SOLID HYDROGEN Theory of the Properties of Solid H2 , HO, and O2

SOLID HYDROGEN Theory of the Properties of Solid H2 , HD, and D2

Jan Van Kranendonk UnNersity of Toronto Toronto, Ontario, Canada

PLENUM PRESS • NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

Van Kranendonk, J., 1924-

Solid hydrogen.

Bibliography: p. Includes index. I. Solid hydrogen. I. Title.

QDl81.HIV3 1982

ISBN-13: 978-1-4684-4303-5

ISBN-13: 978-1-4684-4303-5 001: 10.1007/978-1-4684-4301-1

@ 1983 Plenum Press, New York

546/.2

e-ISBN-13: 978-1-4684-4301-1

Softcover reprint of the hardcover 1 st edition 1983 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

82-18054

* Preface

The solid molecular hydrogens are the simplest and most fundamental molecular solids. Except at ultrahigh pressures on the order of a few mega­bars, where a transition to a metallic, atomic phase is expected, these solids are true molecular crystals in which the molecules retain their identity with properties not too different from those of the free molecules. At energies below the electronic excitation energy, the thermal and spectroscopic pro­perties of these solids are determined by the translational, rotational, and intramolecular vibrational motions of the interacting molecules. The theo­retical analysis of the solid-state properties in terms of the free molecules and the intermolecular interactions forms the main topic of this book. The available detailed knowledge of the properties of the free molecules makes it feasible to carry out this program to a large extent on the basis of first principles, and this is one of the attractive features of these systems.

The solid hydrogens are dominated by quantum effects, the most out­standing property being that the rotation of the molecules is free down to the lowest temperatures, in the sense that the rotational quantum number J characterizing the rotational motion of the free molecules remains a good quantum number in all of the solid-state phases except at ultrahigh pressures. Because of the existence of the metastable ortho and para species of the homonuclear molecules H2 and D 2 , solids composed of these molecules can be regarded as mixtures of molecules in the spherically symmetric J = 0 and the elongated J = 1 states, whereas in solid HD all the mole­cules are normally in the J = 0 state. Since crystals can be grown with any desired ortho-para concentration ratio, the homonuclear solids offer a unique opportunity to study anisotropic interactions and molecular orienta­tion phenomena. Furthermore, the ortho-para transitions induced by the interaction of the nuclear moments of the molecules with the molecular fields give rise to rotation diffusion and ortho-para conversion processes

v

vi Preface

which lead to interesting time effects in the thermal and spectroscopic pro­perties. The solid hydrogens, like the solid heliums, are also translational quantum crystals, the amplitude of the zero-point lattice vibrations being an appreciable fraction of the lattice constant. These lattice vibrations play a crucial role in all the properties of these solids and, because of their effect in particular on the rotational and vibrational properties of the molecules, can be studied much more readily in these solids than in the solid heliums.

Active research on the solid hydrogens has been carried out for over fifty years, and the methods of investigation can conveniently be divided into the following four main areas: 1) measurements of the thermodynamic properties, such as the specific heat and the equation-of-state; 2) nuclear magnetic resonance and relaxation measurements; 3) infrared absorption, and Raman and neutron scattering spectroscopy; and 4) microwave ab­sorption experiments. To help orient the reader and outline the scope of the book, a brief sketch of the history of the subject is presented here. I emphasize that this is only a sketch, from my personal point of view, and I apologize for any undue omissions. More complete references are given at the end of each chapter, and in the review articles quoted here.

In view of the dominant role played by quantum effects, a good point to begin the story is the discovery of the typical quantum-mechanical property of the free rotation of these light molecules in the condensed phases, which was first observed in 1929 by J. C. McLennan and J. A. McLeod 1 in the rotation Raman spectrum of liquid H 2 . In 1930, L. Pauling2 introduced the concept of hindered rotation in solids, and he and F. Simon3 pointed out that the lifting of the threefold degeneracy of the J = 1 rotational state in the condensed phases would lead to a specific heat anomaly at low tempera­tures, which was studied experimentally by K. Mendelssohn, M. Ruheman, and F. Simon.4 Also in 1930, W. H. Keeson et al. s determined the crystal structure of solid para-Hz by X-ray diffraction to be hcp, as confirmed in recent times by a variety of spectroscopic data. In 1931, H. C. Urey, F. G. Brickwedde, and G. M. Murphy6 discovered the heavy hydrogen isotopes HO and O 2 , and their properties were summarized in 1932 in the book by A. Farkas,7 which is now out of print.

In 1939, K. Schafer8 attempted to account for the observed anomalous specific heat in terms of a crystal-field splitting, but he failed to realize that the anisotropic forces of second degree in the orientations of the molecules, which could give rise to such a crystal field, vanish accidentally when sum­med over the twelve nearest neighbors in an hcp lattice. This important property was first pointed out by T. Nagamija and H. Kisi 9 in 1951, and in more accessible form by T. Nakamura10 in 1955 in a paper which marks the beginning of the modern microscopic theory of the solid hydrogens. In this paper, the bulk of the anomalous specific heat was correctly ascribed

Preface vii

to the interaction between the J = 1 molecules, and the main contribution to this interaction was shown to be the electric quadrupole~quadrupole (EQQ) interaction. The resulting simple model, in which the cohesive and lattice vibrational properties are determined by the isotropic interactions, and the molecular orientation phenomena by the EQQ interaction, has survived to the present time as a good first approximation. The main modi­fication required is the phonon renormalization of the EQQ interaction first introduced in 1970 by A. Brooks Harries in a key paper!! on the interac­tions between the molecules in the solid, and by J. Noolandi.!2 The acci­dental vanishing of the crystal field in a rigid hcp lattice is of particular interest in connection with the splitting of the level of a J = 1 impurity in a J = 0 solid, which arises from the phonon renormalization and from anisotropies in the phonon self-energy of the level. I 3.!4 Experimentally this splitting has not yet been determined unambiguously.

The thermodynamic properties are not discussed in detail in this book. except in connection with other properties such as rotation diffusion and pair spectra, but a comprehensive discussion and extensive data and refer­ences are contained in two recent revie\'. articles. IS. I h Mention should be made here in particular of the measurements of the specific heat by J. G. Daunt et al .. of the pressure by H. Meyer et al., and of the equation-of-state by 1. F. Silvera el (/1. Topics ofmos! recent in!ere~t in this general area include the structure of the quadrupolar glass at low temperatures, and the ordering of J = 0 solids discovered in 1981 by 1. F. Silvera and R. J. Wyngaarden 17

in solid deuterium at ultrahigh pressures. The latter work has been made possible by the development of the diamond anvil technique in conjunction with laser spectroscopy, which promises to yield many further interesting results.

The work of Nakamura 10 had its origin in the second major develop­ment in the investigation of the solid hydrogens, the application of NMR spectroscopy, pioneered by J. Hatton and B. V. Rollin!8 in 1949, and by R. Reif and E. M. Purcell 19 in 1953. The work of these authors on the split­ting and shape of the NMR line due to the nuclear spins of the J = 1 mole­cules confirmed the quenching of the rotational angular momenta at low temperatures, which was shown by K. Tomita2o to be of a cooperative nature. This conclusion was consistent with the specific heat measurements carried out in 1954 by R. W. Hill and W. B. A. Ricketson,21 who were the first to observe a i.-type specific heat anomaly indicative ofa phase transition. and who also found that the transition temperature decreases linearly with decreasing concentration X of the J = 1 species. The quenching of the rotational angular momenta and the dominance of the EQQ interaction were also corroborated by the interpretation of the inelastic neutron scattering data by R. J. Elliott and W. M. Hartmann22 in 1967.

Of great importance for the further development in the investigation of

viii Preface

the solid hydrogens was the successful preparation of nearly pure J = 1 solids by D. A. Depatie and R. L. Mills23 in 1968 by the perfection of the technique of preferential adsorption. The mean-field theory of the order­disorder transition for X = 1, assuming EQQ interactions and a fcc lattice, was formulated by H. M. James and J. C. Raich24 in 1967, and the effect of J = 0 impurities on the transition was treated by N. S. Sullivan25 in 1976. The determination of the structure of the ordered phase and of the librational excitations (librons) will be discussed in connection with the spectroscopic properties.

The investigation of the NMR properties, including spin-echo and spin­lattice relaxation processes, has been successfully pursued by H. Meyer et al. in an extensive series of experiments, by J. R. Gaines et al., and by others. The corresponding theoretical development has been carried out by T. Nakamura et al. and others, and in particular by A. Brooks Harris. 26 Some of the highlights of this NMR work are: the discovery of the rotation dif­fusion in solid H2 by L. I. Amstutz, J. R. Thompson, and H. Meyer27 in 1968, and its interpretation by R. Oyarzun and myself2 8 in terms of resonant conversion of ortho-para into para-ortho pairs; the detection and inter­pretation of the striking effect of the irreversible ortho-para conversion process on the NMR spectrum by A. J. Berlinsky and W. N. Hardy29 in 1973, who also extended the theory of the conversion process originally developed by K. Motizuki and T. Nagamija30 in 1956; and the identification of the separate NMR features due to the nine inequivalent nearest neighbor pairs of J = 1 molecules in solid H2 at low X by H. Meyer et al. 31 in 1979. Some of these topics are discussed in this book, but the general theory of the NMR and relaxation properties is not covered in an effort to keep the book to a reasonable size and because excellent texts are available on this topic.

The third major field of investigation is optical spectroscopy, in partic­ular infrared absorption and Raman scattering. This field originated from the work on the collision-induced infrared absorption ofH2 and other homo­nuclear molecules, discovered in 1949 and studied extensively in the gaseous phase over the ensuing three decades by H. L. Welsh et al. 32 It was natural to pursue these investigations to progressively lower temperatures and ultimately to the solid phase. The first infrared spectra were in fact obtained in 1955,33 and the corresponding Raman spectra in 1956. 34 Subsequent work along these lines has been the main source of information about the rota­tional and vibrational energy bands in the solid hydrogens. The theory of these bands and of the corresponding bound-state complexes was developed mainly by myself,35.36 and is discussed in detail in this book. Some high­lights of the experimental and theoretical work in this field are: the discovery of the anomalous intensities in the vibrational Raman spectra of mixed

Preface ix

ortho-para crystals by E. J. Allin et al. 37 in 1965, and its interpretation by H. M. James and myself38 in terms of the delocalization of the bound vi­brational excitations; the discovery by M. Clouter and H. P. Gush39 in 1965 of the change from the hcp to the fcc structure associated with the orienta­tional ordering transition; and the study of the time effects in the infrared spectra due to the rotation diffusion by S. A. Boggs and H. L. Welsh40 in 1973. The spectroscopic approach has also been pursued by various other groups, most notably by W. N. Hardy, I. F. Silvera, and J. P. McTague, who in 1971 studied the phonon Raman spectrum,41 and in 1975 the rotational and librational Raman spectrum of oriented crystals of nearly pure para-D2 and ortho-H2.42 In conjunction with the extensive calculations on the an­harmonic libron interactions by A. Brooks Harris et at.,43 and the inter­pretation of one of the lines in the spectrum as a two-libron band,44 this work provided a firm basis for the Pa3 structure of the ordered phase, orig­inally suggested for EQQ systems by o. Nagai and T. Nakamura45 in 1960. Finally, mention should be made here of the measurement of the phonon dispersion by neutron scattering carried out by M. Nielsen46 in 1973. The phonon properties in these quantum crystals are discussed in some detail in this book, in particular the pair distribution functions relevant for the phonon renormalization.

The most recent field, opened up in 1977 by W. N. Hardy et at., is the measurement of the microwave absorption spectrum by a novel calorimetric method. 47 This technique has yielded extremely accurate data on the energy levels of J = 1 pairs in solid H 2 , and has stimulated much theoretical work48.49 on the finer details of the interactions between the molecules in the solid-state environment, such as the pseudo three-body forces. The pro­perties of clusters of J = 1 molecules in J = 0 solids is one of the most fruitful areas of research and is treated in detail in this book, followed by a discussion of the ordered phase, and of the rotation diffusion process.

The aim of the book is first of all to provide a self-contained account of the theoretical interpretation of the main properties of the solid hydrogens, in particular the spectroscopic properties in a broad sense. The presentation is aimed at the beginning graduate level, the required background being a knowledge of the basic principles of quantum and statistical mechanics, and of some elementary concepts in solid-state physics, such as crystal wave vectors and Brillouin zones. Group theory and many-body techniques are largely avoided.

I hope that the book will not only prove useful to workers in the field as a complement to the existing review articles, but will stimulate others to pursue research in this area. Much obviously remains to be done, and the aim of explaining the main properties of these solids on the basis of first

x Preface

principles is far from realized. Among the many inviting experimental areas I would like to mention are the further exploration of the complete phase diagram, in particular at intermediate concentrations, and at ultrahigh pressures, where the transition to the metallic phase has not yet been found; the vibrational overtone spectra and the ultimate dissociation of the mole­cules; the properties ofHD impurities in solid H2 and O 2, etc. Theoretically, a major challenge is the more precise prediction of the insulator-metal transition in solid H 2, HD, and O2; a more complete treatment of the many aspects of the interaction between the rotational and lattice vibrational motions, and of the double orientational and structural phase transition in ortho-H2 and para-D2; the calculation of the properties of interstellar grains of solid hydrogen, etc.

The book also serves as an introduction to the application of angular momentum and spherical tensor techniques to problems in molecular and solid-state physics. A collection of spherical tensor formulae is contained in the appendix, and together with one of the standard texts on angular mo­mentum theory, the reader should be fully equiped to tackle all problems discussed here.

Last but not least, I hope that the book will appeal to a wider circle of readers as a source of beautiful illustrations of many solid-state concepts. I maintain that the vibrational and rotational energy bands in the pure solids are the simplest and most perfect examples of Bloch states in all of solid-state physics. The fortunate circumstance that the difference between the vibrational frequencies of the ortho and para species is roughly equal to the width of the vibrational bands makes mixed ortho-para crystals equally ideal examples of energy-band impurity problems. This is also true of the mixed rotation-vibration excitations, which give rise to many interesting scattering and bound-state problems, while the rotation diffusion is a unique example of a true random walk process. I hope that in this regard the book will prove useful as a supplementary text in introductory graduate courses in molecular and solid-state physics.

I would like to use this opportunity to express my indebtedness to Professor H. L. Welsh for originally arousing my interest in this field, and for a long and fruitful period of collaboration. I also wish to express my gratitude to the Department of Chemistry at the University of California, Los Angeles, and in particular to Professor W. M. Gelbart, for their warm hospitality during a sabbatical when part of this book was written, and to Professor J. G. Daunt for his interest in this project and his gentle but pursuasive encouragement. Finally, I would like to thank Professor J. E. Sipe for many stimulating discussions and for his help with the proofreading.

Toronto, 1982 J. Van Kranendonk

Preface

References

I. J. C McLcnnan and J. A. McLeod. N(/{1/rc 123.160 (1929). 2 L Pauling. Phrs. Rer 36.430 (J(J.jO)

3. F. Simon. Lry. d. nak!. Nalun!. 9. 260 (1930). 4. K. Mendelssohn. M Ruheman. and I· Simon. Z. PhI'S. (,hcm. 815.121 (1931).

xi

5 W. H. Kcesom. J de Smedt. and H. H Mooy. Proc. Kon. Akad. 1'. Welens. Amsterdam 33. 814 (1930)

6. H C. tJrey. F. C; Bnckwcdde. and G. M. Murphy. Phrs. RCI'. 39.164 (1932). 7 A. Farkas. Orll/(ihydroyclI. parahn/ro1Icll. alld h('([rr hn/roycll. Camhridge Umv. Press

( 1935) X. K. Schafer. Z. Phn. Chem. 842 . .ISO (1939). l). T. Nagamlj<l and H. Kisi. 811.\.I·Clroll· Kellhu 39. 64 (1951).

10. T. Nakamura. Proy. Theo,.. Pln·.\ (Kyoto) 14.135 (1955). II A. Brooks Hams. Phrs. R£'I'. 8 I. I XX I (1970). 12. Noolandl. Call . .I. Phrs. 48. 2031 (1970). 1,\. J. Van Kranendonk and V. F Scars. Call . ./. Phrs. 44. 313 (1966). 14. S. LurYI and J. Van Kranendonk. Call . ./. Phl'.1 57.933 (1979). 15. P. C. Soucrs. Lawrencc Livermore Lahoratory. UCRL-52628 (1979). 16. I. F. Stlvera. Rl'I. Mod. Phys. 52. 3l)3 (1980). 17 I. F Stlvera and R. J. Wyngaarden. PhI'S. RCI'. Lett. 47 . .19 (1l)81) IX . .I Hatton and B. V. Rollin. Proc. Ror Soc. A 199.222 (1949). 1<) R.RclfandE.M.Purcell.Pln·.1 RCI.91.631(1953). 20 K Tomlta.Proc.Phys.Soc.A68.214(1955). 21 R. W Htil and B. W. A. Ricketson. Phil. May. 45. 277 (1<)54) 22. R. J. Elltott and W M. Hartmann. Proc. Phys. Soc. 90. 671 (1967). 2.1 D A. Dcpatlc and R. L. Mills. ReI'. Sci. IlIslrum. 39. 105 (1968). 24. H. M. James and J C Ralch. PhI'S. Rer. 162.649 (1967). 25 N. S. Sulilvan . .I de Phys. (Paris) 37.<)81 (1976). 26. A. Brooh Hams. Phrs. ReI'. 82 . .14<)5 (1<)70). 27 L. I Amstutz . .I R. Thompson. and H. Meyer. Phys. Rcr. LCI!. 21.1175 (1968). 28 R. Oyarllln and J. Van Kranendonk. Call . .I. Phys. 50. 1494 (1972). 2l). W. N. Hardy and A. J. Berlmsky. Phi'S. ReI'. 88.4996.5013 (197.1) . .10. K. Motlzuk! and T. NagamiJa . .I. Phys. Soc. lpn. I I. 93 (1956) . .\ I R. Schweizer. S. Washhurn. H Meyer. and A. B. Harris,.I. Lo\l' Tell1p. Phys. 37 . .109 (1<)79) . .12. H. L. Welsh. M. T. P. Inlerl/(/{. Rl'I. SCience. Phl's. ('hem. 3 . .1.1 (1972) . . '.'. W r .I Hare. E. J. Allin. and H. L Welsh, Phys. R<'1'. 99,1887 (1955) . .14. E. J. Allin, T. Feldman. and H. L. Welsh . .1. Chell/. Phrs. 24. 1116 (1l)56) . .15. J. Van Kranendonk. Phrsi('([ 25.1080 (1959) . .16. J. Van Kranendonk, ('all . .J. Phys 38,240 (1960) . .17. A. H. McKague Rosevaer, G. Whltmg. and E. J. Alltn. Call . ./. Phn 45,3589 (1967). 38. H. M. James and J. Van Kranendonk. Phrs. Rer. 164, 1159 (1967) 39 M. Clouter and H. P. Gush, PhI'S Rer. Li'I!. 15.200 (1965). 40 s. A. Boggs and H. L. Welsh, Call . ./. Phys. 51. 1910 (197.1). 41 I. F. Silvera. W. N Hardy. and J P. McTague. Phys. ReI'. 85, 1578 (1972). 42. W N. Hardy. I. F. Stlvera, and J P. McTague. Phys. Rer. 8 \2, 753 (1975). 43 C F Coli. III and A. Brooks Hams, Phl·s. ReI'. 84.2781 (1971). 44. A. J. Berlinsky and A. Brooks Harm, PhI'S. RCI' 84, 2808 (1971 ). 45. O. Nagai and T. Nakamura. 1'''0[1 '/"eor. Phy.l. (Kyolo) 24, 432 (1960). 46. M. Nielsen. PhI'S. Rer. 87,1626(1973). 47. W. N. Hardy, A J. Berlinsky. and A B. Hams. Call . .I. Phys 55.1150 (1977). 48. A. B. Hams. A. J. Berlinsky. and W N. Hardy, Can . .I. Phys. 55.1180 (1977). 49. S. Luryi and J. Van Kranendonk. Call . ./. Phrs 57, .107 (1979).

* I Contents

1. Properties of Isolated Hydrogen Molecules

1.1 The Adiabatic Approximation. 1 1.2 The Rotation-Vibration States. 4 1.3 The Molecular Multipole Moments and Pol ariz abilities 10 1.4 The Dunham Model 20 1.5 Nuclear Symmetry Species for Homonuclear Molecules 23

References. 27

2. The Intermolecular Interaction

2.1 Definition of the Intermolecular Interactions 30 2.2 The Long-Range Intermolecular Interactions 36 2.3 The Short-Range Intermolecular Interactions 42 2.4 Models for the Pair Potential 45

References. 51

3. Pure Vibrational Excitations

3.1 The fcc and hcp Structures . 53 3.2 Single-Molecule Perturbations. 56 3.3 Vibrational Energy Bands 61 3.4 Localized Vibrational States 67 3.5 The Vibrational Raman Spectrum 75

References. 86

xiii

xiv Contents

4. Rotational Excitations in J = 0 Solids

4.1 Crystal-Field Interactions . . . . . 88 4.2 Pure Rotational Excitations. . . . . . 91

4.2.l The J = 2 Band in an fcc Lattice . 92 4.2.2 The J = 2 Band in an hcp Lattice 96 4.2.3 The Rotational Energy Bands in Solid HD . 98 4.2.4 Multiple Rotational Excitations. . . 102

4.3 Mixed Rotation-Vibration Excitations. . . 105 4.3.1 The S1(0) and Q1(0) + So(O) Manifolds 106 4.3.2 The S1(0) + S1(0) Manifold. . 111

4.4 Rotation Raman and Infrared Spectra. 114 4.4.l Theoretical Foundations . . . 114 4.4.2 Pure Rotation Spectra. . . . 119 4.4.3 Mixed Rotation-Vibration Spectra 126 References . . . . . . . . . . . 128

5. Lattice Vibrations and Elastic Properties

5.l Lattice Vibrations in the Harmonic Approximation 132 5.2 Lattice Vibrations in Quantum Crystals . . . 138

5.2.1 Calculation of the Ground-State Energy 141 5.2.2 Nature of the Elementary Excitations 146

5.3 Elastic Properties and the Anisotropic Debye Model for the hcp Lattice. . . . . . . . . . . .. 156

5.4 Two-Particle Distribution Functions and Correlation Matrices . . . . . 163 References. . . . . . . . . . . . . .. 172

6. Single J = 1 Impurities in J = 0 Solids

7.

6.1 Crystal-Field Interactions in a Rigid Lattice. 6.2 Effect of Static Phonon Renormalization 6.3 Dynamic Crystal-Field Effects. . 6.4 Specific Heat and NMR Properties

References . . . . . . . .

Clusters of J = 1 Impurities in J

7.1 Models for Cluster Distributions. 7.2 Properties of Isolated Clusters. .

o Solids

174 177 181 189 195

198 201

Contents

7.3 Spectroscopy of nn Pairs of J = 1 Molecules. 7.4 Finer Details of the nn Pair Interaction 7.5 Interactions between More Distant Neighbors

References .

xv

204 211 221 223

8. The Ordered Phases

8.1 Orientational and Structural Phase Changes. 226 8.2 The Four-Sublattice Structure of Pure J = 1 Solids 228 8.3 The Order-Disorder Transition in the fcc Solids 235

8.3.1 The Molecular-Field Approximation 236 8.3.2 Effect of Short-Range Correlations 239

8.4 Librons in Pure J = 1 Solids . . . . . 241 8.5 J = 0 Impurities in J = 1 Solids. . . . 246 8.6 Ordering in J = 0 Solids at Ultrahigh Pressures 250

References . 253

9. Rotation Diffusion and Ortho-Para Conversion

9.1 Ortho-Para Conversion Processes . . . . .. 256 9.2 Rotation Diffusion at Small J = 1 Concentrations. 259

9.2.1 Diffusion of Single J = 1 Impurities at Finite Concentrations. . . . . . . . . .. 262

9.2.2 Diffusion of nn Pairs of J = 1 Impurities. 268 9.3 Rotation Diffusion at Small J = 0 Concentrations. 269 9.4 Experimental Results on Rotation Diffusion. 272

References . 275

Appendix A. Spherical Tensor Formalism 277

Appendix B. Lattice Sums 295

Index 297