought he was dealing with dark intervals limiting the luminous bands. The first systematic study of the spectrum of sun light was carried out by German scientist Joseph von Fraunhofer (1787–1826) who compiled an accurate map of it, including several hundred absorption lines (Fig. 10.2). He also evaluated the relative intensity of the lines, reporting this by means of a curve drawn above the spectrum, and marked with capital letters the 8 most intense lines, thereafter called Fraunhofer’s lines. Finally, he studied the spectra of various stars, including Sirius, Castor, and Betelgeuse and found that each one was characterized by a series of particular lines. In 1849, a fundamental step towards understanding the link between elements and spectral lines was made by French physicist Jean Foucault (1819–1868). He found that the bright orange lines, present in the light emitted by carbon burned in an arc lamp, could be observed as dark lines when the sun light passed through a carbon gas. From this observation Foucault deduced that a gas preferentially absorbs the same colours that it emits. Further significant contributions to spectroscopy came in 1853 from Swedish physicist Anders Jonas Angstrom (1814–1874). Angstrom, in a presentation to the Academy of Stockholm, showed how the light emitted during the gas electric discharge was composed of two different spectra superimposed, one characteristic of

Fig. 10.2 Solar spectrum with the absorption lines drawn by Fraunhofer. Wikipedia Public Domain

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the metal, from which the electrodes were made, and the other of the gas contained in the tube. Furthermore, Angstrom proved conclusively that a gas emits rays of the same wavelength as those it absorbs, confirming the Foucault’s results. In the late 1850s, thanks to German scientists Robert Wilhelm Bunsen (1811– 1899) and Gustav Robert Kirchhoff (1824–1887), one had the final demonstration that chemical elements could be identified unequivocally by the spectral lines emitted or absorbed and the birth of spectroscopy as an experimental science. Such results were possible thanks to two inventions they made, a special burner, still known as Bunsen burner, and a newly conceived spectroscope.2 Using these new instruments, Bunsen and Kirchhoff performed a detailed analysis of the spectra of radiation emitted by a variety of chemical substances and showed how from each element a certain group of lines of characteristic wavelengths was emitted. Moreover, they clarified definitively the link existing between absorption and emission lines. Kirchhoff also drew up the three laws of spectroscopy, which summarise the modes of interaction between radiation and matter. The first law states: a hot solid, liquid, or dense gas emits radiation at all wavelengths, giving rise to a continuous spectrum of radiation. The second law states: a hot gas under low pressure emits radiation at a discrete set of isolated wavelengths. The whole spectrum is called an emission-line spectrum. The wavelengths of the emission lines are unique to the type of neutral atom or ionized atom that is producing the emission lines. The third law states: a rarefied gas, kept at a low temperature, placed in front of a source of continuous radiation, gives rise to an absorption spectrum, whose dark lines are at the same wavelengths of its emission spectrum. The fact that every element has a characteristic spectrum of lines proved of great practical importance since it opened up a powerful methodology for identifying substances. As a demonstration, in 1860, Bunsen and Kirchhoff published the spectra of eight elements and identified their presence in various natural compounds. Furthermore, in discovering cesium and rubidium, they also showed how the analysis of spectra made it possible to identify as yet unknown elements. The method revealed itself to be of especially practical significance in astronomy, since it allowed the identification of the elements present in heavenly bodies. In particular, to explain the presence of dark lines in the solar spectrum, Kirchhoff hypothesised that the sun is surrounded by a layer of gases absorbing the light coming from its interior. From a detailed analysis of the dark lines, Kirchhoff concluded that in the sun are present iron, calcium, magnesium, sodium, nickel and chrome. An exhaustive map of solar lines appeared in 1868, drawn up by Angstrom, including more than a thousand spectral lines, whose wavelength he had determined with the precision of a ten-millionth of a millimetre. Earlier, in 1862, Angstrom had also shown that the sun’s atmosphere contains hydrogen. Given spectroscopy’s manifest relevance as a means of investigating the composition of matter, the second half of the nineteenth century saw many detailed studies of 2

The spectroscope is an instrument used in physics and chemistry for the analysis of the wavelength of the electromagnetic radiation emitted by a source.

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Fig. 10.3 Hydrogen absorption lines in the visible range

line spectra of many elements aimed at discovering the laws governing their origin. It was found that some elements had spectra containing hundreds of lines, while others seemed to have very few. A first step for restoring order to the chaotic panorama of data came with the discovery that in many elements the lines could be arranged into series of lines with characteristic properties. The existence of characteristic series of lines—whose knowledge would prove invaluable for understanding atoms’ structure—was particularly evident in hydrogen and in the elements of the periodic table’s first three groups. The discovery’s significance became especially clear when Swiss physicist and mathematician Johann Balmer (1825–1898), in 1885, found an empirical mathematical relationship capable of reproducing the wavelength of the four lines then known of hydrogen [6] (Fig. 10.3). The relationship, simple enough mathematically, is: λ=Λ

n 22

n 22 − n 21

where Λ = 36,456 nm and n1 and n2 are whole numbers. The best agreement with experimental data is obtained positing n1 = 2 and n2 = 3, 4, 5, 6. The discovery showed how emission and absorption of electromagnetic radiation by hydrogen atoms is a regular phenomenon, following an extremely simple algebraic law. This series, for which subsequently were found many more lines, corresponding to values of n2 larger than 6, became known as Balmer’s series. The relationship discovered by Balmer was generalized years later by Swedish physicist and mathematician Johannes Rydberg (1854–1919). Rydberg found a relationship similar to Balmer’s, but more general, making it possible to reproduce the frequencies of many of the series of lines then known and, in particular, of the four series which had been discovered in alkali metals. Rydberg’s formula is: ( ν=R

1 1 − 2 (n 1 + μ1 ) (n 2 + μ2 )2

)

In this expression ν is the frequency of lines, μ1 and μ2 , are two parameters, determined empirically, having a characteristic value for each element and for each series of lines; n1 is a whole number characteristic of the series, while n2 reproduce the lines of the series taking whole values bigger than n1 . A truly surprising aspect is

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the fact that, in the formula, appears a universal constant, R, subsequently named Rydberg’s constant, valid for every element and series considered. Setting μ1 and μ2 equal to 0, n1 = 2 and n2 = 3, 4, 5,… one obtains the Balmer’s series of hydrogen. One further finds that the formula has also a predictive character. In fact, still for hydrogen (and, thus, setting μ1 and μ2 equal to 0) if one posits n1 = 1 and n2 = 2, 3, 4, 5,… one obtains a different series, whose existence was effectively demonstrated by American physicist Theodore Lyman (1874 –1954) some years later, while he was studying the ultraviolet spectrum of hydrogen. In his honour the series was baptized Lyman’s series. If one sets n1 = 3 and n2 = 4, 5, 6, … one obtains another series of hydrogen, observed in infrared by German physicist Friedich Paschen (1865–1947). Years later, in 1922, American physicist Frederick Summer Brackett (1896–1988) demonstrated the existence of the series obtained by taking n1 = 4 and n2 = 5, 6, 7, … In 1924 American physicist August H. Pfund (1879–1949) discovered the series characterized by n1 = 5 and n2 = 6, 7, 8,… The mathematical structure of Rydberg’s formula, as the difference of two terms, valid for the atomic spectra of many elements, led Swiss physicist Walther Ritz (1878–1909) to establish a combination principle, which would prove crucial in the formulation of Bohr’s atomic theory. According to Ritz’s combination principle the frequency of electromagnetic radiation emitted or absorbed by atoms is given by the difference of two terms of the type νn =

R (n + μ)2

where R is Rydberg’s universal constant, n a whole number and μ a parameter, characteristic of the type of atom and the spectral series. The fact such simple formula reproduces so precisely all lines of hydrogen spectrum and that of many other elements was surprising. Many scientists became convinced that this was no accident but must be intimately linked to the structure of atoms. When in the early twentieth century physicists began to envisage the first atomic models, spectra of lines found themselves at the forefront in the list of properties that whatsoever theory should be able to explain in order to be deemed valid!

10.3 Bohr’s Theory The turning point for understanding atomic structure came in 1913 and was due to the Danish scientist Niels Bohr (1885–1962). Bohr proposed [7] an innovative theory for the hydrogen atom, based on the ideas, becoming then ever more widely accepted, of the quantization of physical quantities. The theory addressed many open problems and marked the beginning of a new era in the physics of atoms and in understanding matter. The relevance of Bohr’s theory ever since its appearance can be appreciated by Planck’s comments, some years later, on receiving the Nobel prize [8]:

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If the various experiments and experiences gathered together by me up to now, from the different fields of physics, provide impressive proof in favor of the existence of the quantum of action, the quantum hypothesis has, nevertheless, its greatest support from the establishment and development of the atom theory by Niels Bohr. For it fell to this theory to discover, in the quantum of action, the long-sought key to the entrance gate into the wonderland of spectroscopy, which since the discovery of spectral analysis had obstinately defied all efforts to breach it. And now that the way was opened, a sudden flood of new-won knowledge poured out over the whole field including the neighbouring fields in physics and chemistry”.

In the 1913 theory Bohr considered the simplest possibility, whereby the atom consists of a single electron revolving around a positively charged nucleus in a circular orbit under the effect of the Coulomb attractive force. To develop his theory Bohr introduced the then revolutionary hypothesis that the angular momentum L of the electron is quantized, that is to say, that it can assume only values equal to whole multiples of Planck’s quantum of action h divided by 2π. In formulae: L=n

h ; n = 1, 2, 3, . . . 2π

This hypothesis is equivalent to postulating that only some orbits are allowed for the electron, as opposed to what is stipulated by classical physics. Bohr further postulated that these orbits are stationary, i.e. stable, and that the electron travelling them emits no radiation. Using the possible values of angular momentum, he went on to calculate the possible values of the electron energy, which also depend on n and are quantized. The theory produced a very accurate evaluation of the spectral lines of the hydrogen and ionized helium atoms. It also provided an exact theoretical expression for the Rydberg’s constant. (For more details see More on Bohr’s theory in Appendix C). As result of his work, Bohr concluded that to formulate a quantum theory, that accounted for the stability of atoms found experimentally and the properties of radiation emitted by them, it was necessary to introduce two postulates, that could be enunciated as follows [9]: 1) Among the conceivably possible states of motion in an atomic system there exist a number of so-called stationary states which, in spite of the fact that the motion of the particles in these states obeys the laws of classical mechanics to a considerable extent, possess a peculiar, mechanically unexplainable stability, of such a sort that every permanent change in the motion of the system must consist in a complete transition from one stationary state to another. 2) While in contradiction to the classical electromagnetic theory no radiation takes place from the atom in the stationary states themselves, a process of transition between two stationary states can be accompanied by the emission of electromagnetic radiation, which will have the same properties as that which would be sent out according to the classical theory from an electrified particle executing a harmonic vibration with constant frequency. This frequency v has, however, no simple relation to the motion of the particles of the atom, but is given by the relation hν = E' − E'' where h is Planck’s constant, and E' and E'' are the values of the energy of the atom in the two stationary states that form the initial and final state of the radiation process. Conversely,

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irradiation of the atom with electromagnetic waves of this frequency can lead to an absorption process, whereby the atom is transformed back from the latter stationary state to the former. While the first postulate has in view the general stability of the atom, the second postulate has chiefly in view the existence of spectra with sharp lines. Furthermore, the quantum-theory condition entering in the last postulate affords a starting-point for the interpretation of the laws of series spectra.

It is evident that the second postulate was an implicit adoption of Ritz’s combination principle and thus spectroscopic terms were identified with stationary states of energy. There remained a problem, however. According to Planck’s original quantization hypothesis, an oscillator emits only radiation with frequency equal to its own frequency of oscillation. According to Bohr’s theory, the frequency of electromagnetic radiation, emitted or absorbed, is not equal to the frequency the orbital rotation, but is determined by the difference of energy of two stationary states. Bohr tried to find a correlation between the classical frequency of revolution and the quantum frequency of energy emission, and came to formulate the theorem of correspondence, stating that, for large values of the quantum number n, frequencies of radiation of quantum theory and frequencies of classical orbits become equal.

10.4 Moseley’s Experiments The explanation of spectral lines of hydrogen and ionized helium atoms constituted a prompt and major success for Bohr’s theory. In addition, already in its publication year (1913) further important experimental support came from the measurement of the X-rays spectra carried out by English physicist Henry Moseley (1887–1915). Moseley’s experiments would prove highly significant for understanding atoms’ structure in that, as well as confirming Bohr’s theoretical construct, they offered direct proof for the concept of atomic number and the correctness of the ordering of elements in Mendeleev’s periodic table. As we have seen, the ordering of the elements in the periodic system had been essentially based on increasing atomic weight; thus, the number assigned to each element in the periodic table, the atomic number, reflected such ordering. However, the principle underlying the layout of chemical elements remained a highly debated topic and a source of controversy. We recall that, in his compiling the periodic table, Mendeleev had changed the ordering of a few elements, putting them in more appropriate positions based on their chemical properties, and had left some boxes empty. In 1911, in the article where he had published his results demonstrating the existence, inside atoms, of a massive electrically-charged nucleus, Rutherford had noted how the nucleus’s electric charge was around half the atomic weight, if expressed as a multiple of hydrogen mass. Some months later, Dutch physicist Antonius Johannes van den Broek (1870–1926) suggested (with great foresight) that the number assigned to each element in the periodic table, the atomic number, could

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coincide with the total number of electrons present in the atom and thus with the nucleus’s positive electric charge. When it appeared, Bohr’s atomic theory provided the possibility of solving this problem as well. Indeed, an explicit dependence of the orbit energy on the electric charge Z of the nucleus is foreseen in the theory. Furthermore, a direct method for determining the orbit energy is provided by the postulate according to which the emitted radiation frequency is equal to the difference of energy of stationary states between which the electron transits. Moseley is to be credited with understanding at once that the analysis of X-rays emitted by atoms of different elements could prove the validity of both Bohr’s theory and van der Boek’s hypothesis. Such was Moseley’s aim in measuring the spectra of X-rays undertaken in 1913. As is well known, X-rays were discovered by German physicist Wilhelm Röntgen (1845–1923) in 1895. In that year, while investigating the phenomena which accompany the passing of an electric current through a gas in a Crookes tube at extremely low pressure, Röntgen noted that some mysterious rays were produced when highly energetic cathode rays impacted on the anode. He called them X-Rays to signify their unknown nature. The nature of X-rays remained a theme of debate in the twentieth century’s first decade, until, in 1912, experiments on their diffraction on the part of crystals, suggested and interpreted by German physicist Max von Laue (1879–1960), showed that they consist of electromagnetic radiation of minute wavelength, emitted by matter bombarded by the cathode rays. Thanks to William Henry Bragg (1862– 1942) and William Lawrence Bragg (1890–1971) X-rays rapidly became one of the most powerful tools for studying the structure of matter in the solid state. However, an accepted theory of atomic structure at the time being lacking, the microscopic mechanism leading to their emission was still not understood. Bohr’s theory supplied Moseley with the opportunity of clarifying this aspect as well. In his 1913 experiments, Moseley measured the X-rays emitted by the elements ranging from aluminium to gold in the periodic table. Analysing data, he discovered a systematic relationship, now known as Moseley’s law, between the X-rays frequencies and the square of the atomic number Z of the elements used as a target. This was the same dependence, predicted by Bohr’s theory, relating the orbit energies to the square of the charge Ze of the nucleus. Moseley interpreted his results as demonstrating that, not just in hydrogen, but in atoms in general, electrons are disposed in stationary states corresponding to orbits of increasing radius, characterized by the quantum number n. Furthermore, he inferred that the emission of X-rays is due to the jump of electrons from a more external orbit to an inner one (For more details see X-rays and Mosely’s experiments in Appendix C). Mosely’s measurements and data analysis proved highly relevant to the progress of atomic physics and the understanding of matter. Indeed, they demonstrated the correctness in identifying atomic number with the number of electrons of the atom and with the number of positive charges of the nucleus. They clarified the origin of Xrays as due to the transition of electrons from a more external orbit to a more internal one. Finally, as well as bolstering Bohr’s theory and its revolutionary assumptions on the existence of stationary energy levels, Mosely paved the way for the notion that electrons inside atoms are distributed in shells of characteristic energy and for

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the Aufbauprinzip (building-up principle) introduced by Bohr years later to explain atoms’ structure and the periodic system of elements.

10.5 Experiment of Franck and Hertz Shortly after Moseley’s study into atoms’ structure by means of X rays, direct confirmation of the quantization of the energy of electrons in atoms came via the experiment conducted in 1914 by German scientists James Franck (1882–1964) and Gustav Ludwig Hertz (1887–1975). Franck and Hertz had built an experimental apparatus for studying the mechanisms of energy loss that cathode rays undergo following collisions with the atoms of a gas. Experimenting with mercury atoms at low pressure, they discovered that, in the collisions, electrons always lost only a specific fraction of their kinetic energy, equal to 4.9 eV, independently of initial value of the kinetic energy itself. Furthermore, if the electron kinetic energy was less than 4.9 eV, the collision was elastic, i.e. it occurred without loss of energy by the electron involved. Franck and Hertz realized that the value 4.9 eV corresponded exactly to the energy necessary for the transition of the electron up to the first excited state, according to the scheme of transition between levels postulated by Bohr. To confirm the result they also measured the frequency of electromagnetic radiation emitted by the mercury atoms excited by the collisions and, as expected, found that this corresponded exactly to hv = 4.9 eV. Franck and Hertz’s experiment supplied direct experimental proof that, as Bohr had postulated, atoms absorb energy in precise and definite quantities, equal to the difference in energy between their stationary states and that this is valid not only for the absorption and emission of electromagnetic radiation, but also for the exchange of energy between electrons. According to Planck [8]: Thus, the determination of the so-called resonance potential carried out by J. Franck and G. Hertz, or that concerning the critical velocity is the minimum an electron must possess in order to cause emission of a light quantum or photon by impact with a neutral atom, supplied a method of measuring the quantum of action which was as direct as could be wished for.

10.6 Elliptical Orbits and Space Quantization The success in explaining the origin of hitherto mysterious phenomena, such as spectral lines of the hydrogen atom and the emission of X rays, and experimental confirmation of the quantization principles and the existence of stationary levels of energy, placed Bohr’s theory at the centre of physicist’s attention and the debate on atoms’ structure. Work was begun on the generalization of the theory and the removal of its more restrictive assumptions. In 1916 German theoretical physicist Arnold Sommerfeld (1868–1951) published an extension of the theory, in which the restriction to circular orbits for the electrons was removed and the movement

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in elliptical orbits was taken into account [10]. The new theory, later named BohrSommerfeld’s theory, generalized the concept of quantization and contributed greatly to accrediting it as a key to interpreting phenomena at the atomic scale. A similar theory was drawn up independently by English physicist William Wilson (1875– 1965) in the same period [11]. For an electron moving in an elliptical orbit Sommerfeld found that the stationary states are determined by two quantum numbers: a principal quantum number n determining the energy of the orbit, which can assume all the entire values (n = 1, 2, 3, 4, …) and an azimuthal quantum number k determining the value of angular momentum. The energy of the elliptical orbits is identical to that of Bohr’s circular orbits, has the same dependence on n and does not depend on k. For every value of the principal quantum number n there are n different orbits possible, characterized by distinct values of the angular momentum L, whose azimuthal quantum number k takes on the values 1, 2, 3, …, n. The angular momentum L is, thus, given by: L=k

h ; k = 1, 2, 3, ..., n 2π

It follows that, for every value of energy E n , there are n different possible orbits for the electron, characterized by n distinct values of angular momentum L (a concept referred to as state degeneracy). In reality the electron’s movement occurs in tri-dimensional space, requiring three quantum numbers for its characterisation. Reformulating the theory, Sommerfeld found that the introduction of a third quantum number does not alter the characteristics of elliptical orbits, but introduces a new property of quantized systems: orbits can assume only few specific orientations with respect to a fixed direction in space (space quantization). Since the angular momentum is perpendicular to the orbit’s plane, this is equivalent to affirming that in quantized systems, not only angular momentum’s module, but also its direction in space is quantized. Space quantization enjoys the following property: if one indicates with z a direction in space, the only permitted directions are those for which angular momentum’s components LZ along z satisfy the relationship Lz = m

h 2π

where m is the magnetic quantum number. In Sommerfeld’s theory angular momentum had 2 k distinct directions possible and m took the values: m = ±1, ±2, ±3, . . . , ±k This surprising new property of quantization was demonstrated directly in an experiment conducted by German physicists Otto Stern (1888–1969) and Walther Gerlach (1889–1979) in 1922.

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In conclusion, in Bohr-Sommerfeld’s theory electrons travel in elliptical orbits, characterized by 3 quantum numbers: the principal quantum number n, which determines the orbit’s energy and can assume all the entire values (n = 1, 2, 3, 4,…); the azimuthal quantum number k, which can take n values, corresponding to the whole numbers 1, 2, 3,…, n; the magnetic quantum number m, that can take 2 k values, corresponding to m = ±1, ±2, ±3, . . . , ±k.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Thomson W (1867) On vortex atoms. Proceedings of the Royal Society of Edinburgh 6: 94-105 Nye M J (1996) Before big Science. Twayne Publishers, New York Ihde A J (1984) The development of modern chemistry. Dover Publications, New York Rutherford E (1911) The scattering of α and β particles by matter and the structure of the atom. Philosophical Magazine 21: 669-688 Hunt R (2011) History of Spectroscopy-an essay. Dissertation, University of South Australia White H E (1934) Introduction to atomic spectra. McGraw-Hill, New York Bohr N (1913) On the constitution of atoms and molecules. Philosophical Magazine 26: 476502 Planck M (1920) The Genesis and Present State of Development of the Quantum Theory. Nobel Lecture, Jun 2, 1920 Bohr N (1922) The Structure of the Atom. Nobel Lecture, December 11, 1922. Sommerfeld A (1916) Zur Quantentheorie der Spektrallinien. Annalen der Physik 51: 125-167 Wilson W (1915) The quantum theory of radiation and line spectra. Philosophical Magazine 29: 795-802

Chapter 11

Electrons and Atoms

The success of Bohr’s theory and Sommerfeld’s generalisation in explaining some of the most mysterious properties of atoms, in particular the general characteristics of optical and X-ray spectra, enhanced the credibility of the new concepts on which such theories were based. As a result, the community of physicists engaged in the field started to apply the new theory to explaining the numerous physical properties of atoms, observed experimentally but as yet not understood. Among the most relevant open problems were the configuration of electrons in multielectron atoms (i.e. their electronic structure) and the relation existing between electronic structure and periodicity of elements, as shown by the periodic system. When one passes from the simpler system of a single electron orbiting around the nucleus to the study of more complex systems where a multiplicity of electrons is disposed around a nucleus, it is necessary to find general criteria for choosing between the possible arrangements. In principle, the electrons could be distributed uniformly in space or more or less lumped together; they could form rings or shells around the nucleus; orbits could lie on the same plane or in different ones, etc. Moreover, if electrons are distributed among different orbits, the problem arises as how to determine the number of the more internal electrons and that of the more external ones. At the turn of the 1920s, using inferences obtained from a variety of experiments, scientists began to find answers. At the same time, it began to be evident that the theoretical concepts hitherto developed were inadequate to answer all the open experimental problems. The construction of a new physics, quantum mechanics, stemmed from the attempts to resolve such a situation.

11.1 Electronic Structure and Periodicity of the Elements An essential property that atomic theories under development in the 1920s, had to explain was the periodicity of the elements’ physical and chemical properties. This periodicity had been inferred from the similarity of chemical compounds formed by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_11

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various elements and the concept of valency, developed in the second half of the previous century. In this context, a semi-quantitative rule had also been developed, which would prove particularly important for understanding atoms’ structure, the octet rule. It was an empirical rule able to explain in an approximate way the formation of chemical bonds. The recognition of the relevance of number eight in connection with valency goes back to Mendeleev’s reflections on the periodic table [1]. In 1871 he had published an extended survey of the periodicity of the elements’ chemical and physical properties. Among the properties discussed was, naturally, that of valency concerning which Mendeleev had formulated two rules where the number eight plays a key role. The first affirms that the maximum possible valency of whatever element can never exceed eight. The second establishes a relationship between an element’s maximum valency for hydrogen and its maximum valency for oxygen, affirming that their sum is never greater than eight. Furthermore, if a given element R, belonging to one of the groups from IV a VII of the periodic table, has a hydride1 of the form RH n , its highest oxide2 is necessarily of the form R2 O8 − n . The subject was revived in the early 1900s, together with the development of atomic models elicited by the discovery of the electron. It became clear that the periodicity of physical and chemical properties, in particular valency, must reflect a periodicity in the ordering of electrons and that elements of the same group of the periodic system must have a similar electron ordering. On the subject, in 1902 Richard Abegg (1869–1910), chemistry professor at Breslavia university, referring to the previous Berzelius and Davy’s electrochemical dualism, posited the rule of normal valency and of contra-valency. He postulated that all elements were capable of exhibiting both a maximum electropositive valency and a maximum electronegative valency and that the sum of the two was always equal to eight. The maximum positive valency coincided with the number N of the group to which the element belonged in the periodic table, while its maximum negative valence equalled 8 − N. In an article published in 1904, with reference to a possible unifying theory stating that the ion’s positive charge was due to lack of electrons and its negative charge to excess of them, Abegg gave a simple and prophetic interpretation of his rule [1]: The sum 8 of our normal and contra-valency possesses therefore simple significance as the number which for all atoms represents the points of attack of electrons, and the group-number or positive valence indicates how many of these 8 points of attack must hold electrons in order to make the element electrically neutral.

Abegg’s rule established an explicit connection between valency and possible distribution of electrons. However, he did not supply an explicit model of atomic structure, contenting himself with the vague concept of points of attack (i.e. point of bonding) of the electrons. These characteristics of the valency of the elements were explored explicitly by J. J. Thomson when formulating his atomic model. In the accompanying reflections 1 2

A hydride is a binary compound of hydrogen with another chemical element. An oxide is a binary compound of oxygen with another chemical element.

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Thomson drew up some valuable concepts for constructing atomic models, destined to remain an implicit part of all future electronic theories of valency. Most important of these was the concept that chemical periodicity implies a distribution of atom electrons into shells, with a periodic repetition of the outer shell’s structure, strictly related with the octet concept. In Thomson’s words [2]: We regard the negatively electrified corpuscles3 in an atom as arranged in a series of layers, those in the inner layers we suppose are so firmly fixed that they do not adjust themselves so as to cause the atom to attract other atoms in its neighbourhood. There may, however, be a ring of corpuscles near the surface of the atom which are mobile and which have to be fixed if the atom is to be saturated. We suppose, moreover, that the number of corpuscles of this kind may be anything from 0 to 8, but that when the number reaches 8 the ring is so stable that the corpuscles are no longer mobile and the atom is, so to speak, self-saturated. … Thus we see that an atom may exert an electropositive valence equal to the number of mobile corpuscles in the atom, or an electronegative valence equal to the difference between eight and this number. Each atom can, in fact, exert either an electropositive or electronegative valency, and the sum of these two valencies is equal to eight.

As we see Thomson introduced explicitly the concept that valency of elements can be correlated with the attempt to achieve a saturated shell via the transfer of electrons, and that the stability of noble gases, discovered in the late 1800s by Rayleigh and Ramsay and included as an eighth group in periodic system of elements, may be linked with having a complete external shell. In the first decades of the twentieth century the study of X-rays spectra directly supported the concept of a shell ordering of electrons in atoms. As we have seen, Moseley’s measurements and data analysis had supplied initial evidence for the existence of shells of electrons with characteristic energy, linked to the principal quantum number n of Bohr’s theory. Moseley’s pioneering experiments were taken up by other researchers, in particular by the Swede Manne Siegbahn (1886–1978). By the early 1920s one had a precise map of X lines as a function of atomic number, to be used to trace the energies of the electronic levels inside atoms. As well as radiations K and L, associable with n = 1 and n = 2 in the Bohr’s scheme, in heavier elements and, thus, with a greater number of electrons, were found further radiations, named M, N and O. All radiations turned out to be explicable using the scheme of transitions of Bohr-Sommerfeld’s theory, where an electron in an outer shell jumps into a more internal shell, from which one electron had been previously removed. In this scheme one had a precise correspondence between type of radiation (K, L, M, …) and principal quantum number of atom’s electronic shells, K lines corresponding to radiations emitted by electrons jumping into n = 1 levels, L lines to jumps into n = 2 levels, M lines to jumps into n = 3 levels, and so on, as shown in the diagram of Fig. 11.1.

3

In his writings Thomson always used the term negatively electrified corpuscles instead of electrons.

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Fig. 11.1 Scheme of the transitions which give rise to X radiations

11.1.1 Bohr’s Aufbauprinzip In 1921 Bohr tackled the problem of how one could define a scheme of electronic structure of atoms in agreement with: (a) evidence accumulated supporting a shell ordering of electrons; (b) the periodicity of the elements’ chemical and physical properties, shown in the periodic system; (c) the octet rule [3]. To this end Bohr proposed the Aufbauprinzip (i.e. principle of filling), a hypothetical process where each atom is constructed via pouring electrons into shells of growing energy starting from those of lower energy. A practical way to carry out this filling process is to hypothesize the construction of each atom starting from the atom preceding it in the periodic table of elements and adding a further positive charge to the nucleus and a further orbiting electron. The number of electrons present in each shell is obtained by examining the sequence of groups and periods in the periodic table and by the octet rule. With the exception of the two lightest elements, hydrogen (H) and helium (He), the table shows that each period begins with an element of group I (an alkali metal: lithium (Li), sodium (Na), potassium (K), rubidium (Rb), cesium (Cs)) and ends with an element of group VIII (a noble gas: helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), radon (Rn)). Consequently, assuming that a period is due to the

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progressive filling of an electronic shell, one finds that the number of electrons needed to fill it coincides with the difference between two atomic numbers, that of the noble gas preceding the period in question and that of the noble gas closing the period. Thus, the maximum number of electrons in each shell can be inferred from the distribution of the electrons in the noble gases, whose chemical inertia (here meaning that they bind neither among themselves nor with other elements) is due to having a full and inert octet as outermost shell. Starting from such observations, Bohr constructed a quantitative scheme where every shell contains electrons characterized by the same principal quantum number n and every value of n corresponds to the outermost shell of one of noble gases. The first shell, characterized by n = 1, corresponds to the single shell of helium and is complete when it contains 2 electrons. The second shell, characterized by n = 2, corresponds to the external shell of neon and is complete when it contains 8 electrons. The third shell, characterized by n = 3, corresponds to the outer shell of argon; and so on. Overall Bohr obtained the prospect of shells and of occupation for the noble gases shown in Table 11.1. For all noble gases the outermost shell is made up of an octet, which accounts for their chemical inertia. The maximum number N of electrons in every shell satisfies the relationship N = 2n2 . Starting from the distribution of electrons in noble gases it proved easy to determine the electronic configuration of the other atoms according to the Aufbauprinzip (principle of filling). Let us consider the simplest case of group I elements, the alkali metals. These atoms are obtained by adding one electron to the noble gas preceding them and a positive charge to the nucleus. They are thus characterized as having a single electron in the outermost orbit and all other electrons in internal orbits, which make up full shells, inert from a chemical viewpoint. It follows, therefore, that their optical spectra are due transitions between energy levels of the single electron in the external orbit; this allowed their interpretation through a simple generalization of the Bohr-Sommerfeld theory (strictly valid only for single-electron atoms). The alkaline-earth metals (periodic table’s group II) are obtained by adding one electron and one positive charge to the alkali metals preceding them and are characterized by having two electrons in the outermost orbit and all the others in full shells. Similarly, one may think of constructing the atoms of the other groups, one element after the other. Table 11.1 Prospect of shells and their occupation for noble gases Element Z Helium

Electrons in Electrons in Electrons in Electrons in Electrons in Electrons in shell n = 1 shell n = 2 shell n = 3 shell n = 4 shell n = 5 shell n = 6

2 2

Neon

10 2

8

Argon

18 2

8

8

Kripton 36 2

8

18

8

Xenon

54 2

8

18

18

8

Radon

86 2

8

18

32

18

8

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Bohr’s scheme provided, for the first time, a model for atoms able to explain naturally the periodic system and the similarity of the elements of one same group as regards to chemical behaviour and optical spectra, both properties being linked to electrons in the outermost orbits.

11.2 Fine Structure and Magnetic Field Effects We can summarise the progress made in the early 1920s by saying that the atomic theory, based on empirical rules of quantization introduced by the quantum physics’ pioneers (in primis, Planck, Einstein, Bohr and Sommerfeld), was able to supply an overall understanding of structure of atoms, of properties such as radiation absorption and emission, whether optical or X, the periodicity of the elements’ physical and chemical properties. More thorough examination of these properties showed however the existence of a relevant number of details which still could not be explained, even qualitatively, by using the theoretical concepts developed hitherto. Among these, particularly relevant for understanding matter’s structure and the development of quantum mechanics, were the fine structure of spectra, whether optical or X, and the magnetic field effects on optical spectra. Indeed, the final identification of atoms’ electronic structure would result from the attempt to place in a coherent scheme the experimental results pertaining to such physical effects. By the end of the nineteenth century empirical analysis of optical spectra had shown how many spectral series, if observed with higher resolution, are composed not of single lines but of multiplets, i.e. groups of several lines closely-spaced. The complex of these composite lines was named fine structure of atomic spectra. It was also found that each group of elements had a characteristic fine structure. For example, the spectra of alkali metals were made up of doublets (two closely-spaced lines) and triplets (three closely-spaced lines), those of alkaline-earth metals of series of singlets (single lines) and series of triplets. A fine structure was also present in the spectrum of the simplest of atoms, hydrogen. In fact, American physicist Albert Michelson (1852–1931), studying Balmer’s series with his interferometer, had shown that both the lines, Hα (λ = 6563 A, corresponding, in Bohr’s theory, to the transition from n = 3 to n = 2) and Hβ (λ = 4861 A, corresponding, in Bohr’s theory, to the transition from n = 4 to n = 2), consisted in reality of a couple of very closelyspaced, doublets, separated, respectively, by 0.14 A and 0.08 A [4]. Similarly, it was found that X-rays also had a fine structure. (This would be illustrated together with Sommerfeld and others’ attempts to explain it in the 1920s.) Strictly connected with the problem of fine structure were difficulties in comprehending the evolution of optical spectra in presence of a magnetic field, the so-called Zeeman effect, named after Dutch scientist Pieter Zeeman (1865–1943), Lorentz’s student and the first to reveal it. In fact, in 1896 Zeeman, during experiments aimed at investigating the interaction between radiation and matter, had discovered that the spectral lines were significantly altered by the presence of a magnetic field. This discovery, for which he received the 1902 Nobel prize, would prove fundamental for

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understanding atomic structure and the development of quantum mechanics. More specifically, observing the evolution of the D doublet (yellow lines) of sodium, he had found that, in the presence of a magnetic field, each line grew wider and seemed to consist of three components, the Zeeman triplet. Soon after the discovery, Lorentz showed that this behaviour could be accounted for by the electron theory he had formulated some time before. However, observations later made with higher-resolution apparatus, showed that the effect of the magnetic field was much more complex than this. Often, the lines were split into a number of components different from three; this number varied from line to line and differed also for lines of the same fine structure. For example, in the case of D doublet of sodium, analysed initially by Zeeman, high resolution measurements showed that one of the two lines splits into four components and the other into six. In other cases splitting could be observed in five, six or even nine components. Consequently, in describing the behaviour of lines of different atoms in magnetic field, it had become normal practice to distinguish between normal Zeeman effect (i.e. separation of the line into a triplet, as Lorentz’s theory predicted) and anomalous Zeeman effect (all the other cases). The anomalous Zeeman effect was inexplicable, both by the Lorentz’s classical analysis, or by applying BohrSommerfeld’s quantum theory.4 In the early 1920s attempts to explain the fine structure of optical and X-rays spectra and anomalous Zeeman effect were at the centre of the physicists’ attention. These attempts led to the discovery of a new mind-boggling property of the electron: spin.

11.3 Sommerfeld’s Relativistic Corrections A possible effect contributing to the fine structure of hydrogen was analysed by Sommerfeld at the same time as his generalisation of Bohr’s theory. We should recall that in the scheme postulated by Bohr the frequency of the lines is given by the difference of energy of two stationary states between which the electronic transition occurs. It follows that the presence of a fine structure necessarily requires the existence of a fine structure, i.e. a multiplicity, in the initial or in the final energy level or of both. A multiplicity of states was present in the case of elliptical orbits where, for every value of principal quantum number n, there existed n different states having the same energy, corresponding to orbits characterized by different values of azimuthal quantum number k (i.e. different angular momentum). Sommerfeld showed that the energy is the same only if effects due to relativity are ignored, in particular the relativistic effects due to variation of mass with the velocity.5 Since electrons run in orbits with very high velocity, Bohr, in his early articles had already 4 The Bohr-Sommerfeld theory, applied to calculate the magnetic field effects, reproduces only the normal Zeeman effect. 5 According to special relativity the mass m of a particle moving with velocity v is given by

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noted the need to take into account relativistic effects for more accurate calculations. Consequently, upon introducing elliptical orbits, Sommerfeld also calculated the correction due to relativistic effects and showed how orbits with the same n but different k’s have indeed slightly differing energies. Thus was found a multiplicity of energy levels which could explain the fine structure of optical spectra. In particular, in the case of the n = 2 level of hydrogen, corresponding to the final state of the Balmer series transitions, Sommerfeld obtained an opening in two levels (corresponding to k = 1 and k = 2) with an energy splitting in line with the experimental value of the doublets H α and H β . Sommerfeld’s calculations thus strengthened the hypothesis that relativistic effects underlay the fine structure of optical spectra. However, it would later be discovered, when the problem was solved with the introduction of the electron spin, that this agreement was partly fortuitous and due to the peculiar structure of relativistic effects in the hydrogen atom. Later Sommerfeld tackled the problem of relativistic effects’ relevance in the context of X-radiation and showed that these could explain the fine structure observed in K radiation. Precise analysis had shown that the line K α was in reality a doublet, whose splitting increased with increasing atomic number Z.6 Analysis of energy and intensity of the doublet lines (named K α and K α' ) had led to the conclusion the origin could be due to a splitting of L shell into two sub-shells of different energies, named L 1 and L 2 . Line K α' had been attributed to the transition L 2 = > K and line K α to the transition L 1 = > K. To explain the different energies of L 1 and L 2 , Sommerfeld assumed that L-shell electrons followed orbits similar to those of the hydrogen electron. He calculated their relativistic corrections, finding that their energy depended on azimuthal quantum number k, in analogy with what happened for n = 2 orbits of the hydrogen electron. Sommerfeld also showed that the energy difference between L 1 and L 2 increased as the fourth power of atomic number Z, a result perfectly in line with what had been measured experimentally. However, despite the indisputable success in explaining K radiation’s fine structure, at once it became evident that relativistic effects alone were unable to account for the characteristics of X-rays as a whole. Experimental analysis of L, M and N radiations had proved the presence of a fine structure notably more complex than that of K radiation. The composite line multiplicity could only be explained by admitting a multiplicity of levels larger than the possible values of angular momentum. Indeed, for L shell it was necessary to introduce as well as levels L1 and L2 , a third level L3 ; for M shell a subdivision was needed into five levels (M1 , M2 , M3 , M4 , M5 ) and even into seven levels for shell N. The presence of three sub-shells inside L shell and five sub-shells inside M shell had also been confirmed by X-Rays absorption measurements made in 1913 by French physicist Maurice de Broglie (1875–1960). m0 m= / 1−

v2 c2

where m0 is the particle’s rest mass and c is the velocity of light. 6 Recall that line K had been attributed to transitions between L shell (n = 2 energy levels) and K α shell (n = 1 energy levels).

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It was further found that the lines could be regrouped into two types of characteristic doublets, one behaving like that of K radiation, whose separation grew by the fourth power of atomic number Z and were attributable to relativistic corrections, and a second type, where the line splitting was independent of Z and whose origin remained a mystery.

11.4 A New Quantum Number: Electron Spin In the early 1920s it became clear that the Bohr-Sommerfeld’s theory, even with relativistic corrections included, could not account for the fine structure multiple aspects. As well as the levels multiplicity revealed by analysis of X rays, there still remained the mystery of the anomalous Zeeman effect and the lines multiplicity of optical spectra in elements different from hydrogen. In particular, as we have seen, the series of lines and the fine structure of alkali metals and alkaline-earth metals, had been analysed in detail by Rydberg, Ritz and others in the late nineteenth century. To explain the presence of doublets and triplets, starting form Ritz’s principle of combination, they had had to hypothesise that many terms were not simple, but doubles or triples. Analysis of the main four series of lines (principal, sharp, diffuse and fundamental, see Optical Spectra of Alkali Metals in Appendix C) had led to the establishment of the following multiplicities: the s terms (s initial of sharp) were always single, while terms p, d and f (initials, respectively, of principal, diffuse and fundamental) were double in alkali metals and single and triple in alkaline-earth metals. Since in Bohr’s quantization scheme terms corresponded to energy levels, there turned out to be a multiplicity of energy levels which the Bohr-Sommerfeld’s theory could not explain. On the other hand, given its successes in accounting for the principal characteristics of atoms’ physical properties, its underlying validity could not be questioned. In the early 1920s, to break such dead-lock, was advanced the hypothesis that the anomalies could derive from the existence of a new degree of freedom, not yet identified, to which a new quantum number could be associated, characterizing possible different states with the same values of n and k. This hypothetical quantum number was named internal quantum number. First to suggest this possibility was Werner Heisenberg (1901–1976), who proposed it to explain the fine structure of alkali metals [5]. Heisenberg started from the observation that, leaving fine structure to one side, the series of lines of alkalis were well described by the Bohr-Sommerfeld’s theory, if one assumed they were due to the transitions between stationary states of the outermost electron. He hypothesised that the searched new degree of freedom could be due to a rotational motion of the atom’s core (rumpf in German), composed of the nucleus and all other internal electrons. The core rotation implied the presence of a new quantised angular momentum and, consequently, of an electromagnetic interaction between the orbital motion of the outermost electron and the core’s rotational motion, this leading to a splitting of energy levels. The idea that the fine structure

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could be due to the existence of a further angular momentum would prove substantially correct, even if its origin is not that hypothesised by Heisenberg, but is due, as we will see, to the electron’s spin. The hypothesis of an internal quantum number acquired further credibility when the German Alfred Landé (1888–1976), in 1922, showed how this hypothesis could explain the mysteries of the anomalous Zeeman effect. In his theory Landé assumed the existence of a new angular momentum in the atom, as proposed by Heisenberg. However, he also hypothesised that the internal quantum number was associated with the total angular momentum, sum of the outermost electron’s angular momentum and of the new angular momentum. Landé’s model, based on empirical rules, accounted for all the characteristics of the Zeeman effect and predicted others, subsequently verified, with an accuracy defined as a marvel in the history of science [4]. Still in 1922, Landé showed also how the introduction of an internal quantum number, which he labelled j, and which assumed only two distinct values j = k or j = k − 1, made it possible to accurately classify energy levels giving rise to the fine structure of X radiation. To this end each level was characterized by three quantum numbers, n, k and j. The resulting scheme of levels of K, L and M shells is shown in the Table 11.2 [6]. The classification proposed by Landé was the departure point that, in 1924, allowed English physicist Edmund Stoner (1899–1968) to take another decisive step forward in the tricky construction of a scheme leading to understanding the arrangement of electrons inside atoms. We have seen how Bohr, applying the Aufbauprinzip, correctly determined the number of electrons present in different shells (K, L, M...). However, regarding the distribution of electrons in the sub-shells, lacking reliable criteria to use, Bohr had limited himself to suggesting a uniform distribution, something that would prove substantially mistaken. The problem was solved by Stoner. He surmised that the number of electrons in each sub-shell must be linked to the internal quantum number j and he proposed an ordering whereby this number was twice the j’s value. The final result was the electron distribution shown in the last line of the Table 11.2, which would be proved substantially correct. Despite these undeniable successes, this scheme of quantum numbers was still unsatisfactory. As well as being based on ad hoc hypotheses, justified only empirically, the assumption that the internal quantum number was linked to the core’s Table 11.2 Scheme of the three innermost shells of atoms, with the values taken by the quantum number n, k, j Shell

K

Sub-shell

L

M

L1

L2

L3

M1

M2

M3

M4

M5

n

1

2

2

2

3

3

3

3

3

k

1

1

2

2

1

2

2

3

3

j

1

1

1

2

1

1

2

2

3

Number of electrons

2

2

2

4

2

2

4

4

6

The maximum number of electrons in each sub-shell is also specified

11.4 A New Quantum Number: Electron Spin

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angular momentum seemed questionable. In effect, in the very same period Pauli showed that the core, understood as the sum of the nucleus and all the electrons in completely filled shells, must have a zero angular momentum. As an alternative, in January 1925, he suggested that electrons have, as well as the three classical degree of freedom, a non-classical degree of freedom whose quantum number could take only two values and was correlated to the internal quantum number. Pauli called this new degree of freedom a two-valued quantum degree of freedom, but was unable to specify its physical origin. The situation was clarified definitively in the same year (1925): Two Dutch research students, George Uhlenbeck (1900–1988) and Samuel Goudsmit (1902– 1978), postulated that electrons, besides the orbital motion (associated with the three spatial degrees of freedom), have an intrinsic motion of rotation (spin) around their own axis, just like the planets revolve around their axis in their orbital motion. To think of the electron as a charged particle rotating on its axis is equivalent to postulating that the electron possesses an intrinsic angular momentum (besides the orbital angular momentum) and an intrinsic magnetic dipole momentum. The immediate result is a new contribution to the energy due to the electromagnetic interaction between orbital and spin motions.7 This interaction, named spin–orbit interaction, is the main responsible for the subtle features in the fine structure of atomic spectra. Although modelling the spinning electron as a charged particle revolving around its own axis is useful and has come to be currently used to understand the physical origin of spin–orbit interaction, it should be underlined how it was clear from the outset that the spin angular momentum must be a non-classical angular momentum. Indeed, the laws of classical physics applied to the model predict a behaviour contrasting with the tenets of special relativity. As early as 1903, German physicist Max Abraham (1875–1922) had studied a model in which an electron was simulated as a small rotating sphere with a unit electric charge spread on the surface. He had shown how, to obtain an appreciable angular momentum, it was necessary hypothesise a rotation velocity much higher than the velocity of light. Consulted on this regard by Uhlenbeck himself, in concomitance with the presentation of his idea, Hendrik Lorentz had shown how, to have the correct energy of interaction, the electron’s radius should be much greater than the value compatible with experimental evidence. Later, using quantum mechanics, Bohr definitively showed that spin is an essentially quantum mechanical property of electrons [7]. In the scheme proposed by Uhlenbeck and Goudsmit the spin’s angular h . It enjoys the property of being quantized in space, such momentum S is S = 21 2π as to have only two possible orientations, either parallel or anti-parallel, relative to a chosen direction, and of having an intrinsic magnetic dipole moment, whose value is twice the classical one. Landé’s internal quantum number j, came to be the quantum number identifying the electron’s total angular momentum, the sum of 7

A simple way of understanding this interaction’s origin was devised by Einstein and consists in considering the motion in the electron’s system of reference. In this system the charged nucleus revolves around the electron; this rotation, equivalent to a circulating current, produces a magnetic field acting on the magnetic dipole momentum associated with the electron’s spin.

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orbital angular momentum and spin angular momentum. With these characteristics the electron’s spin can explain quantitatively, not just qualitatively, all the features observed in the fine structure of optical and X-rays spectra, as well as the anomalous Zeeman effect. Thus, its existence was rapidly accepted by the scientific community.

11.5 Wavelike Nature of Matter In the same period, together with these undoubted advances, a new revolutionary property of matter was discovered: wave-particle duality. We have seen how in 1905 Einstein introduced the hypothesis that electromagnetic radiation, as well as wavelike nature, also has corpuscular characteristics and that its energy is quantised. In 1923 the electromagnetic radiation’s corpuscular behaviour was shown experimentally in a direct way by American physicist Arthur Holly Compton (1892–1962). Soon after experimental confirmation of Einstein’s revolutionary idea, an equally revolutionary hypothesis was advanced by French theoretician Louis de Broglie (1892–1987) [8]. It involved the idea, mirroring Einstein’s, that electrons, particles of matter, have a wavelike nature as well as the corpuscular one. According to de Broglie, in the early 1920s the research on microscopic physics presented two prevalent problems: On the one hand was the contradictory nature of light, which exhibited both wave and particle characteristics; on the other was the mysterious behaviour of electrons inside atoms, where only some orbits and some energy values were permitted. To cite de Broglie [9]: …such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics. When I started to ponder these difficulties two things struck me in the main. Firstly, the light-quantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation E = hv which contains a frequency v. 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 eigen-vibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr’s expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. The first objective to achieve had, therefore, to be to establish this correspondence.

According to de Broglie the parallelism between matter and radiation implies that, for both types of particles (electrons or photons), the energy E must be linked to the frequency ν of the associated wave by the relation:

References

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E = hν and the linear momentum p must be linked to the wavelength λ of the associated wave by the relation: p=

h λ

Despite its originality, at the time of its formulation de Broglie’s idea was supported by no experimental evidence and appeared to have no physical reality. It was Einstein who fully appreciated its importance and brought it to scientists’ attention. A series of experiments followed, proving the hypothesis correct and leading to de Broglie winning the Nobel prize five years later, in 1929. Demonstration of de Broglie’s hypothesis required constructing sophisticated apparatus and performing difficult experiments. In fact, it was necessary to highlight wave phenomena with very small wavelength, of the order of a tenth of nanometer, or nearly 5000 times smaller than that of visible light. The first step was taken in 1926, when German physicist Walter Maurice Elsasser (1904–1991) pointed out that matter’s wavelike nature can be demonstrated in the same way as the wavelike nature of X-rays, that is, by irradiating a crystal with a beam of electrons of appropriate energy and analysing the diffraction pattern. On this basis was constructed the experiment of American physicists Clinton Davisson (1881–1958) and Lester Germer (1896–1971), who, in 1927, supplied the first experimental proof of electrons’ wavelike behaviour, putting wave-corpuscle duality on a solid experimental footing. Shortly afterwards, British physicist George Paget Thomson (1892–1975), J. J. Thomson’s son, repeated the experiments, passing the electrons through thin gold foil and obtaining a series of diffraction rings that unequivocally confirmed electrons’ wavelike properties. For their experiments Thomson and Davisson shared the 1937 Nobel Prize. Since 1927, the wave-particle dualism of matter has been considered an established fact by the scientific community. The universality of wavelike properties of matter particles was confirmed experimentally in 1929, when German physicists Otto Stern (1888–1969) and Immanuel Estermann (1900–1973) observed diffraction patterns after experimenting with beams of helium atoms and hydrogen molecules, that is, with composite systems having a mass much greater than that of the electrons.

References 1. Jensen W B (1984) Abegg, Lewis, Langmuir, and the Octet Rule. Journal of Chemical Education 61: 191-200 2. Thomson J J (1914) The forces between atoms and the chemical affinity. Philosophical Magazine 27: 757-789 3. Bohr N (1921) Atomic Structure. Nature 107: 104-107 4. White H E (1934) Introduction to atomic spectra. McGraw-Hill, New York

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5. Pais A (1989) George Uhlenbeck and the discovery of electron spin. Physics Today 12: 34-40 6. Stoner E C (1924) The distribution of electrons among atomic levels. Philosophical Magazine 48: 719-736 7. Pauli W (1946) Exclusion principle and quantum mechanics. Nobel Lecture, December 13 8. De Broglie L (1924) A tentative theory of light quanta. Philosophical Magazine 47: 446-458 9. De Broglie L (1929) The wave nature of the electron. Nobel Lecture, December 12

Part III

Quantum Mechanical Representation of Matter

Chapter 12

Quantum Atoms and Molecules

That described in the previous chapter is, broadly speaking, the state of understanding of the physics of matter in the mid-1920s and the set of concepts, rules and postulates of the theoretical construction, later known as old quantum theory. Despite its successes which granted credit to quantization hypotheses, this theoretical construction showed serious limitations, both practical and conceptual. On the practical front, a very significant example of its inadequacy was the inability to account for the physical properties of the helium atom, the simplest of atoms after hydrogen. Regarding conceptual limitations, we should note that it was based on a series of ad hoc assumptions and postulates introduced to account for the characteristics of matter observed in experiments. A coherent justification of hypotheses and postulates was lacking. The mechanism leading to radiation’s emission and absorption, when the electron transited between two stationary states, remained unclear. Quantization rules seemed arbitrary, also when they proved more effective in their application. The essence of wave-particle duality of both electromagnetic radiation and electrons was still a mystery.

12.1 Quantum Mechanics In the second half of the 1920s, to give this construction a coherent structure, two new theories were drawn up: matrix mechanics, proposed in 1925 by German theoretical physicist Werner Heisenberg (1901–1976) and later developed by him and theoretical physicists Max Born (1882–1970) and Pascual Jordan (1902–1980), and, secondly, quantum wave mechanics, proposed and developed by Austrian theoretical physicist Erwin Schroedinger (1887–1961) in 1926. The two theories, despite the great mathematical difference, were in reality equivalent and produced the same results, as Schrödinger later demonstrated. A more general and abstract formulation, encompassing both wave mechanics and the matrix mechanics, was drawn up by English theoretical physicist Paul Adrien Maurice Dirac (1902–1984) in 1930. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_12

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Together, these theories underpin the new quantum theory or quantum mechanics, tout court. Here we shall briefly illustrate the salient characteristics of quantum wave mechanics, easier to visualize by using general concepts of physics. De Broglie’s postulate stated that the motion of a microscopic particle is driven by the propagation of the associated wave, but did not specify the wave propagation rules. Furthermore, its practical use required that a quantitative relationship be established between the particle properties and the associated wave properties. These questions were answered by the wave equation elaborated by Schroedinger and the interpretation of its wave function developed by Born. The quantum wave equation was devised by Schroedinger in strict analogy with classical wave theory. A classical electromagnetic wave is described specifying a wave function (electric field E or magnetic field H), which evolves in time and space, obeying the wave equation derived from Maxwell’s equations. Similarly, in the quantum wave mechanics, a particle’s motion is described by a wave function Ψ , which evolves in time and space, obeying the wave equation devised by Schroedinger. The relationship between wave-like and particle-like behaviour was clarified by Born, who in 1926 introduced the following postulate to interpret the wave function Ψ : If, at the instant t, a measurement is made to locate the particle associated with the wave function Ψ(r, t), the probability that the particle will be found at the point r is proportional to |Ψ(r, t)|2 . To formulate his postulate, Born used a reasoning similar to that adopted by Einstein to unify wave-like and particle-like aspects of electromagnetic radiation.1 Born described the situation as follows [1]: We describe the instantaneous state of the system by a quantity Ψ, which satisfies a differential equation, and therefore change with time in a way which is completely determined by its form at time t=0, so that its behavior is rigorously causal. Since, however, physical significance is confined to the quantity |Ψ|2 , and to other similarly constructed quadratic expressions, which only partially define Ψ, it follows that, even when the physically determinable quantities are completely known at time t = 0, the initial value of the Ψ-function is necessarily non completely definable. This view of the matter is equivalent to the assertion that events happen indeed in a strictly causal way, but that we do not know the initial state exactly. The motion of particles conforms to the law of probability, but the probability itself is propagated in accordance with the law of causality.

In 1927 Heisenberg introduced the famous uncertainty principle, whereby simultaneous measurement of two conjugate variables, such as position and momentum or energy and time, cannot be performed without a minimal ineliminable amount of uncertainty. Regarding the couple position/momentum, the uncertainty principle states that an experiment cannot determine simultaneously the exact value of a particle’s momentum and the exact value of its position. The measurement’s precision is intrinsically limited such that the uncertainties obey the inequality: 1

Einstein equated the flux I of energy per unit area, carried by the electromagnetic wave, which is proportional to the square of the electric field E, to the average number N of photons of energy hν crossing the unit area per unit time (in formulas: I = N hv).

12.2 Hydrogen

143

Δpx Δx ≥ h/2 where Δpx is the uncertainty associated with momentum’s x component and Δx the uncertainty associated with position’s x component. Analogous relationships hold for the other components. The limitation pertains to the product, not the single component. In principle we can have Δpx = 0, but this implies Δx = ∞. The uncertainty principle’s second part deals with the measurement of energy and time. In this case we have: ΔEΔt ≥ h/2 where ΔE è uncertainty associated with the particle’s energy and Δt is uncertainty associated with time determination. With the advent of quantum mechanics there began a systematic re-examination of the properties of matter by using the new mechanics. The first applications dealt with the problems which were central to the old quantum theory: the structure of atoms and molecules. One started from the simplest system: the hydrogen atom.

12.2 Hydrogen In his first article on wave mechanics Quantisierung als Eigenwertproblem” (Quantization as an eigenvalues problem), published in 1926, Schroedinger, as well as deriving the wave equation bearing his name, applied it to the solution of the hydrogen atom. He found that the wave equation solutions are wave functions Ψ n,l,m , distinguished by three quantum numbers n, l, m. Assuming l = k − 1, they are identical to quantum numbers n, k, m of Bohr-Sommerfeld’s theory. Their significance is also identical: n is the principal quantum number, that can take whole number values from 1 to ∞ (infinity); l is the azimuthal quantum number, which can take only the n values 0, 1, 2, 3, …, n − 1, and is related to the orbital angular momentum L of the electron by the relationship L 2 = l(l + 1) (h/2π)2 ; m is the magnetic quantum number, that can take only the 2 l + 1 values 0, ± 1, ± 2, ± 3, …, ± l, and is related to the component L z of angular momentum along a z direction by the relationship L z = m h/2π. Energy of stationary states corresponding to wave functions Ψ n,l,m depends solely on quantum number n and is identical to that obtained by Bohr: En = −

2π 2 me4 Z 2 Ry ≡− 2 n2 h2 n

Nevertheless, the resulting picture of electronic states is very different from the semiclassical orbits hypothesised by Bohr and Sommerfeld. According to quantum mechanics, for the electron one can only calculate the probability |Ψ(r, t)|2 of finding it at point r at the instant t. This probability has a characteristic distribution for each

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Fig. 12.1 Three-dimensional plot of the electron’s probability density |Ψ 1,0,0 (r)|2 of hydrogen ground state (n = 1, l = 0, m = 0)

of the values of the three quantum numbers n, l, m, which the electron can assume in stationary states. Consequently the electron in the hydrogen atom, instead of being a punctiform object orbiting the nucleus, comes to assume the aspect of a cloud of probability density, equal to |Ψ n,l,m (r,t)|2 , surrounding the nucleus. To each stationary state corresponds a precise distribution of density of probability. By way of example, this is shown schematically in Fig. 12.1 for the hydrogen ground state (n = 1, l = 0, m = 0). In conclusion, although having invalidated the concept of electron as an orbiting particle, quantum mechanics confirmed important aspects of hydrogen envisaged by the old quantum theory. The electron’s stationary states continue to be characterized by the three quantum numbers (n, l, m), which retain their meaning. The energy eigenvalues are identical. Finally, also valid is the concept that emission or absorption of radiation occurs only when the electron changes state and that the energy emitted or absorbed equals the difference in energy between initial and final electronic states. There is, however, an important conceptual difference. In the solution of Schroedinger’s equation, energy and of angular momentum quantisation, as well as the quantum numbers n, l, m, appear naturally as properties of the wave equation solutions. This contrasts sharply with old quantum theory, where quantisation was inserted as a set of ad hoc assumptions. In the same year (1926) that Schroedinger solved the wave equation for the hydrogen atom, Heisenberg and Jordan showed how its fine structure could be explained introducing both relativistic effects and spin–orbit interaction as perturbations (i.e. corrections) to the solutions obtained with the new quantum mechanics [2].

12.2 Hydrogen

145

They demonstrated that each energy level splits into a doublet whose components correspond to the two possible values of the total angular momentum, the sum of the electron’s orbital and spin angular momenta. The two values correspond to the two possibilities: orbital and spin angular momenta being parallel or anti-parallel. Accordingly, the quantum number j associated with total angular momentum takes one of the two values: j = l + 1/2 for parallel momenta and j = l − 1/2 for antiparallel momenta. Electron spin and hydrogen atom fine structure were finally systematized by English physicist Paul Adrien Maurice Dirac (1902–1984). In 1928 he drew up a relativistic quantum theory of the electron. To represent the electronic states he introduced a four-components wave functions, known as Dirac’s spinor. The theory clarified the fine structure’s origin, identifying the nature of corrections to energy values obtained by Schroedinger’s equation. The corrections stem from three physical effects: (1) the variation of mass with the velocity expected in the theory of relativity (the effect highlighted by Sommerfeld); (2) the spin–orbit interaction (consequence of Uhlenbeck and Goudsmit’s spin hypothesis); (3) a further effect, calculated by English theoretical physicist Charles Darwin (1887–1962) (afterwards named Darwin’s term), which apply to s states (l = 0) only. The resulting hydrogen’s energy levels are schematically shown in Fig. 12.2. Hydrogen turns out to have the peculiar property whereby states with different values of l, but the same j, are degenerate (i.e. have the same energy). This is the peculiarity that made Sommerfeld’s calculation of relativistic corrections seem in line with experiments. Fig. 12.2 Diagram (not to scale) of hydrogen’s energy levels: (to left) Schroedinger non relativistic theory; (to right) Dirac relativistic theory. j values (1/2, 3/2, 5/2) are shown as subscripts

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12.3 Helium and Quantum Properties of Multielectron Atoms Investigating the atomic structure of helium, the simplest atom after hydrogen, where two electrons are bonded to a nucleus of positive charge 2e, opened up a new problem: what is the correct quantum–mechanical description of a system containing two or more identical particles? The problem’s solution revealed the existence of new and highly relevant quantum mechanical properties of matter: Pauli’s exclusion principle, particles’ indistinguishability, and exchange interaction. These are properties with no analogy in classical physics and highlight some of the most striking differences between classical and quantum mechanics. Experimental study of helium’s emission lines had shown that its spectrum could be divided into two parts: one, including infrared and visible lines, analysed by Runge and Paschen back in 1896; the other made up of few ultraviolet lines, discovered by Lyman only in 1922. Runge and Paschen’s analysis of the visible and infrared part had shown the presence of two independent systems of lines, one composed of singlets and the other apparently composed of doublets. Each of the two systems comprised the four series: sharp, principal, diffuse and fundamental. Moreover, the analysis showed that no correlation existed between the two systems of lines. Since the presence of two unrelated line systems was not found in any other element, it was initially thought that they signalled the existence of two different types of helium, called orthohelium and parahelium. With time it became clear helium is a single substance. However, until 1925 physicists continued to believe that the two systems of lines derived from two distinct modifications of the helium atoms. In 1926, thanks to the experimental demonstration that the orthohelium lines were not doublets, as believed hitherto, but triplets and thanks also to Heisenberg’s quantum–mechanical calculations of line’s intensities, it became clear that the helium spectrum is analogous to that of the other elements with two electrons in the outermost shell (elements of group II of the periodic table: berillium, magnesium, calcium, …). Thus, one arrived at a first approximation scheme of energy levels of helium and alkaline-earth elements, based on the Aufbauprinzip, where fine structure effects were neglected. The four series of lines were identified as transitions from states in which one electron is excited, while the other remains in its ground state. Accordingly, the dominant interaction felt by the excited electron is the attraction of the nucleus, while, as a first approximation, the effect of the other electron can be thought of as a screening of the nucleus’s charge. The problem becomes similar to that of a hydrogenlike atom and, therefore, the electronic states are characterized by the trio of quantum numbers n, l, m. This made it possible to identify the system’s possible states and the transitions corresponding to the normal series of the spectrum. Normal series correspond to states where an electron becomes excited and takes all the possible values of (n, l, m) while the other electron remains in the lowest energy state. The energy levels and possible transitions of the excited electron are schematically shown

12.3 Helium and Quantum Properties of Multielectron Atoms

147

Fig. 12.3 Energy levels and optical transitions of the helium atom

in Fig. 12.3.2 Notice that, in this scheme and at this level of approximation, each energy level has a high degree of degeneracy, because the energy does not depend on the 2 l + 1 possible values of magnetic quantum number m. The next step consisted in considering fine structure and possible interactions giving rise to it. The first important characteristic to consider was, as already mentioned, that the levels of parahelium are singlets, while those of orthohelium are triplets. The explanation for the existence of singlets and triplets led to the Russell–Saunders’ coupling scheme between angular momenta. Moreover, there were two other notable peculiarities: first, the absence of the triplet state, when the two electrons are both in a 1 s state, as demonstrated by the absence of transitions toward this state in orthohelium; second, the notable energy difference between the homologous singlet and triplet states.3 Explanation of these last two characteristics led to the formulation of the exclusion principle by Pauli and to the discovery, by Heisenberg and Dirac, of the exchange interaction, stemming from the concept of indistinguishability of particles. In 1925 American physicists Henry Norris Russell (1877–1957) and Frederick Saunders (1875–1963) demonstrated empirically [3] that the presence of single and triple levels in atoms with two electrons in the outermost shell (elements of periodic system’s group II) can be explained if one assumes the following scheme for the 2 Remember that the value of the orbital angular momentum is conventionally specified by a letter, instead of the corresponding quantum number, with the following correspondence: s (l = 0); p (l = 1); d (l = 2); f (l = 3). Lower-case letters are used for the orbital angular momentum of a single electron and upper-case letters for the total orbital angular momentum of a system of electrons. 3 States are homologous if they have the same values of n and l.

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interactions between the two electrons and their intensities. The strongest interaction is between the spin motion of the two electrons (interaction spin–spin, whose origin was still unknown), its intensity being much greater than that between the orbital motions (orbit-orbit interaction). This, in turn, is greater than the interaction between the spin motion and the orbital motion of each electron (spin–orbit interaction).4 This difference among the interaction intensities implies that spin total momentum (sum of electrons’ spins) and orbital total momentum (sum of electrons’ orbital angular momenta) are both defined and conserved, and that their mutual interaction is weak. These hypotheses, later named Russell–Saunders coupling scheme, can explain the characteristics of spectral lines of helium (and of group II elements) as follows. Let us consider the effects of the interaction between the spins (spin–spin interaction) of the two electrons. According to quantum mechanics’ rules of composition of angular momenta, we have two possibilities for the spin total momentum: either the spins are parallel (to which corresponds a total spin S = 1) or they are anti-parallel (to which corresponds a total spin S = 0). Supposing now that the spin–spin interaction depends on the relative orientation of the two spins, we end up with energy of S = 0 levels different from that of S = 1 levels, a difference that underlies the distinction between orthohelium and parahelium.5 One sees, therefore, that the hypotheses lead to the scheme of levels and transitions shown in Fig. 12.3. To address the fine structure problem and the origin of triplets and singlets, let us consider the total angular momentum J (J = L + S), sum of the total spin angular momentum S and the total orbital angular momentum L. According to the quantum– mechanical rules of composition of angular momenta, when S = 0 (antiparallel spins) J can have a sole value (J = L), while when S = 1 (parallel spins) J can have three distinct values (J = L − 1, L, L + 1). Assuming an interaction (spin–orbit interaction) between the spin motion and the spatial motion and that this interaction depends on mutual orientation between S and L, it follows that energy levels with the same values of L and S, but different values of J, come to have different energies. This gives rise to singlets (for S = 0) and triplets (for S = 1) in the fine structure. Finally, since in optical transitions the electrons’ spin does not change, there are no transitions between the two types of electronic levels, resulting in two distinct sets of series of lines: those of parahelium (S = 0) and those of orthohelium (S = 1). Russell–Saunders’ coupling scheme, applied also to atoms with more than two electrons in the external shell, immediately appeared to be the right key for interpreting elements’ spectral lines and their fine structures.6 However, it failed to explain one of the peculiarities of the helium spectrum: the absence of the triplet state, when both electrons are in 1 s state, i.e. the absence of the lowest energy state expected for 4

It should be borne in mind that, actually, Russell and Saunders in their article still used the concept of core angular momentum instead of the spin angular momentum, since the spin concept had not yet been introduced. To avoid unnecessary additional difficulties, the exposition of their coupling scheme is done here using modern concepts. 5 As for the interaction between orbital motions (orbit-orbit interaction), it does not intervene since in the normal series one electron remains in state 1 s with zero orbital momentum. 6 In reality this applies only to light elements, i.e. elements with an atomic number up to approximately Z = 40.

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orthohelium. This state, if present, would have the peculiarity of having two electrons with identical values for all quantum numbers n, l, m, s. This peculiarity was noticed by German theoretical physicist Wolfgang Pauli (1900–1958). Suspecting this to be the reason for the missing triplet, he conducted a general survey of the spectral lines, to see if, in other elements and in other conditions, the same thing happened [4]. The survey confirmed that in all cases of missing terms, the electrons would have had identical quantum numbers. This discovery led Pauli, in 1925, to formulate the exclusion principle named after him. Defining an electron’s quantum state as the state characterized by the values of its quantum numbers, the exclusion principle affirms that no two electrons in an atom may be in the same quantum state. This is equivalent to affirming that the quantum numbers values of two or more electrons can never coincide entirely. In the same year, Pauli laid out the principle of indistinguishability of electrons, introducing the concept of identical particles and indistinguishability in quantum mechanics, meaning that the observable results of a quantum mechanical calculation need not depend on how identical particles are labelled. This is equivalent to saying that results of measurement of a physical property must not change if there is a permutation (i.e. an exchange) of particles within the system. Exclusion principle and particles indistinguishability, when translated into mathematical conditions for the wave functions, impact strongly on the form of the wave equation’s possible solutions (For more information see Indistinguishability and exchange interaction in Appendix D). In the case of a multi-electron system, the exclusion principle and particles’ indistinguishability require that wave functions should be antisymmetric for the exchange of two of its particles. This purely quantum mechanical feature has relevant consequences for the properties of atoms and matter in general. Among the most important of these is the emergence of a new form of interaction between electrons, this having no classical analogue and named exchange interaction. This interaction has two important characteristics: (1) its strength is of the same order of Coulomb repulsion between the two electrons and, thus, is much larger than the relativistic corrections to energy levels; (2) in general its action is such as to lower the energy of the triplet state in respect to the singlet’s. Exchange interaction tends therefore to keep the electrons’ spins parallel, a fact of extreme importance for matter’s properties, in particular as a source of magnetism in ferromagnetic substances, as Heisenberg immediately guessed. In helium’s case it explains the origin of the system’s other peculiarity: the notable energy difference between singlet and triplet homologous states. Exchange interaction was discovered independently by Werner Heisenberg and Paul Dirac in 1926.

12.4 Molecules: Covalent Bond By the late 1920s the new quantum mechanics was already being applied to clarifying the nature of chemical bonds, one of the main problems of the structure of matter still open to debate. The moment’s question of priority was the nature of the forces

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allowing equal atoms to bind together to form molecules, as happens for many of the more common elements in gaseous state, such as hydrogen, oxygen, nitrogen, etc. A problem which, as we have seen, had hindered the development of correct atomic and molecular theories. An important step toward solving the problem came with growing acceptance of the hypothesis that a chemical bond results from the transfer and sharing of electrons between the bonding atoms. The first explicit formulation of this hypothesis was contained in a 1916 article by American chemist Gilbert N. Lewis (1875–1946). In this period similar ideas were also supported by other chemists (the Briton Alfred L. Parson, the American W. C. Arsem and the German H. Kauffmann) and physicists (the German Johannes Stark and the Englishman J. J. Thomson) [5]. In his article, discussing the electronic theory of valency and the octet rule in the light of recent discoveries and theories on electrons in atoms, Lewis suggested that two equal atoms can form a bond by sharing electrons to form electron pairs between atoms. The bond he proposed came to be called covalent bond. According to Lewis, for atoms there are two ways to complete the octet: One is the formation of a covalent bond, leading to the octet by sharing electrons of equal atoms or different atoms with the same electronegativity; the second is the formation of an ionic bond, leading to the octet by transfer of electrons from one atom to the other and applied to atoms of different electronegativity. The idea of shared electrons provided an effective qualitative picture of the covalent bond, but real comprehension of the bond’ nature only came with the application of quantum mechanics to solving the problem of molecules formation. First to describe quantum mechanically the covalent bond formation was Danish physicist Øyvind Burrau (1896–1979) in 1927. Burrau applied wave mechanics to calculating the ionized hydrogen molecule, H2 + , the simplest molecule, composed of one electron and two hydrogen nuclei (two protons). The solution of the corresponding Schroedinger’s equation is mathematically complex, but the scheme of reasoning leading to the solution is relatively easy to follow. The basic idea is to calculate the electron’s energy as a function of the distance of the two nuclei. It is found that the wave function of the ground state is such as to entail a progressive increase of the probability of finding the electron between the two nuclei. This increase of negative charge between the two nuclei exerts a growing attraction of the nuclei toward the molecule’s centre, exerting a stabilizing effect. The molecule’s stability is proved by the existence of a minimum of energy at the distance of equilibrium of the two nuclei (For more details see Ionized hydrogen molecule in Appendix D). Demonstration that this molecule made up of three interacting charged particles constitutes, in line with experimental evidence, a stable system, was one of quantum mechanics’ great successes. It showed that the quantum approach to chemical bonds is fundamentally and quantitatively correct. Still in 1927 German physicists Walter Heitler (1904–1981) and Fritz London (1900–1954) carried out the first quantum mechanical calculation for the hydrogen molecule (H2 ). This molecule has two electrons bound to two nuclei and presents, as far as quantum mechanical calculation is concerned, the problems peculiar to systems of identical particles. Heitler and London introduced a general method to attain

References

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approximate solutions of the Schroedinger’s wave equation applicable to molecular problems. This showed how, starting from the two wave functions of the hydrogen atom, each centred on one of the two nuclei, one can construct the molecule’s wave function, tracing the covalent bond formation. Heitler and London’s theory forms the basis of what came to be called the valence bond method. In the same period, thanks to Friedrich Hund, Robert Mulliken, John C. Slater and John Lennard–Jones, a second basic theory was developed to supply a quantum description for chemical bonds. It was called Hund-Mulliken or molecular orbital method.

References 1. Eisberg R, Resnick R (1985) Quantum physics, 2nd edn. John Wiley & Sons, New York 2. Heisenberg W, Jordan P (1926) Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte. Zeitschrift für Physik 37: 263-277 3. Russell H N, Saunders F A (1925) New regularities in the spectra of the alkaline earths. Astrophysical Journal 61: 38-69 4. Born M (1962) Atomic Physics, 7th edn. Dover Publications Inc, New Yok 5. Jensen W B (1984) Abegg, Lewis, Langmuir, and the Octet Rule. Journal of Chemical Education 61: 191-200

Chapter 13

Solid Matter

Among relevant problems unresolved before the advent of quantum mechanics was the nature of electrical properties of solid matter. As we have seen, the problem was an ancient one; by the end of the nineteenth century it had also taken on significant practical implications, given the more widespread use of electricity and the development of new communication technologies. After the discovery of electrons, the problem of understanding the electrical properties was associated with determining the atomic and electronic structures of solid substances. Among the great advances in the investigation of matter made in the early twentieth century was the direct determination of atoms’ ordering in crystalline solids thanks to the advent of X rays. For the first time in history, the spatial arrangement of atoms could be experimentally ascertained. The crystalline lattices of the more common substances were established. This knowledge was of great help when the new quantum mechanics began to be applied to understanding the electronic structure and physical properties of solid matter.

13.1 Atomic Ordering Many of the solid substances aggregate so as to give rise to regular structures, the crystals. Among these, precious stones have enjoyed great popularity since ancient times. Though the properties of crystals have always been a subject of great interest, the first scientific approaches for studying their structure only began in the seventeenth century, in conjunction with the general development of sciences. It is worth recalling that John Kepler (1571–1630), one of astronomy’s fathers, was among the first to formulate a scientific hypothesis for crystal formation. In his Strena seu de Nive Sexangula (A New Year’s Gift of Hexagonal Snow), of 1611, he hypothesised that snowflake crystals’ hexagonal symmetry was due to a regular packing of microscopic spherical particles of water [1].

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The true pioneer of experimental research into crystals’ symmetry, however, was Danish scientist Nicolas Steno (1638–1686). In the 1669, Steno formulated the first law of crystallography, known as Steno’s law or law of constant angles, which affirms that angles between the corresponding faces of a crystal are the same for all samples of the same mineral [1]. Steno’s work was the basis for all successive research into crystal structures. With atomism’s revival in the seventeenth century, various attempts were made to correlate crystal structures to corpuscular theories of matter. For example, English scientist Robert Hooke (1636–1703) in his Micrographia of 1665 affirmed that it is highly probable [2] that all these regular figures that are so conspicuously various and curious,....arise only from three or four several positions of globular particles,.... I could also instance in the figure of Sea-salt, and Sal-gem, that it is composed of a texture of globules, placed in a cubical form....

In the eighteenth century, investigating crystal structures became a highly active research field, also due to widespread interest in the study of rocks and geology in general. Important advances were made. Naturalist Carl Linnaeus (1707–1778) was able to establish a correlation between the various types of salts and different crystalline forms. French chemist Guillaume François Rouelle (1703–1770) held that microscopic observations of the crystallization process of marine salt (common salt) suggest the existence of component particles of cubic form. More generally, he believed that the observed correlation between chemical composition and crystalline form implies that, for every salt, the constituent particles are polyhedrons of definite and constant geometric form. Regarding crystallization’s mechanisms, French chemist Pierre-Joseph Macquer (1718–1784), in his popular Dictionnaire de chymie (1766), introduced the concept of integrant polyhedral molecules and stated these two principles: That, although we do not know the figure of the primitive integrant compound molecules of any body, we cannot doubt but that the primitive integrant molecules of every different body have a constantly uniform and peculiar figure. If...they have time and liberty to unite with each other by the sides most disposed to this union, they will form masses of a figure constantly uniform and similar.

We owe to French scientist Jean-Baptiste Romé de L’Isle (1736–1790) the first attempt to render Linnaeus, Rouelle and Macquer’s ideas more organic and to develop geometrical theories of crystal structures. In his treatises Essai de crystallographie of 1772, and Crystallographie, of 1783, he held that the integrant molecules of bodies each have, according to their own nature, a constant and determined shape and that crystals are classifiable on the basis of their exterior form. Using new data obtained with a goniometer able to measure precisely the dihedral angles of crystal faces, Romé de L’Isle stated his law of constant interfacial angles, asserting that such angles are constant and characteristic of crystals of the same chemical substance. Thanks to Romé de L’Isle crystallography came to have at its disposal a quantitative parameter, the inclination between the characteristic planes ( faces), to be used to classify crystal structures.

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The first general theory concerning crystal structure was formulated by Abbot René Just Haüy (1743–1822), a famous French mineralogist considered Crystallography’s father. In his Essai d’une théorie sur la structure des crystaux (1784) he showed that, for every molecular compound, all crystal faces could be described by superimposing blocks of same shape and size, made up of specific units, constituent (or integrant) molecules, equal to each other in form and composition. The theory stemmed from the insight that the existence of cleavage planes in crystals demonstrates an internal geometric order based on the integrant molecules layout, also explaining the law of constancy of interfacial angles. Haüy’s studies led to the modern conception of crystals as tri-dimensional lattices (crystal lattices), where an elementary cell is repeated indefinitely along three principal directions. In the nineteenth century the theoretical study of crystal structures underwent a notable development, becoming a mathematical branch dedicated principally to investigating properties of symmetry. Thanks to the work of a number of scientists a complete catalogue of possible structures and symmetries was drawn up. An important role was played by French physicist and crystallographer Auguste Bravais (1811–1863). Starting from Haüy’s theory describing crystals as an ordered aggregate of microscopic units (integrant molecules), Bravais introduced the concept of crystal lattice as the set of points in space generated by replicating three non-coplanar primitive translations. He showed that only 14 distinct crystal lattices exist, today called Bravais lattices, grouped into seven crystal systems: cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic and triclinic. In the 1890s Evgraf Fedorov, Arthur Schönflies and William Barlow demonstrated independently that only 230 distinct crystal structures can exist. However, until the advent of X rays there was no way to investigate experimentally crystal structures at a microscopic scale. Any structural hypotheses, even if highly plausible and theoretically well-founded, remained suppositions lacking experimental support. Macroscopic data available were too scant to substantiate the models drawn up. This situation changed radically when German physicist Max von Laue (1879–1960) envisaged using crystals as the diffraction gratings for X-rays, assuming that these had a wavelength comparable to atoms spacing in crystals. Von Laue’s original aim was to verify the wave nature of X-rays, still controversial seven years on from their discovery by Roentgen. In 1912 von Laue, Walter Friedrich and Paul Knipping set up an experiment where an X-ray beam passed through a copper sulphate crystal. They observed on the photographic plate a large number of well-defined spots ordered in a pattern of intersecting circles around the point produced by the central beam. To explain the results, Von Laue worked out a theoretical law linking the diffraction pattern to the shape and dimension of the crystal’s unitary cell. The experiment not only demonstrated X-rays’ wave nature, but had far-reaching consequences, paving the way for the direct determination of the structure of matter in a solid state. For his contributions von Laue received the 1914 Nobel Prize. After von Laue’s pioneering research, the field developed rapidly. Particularly important was the work of English physicists William Henry Bragg (1862–1942) and William Lawrence Bragg (1890–1971), father-and-son. Shortly after von Laue’s

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announcement, they were able to determine the structure of the most common crystals. In addition, Lawrence Bragg envisaged a simple and direct procedure for interpreting diffraction patterns, today known as Bragg law, whereby it is assumed that the observed diffraction peaks are due to the reflection of X-rays off the crystal lattice planes. For their contributions to crystallography the Braggs, father and son, obtained the 1915 Nobel Prize for physics.

13.2 Electrons in Solid Matter At the end of the 1920s a relevant problem to be addressed by scientists with the help of the new-born quantum mechanics was that of electrons in solid matter and the understanding of the electrical properties. This was a wide-ranging subject at the heart of which was the origin of ability to conduct electricity by some substances (metals), while others (insulators) did not conduct and still others (semiconductors) had intermediate properties. To appreciate the special nature of the problem, bear in mind that one needed explaining a difference in electrical conductivity of more than twenty orders of magnitude!1

13.2.1 The Free Electron Gas As was easily predictable, Thomson’s 1897 discovery of electrons had vast and immediate impact on research regarding electricity in matter. In 1900, German physicist Paul Drude (1863–1906) formulated the first organic theory of conduction in metals, in which electrons were identified as the carriers of electric current. In his theory Drude assumed that electrons inside a metal form a gas of free particles; that is, they behave like the particles of a gas in a container; in other words, they are perfectly free to move, have negligible interactions and obey the laws of kinetic theory. There is a major difference with gases, however: the electrons are charged particles and the system must be composed of at least two types of particles of opposite charge to maintain electrical neutrality. Drude’s model assumed that positive charges were

1

A solid-state material’s ability to conduct electric current, conductivity, is specified using Ohm’s law, which defines the resistance R of a bar of material as the ratio between the applied voltage V and the flowing electric current I. R depends on the length L and the section S of the bar according to the relationship R = ρL/S, where ρ is a characteristic quantity of the material called resistivity and is measured in Ohm meter (Ω m). The conductivity σ of a material is the inverse of its resistivity (σ = 1/ρ) and is measured in (Ω m)−1 . Typical values for σ of metals vary in the range 107 –108 (i.e.:10.000.000–100.000.000). Typical values for σ of insulators vary in range 10−14 –10−10 (i.e.: 1/100.000.000.000.000–1/10.000.000.000).

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carried by particles much heavier than electrons.2 (For more details see Classic free electrons gas in Appendix D). Equating metal electrons to a gas of free particles was a bold assumption, since it was difficult to understand how electrically charged particles, strongly interacting with each other due to Coulomb forces, could behave as interaction free. Despite these a priori reservations, Drude’s theory proved notably successful in explaining both electrical and thermal conductivity. In addition, it was able to reproduce with high precision the Wiedemann–Franz law, a then mysterious property of metals conduction, whereby the ratio between electrical and thermal conductivities is a constant independent of the metal considered. In the ensuing years, Drude’s theory was perfected by Hendrik Antoon Lorentz (1853–1928), who introduced the hypotheses that electrons velocities follow the Maxwell–Boltzmann distribution and that the positive particles are fixed inside the metal. Thus, it became possible to explain other properties of metals, such as the thermoelectric phenomena and the absorption and emission of thermal radiation [3]. These successes strongly supported, initially, the assumption of the existence of a free electron gas in metals. Soon, however, a series of problems emerged, creating a climate of uncertainty. The theory was found to be unable to correctly predict the behaviour of the two conductivities (electrical and thermal) as a function of temperature variations. Furthermore, a serious conceptual contradiction was discovered: the existence of a free electron gas was at odds with the behaviour of another basic thermal property of metals, the specific heat. According to kinetic theory, the presence of a free electron gas should also appear in the specific heat measurements, giving a contribution comparable to that coming from the motion of the ions. However, there was no trace of such a contribution in measurements hitherto available. In conclusion, the electrons in metals and the validity of Drude’s theory were among the numerous problems awaiting solution in the early years of the twentieth century and later a test bed for the quantum mechanics in the late 1920s. A first solution to open problems concerning conduction in metals came in 1927, thanks to Pauli and Sommerfeld, who took advantage of recent developments in statistical quantum mechanics due to Fermi and Dirac. In 1926 Fermi and Dirac had tackled independently a conceptually relevant problem linked to Pauli exclusion principle: since every quantum state could be occupied by one particle only, it followed that the statistical law governing the electrons distribution between the possible energy states (Maxwell–Boltzmann distribution) had to be changed so as to satisfy this new condition. Fermi and Dirac had conceived this new distribution law, henceforth named Fermi–Dirac statistics. In the 1927 Pauli showed that the metal para-magnetism could be explained if one assumed the existence in metals of a free electron gas obeying the new Fermi–Dirac statistics. The success of Pauli’s para-magnetism theory added plausibility to the hypothesis that the most significant shortcoming of the Drude-Lorentz theory, lay not

2

Remember that when Drude devised his theory there was no clear evidence regarding the structure of atoms and the existence of nuclei.

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in the hypothesis of a free electron gas, but in the use of Maxwell–Boltzmann distribution to estimate electron energies. Based on this view, in 1927–28, Sommerfeld drew up a quantum theory of the free electron gas, whereby electronic states obey Schroedinger wave equation and electrons are distributed among states according to Fermi–Dirac’s statistics (For more details see Sommerfeld’s free electrons gas in Appendix D). Sommerfeld showed that the electrical and thermal properties of the free electron gas evaluated with the new theory were in line with experimental data. In addition, he solved the enigma of specific heat, demonstrating that, at room temperature, electrons’ contribution to specific heat is so small as to be impossible to measure.

13.2.2 Bloch Waves and Band Theory The success of Sommerfeld’s theory in explaining the main aspects of metal conduction demonstrated that the assumption of the existence of free electrons inside metals must contain elements of truth. However, contextually, it raised a series of new problems of conceptual import: how is it that in metals there are electrons that seem affected by neither the attraction of nuclei, nor the repulsion of the other electrons; how many of these free electrons are there in respect to the total of electrons present; last and above all, why do some substances, the metals, have free electrons and others, the insulators, do not? The first of these questions, the possibility for electrons to move almost freely inside a solid substance, was solved as early as 1928, thanks to Felix Bloch (1905– 1983), a brilliant research student of Heisenberg. He sought to understand why an electron can move from one crystal cell to another without being captured by the charged ions present. His starting-point was the mechanism of covalent bond formation in homo-nuclear molecules. As we have seen in the formation of a hydrogen molecule, when two atoms bind to form a molecule, the outermost electrons are pooled and each electron becomes an electron of the molecule, no longer of single atoms. According to Bloch, the formation of a crystal, where many atoms join to form an ordered lattice, can be seen as the formation of a giant molecule, where each electron is shared by the whole solid. This is equivalent to saying that the electron has the same probability of being in each of the elementary cells of the crystal, acquiring, to all intents and purposes, some freedom to move. To quantify his idea, Bloch assumed that, in the crystal, an electron close to an ion feels the same potential it would feel if the atom were isolated. This assumption implies that the wave functions of electrons in the crystal can be constructed as a linear combination of atomic orbitals, that is to say, as a generalization of the molecular orbitals, that we have seen in the case of the hydrogen molecule. Starting from this assumption Bloch was able to demonstrate that there are as many of these new electronic states as there are elementary cells in the crystal and that each of these states has a slightly different energy (Fig. 13.1). The possible energy values are distributed around the value of energy of the original atomic state. When N becomes

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Fig. 13.1 Schematic of electronic states and energy levels, when equal atoms bind to form molecules and solids. Electronic states (represented by circular orbits) combine to form new states extending to all the atoms. Their number equals the number N of constituent atoms. The original energy level split into as many levels as there are atoms, each with a slightly different energy (shown by the scheme on the right)

very large (ideally tending to infinity) the spacing between energy levels tends toward zero and one speaks of band of energies permitted for the electrons. Bloch also proved the important theorem (now called Bloch theorem), according to which wave functions of electrons in a periodic potential (now known as Bloch waves) are plane waves (i.e. the wave functions appropriate for free particles) modulated by a periodic function with the period of the lattice. In his theory Bloch set up the crystal wave functions as a linear combination of the ground state atomic orbital and obtained a band of allowed energies centred around the energy of this orbital. However, the same procedure can be applied to each atomic energy level, obtaining an energy band for each of these levels. It follows that the energies of the electrons in crystalline substances consist of a sequence of bands of allowed energies separated by intervals (gaps) of forbidden energies (energy gaps or band gaps) (Fig. 13.2). Later on, Bloch calculated several physical properties of metals, demonstrating his approach’s effectiveness and the correctness of his theorem. Among these is the calculation of the conduction properties and the explicit demonstration that, in a perfect crystal, the electric resistance is not due to the presence of static ions, as Fig. 13.2 Schematic showing how atomic energy levels broaden to form the sequence of allowed bands and forbidden gaps in crystalline substances

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supposed by Drude, but to their displacement from the equilibrium position due to vibrations and that it increases with the amplitude of the displacement. Published in a 1929 article Bloch’s theoretical results represented a crucial break-through in comprehending electrons’ behaviour in solid substances, paving the way for the band theory underlying modern quantum theory of solids. Band theory was perfected in the next three years as a result of efforts to explain important aspects still unclear of electrical properties. The relevance of energy bands and band gaps concepts began to be apparent when Heisenberg suggested to Rudolf Peierls (1907–1995), then a research student, that he apply Bloch’s theory to comprehending the anomalies of Hall effect. The effect bearing this name was discovered in 1879 by American physicist Edwin H. Hall (1855–1938). It consists in the appearance of a transverse voltage (Hall voltage) when an electric current flows in a conductor subject to a perpendicular magnetic field. It was initially found that the general features of Hall effect were satisfactorily explained if one extended Drude’s theory, by taking into account the Lorentz force due to magnetic field.3 However, more detailed studies had later shown how, in many materials, Hall effect had anomalous characteristics, in contrast with what the theory predicted. In numerous cases the Hall voltage not only had an anomalous value; it also had an anomalous sign, i.e. it behaved as if current carriers had positive instead of negative charge. This contradicted the concept, then well established, that in solids the only mobile particles were electrons. Not even Sommerfeld’s quantum theory of free electron gas could explain such anomalies. When tackling the problems of electric conduction and of Hall’s effect in light of Bloch’s recent results concerning energy bands, Peierls realised that, from the viewpoint of particle dynamics (i.e. particle response to external forces), the electrons described by Bloch waves differed substantially from the free electrons of Sommerfeld’s theory. Though the states were quantized, the dynamics of Sommerfeld’s electrons remained a classical one, while the dynamics of Bloch’s electrons had completely new features. In presence of an external force, such as an electric or magnetic field, Bloch’s electrons behave as if they have a mass whose value depends on dispersion E(k) (k is the electron wave vector) of the energy band which they belong to. This new mass m*(k), function of wave vector k, was named effective mass. The concept of effective mass proved essential for understanding conduction in solids (For more details see Bloch’s electrons in Appendix D). The most notable difference in behaviour is obtained for an electron positioned at the top of a band where the energy has a maximum value. Acted upon by an external force (for example, an electric field), this electron will decrease its energy (in the normal dynamics its energy will increase) and will behave as if it acquires negative acceleration and velocity. This is equivalent to affirming that the electron responds to the force as if it had a negative effective mass and an electric charge of an opposite 3

In physics, the Lorentz force, named after the physicist Hendrik Antoon Lorentz, is the force acting on a charged particle moving in a magnetic field. The force is proportional to both the charge and the velocity of the particle and is orthogonal to both the direction of the velocity and the direction of the magnetic field.

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Fig. 13.3 Schematic of the movement of electrons in a unidimensional space wherein an electron is removed at the extreme left top site. The presence of an empty site allows electrons to move from right to left, upon application of an electric field. The final result is that the empty site (hole) shifts from extreme left to extreme right and is equivalent to the flow of current of a positive charge travelling from left to right

sign. Peierls realized that these characteristics of Bloch electrons could explain the anomalies of the Hall effect, in particular the origin of the positive charge of the current carriers. To this end he assumed the presence of a fictitious particle endowed with an electric charge and an effective mass both with a sign opposite to that of the electron. This fictitious particle was named hole, when Heisenberg, in a 1931 article, demonstrated that an empty state (i.e. an electronic state missing its electron) near the top of a band fully occupied by electrons behaves as if it were a particle with charge and mass opposite to that of the missing electron. Heisenberg thus concluded [4]: the holes behave exactly like electrons with positive charge under the influence of a disturbing external field ... the electrical conduction in metals having a small number of holes can in every connection be written as the conduction in metals having a small number of positive conduction electrons.

This apparent paradox, where the flow of negative charges, the electrons, produces a current analogous to a flow of positive charges in the opposite direction, can be grasped intuitively by referring to Fig. 13.3.

13.2.3 Metals and Insulators In 1931 the development of band theory was advanced enough to answer the fundamental question of what renders metals and insulators different and gives rise to the enormous difference of conductivity among various solids. It was English scientist Alan Herries Wilson (1906–1995) who highlighted how the energy band concept finally made it possible to solve the mystery. The key reason is linked to Pauli exclusion principle, which establishes that two electrons cannot occupy the same electronic state, and of Fermi–Dirac distribution. As a consequence, electrons in a

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band must occupy different states of increasing energy, starting from the lowest. This distribution gives rise to two qualitatively different results: either the band turns out to be full, meaning the number of electrons is such as to fill all electronic states available in the band itself, or the band is only partially occupied. These two situations differ substantially regarding the possibility of conducting electric current. Indeed, to conduct current, the electrons must be accelerated by the applied electric field and change their energy. This is equivalent to changing state inside the energy band. However, this is possible only if there exist empty states which can be occupied by accelerated electrons, that is, only if the band is not completely filled. If the band is full, this is impossible: the electrons of a full energy band do not conduct electric current! Only when a band is partly empty can its electrons be accelerated and conduct electricity. This is the crucial characteristic of Bloch electrons demonstrated by Wilson. Based on it he was able to affirm that the difference between metals and insulators depends on the state of occupation of their energy bands: in insulators the bands containing electrons are all full while metals have one or more bands partially occupied.

References 1. Wikipedia (2019) X-ray crystallography. https://en.wikipedia.org/wiki/X-ray_crystallography. Accessed 24 October 2019 2. Mauskopf S. (2012) Seeds to Symmetry to Structure. http://xray-exhibit.scs.illinois.edu/Lecture/ HistoryCrystalStructureTheoryshortened.pdf. Accessed 24 October 2019 3. Hoddeson L H, Baym G (1980) The Development of the Quantum Mechanical Electron Theory of Metals:1900-28. Proceedings of the Royal Society A371: 8-23 4. Hoddeson L H, Baym G, Eckert M (1987) The Development of the Quantum Mechanical Electron Theory of Metals:1928-33. Reviews of Modern Physics 59: 287-327

Chapter 14

Semiconductors

By the early 1930s the foundations of the modern concept of matter had been laid. In the process, a formidable tool for the study of nature had been developed: quantum mechanics. The potential of the entire construction is demonstrated by the extraordinary progress that has since been made by physics and chemistry and by the amazing achievements in science and technology. Tracking these developments is beyond this book scope. However, it may be of interest to end this story by briefly illustrating the evolution of a class of materials crucial to modern society: semiconductors. As is well-known, semiconductors form the basis for transistors, microelectronics, optoelectronics and nearly all current information and communications technologies (ICT). Their development is clear proof of quantum mechanics’ power to interpret nature and provide the instruments to govern it. The word semiconductor was first used 1910 by J. Weiss, a doctoral student at Freiburg university, while he was studying electrical conduction of a series of oxides and sulphides, such as iron oxide (Fe2 O3 ) and lead sulphide (PbS) [1]. The name implies his observation that their conductivity is intermediate between that of good conductors, metals, and that of insulators. At that time semiconductors had long been known to form a class of materials distinct from metals not just for the intermediate value of conductivity, but also for a series of peculiar physical proprieties. One of these was conductivity’s behaviour as a function of temperature variations. In 1831, in a study published in Philosophical Transactions of the Royal Society, Michael Faraday, then studying a vast number of chemical compounds, reported that, as temperature varied, the conductivity of silver sulphide (Ag2 S) varied in a way opposite to what Humphry Davy had observed in many metals some years earlier. While in metals resistivity increased as temperature increased (or, equivalently, conductivity decreased as temperature increased), in silver sulphide (and in semiconductors in general) resistivity decreased and conductivity increased. Such behaviour, inexplicable in the light of classical conduction models, remained a mystery until the advent of quantum mechanics. Meanwhile, this circumstance did not prevent experimental study of semiconductors’ physical and chemical proprieties in the various laboratories interested in their use. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4_14

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Another peculiar property of semiconductors, something to prove fundamental for the future development of electronics, was discovered in 1874 by German scientist Karl Ferdinand Braun (1850–1918). While studying electric conduction in two semiconductors, galena (ancient name for lead sulphide (PbS)) and copper sulphide (CuS), he noted that the current’s intensity varied according to the voltage polarisation, positive or negative, applied to metallic contacts. He also observed that the current flowing for a given polarization was systematically greater than that corresponding to the other polarization. Furthermore, the effect was more evident when one of the two metallic contacts was much smaller than the other, in particular if one of the two had a pointed shape. Here was the first observation of signal rectification, the capability of metal/semiconductor junctions to rectify electric current. Nearly twenty years later, in the early stages of radio development, Braun interested himself in the problem of finding an efficient radio wave receiver. To this end, he resumed research into rectification. As a result of these new studies, he invented the semiconductor rectifier diode, historically also called cat’s whisker diode, for the filiform shape of one of the metallic electrodes.1 In a cat’s whisker diode, when one applies the voltage generated by an electromagnetic wave (oscillating between positive and negative values), the current flows only in correspondence with the positive half-wave (rectification effect). The result is an electric current whose average value is different from zero, and which can be detected by direct current meters. Cat’s whisker receivers, characterized by their low cost and simple construction, became the detectors of radio signals mostly used in the first decades of the twentieth century. They represented semiconductors first contribution to communication technologies, enhancing these materials’ popularity within the scientific community. Consequently, when Alan Herries Wilson tackled the problem of the difference between metals and insulators, he was fully aware of the existence of semiconductors as a different class of materials, whose electrical proprieties were peculiar and needed clarification. Wilson found that electrons’ distribution in semiconductors is analogous to that of insulators, i.e. consisting of completely filled energy bands, separated by a gap of forbidden energies from completely empty energies bands. Thus, semiconductors should be defined as insulators. However, it must be borne in mind that the condition of completely filled or completely empty bands can be rigorously achieved only at the temperature of absolute zero (T = 0). At temperatures different from zero, Fermi– Dirac distribution law predicts a non-zero probability that some of the electrons of the highest-energy filled band are transferred into the lowest empty band. This probability, smaller and smaller the larger the energy gap, grows exponentially with increasing temperature. It follows that, in principle, if the temperature is sufficiently high each insulator acquires a certain capability to conduct electric current, in that the bands are no longer completely filled nor completely empty. In practice, since this probability decreases exponentially with the magnitude of the energy gap, if the gap is sufficiently large, the conductivity remains negligible, even at high temperatures. A large gap pertains to solids we commonly define as insulators. However, if 1

For his contributions to the development of radio and electronics, Braun was awarded the Nobel Prize in Physics jointly with Marconi in 1909.

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165

the energy gap is small, the probability of electrons’ transfer from the filled band to empty one can be sufficiently great to give rise a conductivity typical of semiconductors. Moreover, the probability increases exponentially with increasing temperature, resulting in an exponential increase in the number of electrons and holes able to conduct. According to Wilson this explains the mysterious characteristics of semiconductors’ conductivity. In conclusion, as well as clarifying the difference between metals and insulators, Wilson was able to highlight the main feature that distinguishes semiconductors: a semiconductor is an insulator having a small band gap. To further clarify some of the characteristics which make semiconductors so relevant in modern technology we briefly outline the history of the most important of these: silicon.

14.1 Silicon Silicon (Si) is a tetravalent semiconductor (i.e. each atom forms four bonds with nearest neighbours), which crystallizes in diamond cubic lattice (Fig. 14.1). Silicon is not found pure in nature, despite being, after oxygen, the most abundant element in the earth’s crust. First to mention its existence was Lavoisier. After studying numerous oxides, in 1789 he posited that quartz and silica (respectively, crystalline and amorphous silicon dioxide SiO2 ) were compounds of oxygen with an important element yet unidentified. The first to obtain pure silicon was French scientist Joseph Louis Gay-Lussac in 1811, while the first silicon crystals were obtained by German chemist Friedrich Woehler in 1856–58. For a long time silicon’s electrical properties were the object of controversy: Swedish chemist Jons Jacob Berzelius held that it must be a metal, while another eminent scientist of the time, the British chemist Humphry Davy, believed it to be an insulator. Fig. 14.1 Ball-and-stick model of silicon crystal structure. Each atom is at the centre of a tetrahedron, whose vertices are constituted by the four nearest neighbours. (Wikipedia Public Domain)

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Interest in large-scale silicon production began in the second half of the nineteenth century in the context of metallurgy. In fact, it was discovered that adding small quantities of silicon endowed iron alloys with improved properties of hardness, ductility and resistance to corrosion. Thanks to its use in metallurgy, by the end of the century silicon had become commercially available. This silicon, named metallurgical-grade silicon, was a reasonably pure material, with a content of impurities (that is, atoms different from silicon) of the order of some per cent. Silicon entered electronics’ history in 1903, when American engineer Greenleaf Whittier Pickard (1877–1956) built the first silicon cat’s whisker diode. During his research on these detectors, conducted at AT&T, Pickard had tried thousands metal/semiconductor combinations, noting how the best results came using the metallurgical-grade silicon then available [2]. Despite these uses, until World War II interest in silicon and semiconductors remained marginal and limited to a restricted community, mostly of engineers, who used them as signal’s detectors or current’s rectifiers. The principal reason for this limited interest, in particular on the academic community’s part, was due to the poor reproducibility of the measured properties, which very often were contradictory and at odds with each other. Semiconductors were considered systems scarcely definable from the physics viewpoint and so unworthy of attention by eminent scientists. Later it would be understood how this irreproducibility resulted from the fact that, in contrast to what occurs in metals, semiconductor properties critically depend on crystalline quality and the impurity content, these altering the semiconductor behaviour even at extremely low percentages. Hence the need for very pure samples and high crystalline quality. Awareness of how critical this was would not emerge clearly until the radar research carried out in World War II. We owe to the project that developed radar, among the most important American research efforts during World War II and on which vast resources were allocated, the renewed interest in semiconductors generally, but above all in silicon (Si) and germanium (Ge). In the 1930s the USA, Japan and the major European powers, had begun researching the possibility of using electromagnetic waves to detect metallic bodies. Evidence that large metallic objects interfered with radio transmissions had already emerged during experiments in the first decades of the twentieth century. In particular it was noted that radiated electromagnetic waves were partially reflected and that the effect became more pronounced at shorter wavelengths. It was soon clear that the effect could be used to reveal the presence of ships and planes in conditions of low visibility. This possibility had stimulated widespread interest among the military. At the outbreak of World War II, few belligerent nations already possessed some form of radar (this denomination, an acronym of Radio Detection and Ranging, was adopted in the 1940s). It is well-known how deployment of a chain of radar stations for intercepting enemy aircraft allowed England to win the Battle of Britain against the Luftwaffe, in the summer and autumn of 1940. In the next years of the war all the belligerent nations intensified research to improve radar systems. In the USA, in particular, huge investments were made; new projects were launched involving hundreds of scientists and engineers in about twenty universities and more than forty

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167

companies [3]. This enterprise’s first aim was to produce advanced radar systems, with good resolution and minimum load, which could also be installed on planes. It was immediately clear that to reach this objective it was necessary to use electromagnetic waves with wavelengths shorter than one metre. Such waves, later named microwaves,2 had very high frequencies compared to the performance of electronic circuits of the time and posed serious problems for both generation and detection. Regarding receivers, precise analysis of available devices and technologies led to the conclusion that the only devices able to detect a microwave radar pulse were cat’s whisker diodes, used, as we have seen, in the first radio receivers. After experimenting with numerous combinations of semiconductors and metallic contacts, it turned out that silicon, combined with tungsten tip electrodes, was the most satisfactory solution [2]. However, it was evident from the very beginning that for reliability and reproducibility the available cat’s whisker diodes were utterly inadequate for the tasks set by a theatre of war, even if using the best material available, metallurgical-grade silicon. As many contemporaries bore witness, the experience of using a silicon cat’s whisker diode to reveal a microwaves’ signals could equal frustration levels suffered by radio operators in the 1920s, before the use of thermionic valves. The signal very often proved detectable only in some areas, (in jargon hot spots) of the semiconductor chunk, while it was weak or absent in many others. Using the detector required, therefore, a preliminary search for identifying a hot spot. Moreover, as time passed, the signal tended to disappear, necessitating a search for another hot spot. Finally, something even more mysterious happened: Changing hot spot often changed the direction of the current’s flow. Thus, it was obvious that if one aimed at producing reliable, ready-to-use detectors, as the war demanded, the causes for such behaviours needed to be clarified and brought under control. To this end the US radar project devoted substantial resources, material and human, into investigating silicon’s physical and chemical properties, enabling not only the development of efficient radars, but also the future invention of the transistor and the development of modern electronics. It was during the characterisation activities of silicon that the problem’s origin was surmised: cat’s-whiskers diodes’ irregular behaviour and the irreproducibility of semiconductors physical properties resulted from unchecked amounts of impurities; the structure quality and the purity level necessary to observe reproducible physical properties in semiconductors was much higher than that required in metals. This hypothesis’s exactness was definitively confirmed when three Bell Laboratories researchers, Russell Ohl, Jack Scaff and Henry Theuerer, had the idea of trying to further purify commercial silicon, developing an apparatus suitable for this purpose.3 The behaviour of the new materials was effectively more reproducible. Surprisingly, 2

The term microwaves denotes electromagnetic waves with a wavelength spanning between 1 cm and 1 m, and frequencies ranging from 300 MHz to 30 GHz (remember that the relation λ · ν = c holds, where λ is the wavelength, ν is the frequency and c is the speed of light). 3 It was an apparatus for fractional crystallization. Fractional crystallization is a method of refining substances based on differences in their solubility. Applied to silicon, it consists in liquefying silicon, by heating it above its melting temperature (equal to 1414 °C), and slowly cooling it, so that the

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however, they behaved as if two types of silicon existed: one type in which current flowed from the metallic tip to the silicon and another in which the direction of flow was opposite. At times this different behaviour also occurred in the same purified ingot: in samples cut from the ingot’s upper part the current flowed when a positive voltage was applied to the silicon; whereas in samples cut from the lower part the current flowed when the silicon voltage was negative. Ohl and Scaff understood that here was an important discovery and they baptized the two types of silicon, p silicon (p from positive) and n silicon (n from negative), labels universally adopted ever since. This was the first systematic evidence for another characteristic of semiconductors, which would prove essential for the realization of electronic devices: semiconductor doping, i.e. the intentional introduction of impurities into a semiconductor for the purpose of varying its electrical and optical properties in a controlled way. More precise studies into the chemical nature of impurities of the two types of silicon showed that in p-type samples there was a predominance of boron or aluminium, both elements belonging to the periodic table’s group III. Meanwhile in n-type samples the main contaminant was phosphorus, an element belonging to group V. The decisive step for controlling impurities was finally made in 1941, by the research team coordinated by physicist Frederick Seitz and composed of researchers from Pennsylvania University and the Du Pont chemical company [4]. This team perfected a chemical purification process capable of refining silicon to a then unprecedented purity, initially equal to one foreign atom every 10.000 Si atoms and, successively, improving this to one impurity every 100.000 Si atoms. Using these ultra-pure materials it was possible to introduce accurately controlled quantities of group III or V elements to obtain uniform and reproducible Si samples with the desired electrical characteristics, necessary for mass production of microwave detectors. This was one of the first examples of industrialization of the semiconductor doping process. In closing this section, it should also be remembered that the radar project had another legacy which would prove extremely important for the development of microelectronics: having focused the researchers’ attention also on germanium (Ge), a semiconductor until then little investigated and used. Ge is the group IV element located immediately below Si in the periodic table of elements. It had not yet been discovered in 1869, when Dmitri Mendeleev had drawn up the periodic table and predicted its existence calling it eka-silicon. Ge was actually isolated a few years later, in 1885, by the German chemist Clemens Alexander Winkler, who named it after his native land. Germanium has the same crystal structure as silicon and very similar electronic properties, but its melting point is considerably lower (938 C versus 1414 C), making it easier to purify. Attracted by this appealing property, in 1942, the researchers involved in the radar project began to fabricate and experiment with Ge’s cat’s whisker diodes. It was found that Ge diodes, considerably purer than those of Si, were indeed

heavier impurities can settle in the lowest part of the molten Si and the lighter impurities remain floating in the upper part.

14.2 Doping and Microelectronic Devices

169

very promising for use as detectors, especially since they were able to withstand voltages much larger than those of Si without damage.

14.2 Doping and Microelectronic Devices Doping of semiconductors can be considered one of the first examples of engineering materials to obtain a predefined behaviour. It is interesting, therefore, to understand its working. According to band theory, the electrons in silicon (and in germanium and the other semiconductors) are arranged so that they completely fill the valence band, while the next band, the conduction band, is completely empty. One may ask what happens if a phosphorus (P) atom replaces a silicon atom (or a Ge one) in the crystal lattice. Si atoms have four valence electrons, which form bonds with the four nearest neighbors. Like the other elements of group V of the periodic table, phosphorus atoms have five valence electrons. When a phosphorus atom replaces a Si one, only four of its electrons form bonds with the four nearest neighboring Si atoms. The excess electron is very weakly bound and can move easily inside the crystal. At room temperature most of the excess electrons of the phosphorus atoms occupy states of the conduction band. Since electrons in the conduction band can conduct electric current, doping allows varying the semiconductor conductivity at will and in a controlled manner. In the case of p-type doping, which occurs when a boron (B) atom, or another atom of group III of the periodic table, replaces a Si atom, the situation mirrors that of the n doping. Boron, like the other group III elements, has only three valence electrons. Therefore, we are missing one electron for bond forming, which entails the lack of one valence band electron. This situation is defined as the presence of a hole in the valence band. Therefore, p-type doping makes it possible to obtain a semiconductor with positive carriers of electric current. The potentiality of the doping concept cannot be underestimated. For the first time it was possible to achieve the transmutation of a substance regarding its electrical properties. The same substance could be turned into insulator or a good conductor of electricity; current’s carriers could be selected with a positive or negative electric charge. It became evident that by combining regions of different characteristics it was possible to obtain new electronic devices to replace the cumbersome thermionic valves that made up the electronic contraptions of the time. The first momentous step in this direction was the invention of the transistor which kindled the microelectronic revolution. It is no coincidence that the invention of the transistor took place in the USA just two years after the end of World War II. Among the major nations that had fought and won the war, the USA was the only country not to suffer damage on its own territory. Free from the need to use the available resources, both human and financial, for post-war reconstruction, as happened in the rest of the industrialized world, the USA was ready to apply the scientific and technological advances and the new research organization developed during the war to the needs of peacetime

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economy. Among the points of interest there was certainly that of replacing the thermionic devices with less bulky electronic components functioning with reduced energy consumption. The advances in semiconductors’ knowledge during the war, made this goal seem not unrealistic. By Christmas 1947, the goal had been achieved with the invention of the pointcontact transistor, the first transistor to be successfully demonstrated by researchers at the Bell laboratories. It consisted in a germanium rod on which two gold electrodes, acting as point-contacts, were pressed very close together. A third metal contact was soldered to the opposite side of the rod. Its functioning principles were not completely clear but it worked, exhibiting current amplification at a reasonably high frequency. In 1956 the three researchers who had mostly contributed to the invention, John Bardeen, Walter Brattain and William Shockley, were awarded the Nobel Prize for Physics. However, the emergence of the transistor in electronics and the consequent development of microelectronics is linked to the invention of another type of transistor, the bipolar (or junction) transistor. This was conceived theoretically by Shockley in 1948, at the time when he developed his p–n junction’s theory. The p–n junction, also called junction diode, is in itself an important electronic device, which constitutes the semiconductor analogue of the thermionic diode, of historical importance for electronics. A p–n junction is formed when two differently doped regions of a semiconductor, one p-type and the other n-type, come into contact. Similarly to what happens in the case of the thermionic diode and the cat’s-whisker diode, in the p–n junction also there is rectification of the electric current. Simultaneously with the development of the p–n junction theory, Shockley had begun to ponder on the behavior of the device that would be obtained by joining two inverted p-n junctions, which would give rise to a three-layer device. Shockley’s conclusion was that an n-p-n structure (as well as the symmetrical p-n-p structure), properly biased, would exhibit the transistor effect: It meant inventing of the bipolar (or junction) transistor. Its conception resulted from an uncommon ability to schematize and understand the physics of semiconductors. The subsequent realization of the junction diode and the bipolar transistor have confirmed the doping concept’s potential and the understanding of matter at the microscopic scale. With the 1950s came the demonstration of a new concept that would prove to be revolutionary for the development of semiconductor technology and for the evolution towards the microelectronics pervasiveness around us: the integrated circuit. The innovative idea in the concept of integrated circuits was to foresee the possibility that an entire electronic circuit, including the connections between components, could be created simultaneously with the realization of the transistors on the same wafer of semiconductor material. The idea came independently to two researchers from two separate companies, Jack Kilby of Texas Instruments and Robert Noyce of Fairchild Semiconductors. Since their first implementations it was clear that the tendency to an increase in the device density and a decrease in their size was intrinsic to the very concept of

References

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integrated circuit. This miniaturization process, which is one of the aspects characterizing the evolution of microelectronics, goes by the name of large-scale integration. It has led from the first integrated circuits containing a few dozen components to the current chips containing billions of transistors, paving the way for the advent of the current information society. The ability to produce chips with a very high density of devices led to another conceptual innovation in the early 1970s: the advent of the microprocessor (also known as CPU or central processing unit), i.e. the computer on a single-chip. This new integrated circuit, which would revolutionize computers and information technologies, was invented at Intel by three engineers, Federico Faggin, Ted Hoff and Stanley Mazor. The first microprocessor realized, with its 2300 transistors in a silicon area of just 12 mm2 , had the same computing power as the first electronic computer, ENIAC, which occupied an area of 300 m2 and weighed 30 t! In the five decades since their advent, microprocessors have revolutionized our world utilizing intelligent electronics pervasive in all types of machinery, from aircraft to cars, from household appliances to all types of diagnostic instruments: a long way toward the mastering of matter since Aristotle first formulated the theory of the four elements!

References 1. Orton J (2009) Semiconductors and the Information Revolution. Academic Press/Elsevier, Amsterdam 2. Seitz F, Einspruch N G (1998) Electronic genie: The tangled history of silicon. University of Illinois Press, Urbana, Illinois 3. Riordan M, Hoddeson L (1997) Crystal Fire: The Birth of the Information Age. W. W. Norton & Company, New York and London 4. Riordan M, Hoddeson L (1997) The origins of the pn junction. IEEE SPECTRUM June: 46–51

Appendix A

On Gases and Water

Lavoisier’s Experiments Starting in 1772, Lavoisier investigated the combustion of non-metallic substances, such as sulfur and phosphorus, the calcination of metals, such as tin and lead, and the reduction of lead oxides, such as litharge and minium. The experiments with phosphorus were described in the booklet Opuscules Physiques et Chimiques (completed in 1773 and published in 1774). Phosphorus was burned in a glass bell held over mercury in the presence of air. Combustion caused the formation of a residue in the form of a white powder and an increase of mercury level inside the bell. The increase of mercury level was due to the decrease of the air present and demonstrated that, upon formation of the residue, part of the air was absorbed. When combustion ceased spontaneously, the remaining phosphorus could no longer burn in the residual air. Moreover, this residual air did not support normal combustion, as evidenced by the extinguishing of a candle flame.1 The dry white powder,2 which was formed, was heavier than the burnt phosphorus and the weight increase corresponded to approximately one fifth of the weight of the initial air [J. R. Partington, A Short History of Chemistry (Dover Publications, New York, 1989), p. 126]. Discussing his results, Lavoisier wrote that. this weight gain results from the prodigious amount of air which is fixed during combustion and combines with the vapors.

Lavoisier continues [W. H. Brock, The Chemical Tree (W. W. Norton & Company, New York, 1993), p. 102]: What is observed in the combustion of sulphur and phosphorus, may take place also with all bodies which acquire weight by combustion and calcination, and I am persuaded that the augmentation of the metallic calces is owing to the same cause. Experiment has completely 1 2

The residual air was nitrogen only. It was phosphorus pentoxide (P2 O3 ).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4

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confirmed my conjectures: I have carried out a reduction of litharge3 in a close vessel, with the apparatus of Hales, and I have observed that there is disengaged at the moment of passage from the calx to the metal, a considerable quantity of air, and that this air forms a volume a thousand times as great as the quantity of litharge employed. This discovery seems to me one of the most interesting that has been made since Stahl…

The tin and lead calcination experiments were described in a memoir that Lavoisier presented to the Academy in April 1774. The calcination was performed by enclosing the metals in a sealed glass retort, which was weighed before and after the calcination. The sealed system showed no change in weight and for Lavoisier this refuted Boyle’s theory, which had explained the increase in weight of the calcination residue as due to the attachment of ponderable igneous particles from external heating. When, subsequently, Lavoisier opened the neck of the retort, he noticed an entry of air, which he believed took the place of that fixed in the metal during calcination. Consequently, he assumed that the slight weight increase caused by the ingress of air was equal to the weight of the air which had combined with the metal. However, at this stage of his research, Lavoisier does not seem to have realized what was the component of ordinary air involved in combustion and calcination, nor what was the nature of ordinary air. In October 1774 Priestley visited Paris and reported to the Parisian scientific community about his discovery of dephlogisticated air, obtained from mercury red calx and also from red lead. In the following months Lavoisier studied and repeated Presley’s experiments. He made further experiments which convinced him of the existence of dephlogisticated air as air’s purest and eminently respirable part. By the end of this cycle of studies Lavoisier had discovered what happens when mercury and metals in general were heated. He realized that there were two essential steps. When mercury is heated in air to not-too-high temperatures, it calcifies gradually, combining with the air’s active part, which Lavoisier called pure air (Priestley’s dephlogisticated air (oxygen)). In the process, a residual air remained, which did not support combustion and respiration and which Lavoisier initially called mofette atmosphérique (that is, unbreathable atmospheric air). When, in turn, the formed calx was heated to higher temperatures, it decomposed into its constituents, mercury and pure air. Lavoisier obtained definitive experimental confirmation through a famous experiment, described in a memorandum presented in 1777. In this he reproduced the entire calcination/calx-decomposition cycle, determining the exact weight of each component after each passage, and finally showing that, if he reunited the volumes of pure air and mofette atmosphérique released in the cycle, ordinary air was obtained again.

3

Litharge is one of the naturally occurring mineral forms of lead “calx” (lead oxide (PbO)).

Appendix A: On Gases and Water

175

Composition of Water French chemist Pierre Macquer (1718–1784) was the first to notice, in 1777, the formation of moisture on a porcelain held over the flame of inflammable air (hydrogen) [A. J. Ihde, The development of modern chemistry (Dover Publications, New York, 1984), p. 69], but his observation had gone unnoticed. The formation of a dew in vessels where inflammable air had burned was subsequently noticed by Priestley in 1781, while, following Volta’s suggestion, he experimented by igniting the inflammable air by means of electric discharges. He reported the results of these experiments to James Watt and to Cavendish and the latter continued them with his permission. Cavendish’s results are contained in his memoir Experiments on air of 1784. As a first step Cavendish demonstrated that, when inflammable air and ordinary air were detonated in a sealed copper or glass vessel, there was no loss of weight as previously supposed. Subsequently, experimenting with various mixtures of the two airs, he concluded that [J. R. Partington, A Short History of Chemistry (Dover Publications, New York, 1989), p. 138]: When inflammable air and common air are exploded in proper proportions, almost all the inflammable air and near one-fifth of the common air lose their elasticity and are condensed into a dew which lines the glass.

To better examine the nature of this dew, Cavendish burnt the two airs in an experiment in which the burnt airs passed through an 8-foot-long glass tube to condense the dew. He says: By this means upwards of 135 grains of water were condensed in the cylinder, which had no taste nor smell, and which left no sensible sediment when evaporated to dryness; neither did it yield any pungent smell during the evaporation; in short, it seemed pure water.

Finally, he obtained similar results by igniting inflammable air and dephlogisticated air with an electric discharge, similarly to what Priestley had done. The experimental results clearly showed that inflammable air (hydrogen) and dephlogisticated air (oxygen) combined to form water; Cavendish interpreted them on the basis of the phlogiston theory. He says: I think we must allow that dephlogisticated air is in reality nothing but dephlogisticated water, or water deprived of its phlogiston; or, in other words, that water consists of dephlogisticated air united to phlogiston; and that inflammable air is either pure phlogiston, as Dr Priestley and Mr Kirwan suppose, or else water united to phlogiston.4

In light of his recent results Cavendish considered the second alternative to be more likely, i.e. that dephlogisticated air was water minus phlogiston and that inflammable air was water plus phlogiston. Therefore, the reaction, interpreted according to the phlogiston theory, was:

4

We recall that Cavendish, in 1766, had assumed that inflammable air was pure phlogiston.

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in f lammable air (water + phlogiston) + dephlogisticated air (water − phlogiston) ⇒ water So, Cavendish continued to believe that water was an element, as asserted by the Aristotelian tradition, and that the explosive reaction between the two airs was due to a redistribution of phlogiston. However, the memorandum in which he reported his results also has an addendum where he said that the results could also be explained by Lavoisier’s new theory.

Appendix B

On Nineteenth Century Physics

Krönig-Clausius Equation The basic assumptions on which the kinetic theory of gases was built are: (a) the gas consists of a large number of identical microscopic particles (corpuscles) of mass m (similar to material points of Newtonian mechanics), uniformly distributed in space and in constant random movement; (b) the particles are small compared to the distances that separate them and, therefore, the total volume of the particles is negligible compared to the volume of the container; (c) the particles collide with each other and with the walls of the container with elastic collisions; (d) the duration time of a collision is negligible compared to the time interval between two successive collisions; (e) in the time interval between collisions, the particles move with uniform rectilinear motion; (f) the pressure, as a macroscopic quantity that the gas is brought to bear on the walls of the container, is the result of the large number of microscopic collisions of the particles on the walls. Each collision generates a force equal to the change in momentum per unit time.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4

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Appendix B: On Nineteenth Century Physics

The Krönig-Clausius equation is: 1 N mu 2 3

pV =

where p and V are gas pressure and volume, respectively, N the total number of particles, m their mass, u2 their mean square velocity.5 Its derivation is based on the general hypotheses listed above, which make the problem perfectly defined and tractable using elementary mathematical methods. To calculate the pressure the authors adopted the subsequent reasoning. First of all, it is observed that, even though the individual particles have different velocities, their effects on pressure can be evaluated by attributing their average velocity to each of them. Given the macroscopic homogeneity of the gas, one can also assume equal probability for all directions of velocity. Clausius wrote [Clausius R (1957) The Nature of the Motion which we call Heat. Ann. der Physik, 100: pp. 353–80]: According to the previous assumptions it is evident that, in a unit time, each molecule will hit a wall as many times as it will be possible, following its trajectory, to travel from the wall in question to the opposite one and then go back.

Therefore, if we indicate with x the direction perpendicular to two parallel walls and ux the component of the root-mean square velocity in this direction, in the time interval Δt the particle, if it does not undergo collisions, travels a total distance L = ux Δt in his progress back and forth between the two walls. Hence, if d is the distance between the two walls, the number of collisions in the time interval Δt is u x Δt L = 2d 2d and the number of collisions per unit time is ux 2d If N is the number of particles in the container, we have for the total number of collisions per unit time: N

ux 2d

If the particles have mass m, from the conservation of momentum in an elastic collision we obtain that the momentum transferred to the wall in each collision is: 2mu x

5

The mean square velocity of a set of particles is the mean of their velocities squared. The mean square velocity is non-zero even when the mean velocity is zero.

Appendix B: On Nineteenth Century Physics

179

The total momentum transferred per unit time, i.e. the momentum transferred in each collision multiplied by the number of collisions per unit time, is: 2mu x

N ux N mu 2x = 2d d

The force due to these collisions, according to the second law of dynamics, is equal to the change in momentum per unit of time, i.e., in our case, to the total momentum transferred per unit of time. Ultimately, we have for the force exerted on the wall by the particles: F=

N mu 2x d

and for pressure (which is the force divided by the area over which it is exerted): p=

F N mu 2x N mu 2x = = A Ad V

where A is the surface of the container and V = A d is its volume. In the equation the pressure is related to the mean square velocity in the x direction. To proceed we have to relate it to the total mean square velocity. Given the macroscopic homogeneity of the gas and its isotropy (i.e. it behaves identically in all directions) and given the very large number of particles, we can assume that the mean square velocity has the same value in all directions, i.e. u 2x = u 2y = u 2z . Then: u 2 = u 2x + u 2y + u 2z = 3u 2x . Substituting into the previous equation we obtain the Krönig-Clausius equation   2 1 1 2 2 N mu pV = N mu ≡ 3 3 2 where 21 N mu 2 is the total kinetic energy of the particles or vis viva, as it was called at in Clausius’s time.

Kinetic Energy and Temperature Using the equation of state of the ideal gas, it can also be shown that the vis viva is proportional to the gas temperature. Clausius used the modern version of the equation of state which accounts for Avogadro’s law and the absolute temperature, i.e. pV = RT

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where R is the universal gas constant and refers to a mole of gas, i.e. to a number of particles (atoms or molecules) equal to Avogadro’s number NA = 6.03 × 1023 . By comparing the two relations, we get 

1 N A mu 2 2

 =

3 RT 2

We obtain for the average kinetic energy of the single particles (atoms or molecules) 

1 2 mu 2

 =

3 R 3 T ≡ kT 2 NA 2

where k = 1.380 × 10–23 is the Boltzmann constant.

Harmonic Oscillator In physics, a harmonic oscillator is a mechanical system consisting of a particle (considered point-like) of mass m which, when displaced from its equilibrium position, is subject to a restoring force F proportional to the displacement x. The restoring force is, therefore, F = - k x and is called elastic force, k is a positive constant characteristic of the system in question. Examples of macroscopic harmonic oscillators are masses connected to springs, pendulums (at small displacement angles), acoustic systems. Harmonic oscillators are found widely in nature and are exploited in many man-made devices, such as clocks and radio circuits. The harmonic oscillator model is very important in physics, because any mass subjected to forces in stable equilibrium behaves as a harmonic oscillator for small displacements around the equilibrium position. Denoting with x(t) the position of the particle at time t and applying the second law of dynamics ( force equals mass times acceleration) we have for the equation of motion: ma = −kx  m

d2x = −kx dt 2

The solution of this differential equation is of the form: / x(t) = A cos(ω0 t + ϕ) with ω0 =

k m

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181

The mass m performs periodic oscillations of amplitude A around the equilibrium . The position at time t depends on the phase position (x = 0), with period T = 2π ω0 ϕ, which determines the initial position at t = 0. The energy of the harmonic oscillator is: E=

1 2 1 2 1 1 mv + kx = mω02 A2 [sin(ω0 t + ϕ)]2 + k A2 [cos(ω0 t + ϕ)]2 = k A2 2 2 2 2

The energy is constant and oscillates continuously from kinetic to potential and vice versa. If the harmonic oscillator is located in a viscous fluid, there is an additional force, proportional to the velocity v of the particle of the type − γmv, which tends to oppose the motion, and the motion of the oscillator becomes damped. Lorentz oscillator In his “theory of the electron” Lorentz assumed that the atoms and molecules making up matter could be assimilated to microscopic harmonic oscillators, whose particles also had a negative electric charge. Due to the electric charge − e, in presence of an electromagnetic wave of frequency ω the harmonic oscillator is subject to an additional force of the type F = − eE(ω), where E(ω) is the electric field of the wave. The harmonic oscillator performs forced oscillations at frequency ω, with an amplitude that depends on how much ω differs from the oscillator’s own frequency ω0 . From the electromagnetic viewpoint, these oscillating charges behave like tiny Hertzian dipoles, absorbing and emitting electromagnetic radiation.

Measurement of e/m (Charge/Mass) At the Nobel prize awarding ceremony, Thomson illustrated in the following way the method he had used to determine the velocity, charge, and mass of the corpuscles that make up the cathode rays [Thomson J J (1906) Carriers of negative electricity. Nobel Lecture, December 11, 1906]: Measurement of velocity The principle of the method used is as follows: When a particle carrying a charge e is moving with velocity v across the lines of force in a magnetic field, placed so that the lines of magnetic force are at right angles to the motion of the particle, then, if H is the magnetic force, the moving particle will be acted on by a force equal to Hev. This force acts in the direction which is at right angles to the magnetic force and to the direction of motion of the particle. If also we have an electric field of force X, the cathode ray will be acted upon by a force Xe. If the electric and magnetic fields are arranged so that they oppose each other, then, when the force Hev due to the magnetic field is adjusted to balance the force due to the electric field Xe, the green

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patch of phosphorescence due to the cathode rays striking the end of the tube will be undisturbed, and we have Hev = Xe  v =

X H

Thus if we measure, as we can do without difficulty, the values of X and H when the rays are not deflected, we can determine the value of v, the velocity of the particles. Determination of e/m (charge/mass) Having found the velocity of the rays, let us now subject them to the action of the electric field alone. Then the particles forming the rays are acted upon by a constant force and the problem is like that of a bullet projected horizontally with a velocity v and falling under gravity. We know that in time t, the bullet will fall a depth equal to 1 2 gt , where g is the acceleration due to gravity. In our case the acceleration due to 2 the electric field is equal to Xe/m, where m is the mass of the particle. The time t = l/v, where l is the length of path, and v the velocity of projection. Thus the displacement of the patch of phosphorescence where the rays strike the glass is equal to d=

1 Xe l2 2 m v2

We can easily measure this displacement d, and we can thus find e/m from the equation 2d v2 e = m X l2

Appendix C

On Old Quantum Theory

More on Photoelectric Effect In the photoelectric experiments, the behaviour of the photoelectric current (that is, the electric current due to light) is investigated as a function of the voltage applied between cathode and anode at different frequencies and intensities of the ultraviolet radiation. Concerning the behaviour of the photoelectric current as a function of the applied voltage, measured at different light intensities while keeping fixed the light frequency, it shows that the current is saturated for positive voltages and goes to zero for a well defined negative voltage value V 0 . The saturation value depends on the intensity of the light and increases as the intensity increases. Concerning the behaviour of the photoelectric current as a function of the applied voltage, measured at different light frequencies while keeping fixed the light intensity, it shows that, in this case, V0 depends on the light frequency, while the saturation current does not. The modulus of V0 increases as the frequency increases. Considering that the negative voltage creates an electric field between the anode and the cathode, which decelerates the electrons, a current flowing at such voltages means that most of emitted electrons have an appreciable kinetic energy. The existence of a threshold voltage V0 implies that the maximum kinetic energy of emitted electrons is: E max =

1 2 = eV0 mv 2 max

Let us summarize the salient experimental features of the effect. There is a minimum frequency of radiation to observe the effect. Only radiations with frequencies above this threshold value emit electrons. Radiation of lower frequencies does not emit electrons, no matter how intense it is and how long it irradiates the sample. Finally, a characteristic of the photoelectric effect is that the emission of electrons is © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4

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concomitant with the irradiation and no delays are observed between the instant of illumination and the moment of emission. These characteristics of the photoelectric effect cannot be explained using the classical theory of interaction between electromagnetic radiation and matter. First of all, in the classical theory, the energy that the electron acquires from the radiation depends on the electric field of the wave and therefore grows with its intensity and does not depend on the wave frequency. As a consequence, the threshold energy eV 0 (or maximum kinetic energy) should increase as the intensity increases, and it should be possible to generate photoelectrons with radiation of any frequency, contrary to the observations. Furthermore, according to the classical theory, the energy absorbed by electrons should also depend on the irradiation time. Consequently, if the irradiation time is long enough, sufficient energy should be obtained to produce the effect, even with very low intensity radiation. This requires an emission time delay at odds with the observation that, for any intensity, there is simultaneity between illumination and electron emission. Finally, there is the most relevant aspect of the dependence of the photoemission threshold (and kinetic energy) on the radiation frequency, a feature in no way explicable by a classical interaction mechanism. In 1905, in the article entitled On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Einstein showed that all the characteristics of the photoelectric effect could be explained if it was assumed that the energy of electromagnetic radiation is quantized and is related to its frequency according to Planck’s quantization scheme: ε = hν Starting from this hypothesis, Einstein suggested that the kinetic energy E of the photoemitted electron is linked to the electromagnetic wave frequency ν by the relationship: E = hν − W where W is the energy necessary to remove the electron from the metal. For each used frequency, the value of the maximum kinetic energy E max can be determined experimentally by measuring the negative voltage V0 necessary to send the photoelectric to zero, as done by Lenard in his experiments. Then we have for E max : E max = eV0 = hν − W0 where W0 is a characteristic energy of the metal, called work function. It represents the minimum energy needed by an electron to escape the attractive forces that bind the electron itself to the metal. If we plot eV 0 as a function of ν, we obtain a straight line whose slope is h/ e (Planck’s constant divided by the electric charge) and whose cross at eV 0 = 0 determines the work function. Clearly the experimental validation of this relationship

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185

Fig. C.1 Behaviour of the maximum kinetic energy of the electrons as a function of the frequency of the light as measured by Millikan on sodium

would represent a decisive proof in favor of Einstein’s theory and the hypothesis of energy quantization of the electromagnetic radiation. We owe the first experimental verification to American scientist Robert Millikan (1868–1953), who in 1916 published the data obtained on sodium, which demonstrated the linear dependence and provided an estimate of Planck’s constant within an accuracy of 0.5% (Fig. C.1).

More on Bohr’s Theory In the 1913 article Bohr considered the simplest possibility, whereby the atom consisted of a nucleus of positive charge Ze (e is the electric charge) and a single electron revolving in a circular orbit under the effect of the Coulomb attractive force. Setting Z = 1 one obtains the hydrogen atom. Bohr introduced the hypothesis that the angular momentum L of the electron is quantized, that is to say, that it can assume only values equal to whole multiples of Planck’s quantum of action h divided by 2π. In formulae: L=n

h ; n = 1, 2, 3, . . . 2π

Choosing only certain angular momenta was equivalent to postulating that only some orbits were allowed for the electron, as opposed to what was stipulated by classical physics. Bohr further postulated that these orbits were stable (stationary) and that the electron that follows them does not emit electromagnetic radiation. Using the possible values of angular momentum, he calculated the possible values of the electron energy, which depended on n and were quantized. He got: En = −

2π 2 me4 Z 2 h2

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where m is the electron mass. To each n value corresponded a circular orbit whose radius increased with increasing n, as schematized in Fig. C.2. Finally, to explain the spectral lines, Bohr made another and, at the time, more radical assumption, i.e. that electromagnetic radiation is emitted or absorbed by the electron when it jumps from one stationary orbit to another. As we have seen, empirical analysis of the spectral lines had shown that the hydrogen lines’ frequencies were given by the Rydberg formula: 

1 1 ν=R 2− 2 n1 n2



Consistently with this empirical rule, Bohr assumed that the frequency of the spectral lines, multiplied by Planck’s constant, is equal to the energy difference of the two orbits between which the electron jumps: hν = E n 2 − E n 1 =

  2π 2 me4 Z 2 1 1 − h2 n 21 n 22

This relationship coincides with αRydberg’s empirical formula if we set Z = 1 and assume for Rydberg’s constant R: R=

2π 2 me4 h3

This expression provides an exact theoretical expression for Rydberg’s constant R. The theory quantitatively and very accurately predicted all series of hydrogen lines, as depicted schematically in Fig. C.3. Fig. C.2 Schematic representation of Bohr’s atom and the process of electromagnetic radiation’s emission

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187

Fig. C.3 Schematic representation of the transitions that give rise to the series of lines in Bohr’s hydrogen atom. Lyman series: transitions to n = 1 level. Balmer series: transitions to n = 2 level. Paschen series: transitions to n = 3 level. Brackett series: transitions to n = 4 level. Pfund series: transition to n = 5 level

As a further success of the theory, Bohr showed how it made it possible to identify the origin of the lines observed by American physicist and astronomer Edward Pickering years earlier in the spectrum of the star ζ-Puppis, which at the time were still attributed to some form of hydrogen. Bohr showed that by setting in his formula Z = 2, n1 = 4 and n2 = 5, 6, 7, … one obtained frequencies in excellent agreement with those of the series measured by Pickering. His conclusion was that the lines belonged to ionized helium atoms, which consist of a single electron rotating around a nucleus of positive charge equal to 2e.

X-rays and Mosely’s Experiments In 1909, even before the X-rays nature were established, British physicists Charles Barkla and Charles Sadler found that these rays were generally composed of a more penetrating component, hard radiation, and a less penetrating one, soft radiation, components called, respectively, K-radiation (or K-rays) and L-radiation (or L-rays). In his experiments conducted in 1913, Moseley measured both K and L radiations of elements ranging from aluminum to gold in the periodic table. He had

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Fig. C.4 Typical X-ray emission spectrum, showing Kα and Kβ lines as well as the continuous background (Brehmsstrahlung continuum)

a better instrumentation than Barkla and Sadler’s, capable of a greater resolution and exploiting Bragg’s law to determine the wavelengths. Moseley found that Kradiation was typically composed of two lines, which he called K α (the more intense at longer wavelength) and K β (the less intense at shorter wavelength), superimposed on a continuous background (Brehmsstrahlung continuum), as shown by way of example in Fig. C.4. He also found that the L-radiation was composed of several lines very close in frequency. Analyzing the data, Moseley discovered a systematic relationship, now known as Moseley’s law, between the frequencies of the X-rays emitted and the square of the atomic number Z of the elements used as targets. In fact, by plotting the atomic number Z as a function of the square root of the frequency of the X-lines, he obtained a series of straight lines. From these he derived that the relationship was of the type: ν = C R(Z − σ )2 where R is the Rydberg’s constant, while C and σ are two parameters whose value depends on the X-radiation used. In the case of K α lines he found C = 3/4 and σ = 1. Moseley showed how these results agree with Bohr’s theory, which actually predicts a dependence of the energy on the square of the charge Z of the nucleus, if the theory itself is generalized and it is assumed that, in multi-electron atoms, the electrons are disposed in stationary states corresponding to orbits of increasing radius, characterized by the quantum number n. The electrons in states of the same n constitute a shell, as shown schematically in Fig. C.5. Assuming that the Bohr relation holds, the energy of the electrons in the n-th shell is:

Appendix C: On Old Quantum Theory

189

Fig. C.5 Schematic representation of electrons’ shells and the emission process of Kα lines. Upon impact of cathode rays one electron is expelled from the n = 1 shell and a n = 2 electron jumps into the empty level, emitting radiation

En = −

R(Z − σ )2 n2

where the parameter σ takes into account the screening effect on the nuclear charge due to the presence of the other electrons. Finally, Moseley hypothesized that the emission of X-ray radiation is due to the jump of an electron from a more external orbit to an inner one. In K radiation’s case, the process begins with an electron being expelled from the n = 1 shell, as a consequence of the impact of the cathode rays (Fig. C.5). The K α radiation is emitted when an electron of the n = 2 shell jumps to occupy the empty n = 1 level. It follows that Moseley empirical formula for K α lines can be rewritten as:   1 3 1 2 ν = R(Z − 1) − 2 = R(Z − 1)2 12 2 4 this in perfect agreement with the experimental finding. A similar argument applies to L lines. In the case of the most intense line, called L α , Moseley had obtained from the experimental graph the values C = 5/36 and σ = 7.4. Assuming that the L α radiation is emitted when an electron of the n = 3 shell jumps into an empty n = 2 level, the value C = 5/36 is obtained by writing the empirical formula as:  ν = R(Z − 7, 4)2

1 1 − 2 2 2 3

 = R(Z − 7, 4)2

5 36

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Optical Spectra of Alkali Metals Optical spectra of alkali metals were, after those of hydrogen, the simplest to classify. Their analysis led Rydberg to the discovery of his generating formula 1 1 ν = R (n +μ . The spectral lines could be grouped into four series 2 − (n 2 +μ2 )2 1 1) named, for historical reasons, sharp, principal, diffuse, and fundamental series. According to the Rydberg’s formula, the frequency of the lines in a series is obtained as the difference of two terms, a first term, where n1 is a fixed whole number, characteristic of the element, and μ1 a constant characteristic of the series, and a second term, where n2 is a whole variable number, characteristic of the single line, taking values bigger than n1, and μ2 another constant characteristic of the series. Furthermore in 1896, both Rydberg and Anglo-German Arthur Schuster (1851–1934) highlighted some relationships, later known as Rydberg-Schuster’s laws, of great importance for the subsequent series’ rationalization. Based on Rydberg-Schuster’s laws the formulae generating the four series of all alkali metals are:  1 1 ≡ (n1 p −n2 s) shar p serie ν = R − (n 1 + P)2 (n 2 + S)2   1 1 p ≡ (n1 s − n2 p) princi pal series ν = R − (n 1 + S)2 (n 2 + P)2   1 1 ≡ (n1 p − n2 d) di f f use series ν d = R − 2 (n 1 + P) (n 2 + D)2   1 1 f ≡ (n1 d − n2 f) f undamental series ν = R − (n 1 + D)2 (n 2 + F)2 

s

where S, P, D, and F denote the constants μ1 and μ2 in the different series of each element. The last column shows the shorthand notation, currently used in atomic physics text-books. Bohr-Sommerfeld’s theory allowed direct identification of the terms of the spectroscopic series with the electron energy levels: number n1 or n2 of the spectroscopic term was identified with principal quantum number n identifying the energy level, while letters s, p, d and f corresponded, respectively, to the values 1, 2, 3 and 4 del azimuthal quantum number k.6 However, for the orbits of alkali atoms important differences are expected with regards to the Sommerfeld’s theory, built for the hydrogen atom whose single electron is subject to the pure Coulomb potential of the nucleus. In the case of alkali atoms, the outermost electron, whose transitions give rise to the optical spectra, is subject to a complex potential sum of that of nucleus and that of the other electrons in the inner shells. In this potential, states with different 6

Notice that, in the quantum mechanics’ final formulation, a new azimuthal quantum number l has been introduced linked to k by the relation l = k − 1, for which current texts of atomic physics posit that s, p, d and f correspond, respectively, to the values 0, 1, 2 and 3 of the azimuthal quantum number l.

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191

Fig. C.6 Scheme of alkali metals’ energy levels

k and the same n come to have different energies, contrary to what happens for hydrogen, where the energy does not depend on k. In general, the k = 1 states (s terms) are more bound than k = 2 states (p terms), which in turn are more bound than k = 3 states with (d terms), and so on, as shown in Fig. C.6. For greater clarity we examine in more detail the case of lithium (Li). Lithium (Z = 3) is the periodic table’s third element, coming immediately after helium (Z = 2). According to the Aufbauprinzip it is obtained by adding one electron to the helium atom and a positive charge to its nucleus. The result is an atom with two electrons in the full n = 1 shell and the added electron occupying a n = 2 orbit. For n = 2 two orbits are possible, a circular one corresponding to k = 2 (p level) and an elliptical one corresponding to k = 1 (s level). The k = 1 state being that of lower energy is that occupied by the third electron in atom’s stationary state (i.e. state of lowest energy). Optical transitions giving rise to the four spectroscopic series are shown in Fig. C.7. Inspection of the transitions scheme at once reveals that transitions only occur when the azimuthal quantum number of the initial state and that of the final state differs by one unit. Since k is the value of angular momentum in unit h/2π this means that in the transition the electron gains or loses a quantum of angular momentum. Conservation of angular momentum requires that this quantum is transferred to the emitted electromagnetic wave. The fact that an electromagnetic wave is endowed with an angular momentum had already been demonstrated by Lorentz and, independently, by Poincaré and Abraham, in the early years of the twentieth century. The process of electromagnetic radiation emission in spectral series was investigated by Polish physicist Wojciech Rubinowicz (1889–1974). In 1918 he proved the principle of

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Fig. C.7 Scheme of energy levels of lithium outermost electron and allowed optical transitions

selection (or selection rule), stating that only optical transitions occur where the azimuthal quantum number varies by ± 1.

Appendix D

On Matter and Quantum Mechanics

Indistinguishability and Exchange Interaction The requirement that the wave functions of a system of identical particles must satisfy the Pauli exclusion principle and the concept of indistinguishability impacts strongly on the possible solutions of Schroedinger’s wave equation. The concept of indistinguishability implies that the results of a measurement of measurable physical properties cannot depend on how identical particles are labelled. This is equivalent to saying that the results must not change if there is an exchange of coordinates, a permutation, of the particles within the system. The relevance of this concept can be illustrated by considering the example of a system of two identical particles. We have seen that, in quantum mechanics, the measurable physical properties depend on the wave function Ψ , whose square module gives the probability of finding the particles in a certain space position at a given time. If we indicate with Ψ (1,2) the wave function of the two particles (labels 1 and 2 represent the set of spatial and spin coordinates of the two particles) and Ψ (2,1) the wave function where electron 1 has taken the place of electron 2 and vice versa, the indistinguishability of the particles requires that the two probabilities are absolutely identical, i.e.: |ψ(1, 2)|2 = |ψ(2, 1)|2 The relationship is satisfied if Ψ is such that: ψ(2, 1) = ψ(1, 2) ( f unction symmetric for the exchange of the two particles) ψ(2, 1) = −ψ(1, 2) ( f unction anti − symmetric for the exchange of the two particles)

This condition holds in general: the wave functions of a system composed of two or more identical particles must be either symmetric or antisymmetric for the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4

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exchange of two of its particles. Let us see the consequences on a system of electrons, considering the case of helium. We have noted that the dominant interaction for each electron is that due to the screened attraction by the nucleus. On this basis and at this level of approximation, the wave function of each electron is determined by the four quantum numbers n, l, m, s. If we denote by ψn 1 l1 m 1 s1 (1) and ψn 2 l2 m 2 s2 (2) the two wave functions, the symmetric wave function of the system is: ψ S = ψn 1 l1 m 1 s1 (1)ψn 2 l2 m 2 s2 (2) + ψn 2 l2 m 2 s2 (1)ψn 1 l1 m 1 s1 (2), the antisymmetric wave function of the system is: ψ A = ψn 1 l1 m 1 s1 (1)ψn 2 l2 m 2 s2 (2) − ψn 2 l2 m 2 s2 (1)ψn 1 l1 m 1 s1 (2) It is evident that, if n1 , l 1 , m1 , s1 is equal to n2 , l 2 , m2 , s2 and, therefore, ψn 1 l1 m 1 s1 (1) = ψn 2 l2 m 2 s2 (2), ψ A is equal to zero: antisymmetric functions satisfy Pauli exclusion principle ! It follows that the condition that the electrons satisfy Pauli exclusion principle is equivalent to requiring that the wave functions of a system of two or more electrons be antisymmetric for the exchange of two electrons. This purely quantum mechanical property has relevant consequences for the structure of atoms and matter. One of the most important is the emergence of a new form of interaction between electrons, which has no classical analogue and goes by the name of exchange interaction. In helium’s case, it explains the considerable difference in energy between the homologous singlet and triplet states. To understand the working of the exchange interaction let us consider in more detail the wave functions of the two electrons represented as ψn 1 l1 m 1 s1 (1) and ψn 2 l2 m 2 s2 (2). They can be represented as the product of a function of the spatial coordinates and a function of the spin: ψn 1 l1 m 1 s1 (1) = ψn 1 l1 m 1 (r 1 )χs1 and ψn 2 l2 m 2 s2 (2) = ψn 2 l2 m 2 (r2 )χs2 From the Russell–Saunders coupling scheme it follows that the antisymmetric wave function Ψ A of the two electrons can be constructed in two ways: (1) product of a symmetric total spin wavefunction multiplied by an antisymmetric spatial wavefunction; (2) product of an antisymmetric total spin wavefunction multiplied by a symmetric spatial wavefunction. The first case pertains to triplet states, while the second pertains to singlet states. This implies that triplet states have an antisymmetric spatial wave function, while singlet states have a symmetric spatial wave function. ψT ri plet = ψn 1 l1 m 1 (r 1 ) ψn 2 l2 m 2 (r 2 ) − ψn 2 l2 m 2 (r 1 ) ψn 1 l1 m 1 (r 2 ) ψ Singlet = ψn 1 l1 m 1 (r 1 ) ψn 2 l2 m 2 (r 2 ) + ψn 2 l2 m 2 (r 1 ) ψn 1 l1 m 1 (r 2 )

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195

It is evident that the two wave functions give rise to two different probability amplitudes and, consequently, if we consider the contribution to the energy due to the Coulomb repulsion between the two electrons, this must be different for the two types of states. The energy difference between singlet and triplet states arises, therefore, from the requirements of particles’ indistinguishability and can be thought of as a new type of interaction between particles, the exchange interaction, since it is linked to the exchange of particles.

Ionized Hydrogen Molecule The ionized hydrogen molecule is the simplest molecule, composed of one electron and two hydrogen nuclei (two protons). The pattern of reasoning leading to the solution of the corresponding Schroedinger equation can be illustrated as follows. The basic idea is to calculate the electron energy as a function of the distance R of the two nuclei, starting from an initial situation in which they are so distant that neither of them is affected by the presence of the other. Conventionally this corresponds to infinite distance: R = ∞. When R is very large (R = ∞), the system consists of a neutral hydrogen atom H and an ionized hydrogen atom H + . Labeling with A the neutral atom and B the ionized atom, the system is H A + H B + . The electron’s energy coincides with the energy of the ground state of the hydrogen atom (1s state, i.e. n = 1, l = 0). The wave function coincides with that of atom A, which is a 1s function which we can write as: A Φ1s = Φ1s (r − R A )

where r denotes the electron’s position and RA the proton A position. However, it is equally possible that the electron is bound to nucleus B and that the system is H A + + H B. In this case the wave function of the electron coincides with that of the ground state of atom B: B Φ1s = Φ1s (r − R B )

with analogous meaning of symbols. Since both situations have equal probability, the correct wave function must give the same probability for the electron to be near each of the two nuclei. There are two solutions that satisfy this condition:    A  A B B ; ψ− (r ) = Φ1s ψ+ (r ) = Φ1s + Φ1s − Φ1s which are schematically represented in Fig. D.1. The evolution of the wave function when the two nuclei approach each other is also shown on the right side of the figure. As the distance decreases, the probability amplitude ψ+ (r ) increases in the region

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Fig. D.1 Schematic representation of the wave functions ψ + (r) and ψ - (r). At large nuclei’s distance (left) they consist of two unperturbed 1s hydrogen functions. At close distance (right) for ψ + (r) there is a substantial amplitude increase between the nuclei, while it remains small for ψ − (r)

between the two nuclei. This leads to an increased probability of finding the electron in this region, entailing an increase of negative charge between the two nuclei, which results in a greater attraction of the protons towards the molecule’s centre. In the case of the ψ− (r) wave function the probability amplitude remains zero at the molecule’s centre, due to the destructive interference between the two wave A B and Φ1s . The probability of finding the electron between the two nuclei functions Φ1s is minimised. The behaviour of the probabilities |Ψ + (r)|2 and |Ψ − (r)|2 of finding the electron is compared in Fig. D.2. At close nuclei’s distance the Ψ + (r) state exhibits a substantial probability for finding the electron between the two protons, while this is practically zero for the Ψ − (r) state. The behaviour of the energies E + and E − of the two electronic states Ψ + (r) and Ψ − (r) as a function of the nuclei’s distance R is shown in Fig. D.3. The stabilizing effect of the negative charge between the two protons is confirmed by the trend of the energy E + . With decreasing distance the energy decreases, showing that the state becomes progressively more stable, until it reaches a minimum corresponding to the equilibrium distance of the two nuclei in the molecule. For smaller distances the repulsion between the protons becomes dominant, giving rise to a rapid energy increase. The presence of an energy minimum is the characteristic of a stable molecule. As for the energy of the Ψ − (r) state, the minimised probability of finding the electron between the two nuclei entails the predominance of the repulsion between the two protons at all distances, as demonstrated by the increase of E_ as R decreases.

Appendix D: On Matter and Quantum Mechanics

197

Fig. D.2 Schematic representations of the probabilities |ψ + (r)|2 and |ψ − (r)|2 at close protons’ distance

Fig. D.3 Behaviour of the energies E + and E − of the two electronic states as a function of the nuclei’s separation R

In conclusion, we see that the quantum treatment of the problem gives a lower energy solution, corresponding to the existence of a stable H2 + molecule and an excited state involving the dissociation of the molecule. The two states are called bonding and anti-bonding states, respectively.

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Classical Free Electrons Gas The basic assumptions of the free electron gas, as envisioned by Drude, are: 1 The interaction of an electron with other electrons and with ions occurs only when they collide and is negligible in the time interval between collisions. Therefore, apart from collisions and in the absence of externally applied electromagnetic fields, each electron moves with a rectilinear and uniform motion. In presence of external fields, each electron moves following the motion laws of classical mechanics. 2 As in the kinetic theory of gases, collisions are instantaneous events that abruptly alter the velocity of electrons. According to Drude, collisions of electrons with ions are the cause of electrical resistance. 3 It is possible to define a time interval τ, variously known as relaxation time or collision time or mean free time between collisions, such that each electron undergoes on average one collision every interval τ. Time τ is assumed to be independent of electron’ position and velocity. 4 It is assumed that the electron emerges from each collision with a velocity which is not correlated with that it had before the collision. The new velocity is directed randomly and with a value depending on the temperature prevailing in the place where the collision occurred, the hotter the place the greater the speed. In a classical free-electrons gas the energy of a particle is only of the kinetic type and is given by: E=

1 2 p2 mv = 2 2m

where m is the electron mass, v its velocity, and p = mv its linear momentum. Graphically, this relation linking E to v and p (dispersion curve) represents a parabola. The probability for the electron to have a given energy value is provided by the Maxwell–Boltzmann distribution law. When acted upon by external forces, such as electric or magnetic fields, the electron reacts according to the second law of classical dynamics (force equals mass times acceleration): F = ma = m v˙ = p˙ where the superimposed dot means variation per unit time (in mathematical ; p˙ = ddtp ). language: derivative with respect to time, i.e. v˙ = dv dt Based on these assumptions Drude was able to calculate the electrical conductivity σ of metals, which is given by the formula (Drude’s formula): σ =

e2 N τ m

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199

where N is the electron density (number of electrons per unit volume), m and e are the electron mass and charge, and τ is the mean free time between collisions. Conductivity depends linearly on the electron density N, which was not yet known in Drude’s time. Consequently, it was impossible to compare the theoretical expression with the experimental values. However, the experimental comparison could be made with the ratio κ/σ between thermal conductivity κ and electrical conductivity σ , which German physicists Gustav Wiedemann (1826–1899) and Rudolph Franz (1826–1902) had measured back in 1853. They had found that this ratio had approximately the same value in different metals at the same temperature. Since then the constancy of the ratio κ/σ in metals had been called the Wiedemann–Franz law and had represented another mystery awaiting explanation. In 1872 it had been empirically demonstrated by Ludvig Lorenz (1829–1891) that the ratio κ/σ was proportional to the temperature T, for which the Wiedemann–Franz law was summarized by the expression: κ = LT σ where L is a constant of proportionality, called Lorenz number, independent of metal type. Using Boltzmann’s kinetic theory, Drude calculated the heat flux due to electrons in a temperature gradient, and from this obtained the thermal conductivity κ. Both conductivities depend linearly on the electron density N, which therefore cancels out in the ratio κ/σ. Using the expressions for the two conductivities, Drude calculated the Lorentz number L and found a value that showed amazing agreement (later proved fortuitous) with the experimental data. Drude’s theory was the first microscopic derivation of conductivity, both electrical and thermal, and represented the first explanation of the Wiedemann–Franz law.

Sommerfeld’s Free Electrons Gas In Sommerfeld’s quantum theory of free electron gas, electrons become quantum particles described by a wave function, obey Pauli exclusion principle and are distributed in energy according to Fermi–Dirac statistics. Free electron wave functions are plane waves, whose momentum satisfies de Broglie relationship: p=

h λ

The allowed wavelengths λ are obtained from the condition that λ is a submultiple of the length L of the crystal, i.e. L = n λ, where n is an integer. Consequently, the momentum is quantized according to the relationship:

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pn =

h hn = kn = λn L

where kn = 2πL n is the wave vector (also called crystal momentum in solid state h (called h-bar). physics) and  = 2π The kinetic energy of the electrons is also quantized:   pn2 2 kn2 2 2π n 2 En = = = 2m 2m 2m L and only some values are allowed, as shown schematically in Fig. D.4. The most significant difference from the classical free electron gas emerges upon application of the Pauli exclusion principle, asserting that each quantum state can contain only one electron. Denoting with N the total number of free electrons in the gas and assuming a very low temperature (ideally T = 0), the Fermi–Dirac distribution law tells us that the electrons occupy the N lowest energy states. At T = 0, the occupied electronic state of maximum energy is called the Fermi level and has an important role in the physical properties of metals. This distribution of electrons in the allowed energy levels differs significantly from that predicted by Maxwell–Boltzmann statistics. For simplicity, we have considered hitherto a one-dimensional solid. If threedimensional solids are considered, the wave vector becomes a vector quantity k, whose components are: 2π n 1 2π n 2 2π n 3 , , L1 L2 L3 where L 1 , L 2 , L 3 are the lengths of the three solid edges and n1 , n2 , n3 are integers. Similarly, both electron momentum and velocity become vector quantities. Fig. D.4 Energy of the allowed electronic states (represented by dots) for the Sommerfeld’s free electron gas. Ef is the energy of the Fermi level and kf the Fermi wave vector

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201

Although they are quantized, the second law of dynamics continues to hold for Sommerfeld’s free electrons.

Bloch’s Electrons In the case of Bloch’s electrons, the allowed energy values are distributed in a sequence of bands of allowed energies (energy bands), separated by intervals (gaps) of forbidden energies (band gaps). Each band of allowed energies is characterized by its own dispersion relation E = E(k), where k is the wave vector (crystal momentum) (Fig. D.5). The dynamics of Bloch’s electrons depends on the E(k) relation of each band and can differ substantially from that of free electrons, whose dispersion relation is given 2 2 by E(k) = 2mk . It follows that the direct proportionality between momentum and wave vector is no longer valid, nor is that between velocity and wave vector, i.e. p /= k; v /=

k m

Similarly, the second law of dynamics is no longer valid. Investigation of Bloch’s electron dynamics has, however, shown that the electrons behave as if they had a mass whose value depends on the wave vector k and on the dispersion E(k) of the energy band to which they belong. This mass, called effective Fig. D.5 Schematic representation of the dispersion relation E(k) of energy bands separated by band gaps

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mass, is proportional to the inverse of the curvature of the function E(k). Using the effective mass concept, we can therefore formally express the Bloch electron dynamics by the equation: F = m ∗ (k)a where m*(k) designates the effective mass. The dynamics of Bloch electrons can differ greatly from the behavior of classical or Sommerfeld’s free electrons. This is especially true at the top of a band (see the second band of Fig. D.5) where, as k increases, the energy decreases instead of increasing. This implies that, acted upon by an electric field, an electron placed in the maximum of the band acquires negative acceleration and velocity.

Index

A Abegg, Richard, 126 Abraham, Max, 135 Absorption line, 115 Actualization, 12, 25 Aepinus, Franz, 68 Affinity, 59 Agricola, 19 Air, 9, 11 weight, 33 Alchemy, 19 Al-Razi, 21 Amber, 63 Anaxagoras, 11 Anaximander, 7 Anaximenes, 8 Angstrom, Anders Jonas, 115 Anode rays, 98 Apeiron, 8 Aqua fortis, 22 Aqua regia, 22 Arche, 7 Aristotle, 11, 63 Assay, 19 Atomic weight, 82 Atomistic theory, 9 Atoms, 9 Aufbauprinzip, 128 Avicenna, 21 Avogadro, Amedeo, 79 Avogadro number, 94

B Bacon, Francis, 40 Balard, Antoine Jérôme, 84

Balmer, Johann, 117 Bardeen, John, 170 Basson, Sebastien, 31, 39 Becher, Joachim, 46 Béguyer de Chancourtois, Alexander, 84 Bergman, Torbern, 59 Bernoulli, Daniel, 90 Berthollet, Claude-Louis, 60 Berzelius, Jöns Jacob, 72, 80 Biringuccio, 19 Black body, 103 Black body radiation, 103 Black, Joseph, 49, 93 Bloch, Felix, 158 Bohr, Niels, 118, 128 Boltzmann, Ludwig Eduard, 92, 93, 106 Born, Max, 141 Boyle, Robert, 35, 37, 41, 42, 64 Boyle’s law, 35 Brackett, Frederick Summer, 118 Bragg law, 156 Bragg, William Henry, 121, 155 Bragg, William Lawrence, 121, 155 Brattain, Walter, 170 Braun, Karl Ferdinand, 164 Bravais, Auguste, 155 Broek, Antonius Johannes van den, 120 Bunsen, Robert Wilhelm, 116 Burrau, Øyvind, 150

C Cabeo, Niccolò, 64 Calcination, 38, 45 Caloric, 55, 91 Calx, 38, 52, 54

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Evangelisti, The Concept of Matter, History of Physics, https://doi.org/10.1007/978-3-031-36558-4

203

204 Canal rays, 98 Cannizzaro, Stanislao, 83 Carbon dioxide, 49 Carlisle, Anthony, 71 Cathode rays, 96 Cavendish, Henry, 50, 53, 67 Cisternay du Fay, Charles-François de, 66 Clausius, Rudolf, 90, 93 Clément, Nicolas, 93 Combustion, 37, 45, 53, 55 Compton, Arthur Holly, 136 Conservation of matter, 58 Coulomb, Charles-Augustin de, 68 Couper, Archibald, 83 Covalent bond, 150 Crookes’ tubes, 96 Crookes, William, 96 Crystal structure, 154

D Dalton, John, 75 Darwin, Charles, 145 Davisson, Clinton, 137 Davy, Humphry, 72, 80 De Broglie, Louis, 136 De Broglie, Maurice, 132 Definite proportions, 76 Definite proportions, law of , 61 Democritus, 9 Dephlogisticated air, 52, 56 Desormes, Charles Bernard, 93 Diode cat’s whisker, 164 junction, 170 Dioscorides, 6 Dirac, Paul Adrien Maurice, 141, 149 Dobereiner, Johan, 84 Doping, 168 Drude, Paul, 156 Dulong-Petit, law of , 93 Dulong-Petit’s law, 109 Dulong, Pierre Louis, 93 Dumas, Jean Baptiste André, 84

E Earth, 9, 11 Einstein, Albert, 108, 136 Electric discharge, 96 Electrolysis, 71, 81 laws of, 82 Electron, 95

Index Electrostatic generator, 64 Element, 58 Elixir, 20 Elsasser, Walter Maurice, 137 Emission line, 115 Empedocles, 8 Entropy, 92 Epicurus, 9 Equivalent, 60 Equivalent weight, 76 Ercker, 19 Estermann, Immanuel, 137 Ether, 11 Eudoxus, 11 Exchange interaction, 149

F Faggin, Federico, 171 Faraday, Michael, 81, 163 Fermi-Dirac statistics, 157 Fifth element, 20 Fine structure, 130 Fire, 8, 10, 11 Fixed air, 49 Form, 12 Form and matter, 25 Four elements, 12 Four qualities, 12, 20 Franck, James, 122 Franklin, Benjamin, 68 Fraunhofer’s lines, 115 Fraunhofer, Joseph von, 115 Free electron gas, 157

G Galen, 6, 13 Galileo, 33, 39 Galvani, Luigi, 69 Gas, 31 pingue, 32 Gassendi, Pierre, 40 Gay-Lussac, Joseph Louis, 78, 165 Geber, 20 Geissler, Heinrich, 96 Geoffroy, Etienne-Francois, 59 Georg Bauer see Agricola, 19 Germer, Lester, 137 Gilbert, William, 63 Goldstein, Eugen, 96, 98 Goudsmit, Samuel, 135

Index Gray, Stephen, 65 Guericke, Otto von, 34, 64, 96

H Hales, Stephen, 48 Hall, Edwin H., 160 Hall effect, 160 Harmonic oscillator, 93 Hauksbee, Francis, 65 Haüy, René Just, 155 Heisenberg, Werner, 133, 141, 149, 161 Heitler, Walter, 150 Helmholtz, Hermann von, 95, 111 Helmont, Joan Baptista van, 31 Herschel, William, 104 Hertz, Gustav Ludwig, 122 Hertz, Heinrich, 108 Hisinger, Wilhelm, 72 Hoff, Ted, 171 Hooke, Robert, 35, 37 Horror vacui, 34 Hund, Friedrich, 151 Hydrogen, 50

I Iatrochemistry, 23 Inflammable air, 50, 56 Integrated circuit, 170

J Jabir ibn Hayyan, 20 Jeans, James, 107 Jordan, Pascual, 141

K Kekule, Friedrich August, 83 Kepler, John, 153 Kilby, Jack, 170 Kinetic energy, 92 Kinetic theory, 90 Kirchhoff’s Law, 104 Kirchhoff, Gustav Robert, 104, 116 Krönig, August, 90 Krönig-Clausius’ equation, 90

L Landé, Alfred, 134 Laue, Max von, 121, 155 Lavoisier, Antoine, 53, 165

205 Lenard, Philipp, 97, 109, 112 Lennard-Jones, John, 151 Leucippus, 9 Lewis, Gilbert, 150 Leyden jar, 66 Linnaeus, Carl, 154 London, Fritz, 150 Lorentz, Hendrik Antoon, 95, 106, 157 Loschmidt, Joseph, 94 Lucretius, 9 Lyman, Theodore, 118

M Macquer, Pierre-Joseph, 60, 154 Magnesia alba, 49 Mariotte, Edme, 35 Maxwell, James Clerk, 91 Mayow, John, 37 Mazor, Stanley, 171 Melvill, Thomas, 115 Mendeleev, Dmitri Ivanovich, 85, 126, 168 Mersenne, 38 Meyer, Julius Lothar, 85 Michelson, Albert, 130 Microprocessor, 171 Microwave, 167 Mixt, 25 Molecular orbital method, 151 Moseley, Henry, 120 Mulliken, Robert, 151 Multiple proportions, 76 Musschenbroek, Pieter van, 66

N Nagaoka, Hantaro, 112 Natural minima, 13, 25 Newlands, John, 84 Newton, Isaac, 41, 90 Nicholson, William, 71 Noble gas, 127 Nollet, Jean-Antoine, 66 Noyce, Robert, 170

O Octet rule, 126 Odling, William, 84 Oil of vitriol, 22 Orthohelium, 146

206 P Paracelsus, 23 Parahelium, 146 Pascal, Blaise, 34 Paschen, Friedich, 118 Pauli, Wolfgang, 135, 149, 157 Peierls, Rudolf, 160 Periodic table, 85 Perrin, Jean-Baptiste, 112 Petit, Alexis Thérèse, 93 Pfund, August H., 118 Philosopher’s stone, 20, 23 Phlogiston, 45 Photoelectric effect, 108 Pickard, Greenleaf Whittier, 166 Planck, Max, 107 Pliny the Elder, 6 Pluecker, Julius, 96 Pneuma, 8 Pneumatic trough, 48 p-n junction, 170 Potash, 72 Prevost, Pierre, 104 Priestley, Joseph, 51, 68 Prima naturalia, 41 Primary substance, 7, 8, 31 Primitive elements, 8, 11, 21 Proust, Joseph Louis, 61

Q Quantum of action, 108 Quintessence, 11, 15, 20

R Radar, 166 Rayleigh, Lord, 90, 107 Rectification effect, 164 Relativistic effects, 131 Respiration, 37 Rey, Jean, 31, 38 Richter, Jeremiah, 60 Ritter, Johann Wilhelm, 72 Ritz, Walther, 118 Romé de L’Isle, Jean-Baptiste, 154 Röntgen, Wilhelm, 121 Rouelle, Guillaume François, 154 Rowland, Augustus, 95 Russell, Henry Norris, 147 Russell-Saunders’ coupling scheme, 147 Rutherford, Ernest, 113 Rydberg’s constant, 118

Index Rydberg, Johannes, 117

S Sala, Angelo, 39 Saunders, Frederick, 147 Scheele, Carl Wilhelm, 51 Schroedinger, Erwin, 141 Semiconductor, 163 Sennert, Daniel, 31, 39 Shockley, William, 170 Silicon, 165 Slater, John, 151 Soda, 72 Sommerfeld, Arnold, 122, 131, 158 Specific heat, 109 Spectroscopy, 114 Spin, 135 Spin-orbit interaction, 144 Spirit, 32 Stahl, Ernst, 46 Stahl, Georg, 59 Stationary state, 120 Stefan-Boltzmann’s law, 106 Stefan, Josef, 106 Steno, Nicolas, 154 Stern, Otto, 137 Stoner, Edmund, 134 Stoney, George Johnstone, 95

T Thales, 7, 63 Thermal radiation, 103 Thermodynamics, 92 Thomson, George Paget, 137 Thomson, Joseph John, 97, 108, 111, 126 Thomson, William, 111 Torricelli, Evangelista, 33 Transistor bipolar, 170 Transmutation, 13, 14, 24 Tria prima, 23

U Uhlenbeck, George, 135 Uncertainty principle, 142

V Vacuum, 9, 32 Vacuum pump, 34, 96 Valence bond method, 151

Index Valency, 82, 126 Versorium, 64 Vis viva, 92 Volta, Alessandro, 69 Voltaic pile, 69 Vortex atom, 111 W Water, 9, 11 Waterston, John James, 90 Watson, William, 67 Wave function, 142 Wiedemann-Franzlaw, 157

207 Wien, Wilhelm, 98, 106 Wilson, Alan Herries, 161, 164 Woehler, Friedrich, 165 Wollaston, William Hyde, 115

X X-rays spectra, 127

Z Zeeman effect, 130 Zeeman, Pieter, 130