Hydrogen Properties for Fusion Energy [Reprint 2020 ed.] 9780520338401

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Hydrogen Properties for Fusion Energy

Hydrogen Properties for Fusion Energy P. Clark Souers Lawrence Livermore National Laboratory

UNIVERSITY OF CALIFORNIA PRESS BERKELEY

LOS ANGELES

LONDON

University of California Press Berkeley and Los Angeles, California University of California Press, Ltd. London, England Copyright © 1986 by The Regents of the University of California Library of Congress Cataloging in Publication Data Souers, P. C. Hydrogen properties for fusion energy. Includes index. 1. Nuclear fusion. 2. Hydrogen. QC791.S68 1985 539.764 ISBN 0-520-05500-4 (alk. paper)

1. Title. 85-4795

Printed in the United States of America 1 2 3 4 5 6 7 8 9

Contents

Prologue 1. Introduction Hydrogen Fusion Reactions Tritium The Hydrogen Family The Basic Processes in Hydrogen Reviews of Past Work Extrapolating with the Quantum Parameter Rotational Nomenclature

xi 1 1 4 6 7 8 8 12

2. Perfect Gas Thermodynamics Translational Energy Vibrational Energy Rotational Energy Rotational Populations Normal Hydrogen Rotational Property Tables

15 15 17 18 20 24 25

3. Gas Properties at Low Pressures Kinetic Transport Properties Gas Flow through a Pipe Gas Diffusion down a Pipe Thermal Diffusion

32 33 36 40 41

4. Saturated Vapor Pressures and Densities The Phase Diagram Triple Points and Critical Points Solid Vapor Pressures Liquid Vapor Pressures Three-Constant Vapor Pressure Equations Density of the Low-Pressure Vapor Density of the Near-Critical Vapor

45 45 45 49 50 52 54 57

vi

CONTENTS

5. Saturated Pure Component Liquid Properties Density Viscosity Surface Tension Diffusion Coefficients Heat Capacity Thermal Conductivity Sound Velocity Refractive Index

61 61 62 63 65 66 67 70 70

6. Solid Properties The Hexagonal Crystal Structure The Face-Centered Cubic Structure Interstitial Spaces Molar Volumes Sound Velocity Crystal Growing and Quality Crystallite Light Scattering Mechanical Properties Molecular Self-Diffusion Neutral Clusters

73 73 76 77 78 81 81 83 85 88 88

7. Thermal Properties of the Solid The Binding Force Energies of the Saturated Solid Simple Theory of the Solid The Debye Temperature Thermal Conductivity Thermal Resistance at an Interface Thermal Diffusivity Tritium Heat Effects

92 92 93 95 99 100 104 105 105

8. Thermodynamics of the Saturated Phase Boundary Heats of Transformation The Dilute Gas Internal Energy Calibration Enthalpy and Internal Energy Slow Freeze of Dense D-T Gas

110 110 113 114 115 122

9. The Melting Curve Melting Curve Information Heat of Fusion The Lindemann Melting Relation

125 125 128 130

CONTENTS

vii

10. The Pressurized Fluid Pressure-Volume-Temperature Estimated Energies of Fluid D-T Transport Properties

133 133 135 145

11. Gas Interactions with Cold Surfaces Effects on Solid Vapor Pressure Molecular Interaction with the Surface Adsorption Isotherms Physisorption and Hindered Rotation

152 152 154 157 158

12. Hydrogen Solutions Ideal Vapor Pressures Ideal Heats of Transformation First Freezing Temperature Liquid Non-Idealities Phase Separation Possibilities

165 165 166 167 167 169

13. Solution Fractionation and Non-Ideality Equilibrium between Phases Fractionation upon Freezing Non-Ideal Vapor Pressure Effects

171 171 173 178

14. Hydrogen-Helium Mixtures Solubility of Helium in Liquid Hydrogen The Melting Line Fluid Phase Diagram Helium Gas Blocking in a Cryostat Thermal Conductivity of Cold Helium-3 Gas

186 186 191 192 192 198

15. Mixtures with Other Substances Hydrogen-Neon Mixtures with Nitrogen and Oxygen Other Solids in Liquid Hydrogen

199 199 203 203

16. Tritium Decay Decay Constant and Half-Life Heat Output Beta Particle Energy Spectrum Recoil Energy

205 206 208 208 212

17. Range of Tritium Radioactivity Beta Particle Range in Hydrogen

214 214

viii

CONTENTS

Tritium Sample of Finite Thickness—Simple Model Beta Absorption in Wall Materials Radiation from Tritium Decay Absorption of X-Rays High-Energy Electron Beams Electron Scattering from Metal Walls

219 221 221 226 227 228

18. Radiation Damage in Hydrogen The Basic Processes Cross Sections Low-Energy Processes The Background Electron Energy Spectrum

232 232 235 236 238

19. Reactions in Irradiated Hydrogen Kinetic Equations Reaction of H 2 + and H + Electron-Ion Recombination Atom Recombination Hydride Ion Formation Positive Ion Clustering Reactions Negative Ion Clustering Reactions Helium Tritide Ion-Ion Recombination

241 241 242 244 247 248 249 252 252 253

20. Physical Properties of Species in Irradiated Hydrogen The Diatomic Ion The Triatomic Ion Atoms The Hydride Ion Ion Clusters Electronically Excited Molecules Vibrational and Rotational Relaxation

256 256 260 261 263 264 265 267

21. Excited Helium and Muonic Hydrogen Helium Ions and Excited States Helium Tritide Muonic Hydrogen

270 270 275 277

22. Radiation Damage in the Solid Excitons and the Conduction Band Atoms Ions and Electrons

281 281 282 287

CONTENTS

ix

23. The Hydrogen Exchange Reaction Chemical Equilibrium Constants Basic Theory Radiation Effects in the Gas Phase Atoms in the Gas Phase Low-Temperature Catalysts Radiation Effects in the Liquid and Solid

294 294 298 299 304 304 305

24. Rotational Transitions and Quadrupoles Cause of the Rotational Metastability Self-Catalyzed J = 1 J = 0 Transitions J = 1 0 Catalysis J = 1 Quadrupolar Interactions Quadrupolar Thermodynamics

308 308 311 315 317 319

25. Infrared and Raman Spectroscopy Transition Energies Raman Spectroscopy Infrared Spectroscopy Impurity Lines

324 324 325 328 331

26. Nuclear Magnetic Resonance Types and Sizes of Magnetic Moments Relaxation in the Gas and Liquid Relaxation in the Solid Nuclear Spin Polarization Helium-3

336 337 342 344 347 348

27. Electrical Effects Caused by Radiation DC Electrical Behavior AC Electrical Behavior Electron Drift Velocity and Mobility Ion Mobilities Results on D-T Gas and Liquid

352 353 356 357 362 366

Appendix A. Useful Dimensional Relations

369

Appendix B. Hydrogen Permeation in "Impermeable" Materials

370

Index

375

Prologue

This book is intended as a materials science description of hydrogen from 4 K to room temperature, with some lower and higher temperature forays. I arbitrarily take 2 MPa as an upper pressure limit. The first cryogenic applications are just starting in magnetic and inertial confinement fusion. However, the ultimate fusion engineer for whom this book is written has not yet appeared. Perhaps such applications must wait for the natural cryogenics of outer space. I have written this work in the form of a textbook because the assembled hydrogen properties form a mature area of knowledge. There is no point in belaboring obsolete data, and I proceed with the numbers I believe are best. This condemns to obscurity many researchers whose good work has been replaced by better measurements. Because of the textbook form, I have also relegated most current names to the references. I have already discovered that this will displease many people, but the hydrogen field is now too big and the effects of the passage of time too unfair, and this is not a history. One can think of H. L. Johnston of Ohio State University, whose equation of state work was displaced by the National Bureau of Standards, Boulder, who in turn have found part of their work replaced by the group at Kharkov. Or, if we return to 1934, we find the first vapor pressure of deuterium done by G. N. Lewis of Berkeley, who tried so hard to crack the quantum code. His bad luck in the vapor pressure was in being first rather than last. In this book, I was not up to writing all these stories. Perhaps a historian of science will reach this field some day. Now we turn to a few details. If the text says x 10 - 1 8 , it means in units of 10 - 1 8 , not the other way around. Also, I have gone to great lengths to indicate what is estimated, extrapolated, or smoothed, because this often seems not to be done. I have also dropped the "ortho-para" terminology because I have never been able to explain it to my bosses. It is now "odd-J" or J = 1. The most difficult thing I did for this book was to translate the picturesque units of the natives. My favorite people are the gas permeability measurers (Appendix B), from whom at least three different papers on the same substance are required to Socratically arrive at a number. As a result, I have embraced the SI Metric system, especially after the religious experience of discovering that PV really does equal energy. I have no mercy to show on this subject (except, however, for eV and an occasional cm - 1 ). As partial recompense for the lack of history in this book, I would like to thank

xii

PROLOGUE

the following people who sent preprints or special material, or whose comments were incorporated: Dick Alire, Lawrence Livermore National Laboratory, reading of manuscript Randy Bohn, University of Toledo, solid thermal conductivity P. Borgesen and H. Sorensen, Riso National University, Denmark, electron effects on the solid Grady Carney, Allegheny College, ion vibrational frequencies T. E. Cravens, University of Michigan, electron energy loss Ching Fong, University of California (Davis), continuing advice on defects in solids Jim Gaines, Ohio State University, continuing advice on solid thermal and nuclear magnetic properties Walter Hardy, University of British Columbia, energy of the solid Mike Hiza, National Bureau of Standards (Boulder), helium solubility Arnold Honig, Syracuse University, nuclear polarization Jess Hord, National Bureau of Standards (Boulder), H 2 properties Steve Jones, Idaho National Engineering Laboratory, muon-catalyzed fusion Wes Jones, Los Alamos National Laboratory, hydrogen exchange Ken Jordan, Mound Laboratory, tritium half-life Kyekyoon Kim, University of Illinois, liquid properties M. Kinoshita, JAERI (Tokai), non-ideality and distillation I. N. Krupskii, University of Kharkov, solid molar volume The Laser Target Fabrication Group at LLNL, who hounded me into doing D-T thermodynamics: Lee Griffith, Dale Darling, and Tom Bernat, also John Pitts with hydrogen permeation Terry McConville, Mound Laboratory, D-T exchange reactions and ion behavior Duk Poll and Jim Hunt, University of Guelph, continuing advice on infrared physics, defects in solids Paddi Reddy, St. John's University, infrared spectroscopy Hans Roder, National Bureau of Standards (Boulder), density of vapor H. F. Schaefer, University of California (Berkeley), ion clusters Bob Schmieder, Sandia Livermore, optical emission spectra Bob Sherman, Los Alamos National Laboratory, solution properties A1 Sherwood, Lawrence Livermore National Laboratory, continuing advice on solution thermodynamics John Simpson, University of Guelph, specially taken tritium beta decay spectrum Steve Steward, Lawrence Livermore National Laboratory, permeation in metals Bill Stwalley, University of Iowa, atom recombination C. A. Swenson, Iowa State University, solid heat capacity Dean Taylor, Los Alamos National Laboratory, helium in liquid hydrogen David White, University of Pennsylvania, crystal growing Nick Winter, Lawrence Livermore National Laboratory and Orrin Fackler, Rockefeller University, helium tritide calculations Dave Young, Lawrence Livermore National Laboratory, continuing advice on equation of state I would especially like to thank John Simpson, whose superb beta spectrum was done just for this book. I would like to thank my wife Maggie for putting up with the lost time that went into this—2,000 hours, 93 percent of which was my own time. Marcia Litterst did a superb typing job on the manuscript, and although she would rather work for James Michener, I would rather be in Hawaii. Clark Souers Livermore, California May 1984

1. Introduction

Hydrogen Fusion Reactions Controlled hydrogen fusion offers promise as a long-term source of energy. Large experimental machines are under construction now, and commercial reactors are expected in the next fifty years. Although today's machines are the size of medieval cathedrals, their energies provide only a flicker of the fire needed to fuse nuclei. Only heavy hydrogen burns at all, and only for an instant. To create nuclear fusion with our limited earthly energies, we must choose from a handful of reactions. The nuclear reactions that start burning at the lowest temperatures are shown in Fig. l . l . 1 The efficiency of the reaction is given by the cross section as a function of the kinetic energy of the reactants. The most efficient reaction is D + T

He 4 + n + 1.70 TJ/mol (17.6 MeV) 4

(1.1) 2

where D is a deuteron, T is a triton, He an alpha particle, n a neutron, and TJ is 1012 J. (In this chapter, H, D, and T refer to nuclei; in most chapters, these letters will refer to atoms). This reaction, once it goes, really lays out the energy, which comes from the difference in total mass of the reactants and products. It may be calculated from Einstein's E = (AM)c 2 relation, where E is energy, AM the change in molecular weight, and c the speed of light. 3 The molecular weights are listed in Table l . l 4 ' 5 The He 4 and neutron products each carry off equal momentum, so that their share of the total energy is proportional to the molecular weights. The He 4 carries 0.34 TJ/mol (3.5 MeV) and the neutron 1.36 TJ/mol (14.1 MeV). It takes 108 J/mol (1 KeV) just to start off the D-T reaction. This is a lot of energy, as shown by the temperature equivalent (E = RT, where R is the gas constant and T is the temperature) along the top of Fig. 1.1. Present fusion machines can reach 30 MK, but the maximum cross section does not occur until 700 MK. At low excitation energies, the second-best fuel combination is a pair of deuterons: D + D - + T + H + 0.384TJ/mol (3.98 MeV)

(1.2)

and D + D

He 3 + n + 0.314 TJ/mol (3.25 MeV)

(1.3)

2

INTRODUCTION

Temperature (millions of degrees K)

10 2

10

10 3

10 4

Energy (J/mol),

I

I

1

I

10

I

10

I 2

10 3

Energy (keV) Fig. 1.1. The five most easily activated hydrogen fusion reactions (from S. L. Greene, Jr.).

where H is a proton. The two reactions proceed at equal rates. 6 At 30 MK, the D-D reactions are 100 times fewer than those of D-T. The D-D cross section increases with excitation energy, but it does not equal that of D-T until a temperature of 1010 K. Above an excitation energy of 3 GJ/mol (30 keV), the second-best reaction is He 3 + D

He 4 + H + 1.77 TJ/mol (18.3 MeV)

(1.4)

This reaction cannot challenge D-T below 1010 K. Two lower probability reactions are: T + T - He 4 + 2n + 1.09 TJ/mol (11.3 MeV)

(1.5)

3

INTRODUCTION

T A B L E 1.1 BASIC PARAMETERS OF THE HYDROGENS

Species

Molecular weight, M ( x IO"3 kg/mol)

H D T

1.007825 2.014102 3.016050

eH 2 nH 2 HD HT eD 2 nD 2 DT T2

2.015 650 2.015 650 3.021 927 4.023 875 4.028204 4.028204 5.030152 6.032100

He 3 He 4 n

3.016030 4.00260 1.008665

Ii-1'2

(mol/kg)1'2

— —

44.5474 44.5474 38.5841 36.3840 31.5119 31.5119 28.7760 25.751 1 — — —

Triplepoint temp. (K)

Triplepoint pressure (Pa)

Criticalpoint temp. (K)

Criticalpoint pressure (Pa)

none none none

none none none

none none none

none none none

13.804 13.956 16.60 17.70° 18.69 18.73 19.79" 20.62

7030 7200 12370 14 580" 17130 17150 20 080a 21600

32.976 33.19 35.91 37.13" 38.262 38.34 39.42" 40.44"

1.293(6) 1.315(6) 1.484(6) 1.571(6)" 1.650(6) 1.665(6) 1.773(6)* 1.850(6)"

none none none

none none none

3.32 5.20 none

1.16(5) 2.29(5) none

A?ff

(dimensionless) — — —

1.731 1.731 1.471 1.354 1.224 1.224

1.111 1.000 3.08 2.67 —

NOTE: The superscript (a) indicates estimates. The numbers in parentheses are powers of ten.

and He 3 + T - He 4 + D + 1.38 TJ/mol (14.3 MeV)

(1.6)

Equation 1.5 is the third-best reaction at low excitation energies but drops to number four at high energies. Five of the six reactions just listed convert heavy hydrogen (D and T) into helium, a transmutation of the elements. Equation 1.2 is unique in being a rearrangement of hydrogen isotopes. We have presumed above a thermonuclear approach to fusion—namely, that sheer power raises the temperature of hydrogen until the nuclear burn begins. The same equations hold for electronuclear fusion, in which one component is electrically given the necessary activation energy in an accelerator. The hardware is still large, the beam current low, and the need for the same sheer energy still required. There is, finally, muon-catalyzed fusion that can be used for any combination of hydrogen isotopes. The easiest, once again, is D-T, and we may alter Eq. 1.1 to read D + T + n~ - He 4 + n + fi~ + 1.70 TJ/mol (17.6 MeV)

(1.7)

where is the muon. The negatively charged muon takes the place of the electron in forming the ion (/i _ DT) + , which is analogous to D T + . However, the muon mass is 207 times greater than that of the electron, so that the Bohr orbit is 207 times closer to the nucleus. The muon then pulls the two nuclei close enough for fusion, while it itself escapes to be available for another reaction. 7,8 However, muon lifetimes are microseconds, and an

4

INTRODUCTION

accelerator of energy greater than 50 MeV is required to make them in the first place. Despite these problems, this approach is gaining interest in current fusion research. It is clear that D-T is the best possible fuel for hydrogen fusion. Present estimates suggest that a commercial D-T reactor may be running by the year 2030. Beyond then, with improved technology, the D-D and He 3 -D combinations may be used. These manufacture tritium via Eq. 1.2, and worse yet, one needs tritium to manufacture the He 3 . If fusion is an energy source for the centuries, its first period may be the Tritium Age, 2030-2080. Besides providing power on earth, the D-T reaction might be used to drive the first starship. The 3.5 K cold of deep space would naturally favor cryogenic fuel. The conceptual starship Daedalus, supposedly to be ready by 2050, is to be powered with He 3 and D, where the He 3 is somehow to be mined from the atmosphere of Jupiter. 9 The early starships, however, might need to breed their own tritium from an onboard lithium-6 supply. It is estimated that 0.5 million tons of natural lithium could be available early in the next century from pegmatite deposits in North Carolina. 10 We may dream ahead to the ultimate form of fusion: H + H - > D + e+ + v D + H

He 3 +

y

He 3 + He 3 -»He 4 + 2H

(1.8) (1.9) (1.10)

where e + is a positron, v a neutrino, and y a gamma ray. 11 The net reaction from Eqs. 1.8 through 1.10 is 4H

He 4 + 2.58 TJ/mol (26.7 MeV)

(1.11)

This is the hydrogen-fusion-from-seawater reaction that could power the earth for millennia. Unfortunately, the key step—Eq. 1.8—proceeds with a cross section 10~25 smaller than all the other reactions we have considered.12 Our sun is very good at this, but any mass production of this step on earth is hopeless with our present technology. Perhaps in the twenty-second century, a starship will suck hydrogen from deep space and burn it in some fabulous engine. Tritium The two isotopes that nature has most favored for fusion are relatively recent discoveries in science. Deuterium was found in 1931 by a group headed by Harold Urey. 13 The researchers distilled a large quantity of natural hydrogen in order to obtain a small amount of heavy hydrogen. The existence of hydrogen-3 was considered, but the heavier hydrogen was not found. Tritium was created by Lord Rutherford and others in 1934 by starting with deuterium. 14 Accelerated deuterons hitting deuterium produced tritium, as shown by the output of neutrons. The later discovery of tritium's short-lived radioactivity 15 ' 16 explained why it is not found in nature.

5

INTRODUCTION

This completes the hydrogen family, because hydrogen-4 exists only as a transient species, which decays to tritium and a neutron. 17 Let us return to the D - T fuel needed at the present time. Deuterium is cheap, because it occurs in nature in 0.015% abundance (the rest is H). 1 8 The commercial cost is about 250 per gram. Deuterium is not radioactive and it is regularly found in space by spectroscopic observations. Not so with tritium, which is not available on earth, except possibly as a trace-level result of cosmic bombardment in the upper atmosphere. Tritium has never been seen in space; its presence would probably mean that an alien spaceship had piled up. It is, in essence, a man-made isotope, which is manufactured in fission reactors by the reaction Li 6 + n - H e 4 + T

(1.12)

Thus, use of tritium requires the fission technology that fusion is trying to supplant. However, a tritium-breeder reactor can produce a neutron by Eq. 1.1 which produces a new triton by Eq. 1.12. Lithium, however, is not an abundant metal and Li 6 is the rarer isotope, being in only in 7.4% abundance (the rest is L i 7 ) . 1 9 All this makes tritium cost several thousand dollars a gram.* The final problem with tritium— which we shall consider many times in this book—is its radioactivity. It disintegrates by the reaction T - He 3 + + v (1.13) where p~ is a beta particle and v an antineutrino. The decay half-life is 12.3 years. 20 T and He 3 are "mirror nuclei"—one has two neutrons and a proton and the other has the opposite. One of the tritium neutrons turns into a proton. If the neutron were free, the half-life would be 12 minutes. 21 The presence of two other nuclear particles stabilizes the neutron by a factor of 500 000. We pause to summarize the possible nuclear reactions of heavy hydrogen. The D-T reaction is the most efficient, but tritium has to be made in a nuclear reactor and then carefully handled because of its radioactivity. One of its reaction products, the He 4 , is electrically charged and its kinetic energy is directly convertible into electrical energy. The other product—the neutron—is not, and it reacts with nearby materials (like stainless steel) to produce radioisotopes far more dangerous and long-lasting than tritium. The D-He 3 reaction produces two charged particles, and this is the favorite reaction of direct-electrical-energy proponents. No neutron is produced. The direct-conversion technology exists in experimental form, but the desired nuclear reaction is of low efficiency. Worse yet, the He 3 comes by the tedious decay of T, which, in turn, locks the entire technology to fission reactors. This leaves the D-D reaction, which produces one neutron and three charged particles. Two of these particles are T and He 3 , which then burn in the D-T and D-He 3 reactions. Overall, then, we get four charged particles (two He 4 's and two p's) and two neutrons. Thus, * Tritium is available from the Isotope Sales Group, Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830. The average price in 1985 was $1.25 per curie, with there being 10 000 curies in a gram of pure tritium.

6

INTRODUCTION

the neutron problem remains. The tritium must be contained, but at least it is just passing through, as is the He 3 . No fission reactor is necessary, and all the deuterium is available from the ocean. Finally, the D 2 is a pure component with properties that are fairly well known. The D-D reaction would be a sure winner, if only it were more efficient. Let us return to the tritium decay reaction of Eq. 1.13. The mass difference between T and He 3 yields a total decay energy of 1.80 GJ/mol (18.6 keV). This energy can be given, in any combination, to the beta particle and the antineutrino. The latter speeds on its way across the universe, perhaps to be halted by some interstellar cloud or star after uncountable years. Only the beta particle is visible to us, and it appears with a continuous spectrum of energy from zero to the total decay energy. Its mean energy is 0.549 GJ/mol (5.69 keV). There are no gamma rays, so that tritium decay is the least energetic radioactive transformation known. Most but not all of tritium's irritations are the result of the low-energy beta particle, which travels only a short way, wreaking damage as it goes. It is ironic to consider that its partner, the neutrino, will probably be traveling through space for eons to come. As a gas, tritium is much less dangerous than most radioisotopes because hydrogen does not readily mix with the human body. As HTO or T 2 0 , tritium is perhaps 10000 times as dangerous as the element, because the water can easily mix with body fluids.22 The half-life of tritium in the body is about ten days, and this time is said to be shortened by drinking beer in quantity. In any case, tritium must be contained, and, more specifically, kept away from oxygen. This leads to the technology of tritium containment, 23-25 which generally consists of boxes of varying degrees of expense. Tritium handling is generally learned firsthand at one of the tritium laboratories. Should the cost of tritium containment continue to escalate at the present rate, however, humanity may decide to wait for the D-D reaction. There are three ways to package D-T fuel for experimental fusion machines. The first is as gas, usually at room temperature. This is the easiest, although it can require high pressures, and reaction with the vessel walls produces impurities. The gas is used in almost every present method of fueling. The second form of fuel is rarely used—it is hydrogen in a compound, usually a hydride. Present fusion machines barely ignite D-T; the presence of a diluent is usually fatal. Moreover, most hydrides of light metals have unpleasant properties. The third fuel form is frozen D-T. This is dense, relatively pure, exists at low pressure, and hopefully can be cast into particular shapes in particular places. It must be cold, however (4 to 20 K), and its major drawback is the refrigerating equipment. Nevertheless, it is beginning to enter the fusion scene in inertial confinement targets. The Hydrogen Family Hydrogen forms diatomic molecules. The three isotopes (H, D, and T) combine to give six hydrogens: H 2 , HD, HT, D 2 , DT, and T 2 . This sequence is that of their molecular weights, with H 2 the lightest and T 2 the heaviest. The three same-atom

INTRODUCTION

7

hydrogens (H 2 , D 2 , and T 2 ) are often called "homonuclear;" the three differentatom hydrogens (HD, HT, and DT) are called "heteronuclear." The hydrogens H 2 , HD, and D 2 are not radioactive, and we expect a lot of data. Indeed, there is a large quantity of data on H 2 and D 2 , but not for HD. Until recent times, HD had to be synthesized and purified; now only the latter step is necessary. Even allowing for the inconvenience of cryogenic distillation, it is difficult to understand why so little HD data exists. Much of what does exist is thirty years old, and it often does not mesh well with the comparable H 2 and D 2 values. The Ukrainian group at Kharkov is the only current laboratory that measures H 2 , HD, and D 2 together to get good isotopic comparisons. The tritiated hydrogens, HT, DT, and T 2 , have been very unpopular because of their radioactivity. No one has ever prepared HT or DT in bulk, possibly because they were thought to be unstable. As a result, not even their triple-point temperatures have been measured. Studies of hydrogen mixture properties have been unpopular, too. The hydrogens are so similar that excess mixture effects are difficult to measure. On the other hand, the hydrogens are not quantum-mechanical enough (like He 3 and He 4 ) to stir up interest. The most interesting potential combination, that of H 2 and T 2 , involves the handling of radioactive material.

The Basic Processes in Hydrogen We may obtain another point of view on the hydrogen family by considering briefly the various ways that they absorb energy. We order these into four regions according to the magnitude of the energy, starting with the lowest energy and working up: 1. The "solid state" region, at 10 to 100 J/mol (10~4 to 1(T3 eV) or 1 to 10 K. This is represented by the quadrupolar interaction of the electric quadrupole moments of rotationally excited molecules to form an ordered antiferromagnetic state. This low-energy electric interaction takes place between H 2 , D 2 and T 2 molecules, which can be metastably trapped in the first excited rotational state. This field of study has been active in universities for the last fifteen years, and it furnishes the most elegant physics in the hydrogen field. With the exception of thermal conductivity effects, it is probably not too important for most fusion considerations. 2. The "molecular" region, from 100 to 1 000 J/mol (10~3 to 10~2 eV) or 10 to 100 K. Several processes fall into this area: a. Molecular motion and crystal lattice vibration. The kinetic energy possessed

by molecules at normal temperatures causes them to move about. b. Molecular condensation. Hydrogen gas liquifies at temperatures below 33-40 K; it solidifies at 14-21 K. The heat of vaporization is roughly 1 kJ/mol and the heat of fusion about 100 J/mol. The forces that hold hydrogen molecules in the condensed form are weak.

8

INTRODUCTION

c. Molecular rotation. These would be of little interest if H 2 , D 2 , and T 2 did not stick in the first excited rotational state upon cooling, rather than drop to the ground state. This provides endless corrections for instrumental techniques sensitive to rotational levels. 3. The "atomic" region, about 104 to 105 J/mol (0.1 to 1 eV) or 103 to 104 K. Three processes are evident: a. Molecular vibration—the oscillation of the atoms in the molecule relative to each other. Excited vibrational states are more chemically reactive. b. Dissociation—the breakup of molecules into atoms, followed by their recombination. This occurs thermally in hot gas or in the radiation damage caused by tritium. c. Solid state defects—ions and electrons trapped as defects in the irradiated solid. The application of energy in this range will empty the traps and cause recombination. 4. The "ion" region, about 106 J/mol (10 eV) or 10s K. Here, hydrogen ionizes into ions and electrons, caused in tritium's case by its beta particle. The recombining charged species progressively lose their energy of recombination by dropping through processes of ever lower energy. The complete tale of tritium decay begins in the ion region, falls through the atomic to the molecular, and in the solid, to the quadrupolar region.

Reviews of Past Work Hydrogen properties have been summarized almost from the discovery of heavy hydrogen. Many are handy to have around, even though they may be somewhat dated. Three reviews fall into the textbook format. Farkas's excellent book of 1935 was possibly written too soon, 2 6 because the famous Woolley review became the standard reference textbook for almost three decades. 27 A concise, more modern work with considerable heavy hydrogen data is also available. 2 8 , 2 9 Several extensive reviews fall into the handbook category. The American ones are by the National Bureau of Standards in Boulder. 3 0 - 3 2 Computerized lists of all available references are presented, rather than the selected references listed by reviews in the textbook category. The fluid equation of state, which first appears in Woolley's work, grows to extensive and detailed proportions. A Soviet counterpart also exists. 3 3 , 3 4 A recent work by Silvera complements the others nicely. 35 This covers solidstate thermodynamics, potentials, and ordered rotational moments. It summarizes the elegant quadrupolar studies of the past two decades.

Extrapolating with the Quantum Parameter The upshot is that a lot of data exist for the non-tritiated hydrogens and almost none for the tritiated ones. To get ready for D-T work, it is clear that we will have to do a

INTRODUCTION

9

lot of estimating. D-T properties fall into two categories: 1. Properties affected by tritium radioactivity. This includes electrical properties in all phases, defects in the solid, chemical exchange, gas blocking effects caused by He 3 , and probably mechanical properties of the solid. The only known quantity is the tritium decay rate in the sample. The rate at which the sample anneals out the radiation damage is usually unknown and must be measured. If data is not available, one can but guess. 2. Properties not affected by tritium radioactivity. This includes most of them— for example, equation-of-state and transport properties. These may be estimated by taking H 2 and D 2 data (HD if available) and extrapolating to the tritiated hydrogens. The results are probably not bad, although there is no way to be sure without doing the measurement. Most extrapolations are unashamedly empirical. Even so, what function does one use? The hydrogens differ in mass, so that molecular weight is an obvious parameter. These are listed in Table 1.1 with other pertinent parameters. 36 Because the molecules are diatomic, the reduced molecular weight is important in predicting many spectroscopic properties. The reduced molecular weight is

where M a and M b are the molecular weights of the atoms in the molecule. The inverse square root, fi~ 112 , is the function of interest in describing molecular vibration and gas kinetics. Other possible properties are the triple-point and critical-point temperatures and pressures. The triple-point temperatures are the best known of this group. Even so, two of these are estimates. They change slightly with the rotational composition of the hydrogen (note the difference between eH 2 and nH 2 —we shall define the nomenclature in a moment). This makes the 9-, very useful in adjusting phase boundaries of a particular hydrogen in the vicinity of the triple point. For most substances, the ratio of triple-point to critical-point properties are the same. This is the Law of Corresponding States. If P, V, and T indicate pressure, molar volume, and temperature, and the subscript "cp" the critical-point value, then P/P c p , V/V cp , and T/T c p should be the same for every material. This is not true for the hydrogens, because of the presence of quantum mechanical zero-point energy— unreachable kinetic energy possessed even at 0 K. The Quantum Law of Corresponding States adds a fourth parameter: the quantum parameter, A*. This is a dimensionless parameter that defines how great the quantum-mechanical effect is. A large A* means a large effect; zero means no effect at all. The quantum parameter is defined in terms of the Lennard-Jones potential energy well, which describes the forces between two like molecules. 37 This potential energy is (1.15)

10

INTRODUCTION

where r is the intermolecular distance and e and a are constants. The constant £ is the maximum well depth (in J/mol), which is a negative number and which occurs at an intermolecular separation of 21/6

7A

— v=0Zero-point

(3.78)

vibration

Fig. 2.1. Vibrational and rotational energy levels of the isolated T 2 molecule.

17

PERFECT G A S THERMODYNAMICS

Vibrational Energy Molecules vibrate at discrete energies described by the vibrational quantum number v, where v = 0, 1, 2, This is illustrated for an isolated T 2 molecule in Fig. 2.1, which shows the v = 0, 1, and 2 levels on the left hand side. The v = 0 state is the ground state, and we define it to be zero energy. At 0 K, all molecules are in the v = 0 state. This is basically still true even at 300 K. Several thousand degrees are required to thermally produce a significant fraction of excited vibrational states. However, we extensively use infrared spectroscopy on the hydrogens and stimulate these transitions with radiation. The ground state has the vibrational quantum number v = 0, the first excited state v = 1, and so on by ones. The energy (in J/mol) of the v th vibrational level is E(vib, v)* = N c h{v e (v + ! ) - (v e x e )(v + ¿) 2 + (v e y e )(v + ¿) 3 }

(2.6)

The asterisk means that we have not yet adjusted our reference to 0 J/mol for v = 0. N„ is Avodagro's number (6.022 x 1 0 " molecules/mol), which is added to give energy per mol, h is Planck's constant (6.626 x 10~ 34 J • s), and ve, (v e x e ), and (v e y e ) are vibrational spectroscopic constants in Hz. 1 We list the available data in Table 2 . 1 , 2 - 4 with the other hydrogens estimated by the isotope rules. 5 If v = 0, Eq. 2.6 becomes E(zero)* = N

0

h { ^ - ^

+

^

}

(2.7)

This energy is the molecular zero-point energy of vibration. This quantum mechanical energy is built into the vibrating molecule even at 0 K, where there is no thermal energy. This energy is thermally inaccessible. It is also very large, as shown in Table 2.1. For us, it is easier to think of 0 J/mol as the ground state energy, and the thermally accessible energy of the v th vibrational level becomes E(vib, v) = E*(vib, v) - E(zero)*

(2.8)

This is the meaning of the asterisk. TABLE 2.1 VIBRATIONAL SPECTROSCOPIC CONSTANTS OF THE HYDROGENS

Spectroscopic constants (Hz)

H2

HD HTA D2 DT" T2°

(vEXE)

(v e y e )

E(zero)* (kJ/mol)

0(vib)

ve 1.3195(14) 1.1428(14) 1.0776(14) 9.3320(13) 8.5210(13) 7.624 3(13)

3.638(12) 2.720(12) 2.412(12) 1.789(12) 1.468(12) 1.131(12)

2.44(10) 1.58(10) 1.33(10) 8.63(9) 6.57(9) 4.71(9)

25.96 22.53 21.26 18.44 16.85 15.10

5987 5226 4942 4 308 3950 3 551

(K)

NOTE: The molecular zero-point energy E(zero)* is included. 0(vib) is the "temperature" corresponding to the first excited vibrational state. Three hydrogens (a) are estimated.

18

PERFECT GAS THERMODYNAMICS

The first term in Eq. 2.6 describes the harmonic oscillation of the molecules. The next two terms correct for the deviation from harmonic motion by the actual molecules. At room temperature, almost all molecules are in the ground vibrational state. We may use this approximate formula to estimate the small fraction c(v) that are thermally excited to the higher levels even at this low temperature. 6 We have c(v) ~ exp[ — v0(vib)/T]

(2.9)

0(vib) is the "temperature" of the first (v = 1) excited state—namely, E(vib, 1)/R. These temperatures are listed in Table 2.1. We need only this single constant because the vibrational energy levels are equally spaced. F o r H 2 at 300 K, c ( l ) ~ 2 x 10" 9 ; for T 2 , it is 7 x 10" 6 .

Rotational Energy In between the vibrational levels are the molecular rotational energy levels, as seen also in Fig. 2.1. They are labeled by the rotational q u a n t u m number J = 0, 1, 2, . . . , where the series concludes by running into the next higher vibrational level. The energy of the J t h rotational level, as referenced to J = 0 for a given v, is E ( r o t , J ) = N 0 h { B v J ( J + 1) - D e J 2 ( J + l ) 2 } = R 0 ( r o t , J )

(2.10)

where E is the energy in J/mol and the rotational constants Bv and D e are in Hz. Also, 0(rot,J) is the corresponding "temperature" in degrees K for the rotational level. 7 We use temperature because the rotational levels of the v = 0 state are low enough to be significantly populated at our temperatures of interest. N o t e that E(rot, 0) = 0 J/mol, so that we have no zero-point rotational energy with a reference change to worry about. However, we d o have to add the energy of the vibrational level to which the rotational state belongs. Our true 0 J/mol is for the v = 0, J = 0 state. The rotational constants change slightly according to the vibrational state they represent; this is the reason for the v-subscript. We may define three more general rotational constants B e , a e , and ye such that Bv = B e - a e (v + i ) + y e (v + i ) 2

(2.11)

D e can be also broken into v-dependent terms, but these corrections are so small that we shall not use them. Spectroscopic data are available for H 2 , 8 H D , 9 H T , 1 0 D 2 , u D T , 1 2 and T 2 1 3 — p e r h a p s the only example in this book where every hydrogen is represented. These are listed in degrees K and also in H z in Table 2.2. The first three rotational levels of the v = 0 state are also given. The first term on the right in Eq. 2.10 is the rigid rotator energy. F o r this we assume the bond between the two hydrogen atoms is a rigid rod and the molecule turns about an axis at right angles to the rod. This basic term does not produce equally spaced energy levels. Instead, the rotational levels become more widely

19

PERFECT GAS THERMODYNAMICS

TABLE 2.2 MOLECULAR ROTATIONAL SPECTROSCOPIC CONSTANTS IN DEGREES K AND H Z

Degrees K BE H2 HD HT D2 DT T2

87.554 5 65.6630 58.399 43.800 36.528 29.253

A«

TE

4.406 2.806 2.339 1.528 1.168 0.8426

0.0830 0.0468 S 0.0370° 0.0207" 0.0144» 0.0092

DE 0.067 1 0.0379 0.0299 0.0167 0.012 0.0074

Hz BE H2 HD HT D2 DT T2

1.8243(12) 1.3682(12) 1.2168(12) 9.1263(11) 7.6111(11) 6.095 2(11)

—

se c tt. O

C ÛH .2 Q 152 rOn

SO r- 0 vn On m t~- 0 t-; ON — O vo VO 00On — m 0 0 Ö — — vo — o 1, Q has a limiting value of 2. Figure 6.5 shows calculated Q values for transparent aluminum oxide spheres of various radii as a function of the wavelength of incident infrared radiation. 37 The dispersive refractive index varies from 1.09 at 10 f i m to 1.77 at

84

SOLID PROPERTIES

Infrared wavelength (jum) Fig. 6.5. Light scattering efficiencies calculated for transparent aluminum oxide spheres of various radii as a function of infrared wavelength. (From G. N. Plass; see Chap. 6, note 37.)

0.5 fim, with absorptive refractive indexes varying from 5 x 1(T 4 to 10~ 6 . We see that significant scattering occurs when the wavelength and particle size become comparable. If this is so, and the crystallites are tightly packed, then even a thin layer (e.g., where L ~ a ~ X ~ 1 /zm) will significantly scatter light. What will make them so effective is the formation of a solid wall in front of the light. If the same number of crystallites are spread over a volume with space in between, n c decreases and the radiation transmitted will increase. Tritium radiation produces a small amount of light by radioluminescence (see Chap. 22). Should the sample be a clump of microcrystals, this light should bounce around inside the mass until most of it is absorbed.

85

SOLID PROPERTIES

0.2

0.4

Stress under tension (MPa) Fig. 6.6. Stress-strain curves under tension for solid nD 2 at various temperatures. Points A indicate the yield stress; point B is the necking stress. We are in region I—that of rapid initial deformation.

Mechanical Properties Little has been done in this area because of the difficulty of putting moving parts, necessary to mechanical properties measurements, into a cryostat. Almost all the data have been measured on normal H 2 and D 2 polycrystalline samples, with an estimated grain size of 1 to 1.5 mm, under tension. Figure 6.6 shows the "steady-state" stress-strain curves for nD 2 . 3 8 Stress is the pressure that causes the resulting strain, which is here the percent elongation of the sample. These samples are being pulled at both ends and are under tension. The measurements are presumably taken after a few minutes, when the rapid (region I) deformation has taken place but before long-term creep occurs to any great extent. Some well-known features of elasto-plastic deformation can be seen in these curves. 39 From the origin to point A, we are in the linear region, where removal of stress should restore the sample to its original dimensions. In this region, the longitudinal

86

SOLID P R O P E R T I E S

T A B L E 6.5 STRESS-STRAIN PROPERTIES OF F R O Z E N POLYCRYSTALLINE H Y D R O G E N

Necking point

nH 2

nD 2

Temp. (K)

Young's modulus E(MPa)

Yield stress (MPa)

2 3.5 4.2 6 8 10 12

200 100 70 30 20 10 5

0.18 0.20 0.20 0.20 0.18 0.05 0.025

1.4 4.2 8.0 11.6 15.6 16.4

420 270 110 90 80 40

0.42 0.29 0.22 0.16 0.08 0.05

Coefficient of hardening

Stress (MPa)

Strain

(%)

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

1.3 0.6 —

—

—

—

—

—

— —

7 7

0.49 0.47

1.2 1.4

—

—

—

—

Maximum % strain measure 0.3 1.0 1.5 2.0 1.9 3.3 4.2 0.6 1.0 3.0 6.2 13.0 22.3

NOTE: These "steady-state" curves are taken after a "short" equilibration time.

strain e, is related to the stress • 00, all modes are saturated, and the 24.94 J/mol • K is 3R—three degrees of kinetic freedom and three of potential freedom.

98

THERMAL PROPERTIES OF THE SOLID

At the triple-point temperatures of 13.80 K for J = 0 H 2 and 18.69 for J = 0 D 2 , we use V s values of 23.312 and 20.44 /z(m3)/mol.29 We take 0D ~ 100 K for both, if we use the relation C/Vs2 = 9R0D/8, we get E(trans) ~

^

-

(7.14)

We calculate 11 and 39 J/mol for H 2 and D 2 at the triple points. The actual values are 18 and 61 J/mol, so that our simple model is already breaking down. We next consider the thermal pressure, which, from the Debye equation of state, is 30 P(trans) * ^

^ *s

(7.15)

The constant y is the Griiniesen constant and here we set it equal to 2. For H 2 at the triple point, P(trans) = 1.54 MPa. Equation 7.10 gives P(potential) ~ —1.60 MPa. The two together add to about 0, and the true vapor pressure is 7 kPa. Note that 250 MPa terms are subtracted to obtain the potential energy pressure, and MPa terms are subtracted to get the final pressure. The theory has to be fine-tuned indeed to give the right answer. Equation 7.15 applies to a real situation of interest. As discussed in Chapter 6, it takes 5 to 10 MPa to force crystallization at 4 K with guaranteed crystal-to-wall contact. As we go from H 2 to D 2 , it takes more pressure to do this.31 We are squeezing the sample from the triple point to the 0 K molar volume, so that only P(trans) is involved. The heavier the hydrogen, the farther we have to press, because E(trans) is larger. For H 2 and D 2 , we calculate 1.5 and 5.9 MPa, where the equation of state gives 2.4 and 9.0 MPa. 3 2 T 2 is expected to require the highest pressure at about 14 MPa. This is the pressure needed to prevent thermal expansion as the solid warms from 0 K to the triple point without melting. Let us return to the H 2 volume of 10 /¿(m3)/mol. At 0 K , we have already seen that the potential energy resists by requiring 2.4 GPa pressure. Now, let us heat up the solid to the melting point at about 182 K . The Debye temperature is about 460 K, 3 3 so that T/0D ~ 0.40. We use Table 7.2 to estimate E(trans) as being about 1 600 J/mol. Then P(trans) a: 0.32 GPa, close to the actual value of 0.22 GPa. 3 4 Even here, the translational pressure is much less than the potential energy pressure. This simple model only illustrates the thermodynamics of the solid. It is not possible to derive a simple set of constants that makes a really good fit. One reason has to do with the Gruneisen constant, which is defined as

This turns out to be the exponent of our zero-point energy term, and y is 2 because we have used a V s ~ 2 relation for 0 D . In reality, y can be a variable, about 2,35 but such a move completely destroys the simplicity of our model. Also, the V s " 8 / 3 hard-core

THERMAL PROPERTIES OF THE SOLID

99

potential term is completely empirical but happens to give pretty good results. As we shall see in the next section, the solid is not a true Debye lattice and 0 D is not well defined. There are, in effect, two adjustable parameters, 0D and y. Simplifying one complicates the other. For more accurate calculations, a more complicated potential energy than that of Eq. 7.9 is available. 36 The liquid experiences such large volume changes with temperature that simple potential equations do not work at all.

The Debye Temperature The Debye temperature may be obtained and used in several ways. Figure 7.2 shows 0 D for saturated solid J = 0 H 2 . We shall consider the various curves. a. Measured from C P and corrected with the compressibility.31,36 is nearly constant: 2.47 ± 0.05. 39

The value of y

b. From measured C v . 4 0 This converts directly into 9D without a compressibility and should be better than C P data. However, these data were taken under pressure and had to be extrapolated back to the saturation curve. c. From sound velocity and density, using the equation 4 1 (7.17) where h is Planck's constant (6.626 x 1 0 - 3 4 J • s), k is Boltzmann's constant (1.381 x 10" 23 J/K), N 0 is Avogadro's number, and U is the sound velocity. The sound velocity data are listed in Chapter 6. d. From collision-induced infrared spectroscopy.42 It is taken from the difference between the Q! (0,1) and the Q-phonon peaks. The lattice vibration spectrum is reproduced as a sideband to the molecular fundamental vibration-rotation spectrum. e. From P-V-T measurements and reset into the Silvera equation of state. 43 Here 0 D is defined as a function of volume only, and the Gruneisen parameter y is given most of the variability. We finally consider the calculated lattice vibrational modes of solid H 2 , shown in Fig. 7.3. 44 The Debye distribution, with 0 D = 100 K, is set on top. It is evident that hydrogen has a bimodal distribution that differs considerably from the Debye spectrum, so that 0 D is, in reality, not well defined. Figure 7.3 shows why the heat-capacity-derived values have a greater temperature dependence than the others. Heat capacity measurements excite only the lowT/0 d part of the spectrum. There are actually fewer modes there than expected from the Debye spectrum, so that 0 D seems larger than it really is. If the actual density of the modes' spectrum starts catching up with the Debye spectrum at higher temperatures, the apparent value of d D will decrease. Infrared spectroscopy excites all the

100

THERMAL PROPERTIES OF THE SOLID

Fig. 7.2. The Debye temperature 0 D as determined by different means, which are described in the text. Lines a) and b) are the best thermal values for the saturation curve.

modes at once, so that the derived 8D may be said to better represent the entire spectrum. Thermal Conductivity An important thermal property is the solid thermal conductivity. Measured values for saturated solid H 2 , HD, and D 2 are shown in Fig. 7.4 from near 0 to 14 K . 4 5 - 5 1 They are an extreme function of the J = 1 concentration, c(l), which is listed for each curve. The data from early references have not been in agreement, especially at low

101

THERMAL PROPERTIES OF THE SOLID

Temperature (K) Fig. 7.3. The density of the crystal lattice vibrational modes. The simple form is the Debye model, here set with a cutoff frequency of 0 D at 100 K. The carefully calculated spectrum is much more complicated. (Courtesy O. Schnepp and the American Physical Society.)

c(l) values. We believe that the higher values of K for a given c(l) are more likely to be correct. The Fig. 7.4 data should not be taken as exact because of unknown differences in impurities and crystal quality. R. B. Bohn, whose data have resolved many of the discrepancies, points out that the wall-contact problem in thermal conductivity has never been explained. 52 As we discussed in Chapter 6, hydrogen crystals under saturation are likely to pull away from the vessel walls or crack badly. This becomes more likely as the temperature drops to 4 K. Yet all thermal conductivity measurements have been made under saturated conditions, and no one has remarked on any notable difficulties. In fact, the 16-to-16.5 K nD 2 values of 0.38 W/m • K at saturation 53 and at 39 MPa 5 4 are in agreement. There seems to be no wall-contact problem in these thermal conductivity measurements, but it is hard to see why not. Let us return to Fig. 7.4. The highest conductivity occurs with the lowest J = 1 concentration. For c(l) = 0.0034, K is as large as that of room-temperature tungsten, but only at 3.5 K. For c(l) = 0.70 H 2 (almost normal), the curve has a modest rise to a small peak at about 10 K. The D 2 curve fits in at about the right places with the H 2 data. The c(l) = 0.033 D 2 is the normal solid with a peak at 8 K. Some highertemperature data are also available. For saturated nD 2 , K drops from 0.36 to

102

THERMAL PROPERTIES OF THE SOLID

Materials at room temperature

Temperature (K) Fig. 7.4. Thermal conductivity of solid H 2 , HD, and D 2 as a function of the listed J = 1 concentration. Room-temperature conductivities of familiar materials are shown at the right for comparison. 0.31 W / m - K as the temperature increases from 16 to near 18 K . 5 5 For J = 0 H 2 under 8.8 to 20 M P a pressure, K drops from 1.0 to 0.86 W / m - K from 15 to 17 K . 5 6 Let us consider the big conductivity peak of the c(l) = 0.0034 sample. The low temperature rise is caused by the increase in the number of phonons as the temperature increases. The thermal conductivity is K ^ iC s (trans)p s UA

(7.18)

where C s (trans) is the crystal lattice heat capacity, ps the density, U the speed of sound, and A the phonon mean free path. We find for the c(l) = 0.0034 H 2 that only C s (trans) has a large temperature dependence, and this is the source of the initial

THERMAL PROPERTIES OF THE SOLID

103

upswing of K with increasing temperature. At 1.3 K, X ~ 3.0 x 10~4 m, and at the 3.2 K peak, X ~ 1.7 x 10~4 m. In a perfect crystal, X should equal the crystal size, and phonons should bounce from one wall to another. Most sample sizes are tens of millimeters—that is, a hundred times larger than the measured X . We would expect U to be larger in J = 0 H 2 than in nH 2 . It appears, then, that the conductivity is limited by some internal crystallite scattering mechanism even in low-c(l) H 2 . A 10 mm perfect crystal (i.e., with no J = 1 molecules at all) might have a maximum thermal conductivity of 20 000 W/m • K! Now, let us consider the abrupt drop in the thermal conductivity as we continue to heat the c(l) = 0.0034 H 2 sample. For temperatures between 5% and 10% of the Debye temperature, enough phonons have been created that they interfere with one another and impede the flow of heat (called an Umklapp process). In this region, we expect the thermal conductivity to follow the form K~Taexp(0D/2T)

(7.19)

where a is a constant to be determined. 57 An exponential fit of our declining conductivity data yields an exponential temperature constant of about 36 K. We would expect 0 d /2 to be 45 to 50 K, depending on how it is measured. Our data, then, somewhat resemble Eq. 7.19 in their behavior. We recall that they are apparently modified by the scattering mechanism already present. However, the phonon interference process is probably there, as indicated by every sample falling on the same line at temperatures from 12 K on up. It is obvious that J = 1 molecules are superb destroyers of heat flow. Their effect is to reduce X: for c(l) = 0.70 H 2 , X drops from 170 to 70 nm as the temperature increases from 2 K to the triple point. What would be the smallest phonon mean free path imaginable from the data? For nD 2 at 17 K, with K ~ 0.35 W/m • K, X ~ 20 nm. This is still about 6 intermolecular distances, even though every J = 1 molecule has J = 1 neighbors. Even with help from the phonon interference process, the quadrupolar process cannot shut down the flow of heat totally. At 3 K, where phonon interference is not yet important, X is about 100 nm as a result of the quadrupolar process alone, if all molecules are of the J = 1 type. The thermal conductivity of solid T 2 has been crudely measured by nuclear magnetic resonance. 58 The value is 1 ± 0.5 W/m• K from about 8 K to the triple point. The sample began as normal but was probably close to being equilibrium by the end of the run. A similar value is found for D-T solid mixtures. This number is close to the 0.35 to 0.7 W/m • K found for H 2 and D 2 near the triple point. It appears that the tritium radioactivity does not affect the thermal conductivity, at least over several hours of experiment. The reason is that the phonon mean free path is already 100 to 1 000 nm in most samples because of the J = 1 concentration. Not until enough radiation damage occurs to physically disrupt the crystal on a smaller size scale will the thermal conductivity be affected. For tritiated solid hydrogen, we suggest a curve that is based on that of c(l) = 0.70 H 2 . We suggest using

104

THERMAL PROPERTIES OF THE SOLID

K ~ 0.01T 2

0 < T < 8K

(7.20)

This is set to be about 0.7 W/m • K at 8 K. The n.m.r. measurements were too crude to see small changes, but we would expect K to decline somewhat as the temperature increases. K ~ 14T-1'3

10 K < T < Triple/Point

(7.21)

This is set to deliver 0.7 W/m • K at 10 K and 0.35 W/m • K at 17 K.

Thermal Resistance at an Interface We next consider the thermal resistance for solid hydrogen in intimate contact with a cold metal wall. The Debye temperature 0 D of the H 2 is only 100 K, whereas for the wall material, it may be thousands of degrees K. If these materials are ultrasonically excited, there will be an acoustic mismatch at the boundary, because the hydrogen cannot carry high-frequency modes. 5 9 ' 6 0 However, at low temperatures, T < 0 D . The higher frequencies are not thermally excited and a large mismatch is not expected. The heat flow, dH/dt, for a sample with thermal conductivity K, area A, and thickness L is dH/dt = KAAT/L

(7.22)

where AT is the temperature difference causing the heat flow. The thermal resistance Ris R = L/KA

(7.23)

This has been measured for solid H 2 and D 2 under various pressures. 61 No obvious trend emerges and we shall cite one of the higher-resistance results. For H 2 copper near saturation pressures, RA(m 2 • K/W) sa 5.3 x 10" 3 T" 2

(7.24)

as measured from 1.1 to 2.5 K. Similar results are presented to a maximum temperature of 5 K. How important is this thermal resistance? We list RA values in Table 7.3, where the values from 10 to 20 K are extrapolated. Next to it, we list the comparable solid hydrogen values—namely, K - 1 — a s taken from Eqs. 7.20 and 7.21. As listed, column 3 implies a sample thickness of 1 m. We must multiply K _ 1 by the actual sample thickness in order to compare it with the boundary resistance. The samples used in the measurement were only 60 /an thick. We see that we must have an extremely thin hydrogen sample before the boundary and sample resistances are comparable. What could matter more is the solid pulling away from the walls to leave a gas space. In Table 7.3, we list K _ 1 for D 2 gas, as taken from Table 3.1 and extrapolated downward. The extrapolation is dubious and assumes kinetic behavior even at low temperatures. But the point is that a crack that is 1 % as wide as the solid sample will

T H E R M A L PROPERTIES OF THE SOLID

105

TABLE

7.3

T H E R M A L RESISTANCES OF THE S O L I D H Y D R O G E N - C O P P E R W A L L INTERFACE, S O L I D H Y D R O G E N , AND D

Temp. (K)

2

G A S , U S I N G KINETIC VALUES

Solid-wall RA(m 2 • K/W)

1 2 4 10a 15» 20"

5x 1 x 3 x 5x 2x 1 x

Solid K" 1 (m-K/W)

GasK" 1 (m-K/W)

100 25 6.2 1.4 2.4 2.5

10000 2 500 1000 250 140 100

10"3 10"3 10"4 1(T5 10"5 10"5

NOTE: All values are approximate and for rough comparison only. The presence of He 3 in DT would lower the thermal resistance of the gas considerably.

offer comparable thermal resistance, in the simplest analysis. For D-T, this may not be a problem because of the ever-present He 3 that now takes over the heat conduction in the crack (this effect is described in Eq. 14.14).

Thermal Diffusivity The thermal conductivity of solid hydrogen is used for steady-state calculations. For cases where equilibrium is not yet established, we need the thermal diffusivity D (in m 2 /s), where D = K/C s (trans)p s

(7.25)

No measured values exist, so that we must calculate D. The results for c(l) = 0.0034 H 2 show a decline of D from 10" 1 m 2 /s at 3 K to 10" 5 m 2 /s at 12 K. For c(l) = 0.70 H 2 , D is 10" 6 m 2 /s at 3 K, rises to 10~5 m 2 /s at 8 K, and declines to 3 x 10" 6 m 2 /s at 14 K. As an example, the approximate time to come to thermal equilibrium is T ~ L 2 /D

(7.26)

where L is the sample thickness. For a 10 mm crystal with D ~ 10" 8 m 2 /s, T ~ 3 hours—a surprisingly long time.

Tritium Heat Effects Tritium puts out a considerable quantity of radioactive decay heat. Because of this, calorimetry is the standard technique for determining the amount of tritium in a sample. When a gas sample is liquified or frozen, the amount of heat per unit volume is truly surprising. The heat from pure T 2 is 1.954 W/mol (0.324 W/g). The 4 K solid has a density of about 53 400 mol/m 3 for a heat production of 104000 W/m 3 . Pure T 2 has 2.146 x 10 15 Bq/mol (disintegrations/mol • s), which leads to the very hot number

106

THERMAL PROPERTIES OF THE SOLID

at 4 K of 1.15 X 10 20 Bq/m 3 (3 800 Ci/cc).* A sample of 50-50 D-T is half as bad: 0.977 W/mol, 51 000 W/m 3 , and 5.56 x 10 19 Bq/m 3 , as based on an estimated 4 K solid density of 51 800 mol/m 3 . Let us first consider heat flow in one dimension. The top surface of a solid D-T slab at x = L is pinned to the cryostat temperature of T(cold). The rest of the slab hangs down and is perfectly insulated. The bottom face at x = 0 will have the hottest temperature of T(hot). The heat flux at any point x is 62 dT - K — = A0x dx

(7.27)

where A 0 is the internal heat generation in W/m 3 . If K is not a function of T, then we integrate to get T(hot)-T(cold)=^(L2-x2)

(7.28)

For solid D-T, if T(cold) = 4.2 K and the other end is melting for T(hot) = 20 K, and using K ~ 1 W/m • K, L will then be 25 mm. The temperature profile in the sample is quadratic. The experiment in which K was determined for solid tritium had the above onedimensional geometry. 63 The sample was held in glass, which has a thermal conductivity of about 0.1 to 0.15 W/m • K. The glass carried off enough leakage heat to spoil the assumption of perfect insulation. The use of nylon or teflon (0.002 to 0.03 W/m • K) would have been better if a reliable seal to the metal could have been made. The glass tube was surrounded by a vacuum space and a steel secondary container in case the tritium got out. To squeeze in the frozen tritium with good contact, stainless steel walls would be needed for safety, but stainless steel has a thermal conductivity about that of solid tritium and this destroys the validity of this particular geometry. Let us consider a cryogenic cell with a copper roof and otherwise perfect insulation for all other walls. We fill it to a depth of 25 mm with liquid D-T in contact with the copper roof. If we use the 0.1 W/m • K thermal conductivity of liquid H 2 , we calculate a temperature gradient of 156 K! With this condition, the bottom will vaporize and the liquid will stir itself. With convection currents, the liquid's effective K is 10 or 20 W/m • K—maybe more. Liquid tritium, in bulk, is a thermal short circuit in comparison to the solid. The experiment with the tritium in glass showed another interesting feature. 6 4 The cold was at the top, as was the fill tube. As the temperature fell below the freezing point, a plug of the solid formed across the top, blocking the input line and sealing in the liquid under some pressure. The liquid was boiling and delivering its heat to the solid above with near-perfect efficiency. We take x = L to be the position of the cold * A curie (Ci) is 3.7 x 10 1 0 Bq (disintegrations/s) Tritium handlers are no more eager to switch to SI metric than anybody else.

107

THERMAL PROPERTIES OF THE SOLID

block, x = b to be the solid-liquid boundary, and x = 0 to be the bottom of the liquid. We assume the liquid to be so conductive that it instantly moves all its heat to the solid-liquid boundary. The temperature in the solid is now given by T(hot) - T(cold) =

Ao(L 2 X2) 2 k~

+

A

°b(^~X)

(7.29)

where b is the length of the liquid, and L < x < b. For a cylindrical cell, the outer surface will have the temperature T(cold) and the center (ignoring end effects) will be at T(hot). We integrate from the surface at radius a to the center r = 0 to obtain 65 T(hot) - T(cold) = ^

(7.30)

Suppose the cylinder is hollow with an inner radius b and a center material that is a perfect insulator. Now the difference in temperature between the outer (cold) and inner surface is T(ho,)-TMd)

=

A

'ia'-">4K2b''°Wb)»

(7.31)

The outer surface is cooler because it has more area for heat conduction. This geometry uses the tritium as a tube heater. If end losses of heat can be minimized, Eq. 7.31 can be used to measure K. This would require a small temperature sensor in the middle (e.g., a thermocouple) and a cylinder that is long compared to its radii. In spherical geometry, we have T(hot) - T(cold) =

A

°(aJ~ 6K

r2)

(7.32)

where T(hot) is at r and T(cold) is the surface of the sphere at r = a. If the center is hollow at r = b, the temperature difference between the two surfaces is AT = ^ M| aa 2 — b 2 — 2b 3 (-j- — 6K|_ tb a

(7.33)

Only the first term is positive and the next two reduce the temperature difference. This would be an even better way to measure K, if a small nonconducting wire could be run into the center section for a temperature sensor.

Notes 1. C. Kittel, Introduction to Solid State Physics, 4th ed. (John Wiley, N e w York, 1971), pp. 100-103. 2. J. W. Stewart, J. Chem. Phys. 40, 3297 (1964). 3. B. A. Younglove, J. Chem. Phys. 48, 4181 (1968). 4. V. S. Kogan, Y. Y. Milenko, and T. K. Grigorova, Physica 53, 125 (1971).

108

THERMAL PROPERTIES OF THE SOLID

5. J. H. Constable, C. F. Clark, and J. R. Gaines, J. Low Temp. Phys. 21, 599 (1975). 6. G. E. Childs and D. E. Diller, Adv. Cryogenic Eng. 15, 65 (1969). 7. T. Larsen, Z. Physik 100, 543 (1936). 8. See reference in note 1 above, pp. 210-218. 9. J. E. Mayer and M. G. Mayer, Statistical Mechanics (John Wiley, New York, 1940), p. 251. 10. See note 8 above. 1 1 . K . Clusius and E. Bartholome, Z. Physik Chem. 30B, 327 (1935). 12. H. W. Woolley, R. B. Scott, and F. G. Brickwedde, J. Res. Nat. Bureau Standards 41, 379 (1948). 13. H. L. Johnston, J. T. Clarke, E. B. Rifkin, and E. C. Kerr, J. Amer. Chem. Soc. 72, 3933 (1950). 14. R. W. Hill and B. W. A. Ricketson, Phil. Mag. 45, 277 (1954). 15. R. W. Hill and O. V. Lounasmaa, Phil. Mag. 4, 785 (1959). 16. G. Ahlers, Some Properties of Solid Hydrogen at Small Molar Volumes, Lawrence Berkeley Laboratory Report UCRL-10757, Berkeley, Calif. 94720 (1963). 17. G. Grenier and D. White, J. Chem. Phys. 40, 3451 (1964). 18. G. Ahlers, J. Chem. Phys. 41, 86 (1964). 19. J. H. Constable, A. Q. McGee, and J. R. Gaines, Phys. Rev. IIB, 3045 (1975). 20. O. D. Gonzalez, D. White, and H. L. Johnston, J. Phys. Chem. 61, 773 (1957). 21. I. N. Krupskii, A. I. Prokhvatilov, and G. N. Shcherbakov, Soviet J. Low Temp. Phys. 9,42(1983). 22. See note 21 above. 23. I. N. Krupskii, A. I. Prokhvatilov, and G. N. Shcherbakov, Soviet J. Low Temp. Phys. 10, 1 (1984). 24. I. F. Silvera and V. V. Goldman, J. Chem. Phys. 69, 4209 (1978). 25. A. Driessen, J. A. deWaal, and I. F. Silvera, J. Low Temp. Phys. 34, 255 (1979). 26. See note 25 above. 27. See note 8 above. 28. Jahnke-Ende-Lösch Tables of Higher Functions, 6th ed., revised by F. Lösch (McGraw-Hill, New York, 1960), pp. 290-294. 29. See notes 21 and 23, respectively. 30. See reference in note 1 above, 2d edition (1956), pp. 153-155. 31. W. N. Hardy, University of British Columbia, Vancouver, British Columbia, Canada kV6TlW5, private communication (1979). 32. See note 25 above. 33. See note 25 above. 34. See note 25 above. 35. See note 25 above. 36. See note 25 above. 37. See notes 18, 21, and 25 above. 38. V. G. Manzhelii, B. G. Udovidchenko, and V. B. Esel'son, Soviet J. Low Temp. Phys. 1, 384(1975). 39. See note 21 above. 40. J. K. Krause and C. A. Swenson, Phys. Rev. 21B, 2533 (1980). I would like to thank Professor Swenson for a preprint. 41. See note 8 above. 42. H. P. Gush, W. F. J. Hare, E. J. Allin, and H. L. Welsh, Can J. Phys. 38, 176 (1960). 43. See note 25 above. 44. K. N. Klump, O. Schnepp, and L. H. Nosanow, Phys. Rev. IB, 2496 (1970). 45. R. W. Hill and B. Schneidmesser, Z. Phys. Chem., Neue Folge 16, 257 (1958). 46. R. B. Bohn and C. F. Mate, Phys. Rev. 2B, 2121 (1970). 47. J. H. Constable and J. R. Gaines, Phys. Rev. 8B, 3966 (1973). 48. V. B. Kokshenev, Soviet J. Low Temp. Phys. 1, 395 (1975). 49. J. E. Huebler and R. G. Bohn, Phys. Rev. 17B, 1991 (1978).

THERMAL PROPERTIES OF THE SOLID

109

50. R. B. Bohn, University of Toledo, Toledo, Ohio 43606, private communication (1981). 51. B. Y. Gorodilov, I. N. Krupskii, V. G. Manzhelii, and O. A. Korolyuk, Soviet J. Low Temp. Phys. 7, 208 (1981). 52. See note 50 above. 53. D. E. Daney, Cryogenics 11, 290 (1971). 54. L. D. Ikenberry and S. A. Rice, J. Chem. Phys. 39, 1561 (1963). 55. See note 53 above. 56. R. F. Dwyer, G. A. Cook, and O. E. Berwaldt, J. Chem. Eng. Data 11, 351 (1966). 57. R. Berman, "The Thermal Conductivity of Dielectric Solids at Low Temperatures," in Advances in Physics (Taylor and Francis, London, 1953), Vol. II, pp. 109, 117. 58. J. R. Gaines, R. T. Tsugawa, and P. C. Souers, Phys. Lett. 84A, 139 (1981). 59. J. S. Buechner and H. J. Maris, Phys. Rev. Lett. 34, 316 (1975). 60. J. C. A. van der Sluijs and M. J. van der Sluijs, J. Low Temp. Phys. 44, 223 (1981). 61. C. L. Reynolds, Jr. and A. C. Anderson, Phys. Rev. 14B, 4114 (1976). 62. H. S. Carslow and J. C. Jaeger, Conduction of Heat in Solids, 2d ed. (Clarendon Press, Oxford, 1959), pp. 130-132, 191, 232. 63. See note 58 above. 64. See note 58 above. 65. See note 62 above.

8. Thermodynamics of the Saturated Phase Boundary

The saturated phase boundary is the great highway down which we move as a cryostat containing hydrogen cools. At the high-temperature end, it starts at the critical point, where liquid and vapor are the same phase. As we cool, we pass down the liquid-vapor line, through the triple point, and down the solid-vapor line to 0 K. In a real experiment, we may never come close to the critical point, but we will surely hit the saturation curve somewhere. We must now consider the various energies and heats.

Heats of Transformation We have seen that the saturation curve is two-sided, with vapor on one side and either liquid or solid on the other. We may go from the latter to the former phase by supplying the heat of transformation. For solid to vapor, it is the heat of sublimation H(sub), and for liquid to vapor, it is the heat of vaporization H(vap). At the triple point only, the heat of fusion H(fus) equals H (fus) = H(vap) - H (sub)

(8.1)

The heat of transformation is related to the saturated vapor pressure by way of the Clausius-Clapeyron equation: 1 H(vap) = RT 2 z(l - p/p L )dln P s /dT

(8.2)

where z is the compressibility, p the gas density, pL the liquid density, and Ps the pure component vapor pressure. A similar equation exists for H(sub) using ps and Qs. The derivative contains the various constants we used in Chapter 4 to describe the vapor pressures. For the three-component equations (Eq. 4.24), H(vap) = Rz(l - p/pL)[B + CT]

(8.3)

As we approach 0 K, z -»1, pL » p, and H 0 (sub) = RB

(8.4)

111

THERMODYNAMICS OF THE SATURATED PHASE B O U N D A R Y

T A B L E 8.1 TOTAL CALORIMETRIC D A T A AVAILABLE FOR THE HEATS OF FUSION H ( f u s ) AND THE HEATS OF VAPORIZATION

H (fus) (J/mol)

Chap. 8 ref. no.

nH 2 0.06 H 2 eH 2

117 117 117

2 3 4

nD 2 0.022 D 2 0.02, 0.33, 0.8 D 2 nD 2

197 197

5 6

197 199

7 8

Hydrogen

Hydrogen

T(K)

H(vap) (J/mol)

nH 2

14.8 16.3 16.5 17.5 18.9 19.9 20.3

916 919 919 918 913 907 899

eH 2

Ref. 2

H(vap) H(vap) (J/mole)

Chap. 8 ref. no.

Hydrogen

T(K)

eH 2

24.4 26.3 28.1 29.6 31.0 31.8 32.7

859 812 747 614 495 407 244

9

nD 2 0.022 D 2 eD 2

19.7 23.6 24.2 26.8 28.6 30.5 32.5 34.1 35.4 36.6 37.5

1265 1230 1202 1 154 1098 1029 938 830 726 629 502

5 6 9

4

NOTE: "Equilibrium" (e) means the J = 1 distribution at 20.4 K. The number in front of H 2 or D 2 is the J = 1 concentration, c(l).

F r o m the bracketed term in Eq. 8.3, we expect H ( v a p ) to increase linearly with temperature, and it does, at low gas densities. Finally, the 1 — p/pL term takes over a n d H ( v a p ) decreases. A t the critical point, p = pL a n d H ( v a p ) = 0. T h e v a p o r pressures would a p p e a r to offer a g o o d way of obtaining heats of t r a n s f o r m a t i o n , b u t they d o n o t work very well. It is difficult to get d a t a so good that the derivative is also accurate. In general, the only numbers that can be trusted are the calorimetric ones, and these are listed in Table 8 . 1 . 2 - 9 T h e d a t a of Simon a n d Lange, d a t i n g f r o m 1923, are p r o b a b l y the oldest believable experimental work in this b o o k . T h e effect of the first excited rotational state (J = 1) is small. F o r H 2 at 0.1 M P a (20.4 K), the extrapolated value for the n o r m a l f o r m 1 0 is 903 J/mol, as c o m p a r e d to a measured 899 J/mol for the equilibrium f o r m . 1 1 N o difference at all is seen in the heat of fusion for D 2 measured by calorimetry with different rotational populations. 1 2 Let us c o m p a r e the calorimetric a n d vapor-pressure-derived heats. W e use the v a p o r pressures a n d densities of chapter 4 to derive H ( s u b ) a n d H ( v a p ) at the triple point. F r o m this, we calculate H ( f u s ) values, which we c o m p a r e with the calorimetric values. T h e results, in J / m o l , are:

112

THERMODYNAMICS OF THE S A T U R A T E D PHASE B O U N D A R Y

Vapor pressure

nH 2 HD nD 2 T2

H (fus)

Solid H (sub)

Liquid H(vap)

Vapor pressure

Calorimetry

1032 1283 1 523 1638

919 1 112 1282 1455

113 171 241 183

117 159 199 233 (est)

(8.5)

Only the H 2 H(fus) numbers agree, and we see we will have to be very cautious with vapor-pressure-derived data. These problems get worse as the gas density over the liquid increases, and the solid vapor-pressure data are not trustworthy for a derivative at all. In Table 8.2 we list the best heats of sublimation and vaporization that we can T A B L E 8.2 HEATS (ENTHALPIES) OF SUBLIMATION AND VAPORIZATION OF THE PURE-COMPONENT LIQUID HYDROGENS

Heats of sublimation and vaporization (J/mol)

T.P. T.P. T.P.

nH 2

HD

HT"

nD 2

DT"

nT 2

H (sub) H(vap) H (fus)

1028 911 117

1270 1 110 159

1370 1190 178

1470 1270 199

1580 1360 218

1690 1460 233

0 4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

770 850 890 920 960 1000 911* 919 917 906 880 850 810 740 600 380

970 1060 1090 1130 1 170 1210 1240 1260 1110* 1100 1090 1050 1000 940 850 720

1050 1 140 1 180 1220 1260 1290 1320 1350 1 190* 1 190 1180 1150 1100 1040 960 840 690 580

1 150 1240 1270 1310 1350 1380 1420 1440 1470 1270* 1250 1220 1170 1 120 1050 960 840 760 670 570

1250 1340 1370 1410 1450 1490 1520 1540 1560 1 360* 1340 1330 1290 1230 1150 1060 940 860 770 660

1370 1450 1480 1520 1560 1590 1620 1650 1670 1690 1450* 1420 1390 1330 1260" 1 160" 1060" 960" 860" 760"

—

—

—

—

—

—

—

—

—

—

NOTE: The asterisk (*) indicates the transition from solid to liquid. H(fus) is the heat of fusion. H(vap) at the critical point is 0 J/mol. Most numbers are rounded off to 10 J/mol. The heats of sublimation at 0 K contain quadrupolar energy for the three normal species. DT is here molecular, but the same numbers can be used for the D-T mixture. The superscript (") indicates estimates. T.P. indicates the triple point.

113

THERMODYNAMICS OF THE SATURATED PHASE BOUNDARY

come up with for the various hydrogens. Only a small part is based on calorimetry. Many of the higher temperature values are from vapor pressures or estimates. These heats of transformation are enthalpies, because the process takes place at constant pressure. The energy of vaporization is E(vap) = H(vap) - P(V - V L )

(8.6)

The second term represents the expansion of the liquid of volume VL into the vapor of volume V.

The Dilute Gas The vapor above the solid is almost a perfect gas, so that the translational internal energy is 3RT/2 and the translational enthalpy is 5RT/2, where the difference of RT equals PV. These energies are referenced to 0 J/mol for the dilute 0 K gas. We must remember to add the rotational energy E(rot) to each value above to get the total internal energy E or enthalpy H. We recall from Chapter 2 that E(rot) = H(rot). We use only the thermally accessible rotational energy, not the J = 1 residual energy that can be had only with a catalyst. At slightly higher pressures, we may use the second virial coefficients B and C, as described in Chapter 4. We will obtain a non-ideal energy, E(non-ideal), which is negative and will partially offset the translational and rotational energies. At constant volume 13 RT 2 TdB E(non-ideal)=-—|_-

+

1 dCl - - J

^

The total internal energy is then E(gas) = 3RT/2 + E(non-ideal) + E(rot)

(8.8)

The corresponding heat capacity is: Cy(non-ideal) =

RT / dB d2B\ 1 / dC „,d 2 CY 22 — 42 — + T —2^ 1 T —2 4 -T — V V dT *dT J 2V dT ' *dT J

(8.9)

The total heat capacity is the sum of 3R/2, C v (non-ideal), and C v (rot). The non-ideal enthalpy may be obtained by adding PV to the internal energy. We get H(non-ideal) = E(non-ideal) - 0 - l^PV

(8.10)

The enthalpy will be a larger negative number than the energy, because z < 1. The use of the preceding equations is none too good because an accurate derivative dB/dT is difficult to measure well. When densities increase beyond the validity of the two virial coefficients, the situation worsens.

114

THERMODYNAMICS OF THE SATURATED PHASE BOUNDARY

Internal Energy Calibration We still have most of the saturation curve, at higher densities, to go. We must now turn to the equation of state (E.O.S.). The National Bureau of Standards has continuously worked on one for H 2 . 1 4 - 1 8 The P-V-T data necessary for this have been measured but not reported in raw form. Excellent constant volume heat capacity (C v ) data are available from 15 to 90 K and to pressures of 34 MPa. 1 9 The translational internal energy—that is, no E(rot)—may be calculated from 2 0 E(trans) = J [ T ( j ^ - P]dV

(8.11)

E(trans) = J c v ( t r a n s ) d T

(8.12)

or

For the liquid, the saturated heat capacity C s is available as well, 21 and we may calculate 22 E(trans) = J c s ( t r a n s ) d T - j V d V

(8.13)

We shall use the H 2 E.O.S. extensively, 23 although differences of opinion occur even here. Our H(vap) values, derived from the data of Table 8.1, are 5% higher than the E.O.S. values at 26 and 28 K. But most importantly, we have at least some data to support our E.O.S. estimates in the difficult critical-point region. For D 2 , there is much less data, and a National Bureau of Standards E.O.S. also exists. 24 We shall not use this for our work, because of the differences in projecting the phase line that we saw in Chapter 4. For the vapor, the E.O.S. compressibilities are lower than our estimates by 4% at 26 K and 12% at 35 K. This can shift a given internal energy value by 2 K when one integrates Eq. 8.11. Another area of P-V-T difference of opinion occurs deep in the liquid below 30 K. In the absence of data, each of us seeks to span the unknown areas with what we each believe to be the best function. We will now work from the triple point to the critical point in the vapor phase and return to the triple point in the liquid phase. The critical point is a special reference point because E(trans.,c.p.) = 0

(8.14)

where our 0 J/mol reference is the dilute gas at 0 K. If we return to the non-ideal translational internal energy, then E(non-ideal, c.p.) = 3RT/2

(8.15)

It is E(non-ideal) that we shall derive. We use the method of Chapter 4 and fix all the hydrogens to the function we have for H 2 . We use the dimensionless temperature

115

THERMODYNAMICS OF THE SATURATED PHASE B O U N D A R Y

T A B L E 8.3 LIQUID-VAPOR PHASE BOUNDARY POINTS FOR LINES OF CONSTANT N O N - I D E A L TRANSLATIONAL INTERNAL ENERGY, E ( n o n - i d e a l )

Saturation temp (K) E(non-ideal) (J/mol) -10 -50 -100 -200 -300 -400 -500 -600 -700 -750 -795 -800 -900 -1000 -1100 -1120 - 1 150 -1200

Saturation pressure (MPa)

nH 2

nD 2 a

DT a

nH 2

nD 2 a

DT"

17.8 23.8 27.5 30.7 32.1 33.0 b 31.7 31.0 26.8 22.8 14.0C

21.5 28.0 31.9 35.1 36.6 37.6" 38.2 37.3 36.4 36 35 35 32 28 20.5 18.7e

22.4 29.1 32.6 36.1 37.7 38.6" 39.3 38.5 38 37.5 37 37 34.5 31 27 25.5 24 19.8e

0.043 0.24 0.52 0.86 1.1 1.3" 1.1 0.97 0.49 0.21 0.007 e

0.052 0.32 0.63 1.1 1.3 1.5" 1.6 1.45 1.3 1.2 1.05 1.05 0.65 0.30 0.035 0.017 e

0.055 0.34 0.67 1.1 1.4 1.65" 1.7 1.55 1.45 1.4 1.3 1.3 0.87 0.47 0.20 0.135 0.090 0.020 e

— — — — — —

—

—

—

— — — — — —

—

—

—

NOTE: The superscript (") indicates estimates. Points downward to ( b ) are on the vapor side of the phase boundary; points past ( b ) are on the liquid side. The last point ( c ) is the triple point.

variable a of Eq. 4.31. The function ft (Eq. 4.32) is reset in terms of E(non-ideal) instead of compressibility. For nH 2 , for example, E(non-ideal) decreases from 0 for the triple point vapor to —415 J/mol at the critical point to —795 J/mol for the triple point liquid. For nD 2 , the corresponding numbers are 0, —480, and — 1120 J/mol; for DT, they are 0, —490, and —1200 J/mol. We use then the H 2 function E(nonideal) = E(a) to calculate the energies for the other hydrogens. Table 8.3 lists the phase boundary points for nH 2 , nD 2 , and D-T. When we have the non-ideal energy, we can calculate the total internal energy according to Eq. 8.8. Then, the enthalpy is H = E + PV

(8.16)

The PV values for the saturated vapor are listed in Table 8.4 and for the saturated liquid in Table 8.5. Enthalpy and Internal Energy We next calculate the saturated enthalpies. We calculate at each temperature the vapor phase translational enthalpy H(trans gas). By leaving out the rotational

116

THERMODYNAMICS OF THE SATURATED PHASE B O U N D A R Y

TABLE 8.4 SUMMARY OF SATURATED VAPOR P V S OF THE PURE-COMPONENT HYDROGENS

Vapor PV(J/mol) Temp. (K)

nH 2

HD

HTa

nD 2

DTa

T2

T.P.

113

136

145

152

161

167

4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

35 50 67 83 100 114* 128 141 151 159 164 165 163 152 129

35 50 67 83 100 114 130 145* 158 169 176 181 184 181 173 155

35 50 67 83 100 115 130 146* 159 171 181 188 190 191 186 173 163

35 50 67 83 100 115 130 147 161* 173 185 192 197 198 196 188 180 166 152

35 50 67 83 100 115 130 147 162* 174 186 195 202 204 204 199 194 186 176

35 50 67 83 100 115 130 148 163 175* 187 199 206 209 211 208 204 199 192

99

102

104

C.P.

— —

—

—

—

—

—

—

—

86

93

96

NOTE: The upper pressures are over the solid; from the asterisk down, over the liquid. The numbers in parentheses are powers of ten. The triple-point (T.P.) temperatures are: 13.956, 16.60,17.70,18.73,19.79, and 20.62 K. The critical-point (C.P.) temperatures are: 33.18, 35.91, 37.13, 38.34, 39.42, and 40.44 K. The HT and DT values are estimates ("); so are much of the near-critical-point T 2 data. For the D-T mixtures, use the DT values. energy, which varies considerably from one hydrogen to the next, we can smooth the values across the hydrogens. We have H(trans gas) = E(non-ideal) + 3RT/2 + PV

(8.17)

Next, we accept the H(vap) values of Table 8.2 as the best we are likely to get. Then we calculate the liquid enthalpy H*(trans liq) using H*(trans liq) = H 0 (sub) + H(trans gas) - H(vap)

(8.18)

The asterisk is added to indicate that we have changed reference energies: we are using 0 J/mol for the 0 K solid. We may switch between the two types of reference energies using the heat of sublimation at 0 K, H 0 (sub), which we have yet to get. The value of having the two references is that the solid/liquid and vapor energies can be started at the handy value of 0 J/mol in each case. Figure 7.1, in the preceding chapter, shows the relation between the references. All we need do now is add in

117

THERMODYNAMICS OF THE SATURATED PHASE B O U N D A R Y

TABLE

8.5

SATURATED LIQUID P V VALUES

Liquid PV(J/mol) Temp. (K)

H2

HD

HT"

D2

DT a

T2a

1 1 3 5 12 19 28 34 40 51

0 1 2 4 7 10 16 24 30 36 44

1 2 3 5 9 14 21 25 31 38

99

102

104

T.P. 14 16 18

20 22 24

26 28 30 32 34 35 36 37 C.P.

0 1 1 3 5

8

1 1 3 5

8

13 20 30 49

12

19 28 43

86

93

0 1 2 4 6 10 16 24 35 43

96

8

NOTE: All solid PVs are 0 J/mol. The superscript (a) indicates estimates. T.P. is the triple point and C.P. is the critical point.

E(rot) and subtract PV to convert enthalpy to internal energy. The results are listed in Tables 8.6 to 8.9. We should also mention that because H*(liq) lies at the end of a chain of calculations, we have smoothed the results. Most numbers are rounded off to the nearest 5 J/mol. For this reason, the various additions above will not quite add up when compared from table to table. How did we get our energy reference changes, H 0 (sub)? The gas over the solid is essentially ideal, so that its enthalpy is 5RT/2. The solid enthalpy is derived from the saturated heat capacity. We may determine H 0 (sub), then, from the translational enthalpies, using: H 0 (sub) = H*(trans sol) - H(trans gas) + H(sub)

(8.19)

We have changed to solid enthalpies, including the heat of sublimation. We obtain these values: H 0 (sub) (J/mol) nH 2 HD HT nD 2 DT T2

770 970 1050 1 150 1250 1370

(8.20)

118

THERMODYNAMICS O F THE SATURATED PHASE BOUNDARY

T A B L E 8.6 TOTAL ENTHALPY H ( g a s ) FOR THE SATURATED VAPOR

Total gas enthalpy, H (gas) (J/mol) nH 2

HD a

HT a

nD 2 a

D-Ta

nT 2 a

T.P.

290

346

369

384

428

418

4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

87 125 166 208 249 291* 328 355 381 400 410 420 400 340 230

87 125 166 208 249 291 332 372* 400 430 455 470 490 480 450

87 125 166 208 249 292 335 376* 412 444 480 500 520 530 510 470 420

87 125 166 208 249 291 333 374 400* 430 460 480 495 500 490 455 420 390 350

87 125 165 209 251 296 343 390 432* 460 510 540 570 585 595 585 565 540 480

87 125 166 208 249 291 333 374 410 440* 470 500 520 540 540 530 520 500 450

CP.

86

107

220

130

—

—

—

—

—

—

—

—

-

—

176

210

NOTE: The reference energy is 0 J/mol for the dilute 0 K gas. D-T is an abbreviation for 25% nD 2 -50% DT-25% nT 2 . Only the thermally accessible rotational energy is included. The numbers above the asterisk indicate solid in equilibrium; from the asterisk down, liquid. The superscript (a) indicates estimates. T.P. is the triple point and C.P. is the critical point.

We carry these values over to the liquid calculations. They should equal RB, according to Eq. 8.4. Four of the solid enthalpies and internal energies are shown with the superscript "b." This refers to the quadrupolar energy, E(quad), that occurs only in solid homonuclear hydrogens containing the J = 1 form (see Chap. 24). From 4 K to the triple point, this energy amounts to 10 to 14 J/mol for nH 2 , 3 to 4 J/mol in nD 2 , and 10 to 15 J/mol for nT 2 . Because D-T is really 25% n D 2 - 5 0 % D T - 2 5 % nT 2 , it also has an E(quad) of 3 to 5 J/mol. This energy is added to the translational and rotational energies to increase the solid total. H*(sol) = H*(trans sol) + E(rot) + E(quad)

(8.21)

The quadrupolar energy comes out of the potential energy and changes the 0 K heat of sublimation to a larger, corrected value H 0 (sub, quad) = H 0 (sub) + E(quad)

(8.22)

119

T H E R M O D Y N A M I C S O F T H E S A T U R A T E D PHASE B O U N D A R Y

TABLE

8.7

TOTAL ENTHALPY OF THE SATURATED SOLID

H*(sol)

AND LIQUID

H*(liq)

Total solid and liquid enthalpy, H* (J/mol) nH 2 b

HD a

HT 1

nD 2 " b

D-Tab

nT2"'b

T.P. sol T.P. liq

36 153

42 201

53 231

65 264

102 320

96 329

4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

10 13 15 18 23 153* 180 210 245 290 330 380 430 510 620

0 1 2 5 10 19 35 230* 265 310 375 440 520 600 700

0 1 2 5 10 20 35 237* 270 315 380 450 530 620 720 830 890

3 5 7 9 14 23 37 56 280* 330 390 460 525 600 680 765 810 870 930

3 4 6 9 17 29 51 76 320* 370 430 500 590 685 785 895 955 1020 1070

10 13 15 19 24 33 50 69 89 360* 420 480 560 650 750 840 930 1010 1060

C.P.

856

1470

1500

—

—

—

—

—

—

—

— —

1 146

1260

1257

NOTE: The reference energy is 0 J mol for the 0 K solid. The superscript (a) indicates estimates; ( b ) indicates that quadrupolar energy is included in the solid. The numbers above the asterisk are solid; from the asterisk down, liquid. T.P. and C.P. are the triple point and the critical point.

Let us further consider the effects of E(quad) in calibrating our energies. E(quad) lowers the 0 K potential energy, so that nH 2 will be bound by 14 J/mol more than eH 2 . If we reference ourselves to 0 J/mol for the 0 K gas, the 0 K binding energy of nH 2 will be 14 J/mol more negative than that of eH 2 . If we reference ourselves to 0 J/mol for the 0 K solid, we start at the same apparent reference point. But the extra 14 J/mol must be added to all nH 2 energies in the solid above 10 K and in the liquid. For the solid at the triple point, the quadrupolar energy levels are saturated with the 14 J/mol, so that the potential energies of nH 2 and eH 2 are the same. This is why the heats of fusion are also the same. If we reference ourselves to 0 J/mol for the 0 K solid, then both nH 2 and eH 2 appear to start at the same point. But at the triple point, the nH 2 has acquired an extra 14 J/mol. We note that the Silvera solid E.O.S. values of H 0 (sub) are 747 J/mol for J = 0 H 2 and 1104 J/mol for J = 0 D 2 . 2 5 From the preceding discussion, we would expect about 760 and 1 110 J/mol for the J = 0 species. This is close to the Eq. 8.20 value for

120

T H E R M O D Y N A M I C S OF THE SATURATED PHASE B O U N D A R Y

TABLE

8.8

TOTAL INTERNAL ENERGY OF THE SATURATED VAPOR E ( g a s )

Total gas internal energy, E (gas) (J/mol) nH 2

HD a

HT a

nD 2 a

D-T a

my

T.P.

177

210

224

232

267

251

4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

52 75 100 125 150 177* 200 214 230 240 245 255 235 190 100

52 75 100 125 150 177 202 227* 242 260 280 290 305 300 275

52 75 100 125 150 177 205 230* 253 273 300 310 330 340 320 295 255

52 75 100 125 150 176 203 227 240* 255 275 290 300 300 295 265 240 225 200

52 75 100 125 150 181 213 243 270* 286 324 345 370 380 390 385 370 355 305

52 75 100 125 150 176 203 226 247 265* 285 300 315 330 330 320 315 300 260

CP.

0

118

26

—

—

—

—

—

—

—

—

83

—

114

8

NOTE: The reference energy is 0 J/mol for the dilute 0 K gas. The numbers above the asterisk indicate the solid in equilibrium; from the asterisk down, the liquid. The superscript ( s ) indicates estimates. T.P. is the triple point and C.P. is the critical point.

nH 2 , but is 40 J/mol off for n D 2 . This shows that tens of joules/mol uncertainty underlie most of our equation-of-state estimates. We have left the numbers in these tables to the nearest 10 J/mol for putting in a code, but they are hardly that good. We have no way of knowing the accuracy of the algorithm we have chosen. Changing the method can easily move an energy by 100 J/mol in the region just under the critical points. As an example, consider liquid H 2 at 32 K. By setting E(non-ideal) and then H 0 (sub), we work our way to an H*(liq) value of 617 J/mol. But the N.B.S. code gives 653 J/mol, and we obtain 670 J/mol by extrapolating the lower temperature values of H*(liq). The problem stems from trying to match rapidly changing E(non-ideal) numbers with rapidly changing H(vap) values near the critical points. This gives us an unsettling indication of how shaky all these calculated near-critical values are. In Table 8.6, the critical point enthalpies equal PV + E(rot). The wide variation in E(rot) causes the differences from one hydrogen to the next. The much higher critical-point values in Table 8.7 are caused by the change of the energy reference. We can put it into the gas reference by using

121

THERMODYNAMICS OF THE SATURATED PHASE BOUNDARY

T A B L E 8.9 TOTAL INTERNAL ENERGY FOR THE SATURATED SOLID E * ( s o l ) AND LIQUID E * ( l i q )

Solid and liquid total internal energy, E*(J/mol) nH2

b

HD a

HT a

nD2a,b

D-Tab

nT 2 a , b

T.P. sol T.P. liq

36 153

42 201

53 231

65 264

102 320

96 329

4.2 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37

10 13 15 18 23 153* 180 210 240 285 320 365 410 480 570

0 1 2 5 10 19 35 230* 265 305 370 430 510 580 670

0 1 2 5 10 20 35 237* 270 315 375 445 520 605 695 795 845

3 5 7 9 14 23 37 56 280* 330 390 455 515 590 660 735 775 830 880

3 4 6 9 17 29 51 76 320* 370 430 500 585 675 770 870 925 985 1050

10 13 15 19 24 33 50 69 89 360* 420 480 555 640 735 820 905 980 1022

CP.

1368

1396

—

—

—

—

—

—

—

—

770

1053

—

1 164

1 158

NOTE: The reference energy is 0 J/mol for the 0 K solid. The numbers above the asterisks indicate the solid; from the asterisks down, the liquid. The superscript ( a ) inidcates estimates; ( b ) means that quadrupolar energy is included in the solid. T.P. and C.P. are the triple point and the critical point.

H(liq) = H*(liq) - H 0 (sub)

(8.23)

In Table 8.8, the critical-point values equal E(rot) only, and in Table 8.9, H 0 (sub) is added for the change of reference. Let us summarize our method for deriving self-consistent energies. For temperatures up to 20 or 22 K, we know the various heats of the solid and liquid pretty well. Above the solid, we calculate H 0 (sub) using Eq. 8.19. Above the liquid, the corresponding equation is H 0 (sub) = H*(liq) - H(gas) + H(vap)

(8.24)

At higher temperatures, we do not know any of these values very well. We next set H 0 (sub) constant at the value we determined, and smooth the other three in some reasonable way. The tables we end up with themselves constitute an equation of state, which has been made reasonably self-consistent by constant crisscrossing from one

122

T H E R M O D Y N A M I C S OF THE SATURATED PHASE B O U N D A R Y

value to another. This is a constant problem on the saturation curve, where correcting one value changes everything else. The most doubtful parts of these tables are for D-T and T 2 above 30 K.

Slow Freeze of Dense D-T Gas As an example of using our tables, we shall consider a simple model for cooling 25 mol% D 2 - 5 0 % D T - 2 5 % T 2 (D-T). We start with the gas at 4920 mol/m 3 at 35 K, so that it is just ready to liquefy. Now we cool the sample down slowly to 10 K, and it will follow the saturation curve downward. More liquid will condense, and at about 20 K, the liquid will turn into solid. If we start with 1 mol of gas, then the mol fraction that has liquefied, x, is only a function of the gaseous molar volume V, the liquid molar volume V L , and the sample volume (which is the gas molar volume at 35 K). We have, as long as the system is in thermal equilibrium:

A similar equation will hold for the solid mol fraction s below the triple point. Table 8.10 shows the rough results of this model: the mol fraction of liquid or solid and the change in enthalpy for each temperature step. In cooling from 35 to 10 K, a total of 1 800 J/mol is required. As an example of how we calculate the enthalpy for each step, consider the initial drop from 35 to 34 K. We assume that the T A B L E 8.10 RESULTS OF A SIMPLE M O D E L FOR THE SLOW FREEZE OF D - T FROM LIQUID AT 3 5 K

Calculated thermal diffusivity (m 2 /s)

Temp. drop (K)

mol fraction condensed phase

AH enthalpy decrease (J/mol)

Gas

Condensed phase

35->34 34-32 32-30 30-28 28-26 26-24 24-22 22-20 Freeze 20 20-18 18-16 16-14 14-12 12-10

0.194 0.463 0.643 0.767 0.853 0.912 0.951 0.975 0.975 s 0.990 s 0.997 s -Is ~ Is -Is

166 294 256 219 184 146 113 84 213 50 s 35 s 22 s 12s 8s

2.0( —7) 2.0( —7) 2.0( —7) 3.8( —7) 5.1(-7) 7.1(-7) l.l(-6) 1.8( —6) 2.8( —6) 3.8( —6) 9.8( —6) 3.3( —5) I.6C-4) 1.4(-3)

4.6C-8) 5.0( —8) 5.4(-8) 5.8( —8) 6.2( —8) 6.6( —8) 7.3(-8) 7.5(-8) 3.6( —7)! 4.4(-7)! 7.1(-7)! 1.2(-6) ! 2.2(-6)! 4.5(-6)!

NOTE: The superscript ( s ) indicates the solid. The other condensed phase is the liquid.

THERMODYNAMICS OF THE SATURATED PHASE BOUNDARY

123

liquid all forms halfway, at 34.5 K. Naturally, a finer mesh of temperatures would give a better answer. With x = 0.194 at 34 K, we have, for this temperature step: Gas not turned to liquid Gas turned to liquid Liquid condensation Liquid formed

(1 - 0.194)(+ 16 J/mol) (0.194) (8) (0.194) ( - 9 0 0 ) (0.194) ( - 3 2 )

= + 1 3 J/mol = +2 = -175 = -6

- 1 6 6 J/mol

(8.26)

Only the third step involves the heat of vaporization (here a 34.5 K value). The other steps are for the cooling of the gas and liquid. Note that the gas increases in enthalpy, but it is more than offset by the loss of heat of vaporization. If we next assume that our cryostat removes heat at a constant rate, we obtain the results shown in Fig. 8.1. We see that the cooling process goes faster as the temperature decreases because most liquefaction ends, and all other heats are small compared to the heat of vaporization. We also assume that all liquid and solid forms on a single plate in the cryostat, and the thickness of these layers is indicated. The solid is assumed to be theoretically dense rather than being snow. How slow is a slow cool? From elsewhere in this book, we glean the values

Time (arbitrary) Fig. 8.1. Results of the slow freeze model for D-T. All D-T is at the same temperature and heat is removed at a constant rate.

124

THERMODYNAMICS OF THE SATURATED PHASE BOUNDARY

needed to calculate the D-T thermal diffusivities along the saturation curve in the various phases. These are listed in Table 8.10. Note that for the gas, the thermal diffusivity and the molecular self-diffusion coefficient are expected to be the same. This is not true in the other phases, where heat moves faster than the molecules. The time constant T for heat to travel a distance X is roughly T~ X2/D

(8.27)

This will be short in most cases.

Notes 1. F. W. Sears, Thermodynamics, The Kinetic Theory of Gases and Statistical Mechanics, 2d ed. (Addison-Wesley, Reading, Mass., 1959), pp. 119-123. 2. F. Simon and F. Lange, Z. Phys. 15, 312 (1923). 3. K. Clusius and K. Hiller, Z. Physik Chem. 4B, 158 (1929). 4. H. L. Johnston, J. T. Clarke, E. B. Rifkin, and E. C. Kerr, J. Amer. Chem. Soc. 72, 3933 (1950). 5. K. Clusius and E. Bartholome, Z. Physik. Chem. 30 B, 237 (1935). 6. E. C. Kerr, E. B. Rifkin, H. L. Johnston, and J. T. Clarke, J. Amer. Chem. Soc. 73,282 (1951). 7. G. Grenier and D. White, J. Chem. Phys. 40, 3015 (1964). 8. F. Pavese and C. Barbero, Cryogenics 19, 255 (1979). 9. D. White, J. H. Hu, and H. L. Johnston, J. Phys. Chem. 63, 1181 (1959). 10. See note 2 above. 11. See note 4 above. 12. See note 7 above. 13. G. N. Lewis and M. Randall, Thermodynamics, 2d ed., revised by K. S. Pitzer and L. Brewer (McGraw-Hill, New York, 1961), pp. 106-107, 189-191. 14. H. M. Roder, R. D. McCarty, and W. J. Hall, Computer Programs for Thermodynamics and Transport Properties of Hydrogen [Tabcode-11], National Bureau of Standards Technical Note 625, SD Catalog No. C13.46:625 (U.S. Government Printing Office, Washington, D.C., 1972). 15. R. D. McCarty, Hydrogen Technological Survey—Thermophysical Properties (National Technical Information Service, Springfield, Va., 1975). 16. H. M. Roder and R. D. McCarty, A Modified Benedict-Webb-Rubin Equation of State for Parahydrogen-II, National Bureau of Standards Report NBSIR 75-814, Boulder, Colo. 80303 (June 1975). 17. R. D. McCarty, J. Hord, and H. M. Roder, Selected Properties of Hydrogen (Engineering Design Data), J. Hord, ed. (U.S. Government Printing Office, Washington, D.C., 1983). 18. R. D. McCarty, Hydrogen: Its Technology and Implications. Hydrogen Properties, III. Hydrogen Properties, K. E. Cox and K. D. Williamson, ed. (CRC Press, Cleveland, Ohio, 1975). 19. B. A. Younglove and D. E. Diller, Cryogenics 2, 348 (1962). 20. See reference in note 13 above, pp. 65, 106-107, 189-191. 21. B. A. Younglove and D. E. Diller, Cryogenics 2, 283 (1962). 22. See note 20 above. 23. See note 17 above. 24. R. Prydz, The Thermodynamic Properties of Deuterium, National Bureau of Standards Report 9276, Boulder, Colo. 80303 (April 1967). 25. A Driessen, J. A. de Waal, and I. F. Silvera, J. Low Temp. Phys. 34, 255 (1979).

9. The Melting Curve

As we have seen in Fig. 4.1, the phase diagram is divided into two major unconnected areas: the solid and the fluid. In the solid, the molecules assume an arrayed structure; in the liquid, their arrangement has no order. In the solid, in general, the internal energy is a function of the volume only. In the fluid, it is mainly a function of the temperature. The line that divides these two regions is the melting curve. There is no theoretical reason why it cannot run forever—up to very high temperatures balanced by immense pressures.1 Melting Curve Information The higher the temperature goes, the higher the melting pressure. Hydrogen has been frozen at room temperature, but 5.3 GPa is needed to do it. Table 9.1 lists the melting pressures of the hydrogens, some measured and some estimated. 2-9 Note how large the pressure difference is between the triple point and the next listed point. This lowpressure region is further described in Table 9.2, which is built around 20.4 Itcatalyzed eH 2 and eD 2 data. 10 The researchers found a "kink" in the melting curve. For the first degree K above the triple point, we describe the melting pressure Pm at the melting temperature T m by: Pm = P9 + B(Tm - 6)

(9.1)

where P, is the triple-point pressure, 6 the triple-point temperature, and B a constant. Even with all their care, the data within a few tenths K of 6 are uncertain. For the next half to one degree above the Eq. 9.1 region, we use: Pm = P e + C(T m - 0) + D(T m - 0)2

(9.2)

The constants B, C, and D are listed in Table 9.2. The first pair of numbers in the "range" column belongs to Eq. 9.1 and the second pair belongs to Eq. 9.2. We turn next to molar volumes; those of the solid phases are listed in Table 9.3. The data are so scanty that any number related to real work gets an asterisk. The eH 2 numbers 11 have been corrected to agree with the latest triple-point value (see Table 6.3). Other pressure data include eD 2 volumes,12 which cover a wide temperature range but which—like most of the eH 2 data—are extrapolated and smoothed al-

126

THE MELTING CURVE

TABLE 9.1 FREEZING PRESSURES OF THE HYDROGENS

Freezing temp.

Pressure (Pa)

(K)

nH 2

HD a

HT a

nD 2

DT a

nT 2

Triple point

7.20(3)

1.24(4)

1.46(4) A

1.72(4)

2.01(4)"

2.16(4)

15 16 17 18 19 20 22 24 26 28 30 35 40 50 77.4 100 150 200 250 300

3.28(6) 6.72(6) 1.04(7) 1.43(7) 1.84(7) 2.27(7) 3.17(7) 4.14(7) 5.18(7) 6.27(7) 7.42(7) 1.06(8) 1.40(8) 2.20(8) 4.89(8) 7.75(8) 1.59(9) 2.62(9) 3.86(9) 5.30(9)

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

1.4(6) 5.1(6) 9.1(6) 1.3(7) 2.2(7) 3.2(7) 4.2(7) 5.3(7) 6.4(7) 9.6(7) 1.3(8) 2.1(8) 4.9(8)

1.1(6) 5.0(6) 9.3(6) 1.8(7) 2.8(7) 3.8(7) 4.9(7) 6.0(7) 9.2(7) 1.3(8) 2.1(8) 4.9(8)

—

1-1(6)* 5-3(6)" 1.5(7) 2.36(7) 3.37(7) 4.45(7) 5.60(7) 8.74(7) 1.23(8) 2.04(8) 4.8(8)"

—

—

1.2(6) 1.1(7) 2.1(7) 3.0(7) 4.1(7) 5.3(7) 8.4(7) 1.2(8) 2.0(8) 4.8(8)

—

assume same: as H 2

— —

7.6(6)" 1.7(7)" 2.6(7) 3.8(7) 4.9(7) 8.1(7) 1.2(8) 2.0(8) 4.8(8)"

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

NOTE: The superscript (^indicates estimates. The numbers in parentheses are powers of ten.

T A B L E 9.2 EQUATION FOR DESCRIBING THE MELTING PRESSURE OF THE HYDROGENS FOR THE FIRST 1 TO 2 DEGREES ABOVE THE TRIPLE POINT 6

Triple point

eH 2 nH 2 HD" HT" eD 2 nD 2 " DT" T2"

Temp. 0(K)

Pressure P*(Pa)

B (Pa/K)

C (Pa/K)

D (Pa/K) 2

13.80 13.96 16.60 17.70 18.69 18.73 19.79 20.62

7.030(3) 7.200(3) 1.237(4) 1.458(4) 1.713(4) 1.715(4) 2.008(4) 2.160(4)

3.171(6) 3.145(6) 3.59(6) 3.78(6) 4.019(6) 4.01(6) 4.17(6) 4.28(6)

3.038(6) 3.020(6) 3.44(6) 3.62(6) 3.848(6) 3.84(6)

1.118(5) 1.315(5) 1.50(5) 1.74(5) 2.087(5) 2.23(5)

—

—

—

—

(Tm - 0) ranges (K) 0-1.19-2.36 0-0.95-1.97 0-1.00-1.6 0-0.92-1.4 0-0.82-1.20 0-0.76-1.2 0-0.7 0-0.7

NOTE: The B equations are used from T m — 0 = 0 to the center range temperature; the C, D equations are used between the last two temperatures. The superscript (") indicates estimates. The numbers in parentheses are powers of ten.

127

THE MELTING CURVE

T A B L E 9.3 SOLID M O L A R V O L U M E S A L O N G THE M E L T I N G C U R V E

Solid molar volume along freezing curve (/¿(m3)/mol) Temp. (K)

nH 2

eH 2

HD

HT

eD 2

nD 2

DT

nT 2

TP

23.14*

23.31*

21.80

21.2

20.44*

20.40

19.9

19.4

14 16 18 20 22 24 26 28 30 35 40 50 77.4 100 150 200

23.1* 22.5* 22.0* 21.4* 20.8* 20.2* 19.9 19.6 19.3 18.4 17.7 16.6 14.4* 13.2* 11.2*

23.3* 22.6* 22.0* 21.5* 20.9* 20.4* 20.0* 19.6* 19.3* 18.4* 17.7* 16.5* 14.2* 12.9* 10.9* 9.6*

—

21.3 20.7 20.2 19.7 19.3 18.8 18.5 17.6 17.0 15.9 14.2 13.0 11.1 —

—

—

—

—

—

—

—

—

—

—

—

—

—

—

21.1 20.4 19.9 19.4 19.0 18.6 18.3 17.5 16.9 15.8 14.0 12.9 11.0 —

20.2* 19.7* 19.3* 18.9* 18.6* 18.2* 17.5* 16.9* 15.8* 13.7* 12.5* 10.7* 9.4*

20.0 19.5 19.1 18.7 18.3 18.0 17.2 16.6 15.6 13.9* 12.8* 10.9* —

19.6 19.2 18.8 18.4 18.0 17.8 17.0 16.4 15.4 13.8 12.7 10.8 —

—

18.8 18.5 18.1 17.7 17.5 16.7 16.2 15.3 13.6 12.6 10.7 —

NOTE: The asterisks indicate values based on data; all else is estimated—that is, most of it. T.P. indicates the triple point.

ready. The nH 2 and nD 2 data unfortunately cover only the range 77 to 150 K. 1 3 Everything else is estimated. The liquid molar volumes are listed in Table 9.4, and the situation is even worse. The 14 to 22 K H 2 data are really for the equilibrium form, 1 4 and the other data are for the two hydrogens at 77 to 150 K. 1 5 Again, the rest are estimates. These numbers plus solid equation-of-state results 16 have been combined into a larger code which has produced the H 2 and D 2 pressure-volume-temperature results of Table 9.5 17 We tabulate by constant volume at the higher pressures. Volumes ( b ) are those of the solid on the melting curve, and ( c ) are the 0 K solid volumes. The same volumes between ( b ) and ( c ) bear the same relation to each other. For example, 20.84 /z(m 3 )/mol for H 2 and 18.44 /i(m 3 )/mol for D 2 are corresponding points, each halfway between ( b ) and ( c ), and this allows us to estimate a similar point for D-T. We list the pressures needed to keep the given volume at 0 K, the melting temperature, and 0.8 of the way to the melting temperature. We see that the pressure is fairly constant at constant volume. This is because the potential energy of compression is much larger than that of thermal heating. The model also calculates solid internal energies by using an Einstein model. 18 This is a simpler model than the Debye model, because it reduces to a set of uncoupled oscillators, all vibrating at the same frequency. The Gruneison constant and the volume are used to derive the Einstein temperature, which can be incorporated

128

THE MELTING CURVE

T A B L E 9.4 LIQUID MOLAR VOLUMES A L O N G THE MELTING CURVE

Liquid molar volume along freezing curve (^(m 3 )/mol) Temp. (K)

nH 2

HD

HT

nD 2

DT

nT 2

Triple point

26.11*

24.61*

24.0

23.16*

22.6

22.0

14 16 18 20 22 24 26 28 30 35 40 50 77.4 100 150

26.07* 25.00* 24.10* 23.33* 22.66* 22.0 21.4 20.9 20.4 19.5 18.6 17.3 15.1* 13.8* 11.6*

—

—

—

—

23.8 23.0 22.3 21.7 21.1 20.6 20.2 19.2 18.4 17.1 14.9 13.6 11.5

23.7 22.9 22.2 21.6 21.0 20.5 20.0 19.1 18.2 16.9 14.7 13.4 11.4

—

—

—

—

—

—

22.7 22.0 21.4 20.8 20.3 19.9 18.9 18.1 16.8 14.6* 13.3* 11.3*

22.5 21.8 21.2 20.6 20.1 19.8 18.7 18.0 16.7 14.5 13.2 11.2

—

21.6 21.1 20.5 20.0 19.6 18.6 17.8 16.5 14.3 13.0 11.1

NOTE: The asterisks indicate data; all other values are estimates.

into an internal energy. These calculated internal energies for true J = 0 H 2 and D 2 (the molecular rotational ground state) are listed in Table 9.6. The rotational energies have not been added, and the energy reference is 0 J/mol for the 0 K solid. Although the Einstein model may be too simple, these are presently the best available internal energies of the solid.

Heat of Fusion When the fluid condenses to the solid it gives up the heat of fusion, H(fus), in J/mol. By calorimetry, data have been taken for eH 2 from the triple point to 22.6 K. 19 The data fit the relation H ( f u s ) ^ 117 + 6.81 ( T - T m )

(9.3)

We see that H(fus) increases with the melting temperature and pressure. There are no more calorimetry data, and one must turn to the ClausiusClapeyron equation, now refitted for the phase boundary between solid and fluid. dP m dT m

H(fus) T m (V L - V s )

(9.4)

We may calibrate this equation at the H 2 triple point. From Table 6.3, VL — V s is 2.87 /i(m 3 )/mol for eH 2 and 2.97 /x(m3)/mol for nH 2 . From Table 8.1, we obtain

129

THE MELTING CURVE

T A B L E 9.5 EQUATION OF STATE OF SOLID n H 2 , n D 2 , AND D T (ESTIMATED)

Molar volume Mm 3 )/ mol

Melting temp. T m (K)

OK

0.8T m

Tx m

H2

9 10 12 14 16 17.63 18.70" 19.78 20.85 21.92 22.99°

252 197 125 82.2 55.1 40.5 33.2 27.3 22.4 18.3 15.0

3.53 (9) 2.29(9) 1.03(9) 4.88(8) 2.35(8) 1.26(8) 8.11(7) 4.95 (7) 2.74(7) 1.2 (7) 1 (3)

3.77(9) 2.44(9) 1.10(9) 5.20(8) 2.49(8) 1.33(8) 8.57 (7) 5.23 (7) 2.91 (7) 1.3 (7) 6 (5)

3.91(9) 2.54(9) 1.15(9) 5.48 (8) 2.64(8) 1.42(8) 9.20(7) 5.67(7) 3.20(7) 1.5 (7) 2 (6)

D2

9 10 12 14 16 16.28 17.0b 17.72 18.44 19.16 19.88c

244 187 116 73.0 47.2 44.5 38.3 33.0 28.6 24.6 21.3

3.12(9) 1.97(9) 8.32(8) 3.58(8) 1.46(8) 1.26(8) 8.68 (7) 5.49(7) 3.21 (7) 1.47 (7) 1.2 (4)

3.47 (9) 2.21 (9) 9.42(8) 4.09(8) 1.69(8) 1.46(8) 1.02(8) 6.67(7) 4.09(7) 2.10(7) 5.1 (6)

3.63 (9) 2.32(9) 1.00(9) 4.43 (8) 1.87(8) 1.63 (8) 1.16(8) 7.75 (7) 4.95 (7) 2.84(7) 1.1 (7)

DT

9 10 12 14 16 16.63b 17.29 17.95 18.61 19.27c

240 184 112 70.2 45.0 39.4 34.2 29.7 25.9 22.5

3.0 1.9 7.8 3.3 1.3 8.8 5.6 3.4 1.6 3

3.4 2.1 9.0 3.8 1.5 1.1 7.1 4.5 2.4 1

3.6 2.3 9.6 4.2 1.7 1.2 8.5 5.6 3.4 1.6

Pressure (Pa) at

(9) (9) (8) (8) (8) (7) (7) (7) (7) (4)

(9) (9) (8) (8) (8) (8) (7) (7) (7) (7)

(9) (9) (8) (8) (8) (8) (7) (7) (7) (7)

NOTE: The numbers marked ( b ) are the volumes of the solid at the critical-point temperatures. The numbers marked ( c ) are the 0 K volumes. Each number is comparable to the similarly situated numbers for the other hydrogens. The pressures are listed for 0 K, 0.8T m , and T m , where T m is the melting temperature. identical H(fus) values of 117 J/mol. Then, d P m / d T m is calculated to be 2.95 and 2.82 M P a / K . At the triple point, Eq. 9.1 gives d P m / d T m equal to B, the constants listed in Table 9.2. If we compare these H 2 values, the agreement is not good. We have yet another equation-of-state disagreement, which only further experiment can resolve. Luckily, the D - T and T 2 estimates for B are about the same no matter how we do it.

130

THE MELTING CURVE

TABLE

9.6

C A L C U L A T E D T R A N S L A T I O N A L SOLID E N E R G I E S U S I N G A N EINSTEIN M O D E L

Melting temp. T m (K) 252 197 125 82.2 55.1 40.5 33.2 27.3 22.4 18.3 15.0 14.0"

J = 0 H 2 Internal energy (J/mol) at 0K

Tn

Melting temp. T m (K)

7910 5050 1930 480 -210 -500 -610 -678 -719 -740 -747

10 650 7010 2900 980 40 -360 -510 -613 -677 -713 -727 -731

244 187 116 73.0 47.2 44.5 38.3 33.0 28.6 24.6 21.3 18.7b

—

J = 0 D 2 Internal energy (J/mol) at 0K

Tm

5 850 3 350 720 -410 -885 -920 -1000 -1050 -1082 -1098 - 1 104

9410 5 850 2060 270 -540 -610 -750 -860 -932 -986 -1016 -1050

—

NOTE: The bottom lines ( b ) are the triple points.

We have seen that H(fus) for eH 2 increases up to 22.6 K. What does it do at higher temperatures? We see that dP m /dT m increases slightly and that V L — V s decreases slightly with increasing temperature. We expect, then, that H(fus) ~ T m

(9.5)

Estimated heats of fusion are shown in Fig. 9.1, and they are close to linear. Near room temperature, we expect H(fus) values of several thousand joules per mol.

The Lindemann Melting Relation This rule predicts that any solid will melt when its thermal vibrations reach a certain fraction of the intermolecular distance. It says that 2 0 / vibrationalV \ amplitude J intermolecular\ 2 distance J

Tm = M0 D 2 V s 2 / 3

Q

^

where M is the molecular weight, 0 D is the Debye temperature, and G is a constant. All values are on the melting curve. G should be the same for all materials, and it should not change much even in climbing the melting curve. For hydrogen, 0 D ~ V s ~ 2 and T m ~ V s ~ 3 , and we see that G will be fairly constant for all V s . However, the presence of the zero-point energy of the crystal lattice does change this relation. The more quantum mechanical a hydrogen is, the larger V s becomes. G then increases with the quantum parameter A*ff (see Chap. 1) according to 2 1

131

THE MELTING CURVE

Fig. 9.1. The heats of fusion increase with the melting temperature. The results are only approximate, because of their indirect derivation. The DT line is an estimate.

G ~ 360 + 250A e * ff

(9.7)

The zero-point energy does not affect the melting process as much as we might expect. This is because the zero-point motion is a high-frequency process, near the Debye cutoff 0 D , whereas thermal frequencies near the melting point are lower ( T / 0 d ~ 0.2 to 0.4). The two types of motion are largely uncoupled, especially near the triple point. 2 2

Notes 1. D. A. Young, Lawrence Livermore National Laboratory, Livermore, Calif. 94550, private communication, 1981. 2. R. L. Mills and E. R. Grilly, Phys. Rev. 101, 1246 (1956). 3. R. D. Goodwin, Cryogenics 2, 353 (1962).

132

THE MELTING CURVE

4. R. F. Dwyer, G. A. Cook, O. E. Berwaldt, and H. E. Nevins, J. Chem. Phys. 43, 801 (1965). 5. V. V. Kechin, A. I. Likhter, Y. M. Pavlyuchenko, L. Z. Ponizovskii, and A. N. Utyuzh, Soviet Phys. J.E.T.P. 45, 182 (1977). 6. D. H. Liebenberg, R. L. Mills, and J. C. Bronson, Phys. Rev. 18B, 4526 (1978). 7. H. K. Mao and P. M. Bell, Science 203, 1004 (1979). 8. A. Driessen, J. A. deWaal, and I. F. Silvera, J. Low Temp. Phys. 34, 255 (1979). 9. N. G. Bereznyak and A. A. Sheinina, Soviet J. Low Temp. Phys. 6, 608 (1980). 10. See note 9 above. 11. See notes 4 and 8 above. 12. See note 8 above. 13. See note 6 above. 14. R. D. Goodwin and H. M. Roder, Cryogenics 3, 12 (1963). 15. See note 6 above. 16. See note 8 above. 17. D. A. Young and P. C. Souers, A Three-Phase Equation of State Model for the Hydrogen Isotopes, Lawrence Livermore National Laboratory Report UCRL-83829, Livermore, Calif. 94550 (1980). 18. See note 17 above. 19. H. R. Lander, R. F. Dwyer, and G. A. Cook, Adv. Cryogenic Eng. 11, 261 (1965). 20. J. J. Gilvarry, Phys. Rev. 102, 308 (1956). 21. C. Domb, Nuovo Cimento 9 (Series 10), 9 (1958). 22. R. K. Crawford, in "Melting, Vaporization and Sublimation," in Rare Gas Solids, M. L. Klein and J. A. Venables, eds. (Academic Press, London, 1977), Vol. II, pp. 710-713.

10. The Pressurized Fluid

Pressure-Volume-Temperature We are now ready to penetrate into the fluid portion of the phase diagram. We turn to the National Bureau of Standards equation of state (E.O.S.), previously discussed in Chapter 8. Data for H 2 were taken, although not shown in raw form, from 15 to 100 K at pressures up to 35 MPa. 1 Also available are good H 2 and D 2 data from 98 K past room temperature at pressures up to 0.3 GPa in some cases. 2 For D 2 at lower temperatures, the data are scanty. 3 These have been combined into E.O.S.s for H 2 to 100 MPa 4 , 5 and for D 2 to 40 MPa, 6 where most of the higher pressures are extrapolated. Then, fluid data for H 2 and D 2 from 75 to 300 K and 0.2 to 2 GPa were measured. 7 ' 8 All the above were combined with the Silvera solid E.O.S. 9 into a total E.O.S. for H 2 and D 2 . 1 0 We shall try to pull forth the best of all this work for summary in this chapter. We present various pressure-volume-temperature data in the fluid phase. Table 10.1 lists the molar volumes at 77 and 300 K for H 2 and D ^ 1 1 Table 10.2 lists the compressibilities from 77 to 1000 K . 1 2 , 1 3 Tables 10.3 and 10.4 show the least-squares density coefficients for nH 2 and nD 2 gas, as taken directly from the raw data in the range 98 to 423 K. 1 4 These constants A through F fit into the equation z = A + Bp + Cp2 + Dp3 + Ep* + Fp4

(10.1)

where p is density (mol/m 3 ) and z is the compressibility. These constants are not virial coefficients (see Eq. 4.27 for the definition) but are only the result of the computer fit. There is not much difference between H 2 and D 2 at either 77 or 300 K. But as we have seen in Chapter 8, the phase boundaries are different, so that we need a special means of extrapolation in the liquid region. This is shown in the phase diagram of D 2 in Fig. 10.1. Note the extreme bending of the constant volume lines in the liquid region. We have E.O.S. values in the liquid region for H 2 and D 2 , but suppose we wish to extrapolate to D-T (i.e., 25% D 2 - 5 0 % DT-25% T 2 ). We cannot use the Principle of Corresponding States, which says that all P-V-T data may be related just to the critical point. This is not true for the hydrogens, where quantum effects produce different liquid densities and phase boundaries. We therefore relate all our values simultaneously to the three special points shown in Fig. 10.1—the triple point (T.P.), the critical point (C.P.), and the fluid point on the melting curve at the critical

134

THE PRESSURIZED FLUID

TABLE M O L A R VOLUMES OF H

2

10.1

AND D 2 AT LIQUID N I T R O G E N TEMPERATURE AND R O O M TEMPERATURE

Molar volume [/i(m 3 )/mol] 77.36 K

300 K

Pressure (Pa)

H2

D2

H2

D2

3.16(9) 1.78(9) 1.00(9) 5.62(8) 3.16(8) 1.78(8) 1.00(8) 5.62(7) 3.16(7) 1.78(7) 1.00(7) 5.62(6) 3.16(6) 1.78(6) 1.00(6) 5.62(5) 3.16(5) 1.78(5) 1.00(5) 5.62(4) 3.16(4) 1.78(4) 1.00(4) 5.62(3) 3.16(3)

9.25* 10.6* 12.2* 13.9* 16.6 19.2 22.3 26.6 32.9 43.4 64.7 109 196 352 633 1.13(3) 2.02(3) 3.61(3) 6.42(3) 1.14(4) 2.03(4) 3.62(4) 6.43(4) 1.14(5) 2.03(5)

9.00* 10.3* 11.7* 13.3* 16.3 19.0 22.1 25.8 31.3 41.0 60.9 105 192 350 631 1.13(3) 2.02(3) 3.60(3) 6.42(3) 1.14(4) 2.03(4) 3.62(4) 6.43(4) 1.14(5) 2.03(5)

10.0 11.9 14.4 17.7 22.6 29.7 40.9 60.2 94.4 156 265 459 804 1.42(3) 2.51(3) 4.45(3) 7.90(3) 1.40(4) 2.50(4) 4.44(4) 7.89(4) 1.40(5) 2.49(5) 4.44(5) 7.89(5)

9.82 11.8 14.3 17.8 22.7 29.5 40.6 59.9 94.1 155 264 458 803 1.42(3) 2.51(3) 4.45(3) 7.90(3) 1.40(4) 2.50(4) 4.44(4) 7.89(4) 1.40(5) 2.49(5) 4.44(5) 7.89(5)

NOTE: The asterisks indicate the solid phase; otherwise, it is fluid. The pressure is listed in equal increments of log 1 0 P. The numbers in parentheses are powers often. temperature (point "b"). Using these as reference points, we shall map, zone by zone, the values of one hydrogen on to the phase diagram of another. If H 2 and D 2 numbers are both available, the extrapolation is easy, using the quantum parameter as the tool. Table 10.5 shows the constant volume layout we shall use for mapping. For n H 2 , 19.95 /i(m 3 )/mol is point b, 26.11 ¿i(m 3 )/mol is the liquid triple-point volume, and 65.0 /i(m 3 )/mol is the critical-point volume.* The other volumes are selected at *The measured critical-point compressibilities for eH 2 , HD, and eD 2 are 0.309, 0.312, and 0.312, respectively. See H. J. Hoge and J. W. Lassiter, J. Res. Nat. Bureau Standards 47, 75 (1951). We have averaged these to 0.31. Our 65.0 /¿(m 3 )/mol value for nH 2 has been reported from 64.1 /x(m 3 )/mol (see the reference in note 4, Chap. 10) to 66.95 ¿¿(m3)/mol (in H. M. Roder, G. E. Childs, R. D. McCarty and P. E. Angerhofer, Survey of the Properties of the Hydrogen Isotopes below their Critical Temperatures, National Bureau of Standards Technical Note 641 [U.S. Government Printing Office, Washington, D.C., 1973], p. 88).

135

THE PRESSURIZED FLUID

TABLE COMPRESSIBILITIES OF n H

2

10.2

G A S AT R O O M T E M P E R A T U R E A N D A B O V E

Compressibility, z, at given temps. Pressure (MPa)

77.4 K

300 K

400 K

600 K

800 K

1000K

1 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 400 600 800 1000 2000 3000

0.984 0.969 1.006 1.26 1.57 1.88 2.17 2.44 2.69 2.95 3.21 3.47 3.96 4.44 4.91 5.36 5.81 6.87 7.85 9.62 12.7 15.8 19.0 32.0 43.6"

1.006 1.029 1.060 1.123 1.188 1.253 1.318 1.383 1.447 1.510 1.573 1.64 1.77 1.89 2.01 2.13 2.24 2.52 2.77 3.26 4.16 4.99 5.76 9.28 12.4a

1.005 1.024 1.048 1.097 1.147 1.196 1.244 1.293 1.340 1.388 1.434 1.48 1.58 1.67 1.76 1.84 1.93 2.14

1.003 1.017 1.034 1.067 1.100 1.133 1.165 1.197 1.228 1.259 1.290 1.320 1.38 1.44 1.50 1.55

1.002 1.013 1.026 1.051 1.076 1.100 1.124 1.147 1.170 1.193 1.215 1.237 1.27 1.32 1.36 1.40

1.002 1.010 1.021 1.041 1.061 1.080 1.099 1.117 1.135 1.153 1.170 1.187 1.25" 1.29" 1.33" 1.36"

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

NOTE: The superscript (') indicates extrapolations.

special in-between intervals. The triple-point and critical-point temperatures are 13.956 and 33.18 K. The temperatures in between are also selected at definite intervals. Above the critical point, we convert to selected fixed temperatures. Table 10.6 has the same layout with respect to the fixed points for n D 2 . In Table 10.7, we use the same grid for fluid D-T, and we estimate the molar volumes position by position from the preceding two tables. Despite calling it D-T, however, we treat it as a single component with definite phase boundaries (i.e., molecular DT).

Estimated Energies of Fluid D-T We have had a hard enough time estimating P-V-T properties for fluid D-T. Can we do anything with the energies? We shall assume that the H 2 E.O.S. energies 15 are usable for D-T from 150 to 300 K, where isotopic differences presumably are small. From Chapter 8, we have non-ideal energies along the saturation curve. We combine

136

THE PRESSURIZED FLUID

TABLE 10.3 DENSITY COEFFICIENTS FOR n H 2 GAS FROM 9 8 TO 4 2 3 K . T H E MAXIMUM DENSITIES STUDIED ARE:

98 K 26000 MOL/M 3 ; 173 K 28 000 MOL/M 3 ; 273 K 43000 M O L / M 3 ; AND 423 K 34000 T H E CONSTANT A is 0,999 393 IN EQ. 10-1.

MOL/M3.

Temp. (K)

B ( x 10" 5 m 3 /mol)

C ( x IO - 1 0 m 6 /mol 2 )

D ( x 10" 15 m 9 /mol 3 )

E ( x IO" 2 0 m 1 2 /mol 4 )

F (x IO'25 m 1 5 /mol 5 )

98.15 103.15 113.15 123.15 138.15 153.15 173.15 198.15 223.15 248.15 273.15 298.15 323.15 348.15 373.15 398.15 423.15

- 0 . 2 9 8 88 -0.15956 0.079 501 0.267 57 0.502975 0.697891 0.892245 1.078 67 1.20503 1.30184 1.373 29 1.437 55 1.49126 1.537 85 1.56666 1.58544 1.60721

5.03153 5.10673 5.06444 5.18649 5.165 02 4.805 34 4.58822 4.14141 4.051 80 3.875 28 3.885 36 3.561 52 3.229 52 2.945 21 2.89821 2.963 72 2.80783

7.50905 5.12496 3.52091 1.058 7 0.895 809 0.814980 0.99909 3.34473 2.628 19 3.037 34 1.498 89 3.378 36 5.51454 6.692 19 6.18548 4.803 80 5.40620

3.8850 15.7116 22.6460 31.5688 39.000 5 31.643 6 29.2452 17.765 7 18.263 1 14.0256 17.5816 9.243 57 -0.23126 - 4 . 6 2 7 56 -4.08078 -0.44712 -3.65321

54.072 5 30.7131 13.6901 -4.03723 -21.7170 -17.5178 -19.438 1 - 8 . 3 2 3 57 -12.4243 -8.24031 -13.5894 - 4 . 7 5 6 52 6.08900 9.77803 8.70709 4.11035 7.85011

TABLE

10.4

DENSITY COEFFICIENTS FOR n D 2 GAS FROM 9 8 TO 4 2 3 K . T H E MAXIMUM DENSITIES STUDIED ARE: 9 8 K 8 4 0 0 M O L / M 3 ; 1 7 3 K 2 7 0 0 0 M O L / M 3 ; 2 7 3 K 4 2 0 0 0 M O L / M 3 : AND 4 2 3 K 3 3 0 0 0 M O L / M 3 . T H E CONSTANT A i s 0 , 9 9 9 4 2 3 IN E Q . 1 0 - 1 .

Temp. (K)

B ( x 10~ 5 m 3 /mol)

C ( x IO - 1 0 m 6 /mol 2 )

D ( x 10-15 m 9 /mol 3 )

E ( x IO" 2 0 m 1 2 /mol 4 )

98.15 103.15 113.15 123.15 138.15 153.15 173.15 198.15 223.15 248.15 273.15 298.15 323.15 348.15 373.15 398.15 423.15

-0.43423 - 0 . 2 8 5 82 -0.00912 0.18453 0.44258 0.63926 0.838 11 1.01575 1.15182 1.244 59 1.331 13 1.39298 1.445 04 1.48394 1.50950 1.54360 1.55498

4.255 12 4.47783 3.493 66 4.93622 4.62675 4.498 20 4.38716 4.42043 4.105 38 4.183 34 3.63031 3.49076 3.29780 3.21719 3.35011 3.03917 3.28673

24.2406 18.2205 31.9705 -1.0139 0.2128 - 0 . 2 7 7 26 -0.71353 -2.94628 -0.64599 -2.70201 2.428 64 2.76195 3.45391 3.25146 1.06244 3.05640 - 0 . 4 2 3 16

-126.598 -93.6848 -169.138 35.9508 31.3852 34.241 1 34.886 6 42.7673 30.203 9 36.8726 11.7879 8.973 38 5.45008 4.94703 11.4641 3.78841 16.9419

F ( x 10" 2 5 m 1 5 /mol 5 ) — — —

-17.6541 -17.5307 -27.9529 -33.5664 -49.0597 -32.8873 -44.7034 -7.35190 -4.8209 - 1 . 4 8 3 96 -1.202 5 -8.98399 - 0 . 7 5 9 17 -19.1254

137

THE PRESSURIZED FLUID

L

°910T

Temperature ( K ) Fig. 10.1. The pressure-volume-temperature phase diagram of D 2 below lOGPa, from the Young E.O.S. (see Chap. 10, reference 10), showing constant volume lines in /i(m 3 )/mol. The low-pressure solid region is a wild guess. The 23.16 /j(m 3 )/mol line is the fluid triple-point (T.P.) volume. It divides into two sections the tooth-shaped fluid region between the solid-fluid and liquid-vapor lines. C.P. is the critical point. The fluid region to the left of the constant temperature line C.P.-to-b is the most difficult area to describe.

all this in Table 10.8, which lists the starting pressures of various constant volumes of fluid D-T. Our experiment might start at either 77 or 300 K, but it ends on the liquidgas saturation curve. The internal energy and enthalpy on the saturation curves are both given. N o t e that both the saturated liquid and vapor are represented. The thermally accessible rotational energies (for D-T, not D T ) are included, but not the inaccessible J = 1 (first rotational excited state) energies.

3 t3 tí VO i v o M t^ ^ » VD c « -H. rn Tt -- Ä -o "O '3 Sc S u 3 -fi

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*

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—

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a

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e -s

«

r- 00 T* o o 3 1 •o '3 tr

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p oo

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ta

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H

o

r Í N N M N r ó n c ó o o

JS u WD es J3 a

* a M ' t ^ ^ v û û C O O ' H ^ v i m

X

s

C

o

Q 5

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PQ < H mJ < 2 « u

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ü c P

•D a> -Q O

•D

Polyethylene


1, nij = 0 ± 1 transition decreases from two rotational constants with no hindering to one rotational constant when X -» oo. This is the convergent process that led to the adsorption results of Eq. 11.10. The distance to the single m j = 0 sublevel is divergent, because it becomes infinite as X oo. This state cannot exist in two dimensions, but it can exist with only partial hindering. At some point, the hindering will be just enough to make all J = 1, m j = 0 molecules desorb from the surface. With a good infrared spectrometer and a lot of surface, it might be possible to

+1 î +1

o rn Tt -- Tt r-^ oo Ò o m ¿ N N

en +1 +1

O — rnoovoOfNO ©OssOrnOï^in© 6oío\»oío¿oóo¿

o î o II

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a> E to re a O) c 'x £ 0) E

• 0.38 < X 4 < 0 . 6 2 O x4 0.77

O > _CD o

16

20 Temperature (K)

Fig. 12.1. The molar volume mixing parameter B^ for all measured liquid hydrogen solutions. If they were ideal, then By would be 0 J/mol.

out a little more in affecting this property. At 18.8 and 20.3 K, we have these excess viscosities at liquid mol fractions of 0.20-0.80 and 0.50-0.50: nH 2 -nD 2 , - 4 to - 1 0 % and - 3 to - 9 % ; nH 2 -HD, - 4 to - 8 % and - 4 to - 6 % ; and HD-nD 2 , - 4 to — 6% and —3 to —6%. The data are somewhat antisymmetric in X J X J (i.e., they don't really peak at the 50-50 mixture), but this may be experimental error. The H 2 -D 2 effect is clearly larger than the others, and we cannot differentiate between the two measured temperatures. We next derive a viscosity interaction parameter Q j (in J/mol). The results show that Cy consistently gets larger as Xj (the heavier hydrogen) increases, but this may also be an error. For Q j at 18.8 and 20.3 K, we have: H 2 -D 2 , - 5 0 to - 7 5 J/mol; H 2 -HD, - 4 0 to - 6 0 J/mol; and HD-D 2 , - 3 5 to - 5 0 J/mol. We crudely fit these data to the empirical equation, good only at 20 K: Cu

(12.12)

-100{A? f f (i)-A?ff(j)} 0 - 6 th

where we use the effective quantum parameter A*ff. The i hydrogen is here lighter than the j lh , so that the result within the brackets will be positive. For D 2 -T 2 , we estimate Q j to be 39 J/mol; for D 2 -DT and DT-T 2 , we have - 2 5 J/mol.

HYDROGEN SOLUTIONS

169

The non-ideality of the liquid surface tension has been measured for the same three binary combinations that were used for viscosity. 9,10 In every case, the measured value is lower than the expected one. In the case of H 2 - D 2 , it amounts to —3 to — 6% at 20 K. The data do not transform to a well-mannered interaction parameter. A theoretical paper exists. 11

Phase Separation Possibilities A continuing worry in the use of hydrogen mixtures is that they will split into different phases, one rich in the lighter hydrogen and one rich in the heavier. No evidence has ever been seen in the liquid phases for anything but complete miscibility. There is clearly no separation into two solid phases upon freezing, or we would see our single bananas split by eutectic points. This exception to this might be H 2 -T 2 , which has not been studied. The case in the lower temperature solid was made worrisome by some x-ray work, which first indicated phase separation of H 2 - D 2 at 16.4 K 1 2 and then at 4.2 K. 1 3 Subsequent thermal studies since then show no separation from 8 K to the melting point. 1 4 , 1 5 Also, the solution of H 2 and D 2 frozen to 2 to 3 K shows no evidence of phase separation by electron diffraction. 1 6 Nuclear magnetic resonance shows but one signal down to 3 K. However, composite signals were seen between 1.1 and 3 K. 1 7 Even these, however, are not mentioned in the next, more extensive study of low temperature H 2 - D 2 . 1 8 Calculations of the phase-separation temperature yield values of 0.8 to 4 K . 1 9 - 2 2 At these low temperatures, molecular diffusion is so slow that it could be impossible for phase separation, even if energetically favored. 2 3 The upshot of this work is that phase separation appears not to be a problem for fusion-related cryogenics. This is because the temperatures are above 4 K and because D 2 -DT-T 2 is the most similar set of hydrogens that can possibly be mixed. It would be interesting to see what would happen with the solid H 2 -T 2 solution, especially under the energetic effect of the tritium beta particle.

Notes 1. B. F. Dodge, Chemical Engineering Thermodynamics (McGraw-Hill, N e w York, 1941), pp. 133-139. 2. I. N . Krupskii, S. I. Kovalenko, and N. V. Krainyukova, Soviet J. Low Temp. Phys. 4, 564 (1978). 3. M. Lambert, Phys. Rev. Lett. 4, 555 (1960). 4. H. F. P. Knaap, M. Knoester, and J. J. M. Beenakker, Physica 27, 309 (1961). 5. V. N. Grigor'ev and N. S. Rudenko, Soviet Phys.-J.E.T.P. 13, 530 (1961). 6. E. C. Kerr, J. Amer. Chem. Soc. 74, 824 (1952). 7. I. Prigogine, R. Bingen, and A. Bellemans, Physica 20, 633 (1954). 8. N . S. Rudenko and V. G. Konareva, Soviet Phys.-J.E.T.P. 22, 313 (1965). 9. V. N. Grigor'ev and N . S. Rudenko, Soviet Phys.-J.E.T.P. 20, 63 (1965). 10. V. N. Grigor'ev, Soviet Phys.-Tech. Phys. 10, 266 (1965). 11. A. Englert-Chwoles and I. Prigogine, N u o v o Cimento, Suppl. 9, 347 (1958). 12. V. S. Kogan, B. G. Lazarev, and R. F. Bulatova, Soviet Phys.-J.E.T.P. 7, 165 (1958).

170

HYDROGEN SOLUTIONS

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

R. F. Bulatova and V. S. Kogan, Soviet Phys.-J.E.T.P. 21, 89 (1965). M. Simon, Phys. Lett. 9, 122 (1964). D. White and J. R. Gaines, J. Chem. Phys. 42, 4152 (1965). See note 2 above. J. R. Gaines, E. M. de Castro, and J. G. Gaunt, Phys. Lett. 8, 167 (1964). D. S. Metzger and J. R. Gaines, Phys. Rev. 147, 644 (1966). See notes 15, 17, and 18 above. I. Prigogine, R. Bingen, and A. Bellemanns, Physica 20, 633 (1954). W. J. Mullin and H. K. Sarin, Phys. Lett. 29A, 49 (1969). C. E. Hecht and A. M. Sapse, Physica 41, 255 (1969). See note 15 above.

13. Solution Fractionation and Non-Ideality

Equilibrium between Phases In the preceding chapter, we encountered Raoult's Law (Eqs. 12.2, 12.3, and 12.4), which derives an ideal solution vapor pressure P, from pure-component vapor pressures. We write Raoult's Law one more time, now at the three-phase point, where solid, liquid, and vapor are in equilibrium. We have: yi P,

= XiP? = SiQ?

(13.1)

th

For the i hydrogen in the family, y i is the vapor mol fraction, X; the liquid mol fraction, and S; the solid mol fraction. The pure-component vapor pressures are Pover the liquid and Q- over the solid. Because P, / P- ^ Q®, y ^ x ^

(13.2)

This is fractionation—a change in composition at a phase boundary. We will also have to extrapolate some of the vapor pressures beyond their normal range. Equimolar H 2 -D 2 should first freeze at about 0^x101+x404

(13.3)

where 6 l and 0 4 are the triple-point temperatures of H 2 and D 2 . This should be about 16.25 K for the equilibrium species. At this temperature, we are 2.4 K below the triple point of pure eD 2 , and we must extrapolate P 4 over this range. The eH 2 , also, is frozen 2.4 K above its triple point, and we must extrapolate Qj upward in temperature. The phase changes can be confusing in a multicomponent solution. The threephase lines for the binary solution eH 2 -eD 2 are shown in Fig. 13.1. 1-3 We have a double banana composed of vapor, liquid, and solid lines. A tie-line at 16.25 K is shown, and we can read off the mol fractions y 4 (point V), x 4 (L), and s 4 (S). The differences—in the middle of the diagram—are considerable. At each pure component side of the diagram, the banana ends in the triple point for that substance. Also Xl

= 1 - x4

Similar equations exist for y,, y 4 and Sj, s 4 as well.

(13.4)

172

SOLUTION FRACTIONATION AND NON-IDEALITY

Fig. 13.1. Experimental three-phase data for the eH 2 -eD 2 system. Suppose we have a 50-50 liquid solution of e H 2 - e D 2 . T h e 0¡ are 13.80 K a n d 18.69 K , so that we expect the first freezing t e m p e r a t u r e to be at 16.25 K (point 0). W e see that it actually comes at a b o u t 16.36 K in Fig. 13.1—a non-ideal effect. A t 16.25 K , the P® a n d P^ are 2 3 9 6 0 a n d 4 740 Pa, where Ps4 is extrapolated. P, is then 14 350 P a a n d we calculate t h a t y 4 ~ 0 . 1 6 5 — a considerable fractionation. Also at 16.25 K , the QS! a n d Qs4 are 28 510 a n d 3 980 Pa, where Q \ is extrapolated. Then, s 4 ~ 0.595. T h e solid is richer in D 2 t h a n is the liquid because the less volatile D 2 prefers to freeze m o r e t h a n the H 2 does. T h e v a p o r is rich in H 2 because it is the m o r e volatile of the pair. T h e value of P, derived f r o m the solid mol fractions is 13 910 P a — off 3 % f r o m the liquid value. T h e difference is caused by non-ideal effects. T h e threephase " b a n a n a " in Fig. 13.1 is nowhere at c o n s t a n t pressure. The p u r e - c o m p o n e n t

SOLUTION FRACTIONATION AND NON-IDEALITY

173

triple-point pressures are 7030 and 17 130 Pa. In between, the pressure bows upward and reaches a maximum P, value of 17240 Pa for x 4 = 0.89 at about 18.15 K. We see, then, that there is a single triple point only for a solution if we freeze very quickly. Otherwise, fractionation occurs and the three phases are in equilibrium over a range of temperature and composition. Figure 13.2 expands our view of the eH 2 -eD 2 phase diagram, as we add the dimension of pressure. In Fig. 13.1, we are looking downward through the top of the rectangular solid of Fig. 13.2 at the temperature-composition plane. Now, however, our temperatures run in the opposite direction, so that the front and back panels are the familiar pressure-temperature phase diagrams. But this turns the double banana over. We can see now that the three-phase banana indeed has a peak pressure in between the triple points. It is 80% of the way toward the pure eD 2 . The hammocks up in the air in Fig. 13.2 are constant-pressure bananas. With them we can explore the relations between solid and liquid and liquid and vapor. They are simpler, in that we need consider only two phases in each banana. Each banana also ends in a single point on the pure-component phase boundary. Three-component solutions are even more complicated. As shown in Fig. 13.3, the phase diagram becomes a triangular prism with temperature rising vertically and with each edge of the prism being a pure component. Each side of the prism describes two components, so that we have three sets of bananas. Each line of each banana is now just the edge of a surface extending across the interior of the prism. There is a gas, a liquid, and a solid surface. The liquid surface is the closest to being flat; the other two bulge out to some unknown degree. Let us consider the ideal liquid of 25% nD 2 -50% DT-25% T 2 (D-T), which we assume is just starting to freeze. The liquid at point L is in equilibrium with the solid at S and the vapor at V, and they are connected by an isothermal tie-line at an estimated 19.733 K. As soon as the first liquid freezes, the composition of the liquid changes and point L moves. We estimate points S and V to have percent compositions of 23-50-27% and 34-51-15%. Our ideal first freezing temperature 0 is estimated to be 19.733 K. The plane of this constant temperature is ABC in Fig. 13.3. It lies very close to the estimated 19.79 K triple point of molecular DT, but is well away from the triple-point temperatures of pure D 2 and T 2 . No quantitative treatment of this phase diagram has yet been done, and the degree of bulge of the vapor and solid surfaces is only a guess. Fractionation upon Freezing When a solution freezes, more of the heavier, less volatile species is to be found in the solid. The last bit of liquid will be very high in the lighter component. We assume, of course, that the freezing is slow compared to the diffusion of the components through the liquid. The liquid must be well mixed. We next calculate the fractionation of the D-T solution as it freezes, assuming a slow freeze, ideal solution, and ideal gas behavior. The results are listed in Table 13.1 in terms of the amount of the liquid sample remaining. We see that the change of

174

SOLUTION FRACTIONATION AND NON-IDEALITY

Temperature (K) Fig. 13.2. Expanded view of the eH 2 -eD 2 binary system, with pressure now included. S, L, and V refer to the solid, liquid, and vapor phases.

175

SOLUTION FRACTIONATION AND NON-IDEALITY

19

18 DT

Fig. 13.3. Schematic of the three-phase surfaces for nD 2 -DT-nT 2 . Freezing has just begun for a 25%-50%-25% solution of the liquid (point L). The separation between the solid, liquid, and vapor lines is exaggerated for clarity. The arrow shows the path of L as fractionation takes place. composition is roughly linear with the logarithm of the fraction of the remaining liquid. For these closely related hydrogens, the change is not great. Point V moves as shown in Fig. 13.3 to the point of 10 - 6 sample remaining. We have said that the composition does not change if the freezing time of the liquid is short. How short is this? Let us have a film of liquid so thin that the radioactive self-heating does not matter. As a practical matter, we allow no more

176

SOLUTION FRACTIONATION AND NON-IDEALITY

T A B L E 13.1 CALCULATED FRACTIONATION UPON FREEZING AN IDEAL 2 5 % D 2 - 5 0 % D T - 2 5 % T 2 ( D - T ) LIQUID SOLUTION

Liquid mol fraction remain 1 4 ( - 1) 1 ( - 1) 4 ( - 2) 1 ( - 2) 4 ( - 3) 1 ( - 3) 4 ( - 4) 1 ( - 4) 4 ( - 5) 1 ( - 5) 4 ( — 6) 1 ( - 6)

Temp. (K)

D2 x4

DT x5

T2 x6

D2 s4

DT s5

T2 s6

Solid atom fraction deuterium, sD

19.733 19.705 19.664 19.633 19.590 19.559 19.517 19.486 19.444 19.413 19.373 19.343 19.304

0.250 0.266 0.291 0.308 0.335 0.353 0.381 0.400 0.429 0.449 0.479 0.498 0.528

0.500 0.498 0.494 0.490 0.482 0.476 0.465 0.457 0.443 0.433 0.416 0.405 0.387

0.250 0.236 0.215 0.202 0.183 0.171 0.154 0.143 0.128 0.118 0.105 0.097 0.085

0.233 0.248 0.271 0.289 0.313 0.332 0.359 0.379 0.407 0.427 0.456 0.477 0.505

0.501 0.501 0.498 0.495 0.489 0.483 0.474 0.466 0.454 0.444 0.430 0.418 0.401

0.266 0.251 0.231 0.217 0.198 0.184 0.167 0.155 0.139 0.129 0.115 0.106 0.094

0.484 0.498 0.520 0.536 0.558 0.574 0.596 0.612 0.634 0.649 0.670 0.685 0.706

Liquid mol fraction

Solid mol fraction

NOTE: The freezing rate is assumed to be so slow that the liquid is always perfectly mixed. When a small amount of liquid with a given composition freezes, it fractionates to the solid composition on the same line. The numbers in parentheses are powers of ten.

than a 0.1 K temperature gradient across this layer. Its thickness L is L = (2KAT/A 0 ) 1 / 2

(13.5)

where K is the true liquid thermal conductivity (about 0.1 W / m - K from liquid H 2 ) and A 0 is the heat generation (5 x 104 W / m 3 for D-T). We find that L is about 0.6 mm. The time for molecular diffusion across this quiet layer is about L2 7rD

(13.6)

where D ~ 2 x 10~9 m 2 /s for liquid D-T, so that t is 18 s. If we freeze the layer in 2 s, we should have no fractionation. If we take 200 s, the effects should be clear. We note that the thermal diffusivity is about 10~ 7 m 2 /s (see Table 8.10). The heat travels out of the liquid faster than the molecules diffuse. As the layer becomes thicker, it is no longer quiet. If it were on the underside of a plate, then the colder liquid near the plate would sink and convection would occur. On top of the plate, this cannot happen. Instead, the top of the liquid gets hotter until it boils, say at 22 K. It .takes a 3 mm thickness to produce this effect. Once the top boils, the liquid begins to stir itself and hot liquid travels to the bottom to cool off. In either case, the bulk liquid may stir itself fairly well and fractionation may not be a problem. N o one has ever checked.

177

SOLUTION FRACTIONATION A N D NON-IDEALITY

TABLE

13.2

SOLID-VAPOR FRACTIONATION OF EQUIMOLAR D - T ASSUMING AN INFINITELY SLOW FREEZE

Solid mol fraction Temp. (K)

D2

DT

s*

s5

19.73 19 18 17 16 15 14 13 12 11 10 9

0.250 0.274 0.314 0.366 0.434 0.521 0.628 0.751 0.873 0.959 0.993 0.999

0.500 0.504 0.505 0.498 0.475 0.429 0.352 0.244 0.127 0.041 0.007 6(-4)

T2 s6

0.250 0.222 0.181 0.136 0.091 0.050 0.020 0.005 5(-4) l(-5) 2(-8)

2(-12)

Vapor mol fraction

Vapor density (mol/m3)

D2 Y4

DT

T2

y5

y6

126 91.4 57.0 34.1 19.4 10.4 5.2 2.4 0.94 0.31 0.08 0.02

0.347 0.380 0.434 0.499 0.576 0.665 0.763 0.859 0.937 0.982 0.997 1.000

0.483 0.475 0.457 0.427 0.380 0.314 0.230 0.140 0.063 0.018 0.003

0.170 0.145 0.109 0.074 0.044 0.021 0.007 0.001

2(-4)

3(-13)

l(-4)

2(-6) 3(-9)

NOTE: The solid starts as 25% nD 2 -50% DT-25% nT2 at 19.73 K. The numbers in parentheses are powers of ten.

Solid-liquid fractionation might be considerable for the most dissimilar hydrogen plan H 2 - T 2 . For D-T, the problem is not so bad, partly because of the similar nature of the components and partly because real freezing will probably be fairly fast. The fractionation worsens, however, at lower temperatures. This is because the isothermal, pure-component vapor pressures of different hydrogens diverge as the temperature falls. Suppose we have somehow frozen a layer of D-T at 19.73 K. The vapor in equilibrium has the composition 35-48-17% and a density of 126 mol/m 3 . Now we let the sample cool very slowly so that fractionation takes place from the vapor to the solid. The results are shown in Table 13.2. By the time we reach 12 K and about 1 mol/m 3 , the vapor is virtually all D 2 . The last layer of molecules to freeze will be pure D 2 . In a real system, there is always about 1 mol% H in the form of H T and H D . So the last layer of molecules in the real solid will be H D . Again, we consider how slow the freeze must be. We return to Eq. 13.6, now to be used in the gas phase. At 15 K, D ~ 2 x 10" 5 m 2 /s; for L = 0.01 m, t ~ 0.5 second. Clearly, the gas diffuses quickly to the surface where it will freeze. The process of freezing may itself be slowed by the capacity of the cryostat and by the heat flow out of the solid hydrogen. But we see that a "slow" freeze in most circumstances will indeed occur. Further trouble will occur once the solution is frozen if parts of the sample have different temperatures. The solid will sublime and recondense in the colder spots with continued fractionation effects. This is especially serious just below the freezing point, where the solid is " w a r m " and the saturated vapor is dense and capable of carrying considerable mass over a period of minutes or hours.

178

SOLUTION FRACTIONATION AND NON-IDEALITY

Liquid mol fraction, heavier species Fig. 13.4. Non-ideal vapor-pressure data for nH 2 -nD 2 and nH 2 -HD. The Raoult's Law pressures P, (in kPa) are noted.

Non-Ideal Vapor Pressure Effects We now turn to non-ideal vapor pressure effects. If the actual measured pressure is P and the Raoult's Law pressure is P„ then we define the percent non-ideality I as I = 100(P - P,)/P,

(13.7)

Experimental data for nH 2 -nD 2 and nH 2 -HD solutions are shown in Fig. 13.4.4 Each curve is at a constant Raoult's Law pressure. The top H 2 - D 2 curve is at half the pressure of the center H 2 -D 2 curve. This means that the top curve is at a lower temperature. We see that I increases as the temperature (and with it, the pressure) decreases. We note, too, that the two bottom curves are at the same pressure but have different compositions. The H 2 - D 2 non-ideality is more than twice that of H 2 -HD.

179

SOLUTION FRACTIONATION AND NON-IDEALITY

The more dissimilar are two hydrogens in a solution, the greater their non-ideality. Finally, we consider the shape of the curve. We expect zero for X; equal to 0 or 1, and we might expect a symmetric curve with the peak at = 0.5. Instead, the peak is skewed to the side of the heavier hydrogen species. We would expect the non-ideality of D-T solutions to be smaller than that of H 2 HD; perhaps only tenths of a percent. But even this matters in the design of distillation columns to separate the hydrogens. 5 - 6 Let us consider a simple model, 7 which considers only binary effects—that is, interactions between two molecules. We rewrite Raoult's Law—Eq. 12.4—to obtain y i P = «fxiP?

(13.8)

where P, the true pressure, now replaces P,. The constant a, contains all the corrections. These are of two types. The first has to do with the non-ideality of the vapor phase. We use only one virial coefficient to describe this—that is PV = RT + B mix P

(13.9)

where l~R -l-

B ^ Li=ij=i X y . yLT 6 ^ L

(13.10)

where B; and Bj are for the i th and j th components of a multicomponent solution. (Here we assume that the cross-virial coefficient By is the average of Bj and Bj, and also that y ^ = yjyj.) The binary form comes about because the second virial coefficient describes interactions between two molecules. The summation in Eq. 13.10 can be over all six hydrogens. For D-T, we would have: Bmix = ylB 4 + yiB 5 + y | B 6 + y 4 y 5 (B 4 + B 5 ) + y 4 y 6 (B 4 + B 6 ) + y 5 y 6 (B 5 + B 6 ) (13.11) We do not add the third virial coefficient C (which describes three-molecule interactions), as this would give us an extremely complicated and unwieldy expression. The second correction is for the non-ideality of mixing the liquids. We consider the Gibbs free energy difference G E between the actual solution for a given temperature, pressure, and composition, and the ideal solution with the same conditions. The superscript " E " stands for "excess." The partial molal excess free energy of mixing of the i th hydrogen is Gf = (dnxGE/ P,. We see, too, that the liquid interaction part I m is positive and always dominates the negative gas part I g . N e a r the freezing points, I m » I g and the solutions have their

184

SOLUTION FRACTIONATION A N D NON-IDEALITY

T A B L E 13.3 NON-IDEAL VAPOR PRESSURE BEHAVIOR OF VARIOUS HYDROGEN SOLUTIONS

Precent liquid composition

Temp. (K)

Pressure P(Pa)

P-P, (Pa)

% nonideality, I

% gas,

% mix,

I*

In,

50 H 2 50 T 2

20 25 30

5.84(4) 2.20(5) 5.85(5)

3.53(3) 5.20(3) 3.65(3)

6.44 2.43 0.63

-1.9 -2.6 -2.8

8.1 4.8 3.2

50 H 2 50 D 2

20 25 30

6.32(4) 2.41(5) 6.39(5)

2.06(3) 3.22(3) 3.20(3)

3.36 1.36 0.50

-1.2 -1.4 -1.4

4.4 2.6 1.7

50 D 2 50 T 2

20 25 30

2.30(4) 1.24(5) 3.96(5)

1.57(2) 3.60(2) 3.80(2)

0.69 0.29 0.10

-0.16 -0.22 -0.25

0.84 0.50 0.33

25 D 2 50 DT25 T 2

20 25 30

2.25(4) 1.22(5) 3.94(5)

9.30(1) 2.20(2) 2.75(2)

0.42 0.18 0.07

-0.08

0.49 0.29 0.18

-0.11

-0.12

NOTE: Only the H 2 -D 2 system is measured. Equilibrium H 2 and D 2 is assumed. The numbers in parentheses are powers of ten. The actual pressure, P, is always larger than the ideal pressure, P,.

maximum non-ideality. At 30 K, Im ~ Ig and the two terms nearly cancel each other. We see also that the non-ideality of the D-T solution is indeed relatively small. We recommend calculated values for this solution, rather than the measured ones. 1 4 , 1 5 These latter are overly large, in our opinion, and may reflect an error caused by excess He 3 pressure.

Notes 1. N. G. Bereznyak, I. V. Bogoyavlenskii, L. V. Karnatsevich, and V. S. Kogan, Soviet Phys.-J.E.T.P. 30, 1048 (1970). 2. D. White and J. R. Gaines, J. Chem. Phys. 42, 4152 (1965). 3. N. G. Bereznyak, I. V. Bogoyavlenskii, L. V. Karnatsevich, and A. A. Sheinina, Ukr. Fiz. Zh. 19, 472 (1974). 4. R. B. Newman and L. C. Jackson, Trans. Faraday Soc. 54, 1481 (1958). 5. M. Kinoshita, Japan Atomic Energy Research Institute, Tokai, Japan, private communication, 1981. 6. M. Kinoshita, Y. Matsuda, Y. Naruse, and K. Tanaka, J. Nucl. Sci. Technol. 18, 525 (1981). 7. A. E. Sherwood and P. C. Souers, Nucl. Tech. Fusion 5, 350 (1984). A1 Sherwood actually did all the work. 8. J. M. Prausnitz, Molecular

Thermodynamics

of Fluid-Phase Equilibria (Prentice Hall,

Englewood Cliffs, N. J., 1969), pp. 220-228, 485-488. 9. See notes 3 and 4 above. 10. I. Prigogine, R. Bingen, and A. Bellemanns, Physica 20, 663 (1954). 11. H. F. P. Knaap, R. J. J. Van Heigningen, J. Korving, and J. J. M. Beenakker, Physica 28, 343 (1962).

SOLUTION FRACTIONATION AND NON-IDEALITY

185

12. See notes 10 and 11 above. 13. L. V. Karnatsevich, N. G. Bereznyak, and I. V. Bogoyavlenskii, Soviet J. Low Temp. Phys. 1, 311 (1975). 14. R. H. Sherman, J. R. Bartlit, and R. A. Breitmeister, Cryogenics 16, 611 (1976). 15. P. C. Souers, E. M. Kelly, and R. T. Tsugawa, Trans. Amer. Nucl. Soc. 28, 202 (1978).

14. Hydrogen-Helium Mixtures

Hydrogen may be mixed with other gaseous elements. Only in the low-pressure gas are these components mutually soluble. At higher pressures, the elemental differences cause the solution to split apart into immiscible phases (i.e., mixtures). The most important nonhydrogenic impurity in heavy hydrogen is helium. This is because tritium decays into He 3 at a rate of 0.015 mol% day • mol T. Considerable effort is spent cleaning He 3 out of D-T gas, only to have it grow back again. Luckily for us, the hydrogen-helium system is the most studied hydrogen mixture.

Solubility of Helium in Liquid Hydrogen We shall here consider very dilute solutions of helium dissolved in liquid hydrogen. We illustrate this small corner of the phase diagram with the most studied combination, He 4 -in-eH 2 . 1 - 5 This is shown in Fig. 14.1, for solubilities of 0.5 to 10 mol% helium. The freezing line for the liquid is the nearly vertical line to the left. The dashed line is the freezing line for pure eH 2 . We see that the presence of dissolved helium lowers the freezing point, just as salt does in water. The fluid phase lies to the right. The lines shown are for liquid hydrogen containing the indicated'mol% helium. If we lower the temperature at constant pressure, the solubility of the helium will also fall. To maintain a constant solubility, we must increase the pressure considerably as we approach the freezing line. These liquid curves are each in equilibrium with a helium-rich vapor (not shown in Fig. 14.1), and it is this vapor that applies the pressure. We always are on a saturated liquid curve with a saturated vapor above it. The percent of helium in each can vary widely, according to the conditions. Some freezing temperatures and pressures are listed in Table 14.1. Let us first consider the liquid hydrogen. We begin with the non-ideal form of Raoult's Law (Eq. 13.8): yiP

= a iXi P?

(14.1) th

where P is the total pressure, while the subscript i refers to the i hydrogen. Also, y; is the vapor phase mol fraction, X; the liquid phase mol fraction, Pf the pure component vapor pressure, and oq is a number describing the extent of deviation from ideal

187

HYDROGEN-HELIUM MIXTURES

20

25

Temperature (K) Fig. 14.1. Phase diagram for dilute liquid He 4 in eH 2 . The solid freezing line is for this solution; the dashed line next to it is the freezing line for pure eH 2 . The numbers indicate percent solubility of helium in liquid hydrogen. The "0" indicates pure hydrogen.

solution behavior. For true Raoult's Law behavior, a; is 1. The non-ideality parameter for hydrogen, a l s in liquid He 4 -eH 2 solutions is shown in Fig. 14.2.6 The liquid phase freezing line data for He 4 -D 2 from 0 to 2 MPa also fall on the curve. 7 The reason for the apparent leveling-off of a i above 10 MPa is not known. The deviations from ideality for the solvent hydrogen are almost an order of magnitude at this point. Now we turn to the minor constituent, the solute helium. Its non-ideal Raoult's Law is y He P = a„ e x He Pf, e

(14.2)

Here a He represents the deviation from ideal behavior. We cannot accurately use this equation, because most of our temperatures of interest are above the 5.20 and 3.32 K critical points of He 4 and He 3 . 8 To avoid this problem, we define the Henry's Law constant as k He = «HePne = yHeP/xHe

(14.3)

where the expression on the far right contains the measured quantities. The constant k He simply relates the helium solubility to the helium partial pressure in an empirical

188

H Y D R O G E N - H E L I U M MIXTURES

T A B L E 14.1 LIQUID COMPOSITIONS ON THE FREEZING CURVE FOR VARIOUS SOLUTIONS OF HELIUM IN HYDROGEN

P (MPa)

^HE

HE

yHE

X

(MPa)

(Tm - 0.) (K)

He 4 in eH 2

0.25 0.50 0.75 1.0 1.5 2.0 2.5 3.0

0.000 9 0.0019 0.0027 0.003 5 0.005 0 0.005 9 0.0067 0.007 3

40. For the L transition, the mean tritium beta-particle energy is not reached until cesium (Z = 55), and the maximum beta-particle energy is not reached until francium (Z = 87). The M-edge is not reached for the mean beta energy until neptunium (Z = 93). It is always possible for tritium to excite M-transitions, although they may be of low probability.

2 e « o cd « >>B S 8 Ï 2 TS 13 Sä "O S tZ ue es p o

ci m ^ ^ ** ^ I I I I I I I O OOOOOO X X X X X X X m r^ Tt m o ;0 at

D

Element

Atomic number Z

OfJ (0 keV)

1.6 fJ (10 keV)

4 fJ (25 keV)

J"1

(eV) -l

8.(0)

F

Be Al Fe Cu Ag Au U

4 13 26 29 47 79 92

~0.7 ~0.5 0.35 0.35 0.5 ~0.6 ~0.6

0.052 0.18 0.27 0.31 0.40 0.49 0.50

0.049 0.16 0.27 0.31 0.40 0.50 0.52

1.6(15) 6.2(14) 1.6(14) 7.5(13) 1.2(14) 1.2(14) 1.2(14)

2 . 6 ( - •4) 1 . 0 ( - 4) 2 . 6 ( - 5) 1 . 2 ( - 5) 2.0( —5) 2.0( —5) 2.0( —5)

0.69 0.70 0.73 0.73 0.79 0.82 0.84

0.371 0.357 0.315 0.315 0.236 0.198 0.174

NOTE: The numbers in parentheses are powers of ten.

where rja is the probability of scattering back into the container, and ea is the fraction of energy retained by the scattered electrons. We see that some electrons are totally lost in the walls, while some lose a part of their energy. The probability of reflection may be represented by the general empirical equation ii« = ii 0 cxp[(-Iii(i f o )-0.119)(1 - cosa)]

(17.22)

where t\0 is the probability at normal incidence. The value of rjx gets larger as a increases. The more grazing the surface collision, the greater is the probability of being scattered. Equation 17.22 cuts off at 0.888, although there are undoubtedly cases where rjx is as large as 1. Next, we need Approximate numbers are listed for various elements in Table 17.10 for energies of 0 fJ, 1.6 fJ (10 keV), and 4 f j (25 keV). The latter two are comparatively well studied, although the numbers could easily be ±10% off. At the higher energies, the changes in t]0 are small and a linear fit is good enough. At lower energies, r/0 increases and grows to a considerable value for the low-atomic-number elements. The 0 fJ value for beryllium is just a guess—it could be 1. Copper has been studied in some detail, but stainless steel has not. The iron numbers are copper-based estimates to be used for steel. For other elements, we can only interpolate as a function of atomic number. It is important that Table 17.10 not be taken too literally. Note of the data is well defined, and they are liberally smoothed and estimated across the periodic table. Below 1.6 fJ (10 keV), we may use this empirical equation: ito = »to(O)exp(-DE)

(17.23)

where tjo(0) is the value of tj0 at zero energy (column 3 in Table 17.10) and D is a constant listed in the same table. Equation 17.23 can be used across the entire tritium energy spectrum for iron and copper without significant error. The numbers are approximate, because little experimental work has been done at low energy.

230

R A N G E OF TRITIUM RADIOACTIVITY

TABLE

17.11

F R A C T I O N OF THE T R I T I U M B E T A - P A R T I C L E E N E R G Y A B S O R B E D BY A T H I N T R I T I U M SAMPLE OF W I D T H L WITH E L E C T R O N REFLECTION AT THE M E T A L W A L L S

Wall material pL (mol/m 2 ) l(-4) 5(-4) l(-3) 5(-3)

0.01

0.025 0.05 0.10 0.20 0.30 0.50 0.75

Be

Cu

Ag

Au, U

0.02 0.09 0.13 0.26 0.33 0.42 0.51 0.61 0.73 0.80 0.90 0.99

0.02 0.08 0.12 0.27 0.35 0.48 0.59 0.69 0.80 0.86 0.92 0.97

0.02 0.10 0.15 0.31 0.40 0.53 0.62 0.72 0.82 0.87 0.93 0.98

0.03 0.12 0.17 0.35 0.44 0.57 0.67 0.77 0.85 0.90 0.95 0.98

There has been very little work as well on the energy retained by back-scattered electrons. We use our own empirical formula e„ = e « ( 0 ) e x p [ - F ( l - cos a)]

(17.24)

where e a (0) is the retained energy for an angle of incidence of 0°, and F is a constant that is set to make sx = 1 for a = 90°. The constants are listed in Table 17.10. The actual data are for copper, silver, and gold, and the remaining numbers are estimates. If we were to do a really nice model of beta-energy absorption, we would have to follow electrons across the hydrogen sample, watch them bounce off the walls, and track them back through the hydrogen. In a low-density sample, the beta particle might bounce back and forth many times. To do this right, we should also follow beta particles moving in all directions from all parts of the sample. We shall explore this with a simple model. Consider a hydrogen sample of density p with a distance L between two flat vessel walls. This is the pL-sized sample of our earlier calculation. Now, however, we place a single emitting beta particle in the center and give it a single mean path at 45° angles to the walls, so that its wall-towall path distance is 3 1 / 2 . We start a beta particle at the center and follow it, bouncing off the walls, returning over the same track, until it loses all its energy. We use the nine-channel tritium energy spectrum. Sample results are shown in Table 17.11 for this simple model. There is a considerable difference between these results and the simple exponential model of Table 8.2—especially at low px values. The model of this section says that a substantial fraction of energy is deposited in a low p L gas because the beta particles bounce from wall to wall many times. Yet our model is too simple, because we have approximated all beta particles with a single path. Should any experiment be run with

RANGE OF TRITIUM RADIOACTIVITY

231

pL < 0.05, the researcher should probably take the time to run a larger model with a spectrum of paths.

Notes 1. L. M. Dorfman, Phys. Rev. 95, 393 (1954). 2. T. E. Cravens, G. A. Victor, and A. Delgarno, Planet. Space Sci. 23, 1059 (1975). 3. J. Schou and H. Sorensen, J. Appl. Phys. 49, 816 (1978). 4. M. J. Berger and S. M. Seltzer, "Tables of Energy-Losses and Ranges of Electrons and Positrons," in Studies in Penetration of Charged Particles in Matter, National Academy of Sciences Publication 1133, Washington, D.C., 1964, p. 205. 5. J. C. Ashley, C. J. Tung, R. H. Ritchie, and V. E. Anderson, IEEE Trans. Nucl. Sci. NS-23, 1833 (1976). 6. A. F. Akkerman and G. Y. Chernov, Phys. Status Solidi 89B, 329 (1978). 7. P. C. Souers, R. T. Tsugawa, and R. R. Stone, Rev. Sci. Instrum. 46, 682 (1975). 8. A. H. Compton and S. K. Allison, X-Rays in Theory and Experiment, 2d ed. (D. Van Nostrand, Princeton, N.J., 1935), p. 90. 9. CRC Handbook of Chemistry and Physics, 62d ed., R. C. Weast, ed. (CRC Press, Boca Raton, Fla., 1981), pp. E-183 to E-187. 10. M. Green, "The Efficiency of Production of Characteristic X Radiation," in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, V. E. Cosslett, and A. Engstrom, eds. (Academic Press, New York, 1963), p. 185. 11. See note 10 above. 12. G. F. Brewster, J. Amer. Ceram. Soc. 35, 194 (1952). 13. W. H. McMaster, N. Kerr Del Grande, J. H. Mallett, and J. H. Hubbell, Compilation of X-Ray Cross Sections, Lawrence Livermore National Laboratory Report UCRL-50174, Sec. II, Rev. 1, Livermore, Calif. 94550 (1969). 14. See note 2 above. 15. See note 4 above. 16. E. H. Darlington and V. E. Cosslett, J. Phys. 5D, 1969 (1972). 17. E. H. Darlington, J. Phys. 8D, 85 (1975).

18. Radiation Damage in Hydrogen

The most conspicuous feature of tritium is its beta particle. The first thing it usually hits is the hydrogen in the sample itself. In this chapter, we shall consider the damage that it causes in hydrogen. These processes are basic and are essentially the same for externally fired electrons, protons, or any other energetic particle. Only the efficiencies change. The tritium beta particle does not lose its energy in one great crash but in bits and pieces, varying from about 6 aJ (40 eV) down to small values.

The Basic Processes Suppose a femtojoule (keV) electron starts into hydrogen gas. The most basic process that occurs is ionization: H 2 + 2.472 aJ (15.43 e V ) ^ H 2 + + e"

(18.1)

We see the minimum beta-particle kinetic energy to the left that it takes to cause this breakup. 1 The H 2 + ion is unstable and will almost certainly react to form an H 3 + ion, but we leave this story for Chapter 19. The combination of [H 2 + -e~] is called an ion pair, and this is the basic unit in radiation damage. At this point, we encounter our first ambiguity of the sort that enters the many electron damage models. Most models consider only the reaction of Eq. 18.1, but we may also have H 2 + 2.90 aJ (18.08 e V ) ^ H + H + + e"

(18.2)

We here produce one ion pair. Also, we have 2 , 3 H 2 + 5.07 aJ (31.67 e V ) ^ 2H + + 2e"

(18.3)

Here, we produce two ion pairs in one reaction. Table 18.1 shows the results of such a calculation for electrons entering at an initial energy of 16 fJ (100 keV) and degrading all the way to zero energy. 4 To convert the initial electron energy into a number of events, we use the average measured value for high-energy electrons of 5.86 aJ (36.6 eV)/ion pair. 5 We see that only 87% of the ion pairs are computed to be of the H 2 + - e " type. We also have to remember to double the 2H + events for their contribution to forming ion pairs.

233

RADIATION D A M A G E IN H Y D R O G E N

T A B L E 18.1 CALCULATED RADIATION D A M A G E EVENTS IN H 2 G A S BOMBARDED WITH O N E 1 6 f J ( 1 0 0 k e V ) ELECTRON THAT LOSES ALL ITS ENERGY

Processes

Products

Ionization

H2+ + e H + + e~ + H

2(H + + e " )

Excitation

Atom production (sum from above)

H2* H 2 + hv H2* - 2 H H2* - H * + H H 2 * -> 2H*

Number of events

Ion pairs

2 377 341 14

0.87 0.12 0.01

2732

1.00

1788") 740 I 614 [ 28 J

1

3105

1.14

NOTE: There is no further reaction or recombination of the radiation damage products.

The electron that comes loose in ionization by the beta particle is called a secondary electron. If we add up the minimum energies of the basic ionization processes in Table 18.1 an assume that all extra energy goes to the kinetic energy of the secondary electron, we find an "average" secondary energy of 3.4 aJ (21 eV). This gives us a rough physical picture, but it is not accurate. These electrons can go on to cause further ionization, thereby producing tertiary and, in the next step, quaternary electrons. There is no way to tell one electron from another, and any "secondary" electron experiment comes down to what low-energy electrons the equipment is capable of detecting. As the beta particles lose their kinetic energy, the chances of causing ionization in any given collision increase. But for beta particles under 60 aJ (400 eV), a new mechanism of energy loss occurs: molecular excitation to an electronically excited H 2 * molecule. There are two kinds of molecular excitations: to singlet and to triplet levels. A hydrogen molecule has two electrons, each with a spin quantum number of 1/2. These may be combined to yield an overall spin of 0 for a singlet state or 1 for a triplet state. The first excited states are a singlet and a triplet at roughly 1.6 to 1.8 aJ (10 to 11 eV). The singlet de-excites mainly by radiation but the triplet de-excites by dissociation into two hydrogen atoms. This is the main source of "directly" formed atoms. We turn again to Table 18.1. The model here shows 3 170 H 2 * molecules formed as the 16 fJ electron loses all its energy. This is 1.16 H 2 * molecules/ion pair. However, the H 2 * molecules are not stable. The model says that 56% of them probably lose their energy by emission of radiation; these are singlet states. The other 44% of the excited molecules—primarily triplet states—dissociate into atoms in the ground state, H, and various excited atomic states, H*. Excited-state dissociation produces 1.02 H atoms/ion pair. There is a further smaller contribution of

234

RADIATION D A M A G E IN H Y D R O G E N

TABLE

18.2

SUMMARY OF ELECTRON ENERGY LOSS PROCESSES IN HYDROGEN ACCORDING TO VARIOUS MODELS BASED ON EXPERIMENTAL CROSS-SECTION D A T A

Initial electron energy

Total energy per ionization

aJ

eV

aJ/ion pair

eV/ion pair

2.6 2.8 3 4 5 10 15 20 50 100 200 500 1000 2000 3 000

16.2 17.5 18.8 25.0 31.2 62.5 93.8 125 312 625 1250 3125 6250 12 500 18750

42.4 23.4 17.3 11.6 9.84 7.41 6.83 6.61 6.30 6.16 6.06 5.94 5.86 5.86 5.86

265 146 108 72.4 61.5 46.3 42.7 41.3 39.4 38.5 37.9 37.1 36.6 36.6 36.6

Per ion pair Molecular excitations

Direct atoms

22 10 6 4.2 3.1 1.9 1.5 1.5 1.3 1.2 1.1

30 13 8 5 3.5 2.1 1.7 1.5 1.3 1.2 1.1

1 1 1 1

1 1 1 1

Protons — — —

0.015

0.11

0.13 0.16 0.16 0.12 0.11"

0.11" 0.11" 0.11"

0.1 l a 0.11a

NOTE: Each line presents the results after the electron of initial energy has degraded to zero energy. The superscript (a) indicates extrapolations.

0.12 H atoms/ion pair by the higher-energy process of dissociation following ionization. The total is 1.14 "directly formed" H atoms/ion pair. When the incident electron energy falls below 1.5 aJ (10 eV), the electron cannot cause either ionization or molecular vibrations. Below that is excitation of rotational excitations plus collisions which transfer kinetic energy to the target molecules. The sequence of radiation damage processes is shown by the linear energy transfer function shown in Fig. 17.1 in the preceding chapter. 6 , 7 We enter with our highest-energy electron at the far right. Ionization is the main process, but about one-tenth of that produces atoms. The probability for ionization increases as the electron's energy drops, because the electron has longer to interact with the target molecules. The peak is at 15 aJ (100 eV). Near 1.5 aJ (10 eV), ionization dies away but molecular excitation, with roughly half of it leading to dissociation, pops up. At lower energy, this dies away and molecular vibration occurs. Rotational excitation is not shown, but it would occur farther to the left. There have been many computer models that follow the instantaneous electron energy as it degrades by the various processes to zero. 8 - 1 3 The overall agreement is not particularly good. A summarized, smoothed version of all this work is presented in Table 18.2. The electron energies are not instantaneous but are the initial ones in a sequence that ends with zero energy in each case. For example, an electron with

RADIATION DAMAGE IN HYDROGEN

235

100 aJ initial energy produces 100/6.16 = 16.2 ion pairs as it loses all its energy. The total energy per ionization, which includes all ongoing processes, is measurable and constitutes the basic information. 1 4 , 1 5 The "high-energy" asymptote of 5.86 aJ (36.6 eV)/ion pair is perhaps accurate to ± 3%. As the initial electron energy decreases, the efficiency of ion pair production also decreases, so that the energy per ion pair goes up. This dramatically increases when the initial electron energy falls below 5.86 aJ. There is still a small probability of ionization as long as we are above the true molecular ionization energy of 2.47 aJ (15.4 eV). Below 2.47 aJ, no ionization can occur and the energy per ion pair becomes infinite, but atoms can still be produced by direct dissociation. Most ionization models ignore proton formation. As seen in Table 18.2, it is important enough to be considered. 16 The H + / H 2 + ratio never falls below 0.11 except at very low energies. All of the results discussed in this section assume that the number of radiationcreated species is small in number compared to the number of hydrogen molecules— that is, that we do not have a plasma. This is certainly true for tritium in all forms prior to the nuclear fusion. Should the number of electrons become large relative to the number of molecules, the energy deposition results change dramatically, and we are entering the plasma region. 17

Cross Sections The models of electron radiation damage all use experimental data for specific processes. These data are usually given as a cross section (in m 2 /molecule) for a given reaction at an instantaneous electron energy. We recall that the geometric cross section of a hydrogen molecule is its area of about 4 x 10~2° m 2 . Most of the cross sections shown in Table 18.3 are smaller than this, which means that the probability of occurrence is less than 1 in any single collision. The cross sections for ionization are total and include formation of both H 2 + and H + i o n s . 1 8 - 2 0 We see that there is no difference between H 2 and D 2 , and we would expect T 2 to also be the same. The next column is total singlet molecular excitation. We note that a fraction of these excitations do end up as dissociated atoms; the rest de-excite with the emission of radiation. 21 The triplet excitations all de-excite with the ultimate production of atoms. 2 2 , 2 3 One last curiosity that will return to haunt us later is the endothermic reaction to form the hydride ion H~: H 2 + / T + 1.1 a J ( 6 . 6 e V ) - I T + H

(18.4)

where the minimum energy needed is found experimentally. 24,25 It is larger than the 0.595 aJ (3.712 eV) expected from thermodynamics. Above 2.18 aJ (13.6 eV), the atom is excited, H*. Above 2.75 aJ (17.2 eV), both H " and H + can be formed simultaneously. In any case, the cross section for hydride formation is low, with a

236

RADIATION D A M A G E IN H Y D R O G E N

TABLE

18.3

CROSS SECTIONS FOR THE BASIC HIGH-ENERGY PROCESSES OF ELECTRON ENERGY LOSS IN HYDROGEN

Cross section ( x 10 Electron energy aJ

eV

1.8 2.2 2.6 3.0 3.5 4 6 8 10 15 20 50 100 200 500 1000 2000 3 000

11.2 13.8 16.2 18.8 21.9 25.0 37.5 50.0 62.5 93.8 125.0 312.5 625.0 1250 3125 6250 12500 18 750

Total ionization H2

D2

—

—

—

—

0.040 0.20 0.36 0.50 0.84 0.94 0.98 0.95 0.89 0.58 0.28 0.16 0.075 0.042 0.023 0.017

0.046 0.22 0.38 0.51 0.85 0.95 0.98 0.96 0.89 0.59 0.29 0.16 0.075 0.042 0.023 0.017

Total singlet excitation —

0.053 0.080 0.12 0.16 0.19 0.29 0.33 0.31 0.29 0.28 0.18 0.10

20

m2/molecule)

Dissociative singlet excitation — —

0.006 0.026 0.030 0.031 0.041 0.040 0.039 0.032 0.029 0.016 0.010

Total triplet excitation

Hydride ion formation

0.27 0.75 0.72 0.59 0.43 0.32 0.12 0.054 0.029 0.009 0.004

1.2(-4) 1.8(-4) 1.8(-4) 2.1(-4) 2.3(-4) — — — — — —

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

peak value of 3.5 x 1 0 - 2 4 m 2 at 2.27 aJ (14.2 eV). Other values are listed in Table 18.3, and they are small in comparison with the other numbers. The data are for H " , and the values for D~ are lower yet. 2 6 So we write off the hydride ion at this point, only to find it making a surprise reappearance in Chapter 19.

Low-Energy Processes Below 1.5 aJ (eV), the molecules undergo their final low-level excitations. Table 18.4 shows the results of a calculation using 1 600 aJ (10 keV) electrons, which lose all their energy in n H 2 gas at 298 K. 2 7 According to this, one-tenth of the initial electron energy ends up in translational (elastic), vibrational, and rotational energy of the neutral hydrogen molecules. The second column is referenced to the "high-energy" value of 5.86 aJ (36.6 eV)/ion pair. These are "directly formed" excitations, with no ion or atom recombination effects included. Table 18.5 lists the cross sections from which the low-energy models are m a d e . 2 8 " 3 2 The chances of elastic collision (i.e., transfer of energy to the translational energy of the target molecule) are very high.

RADIATION DAMAGE IN HYDROGEN

237

T A B L E 18.4 CALCULATED LOW-LEVEL EXCITATIONS IN n H 2 G A S AT 2 9 8 K

Fraction initial energy

Events/ ion pair

Av = 0, AJ = 0 (elastic)

0.012

Av =

higher

0.005 0.025 0.010

4.2 11.9 2.7

AJ = 0 J = 0->2 J = 1 —» 3 higher

0.036 0.002 0.006 0.002

2.4 0.1 0.4 0.1

0.098

> 21.8

v=

0,

J =

0 -> 1,

0 -> 2 1 ->3

—

NOTE: The gas is bombarded with electrons that are at an initial energy of 1.6 fJ (10 keV), and that lose all their energy. Also, v is the molecular vibrational quantum number, and J is the molecular rotational quantum number.

T A B L E 18.5 CROSS SECTIONS FOR ELASTIC ENERGY LOSS, ROTATIONAL AND VIBRATIONAL EXCITATION OF H 2 BY ELECTRONS

Cross sections ( x 10 v = 0 -> 0

v

Electron energy aJ

eV

0.01 0.025 0.05 0.10 0.25 0.50 1.00 1.25 1.50

0.063 0.156 0.313 0.625 1.56 3.12 6.25 7.80 9.38

Elastic AJ = 0

—

10 11.3 13.1 13.8 11.5 10.3 8.6

J = 0->2

J = 1 ->• 3

0.04 0.10 0.18 0.35 1.0 1.8 1.7 1.4

0.02" 0.06" 0.1 A 0.2 A

1.2

0.6 1.1 1.0 0.8 0.7

AJ = 0

20

m2)

= 0-»l

J = 0 - 2

v = 0 -3

"Total"

—

—

—

—

—

—

—

—

—

—

—

—

—

0.15 0.25 0.15 0.11 0.085

0.13A 0.23A 0.15A 0.10 A 0.07A

0.08 0.14 0.09 0.06 0.04

—

0.012" 0.034 0.023 0.019 —

NOTE: The letters "v" and " J " are the molecular vibrational and rotational quantum numbers. The ( a ) indicates an estimate, where the J = 1 —»• 3 transition is taken to be 60% as strong as the J = 0 -> 2 transition. The "total" cross section for the v = 0 -* 2 transition probably represents a normal H 2 molecule at room temperature.

238

RADIATION DAMAGE IN HYDROGEN

The Background Electron Energy Spectrum The models we have considered literally end with electrons, ions, atoms, and other excitations frozen in position awaiting the next step. Even with this simplicity, there is considerable difficulty in imagining the cloud of secondary and tertiary electrons set Secondary electron energy (eV)

10

1

100

10

500

50

Secondary electron energy (aJ) Fig. 18.1. Measured spectrum of secondary electron energies at all angles from hydrogen gas bombarded by 80 aJ (500 eV) electrons.

RADIATION DAMAGE IN HYDROGEN

239

free by the beta particle. In the model in Table 18.1, we found an average secondary electron energy of 3.3 aJ (21 eV) just after ionization. After molecular excitation, the electrons are down to 1.3 aJ (8 eV). The 1.6 fJ model gives an average energy of 0.55 aJ (3.5 eV) at this point. 3 3 With no model of recombination, we really cannot define the average state of a secondary electron. One practical way is to look at the electrons emitted from a sample. Figure 18.1 shows the secondary electron energy spectrum, averaged over all angles from 80 aJ (500 eV) electrons hitting hydrogen gas. The sample is thin enough that the initial energy does not degrade by more than 10%, which differs from our models in which the primary electron degrades to zero energy. A further problem is that we do not know how to extrapolate below 0.7 aJ (4 eY). Integration of Fig. 18.1 yields an average secondary electron energy of about 2.1 aJ (13 eV). A calculation for a bulk sample of hydrogen shows that the mean emitted electron energy will increase with the initial electron energy to an asymptotic limit of about 4.8 aJ (30 eV) at 2.2 fJ (14 keV). 34 For the mean tritium beta particle, this model predicts emitted secondary electrons with 4.5 aJ (28 eV) energy. The most recent work on secondary electrons from solid hydrogen uses a 45 V bias to collect everything under this energy. 35 We note that the electrons that leave the sample are more energetic than the average. This effect is not included in the simple energyabsorption models of Chapter 17.

Notes 1. G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, 2d ed. (D. Van Nostrand, Princeton, N.J., 1950), pp. 351-352, 360, 363, 459. 2. See note 1 above. 3. G. Herzberg, Atomic Spectra and Atomic Structure (Dover, New York, 1944), p. 11. 4.' K. Okasaki and S. Sato, Bull. Chem. Soc. Japan 48, 3523 (1975). 5. G. N. Whyte, Rad. Research 18, 265 (1963). 6. T. E. Cravens, G. A. Victor, and A. Delgarno, Planet. Space Sci. 23, 1059 (1975). 7. T. E. Cravens, College of Engineering, University of Michigan, Ann Arbor, Mich. 48109, private communication, 1981. 8. See notes 4 and 6 above. 9. W. M. Jones, J. Chem. Phys. 59, 5688 (1973). 10. J. Bergeron and S. Collin-Souffrin, Astron. Astrophys. 25, 1 (1973). 11. R. H. Garvey, H. S. Porter, and A. E. S. Green, J. Appl. Phys. 48, 190 (1977). 12. A. E. S. Green, R. H. Garvey, and C. H. Jackman, Int. J. Quant. Chem., Symp. 11, 97 (1977). 13. D. A. Douthat, J. Phys. 12B, 663 (1979). 14. See note 5 above. 15. D. Combecher, Rad. Research 84, 189 (1980). 16. J.-C. Gomet, Compt. Rend. 287B, 11 (1978). 17. See notes 6 and 10 above. 18. B. L. Schram, F. J. DeHeer, M. J. Van der Wiel, and J. Kistemaker, Physica 31, 94 (1965). 19. D. Rapp and P. Englander-Golden, J. Chem. Phys. 43, 1464 (1965). 20. I. R. Cowling and J. Fletcher, J. Phys. 6B, 665 (1973). 21. W. T. Miles, R. Thompson, and A. E. S. Green, J. Appl. Phys. 43, 678 (1972). 22. S. Chung, C. C. Lin, and E. T. P. Lee, Phys. Rev. 12A, 1340 (1975).

240

REACTIONS IN IRRADIATED HYDROGEN

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

S. J. B. Corrigan, J. Chem. Phys. 43, 4381 (1965). G. J. Schulz, Phys. Rev. 113, 816 (1959). D. Rapp, T. E. Sharp, and D. D. Briglia, Phys. Rev. Lett. 14, 533 (1965). See note 25 above. Seenöte 13 above. R. J. W. Henry and N. F. Lane, Phys. Rev. 183, 221 (1969). R. W. Crompton, D. K. Gibson, and A. I. Mcintosh, Aust. J. Phys. 22, 715 (1969). F. Linder and H. Schmidt, Z. Naturforsch. 26A, 1603 (1971). A. K. Pande and R. S. Singh, Ind. J. Pure Appl. Phys. 15, 493 (1977). A. Klonover and U. Kaldor, J. Phys. 12B, 3797 (1979). See note 13 above. See note 10 above. H. Sorenson, Nucl. Instr. Methods 132, 377 (1976).

19. Reactions in Irradiated Hydrogen

In Chapter 18, we listed the basic ways in which the tritium beta particle, or any other energetic particle, damages molecular hydrogen. The basic products are free to react further or recombine.

Kinetic Equations Most of the reactions we shall consider involve two species. It may be an ion and a molecule or an ion and an electron. In the hydrogen gas, let us consider species A and B, each with the concentration of particles/m3, [A] and [B], Gas kinetic theory predicts a collision frequency, z, leading to a reaction of 1 z = 7td2q[A][B]v

(19.1)

where d is the distance between the centers of the two particles (an encounter distance), q is the probability of reaction (dimensionless), and v is the mean speed of the two interacting particles, as defined by v = (4kT/7t/z)1/2

(19.2)

where m is an averaged mass for the two particles and k is Boltzmann's constant. The term 7td2v is the volume swept out by a moving particle. Then, z is in the units of particles 2 /m 3 • s (or collisions/m3 • s). The interaction area times the probability is the cross section, a: a = Jid2q

(19.3)

If we divide out the concentrations in Eq. 19.1, we have the rate constant, k, in m 3 /particle • s: k = 6 ) - > H - + H

(19.25)

and e~ + H 2 * (high

3

tO-H-

+H

(19.26)

249

REACTIONS IN IRRADIATED HYDROGEN

where Eq. 19.25 is currently the favored one. 5 0 In Eq. 19.25, v is the molecular vibrational quantum number and Eq. 19.26 contains an excited triplet electronic state. For this to work, the tritium beta particle must maintain a constant and large supply of vibrationally excited molecules. At first glance, this does not seem likely, and we seem to have violated our rule not to react two radiation-damage species in low concentration. Specific data are not yet available, however, and we shall have to wait.

Positive Ion Clustering Reactions There is an entire series of heavier positive ions formed by the reaction H 3 + + xH 2

H 3 + (H 2 ) x

(19.27)

All such ions have an odd number of atoms; even-numbered ions are never seen. 5 1 , 5 2 The reaction is exothermic, but not by much, and the reaction energy gets smaller as the ions get larger. Measured room-temperature enthalpies obtained as we progressively add the first four molecules are 0.058 aJ (0.36 eV), 0.020 aJ, 0.018 aJ, and 0.008 aJ. 5 3 The calculated energies are comparable. 5 4 , 5 5 The attractive force for these reactions is the ion's electric charge and the polarizability of the nearby molecules. This force must be greater than the thermal energy of the ions and molecules—that is to say, we need 56 ae 2 /[(47te 0 ) 2 2r 4 ] » kT

(19.28)

where a is the polarizability of the hydrogen molecule (8.9 x 10~ 41 C• m 2 / V-molecule), e the electronic charge (1.6 x 10~ 19 C/charge), e 0 the permittivity constant (8.854 x 10~ 12 C/V • m), and r the ion-molecule distance. For r = 0.25 nm, the attractive energy is about 0.024 aJ (14 kJ/mol) and the room-temperature kinetic energy is about one-sixth of that. The experimental work has made these ion clusters by bombarding solid hydrogen with 4 to 140 aJ electrons. 57 The largest seen is H 4 1 + . Ions with special stability are H 9 + , H 1 5 + , and H 2 7 + , and these correspond to "closed shells" of clustering molecules. From Eq. 19.28 we expect the cluster sizes to increase as the temperature decreases. The equilibrium of these ions, from H 3 + to H u + , has been studied in electron-irradiated hydrogen gas from 100 to 345 K . 5 8 , 5 9 The equilibrium constant K|j(in P a - 1 ) for ions i and j (where j is the next one in the sequence) is given by Ky = K8exp(VT)

(19.29)

The measured constants, K?j and are listed in Table 19.5. They are strictly true only for the narrow pressure range of 270 to 800 Pa. Above this pressure, the equilibrium constants decrease. 60 We also implicitly assume that K^ is independent of the irradiation energy and flux. We average these to obtain

250

R E A C T I O N S IN I R R A D I A T E D

TABLE

HYDROGEN

19.5

EXPERIMENTAL EQUILIBRIUM C O N S T A N T S FOR POSITIVE ION C L U S T E R S AT ABOUT 6 0 0 k P a

Atom numbers ij

Kn

(Pa' 1 )

3,5 5,7 7,9 9, 11

3.7 4.7 3.5 6.0

x x x x

10"11 IO"10 10' 1 0 IO"10

(K)

h

Temp, range (K) measured

4 840 2070 1910 1200

240-435 130-190 115-160 100-120

NOTE: The letters i and j refer to the number of atoms in an ion before and after a molecule is added.

K ^ S x . . - « , ^ )

(19.30)

where Nj is the number of hydrogen atoms in the heavier ion of the pair. 61 This equation does not correct for the special stability numbers but does allow us to extrapolate to heavier ions. For the H 3 + — H s + ion pair at equilibrium, [H5 + ] = [H 3 + ]PK 3 5

(19.31)

We may add in all other ions. For example, the mole fraction of H 7 + in a mixture of H 3 + through H 13 ions is +

C(H 7 ) =

P2K35K57 1 + PK 35 [1 + PK 57 (1 + PK 79 [1 + PK 9 i 1 1 (1 + PK 1 1 1 3 )])] (19.32)

Figure 19.2 shows the calculated equilibrium mol fractions of positive cluster ions in 660 Pa gas from 40 to 400 K. This pressure is the same as that of the equilibrium constant study, so that the results are probably good. Note how, upon cooling, the H 7 + quickly disappears in favor of the more stable H 9 + ion. It is easy to see that the H 3 + ion is predominant only above room temperature. The results, if we extrapolate to other regimes, are very interesting. For T 2 vapor above the liquid and solid, we predict that the T 3 3 + ion is most plentiful from 12 to 21 K and T 3 1 + from 21 to 30 K. 62 We find an estimated maximum ion concentration of 1014 T 3 1 + ions/m 3 at 28 K. This extrapolation does not take into account the special stability of the T 2 7 + ion. If we extrapolate this model upward to higher pressures at 300 K, we find the curves of Fig. 19.3. The H s + ion is expected to predominate at 0.1 MPa and the H 9 + ion at 10 MPa. The effect of high pressure on cluster ion formation has not been experimentally studied, and this extrapolation is questionable.

251

REACTIONS IN IRRADIATED HYDROGEN

100

200

300

400

Temperature (K) Fig. 19.2. Calculated positive ion cluster equilibria in irradiated H 2 gas at 660 Pa. Experimental equilibrium constants are used. (Courtesy P. C. Souers and the Institute of Physics.)

c

o a> > o

a

c o u 03

Pressure (Pa) Fig. 19.3. Estimated positive ion clustering in pressurized room-temperature hydrogen gas. This is a considerable extrapolation of low-pressure data.

252

REACTIONS IN IRRADIATED HYDROGEN

Negative Ion Clustering Reactions In the gas phase, no stable negative cluster ions are known. Both H 2 ~ and H 3 ~ ions have been seen in discharges but are given half-lives on the order of 10~ 15 s. 6 3 There seems no reason why the clustering reaction H " + XH2 - > H " ( H 2 ) x

(19.33)

cannot occur. Calculated binding energies for adding the first six hydrogen molecules are: 0.015 (0.094 eV), 0.012, 0.011, 0.009, 0.007, and 0.007 aJ. 6 4 We recall that kT at room temperature is 0.004 aJ, and we believe that the calculated binding energies may be too large. The hydride ion is 2 to 2.5 times as large as the H 3 + ion. This affects Eq. 19.28 by the inverse fourth power, so that the attractive energy of the (H~) cluster is only expected to be a few percent of that for the H 3 + cluster. The H 5 + ion appears in stable form at about room temperature and we might guess that the H 3 ~ ion will show up in the gas at about 20 K. No one has yet looked, because it is much more difficult to search for negative ions with a mass spectrometer than for positive ones. There is one last case in dense hydrogen. In fluid at 32 K and beyond 1 700 mol/m 3 density, electrons somehow acquire the mobilities of heavy ions (see Chap. 27). 65 This is usually attributed to the "electron bubble." In our simple notation, this is e"+xH2^e"(H2)x

(19.34)

A trapped electron has a high zero-point vibrational energy which is thought to push back the neighboring molecules. The electron, then, is expected to be bulky, like a hydride ion, once the electron is brought to rest. 6 6 , 6 7 In this case, we have at least high density and low temperature to encourage the clustering. No proof that a true electron bubble occurs in liquid hydrogen has yet been given, but most people accept the idea, because no other negative ion species seems reasonable at present. We have all the features of a cluster ion, except that the diffusing electron may pick up new cluster molecules as it moves and "pushes" aside the liquid.

Helium Tritide When a tritium nucleus decays, the He 3 daughter may remain bound to the other tritium atom in the original molecule. T 2 ->(HeT) + + p~

(19.35)

Early experimental work showed about 90% yield for the above reaction in 4 to 19 mPa gas, with an 8% yield of (He 3 ) + and a 2% yield of protons. 6 8 , 6 9 Recent calculations and a reconsideration of the experimental procedure have lowered this value to about 60%. 7 0 For many years, it has been suggested that insufficient He 3 outgasses from liquid tritium (see Chap. 21). One hypothesis is that the helium remains as the helium tritide

REACTIONS IN IRRADIATED

253

HYDROGEN

TABLE

19.6

RECOMBINATION PARAMETERS FOR THE I O N RECOMBINATION R E A C T I O N H

+

+ H ~ WITH T W O

SETS OF P R O D U C T S

Center-ofmass energy

Calculated rate constant (m 3 /ion pair • s)

Cross section ( x 10~ 20 m 2 )

aJ

eV

to 2H

to H 2 + + e"

Total

2.8( —4) l . H - 3) 4.1 ( —3) 0.01 0.02 0.05 0.1 0.2 0.5 1.0

1.7(-3) 6.7(-3) 2.6( —2) 6.2( —2) 0.125 0.312 0.625 1.25 3.12 6.25

240000" 60 000" 16 000" 6 600" 3400 1200 700 430 20 15

300 100 30 12 6.5 4 2 1 0.2

1.5( — 12)a 7.5(—13)" 3.9(—13)" 2.5(—13)" 1.9(—13) 1.0(—13) 8.6(-14) 7.5(—14) 6(-14) 6( —14)

—

NOTE: The first energies correspond to temperatures of 20, 77.4, and 300 K. The superscript (a) indicates estimates.

ion. 71 The helium hydride ion, HeH + , is stable by itself, with a calculated dissociation energy (to He and H + ) of 0.295 aJ (1.843 eV).72 However, the ion is unstable in the presence of hydrogen and reacts according to HeH + + H 2 -» H 3 + + He + 0.489 aJ (3.054 eV) +

(19.36)

+

The H 3 ion is more stable than the HeH ion, and we expect the latter to be destroyed (however, see Chap. 21). The rate constant for this reaction at room temperature is 1.26 x 10~15 m3/molecule • s. 73 Ion-Ion Recombination In the dilute gas, ion-ion reactions go as fast as electron-ion reactions. Table 19.6 shows the cross sections and calculated rate constants for the recombination reaction f 2H H + + H

" - W

+

e -

(19 37)

"

where the upper, diatomic route is the only important channel. 74-76 The cross sections at low energy follow the same a ~ E _ 1 dependence that we found with electrons. The low-temperature cross sections are larger, but with the smaller ion velocities, the rate constants are just about the same as those of electrons and ions. We do not particularly care about the above reaction in tritium but, should hydride ions really be created in abundance, they will recombine with the H 3 + ions. We would expect large rate constants similar to these.

254

REACTIONS IN IRRADIATED HYDROGEN

Above, we considered "electron bubbles in the dense fluid," which act like heavy ions. These recombine with the positive heavy ions, both of unknown mass, with a rate constant of 1.1 x 10~ 14 m 3 /ion pair • s for liquid H 2 bombarded by krypton-85 beta particles at about 20 K. 7 7 This is an order of magnitude below the gas-phase value, but it is surprisingly large considering the density of the liquid medium. The negative species has a mobility of 8.4 x 10~ 7 m 2 / V - s and so is definitely an "ionlike" species.

Notes 1. W. J. Moore, Physical Chemistry, 3d ed. (Prentice-Hall, Englewood, N.J., 1962), pp. 274-278. 2. D. Auerbach, R. Cacak, R. Caudano, T. D. Gaily, C. J. Keyser, J. W. McGowan, J. B. A. Mitchell, and S. F. J. Wilk, J. Phys. 10B, 3797 (1977). 3. H. D. Smyth, Phys. Rev. 25, 452 (1925). 4. T. de Jong, Astron. Astrophys. 20, 263 (1972). 5. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2d ed. (D. Van Nostrand, Princeton, N.J., 1961), pp. 351-352, 360, 363, 459. 6. G. Herzberg, Atomic Spectra and Atomic Structure (Dover, New York, 1944), p. 11. 7. M. E. Schwartz and L. J. Schaad, J. Chem. Phys. 47, 5325 (1967). 8. J. A. Burt, J. L. Dunn, M. J. McEwan, M. M. Sutton, A. E. Roche, and H. I. Schiff, J. Chem. Phys. 52, 6062 (1970). 9. R. C. Raffenetti and K. Ruedenberg, J. Chem. Phys. 59, 5978 (1973). 10. R. Polak, J. Vojtik, and F. Schneider, Chem. Phys. Lett. JJ, 117 (1978). 11. J. J. Thomson, Rays of Positive Electricity (Longmans-Green, New York, 1913). 12. C. H. Douglass, D. J. McClure, and W. R. Gentry, J. Chem. Phys. 67, 4931 (1977). 13. B. G. Reuben and L. Friedman, J. Chem. Phys. 37, 1636 (1962). 14. M. T. Bowers, D. D. Elleman, and J. King, Jr., J. Chem. Phys. 50, 4787 (1969). 15. R. P. Clow and J. H. Futrell, Int. J. Mass Spectrosc. Ion Phys. 8, 119 (1972). 16. G. Gioumousis and D. P. Stevenson, J. Chem. Phys. 29, 294 (1958). 17. See notes 5 through 10 above. 18. See note 5 above. 19. W. M. Jones and D. F. Dever, J. Chem. Phys. 60, 2900 (1974). 20. J. B. A. Mitchell, J. L. Forand, C. T. Ng, D. P. Levac, R. E. Mitchell, P. M. Mul, W. Claeys, A. Sen, and J. W. McGowan, Phys. Rev. Lett. 51, 885 (1983). 21. M. T. Leu, M. A. Biondi, and R. Johnsen, Phys. Rev. 8A, 413 (1973). 22. B. Peart and K. T. Dolder, J. Phys. 7B, 236 (1974). 23. D. Auerbach, R. Cacak, R. Caudano, T. D. Gaily, C. J. Keyser, J. W. McGowan, J. B. A. Mitchell, and S. F. J. Wilk, J. Phys. 10B, 3797 (1977). 24. J. W. McGowan, P. M. Mul, V. S. D'Angelo, J. B. A. Mitchell, P. Defrance, and H. R. Froelich, Phys. Rev. Lett. 42, 373 (1979). 25. M. T. Leu, M. A. Biondi, and R. Johnsen, Phys. Rev. 8A, 413 (1973). 26. M. A. Biondi and S. C. Brown, Phys. Rev. 76, 1697 (1949). 27. See note 24 above. 28. J. M. Warman, E. S. Sennhauser, and D. A. Armstrong, J. Chem. Phys. 70, 995 (1979). 29. D. W. Trainor, D. O. Ham, and F. Kaufman, J. Chem. Phys. 58, 4599 (1973). 30. V. H. Shui and J. P. Appleton, J. Chem. Phys. 55, 3126 (1971). 31. W. N. Hardy, M. Morrow, R. Jochemsen, B. W. Statt, P. R. Kubik, R. M. Marsolais, A. J. Berlinsky, and A. Landesman, Phys. Rev. Lett. 45, 453 (1980). 32. J. M. Yuan, T. K. Lim, and L. H. Nosanow, Phys. Lett. 81 A, 61 (1981). 33. I. F. Silvera and J. T. M. Walraven, Phys. Rev. Lett. 44, 164 (1980).

REACTIONS IN IRRADIATED HYDROGEN

255

34. J. E. Bennett and D. R. Blackmore, Proc. Royal Soc. 305A, 553 (1968). 35. J. E. Bennett and D. R. Blackmore, J. Chem. Phys. 53,4400 (1970). 36. See note 34 above. 37. A. Jablonski, J. Chem. Soc., Faraday Trans. 1 73, 111 (1977). 38. A. E. Kabanskii, Kinetics and Catalysis (Soviet) 20, 8783 (1979). 39. See note 37 above. 40. E. Nicolopoulous, M. Bacal, and H. J. Doucet, J. Physique 38, 1399 (1977). 41. M. Bacal and G. W. Hamilton, Phys. Rev. Lett. 42, 1538 (1979). 42. J. R. Hiskes, A. M. Karo, M. Bacal, A. M. Bruneteau, and W. G. Graham, J. Appl. Phys. 53, 3469 (1982). 43. P. C. Souers, E. M. Fearon, and R. T. Tsugawa, Cryogenics 21, 667 (1981). 44. G. T. McConville, Monsanto Research Corporation, Mound Laboratory, Miamisburg, Ohio 45342, private communication, 1982. 45. J. Callaway, Phys. Lett. 48A, 359 (1974). 46. J. B. Hasted and R. A. Smith, Proc. Royal Soc. 235A, 349 (1956). 47. J. Geddes, J. Hill, M. B. Shah, T. V. Goffe, and H. B. Gilbody, J. Phys. 13B, 319 (1980). 48. V. A. Escaulov, J. Phys. 13B, 4039 (1980). 49. B. Peart, R. A. Forrest, and K. Dolder, J. Phys. 12B, 3441 (1979). 50. M. Bacal, A. M. Bruneteau, W. G. Graham, G. W. Hamilton, and M. Nachman, J. Appl. Phys. 52, 1247 (1981). 51. R. Clampitt and L. Gowland, Nature 223, 815 (1969). 52. A. Van Lumig and J. Ruess, Int. J. Mass Spectrosc. Ion Phys. 27, 197 (1978). 53. K. Hiraoka and P. Kebarle, J. Chem. Phys. 62, 2267 (1975). 54. H. Huber, Chem. Phys. Lett. 70, 353 (1980). 55. S. Raynor and D. R. Herschbach, J. Phys. Chem. 87, 289 (1983). 56. P. Langevin, Ann. Chem. Phys. 28, 316 (1903). 57. See notes 51 and 52 above. 58. See note 53 above. 59. S. L. Bennett and F. H. Field, J. Amer. Chem. Soc. 94, 8669 (1972). 60. See note 53 above. 61. P. C. Souers, J. Phys. 14B, L387 (1981). 62. See note 61 above. 63. W. Aberth, R. Schnitzer, and M. Anbar, Phys. Rev. Lett. 34, 1600 (1975). 64. A. M. Sapse, M. T. Rayez-Meaume, J. C. Rayez, and L. J. Massa, Nature 278, 332 (1979). 65. H. R. Harrison and B. E. Springett, Chem. Phys. Lett. 10, 418 (1970). 66. C. V. Briscoe, S. I. Choi, and A. T. Stewart, Phys. Rev. Lett. 20, 493 (1968). 67. T. Miyakawa and D. L. Dexter, Phys. Rev. 184, 166 (1969). 68. A. H. Snell, F. Pleasonton, and H. E. Leming, J. Inorg. Nucl. Chem. 5, 112 (1957). 69. S. Wexler, J. Inorg. Nucl. Chem. 10, 8 (1959). 70. N. W. Winter, Lawrence Livermore National Laboratory, Livermore, Calif. 94550, private communication, 1983. 71. R. D. Taylor, "Metastable Complexes of Helium-Hydrogen Isotopes," in Proc. 17th Int. Conf. Low Temp. Physics, LT-17 (American Elsevier, New York, 1984), p. EL-10. 72. D. M. Bishop and L. M. Cheung, J. Molec. Spectrosc. 75, 462 (1979). 73. O. J. Orient, Chem. Phys. Lett. 52, 264 (1977). 74. T. D. Gailey and M. F. A. Harrison, J. Phys. 3B, L25 (1970). 75. J. Mosely, W. Aberth, and J. Peterson, Phys. Rev. Lett. 24, 435 (1970). 76. G. Poulaert, F. Brouillard, W. Claeys, J. W. McGowan, and G. Van Wassenhove, J. Phys. 11B, L671 (1978). 77. N. Zessoules, J. Bankerhoff, and A. Thomas, J. Appl. Phys. 34, 2010 (1963).

20. Physical Properties of Species in Irradiated Hydrogen We have considered ions and excitations when the hydrogen molecule is broken up by electrons. In this chapter we consider the physical properties of the radiation damage species. The family of ions may seem confusing, but they fall into the simple arrangement displayed in Fig. 20.1 by way of their potential energy curves. Zero energy is the bottom of the molecular H 2 potential well, and the "size" is here the internuclear distance. All energies of formation are the distances upward from zero to the bottom of the well of whatever species is considered. The depth of each well indicates how stable to dissociation the species is. The "size" at minimum energy for most ions is the internuclear distance. The "size" in the case of the H atom is the Bohr radius and for H~ the wave function maximum. The H 2 7 + ion size is approximate. All these other species are excited varieties of the H 2 molecule. The most stably bound is the H atom, which is placed at the H 2 dissociation energy indicated by the dotted line at 0.717 aJ (4.478 eV). The hydride ion is considered to be an offspring of the atom, and it lies 0.12 aJ below the dissociation energy line. We consider electrons to be energetically "free" in this picture, although the energy to produce the electrons has to come from somewhere. The large size of the hydride ion is evident. The third dotted line from the bottom represents true molecular ionization, at 2.472 aJ (15.426 eV). The resulting H 2 + ion is the most excited state of a family of positive ions, of which the H 3 + is the most stable. The succeeding ion clusters, H 3 + (H 2 ) x , are not shown, but we can imagine a set of shallow potentials extending slightly below the H 3 + dotted line at 2.11 aJ (13.18 eV). The minima would move outward to larger sizes as the positive core adds more neutral molecules. As one example, we show the approximate position of the H 2 7 + well. It dissociates ultimately back to the H 3 + ion. We note that both the H 2 + and H 3 + ions dissociate to the proton. The highest dashed line at 2.90 (18.08 eV) is the lowest-energy birthplace of the proton. All the above energies are for the H 2 family of ions. The heavier mass hydrogens are similar, as shown by a comparative list of energies in Table 20.1.1

The Diatomic Ion This would not appear to be an important family of ions because they quickly react to the triatomic species. But there is a chance that the diatomic species may be found

257

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

0.2 "Size" (nm) Fig. 20.1. Potential energy curves for the hydrogen ion family and the atom. Zero energy is that of ground state H 2 .

T A B L E 20.1 SOME BASIC PROPERTIES OF THE MOLECULAR HYDROGENS WHICH SHOW HOW SIMILAR THEY ARE

Molecular transitions Ionization energy

H2 HD HT" D2 DT a T2"

Dissociation energy

aJ

eV

aJ

eV

Lyman BX (aJ)

2.472 2.475 2.476 2.478 2.479 2.481

15.43 15.44 15.45 15.47 15.47 15.49

0.7175 0.7232 0.725 3 0.7300 0.7326 0.735 6

4.478 4.514 4.527 4.556 4.573 4.591

1.792 1.796 1.797 1.800 1.802 1.804

NOTE: The superscript (") indicates estimates.

Werner C 1

Zero-point energy (aJ)

67.35 58.61 55.4a 48.13 44.0a 39.6a

0.0227 0.019 7 0.018 3a 0.0162 0.0150" 0.013 6"

NOTE: The superscript (a) indicates estimates.

frozen into the solid, and we include some descriptive data. The calculated mean interatomic bond length is about 0.11 nm. 2 This is larger than the 0.074 nm value for the hydrogen molecule, because one of the electrons binding the two nuclei together has been taken away and only one is left to do the work. The H 2 + ion has a single electronic ground state, the X 2 £ g + . There are no excited states that may be used as signatures, and it takes only 0.425 aJ (2.651 eV) to dissociate the ion. 3 Table 20.2 lists the calculated fundamental frequencies in the infrared region for which v = 0 -»• 1 and AJ = + 1 , where v and J are the molecular vibrational and rotational quantum numbers. The zero-point energy of molecular vibration is also given. 4 Only the heteronuclear ions will possess a dipole moment. The calculated values for the v = 0 - » 1, AJ = ± 1 transitions in H D + are about 3.8 x 10~ 34 C - m (1 Debye = 3.336 x 10~ 30 C • m). 5 This is only one-tenth the size of the H D molecule dipole moment, and that is small. Infrared spectroscopy is not a good way to see these ions, although H D + lines have been measured. 6 A better way to see these ions would be electron spin resonance (e.s.r., also called electron paramagnetic resonance). This has never been done because no one has ever collected the necessary 10 14 ions needed for a signal. We will meet e.s.r. again soon, so we shall briefly describe it here. This technique can see unpaired electrons with great sensitivity. A free electron has a magnetic moment characterized by a spin quantum number S = 1/2. The electron has 2S + 1 = 2 magnetic substates of the same energy. In a D C magnetic field, these states are separated in energy, and we may induce a transition between them with microwave radiation. The transition will be from the m s = — 1/2 substate to the m s = +1/2, with Am s = 1. The He 3 nuclear magnetic resonance (n.m.r.) scheme, shown in Fig. 26.2, is analogous to the free electron e.s.r. Because of the difference in mass, n.m.r. occurs in the radio frequency region and e.s.r. in the microwave. In the ions H D + , H T + , and D T + , the electron is virtually free, and we expect one e.s.r. line at

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

259

0.074 0.19

0.30

Fig. 20.2. Positive hydrogen ion clusters. Top: the H 9 + ion. Bottom: two kinds of molecule addition locations to the H 9 + frame. The numbers are calculated bond lengths in nm. (Courtesy S. Yamabe and North-Holland Publishing Co.)

260

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

v(e.s.r.) = 28.025B,'o

(20.1)

where B 0 is the applied D C magnetic field in T and the frequency v is in GHz. We also expect a single line in the case of even-J H 2 + and even-J T 2 + , where the total nuclear spin of 0 does not affect the e.s.r. line. For the other diatomic ions, the nuclear spin is non-zero (see Chap. 26) and the nucleus and electron interact via hyperfine coupling. If the nuclear spin is I, then each electron substate will split into 21 + 1 even finer sublevels. There would appear to be a lot of possible transitions, but the selection rules require that the electron spin quantum number m s and the nuclear spin quantum number m, each change by only ± 1. This limits the number of transitions—and, hence, the e.s.r. lines—to a total of 21 + 1. For odd-J H 2 + , the nuclear spin is 1, and we expect three transitions at 7 v(odd-J H 2 + ) = 28.025B o ;

28.025B o ± 0.935 G H z

(20.2)

The center transition lies at the free electron frequency. Note that the hyperfine splitting is locked in by the electron-nuclear interaction itself, and is not a function of the external magnetic field. In fact, zero-field magnetic e.s.r. would be possible at 0.935 GHz. Odd-J D 2 + and odd-J T 2 + also have a nuclear spin of 1 and will form a spectral triplet. Because of the size of the nuclear magnetic moments (see Chap. 26), we expect the D 2 + triplet to be more closely spaced than that of H 2 + , while the T 2 + triplet is slightly more widely spaced than that of H 2 + . The even-J D 2 + nuclei have a spin of 2 in 5/6 of the states, and this will lead to a quintet of e.s.r. lines. Clearly, e.s.r. could, in theory, give positive identification as to the type of diatomic ion.

The Triatomic Ion The diatomic ion has only one electron to hold together two protons. The triatomic ion is more stable, because two electrons bind three protons. Also, the first electronic orbital is filled and thereby stabilized. The H 3 + ion consists of three nuclei in an equilateral triangle with a measured side length of 0.096 ± 0.005 nm. 8 The dissociation energy to the least energetic products H 2 and H + is 0.785 aJ (4.902 eV), so that the ion is very stable. 9 ' 1 0 The ground state contains two paired electrons and is the singlet . The only excited state appears to be a shallowly bound triplet above the dissociation limit. 1 1 ' 1 2 Because singlet-triplet transitions are forbidden, we cannot do ultraviolet spectroscopy to this state. We effectively have only the ground electronic state. Because the two electrons of the ground state H 3 + are paired, we cannot do electron spin resonance either. One possible signature of these ions is their molecular vibrations. The calculated fundamental frequencies and zero-point energies are listed in Table 2 0 . 3 . 1 3 - 1 6 Although the H 3 + ion has no permanent dipole moment, the asymmetric mode v2 has been spectroscopically seen in 100 Pa hydrogen gas containing only 3 x 10 16 ions/m 3 . 1 7 The permanent dipole moments in the asymmetric ions like H 2 D + should make them stronger candidates for infrared detection. 1 8 The dipole moment of the

261

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

T A B L E 20.3 CALCULATED FUNDAMENTAL INFRARED FREQUENCIES AND ZERO-POINT ENERGIES OF THE TRIATOMIC HYDROGEN IONS

Triatomic ion

VL

V2

V3

Zeropoint energy (aJ)

H3+ + H2D HD 2 + D3+ + H2T HDT + HT2 + D2T+ DT 2 + T3+

104.1 89.94 82.20 69.16 88.77 80.55 78.63 66.16 62.36 57.11

84.36 66.04 58.79 54.74 61.31 54.05 49.29 50.72 47.61 45.27

84.36 69.82 62.12 54.74 67.75 59.69 56.84 51.95 48.51 45.27

0.086 3 0.078 7 0.070 5 0.061 6 0.075 8 0.067 3 0.063 9 0.058 1 0.0544 0.050 5

Frequency (THz)

H 2 D + ion is about 5 x 10~ 3 0 C - m , or a thousand times larger than that of molecular H D . 1 9 , 2 0 It is perhaps surprising that more spectroscopy hasn't been done, but the infrared lines, at least in the gas phase, are very narrow—only 1.5 GHz wide, so that a high-resolution instrument appears to be needed. 21 In denser hydrogen, the lines may be broader. Because of its asymmetric shape, the H 3 + ion has a quadrupole moment, ranging from 2 to 4 x 10" 40 C • m 2 , depending on the orientation. 22 This is comparable to the molecular quadrupole moment of J = 1 H 2 , so that the ion and the rotationally excited neutral molecule should interact. This effect is small and will show its most interesting effects below 4 K.

Atoms Free hydrogen atoms are made as the final step by a number of ionic reactions. Atoms are always present. The atom is just a proton and electron which, in the ground state of energy quantum number n = 1, are 0.0529 nm apart by classical reckoning. The ionization energy is the well-known 2.178 aJ (13.595 eV). The binding of the electron in the state n is the well-known expression "

'

(47Σ 0 ) 2

h N

(20.3)

Here, e 0 is the permittivity of free space (8.854 x 10~ 12 C/V-m), m e is the electron mass (9.11 x 10" 31 kg), e is the electronic charge (1.602 x 10" 19 C), and h is Planck's constant (6.626 x 10~ 3 4 J• s). Because most atoms in our samples are created in excited states, they drop from high n states to lower ones and emit radiation at telltale frequencies. Many of these are listed for the H atom in Table 20.4. 23 The D and T

262

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED H Y D R O G E N

TABLE

20.4

A PARTIAL LIST OF SPECTROSCOPIC TRANSITIONS IN THE HYDROGEN ATOM

Hj

n

OO

8 7

1 1 1 1 1 1 1 2 2 2

6 5 4 3

2 2 2 2

oo

3 3 3

7 6 5 4 3 2

00

9 8

f

Frequency (PHz) Ultraviolet 3.288 3.221 3.197 3.157 3.083 2.923 2.466 0.8222 0.7709 0.755 1 Visible 0.7308 0.6908 0.616 7 0.4568 Infrared 0.3654 0.3248 0.3140

Wavelength (nm)

91.18 93.07 93.78 94.97 97.25 102.6 121.6 364.6 388.9 397.0 410.2 434.0 486.1 656.3 820.4 922.9 954.6

n

i

n

r

Frequency (PHz)

Wavelength (nm)

7 6 5

3 3 3 4 4 4 3 4 4 5 4 5 6 5 5 4 5 6 6

0.298 3 0.2741 0.2339 0.205 5 0.1727 0.1650 0.1599 0.1542 0.1384 0.1315 0.1142 0.098 64 0.091 34 0.09096 0.080 17 0.07400 0.06444 0.06418 0.05847

1005 1094 1282 1459 1736 1817 1875 1945 2166 2279 2625 3039 3282 3296 3 740 4051 4653 4671 5127

00 10 9 4 8 7

oo 6 10

00 9 8 5 7 11 10

NOTE: Initial and final state quantum numbers are ni and n f , where the initial state lies at a higher energy than the final state. PHz is 10 1 5 Hz.

atom values are almost the same. The initial quantum number nj and final quantum number n f are listed. According to n f , the spectral groups have series names: n f = 1, Lyman; 2, Balmer; 3, Ritz-Paschen; 4, Brackett; and 5, Pfund. The first series lies in the ultraviolet, the second in the visible, and the last three are in the near infrared. Higher-order lines in the far infrared have been left out of this list. In each group, the lines are listed downward by a, [}, y, and so on. For example, the Balmer a line at 656 nm is the famous transition that gives time exposures of galaxies their beautiful rose-red color. The Balmer p and y lines, which occur with a lesser probability at 486 and 434 nm, are in the blue-violet region. Also included in Table 20.4 are transitions from the continuum (n; = oo), which would correspond to the radiative recombination of a proton and an electron. However, a third body must be present to take away momentum without damping the radiation. The cross sections for Balmer emission under electron bombardment are a hundred times smaller than those we have listed in Chapters 18 and 19. 2 4 The calculated efficiencies (in photons/ion pair) for 16 fJ (10 keV) electrons in hydrogen are: Balmer a 0.20, /? 0.029, and y 0.009. 2 5 These are "zero-pressure" corrected values as derived from hydrogen below a pressure of less than 0.1 Pa. The Balmer series has been seen in room-temperature tritium gas at 10 and 86 kPa. 2 6 The emission lines are narrow—only 1 to 2 nm wide, as seen at relatively low resolution. The emission is

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

263

definitely quenched as the pressure increases. We may combine our two references to obtain these rough empirical efficiencies in the room-temperature gas. For the Balmer a, the efficiency in photons/ion pair is 0.065 P " 1 / 2

(20.4)

where P is in Pa. We see that the efficiency of 0.20 at low pressures has degraded to 2 x 10~ 4 at 0.1 MPa. For the Balmer /?, we use the same P~ 1 / 2 dependence but change the coefficient to 0.005. For the Balmer y, the coefficient is a very rough 0.002. It is experimentally more difficult to go into the ultraviolet for the Lyman series. The calculated zero-pressure efficiency for the Lyman a is 0.61. 27 For atom densities of 10 20 per m 3 , e.s.r. can and has been used. The unpaired electron interacts with the nuclear spin via the hyperfine coupling, just as we saw for the diatomic ions above. Here, the electron spin is 1 /2 and the nuclear spins of H, D, and T are 1/2, 1, and 1/2 (see Chap. 26). If only 10 14 atoms are present in a cc-sized sample, which is easy to get, then e.s.r. can be used. It is a major tool in the study of free hydrogen atoms. Once again, the nuclear spin interacts with the electron via hyperfine coupling. The nuclear spins of H, D, and T are 1/2, 1, and 1/2, with no rotational complications (see Chap. 26). For the H atom, we generate a telltale doublet at the e.s.r. frequencies o f 2 8 v(H) = 28.025B o ± 0.710 GHz

(20.5)

We obtain two lines, neither at the free electron frequency, which definitely pinpoints the electron as being in the atom. An example of this spectrum is shown in Fig. 22.2. Zero magnetic-field resonance can be done at 1.42 GHz. For the T atom, it is similar: 29 v(T) = 28.025B o ± 0.758 GHz

(20.6)

The D atom, however, displays a triplet at the frequencies o f 3 0 v(D) = 28.025B o ;

28.025B o ± 0.218 GHz

(20.7)

The center line is at the free electron frequency. Once again, the hyperfine splitting is inherent in the atom and is not dependent on the magnetic field.

The Hydride Ion The hydride ion is a single proton plus two electrons. Because of electrostatic repulsion, the two electrons repel each other and spread out so far that the wave function maximum is at 0.16 nm. 3 1 This is twice the bond length of the H 2 molecule. Although the ground state is Is 2 XS, the two electrons do not act like the well-localized Is electrons of the hydrogen atom and molecule. They spead out but retain their paired character, so that electron spin resonance is not possible. The binding energy of the hydride ion is 0.123 aJ (0.766 eV), with the hydrogen atom and the electron the products. 3 2 Because the hydride ion has no permanent

264

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED HYDROGEN

dipole moment, its dissociation cannot be easily seen spectroscopically. The ion may have a shallow excited state, 3 3 ' 3 4 but this too cannot be reached spectroscopically from the ground state. In the gas phase, one can use a mass spectrometer to find hydride ions, but otherwise it carries no special property all its own. It would give rise to a nuclear magnetic resonance signal, but this would be swamped by the signals from the neutral molecules. In any case, n.m.r. is insensitive and a large signal of 10 20 spins would be needed for a measurable signal. In the solid, hydride ions might rattle around and show a far-infrared translational spectrum, as has been seen in hydrogendoped ionic solids. 35

Ion Clusters We recall that neutral hydrogens attach themselves via their polarizability to the H 3 + ion to form H 3 + (H 2 ) X clusters. The first H 2 to add on does not merge into the H 3 + triangle but sits outside. A planar geometry is unstable, 36 and the molecule is believed to sit perpendicular to the plane of the triangle. 37 Two more molecules can add to the other points of the triangle, as shown at the top of Fig. 20.2. We complete the "first" shell of molecules, and this is why the H 9 + ion is especially stable. The bottom of Fig. 20.2 also shows the next two possible locations for hydrogen molecules. There are three possible in-plane locations and six out-of-plane ones. The in-plane locations must be more stable because the H 1 5 + ion is the next special island of stability. Once we have filled the entire "second" shell with nine molecules, we have the stable H 2 7 + ion. All 12 molecules of the H 2 7 + ion superficially resemble the nearest neighbor shell surrounding a molecule in solid hydrogen. For the cluster ion, however, the predicted H 3 + -molecule distances vary according to the positions. Also, the aboveand below-plane molecules do not fit into the interstices of the in-plane molecular arrangement, as is the case in both the h.c.p. and f.c.c. structures. We see at the bottom of Fig. 20.2 that there are three molecules stacked above each other in the positions perpendicular to the H 3 + ion's edges. This is not a close-packed structure. Some work has just begun on the vibrational frequencies of these ions. These values have been calculated: 38 H 3 + 112 and 87 (2 modes) THz H 5 + 134, 111,83, and 82 THz H 9 + 136(3), 108, and 82(2) THz

(20.8)

The H 3 + frequencies are slightly different from those of Table 20.2 because of the uncertainties of calculation. The actual bond lengths and exact orientation have not been experimentally determined, so that any calculated vibrational frequencies are approximate. We have listed above only those values in the near infrared. As the ion grows, it will acquire more and more modes in the far infrared, and these, in the end, may be needed to truly define the ion.

265

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED H Y D R O G E N

TABLE

20.5

THRESHOLD EXCITATION ENERGIES FOR THE LOWEST-LYING ELECTRONIC EXCITED STATES OF THE H 2 MOLECULE

Singlet

Triplet Energy

Name X BlZ?

c'n,, E%+ B'

Q'n, K 'Zg G

i'ng

Energy

aJ

eV

0 1.822 1.988 1.988 2.005 2.218 2.228 2.238 2.241 2.248

0 11.37 12.41 12.41 12.51 13.84 13.91 13.97 13.99 14.03

Name

aJ

eV

1.602 1.904 1.906 2.141 2.239 2.242 2.243 2.247 2.255

10.00 11.88 11.90 13.36 13.97 13.99 14.00 14.03 14.08

—

3

b Zu+ c3nu a 3XL+

e 33 4 d nu g%++ h% 3 i ng

j3Ag

NOTE: Many more states exist from 2.2 aJ (14 eV) to the ionization energy at 2.47 aJ (15.4 eV). The X singlet state is the ground state of H 2 at zero energy.

The structures of the negatively charged ion clusters are considerably more doubtful. The H 3 ~ and H 5 ~ ions are supposed to be linear. 3 9 ' 4 0 As we move to the even more doubtful varieties, we find these predictions: H 7 ~ , triangular and planar; H 9 ", tetrahedral; H n " , molecules above and below the triangle of the H 7 ~ core; and H 1 3 , octahedral. 41 These symmetries are all different, and there is no systematic building-up around a core, as we had with the positive ion clusters.

Electronically Excited Molecules We have just considered the family of ions produced by ionization. An excited molecule need not ionize, however, but can end up in an electronically excited state H 2 *. A list of the lower states is given in Table 20.5. 42 The capital letters refer to singlet states and the lowercase letters to triplet states. The two kinds of states do not mix because of quantum symmetry rules, so that a singlet state de-excites to lower singlet states and finally comes to the ground state X .* The excited triplet states . de-excite to other triplet states and finally arrive at the lowest triplet state, b This cannot de-excite to the X state, and it therefore dissociates, producing two more of the ubiquitous atoms. Thus the b state is an anti-bonding state. Table 20.6 shows *The letter is the special name of the level, the superscript means singlet (" 3 " for triplet), the large Greek letter tells the electronic orbital angular-momentum quantum number (L = 0, n = 1, A = 2), the superscript " + " means the wave function is symmetric to mirror reflection, and the subscript "g" means that the wave function is symmetric to inversion through the center of mass ("u" means it is not). Triplet states de-excite to other triplet states and finally arrive at b 3 E „ .

266

PHYSICAL PROPERTIES OF SPECIES IN IRRADIATED H Y D R O G E N

only levels up to 2.2 aJ (14 eV). Many more are crowded into the narrow interval between this energy and ionization. Table 20.6 lists the known molecular transitions. Those with an arrow " - • " indicate emission when there is a special source of excitation like a tritium beta particle. We see that there are three special regions. The first is in the ultraviolet, and this includes the two best-known transitions: the Lyman band (BX) and the Werner band (C«-»X). The energies for these two special transitions are listed for all the hydrogens in Table 20.1. The second region is in the visible, and there is an almost constant density of lines from 370 to 600 nm. The last region is in the near infrared. The cross sections for Lyman and Werner line emission in "zero-pressure" hydrogen gas bombarded with electrons have been measured and are in the range 3 x 10~ 23 m 2 . 4 3 Calculations for 1.6 fJ (10 keV) electrons show a production of 0.33 Lyman photons/ion pair and 0.38 Werner photons/ion pair. 4 4 We do not know of similar calculations in the visible region, where the lines are solidly packed in. In TABLE

20.6

MOLECULAR TRANSITIONS IN THE ELECTRONICALLY EXCITED H 2 MOLECULE

Transition

Frequency (PHz)

Ultraviolet D" «- X 3.603 D'n~+v

(21.7)

where n and ji are the pion and muon, and v is a neutrino. Only the negatively charged muon is useful in catalyzing hydrogen fusion. The muon thermalizes by collisions in about 100 ps and reacts to form a muonic atom 2 6 , 2 7 fj,~ + D 2 ^ n ~ D + D + e~

(21.8)

where H, D and T here refer to bare nuclei. The muonic atom is electrically neutral. We recall that the binding energy of the hydrogen atom is 2.18 aJ (13.6 eV). The binding energy of the muonic atom is greater by the size of the reduced mass. If this is 1 for the hydrogen atom, it is 186 for ¿u~H, 196 for /i~D, and 200 for T. The ground state binding energy of is 0.427 pJ (2.66 keV) and that of T is 0.434 p j (2.71 keV). The result of this difference is a loss of fi~ D by exchange + T2

/z-T + DT

(21.9)

This fractionation goes too far, so that a preponderance of ¿i~T is created. This has

278

EXCITED HELIUM A N D MUONIC H Y D R O G E N

T A B L E 21.7 BINDING ENERGIES OF M U O N I C HYDROGEN IONS AND ATOMS

Ground state (aJ)

Atom /¿"H

-405.1 -426.7

fi~T

-434.4

v = 0 (aJ) Ion

J= 0

J=1

(H/i-H ) + (H/i~D) + (Hm"T) +

-40.54 -35.49 -34.17 -52.07 -51.13 -58.14

-16.92 -15.4 -15.6 -36.26 -37.20 -46.29

(Dfi-Dy (Pn~T) + (T/x-T) +

J= 2

J= 3

—

—

—

—

—

—

-13.7 -16.5 -27.56

— —

-7.64

v = 1 (aJ)

(Dfi " D) + (D/TT) + 0>"T)+

J= 0

J= 1

- 5.70 - 5.56 -13.4

-0.32 -0.14 -7.19

NOTE: The molecular quantum numbers are v for vibration and J for rotation. H, D and T here refer to here nuclei; aJ is 10~18 J. The ions come apart into muonic atoms and nucleous. only a 5 0 - 5 0 chance of meeting a D nucleus for fusion. Current mixtures of 30% tritium are used to bias the products toward being equimolar in D and T . 2 8 If the tritium beta particle catalyzes the hot-atom equilibrium, 2 9 then considerable molecular D T will form, and we will have a further difficulty in feeding in controlled amounts of D and T . 3 0 The muonic hydrogen atom has excited electronic states, which follow the same 1/n 2 rule as do those of the hydrogen atom. In Table 21.7, we list the total binding energy, from the energy quantum number n to ionization at n = oo. The energy for the transition n = 1 2 would be 1 — 1/2 2 = 0.75 of the listed binding energy. This energy is referenced to 0 J/mol for the dissociated particles, so that we add a minus sign to indicate binding. Instead of ultraviolet emission, muonic hydrogen atoms emit x-rays. 3 1 , 3 2 The first Bohr orbit is also 200 times closer to the nucleus: 52.9 pm for the electron and about 0.26 pm for the muon. The muonic hydrogen atom next reacts to form a metastable molecule. H~T + D 2 -» [(¿TDT) D e ~ e ~ ]

(21.10)

EXCITED HELIUM AND MUONIC HYDROGEN

279

One nucleus of this "diatomic" molecule is a deuteron and the other is the positive /¿"DT ion. The ion is initially in the v = 1, J = 1 state, where v and J are the vibrational and votational quantum numbers. U p o n de-excitation, the muonic ion fuses into H e 4 and a neutron. The muon is now free to begin another reaction sequence. H o w many fusion reactions it catalyzes before it decays after its own lifetime of 2.2 ¡is is one of the major questions now facing this technology. At the moment, about 160 reactions seem probable. 3 3 When the muon finally decays, the products are an electron, a neutrino, and an antineutrino. The binding energies of the diatomic ions are also listed in Table 21.7. 3 4 ' 3 5 The energy reference is the same as for the atoms. However, there are only a few bound levels, as contrasted with the hordes of states in the electronic ion or molecule. The muon has the same magnetic moment and spin (S = 1/2) as the electron, so that muon magnetic resonance is similar to electron spin resonance but the frequency is about 200 times lower. However, muon magnetic resonance will occur at higher frequencies than will nuclear magnetic resonance, for a given external magnetic field.

Notes 1. R. W. Schmieder, J. Opt. Soc. Amer. 72, 593 (1982). 2. P. C. Souers, E. M. Fearon, R. T. Tsugawa, and G. H. Smith, Lawrence Livermore National Laboratory, Livermore, Calif. 94550, unpublished data. 3. E. E. Benton, E. E. Ferguson, F. A. Matsen, and W. W. Robertson, Phys. Rev. 126, 206 (1962). 4. G. Herzberg, Atomic Spectra and Atomic Structure (Dover, New York, 1944), pp. 64-66. 5. See note 3 above. 6. C. E. Moore, Atomic Energy Levels, National Bureau of Standards Report NSRDSNBS 35, SD Catalog No. C13.48:35/V.I (U.S. Government Printing Office, Washington, D.C., 1971), Vol., I. pp. 1-7. 7. See note 6 above. 8. See note 6 above. 9. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979), pp. 292-298. 10. See note 1 above. 11. F. J. Soley and W. A. Fitzsimmons, Phys. Rev. Lett. 32, 988 (1974). 12. See note 11 above. 13. I. Dabrowski and G. Herzberg, Trans. New York Acad. Sci., Series II, 38, 14 (1977). 14. D. M. Bishop and L. M. Cheung, J. Mol. Spectrosc. 75, 462 (1979). 15. N. Winter, Lawrence Livermore National Laboratory, and Orrin Fackler, Rockefeller University, New York, N.Y. 10021, private communication, 1983. These calculations were done as part of the neutrino mass experiment being carried out currently at LLNL under the direction of O. Fackler. 16. See note 15 above. 17. A. H. Snell, F. Pleasanton, and H. E. Leming, J. Inorg. Nucl. Chem. 5, 112 (1957). 18. S. Wexler, J. Inorg. Nucl. Chem. 10, 8 (1959). 19. See note 15 above. 20. See notes 13 and 14 above. 21. G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2d ed. (D. Van Nostrand, New York, 1950), pp. 141-145. 22. R. R. von Frentz, K. Luchner, H. Micklitz, and V. Wittner, Phys. Lett. 47A, 301 (1974).

280

EXCITED HELIUM AND MUONIC HYDROGEN

23. J. Gray and R. H. Tomlinson, Chem. Phys. Lett. 4, 251 (1969). 24. R. D. Taylor, "Metastable Complexes of Helium-Hydrogen Isotopes," Proc. 17th Int. Conf. Low Temp. Physics, LT-17 (American Elsevier, New York, 1984), p. EL-10. 25. S. Raynor and D. R. Herschbach, J. Phys. Chem. 87, 289 (1983). 26. J. D. Jackson, Phys. Rev. 106, 330 (1957). 27. Yu. V. Petrov, Nature 285, 466 (1980). 28. S. Jones, Idaho National Engineering Laboratory, Idaho Falls, Idaho 93415, private communication, 1983. 29. P. C. Souers, E. M. Fearon, and R. T. Tsugawa, J. Phys. 15D, 1535 (1982). 30. See note 28 above. 31. A. Placci, E. Polacco, E. Zavattini, K. Ziock, C. Carboni, U. Gastaldi, G. Gorini, and G. Torelli, Phys. Lett. 32B, 413 (1970). 32. B. Budick, J. R. Toraskar, and I. Yaghoobia, Phys. Lett. 34B, 539 (1971). 33. See note 28 above. 34. L. Bracci and G. Fiorentini, Phys. Reports 86, 170 (1982). 35. L. I. Ponomarev, "Mesic Atomic and Mesic Molecular Processes in the Hydrogen Isotope Mixtures," in 6th International Conference on Atomic Physics Proceedings, R. Dambrug, ed. (Plenum Press, New York, 1978), pp. 182-206.

22. Radiation Damage in the Solid

Little has been done in the area of radiation damage in the solid because most people have concentrated on the purest crystals possible. This promises to be one of the fastest growing areas over the next few years. Excitons and the Conduction Band Some energy loss spectra of solid H 2 and D 2 have been measured and are shown in Fig. 22.1. There are four sets of curves on the right that show similar behavior: a big peak at 3.1 PHz (13 eV) and another one at 4.2 to 4.6 PHz (17 to 19 eV). These are absorption curves for ultraviolet radiation in solid samples at 2 to 6 K. 1 - 3 The peak at 3.4 PHz (14 eV) is obtained by measuring the energy lost by 5.9 fJ (37 keV) electrons passing through frozen films 10 nm thick.4 The energy lost is from whatever probing beam is used. We assume that this energy is added to the electrons in their valence band. Calculations have been carried out for J = 1 H 2 in the f.c.c. structure, because it is easier than handling the h.c.p. structure. 5 ' 6 We take the valence band to be at zero energy, although it is up to 0.15 aJ (1 eV) wide. The jump from the valence band to the lowest point in the conduction band is 2.4 to 2.5 aJ (15 to 15.5 eV). This is the same as the ionization energy of the free molecule. The conduction band is broad—about 0.37 aJ (2.3 eV) wide. The "C" in Fig. 22.1 shows the expected location of the conduction band. Above this come higher conduction bands at 3.2 aJ (20 eV), 3.8 aJ (24 eV), and 4.2 aJ (26 eV). These may cause the higher energy losses seen in Fig. 22.1. The analog in the solid of the excited molecule H 2 * states will be the exciton bands. The lowest calculated excitons represent the free molecule levels B and c 3 n u , and they should appear at 1.3 to 1.5 aJ (8.4 to 9.3 eV).7 They are not evident in Fig. 22.1. Another calculation yields the result 2.15 aJ (13.5 eV), which looks better.8 In tritiated hydrogen, we expect the daughter element helium-3. This will have excited atomic states that could produce exciton bands in the crystal. No one has yet studied helium, but xenon data are available to give us a clue. In the isolated xenon atom, the two lowest transitions are from the 5p6 'S ground state to the 5p56s 2 P 3/2 and 5p56s 2 P 1/2 states. These transitions occur at 1.3 aJ (8.4 eV) and 1.5 aJ (9.5 eV).9 These same transitions are the two major peaks seen in Fig. 22.1 for 50 ppm xenon in

282

RADIATION DAMAGE IN THE SOLID

Energy loss (eV)

12

16

2.0

2.5

Energy loss (aJ) Fig. 22.1. Measured energy loss peaks in solid hydrogen. The region C is the calculated conduction band.

solid D 2 . 1 0 The forces in the solid increase the transition energies to 1.5 aJ (9.1 eV) and 1.7 aJ (10.6 eV). Atoms

Hydrogen atoms are a major product of the various reactions following radiation damage. The technique used most often is electron spin resonance (e.s.r., see Chap. 20). Figure 22.2 shows a sample run. 1 1 We recall that a large hyperfine splitting causes the H atom lines to appear as a doublet to either side of the free electron e.s.r. frequency. This is the signature of the H atom. The four-line pattern in the center is unexplained but could be caused by oxygen impurities. 1 2 ' 1 3 Figure 22.2 shows another interesting feature. The two H lines are displaced to lower frequencies relative to the free electron line. From chapter 21, we would expect an exact splitting of ± 7 1 0 MHz for the H atom. Instead, we find a definite asymmetry in all the data, which is an effect of the solid s t a t e . 1 4 - 1 6 Table 22.1 lists the asymmetries of the measured atom lines in the solid state. Now, e.s.r. has been used to measure the number of atoms in the solid. The data, in parts per million (ppm), are listed for 4.2 and 1.4 K in Table 22.2. One experiment studied 1% tritium in solid D 2 , at a dose rate of 1 kJ/m 3 • s. 17 The other employed a 26 fJ (160 keV) electron gun, running at the much higher dose rate of 2.4 MJ/m 3 • s. 18

283

RADIATION D A M A G E IN THE SOLID

Free electron

0.316 93 T

I I I

J

1 0.289 60 T

~TY|Y"V I I I I I I i

r

0.340 50 T

DC magnetic field (T) Fig. 22.2. Electron spin resonance spectrum of hydrogen atoms in 4.2 K eH 2 four days after an electric discharge over the surface. Approximately 4 x 1021 atoms/m 3 were present. The e.s.r. frequency was 8.882 GHz. The center portion may be due to an oxygen impurity (from J. R. Gaines; see Chap. 22, reference 11). TABLE 22.1 ASYMMETRY IN THE HYPERFINE SPLITTING OF HYDROGEN ATOMS IN SOLID HYDROGEN

Free electron frequency to (in MHz)

Hydrogen H D T

Lowerfrequency line -768 -726 -766 -220 -215 -111

Center line — — —

-5 -16 —

Higherfrequency line

Chap. 22 ref. no.

+ 658 + 695 + 661 + 216 + 221 + 741

15 16 11 15 16 16

The results, however, are comparable. The highest dose of 600 MJ/m 3 is about 15 kJ/mol, or the amount of energy put out by D-T in 3.3 hours—not much. It is not clear, then, that we have really reached an equilibrium atom density. Nor are we sure of the temperatures in the irradiated samples. We report only what the temperature

284

R A D I A T I O N D A M A G E IN T H E S O L I D

TABLE

22.2

A T O M DENSITIES M E A S U R E D IN I R R A D I A T E D S O L I D H Y D R O G E N BY ELECTRON S P I N R E S O N A N C E

ppm atoms in solid hydrogen 160 keV electrons on H 2 , ref. 18

Absorbed dose (J/m 3 )

1%T in D 2 , 4.2 K (ref. 16)

4.2 K

5(5) 1(6) 3(6) 1(7) 3(7) 1(8) 3(8) 6(8)

0.22 0.48 1.8 7.1 8.4 9.2 26 29

0.45 1.1 3.5 8.6 14 17 19

—

1.4 K —

0.53 1.3 4.1 11 27 47 61

NOTE: The numbers in parentheses are powers of ten. One part per million equals 2.6 x 10 22 atoms/m 3 in H 2 and 3.0 x 10 22 molecules/m 3 in D 2 (see Chap. 22 references as noted).

sensor says. To convert ppm to atoms/m 3 , we multiply by 2.6 x 10 22 for H 2 and 3.0 x 10 22 for D 2 . Atom density data is summarized in Fig. 22.3. If we irradiate with a beam to some atom density, N i5 we can then turn the beam off and watch the e.s.r. signal decay. The decay of the atom density N is described by ^ = - k N dt

2

(22.1)

with the solution N

N;

1

1 + Njkt

(22.2)

where k is the recombination rate constant (in m 3 /atom • s). The 1/e time, T (time to 0.368 of the original signal), is 1/717

Two sets of solid H 2 data exist. One covers the range 6.75 to 8.05 K and is derived from the 26 fJ electron beam work. 1 9 The other covers 1.4 to 5.4 K and is obtained from radio frequency dissociation and condensation. 2 0 ' 2 1 The first reference provides the rate constant 2 2 k = 4.1 x 10~ 14 exp(— 195/T) 6.75 < T < 8.04 K and the second reference provides the rate constant ratio, R, of

(22.4)

285

RADIATION D A M A G E IN THE SOLID

T A B L E 22.3 COMPOSITE OF ATOM RECOMBINATION D A T A IN IRRADIATED SOLID H 2

Temp. (K)

Rate constant, k (m 3 /atom • s)

8.05 6.75 5.4 4.8 4.2 3.0 2.0 1.4

1.2( —24) 1.2( —26) 3.5( —28)a 3.8(—29)" 8.3( —30)a 4.7( —30)" 3.0 ( — 30)" 1.5(—30)a

Time constant,

Atom density, N;

T(S)

(per m 3 )

14a 360a

9.0(22) 1.2(24)

—

—

—

—

3.8 x 104

3.0(24)

a

(ppm) 4 50 —

— —

120a

—

—

—

—

—

6.5(25)"

3(— 13)a l(-15)a

—

—

104

Diffusion coefficient (m 2 /s)

2600a

6(— 19)a — —

4( —20)a

NOTE: The superscript (a) indicates a calculation. The numbers in parentheses are powers of ten.

R = 5.5 x 10 8 exp(—95/T) 4.8 < T < 5.4 K 23

(22.5)

where R = 1 at 4.72 K. Note that the exponents are different by a factor of two, but this could be caused by error in the narrow temperature ranges used here. We now extrapolate Eq. 22.5 upward to 6.75 K and attach it to Eq. 22.4. This produces the recombination rate constants listed in Table 22.3. By extrapolating with Eq. 22.5, we produce the largest possible values of k in this estimate. We assume no magnetic field effect on the rate constants. We also derive as many time constants and atom densities as the available data permit. The estimate of 2 600 ppm atoms at 1.4 K agrees with the researchers' estimate of 1000 to 10000 ppm. 2 4 Measured values of 3.8 x 104 s and 5.0 x 10 23 atoms/m 3 at 4.2 K produce a rate constant of 5 x 10~ 29 m 3 /atom-s—about six times larger than the value estimated in Table 22.3. These 4.2 K numbers do not fit with the higher temperature data taken in the same study, and the reason is unknown. 2 5 Another experiment to consider is an attempt to store a maximum number of atoms in irradiated H 2 at 0.3 to 0.8 K. 2 6 The author claimed an unexpectedly large sample magnetization between 0.30 and 0.37 K and postulated a ferromagnetic array of atom moments. He did not present a ppm atom number, but instead described his results in terms of atom "storage times" and "trigger temperatures." This led to an extensive theory of mobile and trapped atoms in an effort to explain the low temperature results. 27 Calculated H atom densities ranged from 8000 ppm at 0.1 K to 0.1 ppm at 1 K, in poor agreement with the actual data. There is an isotope effect in atom recombination. It is easy to trap and hold atoms in frozen D 2 , but atoms recombine much more quickly when H 2 is present in 75% quantity or more. 2 8 This effect is not obvious in the data of Table 22.2, but the H 2 dose rate is 2400 times higher than that used for the D 2 . We shall meet this isotope effect again when we consider charged defects in the solid.

286

RADIATION DAMAGE IN THE SOLID

If we take Eq. 19.19 for gas-phase recombination and add a crystal density of 2.6 x 10 28 molecules/m 3 to remove the third-body effects, the resulting pseudosecond-order rate constant is about 2 x 10~ 16 m 3 /atom• s. This is many orders of magnitude larger than the recombination rate in the solid and shows that atomic diffusion is the critical factor. We return to Eq. 19.5, which we rewrite as D = Ak/3o-

(22.6)

where D is the atomic diffusion coefficient, A the mean free path, and J = 0 Transitions The even-J and odd-J species of H 2 (or D 2 and T 2 ) do not easily react with each other. Over a period of time, however, something must bring the metastable rotational states into equilibrium. Almost anything that provides an unpaired electron can do it: paramagnetic impurities (including oxygen), electric discharges, atoms formed by thermal dissociation, metal catalysts, and radiation. The slowest catalyst is the hydrogen itself under conditions where none of the above are around to help. The most extreme example is the isolated H 2 molecule, never found in the real world. If the nuclear spin I (total) were 0 instead of 1, this molecule would sit forever in the J = 1 state even at 0 K. However, the interaction of J and I(total) is expected to slowly allow the J = 1 - » J = 0 transition to take place. The calculated rate constant for H 2 is about 10~ 20 (mol fraction • s) _ 1 , and because it is a first-order reaction, the time constant is about 10 12 years! 7 Real H 2 molecules have many neighbors, and as the density increases, the intermolecular forces become stronger. The most studied example of "natural conversion" is in the liquid and solid. The gas is not considered here because the vessel wall is expected to be the decisive catalyst. No one has ever studied the high-density fluid. In solid or liquid H 2 , the interaction causing the J = 1 de-excitation is that of the J = 1 and I(total) = 1 magnetic moments of neighboring molecules. 8,9 The energy of two magnetic dipoles, ignoring the directional effects, is (24.7) where n 0 /4n = 10" 7 N/A 2 , an SI metric magnetic constant, /X; and n i are the magnetic moments, r is the distance between the dipoles, and N 0 is Avogadro's Number. For H 2 , the nuclear magnetic moment is about 14 x 10~ 27 J/T (see Chap. 26) and the two nuclei are about 0.074 nm apart. The energy is about 30 /iJ/mol—not very much. Suppose a neighboring J = 1 molecule approaches, with its total nuclear magnetic moment of 28 x 10 - 2 7 J/T. It interacts across 0.31 nm with one nuclear spin of the first molecule and across 0.39 nm with the other. The energy difference is 0.4 ¿tJ/mol—less than the interaction within the molecule itself. Nevertheless, this asymmetric outside force is enough to gradually uncouple the nuclear spins and cause the J = 1 0 transition. The transition, then, is not thermodynamically expected, but in quantum mechanics, there is always a probability that it can happen anyway. (A free electron moment nearby leads to an energy of 260 /iJ/mol and the nuclear moments are easily decoupled.) The next question is: where does the J = 1 - » 0 energy go? For H 2 , this energy (expressed as a temperature equal to E/R) is 170 K, but the highest energy crystal

312

ROTATIONAL TRANSITIONS AND QUADUPOLES

lattice vibration, the Debye temperature dD, is only about 100 K . The lattice cannot take the extra energy in one shot: it takes two, at least. A two-phonon process will be about 10 times slower than a one-phonon interaction. 1 0 F o r liquid 30 K H 2 , 0 D is about 50 K , and three-phonon processes, another decade slower, will be needed for de-excitation. The mechanisms in T 2 should be basically the same as in H 2 . However, the T 2 J = 1 - » 0 energy is 58 K , while 9 D for the solid is again 100 K . F o r the liquid at 30 K , 9 d is 65 K . So T 2 can de-excite by one-phonon processes in both the solid and liquid, and it should convert 10 to 100 times as fast as H 2 . T 2 also has its beta particle, which will add to its speed of conversion. D 2 will be somewhat different. The deuteron magnetic moment is 3 0 % as large as that of the proton. The J = 1 -»• 0 reaction rate goes as the fourth power o f the magnetic moment, so that we expect D 2 to be 100 times slower than H 2 . n The D 2 J = 1 -> 0 energy is 86 K and the Debye temperature at 0 K is 100 K , so that onephonon processes should be possible in the solid. This should retrieve a factor o f 10 for D 2 in relative reaction rate. In addition to the I(total)-J interaction, D 2 has another mechanism for de-excitation. Because I > 1/2, the D nucleus has a nonspherical electric charge distribution described by the quadrupole moment, Q n , of 2.2 x 10~ 5 0 C - m 2 . 1 2 * The molecule also possesses a quadrupole moment, Q, about ten orders o f magnitude larger than that o f the nucleus. The molecule also has a nonspherical distribution of charge, with the electrons here being the major factor. All molecular states possess quadrupole moments, even the J = 0 states. 1 3 However, quantum symmetry rules forbid J = 0 molecules to interact, so that the quadrupole moment is not important. The J = 1 molecules interact readily and their quadrupole moments are very obvious. This interaction is the feature that has dominated most low-temperature hydrogen work in universities for the last two decades. F o r v = 0, H 2 , H D , and D 2 in the J = 0, 1, and 2 states, the quadrupole moments are each about 2.2 x 10~ 4 0 C • m 2 . The quadrupole moment is an extreme function of the vibrational state o f the molecule. 1 4 , 1 5 We return to J = 1 D 2 , for which the nuclear magnetic moment is 8.66 x 1 0 - 2 7 J/T. 1 6

Using Eq. 24.7, we find that the approximate energy between two nuclear

moments 0.35 nm apart is about

100 nJ/mol. Ignoring the angle-dependent

effects, the interaction energy between the nuclear and molecular quadrupoles is approximately 1 7 E(quad) ~

N° ^ 4ti£ 0 20r

(24.8)

* T h e quadrupole moment in SI units is in C - m 2 . In the literature, it may appear in the cgs units of e s u - c m 2 , and we multiply it then by 3 . 3 3 5 6 x 1 0 " 1 4 to obtain C - m 2 . It may also appear in cm 2 , and we must multiply then by 1.602 1 x 10~ 2 3 . In this case, the charge equal to 4.803 x 1 0 " 1 0 esu has been left out. Finally, in atomic units, we multiply by 4.486 x 1 0 " 4 O . W e use the quadrupole moment definition of Karl and Poll (see Chap. 24, note 13), which makes Q half the size of all earlier reported values. All data before 1967 must be divided by 2, and all data after that must be checked. W e may compare the above terminology with the electric dipole moment, in Debyes. One Debye equals 1 0 " 1 8 esu-cm, where the cm is sometimes left off. W e may then multiply again by 3 . 3 3 5 6 x 1 0 ~ 1 2 to obtain the SI unit in C - m .

313

ROTATIONAL TRANSITIONS A N D QUADUPOLES

where e0 is the SI metric permittivity constant of 8.854 x 1CT12 C/V • m. For the D 2 case, we obtain an energy of about 0.3 nJ/mol for a distance of 0.35 nm. A detailed calculation shows the quadrupolar interaction to be about half as effective in J = 1 0 as the magnetic dipolar interaction. 18 Now we turn to the actual rates. The H 2 de-excites when two molecules approach one another so that the reaction is bimolecular. If the fraction of molecules in the J = 1 state is c(l), the rate equation is dc(l) , „ / m2 = —kc(l) dt

(24.9)

where k is the rate constant in (mol fraction-s) - 1 . We assume that at equilibrium c(l) = 0, so that we do not worry about the J = 0 1 rate. This is a good assumption, but it is not quite true. The solution is ^ = — ^ c(l) 0 1 + c(l) 0 kt

(24.10)

where c(l) 0 is the initial concentration. We note that the answer is dependent on the starting condition. At one time constant, c(l) is 0.368 of the initial concentration (by analogy with the first-order time constant), so that T is

c(l)ok

P4.ll)

Table 24.1 lists H 2 rate data in the liquid and solid. 1 9 - 2 2 The time constant is referenced to nH 2 —that is, c(l) 0 = 0.75 at time zero. We note that T increases as H 2 is heated on up the saturation curve. This is because 0D decreases and higher-order phonon processes are needed to remove the energy. Conversely, increasing pressure at 4.2 K speeds up the reaction. Pressure stiffens the crystal lattice and 0D increases. The reason for the discrepancy in the two 4.2 K sets of data is unknown, but the trend is the same. The second set also possesses a small peak in the rate at 65900 mol/m 3 , and the reason is not known. We turn now to D 2 . The rate equation is: dt

= - k l C ( l ) 2 - k 2 c(l)[l - c(l>]

(24.12)

where the rate constants are k t and k 2 in (mol fraction -s) - 1 . The constant kx describes the interaction of two J = 1 molecules through both the nuclear dipolar and quadrupolar interactions. The constant k 2 describes the interaction of a J = 1 molecule with neighboring J = 0 molecules via the nuclear quadrupole interaction. We recall that 5/6 of the J = 0 molecules have I(total) = 2 and that they therefore possess a nuclear quadrupole moment. Once again, the J = 0 -* 1 back reaction is ignored. The solution is cOMk. + ^ - k . M i ) ] = exp( — k 2 t) c(l) 0 [k 2 + (k t - k 2 )c(l) 0 ]

(24.13)

314

ROTATIONAL TRANSITIONS AND QUADUPOLES

TABLE N A T U R A L CONVERSION R A T E S FOR THE J =

24.1

1 - * 0 T R A N S I T I O N IN S O L I D A N D L I Q U I D H

2

AND

D2 Temp. (K)

Conditions

H7

4.2 13.7 14.2 20.4

H?

4.2

Solid under pressure

H7

4.2

D,

4.2 20.4

| Saturated 1 solid (Saturated 1 liquid

Density (mol/m 3 )

Rate constant (mol fraction • s)"1

43400 43100 38 200 35200

Chap. 24 ref. no.

-6) -6) -6) -6)

135 144 180 201

19,20

44200 52600 61000

5.06 - 6 ) 8.31 - 6 ) 12.7 - 6 )

126 77 50

21

Solid under pressure

43400 53 300 58 500 61600 65 900 67600 69400 71 500 73 700

10 17 22 21 25 22 31 30 38

64 37 29 30 25 29 21 21 17

22

Saturated solid Saturated liquid

50100 42300

4.72 4.42 3.53 3.16

1/e time, T (hours)

1.6 1.5 1.0 0.92

-6) -6) -6) -6) -6) -6) -6) -6) -6) -7)1 -7)1 -7)1 -7)1

1900

19, 20

3100

NOTE: The D 2 rate constants are k x (upper number) and k 2 (lower). The numbers in parentheses are powers of ten.

Some values of the rate constants for D 2 are listed in Table 24.1. The time constant to 0.368 of the original J = 1 fraction is determined from nD 2 , where the time zero value of c(l) 0 = 0.333. From Table 24.1 , we see that k 1 ~ k 2 , so that -^Uexp(-k2t) c(l) 0

(24.14)

t~2.717/k2

(24.15)

where

The J = 1 —> 0 reaction for D 2 will appear to be first order, although it is not. Under comparable conditions, the D 2 transition is an order of magnitude slower than the H 2 transition, in agreement with the qualitative predictions. New de-excitation mechanisms appear in mixtures. The rate constant for H 2 in HD at 4 K is about 1.2 x 10~6 (mol fraction • s) - 1 . 2 3 This is slower than that of pure H 2 but still fast when one considers that the HD is almost all in the J = 0 state.

ROTATIONAL TRANSITIONS AND QUADUPOLES

315

However, the H and D nuclei are not coupled, so that the H nucleus can de-excite nearby H 2 molecules via a two-phonon process. A second process exists that is about one-fifth as likely. An H 2 molecule drops from J = 1 to 0 while pumping up an H D molecule from J = 0 to 1. This can occur because the J = 1 - » 0 energy is less for H D than for H 2 , so that the difference can be absorbed by the lattice as a single phonon. The J = 1 H D molecule can de-excite instantly because there is no metastability. A J = 1 D 2 molecule does not have enough rotational energy to pump up an H D molecule: it must de-excite by interaction with the D nucleus in HD. However, the process involves a single phonon with the factor of 10 increase over the two-phonon rate. The calculated H 2 and D 2 de-excitation rate constants in solid H D are 2.5 x 10~6 and 1.3 x 10~6 (mol fraction • s) _ 1 . 2 4 Finally, we note an interesting piece of speculation regarding the lattice spectrum. Phonon spectra for solid hydrogen are believed to have a fairly abrupt cutoff at high energies (see Fig. 7.3). Because of difficulties in explaining J = 1 0 rates, it is suggested that the phonon spectrum may be smeared out in a long tail at high frequencies. 25

J = 1

0 Catalysis

The literature on the subject of J = 1 -» 0 catalysis is vast, and the topic has literally become an industrial process. 2 6 ' 2 7 To catalyze liquid hydrogen rocket fuel and get the reaction over with, the paramagnetic nickel-silica Apachi catalyst was developed. 2 8 , 2 9 One can drop a few pellets of this into the cryogenic cell and, given good circulation of hydrogen, easily de-excite the sample. Many people have their personal favorite paramagnetic oxide also. There are two mechanisms for J = 1 0 catalysis. At room temperature, most metals and their oxides also work well as catalysts because hydrogen adsorbs onto them as atoms. When the atoms recombine, the new molecule may have a different J. A second mechanism works at any temperature, even low ones, where hydrogen adsorbs as a molecule onto metals. The presence of a paramagnetic substance—that is, one with unpaired electrons—does the job. The inhomogeneous magnetic field of the impurity atom decouples the paired nuclear spins of the hydrogen molecule. When the paramagnetic species diffuses or fluctuates out of sight, the hydrogen can recouple to a different J state. The lowest temperature generally used is 20.4 K, where hydrogen is still liquid and can flow over the catalyst. For both gas and liquid, catalysis is a zero-order reaction. 3 0 , 3 1 It, therefore, proceeds linearly until so little J = 1 hydrogen is left that it takes a long time for it to diffuse to the catalyst. Then the reaction "tails" away from its linear behavior. Some reaction rates, in mol/m 2 • s (of catalyst) are listed in Table 24.2. 3 2 , 3 3 None of these catalysts are well characterized, and the numbers should be considered approximate. "Unsupported" means there is no substrate onto which the catalyst is deposited in order to achieve finer particle size. The listed numbers are helpful in estimating how long catalysis will take. Suppose we have 0.02

316

ROTATIONAL TRANSITIONS A N D QUADUPOLES

T A B L E 24.2 CATALYSIS RATES FOR J =

1 TO J = 0 H 2 CONVERSION AT 2 0 K

Reaction rate (mol/m 2 • s) Alumina, "pure" Cu, reduced 630 K Ni, as received l - 2 % C r 2 0 3 on A1 2 0 3 G d 2 0 3 , unsupported 1-2% F e 2 0 3 on A1 2 0 3 , glass C e 2 0 3 or N d 2 0 3 , unsupported Ni, reduced 770 K Au, reduced 470 K F e 3 0 4 , unsupported 20% C r 2 0 3 on A1 2 0 3 5.3% Ni, 0.2% ThO z on A1 2 0 3 18% M n 0 2 on S i 0 2 0.5% Ni on A1 2 0 3 Ni, oxide covered MnO z , hydrous, unsupported F e 2 0 3 , hydrous, unsupported Fe, reduced 760 K Fe, oxidized 300 K

1( - 9 ) * 6( - 8 ) 7( - 8 ) 1( - 7 ) * 1( - 7 ) 2( - 7 ) * 2( - 7 ) 3( - 7 ) 4( - 7 ) 4( - 7 ) 5( - 7 ) 6( - 7 ) 8( - 7 ) 1( - 6 ) 2( - 6 ) 2( - 6 ) 3( - 6 ) 4( - 6 ) 1( - 5 )

NOTE: The asterisk is for liquid H 2 ; all others are for the gas at 0.1 MPa.

mol nH 2 containing 0.014 mol J = 1 H 2 with 1 m 2 surface area of granular hydrous ferric oxide, which has a listed rate of 3 x 10~6 mol/m 2 • s. The approximate conversion time is estimated to be 1.4 hours. Of special interest to us is the effect of radiation. Little work has been done on this, because it is so easy to catalyze with paramagnetic materials. Gamma radiation on H 2 at 1 to 15 MPa shows conversion at 200 and 300 K but, oddly, not at 77 K. The postulated reaction is the same ionic chain reaction that occurs for the exchange reaction in the gas phase. 3 4 We here break up the hydrogen as we did by chemisorption on metals at room temperature, although this process is ionic rather than atomic. In solid hydrogen, we expect radiation to produce hydrogen atoms, which are paramagnetic and can catalyze the J = 1 0 reaction. The main question is how fast the atoms recombine back to the molecule. Table 24.3 shows measured 1/e reaction times in liquid and solid tritiated hydrogen. 3 5 - 3 7 For pure T 2 , we estimate the natural time constant to be about ten times shorter than that of H 2 , because singlephonon processes are possible. For solid and liquid tritium, we estimate natural time constants of about 7 and 9 hours. We see that the radiation probably has no effect in the liquid, but the catalysis becomes stronger in the solid as the temperature falls. This agrees with our observation in Chapter 23 that catalytic hydrogen atoms are frozen into the irradiated solid as the temperature decreases. The tritium data of Table 24.3 must be taken as qualitative. The first two points were taken from small samples, but the remaining samples had a size of 1 /3 cc and

317

ROTATIONAL TRANSITIONS A N D QUADUPOLES

T A B L E 24.3 THE J =

1

0 REACTION TIME IN VARIOUS TRITIATED HYDROGEN MIXTURES

J = 1 -> 0 hydrogen

Hydrogen mixture

Phase

t2

98T 2 -2DT

Solid

Mean temp (K)

1/e time (hours)

4.2 4.3 10 11 13 18 19 22

0.28 1.5 1.2 2.6 3.9 6.9 7.2 8.2

Liquid

Chap. 24 ref. no. 35 36 37

had tritium heating problems. The series from 10 to 22 K was done by nuclear magnetic resonance with a slender glass sample tube so that the temperature gradient was 5 K from the copper block to the sample midpoint. We may combine Eqs. 19.6 and 23.27 to obtain 97tar[T] where D is the diffusion coefficient of a hydrogen atom in the solid, a is intermolecular distance (0.35 nm), and [T] is the atom density, which we set to 3 x 1022 atoms/m 3 (10 ppm). For x equal to an hour, D ~ 10~19 m 2 /s. We also recall that free electrons are present that could aid in the J = 1 -»0 reaction, but their numbers are believed to be at least an order of magnitude below the atom density. In solid tritium, the J = 1 infrared line Q^O, 1) + S 0 (l) drops to c(l) = 0.03 and stays there. This could mean a sample temperature of a little over 10 K. If T ~ 1.5 hours, then the fraction of molecules undergoing the J = 1 -* 0 reaction is about 6 x 10"6. But a fraction of 6 x 10~7 molecules is being ionized every second. Another factor of 10 to 20 may be added to the latter number for atom formation and the chain length of the exchange reaction. This means that the fraction of molecules being formed with J = 1 can be comparable to those undergoing the J = 1 —> 0 deexcitation. We have then a radiation-induced J = 1 formation effect, caused by the recombination of "hot" fragments. At some cold temperature yet to be determined, the fraction of J = 1 hydrogen is no longer an adequate thermometer. J = 1 Quadrupolar Interactions38 The J = 1 H 2 and D 2 molecules have electric molecular quadrupole moments that readily interact at close distances. Because these J = 1 levels are stable in the absence of a catalyst, it is easy to study the interaction between J = 1 molecules. The approximate energy between two molecules is

318

ROTATIONAL TRANSITIONS AND QUADUPOLES

We use an intermolecular distance of 0.35 nm and a molecular quadrupole moment of 2.2 x 10~4° C • m 2 . We obtain an energy of about 2.5 J/mol between two quadr u p l e s . Somewhere below 10 K, we may expect the quadrupolar energy to exceed the thermal energy, and the quadrupoles then order themselves into a lattice. For a J = 1 concentration greater than 0.5, H 2 and D 2 (and probably T 2 ) will change to the ordered f.c.c. structure. To be honest, this phenomenon does not appear to be of any use in the fusion business, yet. But nuclear spin polarization rose recently from a curiosity to a gripping topic of interest within the space of a single year, and one hesitates to place any property off-limits. In any case, a vast literature on quadrupolar interactions has been compiled in the last twenty years. A dipole is a combination of a positive and a negative charge. A simple quadrupole could consist of three charges on a line—a positive charge + e on each end and a charge — 2e in the middle. Quadrupoles, like dipoles, are electrically neutral overall. When one observes an electric (or magnetic) charge of arbitrary shape from far away, the shape resolves into certain simple geometric forms. First in simplicity comes the spherical single charge, then the dipole, then the quadrupole. Dipoles align end to end, with the positive charge of one dipole next to the negative charge of its neighbor. Quadrupoles align at right angles with the positive end of one pointing at the negative midsection of another. 3 9 The simplest case is two J = 1 molecules at the solid intermolecular distance from each other. Figure 24.1 shows the calculated spectrum for two J = 1 D 2 molecules as the forces are progressively turned on. 4 0 ' 4 1 First, there is no interaction and we show the molecules at "zero" energy for convenience (ignoring the energy that the J = 1 level lies above the J = 0 ground state). Each molecule has three magnetic substates: nij = —1,0, and + 1. If we turned on just a DC magnetic field, these sublevels would split apart with the mj = 0 level below zero energy and the mj = + 1 level above. In the quadrupolar interaction, the three sublevels of each molecule interact with the three of the other molecules for a total of nine. The b) part of Fig. 24.1 shows how the quadrupolar force distributes these nine levels. The total energy splitting is 7.7 J/mol for D 2 . 4 2 For T 2 , we might estimate a total splitting of 11 J/mol. The final force shown is the smaller "anisotropic potential." This arises from the fact that J = 0 molecules are not perfectly spherical, and a correction as to the direction of molecular alignment must be added. When the two J = 1 molecules are placed in an actual crystal with J = 0 surroundings, this small, electric crystal field force further splits the lines slightly. When three J = 1 molecules interact, we have 3 3 = 27 substates, all more closely spaced in energy than with nine substates. When N J = 1 molecules interact, we have 3N substates, forming literally an energy band around the zero energy. The quadrupoles in pure J = 1 hydrogen align in the structure shown in Fig. 24.2. The structure is face-centered cubic (f.c.c.), and four interpenetrating sublattices of oriented molecules are arranged in it. 43 The quadrupoles are as much at right angles as they

319

ROTATIONAL TRANSITIONS AND QUADUPOLES

CM I

a) k. 3 CO a> a E a>

O

>• o> w w c 111

Mechanism t u r n - o n Fig. 24.1. The energy level splitting between two J = 1 D 2 molecules for a) no interaction, b) turn-on of electric molecular quadrupolar force, and c) subsequent turn-on of anisotropic potential, i.e., crystal electric field. The number of magnetic substates is listed. (Courtesy I. F. Silvera and the American Physical Society.)

can manage in three dimensions. There is no net electric moment, and the sample is like an antiferromagnet. 44 Pure J = 1 H 2 would order itself at about 3 K, J = 1 D 2 at 4 K, and J = 1 T 2 would be expected at about 4.7 K. Beta decay would, of course, be expected to destroy the ordering through J = 1 —»• 0 conversion.

Quadrupolar Thermodynamics There is a small amount of energy involved in the quadrupolar interaction, whether in the ordered phase or not. This energy looks impressive in the heat capacity for saturated solid H 2 , as shown in Fig. 24.3. 45,46 The spike is the ordering transition, but at higher temperatures, an additional heat capacity above the c(l) = 0 curve is evident. An overall model of the quadrupolar thermodynamics of many molecules is complex. 47 ' 48 We shall simplify it by using a simple two-energy-level model. 49 We assume the nij = 0 sublevel to be the ground state with the two mj = ± 1 sublevels 4.5 K higher. The partition function is

320

ROTATIONAL TRANSITIONS AND QUADUPOLES

Fig. 24.2. The ordered form of pure J = 1 hydrogen. The four sublattices of the f.c.c. crystal are numbered. The molecular orientations along the four body diagonals are indicated by the axes of the diatomic dumbbells. (Courtesy J. C. Raich and the American Physical Society.)

Q = 1 + 2exp(-4.5/T)

(24.18)

The " 2 " is present because of the two substates. Our oversimplified model gives us a simple form for the quadrupolar heat capacity, C(quad). We must make the final adjustments empirically from the available experimental d a t a . 5 0 - 5 8 We have: C ( q u a d ) ^ 340

exp(-4.5/T) c(l) 1 . 5 T 2 [1 + 2 e x p ( - 4 . 5 / T ) ] 2

(24.19)

ROTATIONAL TRANSITIONS AND QUADUPOLES

321

o E

o

CD

a TO

u

TO

a> "o O I-

Temperature (K) Fig. 24.3. The total crystal lattice heat capacity of solid nH 2 as a function of the J = 1 concentration, c(l). The J = 1 dependent heat capacity is caused by interaction of the electric molecular quadrupole moments. The transition to the ordered phase is indicated by the thermal spikes. The dashed lines are estimated. (Courtesy R. W. Hill and Taylor & Francis Ltd.)

322

ROTATIONAL TRANSITIONS AND QUADUPOLES

This gives us a peak at about 1.7 K, and we are well d o w n the tail by the time we come to 4.2 K, our usual starting temperature in fusion applications. The energy tied up in the quadrupolar interaction is the integral of Eq. 24.19. E(quad) ^ 7 5 c ( l ) ' For c ( l ) = 0.75 and T

-f^exp^—4J/T)

< 2 4 ' 2 °>

oo, E(quad) ~ 16 J/mol, which is not much. At 4.2 K, 60%

of E(quad) has been attained, because we have passed well beyond the peak value of C(quad). Equation 24.19 does not include the spike that signals the low-temperature h.c.p.-to-f.c.c. ordering transition. This spike amounts to about only 2 to 4 J/mol for c ( l ) = 0.65 to 0.74, 5 9 so that only about 15% of the quadrupolar energy appears to be in the spike. At 4.2 K, 10 K, and the triple point, the estimated quadrupolar energies are: n H 2 , 10, 14, and 14 J/mol; n D 2 , 3, 4, and 4 J/mol; n T 2 , 10, 14, and 15 J/mol; and 2 5 n D 2 - 5 0 D T - 2 5 n T 2 , 3, 4, and 5 J/mol. The quadrupolar energy comes out of the potential energy and adds onto the heat of sublimation at 0 K. At 4 K and above, it does not affect the heat of sublimation. This is the same quadrupolar energy that we dutifully added in Chapter 8. Notes 1. D. Böhm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951), pp. 493495. 2. C. A. Coulson, Valence, 2d ed. (Oxford University Press, New York, 1961), pp. 116119. 3. G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, 2d ed. (D. Van Nostrand, Princeton, N.J., 1961), pp. 69-70, 76-78, 130-141. 4. See note 3 above. 5. N. F. Ramsey, Phys. Rev. 85, 60 (1952). 6. A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961), pp. 223-232,316-319. 7. J. Raich, Ortho Para Transition in Molecular Hydrogen, Ph.D. thesis, Iowa State University, Ames, Iowa (1963); University Microfilms No. 64-3889, Ann Arbor, Mich. 8. K. Motizuki and T. Nagamiya, J. Phys. Soc. Japan 11, 93 (1956). 9. A. J. Berlinsky and W. N. Hardy, Phys. Rev. 8B, 5013 (1973). 10. See note 8 above. 1 1 . K . Motizuki, J. Phys. Soc. Japan 12, 163 (1957). 12. R. V. Reid, Jr. and M. L. Vaida, Phys. Rev. 7A, 1841 (1973). 13. G. Karl and J. D. Poll, J. Chem. Phys. 46, 2944 (1967). 14. See note 13 above. 15. A. Birnbaum and J. D. Poll, J. Atmos. Sei. 26, 943 (1969). 16. R. G. Barnes, P. G. Bray, and N. F. Ramsey, Phys. Rev. 94, 893 (1954). 17. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley, New York, 1954), p. 27. The old, twice-as-large quadrupole moment is used. 18. K. Motizuki, J. Phys. Soc. Japan 17, 1192 (1962). 19. Y. Y. Milenko and R. M. Sibileva, Ukr. Fiz. Zh. 19, 2008 (1974), in Russian. 20. Y. Y. Milenko and R. M. Sibileva, Soviet J. Low Temp. Phys. 1, 382 (1975). 21. G. Ahlers, J. Chem. Phys. 40, 3123 (1964). 22. S. A. Dickson and H. Meyer, Phys. Rev. 138A, 1293 (1965). 23. W. N. Hardy and J. R. Gaines, Phys. Rev. Lett. 17, 1278 (1966).

ROTATIONAL TRANSITIONS AND QUADUPOLES

24. 25. 26. 27. 28. 29.

323

K. Urano and K. Motizuki, Solid State Commun. 5, 691 (1967). See note 9 above. G. E. Schmauch and A. H. Singleton, Indust. Eng. Chem. 56, 20 (1964). G. E. Schmauch and A. H. Singleton, Cryogenic Eng. News 1, 43 (1966). See note 27 above. Available from Houdry Division of Air Products and Chemicals, Inc., Philadelphia,

Pa. 30. C. M. Cunningham and H. L. Johnston, J. Amer. Chem. Soc. 80, 2377 (1958). 31. J. T. Kummer, J. Phys. Chem. 66, 1715 (1962). 32. See note 30 above. 33. D. H. Weitzel, J. W. Draper, O. E. Park, K. D. Timmerhaus, and C. C. Van Valin, Adv. Cryogenic Eng. 2, 12 (1960). 34. J. W. Conant, F. J. Edeskuty, J. E. Huston, and F. V. Thorne, Cryogenics 15, 12 (1975). 35. E. W. Albers, P. Harteck, and R. R. Reeves, J. Amer. Chem. Soc. 86, 204 (1964). 36. R. Frauenfelder, F. Heinrich, and B. Olin, Helv. Phys. Acta 38, 279 (1965). 37. J. R. Gaines, R. T. Tsugawa, and P. C. Souers, Phys. Rev. Lett. 42, 1717 (1979). 38. For an excellent review, see I. F. Silvera, Revs. Mod. Phys. 52, 393 (1980). 39. A. B. Harris, J. Appl. Phys. 42, 1574 (1971). 40. T. Nakamura, Progr. Theor. Phys. 14, 135 (1955). 41. See reference in note 38 above, pp. 407-410. 42. H. Miyaki, Progr. Theoret. Phys. 40, 1448 (1968). 43. J. Felsteiner, Phys. Rev. Lett. 15, 1025 (1965). 44. See note 39 above. 45. R. W. Hill and B. W. A. Ricketson, Phil. Mag. 45, 277 (1954). 46. G. Ahlers and W. H. Ortung, Phys. Rev. 133A, 1642 (1964). 47. R. J. Roberts and J. G. Daunt, J. Low Temp. Phys. 6, 97 (1972). 48. R. J. Roberts, E. Rojas, and J. G. Daunt, J. Low Temp. Phys. 24, 265 (1976). 49. W. H. Orttung, J. Chem. Phys. 36, 652 (1962). 50. See notes 45 through 49 above. 51. K. Clusius and E. Bartholome, Z. Physik Chem. 30B, 327 (1935). 52. E. C. Kerr, E. B. Rifkin, H. L. Johnston, and J. T. Clarke, J. Amer. Chem. Soc. 73, 282(1951). 53. O. D. Gonzalez, D. White, and H. L. Johnston, J. Phys. Chem. 61, 773 (1957). 54. R. W. Hill and O. V. Lounasmaa, Phil. Mag. 4, 785 (1959). 55. G. Ahlers, Some Properties of Solid Hydrogen at Small Molar Volumes, Lawrence Berkeley Laboratory, Report UCRL-10757, Berkeley, Calif. 94720 (1963). 56. G. Grenier and D. White, J. Chem. Phys. 40, 3451 (1964). 57. R. J. Roberts and J. G. Daunt, Phys. Lett. 33A, 353 (1970). 58. R. J. Roberts and J. G. Daunt, J. Low Temp. Phys. 16, 405 (1974). 59. See notes 45 and 46 above.

25. Infrared and Raman Spectroscopy

Two types of spectroscopy appear often in the analysis of cryogenic hydrogen: Raman and infrared.1 Both look at the vibrational-rotational motion of the hydrogen molecules. Both methods can be empirically calibrated to tell the numbers of molecules giving rise to the signal. We shall not discuss here the specifics of such spectroscopy but only the major transitions involved. Transition Energies The basic selection rule for these transitions is2 Av = 0, ± 1 ; A J = 0, ± 2

(25.1)

and J is the rotational quantum number. For Raman spectroscopy, the molecule undergoes an electronic transition by laser excitation. The vibrational-rotational transitions are added or subtracted from the basic optical frequency. Directly stimulated Av = 0 spectroscopy involves no vibrations and can be stimulated in the microwave region. Infrared spectroscopy has a v = 0 1 transition for the fundamental, but the weaker overtones, such as v = 0 -» 2, can also be excited. The A J = ± 2 rule is a result of the quantum symmetry of the diatomic molecule. We might expect the rule to be broken for heteronuclear hydrogens, such as HT, which have an electric dipole moment. This moment, however, is very small and gives rise only to a small, narrow AJ = + 1 line that can be seen only with high-resolution techniques.3 All the hydrogens, therefore, obey the basic rules of symmetry for Raman and infrared spectroscopy. Much weaker A J = ±4 transitions are also possible.4 In Chapter 2, we considered the spectroscopic constants and how they may be combined to give vibrational and rotational energies. We summarize the more important formulas here. For the fundamental vibrational transition, the frequency will be v(v = 0 - 1) = ve - 2(v e x e ) + ^(v e y e )

(25.2)

with all quantities in Hz. The constants on the right are listed in Table 2.1. For the first overtone, we have:

325

INFRARED AND RAMAN SPECTROSCOPY

T A B L E 25.1 G A S - P H A S E Q I ( 0 ) A N D Q 2 ( 0 ) T R A N S I T I O N S FOR THE H Y D R O G E N S

Frequency, for J = 0 Units

v(v = 0 - » 1)

v(v = 0

2)

Hz

H2 HD HTA D2 DTA T2

1.247 5(14) 1.0889(14) 1.0298(14) 8.9770(13) 8.229 5(13) 7.3996(13)

2.424 5(14) 2.1249(14) 2.012 5(14) 1.7604(14) 1.6171(14) 1.457 7(14)

cm-1

H2 HD HT" D2 DTA T2A

4161 3632 3435 2994 2745 2468

8087 7088 6713 5872 5394 4862

These correspond to pure vibrational fundamental and first overtone transitions in the infrared. The superscript ( a ) indicates estimates. NOTE:

v(v = 0 -» 2) = 2ve - 6(v e x e ) + ^ ( v e y e )

(25.3)

For molecules in the v = 0 state, the rotational transitions are v(J = 0 —• 2) = 6B 0 - 36D e

(25.4)

v(J = 1 - 3) = 10Bo - 140De

(25.5)

and

with the rotational constants listed in Table 2.2. In the v = 1 state, we replace B 0 with B l 5 which is a little smaller. D e is too small to be worth adjusting according to the vibrational state. The resulting frequencies of greatest use are listed in Tables 25.1 and 25.2. The former shows the v = 0 -> 1 and v = 0 - » 2 frequencies with no rotation; the latter ignores vibration and tells how much more has to be added for J = 0 - » 2 and J = 1 3 rotational transitions.

Raman Spectroscopy Raman spectroscopy is the more expensive method because of the laser and the need to observe the weak sidebands with the vibrational-rotational information in them. However, the Raman results are simpler than those of infrared spectroscopy because only one hydrogen molecule is involved at a time. An individual molecule responds to the electric field of the laser light, 5 and the resulting signal is proportional to the number of molecules present. The molecules do not interact with each other very

326

I N F R A R E D A N D R A M A N SPECTROSCOPY

T A B L E 25.2 GAS-PHASE ROTATIONAL ENERGIES AS MEASURED FROM THE Q O ( 0 ) AND Q ^ O ) LINES

For v = 0 Units

For v = 1

Av(J = 0 - 2 ) = 6B0

Av(J = 1 - 3 ) = 10BO

Av(J = 0 - 2 ) = 6B!

Av(J = 1 - 3 ) = LOBJ

Hz

H2 HD HT D2 DT T2

1.067(13) 8.035(12) 7.156(12) 5.381(12) 4.494(12) 3.605(12)

1.779(13) 1.339(13) 1.193(13) 8.968(12) 7.490(12) 6.008(12)

1.014(13) 7.696(12) 6.873(12) 5.195(12) 4.352(12) 3.502(12)

1.690(13) 1.283(13) 1.145(13) 8.658(12) 7.253(12) 5.836(12)

cm - 1

H2 HD HT D2 DT T2

356 268 239 179 150 120

593 447 398 299 250 200

338 257 229 173 145 117

564 428 382 289 242 195

NOTE: These can be excited as pure rotational transitions in the microwave region or added onto the vibrational numbers of Table 25.1 to produce infrared vibration-rotation transitions. much or with the crystal lattice. This results in narrow line widths and generally an absence of sidebands caused by phonon interactions. Figure 25.1 shows a calculated Raman spectrum of all six hydrogens at 4 kPa at room temperature. 6 The fact that intensity is proportional to the number of molecules makes it possible to see the spectrum in low-pressure gas. The hydrogen lines are shifted by the inverse square root of their reduced masses. Because the lines are so narrow, the different hydrogens are easily resolved. The narrowness of the lines also requires a more expensive spectrometer for resolution. All these lines are called Q l s which indicates the transitions v = 0 — 1 and AJ = 0. At room temperature, many of the rotational levels are thermally populated and are included in transitions. The " 6 " bump indicates J = 6 — 6 and is the Qi(6) line. The extreme resolution in obtaining all these levels is the reason why Raman spectroscopy is used to obtain the spectroscopic constants in the first place. At cryogenic temperatures, we expect only the Q, (0) and Q i ( l ) lines to start the series. Two other lines will appear. One is the Qj(0), which corresponds to the v = 0 — 1, J = 0 — 2 transition in a single molecule. The other, the Q i ( l ) , corresponds to the v = 0 — 1 , J = 1 — 3 transition. We expect the S^O) line to lie six rotational B constants to the higher frequency side of Qi(0), and we expect S x ( l ) to lie ten B constants from Q i ( l ) . We can obtain the J = 1 composition by a measurement of either the Q- or S-pair of lines. The pure rotational S o (0) and S 0 ( l ) pair, with no vibrational excitation, can also be stimulated in the microwave region. The gas phase Raman frequencies are listed in Tables 25.1 and 25.2: the former shows the (^(O) and Q 2 (0) frequencies, while the latter lists the additional frequencies

327

INFRARED AND RAMAN SPECTROSCOPY

Wavelength (nm) Fig. 25.1. Calculated positions for the Qj-lines in the Raman spectrum of a mixture of gaseous hydrogens at 4 kPa at room temperature. The laser wavelength is 488 nm (from Setchell and Ottesen; see Chap. 25, reference 6).

to be added to obtain the J = 0 2 and J = 1 —»3 lines. In the liquid and solid, the crystal forces affect the Raman lines somewhat, shifting them to lower frequencies by about 0.3 THz. 7 The lines are also slightly broadened. Raman spectroscopy can be a delicate tool. The transitions between two J = 1 H 2 molecules in a J = 0 crystal—the same crystal-field split-levels shown in the c part of Fig. 24.1—can be measured using this method. 8 Raman spectroscopy has also been used to study the free rotation of hydrogen molecules in the solid. We have noted that they rotate without hindrance or, as is

328

INFRARED AND RAMAN SPECTROSCOPY

often said, "J is a good quantum number." We saw in Chapter 22 that only a minor hindrance occurs even for molecules thought to be next to trapped electrons. Under enough pressure, however, the molecules should be forced close enough together to inhibit rotation and mix the rotational states into new combinations. 9 The onset of such effects has been seen using Raman spectroscopy on solid J = 0 D 2 at 29 GPa. 1 0

Infrared Spectroscopy Infrared spectroscopy is the cheaper way to go, because low resolution is adequate and the spectroscopy involves only the direct passage of an incoherent beam through the sample. One drawback is that infrared spectroscopy measures too much. This arises because individual molecules do not undergo transitions. Their environment is almost the same in every direction around them, and the various fields cancel. 11 However, two species can interact and each undergo a transition. In fact, infrared spectroscopy in hydrogen is a forbidden process until molecular collisions produce the electron cloud distortions that allow the transitions. The double transitions with infrared spectroscopy mean that the absorption intensity goes as the square of the molecular density. There is no signal at all from low-pressure gas unless a very long path length is used. In the solid, about 0.1 to 1 ¿¿(m3) of sample is needed, just as with Raman spectroscopy. The presence of double transitions adds quickly to the complexity. The v = 0, AJ = 0 line is given the same Qi(0,1) name as we found previously with Raman spectroscopy. The interactions are much stronger between molecules and the two Raman lines are here usually merged into a single broad one. This line is shown for solid T 2 in Fig. 22.4. 12 The transitions are Q^O) + Q 0 (l), Q i ( l ) + Q o (0), and Q i ( l ) + QoO), where the double-zero transition is forbidden. The first term represents the molecule that has the vibrational excitation; the second term represents the neighboring molecule that undergoes only a rotational transition. The Raman S^O) transition becomes an infrared Q^O, 1) + S o (0) double transition, also shown in Fig. 22.4. The Raman S ^ l ) becomes the infrared Q^O, 1) + S 0 (l) transition. These big broad peaks come at roughly six and ten times the B rotational constant away from the Q^O, 1) line. After each of the main lines comes another type of double transition—that involving phonons. The Q-phonon peak, shown in Fig. 22.4, is a Q^O, 1) transition on a molecule plus the absorption of energy from the beam into the crystal lattice vibrations. The Q-phonon band is a representation of the phonon spectrum. Similar phonon peaks follow the Qi(0,1) + S o (0) and Qi(0,1) + S 0 (l) peaks. These phonon sidebands are not as representative of the true phonon spectrum as is the Q-phonon band. The phonon peaks appear only to the high-frequency side because of the low temperature. The crystal lattice has no energy to contribute to the transition. If we had a condensed phase at high temperature, we would expect to see phonon bands as well to the low-frequency side. The full horror of the infrared spectrum becomes evident when the hydrogens are mixed. The Qi(0,1) + S 0 (l) line for T 2 blossoms into a triplet when DT and D 2

329

INFRARED AND RAMAN SPECTROSCOPY

Frequency (cm" 1 )

2500

2700

2900

Frequency (THz) Fig. 25.2. Collision-induced infrared spectrum of solid 25% D 2 - 5 0 % D T - 2 5 % T 2 at 18 K. The peak numbers are explained in Table 25.3.

are added. This is shown as lines 3, 4, and 5 for solid equimolar D-T at 18 K in Fig. 25.2. 13 The Q^O, 1) part belongs to T 2 , but the S 0 (l) parts belong to T 2 , DT, and D 2 , in order. The line numbers in Fig. 25.2 are explained in Table 25.3, which lists the peak frequencies seen or expected in various combinations of solid hydrogens. 1 4 - 1 7 Note that the frequencies are slightly shifted from the gas-phase values.

330

I N F R A R E D A N D RAMAN SPECTROSCOPY

TABLE 25.3 P E A K FREQUENCIES OF COLLISION-INDUCED F U N D A M E N T A L I N F R A R E D PEAKS FOR VARIOUS SOLID H Y D R O G E N S

Peak frequency THz nH 2

HD

HT

nD 2

CHOKQOCO.L) Qi(0) + QoO) Q-phonon QI(l) + So(0) Qi(0) + s O (0) S(0)-phonon Qi(l) + S 0 (l) QI(0) + S o (l) S(l)-phonon QI(0) Q r phonon # 1 #2 Q,(0) + SO(0) S(0)-phonon # 1 #2 Q ^ + SOO)* S(l)-phonon # l a #2" Q,(0) Q-phonon # 1 #2 CHW + SOIO) S(0)-phonon # 1 #2 Q.(0) + s o ( i ) S(l)-phonon # l a #2" QiO) + Qo(0, i) Qi(0) + QoO) Q-phonon Q,(0) + SO(0) S(0)-phonon Q 1 (0,l) + S o (l) S(l)-phonon

124.3 124.5 126.6 134.8 135.0 136.3 141.9 142.0 143.0 108.7 110.4 111.6 116.7 117.9 119.3 122.0 122.9 124.1 102.8 104.3 105.6 110.1 111.2 112.4 114.4 115.4 116.3 89.49 89.55 91.59 94.88 96.35 98.51 99.92

Peak frequency

1

No.

4146 4153 4223 4497 4504 4547 4732 4738 4770 3625 3684 3722 3894 3934 3980 4068 4101 4138 3429 3479 3523 3671 3710 3748 3816 3848 3879 2985 2987 3055 3165 3214 3286 3333

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

cm

D 2 -DT-T 2 QI(O, I)T 2 Q-phonon T 2 QI(0> 1)T2 + S o (0)T 2 QI(0,1)T 2 + SO(0)DT Qi(0> 1)T2 + S„(0)D2 Q ^ i y r . + SOO)^ Qi(0) DT Qi(o, I)T 2 + s„(I)D 2 Q-phonon DT QI(0)DT + SO(0)T2 Q,(0)DT + SO(0)DT Q!(0)DT + SO(0)D2 QI(0)DT + S 0 (1)T 2 QI(O, 1)D 2 Q . ^ D T + SOI^DJ Q-phonon D 2 Q.CO^D. + QOWT, Qi(0» 1)D2 + QO(0)DT Q,(0,L)D 2 + QO(0)D2 Qi(0» 1)D2 + Q 0 (1)T 2 Qi(0> 1)D2 + Q 0 (l)DT a QI(O, i)D 2 + Q 0 (1)D 2

1

THz

cm

73.69 75.49 77.23 78.16 79.03 79.65 82.02 82.62 84.00 85.59 86.55 87.42 88.05 89.52 90.96 91.62 93.03 93.98 94.85 95.51 96.83 98.42

2458 2518 2576 2607 2636 2657 2736 2756 2802 2855 2887 2916 2937 2986 3034 3056 3103 3135 3164 3186 3230 3283

NOTE: The numbers for D 2 -DT-T 2 refer to Figs. 25.2 and 25.3. The (a) numbers are calculated. Two values appear for the heteronuclear hydrogen phonon bands because of the "holes."

The Q^O, 1) + S o (0) and Q i(0,1) + S 0 ( l ) lines seem to be the best for measuring both hydrogen and J = 1 concentrations. 18 The results used from peak heights are as good as the fancier ones determined from areas, which shows that the quantitative analysis has a long way to go. The Q^O, 1) peak has strange and unknown J = 1-related factors in it and should not be used until more is learned. Figure 25.3 shows the mercifully simpler spectrum of molecular D T itself. 19 The

INFRARED AND RAMAN SPECTROSCOPY

331

Frequency ( c m - 1 )

2600

2800

3000

3200

c o c (0 -U a>

80

90

100

Frequency (THz) Fig. 25.3. Collision-induced spectrum of 96% molecular DT. About 2 to 3% D 2 is present. The numbers refer to assignments in Table 25.3 (from P. C. Souers).

compound is made by the reaction of lithium tritide and deuterated methanol. The yield is about 96%, with 2 to 3% D 2 impurity being clearly present. The numbered peaks are those listed in Table 25.3. As we described in Chapter 23, the DT breaks into D 2 and T 2 with a time constant of perhaps 120 hours. After 3 hours, the T 2 and D 2 lines are indeed seen to grow.

Impurity Lines Modern cryogenic distillation facilities allow production of comparatively pure tritium. An analysis of just-produced 99.54 mol% tritium is shown in Table 25.4. 20

332

I N F R A R E D A N D R A M A N SPECTROSCOPY

T A B L E 25.4 ANALYSIS OF HIGH-PURITY TRITIUM JUST AFTER PRODUCTION BUT BEFORE STORAGE

Mol%

99.108 0.635 0.078 0.063 0.052 0.023 0.015 0.013 0.003 0.002 0.002 0.002 0.002 0.001

T2 DT N2 HT D 2 0 or HTO HD T2O H DO Ar

DTO CT4

co2 O2 D2

99.999

TABLE 25.5 PRODUCTION OF TRITIATED METHANE FROM TYPE 3 0 4 STAINLESS STEEL IN CONTACT WITH

124 kPa (930 Torr) 92% T 2 -8% DT AT 298 K As received, rinsed

As received, degreased

Machined, rinsed

Time (days)

Methane (ppm)

Time (days)

Methane (ppm)

Methane (ppm)

0 1 7 14 28 50 113

40 740 1240 1690 2430 2970 3 780

0 1 7 15 29 64 141

42 186 487 652 747 857 996

42 96 221 265 297 308 342

NOTE: The vessel volume is 293 ß(m 3 ) and the internal surface area is 0.0353 m 2 , for an areato-volume ratio of 120 m" 1 .

Unfortunately, this gas is not stable. The tritium, by way of its radioactivity, interacts with the storage container walls to produce impurities. What impurities and how much is produced depends on the type of wall material and its cleanliness. Steel is the usual vessel material and, unfortunately, it is quite reactive. One common impurity is tritiated methane, which is formed by the reaction of tritium with the carbon in the stainless steel. Table 25.5 shows data taken on 96% tritium stored in 304 stainless steel at 124 kPa at room temperature. 2 1 The internal

333

I N F R A R E D A N D R A M A N SPECTROSCOPY

T A B L E 25.6 IRRADIATION-INDUCED OUTGASSING IN m o l FRACTIONS FROM VESSEL WALLS FOR 9 9 . 7 % T 2 AT

0°C

Gas HT

Methane

Time (days)

Untreated commercial 304 SS

50 100 150 200 250 300

5.4 6.8 7.9 9.1 10.1 10.9

50 100 150 200 250 300

0.93 1.4 1.6 1.9 2.1 2.2

AND

0.1 MPa (1 atm)

Cadmiumplated, electrocleaned 304 SS 0.77 0.91 1.1 1.15 1.2

Pyrex glass 0.68 0.78 0.81 0.85 —

Electropolished 0 2 discharge 304 SS 0.52 0.61 0.70 0.75 0.78 —

—

0.009 0.013 0.018 0.015 0.013 —

um.; 0.020 0.021 0.022 —

0.025 0.031 0.033 0.032 0.033

—

—

Burnished aluminum

Drypolished 304 SS

0.14 0.15 0.16 0.16 0.16

0.05 0.05 0.05 0.06 0.08

—

0.025 0.046 0.066 0.065 0.067 —

—

0.013 0.022 0.031 0.031 0.032 —

NOTE: "SS" is stainless steel.

area-to-volume ratio, (A/V), is 120 m - 1 . The methane production is very much dependent on the history of the steel. The degreasing was done with trichlorethylene in an ultrasonic bath and the rinsing was done with Freon. The machining was done dry to remove 0.25 mm from the surface. All samples were baked in vacuum at 430 K for 12 hours. An even wider assortment of outgassing results is listed in Table 25.6. 22 Here, 0.1 MPa (1 atm) of 99.7% tritium gas is stored at 0°C for one year. Both HT and tritiated methanes are measured. The H, of course, comes out of the vessel walls. We see that untreated stainless steel is filthy. Cleaning it by any means—or going to Pyrex or aluminum—helps considerably. The presence of adsorbed organics is especially noxious in forming impurities. No single cleaning procedure or material listed here is best in reducing both H T and methane. Clearly, the art of confining and retaining pure tritium gas has a long way to go. Our suggestion for how to extrapolate these data to other steel vessels is to take the methane concentration at some time to be proportional to (ppm methane) ~ P T (A/V)

(25.6)

where P x is the partial pressure of tritium. One has to decide, of course, as to the category of cleaned surface to be used. These impurities can indeed produce confusing peaks in the vibration-rotation region of the condensed hydrogens. The impurities are frozen and give infrared lines characteristic of the solid. We list these common impurities in Table 25.7. 23 ~ 28 Water

334

INFRARED AND RAMAN SPECTROSCOPY

T A B L E 25.7 INFRARED FREQUENCIES OF THREE COMMON IMPURITIES IN D - T

Infrared peak frequency (THz) Compound CD 2 T 2 CT 4 CDT 3 CO CD 3 T CD 2 T 2 CDT 3 CD 2 T 2 CD 4

Vibrational mode V 3 C(B 2 )

V3(F2)

v 3b (E) VL V3a(A1)

V3A(Al) V3a(A,)

co2

v^BJ v 3 (F 2 ) v 3b (E) v3 v3

co2 co2

2V2 + v 3 Vl + v3

CDJT 13

CO 2

In liquid D-T —

57.92 —

64.07 64.61 65.65 66.55 —

67.42 —

68.35 70.45 70.63 70.90* 108.0 111.2

Alone as gas 58.01 58.07 58.13 64.25 64.91 66.04 66.85 67.48 67.72 67.75 —

70.42 110.4 112.0

NOTE: The C 0 2 values represent peaks at 100 and 1 000 ppm plus a shoulder (*) at 1 000 ppm solid particles in liquid D-T.

does not appear because the hydrogen-bonding forces in ice are so great that the infrared lines are broadened out of s i g h t . 2 9 - 3 2

Notes 1. These special types of spectroscopy with hydrogen were first developed experimentally by H. L. Welsh and theoretically by J. Van Kranendonk, both of the University of Toronto. 2. G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, 2d ed. (D. Van Nostrand, Princeton, N.J., 1961), pp. 86-90. 3. A. Crane and H. P. Gush, Can. J. Phys. 44, 373 (1966). 4. R. D. G. Prasad, M. J. Clouter, and S. P. Reddy, Phys. Rev. 17A, 1690 (1978). 5. E. U. Condon, Phys. Rev. 41, 759 (1932). 6. R. E. Setchell and D. K. Ottesen, The Potential Use of Raman Spectroscopy in the Quantitative Analysis of Hydrogen Isotopes, Sandia Laboratories, Report SAND 74-8644, Livermore, Calif. 94550 (1975). 7. S. S. Bhatnagar, E. J. Allin, and H. L. Welsh, Can. J. Phys. 40, 9 (1962). 8. I. F. Silvera, W. N. Hardy, and J. P. McTague, Phys. Rev. 4B, 2724 (1971). 9. W. England, J. C. Raich, and R. D. Etters, J. Low Temp. Phys. 22, 213 (1976). 10. I. F. Silvera and R. J. Wijngaarden, Phys. Rev. Lett. 47, 39 (1981). 11. H. P. Gush, W. F. J. Hare, E. J. Allin, and H. L. Welsh, Can. J. Phys. 38, 176 (1960). 12. P. C. Souers, E. M. Fearon and R. T. Tsugawa, Lawrence Livermore National Laboratory, Livermore, Calif. 94550, unpublished data. 13. See note 12 above. 14. See note 11 above.

INFRARED AND RAMAN SPECTROSCOPY

335

15. A. Crane and H. P. Gush, Can. J. Phys. 44, 373 (1966). 16. P. C. Souers, D. Fearon, R. Garza, E. M. Kelly, P. E. Roberts, R. H. Sanborn, R. T. Tsugawa, J. L. Hunt, and J. D. Poll, J. Chem. Phys. 70, 1581 (1979). 17. P. C. Souers, J-. Fuentes, E. M. Fearon, P. E. Roberts, R. T. Tsugawa, J. L. Hunt, and J. D. Poll, J. Chem. Phys. 72, 1679 (1980). 18. P. C. Souers, E. M. Fearon, and R. T. Tsugawa, J. Phys. 15D, 1535 (1982). 19. See note 12 above. 20. Sample from E. I. DePont de Nemours & Company, Savannah River Plant, Aiken, S. C. 29801, Batch reference number 8H2PL-2, November 15, 1984. 21. G. A. Morris, in Proceedings of the 24th Conference on Remote Systems Technology, Washington, D.C., November 1976 (American Nuclear Society, Hinsdale, 111. 60521, 1976), p. 18. 22. J. T. Gill, J. Vac. Sci. Technol. 17, 645 (1980). 23. L. H. Jones and R. S. McDowell, J. Mol. Spectrosc. 3, 632 (1959). 24. C. Chapados and A. Cabana, Can. J. Chem. 50, 3521 (1972). 25. See reference in note 2 above, pp. 62, 274. 26. G. J. Jiang, W. B. Person, and K. G. Brown, J. Chem. Phys. 62, 1201 (1975). 27. D. A. Dows and V. Schettino, J. Chem. Phys. 58, 5009 (1973). 28. P. C. Souers, E. M. Fearon, R. L. Stark, R. T. Tsugawa, J. D. Poll, and J. L. Hunt, Can. J. Phys. 59, 1408(1981). 29. T. A. Ford and M. Falk, Can. J. Chem. 46, 3579 (1968). 30. P. Thirugnanasambandam and S. Mohan, J. Chem. Phys. 61, 470 (1974). 31. Yu. S. Makushkin and O. N. Ulenikov, Opt. Spectrosc. (USSR) 40, 534 (1976). 32. J. E. Bertie and F. E. Bates, J. Chem. Phys. 67, 1511 (1977).

26. Nuclear Magnetic Resonance

Nuclear magnetic resonance (n.m.r.) is a major tool for studying hydrogen. A sample is placed in a DC magnetic field and each kind of nucleus with a magnetic moment absorbs radio-frequency energy of a particular frequency. For example, at 1 T (10,000 gauss) we find resonance frequencies of H D T He 3

42.6 6.5 45.4 32.4

MHz MHz MHz MHz

(26.1)

These are the light isotopes of the fusion business. We can distinguish between each one by its frequency. We are lucky in that, of all n.m.r. nuclei, tritium is the most sensitive, hydrogen is the second most sensitive, and He 3 is fourth (F 1 9 is third). By contrast, important nuclei like He 4 and O 1 6 have no nuclear magnetic moments and cannot be seen by n.m.r. at all. There is a penalty for the specific knowledge as to the nucleus involved, and it is sensitivity. At 4 K, 10 19 H nuclei can be seen in a typical sample of about one cc with 4:1 signal-to-noise. 1 The signal decreases with increasing temperature so that 1021 nuclei are required at room temperature. In contrast, the much larger magnetic moment of the electron makes samples of only 102° electrons/m 3 visible by electron spin resonance (e.s.r.), which is the microwave analog of n.m.r. However, in most cases, it is not possible to distinguish between different e.s.r. signals because the electron resonance is usually at the same frequency. Now, n.m.r. in hydrogen is usually studied with a pulsed spectrometer, which delivers a microsecond burst of radio-frequency energy. Following a tt/2 pulse, the transmitter turns off and the sample rings with energy of the same frequency. This ringing often decays exponentially and is called a free induction decay. Its height at time zero is a measure of the number of nuclei present in the sample, as long as a calibration standard is used. The excitation put into the nuclei dies away with two relaxation times. One is the longitudinal (or spin-lattice) relaxation time T x , which measures the loss of nuclear magnetic energy. The other is the transverse (or spin-spin) relaxation time T 2 (not to be confused with the symbol for molecular tritium), which measures the loss of phase

337

NUCLEAR MAGNETIC RESONANCE

1

1

|

CM

Ia>~

E 5 c o

10- 2

CD

X

re a) a> (A a> > T 2 . From 4 K to about 10 K, the relaxation time T j is independent of frequency. From about 10 K to the melting point, T t is a function of frequency because the hopping frequency of diffusing molecules in the solid is close to the n.m.r. frequency. In the region marked "a," the top T t curve is for 29 to 30 MHz and the bottom curve is for 5.5 MHz. The dashed lines in the liquid region are our guess as to what might happen if the solid did not melt but just got hotter. The point " b " would be the solid T t minimum, much like the one we saw in the gas. Here, instead, we have a molecular diffusion hopping frequency v h , and the minimum occurs when vh ~ v 0 . For vh < v 0 , we see T 2 beginning to drop away from T t . The molecules are beginning to "freeze" into the lattice, and the nuclear moments, having lost their thermal motion, now strongly see nearby magnetic moments. This decreases T 2 . Below 12 K, the free induction decay is no longer exponential but looks more like a Bessel function. It may be roughly represented by the equation F(t) = exp[ —a 2 t 2 /2](sin bt/bt)

(26.6)

where a and b are constants, and (bt) in the sine function must be in radians. 3 3 Below 4 K, the free induction decay splits into complicated shapes. T h e T i value for solid nD 2 in the frequency-independent region is about 8 s 3 4 — much longer than the comparable 0.25 s for n H 2 . The reason lies in the relative strength of the nuclear magnetic moments. In the case of HD, the two nuclei are not relaxed by each other, as was the case for H 2 and D 2 . For pure HD, T t should be very long, but the presence of J = 1 H 2 impurities provides an excellent mechanism for relaxing the H D protons. Likewise, J = 1 D 2 impurities relax the deuterons in HD. Figure 26.6 shows another summarized view, this time of T t measured in solid hydrogen at 4.2 to 5 K. The exception is tritium at 9 K, for which these are the only available data. The data in Fig. 26.6 are shown as a function of the fraction of J = 1 hydrogen in the sample. Included are: protons in solid H 2 3 5 and H D , 3 6 - 3 8 D in solid D 2 3 9 and H D , 4 0 and T in solid tritium, T 2 . 4 1 The J = 1 concentration in each case is for H 2 in H 2 and HD, D 2 in D 2 , and T 2 in T 2 . The shape of the T x curve in Fig. 26.6 resembles that for the gas, where c(l) instead of density lies along the x-axis. The nuclear moments are relaxed by the fluctuating magnetic fields of nearby nuclear and rotational magnetic moments, with the latter being modulated by the fields of the electric molecular quadrupole moments. 4 2 , 4 3 The frequency spectrum of these fluctuations is a flat one extending from zero to some maximum cutoff frequency, vm. As c(l) increases, the number of modes in the fluctuation spectrum increases and vm becomes larger. If c(l) is very small, then vm < v 0 , and the few magnetic moments present are ineffective in relaxation. For vm > v 0 , which occurs for c(l) > 0.1, the fluctuating fields can relax the nuclei, and we find that I ^ is much shorter than for the very low-c(l) case. However, as vm increases, the fraction of the spectrum near v0 decreases, so that the mechanism becomes less effective. For high c(l), T, increases as c(l) 1 / 2 . 4 4 The maximum density

346

NUCLEAR MAGNETIC RESONANCE

E c o CD

X ra

ra c 3

"5> c o

lO"*

10"

2

Fraction J=1 Hydrogen, c(1 ) Fig. 26.6. The n.m.r. longitudinal relaxation time as a function of the J = 1 concentration in solid hydrogen at 4 to 5 K. The T 2 is measured at 9 K. The nucleus being observed with n.m.r. is shown. In HD, the controlling impurity is J = 1 H 2 . Curve " a " is taken at 30 MHz and " b " at 5.5 MHz.

NUCLEAR MAGNETIC RESONANCE

347

of states near v0 occurs when vm ~ v 0 . This is the region of the minimum near c(l) = 0.01. From the minimum to c(l) ~ 0.2, ~ c(l) 5 / 3 . 4 5 In the high-c(l) region, T t appears to be n.m.r. frequency-independent, 46 as long as we stay below the temperature where the molecular hopping frequency is comparable to v0 (T < 10 K). Everything about the T 1 minimum, however, will be frequency-dependent. For c(l) values below the T\ minimum, much work remains to be done. T j is some function of the magnetic field, perhaps as the one-third to first power. 47 The reason for the change of slope in the H D line in Fig. 26.6 is unknown.

Nuclear Spin Polarization The H D example in the preceding section leads us to a topic of current interest in hydrogen fusion. It has been suggested that the D-T fusion cross section would increase if the D and T magnetic moments are parallel to each other, i.e., spinpolarized. 4 8 ' 4 9 A subsequent calculation indicates that solid D-T may not depolarize, even in the hostile environment of a nuclear burn. 5 0 No one has prepared spinpolarized D-T, and we must turn back to the H D case for guidance. We find that H D has been polarized in the solid state by two methods. Both methods require pare HD, which is made by the chemical reaction LiAlH 4 + 4 D 2 0 - LiOD + Al(OD) 3 + 4HD

(26.87)

If run at 273 K, a 99% yield of H D is obtained, with the remainder being H 2 and D 2 . 5 1 It is further purified by cryogenic distillation in a Stedman column. 5 2 ' 5 3 Fairly pure H D is now commercially available. The first spin-polarization method is the "brute-force approach," which uses a high magnetic field and a low temperature to polarize the nuclei. Purified H D with c(l) ~ 10" 4 and T t ~ 3 s is placed in a 15 T field at 10 mK for one to two days. The sample is now polarized, but the J = 1 H 2 impurities make T t too short. The sample is then "parked" at 1 to 4 K and 0.3 T, where the J = 1 - » 0 transition takes place over several weeks. A 90% proton polarization is obtained and T t > 105 s. The protons can also transmit their polarization to the deuterons via dipolar interactions of nuclei on different molecules. 54 The second method is the "solid-state effect." In one version, the n.m.r. radio frequency is used to partially polarize the protons, which then transmit the polarization to the deuterons via dipolar coupling. Once polarized, the deuterons are very stable. For a proton T, of 12 minutes, the corresponding deuteron T j is 10 hours. 5 5 Another version of the "sold-state effect" involves doping H D with paramagnetic oxygen molecules and then irradiating to form hydrogen atoms in the solid HD. Microwave radiation is used to pump near the atom's e.s.r. resonance, which then aligns the nuclei, by way of the weak dipolar interaction. The oxygen is present to lower the electron's Ti so the same atom can be used to align many nuclei. Once again, no J = 1 H 2 can be present since the nuclear magnetic T t must be long. This work was done at 4 K and at modest magnetic fields of about 1 T. The

348

NUCLEAR MAGNETIC RESONANCE

nuclear polarizations obtained were 3.75% for the protons and 0.3 to 0.4% for the deuterons, with no improvement noticed as the temperature was lowered from 4.2 to 1.3 K. 5 6 DT, of course, brings its special problems in trying to spin-polarize the nuclei. One is the self-heating caused by the tritium, and the other is radiation damage. This latter would automatically produce atoms and possibly even remove the necessity for oxygen in the microwave solid-state approach. The worst problem is that the damage products shorten the nuclear A sample of H D was held for 10 days until the proton T x was 5 000 s. The Brookhaven 28-GeV proton beam then reduced T x to 250 s after 1 hour. After 10 days of annealing at 4 K, T x climbed back to 2000 s. 57

Helium-3 The daughter of tritium is He 3 , the possessor of the fourth strongest n.m.r. signal. Because He 3 is an isolated atom, we expect it to interact only by collisions with other species. In helium of infinite extent, we expect T t = T 2 ~ 8 x 10 4 T 3 / 2 /p

(26.8)

Here, only the collision frequency with other helium atoms matters. Should another gas, such as hydrogen, be mixed in, the same equation is expected, with p now the total gas density. 58 Figure 26.7 shows various measured T x values. 5 9 - 6 2 The equation predicts 8 x 109 s for the room-temperature, 0.05 mol/m 3 point. The experimental result is 105 times lower. All the measured points are low. This could be caused by paramagnetic oxygen impurities at temperatures above 77 K or by collision with the vessel walls. All the data shown in Fig. 26.7 are for glass bulbs of varying sizes, and all of the data were taken at low magnetic fields (i.e., 0.001 to 0.1 T), as compared to customary 1 T n.m.r. work. The samples show both increases and decreases with field according to the wall material. In the relation (l/Ti)

s

o b

,

= (1/Ti) +

(1/TOwa..

(26.9)

(Ti)obs' the observed longitudinal relaxation time, T, the true, bulk one from Eq. 26.8, and (T 1 ) wall the wall time. This latter time apparently dominates in all experiments so far. Even the good-looking 0.5 and 1.1 mol/m 3 curves in Fig. 26.7 are considered by their authors to be dependent not on gas density but on wall properties. For walls that are "dirty" with paramagnetic impurities, relaxation is thought to depend only on how long it takes for a molecule to cross the bulb. In this case, (Ti)Waii is expected to be proportional to the radius of the bulb squared and to the density of the gas in it. For the case of the "clean" paramagnetic wall, an atom can bounce off it many times before energy loss occurs. Then, (T, ) wall is expected to be proportional to the bulb radius and to the square root of the temperature, but not to the gas density. These theories have been tried with indifferent success for dense gas

349

NUCLEAR MAGNETIC RESONANCE

O) c

o cc

• • - / / 0

7s—

6 000

0

2

_ /

1 200

100

200

300

Temperature (K) Fig. 26.7. The n.m.r. longitudinal relaxation time T, for He 3 in glass bulbs. The gas density in mol/m 3 is shown.

in Pyrex bulbs at 4.2 K and 0.11 T. T t increased with density from 1 200 to 3 200 mol/m 3 , where it peaked. At higher densities, up to the highest measured value of 23 000 mol/m 3 , T t decreased with increasing density. The entire T j curve also rose to larger values after each "cleanup" of the glass surface. Two other cryogenic runs to about 40000 mol/m 3 showed T x to increase with gas density to the 0.6 to the first power. 6 3 , 6 4 More information is hidden in the case of metal tritides, which undergo radiation damage with a constant buildup of internally trapped H e 3 . 6 5 , 6 6 With increasing radiation dose, both relaxation times T t (order of seconds) and T 2 (ms) increase. This is taken as evidence that the He 3 starts as an interstitial atom and slowly forms ever larger bubbles. Not enough is known of the bubble properties to correlate the relaxation times. The unfulfilled promise of the n.m.r. properties of He 3 is left for future research.

Notes 1. J. R. Gaines, Ohio State University, Columbus, Ohio 43210, private communication, 1983. 2. P. C. Souers, T. Imai, T. S. Blake, R. M. Penpraze, and H. R. Leider, NMR and Electron Microscopy Studies on Irradiated Lithium Hydride, Lawrence Livermore National Laboratory Report UCRL-71835, Livermore, Calif. 94550 (1969). 3. Y. S. Oci, H. Rictering, and H. D. Wiemhofer, Ber. Bunsenges Phys. Chem. 83, 463 (1977).

350

NUCLEAR MAGNETIC RESONANCE

4. R. G. Barnes, P. G. Bray, and N. F. Ramsey, Phys. Rev. 94, 893 (1954). 5. N. F. Ramsey, Jr., Phys. Rev. 58, 226 (1940). 6. N. J. Harrick and N. F. Ramsey, Phys. Rev. 88, 228 (1952). 7. K. Lee and W. A. Anderson, "Nuclear Spins, Moments and Magnetic Resonance Frequencies," in CRC Handbook of Chemistry and Physics, 62d ed., R. C. Weast, ed. (CRC Press, Boca Raton, Fla., 1982), pp. E67-E71. 8. A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961), p. 2. 9. C. G. Havens, Phys. Rev. 43, 992 (1933). 10. J. Rychlewski and W. T. Raynes, Mol. Phys. 41, 843 (1980). 11. W. N. Hardy, Can. J. Phys. 44, 265 (1966). 12. R. G. Dorothy, Ph.D. thesis, University of British Columbia, Vancouver, B.C., Canada (1967); reported in K. E. Kisman and R. L. Armstrong, Can. J. Phys. 52, 1555 (1974). 13. M. Lipsicas and A. Hartland, Can. J. Phys. 40, 382 (1962). 14. See note 12 above. 15. E. Tward and R. L. Armstrong, Can. J. Phys. 46, 331 (1968). 16. M. Bloom, Physica 23, 237 (1957). 17. M. Lipsicas and M. Bloom, Can. J. Phys. 39, 881 (1961). 18. K. E. Kisman and R. L. Armstrong, Can. J. Phys. 52, 1555 (1974). 19. W. P. A. Hass, N. J. Poulis, and J. J. W. BorlefTs, Physica 27, 1037 (1961). 20. M. Lipsicas and A. Hartland, J. Chem. Phys. 44, 2839 (1966). 21. W. P. A. Hass, G. Seidel, and N. J. Poulis, Physica 26, 834 (1960). 22. See note 16 above. 23. M. Bloom, Physica 23, 378 (1957). 24. G. Seidel, W. P. A. Hass, and N. Poulis, Proc. Int. Conf Low Temp. Physics, LT-7, 73 (1961). 25. C. E. Miller, W. J. Alspach, and T. M. Flynn, Cryogenic Eng. News 1, 66 (1966). 26. C. E. Miller and M. Lipsicas, Phys. Rev. 176, 273 (1968). 27. C. J. Fisher and J. W. Riehl, Physica 66, 1 (1973). 28. See note 19 above. 29. L. I. Amstutz, H. Meyer, S. M. Myers, and D. C. Rorer, Phys. Rev. 101, 589 (1969). 30. F. Weinhaus and H. Meyer, Phys. Rev. IB, 291A (1973). 31. M. Bloom, Physica 23, 767 (1957). 32. R. S. Rubins, A. Feldman, and A. Honig, Phys. Rev. 169, 299 (1968). 33. See reference in note 8 above, p. 120. 34. F. Weinhaus, S. M. Myers, B. Maravaglia, and H. Meyer, Phys. Rev. 3B, 626 (1971). 35. See note 30 above. 36. W. N. Hardy and J. R. Gaines, Phys. Rev. Lett. 17, 1278 (1966). 37. R. B. Bernardi, Proton Spin-Lattice Relaxation in Solid Hydrogen Deuteride over the Temperature Range 0.44 to 4.2°K, Ph.D. thesis, Syracuse University, Syracuse, N.Y. (1972); University Microfilms, Ann Arbor, Mich. 48106, No. 73-7704. The data have been adjusted to 0.15 T by A. Honig and H. Mano of Syracuse University. 38. A. Honig and H. Mano, Syracuse University, Syracuse, N.Y. 13210, private communication, 1983. A. Honig kindly provided special data for Fig. 26.6. 39. See note 34 above. 40. See note 38 above. 41. J. R. Gaines and P. C. Souers, unpublished data taken at Lawrence Livermore National Laboratory, Livermore, Calif. 94550. 42. See note 30 above. 43. C. P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963), pp. 138-142. 44. T. Moriya and K. Motizuki, Progr. Theoret. Phys. 18, 183 (1957). 45. C. C. Sung, Phys. Rev. 167, 271 (1968). 46. See note 30 above. 47. See notes 32 and 36 above.

NUCLEAR MAGNETIC RESONANCE

351

48. R. M. Kulsrud, H. P. Furth, E. J. Valeo, and M. Goldhaber, Fusion Reactor Plasmas with Polarized Nuclei, Princeton Plasma Physics Laboratory Report PPPL-1912, Princeton, N.J. 08544 (1982). 49. Physics Today (August, 1982), pp. 17-19. 50. R. M. More, Phys. Rev. Lett. 51, 396 (1983). 51. A. Fookson, P. Pomerantz, and E. H. Rich, Science 112, 748 (1950). 52. A. Fookson, P. Pomerantz, and E. H. Rich, J. Res. Nat. Bureau Standards 47, 31 (1951). 53. A. Fookson, P. Pomerantz, and S. Rothburg, J. Res. Nat. Bureau Standards 47, 449 (1951). 54. A. Honig, Syracuse University, Syracuse, N.Y. 13210, speech given at Lawrence Livermore National Laboratory, Livermore, Calif., June 1983. 55. A. Honig and H. Mano, Phys. Rev. 14B, 1858 (1976). 56. J. C. Solem, Nucl. Instr. Methods, 117, 477 (1974). 57. See note 54 above. 58. R. Chapman and M. G. Richards, Phys. Rev. Lett. 33, 18 (1974). 59. See note 58 above. 60. W. A. Fitzsimmons, L. L. Tankersley, and G. K. Walters, Phys. Rev. 179,156 (1969). 61. R. S. Timsit, J. M. Daniels, and A. D. May, Can. J. Phys. 49, 560 (1971). 62. J. G. Ganiere, Helv. Phys. Acta 46, 147 (1973). 63. See note 61 above. 64. B. N. Esel'son, V. A. Mikheev, V. A. Maidanov, and N. P. Mikhin, Soviet J. Low Temp. Phys. 7, 466(1981). 65. H. T. Weaver and W. J. Camp, Phys. Rev. 12B, 3054 (1975). 66. R. C. Bowman, Jr., Nature 271, 531 (1978).

27. Electrical Effects Caused by Radiation

Pure hydrogen is about as "hard" an insulator as one can find. For liquid H 2 at 19.3 to 19.5 K as measured in a 9.4 mm gap, the D C electrical conductivity g (in S/m or [i2-m] - 1 ) is 1 g ~ 2.8 x 10~ 14 /E

(27.1)

where E is the electric field in V/m. This is such a small value that almost 5 liters of liquid were used in the experiment. At 3 000 V/m, about half the maximum used, g is only about 10 - 1 5 S/m. The basic equation for direct current is g = Ne(u + + u _ )

(27.2)

where N is the density of charges per m 3 , e the electronic charge (1.602 x 10~ 19 C/charge), u+ the mobility of the positive charge carrier in m 2 /V • s), and u_ the mobility of the negative charge carrier. If electrons are present, then u e = u_ » u+, and we may ignore the positive ion term. If we use an electron mobility of about 10" 3 m 2 / V - s in the liquid, then N at 3000 V/m is only about 60000 electrons/m 3 . The same number of ions are also present but presumably do not affect the conductivity, so that we may say that we have 60 000 ion pairs/m 3 as well. The charge density above is incredibly low. It means that the average cubic centimeter does not have a charge in it. However, it is difficult to see how thermal ionization at 20 K could produce any charges, because 2.5 aJ (16 eV) energy is needed. It was concluded that cosmic radiation probably produces the conductivity seen in liquid H 2 . 2 The form of Eq. 27.1 is also peculiar. Usually g increases with E, not the other way around. The interpretation is that the few charges formed are quickly pulled to the electrical plates by the electric field, so that the sample is literally sucked dry. It might be expected that a little radiation goes a long way in hydrogen. For liquid D 2 at 19.4 K, the conductivity is g ~ 1.6 x 10" l o /E

(27.3)

This is four orders of magnitude larger than that of H 2 ! It was assumed that a small tritium impurity—only 1 part in 10 9 —caused this excess conductivity. The D 2 sample was apparently not truly pure, although it was thought to be so at the start of the experiment.

353

ELECTRICAL EFFECTS CAUSED BY RADIATION

We shall consider only "low" electric fields in this chapter. This means that only ions and electrons formed by radiation damage processes enter into the current. We do not have avalanching of electrons.

DC Electrical Behavior Let us consider a sample of hydrogen gas between two flat plates a distance L apart. Some source of radiation—either electromagnetic or energetic particles—causes uniform ionization in the gas, where the number of ion pairs produced per m 3 • s is dN/dt. The ion pairs recombine with a rate constant k, in m 3 /ion pair • s—that is, according to the reaction dN — =-kN2 dt

(27.4)

We now turn on a "weak" DC electric field. The electrical conductivity is3 ( \ dN\1/2 g = (u++u_)e(^—J

(27.5)

where we list the mobilities of the positive and negative species separately. "Weak" means that the electric field is not strong enough to perturb the recombination reaction. We next consider a "strong" electric field. The field is still "low," in that no avalanching of electrons occurs, so that "strong" means that the field pulls every electric charge to the plates before recombination occurs. In this case, we have: g6 =

eL dN E dt

v(27.6) ;

The mobility does not appear here because there is nothing to halt the charges in their journey across the sample. The distance L is included as a measurement of the volume of sample. For the case in between Eqs. 27.5 and 27.6, we have 8

=f { ( l

+

^ P )

,

'

i

- > }

(27.7)

where y = (u++u_)2E2/2kL

(27.8)

With the combination of recombination plus electrical "sweep-out," the steadystate density of ion pairs (per m 3 ) becomes

354

ELECTRICAL EFFECTS CAUSED BY RADIATION

All the above are for a DC electric field. They also assume that the charge density N is so low that it does not distort the applied electric field E—that is, that dV/dx is constant across the sample, where V is the applied potential. If this is not true, then the sample is said to be "space charged." A rough condition for a space charge is (27.10)

eL

where the constant e0 = 8.854 x 10~12 C/V • m, the permittivity of free space. 4 If space charging is present, we must write V/L as the average field E across the gap. We are very likely to have a space-charged system when L is big (i.e., for a thick sample). In this case, the charges distort the applied electric field to a large degree. What results is a "skin" layer of positive charge near the cathode and a layer of negative charge near the anode. This produces large electric fields near the plates. Between these zones, in the center of the sample, the external electric field has been neutralized almost to zero. Here, ion pairs are born and destroyed by recombination, scarcely knowing that the electrical plates are nearby. If we have positive and negative ions of equal mobility formed by the radiation, the "skin layer" near each plate has a thickness X of about 5 , 6

Note that the mobility does not enter into the equation for the skin layer. For the electron-positive ion case, where u_ = ue » u + = uj5 1(Cath

°de)-e(d|dt)

(27 12)

'

and l(anode) ~ — l(cathode) ue

(27.13)

The positive ions fill up a much larger zone near the cathode than vice versa. This is because the more mobile electrons are sucked out of the skin layer toward the anode. There is no need for charge to be conserved within the sample. There is a need for charge conservation in the entire sample cell, however, and charge flows in or out of the plates to achieve the proper balance. The space-charge-limited conductivity is obtained by solving quadratic equations. For the electron and the positive ion, where u e » Uj, we have 7,8 1 / 2 2

e (dN/dt)

1/2

[\4e0nJ

_k^l 2 ue J g

+

E2L ey

t

|

2L(dN/dt) 1/2

-) -1 - l \ g= v (27.14)

where y is defined in Eq. 27.8. Equation 27.14 is an approximate solution. For the

355

ELECTRICAL EFFECTS CAUSED BY RADIATION

case of positive and negative ions with identical mobilities u i ; we obtain a similar equation 9 1/2

e 2 (dN/dt) 3 / 2 [\8fioU,-J

k 1 / 2 l>g 2, + E 2 L 2u, j 6 ey

t +

2L(dN/dt) j " ' _

[

|~'e_

v

(27.15) This is an exact equation. The rate constants, k, will be particular to either the electron-ion or ion-ion case, although they are about the same (they are listed in Tables 19.3 and 19.6). V is the total potential drop across the sample, so that E = V/L. Equations 27.14 and 27.15 reduce to Eq. 27.7 when the quadratic space-charge term is small. The equations above are actually derived for the current density, j, in A/m 2 . We here convert to electrical conductivity, by the use of gE = j, so that g is an "averaged" value. A similar "skin layer" exists in a hot plasma, where all electrical conductivity comes from a thin layer near the plates. The center of the plasma is opaque to electromagnetic radiation. This thickness of this layer is 10

where k here is the Boltzmann constant (equal to 1.38 x 10 2 3 J/K—different from k, the recombination rate constant). However, a necessary condition is that kT > eV

(27.17)

in other words, that the charged particles have a thermal kinetic energy much greater than the imposed electrical energy. This is true for a few of the particles in tritiumirradiated hydrogen, especially the just-created beta particle itself. However, many of the charged particles will be thermalized, and tritium does not have enough of them to really be called a plasma. Let us return to our sample of irradiated gas with no space charge and a D C electric field. The DC capacitance C will be C = ^

(27.18)

where K is the dielectric constant and A the area of an electrical plate. The dielectric constant is 1 if no sample is there. When unirradiated hydrogen is present, there is an additional dielectric constant AK 0 caused by the polarizability of the molecules (i.e., the distortion of the electrons in the field). This dielectric constant is a function of density only, and we may write it with this equation, obtained for the gas and liquid: 11 - 12 1 + A K 00 =

1 + 4 048 * 1 0 ? 1 - 2.024 x 1(T 6 "

(27.19) v '

356

ELECTRICAL EFFECTS CAUSED BY RADIATION

where p is the density in mol/m 3 . For liquid or solid hydrogen, where p ~ 5 x 104 mol/m 3 , K ~ 1.34—not very big. The presence of anything caused by radiation damage is not included. In tritiated hydrogen with no applied potential, there will be a constant density of positive and negative charges. There is no reason to believe that they form any kind of bound system other than recombining back to being molecules. As all electric charges are expected to be free and mobile, we do not expect a capacitance to be associated with them, providing the sample is not space charged. If the sample is space charged, then excess electric charge permanently resides near each plate. Suppose we have two ions of equal mobility so that the D C charge layer thickness X is the same. For the DC case, we assume a charge density difference of N charges/m 3 —the same as the steady-state ion-pair density—at each plate, and a difference of zero at the distance X into the sample. Beyond X , in the interior of the sample, we have equal numbers of positive and negative charges, so this region does not contribute to the capacitance. In the plate region, we assume the charge density difference to be linear. This is not too far off, although the actual distribution is a complicated exponential function. 1 3 In our simple model, the total charge difference between the plates is eNAA/2. The excess dielectric constant due to space charging, AK, is then (27.20) We obtain N from Eq. 27.9 using the average value of E. In the ion-ion case, the charge layer in the gas near each electrode will be of the same size, although of opposite charge. Added together, there is charge neutrality for the sample. For the electron-ion case, there will be more positive charges than negative. However, the sample cell as a whole will be neutral, so that more negative charge must appear on the anode. We therefore substitute /(cathode) from Eq. 27.12 in Eq. 27.20. This doubles the size of X and AK in the case of electrons as compared to negative ions, providing that the recombination rate constants are the same.

AC Electrical Behavior Let us now turn to the AC behavior. A simple model of a hydrogen sample is a circuit with two parallel arms. One arm has a resistance R, and the other arm has a capacitance C and a second resistor R c in series. The complex conductivity g* for this simple geometry is: (27.21) where co is the angular frequency in radians/s (equal to 2nv, where v is the frequency in Hz), and i is the square root of minus one. The real part is what we have been calling the electrical conductivity so far in this chapter. It is in phase with the applied

ELECTRICAL EFFECTS CAUSED BY RADIATION

357

potential. It equals L/AR for the DC case (co = 0) and L(R + R C )/ARR C for co oo. We see that the conductivity has increased with frequency, because current now passes through C and R p . The imaginary part is 90° out of phase with the applied potential, but it represents real current. The capacitance is obtained from g*A/icoL. It equals C for the D C case and declines to zero at high frequency. To understand AC behavior, we must compare the time that it takes for charges of a particular type to cross the sample gas. This time is L / u ± E. We compare it to the time for half a cycle of the AC signal, t 1 / 2 . If L / u ± E « t1/2

(27.22)

for both electrons and ions, then all charges can follow the changes in AC field. We expect " D C behavior" for both the electrical conductivity and the dielectric constant. Of course, the charge distribution rises and falls with the AC signal, but the timeaveraged electrical values should be like those of the D C case, where the potential is the root-mean-square AC value. As we increase the frequency and t 1 / 2 decreases, the two sides of Eq. 27.22 become comparable. Now the charge distribution cannot accurately follow the AC field. The first species to feel it will be the low-mobility ions. Their charge density near the plates will drop to zero and will peak instead somewhere near l¡2—that is, about halfway into the DC skin layer. 1 4 ' 1 5 When L/u;E ~ 10t 1/2

(27.23)

the ions no longer respond at all to the AC field. The space charge can no longer set up. The electrons, however, can still be space charged, because their mobility is much higher than that of the ions. The electrons dominate the conductivity anyway, so that not much overall difference may appear in the sample. When Eq. 27.23 becomes true even for electrons, then all space-charge effects are at an end. The high-frequency limits revert to the simple Eqs. 27.5 through 27.8, with no space charge. The measured dielectric constant drops and the electrical conductivity rises. This is the normal AC phenomenon of a capacitor, which blocks lowfrequency signals but passes high-frequency ones as current.

Electron Drift Velocity and Mobility The electron is the most mobile of all carriers and will often carry all of the current, even with an equal or larger number of ions present. The electron mobilities are not fixed to the temperature because imposing an electric field puts energy into the electrons, and they can be "hotter" than the material around them. We consider the usual case, where the electric field produces a drift velocity we that is small compared to the thermal velocity v e , as calculated from kinetic theory. The electrons, if present, bounce randomly about at high speeds. The external electric field imposes a slow net drift in the field direction. We assume that the electrons collide with molecules, as expected from random kinetic behavior. The time between collisions for an electron is

358

ELECTRICAL EFFECTS CAUSED BY RADIATION

i. In between collisions, the electric field accelerates the electron from zero to eE/m e m/s 2 , where m e is the electron mass (9.11 x 1 0 - 3 1 kg). We integrate to obtain the average velocity we of eEr we = —

(27.24)

The time between collisions, T, equals the mean free path Ae divided by the thermal velocity v e . From Chapter 3, we use the gas kinetic equation Ae =

1

3.09 x 10" 5

2 ' 7td N„p

p

(27.25)

where d is a collision diameter, N„ is Avogadro's Number, and p is the gas density. We use 0.11 nm for the distance between the electron and molecule at the instant they touch, where the electron has a zero size in comparison with the molecule. The gas density p is used because the electron is colliding with molecules, which are more numerous than electrons. The thermal velocity is, from kinetic theory, 8RT V ' 2

~ 6200T 1 ' 2

(27.26)

We combine all the parts to obtain 440 E

(27-27)

We note that kinetic theory predicts that wc will increase as the temperature decreases. Also, we is a function of E/p, the electric field divided by the density (in V • m 2 /mol). The mobility is, for electrons, we 440 i f ~ pT 172

Ue =

(27

"28)

When should we expect these kinetic equations to hold? We must have a low density so that the electrons collide with single molecules rather than pairs. We also require the electric field to be low enough that it puts no appreciable energy into the electrons—that is 16 eE!N 0 « RT

(27.29)

E - « 2.8T

(27.30)

Using Eq. 27.25, we obtain P

at 300 K, E/p « 850 V-m 2 /mol. In the true kinetic region, we expect

ELECTRICAL EFFECTS CAUSED BY RADIATION

359

E/p ( V - m 2 / m o l ) Fig. 27.1. The variation of the "constant" A with E/p for nH 2 gas. Only for E/p < 5 V • m 2 /mol is A constant and does the kinetic theory hold.

to be a constant. Figure 27.1 shows the results for low-density n H 2 gas. W e see that it obeys the kinetic theory only for E/p < 5 V - m 2 / m o l — o r roughly a factor of 100 below the value in Eq. 27.30. W e calculate values for A of 23 a n d 46 mol/V • m • s at 300 a n d 77 K . 1 7 - 2 0 T h e low E/p values in Fig. 27.1 are 26 a n d 60 mol/V • m -s, which are close. T h e d a t a d o n o t quite show the expected T 1 / 2 dependence. Figure 27.2 shows the displayed values of w e f o r n H 2 gas as a function of E/p a n d density. 2 1 W e note the t e m p e r a t u r e dependence to the right. A s E/p increases, the t e m p e r a t u r e dependence is washed away. This occurs when w e ~ v e — t h a t is, when the two sides of Eq. 27.29 are c o m p a r a b l e . W e note that this behavior indeed occurs for E/p > 850 V - m 2 / m o l . T a b l e 27.1 lists zero-pressure w e d a t a for three hydrogen gases. W e m a y use the 293 a n d 76.8 K n H 2 d a t a to extrapolate to other temperatures. Between 76.8 K a n d a t e m p e r a t u r e T , we try this relation at constant E/p:

360

ELECTRICAL EFFECTS CAUSED BY RADIATION

10

0 - 100 mol/m 3

10^

/ 1 1 000 102

1

10 4

E/p (V-m 2 /mol) Fig. 27.2. The electron drift velocity of nH 2 as a function of E/p and density.

w e (T) =

/76 8 \ c — ) w e (76.8 K)

(27.32)

For low E/p, we expect c to be 0.50, as predicted by kinetic theory, although it is not quite that in the actual data. The value of c becomes smaller as E/p increases. Table 27.1 lists the c values derived from the two hydrogen columns. The asterisk values of 0.500 are defined as the low-temperature limit. We may now estimate w e at any other temperature. For example, for D 2 at 20 K and E/p = 100 V-m 2 /mol, we = (76.8/20) 0 1 1 5 (2 930) = 3420 m/s. Table 27.1 refers strictly to zero-density gas, but in practice we may use it in the range of 0 to 100 mol/m 3 with only a probable 1 to 2% error. We may even use it in the 1 000 mol/m 3 range with a maximum error of about 16%. 2 2 We further note in Table 27.1 that the nH 2 and nD 2 data vary by as much as 24%. The reason is the excitation by the conduction electrons of rotational J = 0 2 and J = 1 -» 3 transitions in the molecules undergoing collisions. The two hydrogens have different J = 0 and J = 1 concentrations and therefore different mobilities. The H 2 has a larger rotational energy splitting than the D 2 . Further analysis shows that increasing E/p excites more 3 = 2 transitions in H 2 , whereas the D 2 rotational levels have been saturated. Even at the highest E/p value of Table 27.1, the H 2 levels are apparently not saturated. 2 3 This means that we should use D 2 , not H 2 , to estimate D T and T 2 and their mixtures. We have not yet mentioned the word "mobility." The reason is that we traces out a unique curve with E/p at zero pressure. Table 27.1 is general and may be applied to

361

ELECTRICAL EFFECTS CAUSED BY RADIATION

T A B L E 27.1 ZERO-DENSITY DRIFT VELOCITIES IN HYDROGEN GAS

Electron drift velocity, w c (m/s) E/p (V • m 2 /mol) 0.50 0.75 1.0 2.5 5.0 7.5 10 25 50 75 100 250 500 750 1000 2500 5000 7 500 10000 12 500 15000 17500 20000

nH 2 293 K 13 20 26 65 130 195 260 630 1 120 1640 2100 3 950 5 780 6840 7 760 11900 16900 21200 25200 28 700 32100 35 900 39500

nH 2 76.8 K 30 45 60 145 275 390 490 945 1520 2010 2450 4360 6190 7280 8170 12100 17100 21300 25200 28800 32100

nD 2 76.8 K — — — —

290 410 520 1 120 1880 2460 2930 4480 5 560 6380 7150 10 700 15 200 — — — —

—

—

—

—

Power c 0.500 0.500 0.500 0.500 0.500 0.500 0.475 0.303 0.228 0.152 0.115 0.075 0.051 0.047 0.038 0.013 0.0089 0.003 5 0 0 0 0 0

NOTE: The c values of 0.500 are defined as the low E/p limit. The nD 2 values should be used for DT and T 2 under the same conditions.

any experiment. In a particular experiment, we convert to the electron mobility ue (in m2/V • s) with the equation

We see that u c is not a constant but depends on the electric field used in a given experiment. Also, in the zero-pressure region, ue is inversely proportional to p—a consequence of the kinetic theory. As an example of the magnitudes of the variables, consider nH 2 gas at 77 K and E/p = 50 V-m 2 /mol. 2 4 At densities of 5.5, 1 730, and 11 000 mol/m3, we is 1 520, 1 400, and 14 m/s; the mobility ue is 5.5, 0.016, and 2.5 x 1 0 - 5 m 2 /V- s. We then see that ue generally has a wider variability than w e . The decrease of we with density spoils the unique relation between we and E/p. At this point, most people abandon we and go to u e , often forgetting to list the appropriate E/p value. In the above example, ue decreases with increasing density by

362

ELECTRICAL EFFECTS CAUSED BY RADIATION

about 15% from 5 to 2000 mol/m 3 . But at 11 000 mol/m 3 , u e is ten times lower than expected from the kinetic theory. These effects are attributed to the electron scattering from two or more molecules at o n c e . 2 5 - 2 7 There is little cryogenic data. For dense H 2 gas at 26 to 32 K, the electron mobility is roughly 0.036 m 2 /V • s at 1 700 mol/m 3 and 0.014 m 2 /V • s at 2 500 mol/m 3 . Between 1 700 and 3 400 mol/m 3 at 32 K, the electron combines with nearby molecules to form an ionic species. 28 A final point of interest is the barrier energy E B . This is the energy needed to pull an electron from a metal wall or electrode and to place it in the conduction band of the fluid—that is, to have it running loose. For 46 K H 2 and D 2 fluid, this is E b ~ 3.65p

(27.34)

where E B is in J/mol. A value greater than 190 kJ/mol (2 eV) has been obtained for liquid H 2 at 15 K, where the density is 38 000 mol/m 3 . 2 9 The energy E B is less than the metal wall's work function, which is the energy needed to remove the electron completely from the metal and into the vacuum. The energy barrier of height E B occurs at a distance xB from the metal into the fluid sample. If an electron has less energy than E B , it will bounce back from the barrier into the metal wall. Once over the barrier, it will be accelerated by the externally applied electric field E. The barrier is formed by the electric field's polarization of the sample, which has a dielectric constant K. The charges within the sample set up a secondary field that weakly repels entering electrons. The barrier distance

(27.35) where L is the distance between the plates or walls. Let us use L = 2.5 x 10~ 4 m and E = 2 800 V/m from an upcoming D-T experiment. For K = 1.3, xB is 0.22 /im into the fluid. Note that xB does not depend on the density, except to the extent that K does.

Ion Mobilities We expect generally the same behavior for an ion as with the electron. The ion, however, is perhaps 5 000 times heavier and has four times the collision cross section. This latter comes about because we expect the collision of two particles, each about 0.22 mm in diameter, instead of one plus a "zero-size" electron. Thus, the ion mobility Uj should be 4 x (5 000) 1/2 ~ 300 times less than u e at the same E/p value. Equation 27.30, adapted for ions, becomes, as the criterion for thermalization, - « P

11.2T

(27.36)

At 300 K, this is 3 400 V • m 2 /mol. We use kinetic equations, except that we require

363

ELECTRICAL EFFECTS C A U S E D BY RADIATION

T A B L E 27.2 ZERO-FIELD MOBILITY D A T A FOR IONS IN HYDROGEN G A S

Ion

Reduced mass//* (kg/mol)

H+ D+ H3+

6.719(—4) 1.343( —3) 1.209( —3)

D3+ H5+

2.417( —3) 1.440( —3)

H" D~

6.719(—4) 1.343 ( - 3 )

Constant

(K)

A (X 10~2)

300 300 300 293 273 300 293 273 194 300 300

3.20 3.31 3.03 2.97 2.86 3.07 3.07 2.95 2.45 8.61 8.52

Temp.

NOTE: The numbers in parentheses are exponents.

the reduced molecular weight fi P =

i l — + 77 ^ion

^^molecule

(27.37)

which includes the motion of both species. 31 We recall the same reduced mass in describing the kinetic theory of gas diffusion (see Chap. 3). We obtain the kinetic relation A

'

Ï7TW p(fiT)

2

(27.38)

where A is a constant listed for the best zero electric field, low gas density data in Table 27.2. 3 2 ' 3 3 We note that A is approximately constant for the positive ions but that the hydride ions have values almost three times as large. The gas-phase ion mobilities do not change much when E¡p does become large. For H 3 + ions in low-density H 2 gas at 273 K, the product u¡p is 0.050 mol/V • m • s from E/p of 0 to 20 000 V • m 2 /mol. It increases to 0.064 at 60000, reaches a peak of 0.071 at 85 000, and declines to 0.068 mol/V • m • s at 120 000 V • m 2 /mol. The mobility of 304 K H + ions decreases only 22% from 20000 to 180000 V - m 2 / m o l , while that of 313 K H 5 + ions increases 8% as the density increases from 12000 to 58000 V-m2/mol.34 We digress a moment to note that the mobility of a thermalized species—usually an ion—may be related to its diffusion coefficient D by means of the Nernst equation D = kTu/e

(27.39)

where k is again the Boltzmann constant and not the recombination rate constant. 3 5

364

ELECTRICAL EFFECTS CAUSED BY RADIATION

TABLE

27.3

IONIC C A R R I E R M O B I L I T Y IN L I Q U I D A N D S O L I D H

Phase Fluid

Temp. (K) 30-32

Density (mol/m 3 ) or pressure (kPa)* 1700 3400 5100 8000

2

Ion mobility, u i (m 2 /V• s) Negative 3(-5) 1.2(-5) 7(-6) 5(-6)

Positive

— — —

Liquid

15 16.5 17.8 15.1-16.9

12.7* 25.4* 42.7* 101*

3.7(-7) 5.4(-7) 5.8( —7) 7.6(-7)

1.8(-7) 2.7( —7) 2.5(-7) 4.9(-7)

Liquid

15 18 20 22 24 26

7.2* 46.0* 89.9* 158* 257* 393*

5.0( —7) 6.8( —7) 8.0( —7) 9.2( —7) 9.7 ( - 7 ) 9.7( —7)

4.2( —7) 6.1(-7) 7.4(-7) 8.7( —7) 10.0( —7) 11.3( —7)

Solid

12.1 12.5 13.0 13.5 13.7

3.7(—10) 4.6(—10) 6.0(—10) 7.2(—10) 8.0(—10)

— — — — —

— — — — —

NOTE: The numbers in parentheses are powers of ten. In column 3, the numbers without asterisks are densities; those with asterisks are pressures.

We next consider the liquid and solid data, which are summarized in Table 27.3. The first entry is for dense 30 to 32 K fluid H 2 . 3 6 The approximate mobility of the ion is Uj

~

0.042 P

(27.40)

where the density range is 1 700 to 8 000 mol/m 3 . Also present in this dense fluid are free electrons with mobility that varies from 0.1 m /V • s at 1 200 mol/m 3 to 0.01 m 2 / V • s at 2 800 mol/m 3 . Between 1 500 and 3 000 mol/m 3 , the electrons disappear linearly as the fluid density increases, and the low-mobility species grows in at the same rate. At 2400 mol/m 3 , the two species are in approximately equal abundance. Because the ions grow from the electrons, they are assumed to be electron bubbles, a species first postulated in liquid helium. 3 7 ' 3 8 Such a bubble is a trap in which the electron comes to rest in the liquid, much like the trapped electron in the solid in Chapter 22. The electron's zero-point energy of vibration in its cell in the liquid is so high that it pushes back the surrounding molecules. The resulting species resembles an ion in being charged and massive. It is unlike an ion in that the central charge does not carry

365

ELECTRICAL EFFECTS CAUSED BY RADIATION

the added molecules but pushes its way through the liquid. The molecules on the interior wall then presumably change constantly. There is no conclusive evidence, however, that such a species actually exists in liquid hydrogen. This density-induced transformation of the electron is also well documented in the case of cold, dense helium gas. 3 9 , 4 0 Here, the electron appears to continuously transform into the bubble species. It is unclear, then, whether the electron mass changes continuously or in a single jump as it falls into the potential well of the trap. Table 27.3 also contains liquid hydrogen values. 4 1 , 4 2 We note that the positive and negative ions are comparable in their mobility. Finally, Table 27.3 shows ion mobilities in solid H 2 , where the ion is thought to be negative. Nothing is known about it, except that it is not a free electron. Conduction electrons in the solid rare gases have these sorts of values: argon, u e ^ 0.1 m 2 /V • s at E = 104 V/m and u e ~ 0.01 m 2 / V - s at 106 V/m; krypton, 0.4 at 103 and 0.001 at 107; and xenon, 0.4 at 103 and 0.000 6 m 2 /V • s at 107 V/m. 4 3 The electric field for the data in Table 27.3 is not given, 44 but it is probably low. We see, then, that the H 2 solid data are at least seven

100

1 000

Gas density (mol/m 3 ) Fig. 27.3. The electrical conductivity (g) and dielectric constant (K) for D-T gas at 20 to 26 K for two tritium compositions. The distance between the electrical plates is 0.25 mm and a 1 V peak-to-peak, 1 592 Hz signal on a capacitance bridge is used.

366

ELECTRICAL EFFECTS CAUSED BY RADIATION

orders of magnitude too low to represent conduction electrons. The true nature of this ion-like carrier is yet to be determined.

Results on D-T Gas and Liquid Some electrical conductivity and dielectric constant data for the gas and liquid have been measured at 1 592 H z . 4 5 - 4 7 The 1 V peak-to-peak signal across a 0.25 mm gap produces an average electric field of 2 830 V/m. Figure 27.3 shows D-T gas results at 20 to 26 K for two compositions. The sample is space charged, so that the electrical conductivity is reduced by about an order of magnitude. The dielectric constant is caused by charge accumulating near the electrical plates. Figure 27.4 shows the calculated ion pair density from both the gas and liquid experiments for three compositions of tritium. The density is not high—about one

Closed 295 K Open 2 0 - 2 6 K 10 14 10^

10« Sample density (mol/m 3 )

Fig. 27.4. The steady-state ion-pair density in gas and liquid D-T samples. The conditions are the same as those of Fig. 27.3. (Courtesy P. C. Souers and the American Institute of Physics.)

ELECTRICAL EFFECTS CAUSED BY RADIATION

367

ion pair per 10 10 molecules (as contrasted with 1 ppm estimated from spectroscopy for the solid at 6 K). We note, too, that the ion pair density goes roughly as the halfpower of the tritium concentration at a given density. This is expected because each ionization produces two charged particles. If we solve for N in Eq. 27.4, where dN/dt is proportional to the tritium mol fraction, we obtain the half-power result. Analysis shows that the low-density gas samples are not space charged and correspond to the "strong" case of Eq. 27.6. At most of the densities, the samples are expected to be space charged. Above 100 to 300 mol/m 2 , the 1 592 Hz signal cannot affect the ions. The electrons, which dominate the electrical behavior, remain space charged, so that the data agree reasonably with the equations derived for electrical conductivity and excess dielectric constant. The presence of the space charge suggests that negative ions cannot be present in great quantities in the gas. The liquid results are more uncertain. Electrons have a high enough mobility that a space charge should be present. We find that it is, except that the measured electrical conductivity is one to two orders of magnitude below that calculated. Ions, on the other hand, are so slow that no space charge should be present at 1 592 Hz, and the conductivity should be higher. Perhaps the liquid contains both electrons and negative ions. 4 8

Notes 1. W. L. Willis, Cryogenics 6, 279 (1966). 2. See note 1 above. 3. A. Von Engel, Ionized Gases, 2d ed. (Oxford, Clarendon Press, 1965), pp. 6-10. 4. See note 3 above. 5. J. J. Thomson and G. P. Thomson, Conduction of Electricity through Gases, 3d ed. (Dover, New York, 1969); pp. 193-206. Our Eq. 27.14 is No. (22) on p. 205. The Thomsons' first term must be divided by 4ne0 to bring it into SI metric units. Their linear term has been modified to account for a finite gap width as expressed by Eqs. 27.7 and 27.8. The details are given in the reference in note 8 below. 6. Leonard B. Loeb, Basic Processes of Gaseous Electronics (University of California Press, Berkeley and Los Angeles, 1960), pp. 634-645. Our Eq. 27.15 is No. (7.36) on p. 642. The modifications are similar to those described in the reference in note 5 above. See the reference in note 8 for more detail as to the derivation. Loeb's equation contains an error of a factor of 2 because of improper averaging of the electric field in the surface layer. Correction of this error produces the "8" in our Eq. 27-15. 7. See note 5 above. 8. P. C. Souers, E. M. Fearon, and R. T. Tsugawa, J. Vac. Sci. Technol. 3A, 29 (1985). This contains the error mentioned in note 6. 9. See notes 6 and 8 above. 10. J. R. Reitz and F. J. Milford, Foundations of Electromagnetic Theory (AddisonWesley, Reading, Mass., 1960), pp. 271-272. 11. J. W. Stewart, J. Chem. Phys. 40, 3297 (1964). 12. J. H. Constable, C. F. Clark, and J. R. Gaines, J. Low Temp. Phys. 21, 599 (1975). 13. H. A. Hoyen, Jr., J. A. Strozier, Jr., and C. Y. Li, Surf. Sci. 20, 258 (1970). 14. See note 13 above. 15. G. Jaffe, Phys. Rev. 85, 354 (1952). 16. See reference in note 6, pp. 142-145. 17. J. L. Pack and A. V. Phelps, Phys. Rev. 121, 798 (1961).

368

ELECTRICAL EFFECTS CAUSED BY RADIATION

18. J. J. Lowke, Aust. J. Phys. 16, 115 (1963). 19. R. Grunberg, Z. Naturforsch. 23 A, 1994 (1968). 20. A. G. Robertson, Aust. J. Phys. 24, 445 (1971). 21. See notes 17 through 20 above. 22. A. Bartels, Appl. Phys. 8, 59 (1975). 23. R. W. Crompton and A. G. Robertson, Aust. J. Phys. 24, 543 (1971). 24. See note 22 above. 25. W. Legier, Phys. Lett. 31A, 129 (1970). 26. V. M. Atrazhev and I. T. Iakubov, J. Phys. 10D, 2155 (1977). 27. G. L. Braglia and V. Dallacasa, Phys. Rev. ISA, 711 (1978). 28. H. R. Harrison and B. E. Springett, Chem. Phys. Lett. 10, 418 (1970). 29. W. D. Johnson and D. G. Onn, J. Phys. 11C, 3631 (1978). 30. See note 29 above. 31. See reference in note 5, pp. 156-158. 32. E. Graham IV, D. R. James, W. C. Keever, D. L. Albritton, and E. W. McDaniel, J. Chem. Phys. 59, 3477 (1973). 33. M. T. Elford and H. B. Milloy, Aust. J. Phys. 27, 795 (1974). 34. M. Saporoshenko, Phys. Rev. 139A, 349 (1965). 35. W. J. Moore, Physical Chemistry, 3d ed. (Prentice-Hall, Englewood Cliffs, N. J., 1962), p. 766. 36. See note 28 above. 37. M. H. Cohen and J. Jortner, Phys. Rev. 180, 238 (1969). 38. T. Miyakawa and D. L. Dexter, Phys. Rev. 184, 166 (1969). 39. J. L. Levine and T. M. Sanders, Jr., Phys. Rev. 154, 138 (1967). 40. H. R. Harrison and B. E. Springett, Phys. Lett. 35A, 73 (1971). 41. G. E. Grimm and G. W. Rayfield, Phys. Lett. 54A, 473 (1975). 42. P. G. LeComber, J. B. Wilson, and R. J. Loveland, Solid State Commun. 18, 377 (1976). 43. W. E. Spear and P. G. LeComber, "Electronic Transport Properties," in Rare Gas Solids, M. L. Klein and J. A. Venables, eds. (Academic Press, London, 1977), Vol. II, pp. 1131-1132. 44. See note 42 above. 45. See note 8 above. 46. P. C. Souers, E. M. Fearon, J. H. Iwamiya, P. E. Roberts, and R. T. Tsugawa, Cryogenics 20, 247 (1980). 47. P. C. Souers, E. M. Fearon, and R. T. Tsugawa, Cryogenics 21, 667 (1981). The analysis in this reference and in the reference in note 46 is wrong, but the data are good. See the reference in note 8 plus this chapter for the new space-charge analysis. 48. See note 8 above.

Appendix A. Useful Dimensional Relations

Pa = J/m 3 = N/m 2 = kg/m • s 2 = A • T/m = C • V/m 3 J = P a - m 3 = N - m = kg-m 2 /s 2 = C - V = F - V 2 = A - T - m 2 N = P a - m 2 = kg-m/s 2 = C - V / m = A - T - m kg = Pa • m • s 2 = J - s 2 / m 2 = C - V - s 2 / m 2 = V 2 - s 3 / Q - m 2 = A - T - s 2 V = A - Q = J/C = C - Q / s = J - Q / T - m 2 = kg-m 2 /C*s 2 C = A - s = V-s/Q = F- V = kg- m 2 /V • s 2 = A • T • m 2 /V T = H - A / m 2 = Wb/m 2 = J / A - m 2 = N / A - m = kg/A-s 2 S = Í2" 1 A = ampere (electric current)

N = newton (force)

C = coulomb (electric charge)

Pa = pascal (pressure)

F = farad (capacitance)

S = siemen (electrical conductance)

H = henry (inductance)

s = second (time)

J = joule (energy) kg = kilogram (mass) m = meter (distance)

T = tesla (magnetic induction) V = volt (electric potential) Wb = weber (magnetic flux) Q = ohm (electrical resistance)

Appendix B. Hydrogen Permeation in "Impermeable" Materials

The most important feature of life with tritium is the need to contain it. There is a continuing interest in materials that are "impermeable" to hydrogen. Because this question comes up so often, this section is included. Consider a flat slab of metal with surface area A and thickness L. Hydrogen gas is at a high pressure, P h i , on one side and a low pressure, P l0 , on the other. The pressure difference forces hydrogen to permeate through the slab from the high- to the low-pressure side. The basic permeation equation is:

metal:

S

=I

r^(Phil/2~Piol/2)

(B1)

where v/dt (in mol/s) is the gas flow out the low-pressure side and K (in mol/m • s • Pa 0 5 ) is the permeability. K is a constant specific to hydrogen in a given material at a given temperature. The half-power of the pressure in Eq. B.l indicates dissociation of the hydrogen inside the metal ¿H2 - H

(B.2)

The hydrogen diffuses through the metal as an atom. In glass, the hydrogen diffuses as a molecule. The permeation equation is glass: ^ = ^ ( p h i - p . c )

(B.3)

Here, K has units of mol/m • s • Pa. Permeability is the product of two components, the diffusion coefficient D (m 2 /s) and the solubility S (mol/m 3 • Pa 1 / 2 in metals and mol/m 3 • Pa in nonmetals)— that is K = D-S

(B.4)

where D describes how fast the hydrogen atoms or molecules hop through the material, and S tells how many of them are hopping. S is the pressure-dependent term. For the material to be "impermeable"—that is, for K to be small—both D and

371

APPENDIX B

T A B L E B.L PREMEABILITIES OF H Y D R O G E N " I M P E R M E A B L E " M A T E R I A L S LISTED IN O R D E R OF D E S C E N D I N G EXTRAPOLATED PERMEABILITY AT 3 0 0 K

Material permeated

Hydrogen

Fe 89 Fe-11 Ge Ni Co (a) Stainless, 304S Co(fi) Ag Cu Silica glass Pt Stainless, 309S Mo Pyrex glass 95 Cu-5 Sn 91 Cu-7 Al-2 Fe 50 Cu-50 Au M = 15 glass Stainless, oxide Be A1 M = 30 glass Au W Ge Si /J-SiC

H2 H2 H2 D2 H2 D2 H2 H2 H2 H2 D2 H2 H2 H2 H2 H2 H2 D2 T2 T2 H2 D2 H2 H2 H2 T2

K0 (mol/m-s-Pa 1 / 2 or mol/m • s • Pa) 4.1(-8) 3.7( —8) 4.0(-7) 3.8( —8) 2.0( —6) 6.3( —9) 3.4(-8) 8.4(-7) 3.4(-17)-T 1.2C-7)

1.5(-7) 2.3( —7) 3.6(—17)-T 6.6(

—7)

4.0( —8) 6.5( —6) 6.1(-17)-T 2.8( —5) 5.8(—14) 5.8(-5) 2.5(-16)-T 3.1 ( — 6) 7.8( —7) 1.2(-5) 1.4(-5) 1.8( —10)

0k (K) 4200 5030 6600 7750 8 660 6850 7350 9 320 3600 8 500 8 500 9710 4590 9810 8 850 11500 6080 13900 2200(?) 14 800 8 550 14 800 17000 24000 27000 55600

Extrapolated K at 300 K and 1 Pa (mol/m • s) 3 2 1 2 6 7 8 3 6 6 7 2 2 4 6 1 3 2 4 2 3 1 2 2 1 6

-14) -15) -16) -19) -19) -19) -19) -20) -20) -20) -20) -21) -21) -21) -21) -22) -23) -25) -17) -26) -26) -27) -31) -40) -44) -91)

Measured temp. range App. B ref. no. (K) 375-850 570-1070 300-775 670-820 370-570 470-670 730-980 470-700 300-1000 540-900 520-720 500-1700 300-720 560-640 620-840 370-570 550-720 370-700 670-1170 420-520 550-720 500-900 1100-2400 1040-1200 1240-1485 700-1570

2* 3 4-6* 7* 8-10 7* 11 12* 13-19 20* 21 22* 13-19 12 12 12 13-19 23 24* 25* 13-19 26* 27* 28* 28* 29

NOTE: The position of the Be is a guess and the oxidized stainless steel is a perfect oxide layer value. The asterisk indicates that the value was selected as "best" in the review in reference 1 of Appendix B. All metals and SiC follow the Pa 1 ' 2 dependence; all else is presumed to follow the Pa form.

S must be small. This eliminates a large number of metals that are reactive with hydrogen, such as titanium, zirconium, vanadium, niobium, palladium, and all rare earth and actinide elements. For these, S is t o o large, regardless of h o w small D may be. We list the permeabilities of 26 different materials in Table B . l . 1 - 2 9 The metals and the silicon carbide (SiC) follow the P 1 / 2 dependence of Eq. B . l . For them, we use the temperature equation K = K 0 e x p ( —0/T) where K 0 and 6 are constants. For the glasses, we use the form

(B.5)

372

APPENDIX B

K = KoTexp(-0/T)

(B.6)

which has been proposed, actually, only for silica glass. 3 0 The only data included in Table B.l are those in which K was measured directly or in which D and S were measured in the same experiment. The combination of D and S terms from different sources is not dependable. Note, too, that the temperature range measured is above room temperature in every case. T h a t is because most researchers heat up their samples until permeation is easy to measure. The more impermeable the sample, the hotter one must go. One cannot extrapolate to room temperature with any degree of certainty. Having said this, we d o it anyway because some engineers have no other choice at present. The materials are arranged by decreasing, extrapolated 300 K permeability. The measured equation for beryllium gives an absurdly high room-temperature value, and we have guessed its position from the temperature range used in the measurement. The permeabilities in Table B.l marked with an asterisk have been selected as the " b e s t " values following an exhaustive review of the literature. 3 1 Other values, however, do not necessarily represent a detailed investigation. This is especially true of the vast literature in stainless steels. We give only illustrative values, and one must be especially careful in extrapolating this data to room temperature. The oxidized value at 2 x 10~ 2 5 mol/m-s at 1 Pa represents a sample with a perfect hydrogenblocking oxide layer. Often these layers crack, because they are thermally mismatched with the steel, so that good reproducibility is difficult to obtain. 3 2 We next consider glass, which has two kinds of components: (1) the network formers S i 0 2 , B 2 0 3 , and P 2 0 5 , which hold the glass together and are all assumed to have the same high permeability, and (2) the non-network formers, which break u p the lattice but add other desirable physical properties to the glass. These include N a 2 0 , CaO, K 2 0 and A 1 2 0 3 . The percent of non-network formers we call M, and we assume that K decreases as M increases, no matter what the non-network former is. This scheme works, as long as silica is the main ingredient and for M < 30%. The permeability of all silicate-soda lime glasses may then be estimated b y 3 3 K ~ [3.4 + (8 x 10~ 4 )M 3 ] x 1 0 " 1 7 T - e x p

-(3 600 + 165M)

(B.7)

Only sparse data exist for high-M glasses. We note that window and bottle glass have M ~ 25%. We finally note that we may allow for different hydrogens by changing the mass. If we go from H 2 to T 2 , then we divide the H 2 permeability by the square root of 3. In extrapolating to room temperature, the probable error is so high that this minor correction is not necessary.

Notes 1. S. A. Steward, Review of Hydrogen Isotope Permeability through Resistant Materials, Lawrence Livermore National Laboratory Report UCRL-53441, Livermore, Calif. 94550 (1983).

APPENDIX B

373

2. O. D. Gonzalez, Trans. Metall. Soc. AIME 239, 929 (1967). 3. R. A. Ryabov, V. I. Salii, P. Geld, and M. L. Tesler, Soviet. Mater. Sei. 12, 444 (1976). 4. Y. Ebisuzaki, W. J. Kass, and M. O'Keeffe, J. Chem. Phys. 46, 1378 (1967). 5. W. M. Robertson, Z. Metallkd. 64, 436 (1973). 6. M. R. Louthan, Jr., J. A. Donovan, and G. R. Caskey, Jr., Acta Metall. 23, 745 (1975). 7. G. R. Caskey, Jr., R. G. Derrick, and M. R. Louthan, Jr., Scr. Metall. 8, 481 (1974). 8. K. F. Chaney and G. W. Powell, Met. Trans. 1, 2356 (1970). 9. G. W. Powell, J. D. Braun, K. F. Chaney, and G. L. Downs, Corrosion 26, 223 (1970). 10. W. G. Perkins, J. Vac. Sei. Technol. 10, 543 (1973). 11. Y. I. Zvezdin and Y. I. Belyakov, Soviet Mater. Sei. 3, 255 (1967). 12. D. R. Begeal, J. Vac. Sei. Technol. IS, 1146 (1978). 13. P. C. Souers, I. Moen, R. O. Lindahl, and R. T. Tsugawa, J. Amer. Ceram. Soc. 61, 42 (1978). 14. J. E. Shelby, J. Appl. Phys. 45, 2146 (1974). 15. W. G. Perkins and D. R. Begeal, J. Chem. Phys. 54, 1683 (1971). 16. J. L. Barton and M. Morain, J. Non-Cryst. Solids 3, 115 (1970). 17. H. M. Laska, R. H. Doremus, and P. J. Jorgenson, J. Chem. Phys. 50, 135 (1969). 18. R. W. Lee and D. L. Fry, Phys. Chem. Glasses 7, 19 (1966). 19. R. W. Lee, J. Chem. Phys. 38, 448 (1963). 20. Y. Ebisuzaki, W. J. Kass, and M. O'Keeffe, J. Chem. Phys. 49, 3329 (1968). 21. W. A. Swansiger, R. G. Musket, L. J. Weirick, and W. Bauer, J. Nucl. Mater. 53, 307 (1974). 22. W. T. Chandler and R. J. Walter, "Hydrogen Effects in Refractory Metals," in Refractory Metal Alloys, I. Machlin, R. T. Begley, and E. D. Weisert, eds. (Plenum, New York, 1968), pp. 197-249. 23. W. A. Swansiger, "Permeation of Tritium and Deuterium through 21-6-9 Stainless Steel," in Radiation Effects and Tritium Technology for Fusion Reactors, Proc. Int. Conf, Gatlinburg, TN, October 1-3, 1975 (CONF-750989, National Technical Information Service, U.S. Department of Commerce, Springfield, Va.), Vol. IV, p. 401. 24. P. M. S. Jones and R. Gibson, J. Nucl. Mater. 21, 353 (1967). 25. H. Ihle, U. Kurz, and G. Stöcklin, "The Permeation of Tritium through Aluminum in the Temperature Range of 25 to 250°C," in the reference in note 23 above, Vol. IV, p. 414. 26. G. R. Caskey, Jr. and R. G. Derrick, Scr. Metall. 10, 377 (1976). 27. E. A. Aitken, H. C. Brassfield, P. K. Conn, E. C. Duderstadt, and R. E. Fryxell, Trans. Metall. Soc. AIME 239, 1565 (1967). 28. A. Van Wieringen and N. Warmoltz, Physica (Utrecht) 22, 849 (1956). 29. R. A. Causey, J. D. Fowler, C. Ravanbakht, T. S. Elleman, and K. Verghese, J. Amer. Ceram. Soc. 61, 221 (1978). 30. J. E. Shelby, J. Amer. Ceram. Soc. 54, 125 (1971). 31. See note 1 above. 32. See note 23 above. 33. See note 13 above.

Index

All entries refer to pure component hydrogen unless noted. Entries where the phase is not mentioned refer to the low pressure gas phase. We list generally by property, and all six hydrogens are usually represented under each heading. For fast reference, we include DT and D-T sections, which list commonly used property values, generally as listed in tables. Most other hydrogens will also be at these listings. Absorption constants of beta particles. See Tritium beta particle range Accommodation coefficient, 155-156 Actinide elements, reactivity and permeation, 371 Adsorption, of molecules, 152-163 Air electron energy loss in, 221 solid thermal conductivity comparison with, 102 x-ray absorption in, 227 Alkali halides, color centers, 288 Alpha particles boson nature of, 309 exchange equilibrium and, 301 Aluminum (alumina) electron back-scattering from, 229 electron energy loss in, 221-223 f.c.c. structure of hydrogen on, 167 heat-irradiated hydrogen on, 153 impurities from, with tritium, 333 permeability through, 371 wetting of, 65 x-ray absorption in, 227 x-ray emission from, 225 Aluminum oxide (alumina) adsorption on, 158-160, 162 catalyst for exchange, 305 J = 1 -»0 catalysis by, 316 light scattering from, 83-84 permeability effect in glass, 372 separation factor on, 160, 162 Analysis, of pure tritium, 332 Anisotropic potential, 318-319 Annealing of crystals, 82 Antineutrinos from neutron decay, 205

from tritium beta decay, 5, 205, 208-212 Antisymmetric states, 309-310 Apachi, catalyst for J = 1 -> 0 transition, 315 Argon clusters of, 89 electron mobility in, 365 hydrogen layer on, 152-153 in pure tritium, 332 quantum parameter, 10 reduced diffusion coefficient, 11 Atoms (hydrogen) Balmer lines, 262-263 beam of, on copper surface, 155-156 density in solid hydrogen, 282-286 from diatomic reaction, 243 from electron-ion recombination, 244-245 electron spin resonance, 263 electronic energy levels, 261-262 emission frequencies, 262-263 in exchange reaction, 304 excited from electron bombardment, 233-235, 253 from excited molecules, 233, 236, 265 high temperature concentration of, 304 from hydride ion dissociation, 263 from ion-ion recombination, 253 per ion pair, direct, 234 per ion pair, total, 245 J = 1 0 catalysis in solid, 316-317 Lyman lines, 262-263 from muon-hydrogen reaction, 277 muonic atoms, 277-278 nuclear magnetic moments of, 338 permeation in metals, 370 production in tritium beta recoil, 275 from proton-hydrogen reaction, 244 recombination, 247-248

376

INDEX

size, 257, 261 in solid hydrogen, 284-286, 341 in solid neon, 286-287 Atoms of helium. See Helium Azeotrope line, of neon-hydrogen, 201-202 Balmer series, 262-263 Barrier, electrical, 362 Beryllium electron back-scattering from, 229-230 electron energy loss in, 221-223 heat-irradiated hydrogen on, 153 permeability through, 371-372 x-ray emission from, 225 Beta decay. See Tritium Beta particle. See Tritium beta particle Binary hydrogen solutions, 171-174, 178, 182-184 Binding energy. See Energy, binding Binding force in solid, 92 Boiling, caused by decay heat, 176 Bond length in hydrogen, 19-20 Boron oxide, permeability effect in glass, 372 Bosons, 309 Brackett series, 262 Bremsstrahlung (x-rays) from high energy electrons, 228 from tritium beta particles, 223-226 Brute-force, nuclear polarization and, 347 Bubbles (see also Electron bubbles) of He 3 , nuclear magnetic resonance of, 349 of hydrogen, nuclear magnetic resonance of, 337 Calcium, x-rays from, 223-224 Calcium oxide, permeability effect in glass, 372 Calorimetry, heats of transformation from, 111

Capacitance, 355-357 Carbon. See Graphite Carbon dioxide infrared lines, 334 in pure tritium, 332 solid particles in liquid hydrogen, 203-204 from stainless steel, 303 Carbon monoxide infrared lines of, 334 solid particles in liquid hydrogen, 203-204 from stainless steel, 303 Catalysts in atom recombination, 248 for exchange, 301, 304-305 for J = 1 -> 0 transitions, 24-25, 27-30, 311, 315-317

self, for heteronuclear hydrogens, 25, 28 unsupported, definition of, 315 Cerric oxide, J = 1 -> 0 catalysis by, 316 Cesium, x-ray edge, 224 Chain length, in exchange reaction, 300-301 Chemical exchange. See Exchange reaction Chromatography, 159, 162 Chromium oxide (chromia) adsorption on, 158 catalyst for exchange, 301, 305 J = 1 - » 0 catalysis by, 316 Clausius-Clapeyron equation liquid-solid, 128 liquid-vapor (and solid-vapor), 49, 110 liquid-vapor solutions, 166 Closed shells, of ion clusters, 249 Cluster ions. See Ion clusters Clusters (neutral), 88-89 Cobalt, permeability through, 371 Coefficient of hardening, 86 Compressibility of gas in Clausius-Clapeyron equation, 49, 110 at critical point, 58 definition of, 54 gas density and, 133, 136 gas pressure and, 135 and non-ideal energy, 113 along saturation curve, 58 virial coefficients and, 55-56, 133 Compressibility of solid, heat capacity and, 99 Conduction band of fluid, 362 of solid, 281-282, 288 Consolute point, of neon-hydrogen, 200-202 Constant pressure gas, cooling curves, M S MS Constant volume gas, cooling curves, 143144 Contact angle, of liquid, 65 Convection in liquid, 67-68 Copper atom beam on, 155-156 crystal seed spike, 82 electron back-scattering from, 228-231 electron energy loss in, 221-223 electron-irradiated hydrogen on, 154 heat-irradiated hydrogen on, 153-154 J = 1 -> 0 catalysis by, 316 permeability through, 371 solid thermal conductivity comparison with, 102 sputtered hydrogen on, 154 thermal resistance with solid, 104-105 tritium self-heating in, 106 x-ray emission from, 225

377

INDEX

Copper alloys, permeability through, 371 Corresponding states, 9 Cosmic radiation and electrical conductivity, 352 Coulomb potential, 288 Cracks, thermal conductivity and, 104-105 Creep, of solid, 85-88 Critical point compressibility at, 58 energy, 114-115, 120-121 enthalpy, 118-119 as fixed point, 45-49 of helium-hydrogen mixtures, 192-193 non-ideal energy, 113-116, 120 phase diagram mapping and, 133 pressure, 3, 48 pressure, of helium, 3 PV, 116-117 temperature, 3, 48 temperature, of helium, 3 vapor density at, 57 vapor pressure at, 52 Cross section for atom formation, 236 for atom recombination, 286 for atoms in exchange reaction, 304 Balmer atom emission, 262 for crystallite light scattering, 83 for diatomic ion-hydrogen reaction, 243-244 for elastic collisions, 237 geometric, of hydrogen atom, 286 geometric, of hydrogen molecule, 32, 235 hydride ion formation, 236 ion-electron recombination, 246 ion-ion recombination, 253-254 for ionization of hydrogen, 235-236 in kinetic gas theory, 241 Lyman (molecular) emission, 266 for molecular excitation, 236 for rotational excitation, 237 for vibrational excitation, 237 Werner emission, 266 Crystal field (anisotropic potential), 318-319, 327 Crystal growing, 81-82 Crystal lattice vibrations atom recombination and, 248 calculated spectrum, 99, 101 Debye spectrum, 97, 99, 101 heat capacity of liquid and, 66-68 infrared spectrum and, 99-100, 328-330 J = 1 - » 0 effects, 311-315 phonon mean free path, 68-69, 102-103 Raman spectroscopy and, 326 smeared end-point, 315

thermal energy in, 95, 97-98 zero-point energy, 93-98 Crystal structure face-centered-cubic, 75-76, 79, 191 hexagonal, 73-75, 78-80 interstitial spaces in, 77-78 Crystallite light scattering, 83-84 phonon scattering in, 103 C T 5 + ion, 303 Curie, definition of, 206 Daedalus starship, 4 Debye model, 94-95, 97-99, 101, 127 Debye temperature definition, 94-95, 101 Gruneison constant and, 98-99 in J = 1 -»• 0 transition, 312-313 Lindemann melting relation and, 130-131 mass relation, 97 measurements of, 99-100 in model of solid, 94-99 solid thermal conductivity and, 103 in zero-point energy, 95 Decay constant for tritium, 206 Defect-pinning, 87 Density (includes molar volume, which is the inverse) coefficients for, 133, 136 of constant pressure gas, 145 of constant volume gas, 144 critical point vapor, 57-58 Debye temperature from, 99 electron mobility effect, 358-362 gas at high temperature, 135-136 gas at 77 K, 134-135, 141, 143 gas at 300 K, 134-135, 141, 143 ion mobility effect, 362-364 of liquid hydrogen solutions, 167-168 liquid along melting curve, 127-129 molar volume mixing parameter, 168 of neon-hydrogen liquid, 203 nuclear magnetic relaxation of gas and, 342 phase transformation and, 76 saturated liquid, 61-62 saturated solid, 78-80 saturated vapor, 57 of solid hydrogen solutions, 167-168 solid along melting curve, 125-127 solid under pressure, 129 solid thermal conductivity and, 102 solid thermal diffusivity and, 105 triple point liquid, 61-62 triple point solid, 80 triple point vapor, 57

378 Deuterium discovery, 4 Diamagnetism, 339 Diatomic ions (hydrogen) dipole moment, 258 dissociation energy, 258 electronic energy levels, 258 electron spin resonance, 258, 260 infrared frequencies of, 258 from ionization of hydrogen, 232-233 reaction with atom, 244 reaction with hydrogen, 243-244, 300 recombination with electrons, 244-247 size, 257-258 from triatomic ion, 248 vibration-rotation levels, 258 zero-point energy, 258 Dielectric constant definition, 355 gas and liquid, 355-356, 365-366 space charge effect, 356, 366 Diffusion coefficient of atoms in solid, 285-286, 317 for exchange reaction in solid, 241-242, 306 gas at constant pressure, 145 gas at constant volume, 144 gas down pipe, 40-41 gas at high pressure, 149-150 gas at low pressure, 34-35 in kinetic gas theory, 241-242 mobility (electrical) relation, 363 relation to mean free path, 34 relation to permeability, 370 relation to rate constant, 241-242 of saturated liquid, 65-66, 124 of saturated solid, 88, 124 of saturated vapor, 122, 124 solid phase separation and, 169 Diffusivity. See Thermal diffusivity; Diffusion coefficient Dimensional relations, 369 Dipole moment of diatomic ion, 258 of heteronuclear hydrogen, 258, 261, 324 of hydride ion, 263-264 quadrupole comparison with, 318 spectroscopy and, 324 transient molecular, 92 of triatomic ion, 260-261 Dissociation of adsorbed molecules, 163 of diatomic ion, 258 from electron bombardment, 233-234 of helium tritide, 275 of hydride ion, 263-264 of hydrogen, 257

INDEX

linear energy transfer function for, 215-217, 234 of triatomic ion, 260 Drift velocity of electron. See Velocity (drift), of electron Drop size, 63-64 Droplet oscillations, 64 DT (specific molecular properties only, as listed in tables; most other hydrogens are at the same places) critical point parameters, 3, 49 decomposition of, by tritium, 305-306 density of gas at critical point, 57 density of gas at triple point, 57 exchange reaction of, 305-306 heat capacity of perfect gas, 28 heat capacity of saturated liquid, 68 heat capacity, total of saturated solid, 96 infrared lines, 329-331 molecular weight, 3 nuclear magnetic moments, 338 quantum parameter, 3 Raman lines, 325-327 rotational energy of perfect gas, 27 rotational population, 20-21, 23, 26 spectroscopic constants, 17, 19 synthesis of, 331 triple point parameters, 3, 48 vibration-rotation transitions, 325-327, 329-331 virial coefficients, 55-56 D-T (equimolar three-component solution; includes properties interchangeable with DT; numbers refer to tables only; most other hydrogens are at the same places) constant pressure paths in fluid, 145 constant volume paths in fluid, 141, 144 convection in liquid, 68 density of critical point fluid, 57 density of fluid under pressure, 140 density of liquid on freezing curve, 128 density of saturated liquid, 61-62 density of saturated solid, 80 density of saturated vapor, 57 density of solid on freezing curve, 127 density of solid under pressure, 129 density of triple point fluid, 57 dielectric constant, 365-367 diffusion coefficient of constant pressure fluid, 145 diffusion coefficient of constant volume fluid, 144 diffusion coefficient at low pressure, 35 diffusion coefficient of saturated vapor, 122 electrical conductivity, 365-367

INDEX

energy of fluid, 141, 143-144 energy, total, of saturated liquid, 121, 141 energy, total, of saturated solid, 121 energy, total, of saturated vapor, 120, 141 enthalpy, change upon freezing, 122 enthalpy, total, of saturated liquid, 119, 141 enthalpy, total, of saturated solid, 119 enthalpy, total, of saturated vapor, 118, 141 equation of state of fluid, 140 exchange equilibrium constant, 294-297, 299 fractionation upon freezing, 173, 175-177 freezing pressures and temperatures, 126, 129 heat capacity of perfect gas, 28 heat capacity of saturated liquid, 68 heat capacity of saturated solid, 95 heat of fusion, 112 heat of sublimation, 112 heat of sublimation at 0 K, 117 heat of vaporization, 112 helium-3 solubility in, 186-191 infrared impurity lines in, 334 infrared lines, 330 ion pair density in, 366-367 molar volume (see Density) non-ideal energy of saturated gas and liquid, 115 non-ideal vapor pressures, 178-184 PV of liquid, 117 PV of vapor, 116 rotational energy of perfect gas, 28 separation factor in chromatography, 159-160, 162 77 K starting points to phase boundary, 141, 144-145 slow freeze and thermal diffusivity, 122-124 sound velocity of saturated liquid, 70 sound velocity in saturated solid, 81 surface tension, 63-64 thermal conductivity of constant pressure fluid, 145 thermal conductivity of constant volume fluid, 144 thermal conductivity of fluid under pressure, 146, 148 thermal conductivity at low pressure, 33-35 thermal diffusivity of gas (see Diffusion coefficient) thermal diffusivity of liquid and solid, 122, 124 300 K starting points to phase boundary,

379 141, 144-145 vapor pressure of, 52, 54 viscosity of constant pressure fluid, 145 viscosity of constant volume fluid, 144 viscosity of fluid under pressure, 146-147, 150 viscosity at low pressure, 33, 35-36 viscosity of saturated liquid, 62-63 volume (see Density) Einstein model of solid, 127-128, 130 Einstein temperature, 127 Elastic collisions, electrons on hydrogen, 236-237 Elasto-plastic deformation, 85 Electrical conductivity AC effects, 356 DC equations, 353-355 of D-T, 365-367 mobility relation, 352 of pure liquid, 352 space charge and, 353-355, 357, 366-367 Electron beam atoms in solid from, 282, 284-285 exchange equilibrium and, 301 high energy into hydrogen, 227-228 on solid hydrogen, 154, 282, 284-285 Electron bubble in liquid, 252, 364 in liquid helium, 364 in solid, 287-288 Electronically excited hydrogen. See Hydrogen, excited molecules Electronic transitions, nomenclature, 265 Electron paramagnetic resonance. See Electron spin resonance Electrons (see also other Electron headings) beta particles (see Tritium beta particle) electrical properties (see Electrical conductivity; Mobility; Space charge; Velocity of electrons) indistinguishability of, 308 magnetic moment, 337-338, 341 from muon-hydrogen reaction, 277 produced by beta collision (see Ion pair; Ionization) recombination with ions {see Recombination) secondary, 233, 238-239 trapped in solid (see Electron bubble) from tritium beta recoil, 275 Electron scattering from metal walls, 228-231 Electron spin resonance of atoms, 263, 339 of atoms in solid, 282-284, 286

380 of diatomic ions, 258, 260 energy levels of, 258 nuclear magnetic resonance and, 336, 339 sensitivity of, 336 zero field, 339 Energy (refers to pure component hydrogen unless noted) of adsorption, 153, 158-159 atom-electron reaction, 248 atom recombination, 247 barrier, electrical, 362 binding of nuclei, 205 of binding in solid, 93-94 center of mass, 242 of cluster ions (negative), 252 of cluster ions (positive), 249 comparison of magnitudes, 7 - 8 to conduction band in solid, 281-282, 287 coulomb potential, 288 diatomic ion-electron recombination, 244 diatomic ion-molecule reaction, 243 dissociation of diatomic ion, 258 dissociation of helium tritide, 275 dissociation of molecule, 257 dissociation of of triatomic ion, 260 of electron energy loss in hydrogen, 214-217, 232-239 enthalpy relation, 115 excess free energy of solution, 179-182 exchange in binding, 308 of excitons in hydrogen, 281 -282 free energy and Clausius-Clapeyron equation, 49 freezing parameters, 122-124 of fusion reactions, 1 - 4 h.c.p.-to-f.c.c. transition, 76 helium tritide-hydrogen reaction, 253, 277 helium tritide recoil, 275 ionization of helium, 271 ionization of molecule, 232, 234-235, 256-257 per ion pair, 232, 235-236 in Joule-Thomson expansion, 142-143 Lennard-Jones potential, 9 - 1 0 of liquid surface, 64 Lyman (molecular) band, 266 between magnetic dipoles, 311 muonic atom, binding, 277-278 muonic ion, binding, 278 non-ideal at critical point, 114 of non-ideal fluid under pressure, 141-143 of non-ideal gas, 113 non-ideal, of saturated vapor 115 in nuclear magnetic resonance, 339-340 partial molal free energy of solution, 180-181

INDEX

potential energy at 0 K, 96, 117 proton-hydrogen reaction, 244 quadrupolar, 79, 95-96, 118-119, 312, 317-322 recoil from tritium decay, 212-213 rotational, in calculation of enthalpy, 118 rotational, in calculation of total gas energy, 113 rotational, hindered, 159-161 rotational, for perfect gas, 15-16, 18-31 solid energy and pressure, 96, 98 square potential, 288 of sublimation, 93-94 surface, of liquid, 64 total, for fluid under pressure, 135, 137, 141 total, of gas cooled at constant volume, 141, 144 total, for perfect gas, 15 total, for saturated liquid, 121, 141 total, for saturated solid, 121 total, for saturated vapor, 120, 141 translational, at critical point, 114 translational, for gas, 15, 114 translational, for saturated solid, 95, 9 7 98 translational, for solid under pressure, 98 triatomic ion-electron recombination, 245 of triatomic ion-xenon reaction, 300-301 tritium beta absorption in hydrogen, 214-220 tritium beta absorption in various materials, 221-223, 230 tritium beta, emitted, 220 tritium beta maximum, 209, 211 tritium beta mean, 208, 214 tritium beta spectrum, 208-212 tritium recoil, 212-213 valence to conduction band, 281-282, 287 of vaporization, 113 vibrational, 15, 17-18 Werner transition, 266 of x-rays, 221, 223-227 zero-point of clusters, 89 zero-point of diatomic ions, 258 zero-point for helium tritide ion, 276 zero-point liquid or solid, 9, 93-97 zero-point of molecule, 17 zero-point of molecule, in exchange equilibria, 295 zero-point of molecule against a surface, 158 zero-point of trapped electron, 252 zero-point of triatomic ions, 261

381

INDEX

Enthalpy (see also Heats) change upon freezing D-T, 122 of constant pressure gas, 145 of constant volume gas, 141, 144 energy relation, 115 excess, of hydrogen solutions, 179, 181-183 of ion cluster formation, 249 non-ideal, of gas, 113 rotational for perfect gas, 21 total for perfect gas, 29-30 total for saturated liquid, 119, 141 total for saturated solid, 119 total for saturated vapor, 118, 141 translational, of perfect gas, 16 zero energy reference points, 116 Entropy, excess, of hydrogen solutions, 179 Equation of state of fluid under pressure, 133-145 of solid underpressure, 127, 129 Equilibrium constant for chemical exchange, 294-297, 299, 302 for positive ion clusters, 249-250 Equilibrium hydrogen, definition, 13, 27-28 Ethane, gel with liquid hydrogen, 204 Eucken equation, 33 Eutectic points, of neon-hydrogen, 200-201 Evaporation, of solid, 154-155 Even-, odd-J nomenclature, 12 Excess helium pressure. See Helium, blocking effect Exchange reaction atomic, in gas phase, 304 catalyzed, 301-302, 304-305 chain length, 300-301 chemical equilibrium constants, 294—297, 299 hot atom equilibria, 301-302, 305-306 kinetics of, 298-299 muon-catalyzed, 277 stainless steel walls in, 302 time contants of, 299, 303, 306 triatomic ion in, 299-301 Excited hydrogen molecules, H 2 * emission lines from, 266 energy levels of, 257, 265 excitons in solid, 281-282, 287 notation for, 265 polarizability and, 92-93 singlet-triplet symmetry, 265 from tritium beta particle, 233-234, 236 Excitons hydrogen atom in neon, 286-287 hydrogen atom in xenon, 287 hydrogenic model for, 287 in solid hydrogen, 281-282

Extrapolation, of properties, 8-12 Extrusion of solid, 88 Face-centered-cubic structure on f.c.c. metals, 76, 167 geometry of, 75-76 in helium-hydrogen mixtures, 191 ion clusters and, 264 interstitial spaces in, 76-77 transition to, 74, 76 Falling liquid film condenser, 196 Fermi theory of beta decay, 208-211 Fermions, 309 Ferric oxide, J = 1 -> 0 catalysis by, 316 Finite thickness tritium model, 219-220 Fixed points. See Critical point; Triple point Fluorescence (x-ray), 223, 225-226 Fluorine-19, nuclear magnetic resonance of, 336 Fractionation of hydrogen solutions, 171-177 in muon-catalyzed fusion, 277-278 for slowly freezing D-T, 173, 175-177 thermal diffusion of the gas and, 40-41 Francium, x-ray edge, 224 Free induction decay, 336 Freezing. See Melting Freezing point. See Triple point; Melting curve Fusion. See Hydrogen fusion; Heat, energy of Gadolinium oxide, J = 1 0 catalysis by, 316 Gamma radiation, on J = 1 0 transition, 316 Gas flow through pipe, orifice, or hole, 3638 Gas law, 32, 54 Geometric mean, rule, 52 Germanium atom recombination on, 248 permeability through, 371 Glass (see also Pyrex; Soda-lime glass) hydrogen adsorption on, 157 impurities from, with tritium, 333 permeability through, 370-372 solid thermal conductivity comparison with, 102 thermal transpiration and, 38-40 tritium heating in, 106, 317 tritium vapor pressure in, 51 x-rays from, 223-224, 226 Gold electron back-scattering from, 229-230 electron energy loss in, 221-223

382 electron-irradiated hydrogen on, 154 heat-irradiated hydrogen on, 153 J = 1 -> 0 catalysis by, 316 permeability through, 371 x-ray absorption in, 227 x-ray emission from, 225 Graphite electron energy loss in, 221-223 hydrogen adsorption on, 157 x-ray emission from, 225 Gruneison constant, 98-99, 127 Gyromagnetic ratio, 339 H + . See Protons H~ ion. See Hydride ion H2*. See Excited hydrogen molecules H 2 + ions. See Diatomic ions H 3 + . See Triatomic ions H 5 + , H 9 + , H 2 7 + , etc. See Ion clusters, positively charged Hj~, H s ~, etc. See Ion clusters, negatively charged Hafnium tritide, half-life from, 206-207 Half-life of muons, 279 of neutrons, 205 of pions, 277 of tritium, 206-207 Hardness, radiation and, 88 He2* (diatomic helium), 272, 274-275 Heat of adsorption, 159 excess, of mixing of solutions, 181-183 of fusion along melting curve, 128-131 of fusion at triple point, 49, 110-112 of sublimation, 49, 110-112, 117 of sublimation of ideal hydrogen solutions, 166 of sublimation at 0 K, 116-121, 322 from tritium decay, 25, 105-106, 176, 208, 348 of vaporization, 49, 110-113, 116, 121 of vaporization, of solutions, 166 of XeD + formation, 301 Heat capacity non-ideal, of gas, 113 for ordering transition, 320-322 quadrupolar, 96, 320-322 rotational, for perfect gas, 15, 21-23, 2831,33 saturated liquid, 66-68 saturated solid, 96 solid thermal conductivity and, 102 solid thermal diffusivity and, 105 total for low pressure gas, 33-34, 36 total for solid, 95-96

INDEX

translational, energy relation, 114 translational, for perfect gas, 16 vibrational, for pefect gas, 15 Helium-3 blocking effect in cryostat, 40-41, 184, 192-197 bubble break-up in cryostat, 196-197 critical point pressure, 3 critical point temperature, 3 crystallization effect on hydrogen, 82 electronic energy levels, 270, 272 emission spectrum, 270-275 f.c.c. form of hydrogen solution with, 191 fluid mixtures with hydrogen, 188-190 formation from tritium, 5, 205 fusion reactions with, 2-5 heat conduction with, 105 ionization energy, 271 in K-orbit capture, 207 quantum parameter, 3 nuclear binding energy, 205 nuclear magnetic moment, 338-339 nuclear magnetic resonance of, 258, 348-349 nuclear magnetic resonance frequency, 336, 338 rate of formation, 186, 206 reduced diffusion coefficient, 11 removal from hydrogen, 196 selection rules for, 270 singlet-triplet symmetry, 270 solubility in liquid hydrogen, 188-190 thermal conductivity of gas, 104-105, 198 tritium vapor pressure and, 51 vapor pressure of, with hydrogen, 188— 190 Helium-4 atom recombination on liquid, 247 critical point pressure, 3 critical point temperature, 3 electron bubbles in, 364 emission spectrum, 270-275 exchange reaction and, 301 fluid mixtures with hydrogen, 186-193 from fusion reactions, 1-4 from Li 6 , 5 melting line, with hydrogen, 191-192 molecular weight, 3 nuclear magnetic resonance of, 336 quantum parameter, 3, 199 thermal diffusion with hydrogen, 42 triple point pressure, 3, 199 triple point temperature, 3, 199 vapor pressure of, 188 vapor pressure with hydrogen, 186-191 Helium dimer. See He 2 *

383

INDEX

Helium hydride. See Helium tritide ion; Helium tritide molecule Helium ions electronic energy levels, 271-273 emission lines, 271-272, 275 production in tritium beta recoil, 275 Helium tritide ion dissociation energy of, 275 electronic energy levels, 275 emission line from, 277 infrared spectrum of, 277 ion clusters with, 277 production in tritium beta recoil, 252, 275 reaction with hydrogen, 253, 277 recoil energy of, 275-276 recombination and, 207 rotational energy levels, 275-277 spectroscopic constants, 276 vibrational levels, 275-277 zero-point energy, 276 Helium tritide molecule, 277 Henry's law constants, helium-hydrogen, 187-191 Heteronuclear hydrogen, definition, 7 Hexagonal-close-packed structure geometry of, 73-75 h.c.p. substrate stabilization, 79-80 interstitial spaces in, 77-78 ion clusters and, 264 J = 1 effect on, 73-74, 318-321 molar volume of, 78-81 spin glass phase, 73 Hindered rotation of molecule, 159-161 Homonuclear hydrogen, definition, 7 Hot atom equilibrium in exchange reaction, 301-302, 305-306 in J = 1 molecule formation, 317 Hydride ion binding energy of, 263 dipole moment of, 263-264 electronic energy levels, 264 far infrared of, 264 formation of, 235-236, 248-249 nuclear magnetic resonance, 264 proton reaction, 253 size, 257, 263 in solid, 289, 291 Hydrides, as fusion fuel, 6 Hydrocarbon gels, 204 Hydrogen atoms. See Atoms Hydrogen deuteride nuclear polarization of, 346-348 synthesis of, 347 Hydrogen-4, 5 Hydrogen fusion cross sections, 2

heats of reaction, 1 - 4 muon catalyzed, 3-4, 277-279 Hydrogen in pure tritium, 332 Hydrogen tritide impurity in tritium, 332-333 vapor pressure of D 2 and, 10, 52-53 Hyperfine splitting (coupling) of atoms, 263, 282-283, 286, 339 of diatomic ion, 260 Impurities in tritium, 332-334 Inconel heat-irradiated hydrogen on, 153 wetting of, 65 Infrared spectroscopy Debye temperature from, 99 in exchange reaction, 305 explanation of spectrum, 328-331 frequencies of, 324-325, 329-331 for hindered adsorbed molecules, 160, 162 hydride ion in solid, 264 impurity lines in, 334 J = I —• 0 transition and, 317 radiation-induced lines in solid, 287-292 selection rules for, 324 of solid, 287-292, 328-329 triatomic ions and, 261 Intermolecular distance in solid. See Crystal structure Internal energy. See Energy Interstitial positions, in solid, 77-78 Ion clusters, negatively charged formation of, 252 structure of, 265 Ion clusters, positively charged exchange reaction and, 303 formation of, 249-251 mobility of, 363 size, 257, 259, 264 structure, 259, 264 vibrational frequencies, 264 Ionization cross section for, 236 from electron bombardment, 232-235 energy of, 232 of helium atom, 271 linear energy transfer function for, 215-217, 234 Ion mobility, 362-366 Ion pair definition, 232 density of, in D-T, 366-367 energy of formation, 232, 234-236 Iron catalyst for exchange, 304 electron back-scattering from, 229

384 electron energy loss in, 221 J = 1 -> 0 catalysis by, 316 permeability through, 371 solid thermal conductivity comparison with, 102 x-ray absorption, 227 Iron-germanium, permeability through, 371 Iron oxide (Fe 3 0 4 ), J = 1 —»• 0 catalysis by, 316 Isospin, 310 AJ = ± 4 transitions, 324 Joule-Thomson expansion, 141-143 Kinetic gas region, 32, 36, 39 K-orbit capture of beta particles, 207 Krypton electron mobility in, 365 exchange reaction and, 301 hydrogen layer on, 79, 152-153 quantum parameter, 10 reduced diffusion coefficient, 11 Krypton-85, beta particles in liquid hydrogen, 254 Kurie plot, 210-212 Lead electron energy loss in, 221 heat-irradiated hydrogen on, 153 Leak, gas through, 36-38 Lennard-Jones potential, 9-10, 96 Light scattering. See Crystallite light scattering Lindemann melting relation, 130-131 Linear energy transfer function of high energy electrons in hydrogen, 227-228 of tritium beta particles in hydrogen, 215-217, 234 of tritium beta particle in various materials, 221-222 Liquid interaction parameter, 180-182 Lithium-6 for starship, 4 in tritium production, 5 Lithium hydride, hydrogen bubbles in, 337 Lithium iodide, electron energy loss in, 221 Lithium tritide, DT synthesis from, 331 Longitudinal relaxation. See Nuclear magnetic relaxation, T, Lucite, electron energy loss in, 221 Luminescence of solid, 84 Lyman series (atomic), 257, 262-263 Lyman series (molecular), 266

INDEX

Magnesium fluoride, heat-irradiated hydrogen on, 153 Magnetic dipolar interaction, 311 Magnetic susceptibility, 339-341 Magnetization, nuclear, 339 Manganese oxide, J = 1 —»0 catalysis by, 316 Mass, effective, 158 Mass-energy relation, 1, 205 Mean free path of atoms in solid, 286 of electrons in gas, 358 of gas molecules, 32, 34, 241 of phonons in liquid, 68-69 of phonons in solid, 102-103 relation to rate constant, 241 Mechanical properties, 85-88 Melting curve of clusters, 89 of helium-hydrogen mixtures, 191-192 nuclear magnetic relaxation and, 344-345 of pure component hydrogens, 125-131 Meniscus, 65 Mercury-203, acceleration and beta decay, 207 Mesic hydrogen. See Muonic hydrogen Methane exchange reaction and, 303 infrared lines of tritiated, 334 production in tritium, 332-333 in pure tritium, 332 solid particles in liquid hydrogen, 203-204 Methanol, and DT synthesis, 331 Mirror nuclei, 5, 205 Mixtures (Hydrogen-hydrogen combinations are assumed to be solutions; see specific property desired) carbon dioxide-hydrogen, 203-204 carbon monoxide-hydrogen, 203-204 helium-hydrogen, 186-197 methane-hydrogen, 203-204 neon-hydrogen, 199-203 nitrogen-hydrogen, 203 oxygen-hydrogen, 203 phase separation of hydrogen, 169 water-hydrogen, 203-204 Mobility diffusion coefficient relation, 363 drift velocity relation, 358, 360 electrical conductivity relation, 352 of electrons, 357-361 of ions, 362-366 Molar volume. See Density Molecular bond distance, 20 Molecular gas region, 32, 36-39 Molecular rotation. See Rotation, molecular; Rotational population

INDEX

Molecular shape, 73 Molecular velocity. See Velocity (thermal), of molecules Molecular vibration. See Vibration, molecular Molecular weight of diatomic hydrogen, 3 of helium, 3 of hydrogen atoms, 3 Monolayer, definition of, 152 Muonic hydrogen diatomic ion, 278-279 metastable molecule, 278-279 monatomic atom, 277-278 x-rays from, 278 Muons magnetic moment and nuclear spin, 279 magnetic resonance frequency, 279 mass, 3, 277 from pion decay, 277 reaction with hydrogen, 277 National Bureau of Standards equation of state (fluid), 114, 120, 133, 135, 138, 145-149 Necking, 85-86 Neodymium oxide, J = 1 - » 0 catalysis by, 316 Neon density of liquid, 203 density of liquid hydrogen solutions, 203 fluid mixtures with hydrogen, 199-203 with hydrogen atoms in, 286-287 quantum parameter, 10 solid mixtures with hydrogen, 200-201, 203 sound velocity of liquid, 203 sound velocity of liquid neon-hydrogen, 203 triple point of, 199 vapor pressure, 199 Neptunium, x-ray edge, 224 Network formers, 372 Neutrinos (see also Antineutrinos) from fusion reactions, 4 from pion decay, 277 Neutron decay of, 205 from fusion, 1 - 3 magnetic moment of, 338-339 molecular weight, 3 Nickel atom recombination on, 248 catalyst for exchange, 301, 305 J = 1 -> 0 catalysis by, 315-316 permeability through, 371

385 Niobium, reactivity and permeation, 371 Nitrogen exchange reaction and, 301 in films of hydrogen, 155 in pure tritium, 332 solubility in liquid hydrogen, 203 Non-ideality energy-enthalpy relation, 113 energy of gas and, 115 energy of gas at critical point and, 114-115 of helium-hydrogen mixtures, 186-191 of liquid molar volumes, 167-168 of liquid surface tension, 169 of liquid viscosity, 168 of vapor pressure, 178-184 Non-network formers, 372 Normal hydrogen definition, 12-13, 24 energy calculation for, 24-25 Nuclear magnetic moment in J = 1 -* 0 transition, 311 nuclear spin and, 337-340 quadrupole moment interaction with, 312-313 sizes of, 338 Nuclear magnetic relaxation bubble sizes and T 2 , 337 definition of, 336-337 helium-3 T ( , 348-349 nuclear spin polarization and, 347-348 radiation damage o n T , , 348 in gas, 341-344 T t in liquid, 341-344 T , minimum, 341-342, 345-347 T ! in solid, 344-348 T 2 in gas, 341-342 T 2 in liquid, 341-342, 344-345 T 2 in lithium hydride, 337 T 2 in solid, 344 wall effects on He 3 , 348-349 Nuclear magnetic resonance (see also Nuclear magnetic relaxation) energy levels in, 339-340 frequencies of, 336, 338 of gas and liquid, 341-344 of helium-3, 348-349 of hydride ion, 264, 337 of hydrogen bubbles, 337 in J = 1 ->0 study, 317 magnetization, 339 of solid, 103, 344-348 thermal conductivity from, 103 Nuclear spin. See Nuclear magnetic moment Nuclear spin polarization, 347-348 Nylon, tritium self-heating in, 106

386 Optical absorption in solid, 83-84, 289 Ordered phase, 73-74, 319-332 Orifice, gas flow through, 36-38 Ortho-para effects. See Rotational population Ortho-para nomenclature, 12 Oxygen electron spin resonance effect, 282-283 exchange reaction and, 301 in helium-3 n.m.r., 348 nuclear magnetic resonance of, 336 in nuclear polarization, 347 in pure tritium, 332 solubility in liquid hydrogen, 203 from stainless steel, 303 tritium containment and, 6 x-rays from, 226 Palladium for He 3 removal, 82, 196-197 reactivity and permeation, 371 Paramagnetic susceptibility, 340-341 Partition function equilibrium constant from, 295 for quadrupolar interaction, 320, 322 rotational, 20-24 rotational, hindered, 160 Pegmatite, lithium from, 4 Penetration depth, 217 Perfect gas law, 32, 54 Permeation through metals, 370-371 Pfund series, 262 Phase diagram, general description, 45 Phase separation, 169 Phase transformation in solid, 73-74, 76 Phonons. See Crystal lattice vibrations Phosphorous pentoxide, permeability effect in glass, 372 Pion, decay to muon, 277 Pipe gas diffusion down, 40-41 gas flow through, 36-40 Plasma, 46, 355 Platinum catalyst for exchange, 305 permeability through, 371 Poisson ratio, 86 Polarizability of hydrogen, 92 ion clusters and, 249 ions in the solid and, 289 magnetic susceptibility and, 339 refractive index and, 92 solid binding energy and, 92-93 surfaces and, 158

INDEX

Polarized light and crystals, 82 Polyethylene electron energy loss in, 221 hydrogen adsorption on, 157 Potassium oxide, permeability effect in glass, 372 Potential energy of hindered rotation, 160-161 Lennard-Jones, 9-10, 96 simple form for solid, 94-96 Pressure. See Equation of state; Vapor Pressure Pressure on helium emission lines, 275 Probability of reaction, 241 Protons hydride ion reaction, 253 hydrogen reaction with, 244 from ionization of hydrogen, 232-235 mobility of, 363 production in tritium beta recoil, 275 from triatomic atom dissociation, 260 PV at critical point, 113,116-117 energy-enthalpy relation, 115 of saturated liquid, 117 of saturated vapor, 116 P-V-T. See Equation of state Pyrex (see also Glass) atom recombination on, 248 exchange reaction in, 301-303 heat-irradiated hydrogen on, 153 in helium-3 n.m.r., 348-349 impurities from, with tritium, 333 permeability through, 371 Quadrupolar interaction in J = 1 -» 0 transition, 313 between J = 1 molecule and deuteron, 312-313 between J = 1 molecules, 317-319 nuclear magnetic relaxation and, 345 solid molar volume and, 79-80 thermodynamics of, 319-322 Quadrupole moment of deuteron (nucleus), 312 of molecular hydrogen, 312, 318, 345 Quantum parameter definition of, 10 effective, 3, 10 of helium, 3, 11, 199 of hydrogen, 3, 11 hydrogen solutions and, 181 Lindemann relation and, 130-131 liquid diffusion and, 11 as mass function, 158

INDEX

of neon, 199 of rare gases, 10-11 Quartz, atom recombination on, 248 Radiation crystal shape and, 82 vapor pressure and, 152-154 Radiation induced lines in solid, 287-292 Raman spectroscopy explanation of spectrum, 325-327 frequencies of, 324-327 selection rules for, 324 Range of beta particle (see Tritium beta particle, range) of high energy electrons, 227-228 Raoult's law definition of, 165-166, 171 for helium-hydrogen mixtures, 186-187 for neon-hydrogen mixtures, 202 non-ideal corrections to, 178-184 Rare earth elements, reactivity and permeation, 371 Rare gas clusters, 89 Rate constant atom recombination in gas, 247 atom recombination on walls, 247 atom recombination in solid, 284-286 for diatomic ion-hydrogen reactions, 243-244 electrical conductivity and, 353 in exchange reaction, 298-300, 304-305 ion-electron recombination, 245-246 ion-ion recombination, 253 for J = 1 0 transition, 313-316 in kinetic gas theory, 241-242 Recoil, in tritium decay, 212-213 Recombination of atoms in gas, 247 of atoms in solid, 284-286 of atoms on walls, 247 electrical conductivity and, 353 of helium ion, 271 ion-electron, 244-247 ion-ion, 253-254 of proton, 262 vibration-rotation and, 267 Recombination efficiency, of atoms, 248 Reduced mass. See Reduced molecular weight Reduced molecular weight of helium tritide, 275 of hydrogen, 3, 9, 20, 34 Refractive index of gas, 70, 92

387 in light scattering, 83-84 of saturated liquid, 70, 92 of solid, 70, 83 Regular solution, 179 Resonance states in atomic recombination, 267 Reviews on hydrogen, 8 Ritz-Paschen series, 262 Rotation of helium tritide ion, 275-277 Rotation of molecular hydrogen calculation of transitions, 324-325 energy of, 13-16, 18-31 energy levels of, 16, 18-19 enthalpy of, 21, 29-30 excitation by electrons, 236-237 heat capacity of, 15, 21-23, 28-31 hindered on a surface, 159-161 infrared spectroscopy and, 325, 328-331 magnetic moment of, 337-339 ortho-para nomenclature, 12 pressure effect on, 327-328 pumping by neighbors, 315 pure rotational spectroscopy, 324 quantum number definition, 18 Raman spectroscopy and, 325-328 relaxation times, 268 resonance states in atom recombination, 267 symmetry of states, 309-311 in two dimensions, 159-161 Rotation, of muonic hydrogen, 278-279 Rotational constant of molecule, 18-20, 159-161 Rotational energy. See Energy, rotational Rotational population adsorption effect, 158-163 calculation of, 20-24, 26 creep, effect on, 87 crystallization, effect on, 82 crystal structure effect, 73-74 electrical pumping of, 360 J = 1 -»0 transition, catalyzed, 315-317 J = 1 —»• 0 transition, self-catalyzed, 311-315 metastability of (ortho-para effect), 308-311 nuclear magnetic relaxation effect, 342-347 pressure effect on, 328 solid density effect, 79-80 solid thermal conductivity effect, 101— 104 vapor pressure effect, 49-51 Rotational selection rules with hindering, 160-162

388 in spectroscopy, 324 thermal, 29 Saturated, definition, 45 Schmidt coefficient, 146, 150 Secondary electrons, 238-239 Selection rules for helium atom, 270 for infrared spectroscopy, 324 for Raman spectroscopy, 324 rotational, hindered, 160-162 rotational, thermal, 29 vibrational, 324 Separation factor, 159-160, 162 Shear stress, of solid, 88 Silica (Silicon dioxide) atom recombination on, 248 permeability through, 371-372 x-ray absorption in, 227 Silicon permeability through, 371 sticking coefficient on, 155-156 x-ray emission from, 223-224, 226 Silicon carbide, permeability through, 371 Silicon-lithium, beta spectrum, from 208, 210 Silver electron back-scattering from, 229-230 electron energy loss in, 221-223 heat-irradiated hydrogen on, 153 hydrogen adsorption on, 157 permeability through, 371 x-ray emission from, 225 Silvera equation of state (solid), 98-99, 119, 127 Singlet states cross sections for, 235-236 in diatomic ion, 260 in excited hydrogen molecule, 235-236, 265 of helium, 270 notation for, 265 in triatomic ion, 260 Size, of hydrogen species, 257 Skin layer in plasma, 355 in weakly ionized gas, 354 Slush hydrogen, 204 Snow formation, 81, 88, 123, 154 Soda-lime glass permeability through, 371-372 x-ray emission from, 223-224 Sodium chloride, solid thermal conductivity comparison with, 102 Sodium iodide, electron energy loss in, 221 Sodium oxide, permeability effect in glass, 372

INDEX

Solid state effect. See Nuclear spin polarization Solubility, in hydrogen of carbon dioxide, 203-204 of carbon monoxide, 203-204 of helium-3, 188-190 of helium-4, 186-191 of methane, 203-204 of neon, 200-202 of nitrogen, 203 of oxygen, 203 permeability in materials and, 370 of water, 203-204 Solutions, ideal property estimation, 165-167 Sound velocity Debye temperature from, 99 in gas at low pressure, 36 longitudinal, in solid, 81 of saturated liquid, 70 of saturated solid, 81 solid thermal conductivity and, 102 transverse, in solid, 81 Space charge, 354-356, 366-367 Spectroscopic constants of helium tritide, 276 of hydrogen, 17-19, 324-326 Spin glass, 73 Spin-lattice relaxation. See Nuclear magnetic relaxation (T,) Spin-spin relaxation. See Nuclear magnetic relaxation (T 2 ) Sputtering by electrons, 154 Square potential well, 288 Stainless steel electron back-scattering estimates, 229 exchange reaction and, 302-303 heat-irradiated hydrogen on, 153 impurities from, in tritium, 332-333 methane production from, 332-333 neutron activation of, 5 oxidation effect on permeation, 371-372 permeability through, 371-372 thermal conductivity measurement and, 106 tritium self-heating in, 106 wetting of, 65 Sticking coefficient, 154-156 Stopping power, of hydrogen on beta particle, 215 Strain, on crystals, 85-87 Strain hardening modulus, 86 Stress, on crystals, 85-88 Supercooling, 89 Surface tension of clusters, 89

INDEX

of hydrogen solutions, 169 of pure component hydrogens, 63-64 Symmetric states, 308-310 T [ . See Nuclear magnetic relaxation T 2 (time constant). See Nuclear magnetic relaxation Teflon adsorption on, 157 atom recombination on, 248 tritium self-heating with, 106 wetting of, 65 Temperature scale, 47, 50-51 Tertiary hydrogen solutions, 173, 175 Thermal conductivity of constant pressure gas, 145 of constant volume gas, 144 of fluid under pressure, 144-146, 148 of low pressure gas, 33-35 phonon mean free path and, 68-69, 102-103 of saturated liquid, 67-69 of saturated solid, 100-104, 154 Thermal diffusion of low pressure gas, 41-42 Thermal diffusivity of gas (see Diffusion coefficient) of liquid under pressure, 149 of saturated liquid, 122 of saturated solid, 105, 122 Thermal resistance, solid-wall, 104-105 Thermal transpiration, 38-40 Thermal velocity of electron. See Velocity (thermal), of electrons Thorium oxide, J = 1 —>• 0 catalysis by, 316 Three-body recombination in gas, 247, 262, 267 in solid, 286 Three-phase lines, 171-173, 175 Time constant (1/e-time; see also Half-life) for AC electrical response, 357 for atomic recombination in solid, 284-285 damping of droplet oscillation, 64 diffusion coefficient relation, 306, 317 diffusion in freezing, 176 for electron collisions in gas, 358 for exchange reaction, 303, 306 for gas diffusion in a pipe, 40 for heat flow in crystals, 105 for heat flow in fluid, 124 for helium diffusion in cryostat, 196-197 for helium forbidden transitions, 270 helium tritide molecule lifetime, 277 for J = 1 0 transition, catalyzed, 317 for J = 1 -> 0 transition, self-catalyzed, 311, 314

389 nuclear relaxation (see Nuclear magnetic relaxation) rate constant relation, 299 for rotational relaxation, 268 for tritium decay, 206 for vibrational relaxation, 267-268 Titanium reactivity and permeation, 371 wetting of, 65 Titanium tritide, half-life from, 206 Toluene, electron energy loss in, 221 Transition gas region, 32, 36-40 Transpiration, thermal, 38-40 Transport properties. See Diffusion coefficient; Thermal conductivity; Viscosity Transverse relaxation. See Nuclear magnetic relaxation, T 2 Triatomic ions (hydrogen) from diatomic ions, 242-244 dipole moment, 260-261 dissociation energy of, 260 electronic energy levels, 260 in exchange reaction, 300 infrared of, 260-261 mobility of, 363 from protons, 244 quadrupole moment, 261 recombination with electrons, 245-246 shape, 260 size, 257, 260 vibration-rotation levels, 261 xenon reaction with, 301 zero-point energy, 261 Triple point of clusters, 89 energy, for saturated liquid, 121 energy, for saturated solid, 121 energy, for saturated vapor, 120 enthalpy of liquid, 119 enthalpy of solid, 119 enthalpy of vapor, 118 first freezing of hydrogen solutions, 167, 171 as fixed point, 45-48 non-ideal gas energy, 115 phase diagram mapping and, 133 pressure, 3, 47-48, 126 pressure, of neon, 199 PV of liquid, 117 PV of vapor, 116 temperature, 3, 47-48 temperature, of helium, 199 temperature, of neon, 199 Triplet states atoms from, 233, 236, 265

390 cross sections for, 235-236 in diatomic ions, 260 in excited hydrogen molecules, 235-236, 265 of helium, 270 notation for, 265 in triatomic ions, 260 Tritium analysis of pure, 332 beta particle (see Tritium beta particle) cost, 5 danger as HTO, 6 decay constant, 206 decay products, 5, 205 discovery, 4 forms as nuclear fuel, 6 half-life, 206-207 heat produced, 25, 105-106, 176, 208, 348 K-orbit capture, 207 mean decay energy, 208, 214 methane production from, 332-333 nuclear binding energy, 205 production of, 5 recoil energy of, 212-213 Tritium beta particle absorption in hydrogen and metal walls, 230 bremsstrahlung production from, 223226 energy of emitted, 220 energy per ion pair in hydrogen, 232, 235-236 energy loss mechanism in hydrogen, 232-239 energy spectrum of, 208-212 J = 1 0 catalysis by, 316-317 maximum energy of, 211 mean energy of, 208, 214 mechanical properties effect, 88 ordering destruction by, 319 phase separation effect, 169 range in hydrogen, 214-219 range in various materials, 221-223 vapor pressure effect, 154 wall scattering (see Electron scattering from metal walls) x-ray fluorescence by, 223-226 Tungsten permeability through, 371 solid thermal conductivity comparison with, 101-102 x-rays from, 223 Umlclapp process, 103 Uncertainty principle, 19, 94 Units, 369

INDEX

Uranium electron back-scattering from, 229-230 electron energy loss in, 221 in tritium purification, 196 tritium vapor pressure and, 51 x-ray emission from, 225 Valence band, 281 Vanadium reactivity and permeation, 371 x-ray edge, 224 Van der Waals force, 73 Vapor pressure of clusters, 89 of helium-4, 188 of helium-hydrogen mixtures, 186-191 HT-D 2 difference, 10, 52-53 of ideal hydrogen solutions, 165-166 of liquid, 50-52, 54, 115, 165-166 of neon, 199 non-ideal effects, 178-184 simple solid model and, 98 of solid, 49-50, 54, 165 of solid under radiation, 152-154 Velocity (drift), of electrons, 358-361 Velocity (thermal), of electrons, 241, 246, 358-359 Velocity (thermal), of molecules, 34, 38, 241, 247 Vibration, of helium tritide, 275-277 Vibration, of molecular hydrogen calculation of transitions, 324-325 energy of, 15, 17-18 energy levels of, 16-17 excitation by electrons, 236-237 in hydride ion formation, 248 quadrupole moment and, 312 quantum number definition, 17 relaxation times, 267-268 states in atom recombination, 267 Vibration, other of diatomic ions, 258 of helium tritide, 275-277 of hydride ions in lattice, 264 of ion clusters, 264 of muonic hydrogen, 278-279 of triatomic ions, 260-261 Virial coefficients definition of, 54-55 density and pressure relation, 56 of hydrogen solutions, 179 of pure hydrogens, 55 Viscosity of constant pressure gas, 145 of constant volume gas, 144 of fluid under pressure, 144-147, 150

391

INDEX

of liquid solutions, 168 of low pressure gas, 33-36 of saturated liquid, 62-63 Viscosity interaction parameter, 168 Viscous gas region. See Kinetic gas region Volume. See Density Volume interaction parameter, 167-168 Walls atom recombination on, 247-248 bubbles in cryostat and, 197 bubbles and nuclear relaxation, 337 electrons electrically from, 362 exchange catalysis by, 301-303 gas flow in pipes and, 38 He 3 n.m.r. and, 348-349 solid contact with, 82, 104-105 thermal resistance at, 104-105 thermal transpiration and, 39 Water danger of tritium in, 6 electron energy loss in, 221 in films of hydrogen, 155 infrared lines, 332-333 in pure tritium, 332 solid particles in liquid hydrogen, 203-204 solid thermal conductivity comparison with, 102 Werner band, 257, 266

Wetting, by liquid, 65 Wohl's method, 180 XeD + ion, 301 Xenon electron mobility in, 365 electronic structure of, 281 exciton of, in deuterium, 281-282 exciton with hydrogen atom in, 287 hydrogen layer on, 152 ionic character of exchange and, 300-301 quantum parameter, 10 reaction with triatomic ion, 301 reduced diffusion coefficient, 11 X-ray absorption by hydrogen, 227 by various materials, 227 X-ray production. See Bremsstrahlung; Fluorescence; Muonic hydrogen; Tritium beta particle Yield stress, 85-86 Young's modulus, 86 Zeolite, hydrogen adsorption on, 157, 162 Zero-point motion. See Energy, zero-point Zirconium reactivity and permeation, 371 x-ray edge, 224