Progress in Low Temperature Physics, Vol. 10

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PROGRESS IN LOW TEMPERATURE PHYSICS X

This Page Intentionally Left Blank

PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

D.F. BREWER Professor of Experimental Physics, Dean of the School of Mathematical and Physical Sciences, University of Sussex, Brighton

VOLUME X

1986 NORTH-HOLLAND AMSTERDAM. OXFORD. NEW YORK .TOKYO

0Elsevier

Science Publishers B.V., 1986

All righ1.r r1veriv.d. No purr of this publication may be reproduced. stored in a retrieval sysrem, or Irunsnrirred. in any form or by any means, electronic. mechanical, photocopying, recording t i i ofherwise, wirhout rhe prior permission of rhe publisher. Elsevier Science Publishers H . V . (North-Holland Physics Publishing Division), P.O. Box 103. loo0 AC Amsrerdutn. The Nerherlunds. Spccial rc,jiularions for readers in the USA: This publication has been registered with the Copyright Clearance Cenrer Inc. (CCC). Salem. Massachuetts. Informarion can be obtained from (he C'CC' abour conditiont under which photocopies of parts of this publication may be made in the USA. :111 olhrr c,opwighi yuesrions. including photocopying outside of rhe USA, should be referred l o rhe puhli.dicr.

ISBN: 0 444 86986 7

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ELSEVIER SCIENCE PUBLISHERS B.V. P.O. B O X 103 1MK)AC AMSTERDAM T H E NETHERLANDS S0I.E DISTRIBUTORS FOR THE USA A N D CANADA:

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(Revised for vol. 10) Gorter, C.J. (Cornelius Jacobus) Progress in low temperature physics. (Series in Physics) Vol. 10 edited by D.F. Brewer. Vol. 10 has imprint: Amsterdam; New York: North-Holland; New York, N.Y.: Sole distributors for the U.S.A. and Canada: Elsevier Science Pub. Co. has imprint: Amsterdam, North-Holland Pub. Co.; New York, lnterxience Publishers. I. Low Temperatures. 1. Brewer, D.F. (Douglas Forbes) 11. Title. 111. Series. QC278.G6 536.56 55-14533 PRINTED IN THE N F n l E R l ANDS

PREFACE

In this tenth volume of Progress in Low Temperature Physics I have, partly, at least fulfilled the intention I expressed in Volume IX, to widen the scope of the articles. The quantum fluids theme still tends to dominate, as must be expected in Low Temperature Physics. Hence the article by Fetter on vortices in superfluid 3 He, which bear both similarities and striking differences to those in superfluid 4He. Hence also the article by Rainer, though of a very different nature, on the ab initio calculation of the transition temperatures of superconductors; it is interesting to recall the history of the perhaps more difficult prediction of the superfluid transition temperature of another Fermi fluid, liquid ’He, which was first much too high and finally an order of magnitude too low. Silvera and Walraven are particularly well known not only for their original experiments on spin-polarized atomic hydrogen but also for their persistent belief in the possibility of observing it, over a period of many years which saw discouraging failures. They were not entirely alone in their faith, but they seem to have pursued their aim against experimental difficulties with notable single-mindedness and conviction. Finally, Dahm’s article on charge motion in solid helium: not so obviously a quantum fluid, but a system in which the behaviour is a typical example of low temperature phenomena. As this volume goes to press, we have heard of the sad and untimely death of Professor John Wheatley, who contributed to both Volume VI and Volume VII of this series. He was a man of enormous energy and commitment, and had the unusual distinction of being awarded both the London and the Simon Prize. He is a great loss to Low Temperature Physics. As usual, I thank colleagues for their views on topics of particular current interest in low temperature physics, those authors who wrote the articles, and the publishers, particularly Professor Peter de Chitel, for their help. Sussex, 1986

D.F. Brewer


CONTENTS VOLUME X Preface,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Ch. 1 . Vortices in rotating superfluid 3He, A . L . Fetter . . . . . . . . . . . . . .

1

1. Introduction..

...

............................................... ........

3 4

3. General propertie 4. Vortices in supe

5.2.1. Singular and nonsingular textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Textures in low magnetic fields. . . . . . ...................... 5.2.3. Textures in high magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. NMR in rotating 'He-A. . . . . . .................................. on schemes ........................... 5.4. Other 6. Discussion. ..... Note added in ..... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 56 58 63

70

Ch. 2. Charge motion in solid helium, A . J . Dahm . . . . . . . . . . . . . . . . . 73 1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2. Background.. . . . . . . . . . . ......................... 2.1. Description of vacancies and isotopic impurities in solid h 2.2. Structure of ions . . . . . . . . 79 2.2.1. Structure of ions in liquid helium ............................. 79 2.2.1.1. Positive ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2.1.2. Negative ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2.2. Ion structures in solid helium. . . . ..................... 81 ..................... 82 2.2.2.1. Positive ions . . . . . . . . . . ..................... 82 2.2.2.2. Negative ions . . . . . . . . . ..................... a4 2.3. Experimental methods of measuring mob 2.3.1. Spacecharge limited currents . . . . . . . . . . . . . . . 2.3.2. Transient space-charge limited currents . . . . . . . . . . . . . . . . . . 86

vii

...

\’Ill

CONTENTS

2.3.3. Time of flight techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4. Comparison of techniques and the effect of crystal str mobility measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Ionic mobilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Early investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Mobility measurements . . . . . . . . . . . . . . . . . ....................... 3.3. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................... 3.3.1. Negative ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1. Vacancy-assisted mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2. Surface adatom diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Positive ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1. Hole hopping . . . .......................... 3.3.2.2. Vacancy diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Species-independent mechanisms . . . . . . . . . ................. ................. 3.3.3. I . Phonon-assisted ion motion . . . . . . 3.3.3.2. Motion by a plastic flow mechanism . . . . . . . . . . . . . . . . . . . . 3.3.3.3. Vacancy-wave scattering of charged defects . . . . . . . . . . . . . . 3.3.3.4. Ions as quasiparticles . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.5. Bound ion-vacancion complexes . . . . . . . . . . . . . 3.4. Ion velocities in large electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Related phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Ion-dislocation ring complexes . ....................... 3.5.2. Motion of charged grain bounda ....................... 3.5.3. Test for the collapse of the negative-ion cavity . . . . . . 3.5.4. Search for vacancy waves . . . . . . . . . . . . . . . . . . . . . . . 3.5.5. Motion of charged droplets of liquid helium through the solid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6. Theory for positive-ion mobility in a dilute “He-’He solution . . . . . . 3.5.7. Theory for negative ions in an rf electric field . . . . . . . . . . . . . . . . . . . 4 . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. General comments . . . . . . . . . . . . . . . . . ............. 4.2. Helium three . . . . . . . . . . . . . . . ............. 4.3. Helium four . . . . . . . . . . . . . . . 5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .............

.

Ch . 3. Spin-polarized atomic hydrogen. I .F . Silvera and J . T . M . Walraven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 1.1. CJeneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.?. Rose statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 2 . Single-atom properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 2.1. Hyperfine energies and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2, Electron spin polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. ESR and NMR transitions . . . . . . . . . . . . . ....................... 3 . Interatomic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 90 91 91 92 101 101 101 104 106 106 106 108 108 108 108

114 116 116 11Y

123 123

124 125

125 12Y 132 134

139 147

149 153 156 1.56 164

162 163

CONTENTS 4. Single-atom interactions with helium surfaces . . . .

ix

. . . . . . . . . . . . . . . . . . 174 176

. . . . . . . . . . . . . . . . 177 181 182 . . . . . . . . . . 185 . . _ . . . . . . . 188 ........... 5. Experimental developments. . . . . . . . . . 189 189 189 ......................... 191 192 ....................... 192 e, vapor compression and 5.1.4.2. Wall confinement, wal 194 thermalization 5.1.5. Detection of H1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1% 196 5.1.5.1. Bolometric detection. . . . . , . , . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 200 200 5.2.1. Effective rate constants . . . . . 5.2.2. Notation for rate constants . . ............ 20 1 201 5.2.3. General equations . . . . . . . . 203 5.2.3.1. Thermal escape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.2.3.2. Decay of H i . . . . . . . 203 204 205 208 5 . 5 . Two-body surface 209 5.5.1. Deuterium . . . . . . . . . 209 209 . . . . . . . . . . . . . . . 213 5.5.2.2. The value of K ’ .. . . . . . . . . . . . 214 ......................... 217 218 218 220 ..................... 5.8.2. Impurity relaxation. . . . . . . . . 5.8.3. Relaxation and the boson nat . . . . . . . . . . 222 223 5.9. Electronic relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. Nuclear magnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 227 ...................... 5.11. Electron spin resonance . . . . . 229 235 6.1. Recombination , , , ..._.............. . . . . . . . . . . . . 236 6.1.1. Resonance r 239 6.1.2. Second-orde 24 1 24 1 255 257 259 6.1.3. Third-order recombination. . . . . .

CONTENTS

X

6.1.3.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.2. Exchange recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 262 6.1.4. Dipolar recombination . . . . . . . . . . . ..................... 280 6.1.4.1. bb-He recombination . . . . . . . . . . . . . . . . . . . . . 280 6.1.4.2. The KVS mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 6.1.4.3. The KVS mechanism on the surface . 290 6.1.5. Relationship to phenomenological experimental rates . . . . . . . . . . . . 294 6.2. Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.2.1. introduction-Volume and surface processes . . . . . . . . . . . . . . . . . . . 297 6.2.2. Spin-exchange relaxation . . . . . . . . . . . . . . . . . . 303 6.2.2.1. Relation to the rate equations . . . . . . . . . . . . . . . . . . . . . . . 309 6.2.3. Dipolar relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 6.2.3.1. Dipolar relaxation-general ..................... 312 6.2.3.2. Nuclear spin relaxation in the bulk ga 314 6.2.3.3. Nuclear spin relaxation on the surface 317 6.2.3.4. Electronic spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7 . Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 7.1. Ouantum theory of corresponding states . . . . . . . . . . . . . . . . . . . 7.2. Ground-state calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. I . The boson case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 7.2.2. The fermion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8 . Many-body static and dynamic magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . 334 8.1. Static magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. Noninteracting gases . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. The weakly interacting Bose gas . . . . . . . . . . . . . . . . 8.2. Dynamical properties: spin-waves . . . . . . . . . . . . . . . . . . . . 8.2.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Nuclear spin-waves in H i $ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9 . Many-body effects on the surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -346 9.1. Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.2. Two-dimensional supertluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Y.3. Hydrodynamic modes of two-dimensional 352 355 10. Prospects for spin-polarized hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 1 0 . 1 . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 10.2. Goals in the study o f quantum fluids. . . . . . . . . . . . . . 358 10.2.1. Compression of bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 10.2.2. Traps for low-field seekers . . . . . . . . . . . . . . . 362 10.2.3. Traps for high-field seekers . . . . . . . . . . . . . . . 363 10.2.4. Two-dimensional superfluidity .......................... 365 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ch . 4 . Principles of ah initio calculations of superconducting transition temperatures. D . Rainer . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Many-body aspects diagram analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Technical preliminaries notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 377 379

.

.

CONTENTS

xi

2.2. Classification and calculation of self-energy diagrams. . . . . . . . . . . . . . . . . . . . 383 3.1. Bloch-function representation ........................ 4. Band-structure theory and the electron-phonon interaction . . . . . . . . . . . . . . . . . . 4.1, The electron-phonon coupling parameters 5 . Strong-Coupling Theory of the transition temperature . . . . . . . . . . . . . . . . . . . . . . . 5.1. The linearized Eliashberg equ 5.2. Calculation of T , from Eliashberg's equations . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion ...................................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 394 40 1 402 41 1 412 411 419 42 1

Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,439

3. The low-energy equations

.............................................


CONTENTS OF PREVIOUS VOLUMES

Volumes Z-VZ, edited by C.J. Gorter

Volume I (1955) The two fluid model for superconductors and helium 11, C.J. Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of quantum mechanics to liquid helium, 11. R.P. Feynman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Rayleigh disks in liquid helium 11, J.R. Pellam . . . . . Oscillating disks and rotating cylinders in liquid helium IV. 11, A.C. Hollis Hallett . . . . . . . . . . . . . . . . . . . . . . . . . The low temperature properties of helium three, V. E.F. Hammel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Liquid mixtures of helium three and four, J.M. Beenakker and K.W. Taconis . . . . . . . . . . . . . . . VII. The magnetic threshold curve of superconductors, B. Senn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. The effect of pressure and of stress on superconductivity, C.F. Squire . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Kinetics of the phase transition in superconductors, T.E. Faber and A.B. Pippard.. . . . . . . . . . . . . . . . . . Heat conduction in superconductors, K. Mendelssohn X. XI. The electronic specific heat in metals, J.G. Daunt . XII. Paramagnetic crystals in use for low temperature research, A.H. Cooke . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII. Antiferromagnetic crystals, N.J. Poulis and C.J. Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV. Adiabatic demagnetization, D. de Klerk and M.J. Steenland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv. Theoretical remarks on ferromagnetism at low temperatures, L. Nkel . . . . . . . . . . . . . . . . . . . . . . . . . XVI . Experimental research on ferromagnetism at very low temperatures, L. Weil . . . . . . . . . . . . . . . . . . . . . . . . . XVII. Velocity and absorption of sound in condensed gases, A. van Itterbeek. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII. Transport phenomena in gases at low temperatures, J. de Boer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.

xiii

1-16 17-53 54-63 64-77 78-107 108-137 138-150 151-158 159-183 184-20 1 202-223 224-244 245-272 272-335 336-344 345-354 355-380 381-406

xiv


Volume I t (1957)

I.

Quantum effects and exchange effects on the thermodynamic properties of liquid helium, J . de B o e r . . Liquid helium below 1"K, H.C. Kramers . . . . . . . . . . 11. Transport phenomena of liquid helium I1 in slits and 111. capillaries, P. Winkel and D.H.N. Wansink . . . . . . . . Helium films, K.R. Atkins. . . . . . . . . . . . . . . . . . , . . . IV. Superconductivity in the periodic system, V. B.T. Matthias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron transport phenomena in metals, VI. E.H. Sondheimet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Semiconductors at low temperatures, V.A. Johnson and K. Lark-Horovitz. . . . . . . . . . . . . . . . . . . . . . . . . . VIII. The De Haas-van Alphen effect, D. Shoenberg . . . . IX. Paramagnetic relaxation, C.J. Gorter . . . . . . . . . . . . . X. Orientation of atomic nuclei at low temperatures, M.J. Steenland and H.A. Tolhoek . . . . . . . . . . . . . . . Solid helium, C. Domb and J.S. Dugdale. . . . . . . . . . XI. XI1 . Some physical properties of the rare earth metals, F.H. Spedding, S. Legvold, A.H. Daane and L.D. Jennings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII. The representation of specific heat and thermal expansion data of simple solids, D. Bijl . . . . . . . . . . . XIV. The temperature scale in the liquid helium region, H. van Dijk and M. Durieux.. . . . . . . . . . . . . . . . . . .

1-58 59-82 83-104 105- 137 138-150 151-186 187-225 226-265 266-291 292-337 338-367 368-394 395-430 431-464

Volume III (1961) Vortex lines in liquid helium 11, W.F. Vinen . . . . . . . Helium ions in liquid helium 11, G. Careri.. . . . . . . . The nature of the A-transition in liquid helium, M.J Buckingham and W.M.Fairbank . . . . . . . . . . . . . IV. Liquid and solid 'He, E.R. Grilly and E.F. Hammel 3 He cryostats, K.W. Taconis . . . . . . . . . . . . . . . . . . . . V. VI. Recent developments in superconductivity, J. Bardeen and J.R. Schrieffer . . . . . . . . . . . . . . . . . . VII. Electron resonances in metals, M.Ya. Azbel' and I.M. Lifshitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Orientation of atomic nuclei at low temperatures 11, W.J. Huiskamp and H.A. Tolhoek . . . . . . . . . . . . . . . IX. Solid state masers, N. Bloembergen . . . . . . . . . . . . . .

I.

11. 111.

1-57 58-79 80- 112 113- 152 153-169 170-287 288-332 333-395 396-429


X. XI.

The equation of state and the transport properties of the hydrogenic molecules, J.J.M. Beenakker . . . . . . . Some solid-gas equilibria at low temperatures, Z. Dokoupil .................................

xv

430-453 454-480

Volume N (1964)

Critical velocities and vortices in superfluid helium, V.P. Peshkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Equilibrium properties of liquid and solid mixtures of helium three and four, K.W. Taconis and R. de Bruyn Ouboter.. . . . . . . . . . . . . . ........... 111. The superconducting energy gap, D.H. Douglas, Jr. and L.M. Falicov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anomalies in dilute metallic solutions of transition IV. elements, G.J. van den B e r g . . . . . . . . . . . . . . . . . . . . Magnetic structures of heavy rare-earth metals, V. Kei Yosida.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic transitions, C. Domb and A.R. Miedema VI . VII . The rare earth garnets, L. Neel, R. Pauthenet and B. Dreyfus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Dynamic polarization of nuclear targets, A. Abragam and M. Borghini.. . . . . . . . . . . . . . . . . . Thermal expansion of solids, J.G. Collins and IX. G.K. White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 1962 3He scale of temperatures, T.R. Roberts, X. R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde I.

1-37 38-96 97- 193 194-264 265-295 296-343 344-383 384-449 450-479 480-514

Volume V (1967)

I. 11. 111.

IV. V.

The Josephson effect and quantum coherence measurements in superconductors and superfluids, P.W. Anderson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissipative and non-dissipative flow phenomena in superfluid helium, R. de Bruyn Ouboter, K.W. Taconis and W.M. van Alphen. . . . . . . . . . . . . . Rotation of helium 11, E.L. Andronikashvili and Yu.G. Mamaladze . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of the superconductive mixed state by neutrondiffraction, D. Gribier, 3 . Jacrot, L. Madhav Rao and B. Farnoux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiofrequency size effects in metals, V.F. Gantmakher . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-43 44-78 79-160 161- 180 181-234

xvi

VI. VII.


Magnetic breakdown in metals, R.W. Stark and L.M. Falicov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic properties of fluid mixtures, J.J.M. Beenakker and H.F. P. Knaap . . . . . . . . . . . . .

235-286 287-322

Volume VI ( I Y70)

I

Intrinsic critical velocities in superfluid helium, J.S. Langer and J.D. Reppy . . . . . . . . . . . . . . . . . . . . 11. Third sound, K.R. Atkins and I. Rudnick . . . . . . . . . Experimental properties of pure He3 and dilute solu111. tions of He3 in superfluid HeJ at very low temperatures. Application to dilution refrigeration, J.C. Wheatley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I v. Pressure effects in superconductors, R.I. Boughton, J.L. Olsen and C . Palmy . . . . . . . . . . . . . . . . . . . . . . . V. Superconductivity in semiconductors and semi-metals, J.K. Hulm, M. Ashkin, D.W. Deis and C.K. Jones VI. Superconducting point contacts weakly connecting two superconductors, R. de Bruyn Ouboter and A.Th.A.M. de W a e l e . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Superconductivity above the transition temperature, R.E. Glover I11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Critical behaviour in magnetic crystals, R.F. Wielinga IX . Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities, G.R. Khutsishvili The international practical temperature scale of 1968, X. M. Durieux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-35 37-76

77-161 163-203 205-242

243-290 29 1-332 333-373 375-404 405-425


xvii

Volumes VII-IX, edited by D.F. Brewer

Volume VII (1978) 1. Further experimental properties of superfluid 'He, J.C. Wheatley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Spin and orbital dynamics of superfluid 3He, W.F. Brinkman and M.C. Cross.. . . . . . . . . . . . . . . . . . . . 3. Sound propagation and kinetic coefficients in superfluid 3 He, P. Wolfle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The free surface of liquid helium, D.O. Edwards and W.F. Saam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Two-dimensional physics, J.M. Kosterlitz and D.J. Thouless. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. First and second order phase transitions of moderately small superconductors in a magnetic field, H.J. Fink, D.S. McLachlan and B. Rothberg Bibby . . . . . . . . . . . . . . 7. Properties of the A-15 compounds and one-dimensionality, L.P. Gor'kov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Low temperature properties of Kondo alloys, G. Gruner and A. Zawadowski. . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . 9. Application of low temperature nuclear orientation to metals with magnetic impiirities, J. Flouquet . . . . . . . . . . .

1-103 105-190 191-281 283-369 371-433 435-516 517-589 591-647 649-746

Volume VIIZ (1982) 1. 2. 3. 4.

Solitons in low temperature physics, K. Maki . , . . . . . . . . Quantum crystals, A.F. Andreev . . . . . . . . . . . . . . . . , . . . Superfluid turbulence, J.T. Tough . . . . . . . . . . . . . . . . . . . Recent progress in nuclear cooling, K. Andres and O.V. Lounasmaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-66 67-132 133-220 221-288

Volume fX (1986) 1. Structure, distributions and dynamics of vortices in helium 11, W.I. Glaberson and R.J. Donnelly . . . . . . . . . . . . . . . . 2. The hydrodynamics of superfluid 3He, H.E. Hall and J.R. Hook ...................................... 3. Thermal and elastic anomalies in glasses at low temperatures, s. Hunklinger and A.K. Raychaudhuri . . . . . . . . . .

1-142 143-264 265-344


CHAPTER 1

VORTICES IN ROTATING SUPERFLUID 3He* BY

Alexander L. FETTER Institute of Theoretical Physics, Department of Physics Stanford University, Stanford, C A 94305, USA

*Research supported in part by the National Science Foundation, under Grant No. DMR-81-18386.

Progress in Low Ternperorwe Physics, Volume X Edited by D.F. Brewer 0Elsevier Science Publishers B.V., 1986 1

Contents I . Introduction. . . . . . . . . . . . . . . . . 2. Vortices in rotating superfluid 'H 3. General properties of superfluid 4. Vortices in superfiuid 'He-B.. . . 4.1. Structure of an individual vortex line. . . . . . ..............

4.3. NMR in a rotating container 5. Vortices in superfluid 'He-A . . .

5.3. NMR in rotating 'He-A , . 5.4. Other detection schemes . . . . . . . . 6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

2

1. Introduction

Since the original suggestion by Onsager (1949) and Feynman (1955) that vortices in superfluid 4He should be quantized, experimentalists and theorists have studied their properties in great detail. It is particularly appropriate that this series reviews the current status of vortices in rotating superfluid 3He, because Feynman’s original article appeared in the first volume of this series. Subsequent volumes have also contained many related articles (Vinen 1961, Peshkov 1964, Anderson, 1967, Andronikashvili and Mamaladze 1967, Langer and Reppy 1970, Tough, 1982). However, all these articles have considered only 4He or superconductors, and the present review will deal rincipally with the newer and more exotic case of vortices in superfluid He. The essential content of Feynman’s suggestion involves two specific points. First, the idea of some sort of order parameter that characterizes the state of the superfluid condensate. In the familiar case of superfluid 4 He or a superconductor, this order parameter is a single complex amplitude. For superfluid 3He, the order parameter is much more complicated, which accounts for the subject’s fascination. In general, the order parameter varies smoothly in space, but it may have singularities or defects, where its value is not defined. The vortex core will be seen as a particular example of a line defect, which connects it directly to one area of current and active research in modern condensed-matter physics (Mermin 1979). Second, the idea that the order parameter must be a single-valued function of position; in particular, it must return to the same value if one traverses a closed path in the fluid. As will be seen below, this requirement immediately yields the quantization of circulation in superfluid 4He, and the corresponding quantization of magnetic flux in a superconductor. Similar properties hold for superfluid 3He, but they are considerably more complicated than those for 4He. During the 1960s, many important experiments clarified the properties of vortices in 4He. By the early 1970s, however, the subject became less active, and many low-temperature experimentalists turned to superfluid 3 He after its discovery in 1972 at Cornell (Osheroff et al. 1972). Correspondingly, theorists concentrated on the new superfluid, which offered unique possibilities. Although the question of vortices and rotating

s

3

4

A.L. FETTER

[Ch. 1 , 02

superfluid 3He had been raised soon after the original discovery, it was not until the early 1980s that such experiments were performed at the Helsinki University of Technology. Even the first preliminary data showed a much richer variety of phenomena than had been generally expected, and close interplay between theory and experiment has been essential in unravelling this fascinating subject (which is now largely, but not wholly, understood). The present review will concentrate on the static properties of vortices, which have been explored in considerable detail. In contrast, the dynamical behavior has not been widely studied experimentally or theoretically. This situation is similar to that for rotating 4He, where the clear picture of equilibrium properties contrasts with the uncertainty concerning nonequilibrium phenomena. Even the simplest question of vortex nucleation remains unanswered in detail, and it will doubtless require considerable effort before similar problems concerning 3He are solved. This review will start with a brief summary (section 2) of the properties of vortices in 4He to provide a basis for comparison with the more complicated situation in 3He. Section 3 reviews the relevant general properties of this superfluid, including the order parameter and the Ginzburg-Landau (GL) description. Section 4 then describes the vortices in the B phase, which are more like to those in superfluid 4He; in particular, they both have irrotational flow outside the vortex core. Vortices in the A phase are considered in section 5 , where the question of nonsingular and singular vortices is studied in detail. 2. Vortices in rotating superfluid 4He

When liquid ‘He is cooled below T, = 2 . 2 K , it undergoes a phase transition to a new ordered state that acts like a mixture of two interpenetrating components-the normal fluid with density p, and finite viscosity, and the superfluid with density p, and zero viscosity. This behavior represents a second-order transition, in which the order appears continuously as the temperature falls below T,. It reflects the onset of Bose condensation, in which a finite fraction of the particles starts to occupy the lowest single-particle state. As in an ideal Bose gas, this ordering is best characterized by an effective single-particle wave function T(r) = lly(r)l exp[i@(r)], which here becomes the order parameter. Since this function is complex, it has both a magnitude I T1 and a phase @. For a bulk uniform system, the magnitude is fixed by the temperature, but the phase remains a free parameter. Thus the order parameter in superfluid ‘He has, in effect, one scalar degree of freedom, analogous to the angle that specifies the position on a circle.

Ch. 1 , 62)

VORTICES IN ROTATING SUPERFLUID 3He

5

In a uniform sample, the phase is a spatial constant, and all values from 0 to 27r are equivalent, producing a set of degenerate states. Slow spatial variations in @constitute low-energy excited states, similar to spin waves in a ferromagnet. In analogy with the usual quantum mechanics of oneparticle states, the free-energy density associated with nonuniform states is taken in the form IVFI’, which has contributions both from changes in the magnitude (Vl’PI)’ and in the phase I?P1*(V@)’. In the particularly simple case of a plane wave, where ly 0: exp[ik * r] the phase is just k. r. Thus the velocity associated with the ordered state (namely the superfluid velocity u s )obeys the relation us= p/m, = hk/m, = (h/m,)V@. This last form holds in general, leading to the identification us= (fL/m,)V@

.

(2.1)

Thus us is determined by the local properties of the order parameter. An equivalent and useful viewpoint is to consider two nearby points at r and r + Sr. The change in phase in moving from one point to the other is proportional to Sr, with the coefficient defining us through

The usual relation for the gradient of a scalar function then reproduces eq. (2.1). Note that the free-energy density also has the expected form, with a contribution ;p,u:, where p, is proportional to (P(’. Equation (2.1) has several important consequences: (1) In any local region where V@ is bounded, the superfluid velocity is irrotational curl us= 0 ,

(2.3)

the only exception being singularities where the phase is not defined, (2) The superfluid circulation about some closed path C is h/m, times the change in phase about C . Since the order parameter must be single-valued throughout the fluid, this change in phase must be an integer p times 27r, leading to the familiar quantization of superfluid circulation in units of a fundamental quantum K = h/m, = cm’/s,

where p is any positive or negative integer, or zero. Equivalently, p is the winding number of the phase angle associated with the contour C ;if p differs from 0, then @ must have a singularity somewhere in C.

[Ch. 1 , 82

A.L. FE’ITER

6

The simplest example is to take @ = ~ I # J + Go, where # is the azimuthal angle of cylindrical-polar coordinates. From eq. (2. l ) , the corresponding superfluid velocity becomes us= ( pfi /m4& ,

(2.5)

which characterizes a p-fold quantized vortex line on the symmetry axis. This configuration is generally visualized by drawing circular streamlines of the velocity field, but a more precise representation is to draw a field of unit vectors oriented at the local angle @ with respect to some fixed direction. Figure 1 illustrates this situation for the cases p = 2 1 ; note that an additive constant does not affect us, so that all three cases for p = 1 represent the same flow despite their very different appearances. In the presence of such a quantized vortex, the free-energy density is proportional to the quantity I?PI’(plr)’. Since us becomes infinite as r - 0, the magnitude of the order parameter \!PI must vanish at that point, and the balance between the various energies determines the coherence length (, which is the characteristic size of the vortex core. For a singly quantized vortex with p 2 = 1, this magnitude increases linearly inside the region r 6 (, and assumes its asymptotic value for r + m . The total energy associated with the vortex is largely the kinetic energy of circulating superflow; use of the approximate energy density f, = p p f leads to the energy per unit length of vortex line mps(pfrlm4)2ln(a/(), where the logarithmic divergence has been cut off at 6 (near the core) and at some

4

/I\

\I/ (Cl

(d l

Fig. 1. Phase-angle representation of vortices, (a), (b) and (c) all represent p = +1 vortices, (d) represents a p = - 1 vortex (Fetter et al. 1983, reprinted by permission of the authors and the American Physical Society).

Ch. 1 , $21

VORTICES IN ROTATING SUPERFLUID 'He

7

large distance a. It is clear that the order parameter is not well-defined at the center of the vortex, since the phase @depends on how one approaches the axis. Thus this state constitutes a singularity or defect, associated with the node in the order parameter. In 4He, the simple complex scalar form of 'Pensures that it must vanish at the center, because there are no other ways to avoid a divergent energy. In superfluid 3He, in contrast, the much more complicated form of the order parameter allows many different possible forms for the vortex core. Vortices in superfluid 4He can be created in various ways, such as the passage of heat currents o r ions through the sample. In the present context, however, the most important method is to rotate the fluid. In this case, the normal fluid executes solid-body rotation with

where d2 is the angular velocity of the container; this normal flow has vorticity 2a,and it is readily proved to minimize the associated free energy (see below). In contrast, the superfluid must remain irrotational and hence, cannot assume the same equilibrium flow as the normal fluid. Instead, it forms an array of singly quantized vortex lines, aligned parallel to 0. To verify these assertions, consider a rigid container rotating with angular velocity 0. Since the walls are microscopically rough, a true equilibrium state can be described only in the rotating frame, since, otherwise, the walls constitute a time-dependent perturbation. In this noninertial frame, the free-energy density f differs from that in the laboratory by a term whose integral is a - L , where L is the total angular momentum. If f is the original free-energy density [it contains the hydrodynamic contributions k(pnu: + p,uf), as well as any additional quantum mechanical contributions arising from (VlPI)'], then

f=f-a-(rxj),

(2.7)

where the total current is

It follows directly that the minimum of f with respect to u, occurs for solid-body rotation. Simple manipulations allow f to be rewritten as

where additional (nonhydrodynamic) terms have been omitted. The last

x

[Ch. 1, 12

A.L. FEITEK

term here leads to an additive constant that depends only on the shape and angular velocity of the container; it will henceforth be omitted. The minimization of the free energy f’ with respect to the superfluid velocity is equivalent to minimizing the mean-square difference (u, - u , , ) ~ ; equivalently. eq. (2.6) shows that us- u, is just the transformed superfluid velocity as seen in the rotating frame. Since u, is irrotational, this quantity cannot vanish everywhere, and the superfluid instead mimics solid-body rotation by forming a uniform array of rectilinear vortices parallel to the rotation axis. If Cis a contour encircling Nc vortices in an area A c, then eqs. (2.1) and (2.3) show that

f

(2.10)

u;ds=p~N,.

To minimize the free energy in the rotating frame, eq. (2.10) must also equal the line integral of u, around the same contour. By Stokes’ theorem, this quantity is just 2f2A,-, which then gives the mean vortex density (2.11) proportional to the angular velocity R. Experiments on rotating 4He indicate that only singly quantized vortices occur ( p = l), with a vortex density about 2000cm-’ for an angular speed of 1 rad/s. Thus the intervortex spacing (of order 100pm for this typical rotation speed) far exceeds the core size (a few A ) for any conceivable experiment. For sufficiently slow rotation, the intervortex separation can exceed the linear dimensions of the container, and the superflow then remains irrotational everywhere. Indeed, for a slowly rotating circular cylinder, the superfluid is stationary, since the rotating walls exert no force on the nonviscous component. This situation persists up to a critical angular velocity l&, [ = ( h / m , R ’ ) ln(R/() for a circular container of radius R]. Above R,, , the system makes a sequence of discrete transitions between states with increasing number of vortices, and, for R @, , it behaves like an unbounded fluid. Since Ocl is usually very small, this latter limit is the most important in practice. It also follows quite generally that l2 4 O,, (=h/2m45’ = 10” rad/s), defined as the angular velocity where the vortex cores start to overlap. The preceding considerations are independent of the particular arrangement of the vortices. In an infinite system, they form a lattice that rotates rigidly with the container (this rigid motion must hold for any equilibrium configuration. since, otherwise. the mutual friction bctween the normal

*

Ch. 1. 92)

VORTICES IN ROTATING SUPERFLUID 3He

Y

and superfluid components produces dissipative drag forces on the vortices). Tkachenko (1965) has studied the phase function @ of the order parameter for such lattices. Since each vortex moves with the local superfluid velocity at that point, this overall rotation in the lab frame implies that neither V@(= u s ) nor @ itself can be a periodic function of position. The net rotation can be eliminated, however, by transforming to the rotating frame, where the lattice appears stationary. Hence the quantity us- u, is a periodic function throughout the vortex lattice, and it is sufficient to consider a unit cell containing only a single vortex (assuming a Bravais lattice of p-fold quantized vortices). If A, is the area of the cell, it follows from eq. (2.11) that the function

V@

- (.rrp/A,)z^ X

r = (m,/h)(u,

(2.12)

- u,)

is periodic; it depends solely on the properties of @. As a corollary, the free-energy density in the rotating frame also is a periodic function with the symmetry of the lattice, and it suffices to treat only a single unit cell. For simplicity, consider an equivalent circular Wigner-Seitz (WS) cell with area A, = nV-’= .rrpfi/m,R = p ~ l 2 R,

(2.13a)

and radius

Inside this cell, the true relative velocity is approximated by an axisymmetric flow us- u, = ub, where -1

u = fip/m4r - Rr = (hp/rn,)(r

- r/a,2).

(2.14)

Note that u = O on the boundary of the WS cell. The resulting hydrodynamic contribution to the free energy per unit cell is easily found to be

F,

=

I

d2r f = .rrps(ph/m4)2[ln(a,/~) -

$1

C

= ( p , p2K2/47r)[ln(ac/() -

$1 ,

(2.15)

where the integral has been cut off at r = ( to avoid a divergence in the core. Tkachenko (1965) has shown that the exact result for square and triangular lattices differs by only.smal1 corrections, and that the triangular lattice is the stable configuration.

10

A.L. FETTER

[Ch. 1. $3

3. General properties of superfluid ’He

In the boson system ‘He, the order parameter is related to the condensate amplitude. Such a direct interpretation cannot apply to fermion systems because of the Pauli principle. Nevertheless, the evident similarities between superfluid ‘He and metallic superconductors stimulated the introduction of corresponding effective order parameters to characterize the condensation of superconducting electrons. The precise meaning of this order parameter became clear with the discovery of Cooper pairing and the subsequent BCS theory of superconductivity. Under the influence of a weak attractive interaction, two particles near the Fermi surface in a degenerate Fermi system can form a bound state (the Cooper pair). In metals, the spin $ electrons combine in a singlet state (total spin zero) [except, perhaps, for certain exotic “heavy fermion” systems, where triplet, p-wave pairing has been suggested (see, for example, Stewart 1984)j. The antisymmetry of the electron wave function then requires that the pairing occurs in even relative orbital states, and all experiments suggest that the pairing is, in fact, in s-waves. As a result, the amplitude for the Cooper pairs in metals has no internal degrees of freedom (since L and S both are zero) and can be represented by a single complex function, closely analogous to that for ‘He. Although the detailed behavior of superconductors is complicated by the presence of the electronic charge, one class of superconductors (type-11) indeed behaves very much like superfluid ‘He. Since particle currents in a superconductor are also electrical currents, the resulting vortices are magnetic, with the quantization of circulation replaced by the quantization of magnetic flux. Furthermore, vortices in superconductors appear in response to an applied magnetic field, which is the electromagnetic analog of rotation in the neutral system (as is familiar from Larmor’s theorem). Many experiments have demonstrated that these flux lines generally form triangular arrays, as is the case for 4He. One important difference. however, is the coherence length, which is typically hundreds of 8, in type-I1 superconductors, instead of the few A in ‘He. Since the flux-line density is proportional to the applied magnetic field [see eq. (2.11) for the similar effect of angular velocity], the cores of the flux lines will overlap when the intervortex separation is comparable to the coherence length; such fields, which are readily attainable in many cases, drive the system normal. In contrast, the corresponding angular velocities for rotating ‘He are of order 10’’ rad/s. so that those vortices always constitute a lowdensity system. The discovery of new ultra-low-temperature phases of 3He in 1972

Ch. 1, 83)


11

(Osheroff et al. 1972) stimulated an enormous effort to identify them and to understand their properties (see, for example, Leggett 1975, Wheatley 1975, Anderson and Brinkman 1978, Lee and Richardson 1978). Below 2.7 mK, the system entered a new phase, accompanied by a specific-heat anomaly reminiscent of that seen in metallic superconductors. Since the interatomic potential between He atoms has a strong short-range repulsive core, Cooper pairing was not expected in relative s-waves, and various authors had suggested either p-wave pairing or d-wave pairing. The observed magnetic properties of the new phases could not readily be explained by singlet Cooper pairs, however, so that triplet pairing (S = 1) was rapidly accepted. The Pauli principle then requires an odd orbital state, and all present evidence confirms that superfluid 3He consists of triplet p-wave Cooper pairs, with S = 1 and L = 1. The presence of internal quantum numbers associated with the unit spin and orbital angular momenta means that superfluid 3He has a much richer structure than that found in superconductors. This difference is already evident in the appearance of two distinct bulk phases (A and B); it will be particularly important in understanding the rotational behavior of the two phases. To characterize the order parameter of superfluid 3He, it is convenient to introduce the amplitude for a Cooper pair, which now must have nine independent complex elements to account for the internal degrees of freedom. Although it is sometimes useful to couple the two angular momenta to form states of definite total angular momentum ( J = 0 , 1 , 2 ) , physically relevant perturbations, such as an external magnetic field, rotation, or boundaries, all tend to introduce preferred directions that destroy spatial isotropy. Thus it will be more convenient to retain a Cartesian basis for the separate unit vectors S and L, introducing a 3 x 3 matrix A * , , whose first and second indices refer to spin and orbital coordinate axes, respectively. Near T,, which depends on the pressure, it is natural to rely on a Ginzburg-Landau (GL) formalism, which has the advantage of describing both the A and B phases. The dominant terms in the free-energy density are invariant under separate rotations of the spin and orbital coordinates, which greatly restricts the allowed form. There is one bulk quadratic contribution

f, = - ( Y A ; ~ A,~ ;

(3.la)

where LY = ;N(O)(l- T / T , ) , N ( 0 ) is the density of states of one spin projection at the Fermi surface, and repeated dummy indices are summed from 1 to 3. In contrast, there are five bulk quartic contributions

A.L. FETTER

12

[Ch. 1 . 83

p , are a set of pressure-dependent

(but temperature-independent) parameters that are only partially determined by experiments. They differ from the weak-coupling values

by strong-coupling corrections that fix the overall phase diagram and several thermodynamic properties of the phase transition. Minimization of the bulk free energy for a uniform system shows that the bulk order parameter has a magnitude (A1= A = ( c r / B ) ” ’ , where denotes some linear combination of the five quartic parameters. In addition. the free-energy density has a term associated with spatial variations in the order parameter

B

where K, are a set of constants. In the weak-coupling limit, they all have the same value K = N ( O ) t i with 5, the BCS coherence length (pressure dependent and of ordei 200 A). Since strong-coupling corrections are thought to be negligible, only the single parameter K will be used here. The balance between the bending energyf, and the bulk quadratic energy fi determines the G L coherence length [( T ) = ( K / a ) ” ’

.

(3.4a)

which can easily be rewritten

t(T) = (

&(1 - T / T c ) - ’ ! 2 ,

(3.4b)

It plays an important role in setting the characteristic length scale of the various vortex structures. Two other contributions to the free-energy density must be included. One arises from the weak magnetic-dipole interaction between the ’He nuclei, enhanced by the quantum coherence of the Cooper pairs (Leggett 1973, 1974). Near T , , a direct calculation (see, for example, Brinkman and

Ch. 1, 841

VORTICES IN ROTATING SUPERFLUID ’He

13

Cross 1978, but note the slightly different notation) gives the expression

where g, is a temperature-independent dipole coupling constant. Note that the first two terms of eq. (3.5) explicitly couple the spin and orbital indices, so that f, is not invariant under separate rotations of the spin and orbital coordinate systems, unlike the previous bulk contributions to the freeenergy density. The balance between fk and fd defines the dipofe length

L,

=(

K/g,y,

(3.6)

cm; it sets the characteristic scale for deviations from the of order bulk order parameter caused by f,. The other important term in the bulk free-energy density is the magnetic contribution arising from the reduced susceptibility in the superfluid relative to that in the normal Fermi liquid. A detailed calculation near T, gives the result (3.7) where g, is a temperature-independent magnetic coupling constant. Comparison of eqs. (3.5) and (3.7) defines the characteristic magnetic dipole field (3.8) which is of order 25 G. For fields H %- H,, it follows that f, dominates f,.

4. Vortices in superfluid 3He-B

To understand the behavior of vortices in the B phase, it is essential to consider first the detailed form of the order parameter, both asymptotically and near the center of an isolated vortex line. All current calculations of this structure have relied on the GL formalism, or extended versions (Passvogel et a!. 1982) valid to next order in T, - T. This matter is discussed in section 4.1. The next question (section 4.2) is the overall texture of the B phase in a stationary container, and how that texture is affected by rotation (namely, by the presence of vortices). Since the only currently available method for detecting vortices relies on NMR, it is necessary to consider the interaction of an external magnetic field with a

A.L. FETTER

14

(Ch. 1, 84

single vortex (section 4.1) and also its effect on the overall texture (section 4.2). Section 4.3 then treats the NMR of the rotating B phase. 4.1. STRUCTURE OF

AN INDIVIDUAL VORTEX LINE

In the bulk uniform B phase, the equilibrium order parameter is simply the unit matrix or any matrix obtained from it by a relative rotation of the spin and orbital coordinates. Among the various ways to specify a rotation in three-dimensional space, it is simplest for the B phase to introduce the axis ri and the angle 8 of the equivalent single rotation. Given these parameters, the actual rotation matix has the explicit form

R P i = SPi cos 8 + n , n i ( l

- cos 8 ) -

~,~ sin, 8i ,,

(4.1)

and the corresponding bulk order parameter becomes

A,; = A(T)R,, .

(4.2)

Here A(T) is a temperature-dependent amplitude, given in the GL regime by

where pij denotes the sum of the corresponding p's [see eq. (3.1b)l. Any order parameter of the form (4.2) is a local minimum of the bulk GL free energy, obtained by integrating eq. (3.1) over the sample. Thus the B-phase order parameter is massively degenerate, and various additional small terms in the free-energy density are needed to determine its actual form. In this context, the most important is the dipole energy density fd in eq. (3.5). For a bulk uniform sample, minimization of fd fixes the rotation angle 8 to be arccos(- ) = 104", but the unit vector ri remains undetermined. This question will be studied in section 4.2, in connection with the determination of the li texture in a container. In the presence of slowly varying superflow, the order parameter for the B phase acquires a phase factor exp(i@), analogous to that for superfluid 4 He (compare section 2). Thus the corresponding order parameter becomes

a

A P i= A(T) ei@'RPi

.

(4.4)

Since the effective mass of a Cooper pair is 2m3, the superfluid velocity is given by

Ch. 1 , 541


us=(h/2m3)V@ .

15

(4.5a)

Substitution into the kinetic-energy density [eq. (3.3)] then identifies the B-phase superfluid density as p, = (2m3/fi)210KAi,

(4.5b)

where the rotation matrix has been taken as a spatial constant on the scale of variations in the phase @ (the additional terms associated with gradients of i will be considered in section 4.2). In direct analogy with the arguments used to obtain eqs. (2.3) and (2.4), the representation (4.5a) of 2m,u,lh as the gradient of the phase has two immediate corollaries: (1) The superfluid velocity in the B phase is irrotational apart from singularities, and (2) the cm2/s. When circulation is quantized in units of u = h/2m3~ 0 . 6 X7 applied to bulk rotating 3He-B, the same arguments as in eq. (2.11) imply that p-fold quantized vortices form a regular lattice with areal density

n, = 2Rlpu = 2rn3R/.rrph,

(4.6a)

and an equivalent circular (WS) unit cell of radius [compare eq. (2.13)]. a, = ( p K / 2 7 r ~ ) ' / ~( = p h / 2 m , ~ ) ' / *.

(4.6b)

Since the quantum of circulation in the B phase is comparable to that for 4He, the intervortex separation Q, in rotating superfluid 'He is also =lo0 pm for typical rotation speeds of order 1rad/s. For containers with radii of order 1cm, the lower critical angular velocity R,, is again very small [compare the paragraph below eq. (2.11)], so that the inequality R C , 4R applies to most experiments on 'He-B. The ultimate aim is to study an array of vortices in the rotating B phase. As will be seen, the vortex-core radius is of order ( ( T ) ~ 0 . 0 pm, 5 much smaller than the intervortex separation. In contrast, the characteristic size for spatial variations in the unit vector i is of order 0.05cm [see eq. (4.23)], much larger than the same intervortex separation. This hierarchy of lengths suggests an important simplification: consider a single vortex by itself, and impose an asymptotic boundary condition on the order parameter to represent the behavior at the boundary of the unit cell in the rotating array. Since the unit vector is constant on this scale, the associated matrix R can (for the remainder of section 4.1) be replaced by the unit matrix, because an overall constant rotation has no effect on the free energy. Such an order parameter in the B phase is known to represent Cooper pairing with total angular momentum J=O (Balian and Werthamer 1963), as

A.1.. FEITER

I6

(Ch. 1, 24

expected from its isotropic form; any rotation induced by a general R then introduces components with J = 1 or 2 . It is natural to introduce cylindrical-polar coordinates ( r , 4, z), with the center of the vortex on the axis. Far from the vortex core, the matrix structure of the order parameter should approach that of the bulk. with a , circulating superfluid velocity of magnitude u, = p h / 2 m 3 r= p ~ / 2 7 r rrepresenting a p-fold quantized circulation. Just as in section 2, this condition requires that the phase @ reduces, asymptotically, to p 4 , leading to the boundary condition at infinity

Although the asymptotic superflow about a vortex in the B phase resembles that for a vortex in superfluid 'He. the tensor structure of the full order parameter introduces many new complicating features that are absent for a complex scalar order parameter. In particular, it is essential to consider various possible choices for the unit-vector basis used in constructing the order parameter itself. As described in section 3, the matrixA is projected on one set of Cartesian axes for the spin and another (in principle, independent) set for the orbital angular momentum. Here, the presence of the unit tensor in eq. (4.7) means that these axes, in fact, coincide. As noted by Ohmi et al. (1983) and by Salomaa and Volovik (1983b). this Cartesian set is inconvenient for studying a vortex, and it is preferable to use eigenstates of S, and L z , introducing the complex unit vectors

that satisfy the orthonormality relations (4.8b)

To reexpress the order parameter A,, in this new basis set, it is helpful to think ol an intrinsic tensor A, whose components in the original Cartesian basis are just the set of scalar products with the Cartesian basis vectors

Equivalently, the tensor A follows by inverting this relation

.

A =i,,ApIil

(4.9b)


Ch. 1. 841

17

where a summation convention is again implied. The projection onto any other basis set now follows directly by taking appropriate scalar products. Virtually all studies of vortices in the B phase have assumed axisymmetry, since such structures are likely to have the lowest free energy. Physically, an axisymmetric order parameter must look locally the same at any point around a circle of fixed radius from the center. In this connection, the eigenvectors in eq. (4.8) are not axisymmetric, since they involve preferred directions in the horizontal plane. Thus, it is natural to introduce a related set of unit vectors, defined in terms of the unit vectors ( i ,4, i) in cylindrical-polar coordinates, (4.10)

This set indeed embodies the appropriate axisymmetry; it is simple to verify that the unit vectors in eqs. (4.8) and (4.10) are related by a phase factor (4.11)

where the index A is +, 0, or -. In terms of this new basis set, it has become conventional to introduce the components of the order parameter through the relations A = A 6A CAP i.P e i p * .

(4.12)

Use of the orthonormality conditions, 6:. 6, = & - A ' A

.

.

Ep

=a,,

,

(4.13)

and eq. (4.9) readily gives

(4.14) for CAFin terms of the original Cartesian components of the order parameter. In particular, the asymptotic boundary condition (4.7) now becomes

which reflects the I = 0 character of the bulk order parameter. Note that only three components of the tensor C have nonzero values at infinity, so that the other six components are associated with the core. As a result, the

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ICh. 1. $4

core structure for a vortex in the superfluid B phase is far more complicated than in superfluid 'He. In principle, each of the nine components CAW could still depend on the full set of variables ( r , 4, z), but it is reasonable first to specialize to an axisymmetric structure, in which C depends only on r , independent of the azimuthal angle or the distance z along the vortex axis. Analysis of even this simplest case will become quite intricate. The first step is to rewrite the GL free energy in terms of the new components. Use of eqs. (3.1), (4.9) and (4.14) yields the bulk terms directly (4.16a)

In contrast, the spatial derivatives in fk [eq. (3.3)] complicate the associated transformation. It is simplest to use the intermediate set of unit vectors GA, since they commute with the gradient operators, and a straightforward calculation with eq. (4.11) eventually gives

The second terms (proportional to r -') represent the centrifugal barriers for the various components; it is not difficult to see that they imply the following behavior near the vortex axis

c,,

(x

- A --PI

(r+O).

(4.18)

In general. different components behave differently as r - 0 ; indeed, some components remain finite at the center of the vortex, indicating the presence of a supertluid core. If p = 1 , for example, the components C+,, and C,, can have nonzero values at the origin, whereas, for p = 2 , C , can have similar behavior. Once again, the rich and complicated core structure in the B phase contrasts with the much simpler case of superfluid 'He, where the one-component order parameter for a p-fold quantized vortex line vanishes like rlP' as r + 0 . +

Ch. 1. 541


19

For axisymmetric vortices of the sort considered here, Salomaa and Volovik ( 1983b, 1985) have introduced a convenient group-theoretic classification based on the behavior under various discrete symmetries. There are five distinct classes: The most symmetric vortex (their o vortex) has only five nonzero amplitudes CAF(those with A + p even), which are real. For the u vortex, the same amplitudes may be complex. In contrast, none of the amplitudes vanishes identically for the three remaining types. The v vortex has nine real amplitudes; the w vortex has real amplitudes for A + p even but imaginary ones for A + p odd; for the final type (the uvw vortex), all amplitudes can be complex. It is striking that even the simplest of these vortices has a relatively complicated structure involving five distinct, but coupled, amplitudes. Thus approximate schemes with fewer amplitudes or fewer independent functions (Passvogel et al. 1983) can never represent exact solutions; they can, of course, be considered as trial solutions, and the variational basis of the G L equations then implies that the corresponding free energy would be an upper bound to the true value. Each of the exact vortex types represents a possible equilibrium solution of the bulk G L equations, obtained as the Euler-Lagrange equations of the bulk free-energy density in eqs. (4.16) and (4.17). The o vortex, for example, gives rise to five coupled nonlinear ordinary differential equations for the five real radial functions. It is necessary to solve these G L equations for each of the different vortex types. Unfortunately, the solutions of the G L equations merely represent stationary points of the free energy, so that it is also necessary to consider whether each type is indeed a local minimum of the free energy, and, in addition, which type actually represents the absolute minimum. For the weak-coupling parameters pi in eq. (3.2), thought to describe the low-pressure part of the phase diagram near T,, detailed numerical work (Salomaa and Volovik 1983b, 1985, Passvogel et al. 1984) has shown that the v vortex (with nine real amplitudes) has the lowest free energy of all singly quantized axisymmetric vortices. As expected, the free energy per unit length has the form of eq. (2.15) for p = 1, with appropriate redefinitions of p, and the circulation quantum [see the discussion involvingeqs. (4.5) and (4.6), but note that this calculation for the B phase omits the contribution of the normal fluid]. Figure 2 shows the resulting solutions, normalized to 1 at infinity, with the radial distance scaled by the G L coherence length (( T). As expected from the previous analysis, only three components have nonzero values far from the vortex core, and only the components C,, and C,, remain finite at the center of the vortex. For the o vortex, the five nonzero components look qualitatively like those for the v vortex, but this solution turns out to be unstable in the weak-coupling regime, being only a saddle point; furthermore, there is no evidence of stability for the remaining three complex vortex types.

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-0.5 :---0

I

I

2.5

5

[Ch. 1 , 84

I

10

I

25

-

Fig. 2. Order parameter for the v vortex in ’He-B. For the o vortex, the top five functions are qualitatively similar, and the bottom four are absent (Salomaa and Volovik 1983b. reprinted by permission of the authors and the American Physical Society).

It is also interesting to consider axisymmetric vortices with other values of p . For liquid ‘He, the scalar form of the order parameter ensures that different multiply quantized vortices still have the same qualitative core structure. Thus the free energy of a p-fold quantized vortex scales approximately with p2 [see eq. (2.15)], whereas the area per unit cell of the vortex lattice increases only linearly with p [see eq. (2.13a)l; the resulting free energy per unit area is essentially proportional to p and is minimized for p = 1, indicating that only singly quantized vortices are expected to occur in rotating ‘He. In contrast, singly and doubly quantized vortices in 3 He-B have intrinsically different core structures, which precludes such a simple scaling argument. For doubly quantized vortices, Salomaa and Volovik (1984, 1985) find that only the o type is stable in the weak-coupling regime. In addition, a lattice of such doubly quantized o vortices has a higher free energy per unit volume than one of singly quantized v vortices

Ch. 1, $41

VORTICES IN ROTATING SUPERFLUID ‘He

21

for angular velocities less than about 160 rad/s ( ~ 1 6 0 rpm). 0 Since current experiments typically operate at about 1 rad/s, observation of doubly quantized vortices in the B phase is not anticipated soon. The quartic term in the G L free energy [eq. (3.lb)l depends on the five parameters pi that include the strong-coupling corrections to the weakcoupling BCS theory. Since the weak-coupling parameters in eq. (3.2) imply that only the bulk B phase is stable, it is evident that the strong-coupling corrections must be important in studying the behavior at elevated pressure, especially near the polycritical point where the A and B phases coexist at T,. Although certain combinations of the p’s determine various physical quantities near T,, they are not known individually with any precision. The early spin-fluctuation model for the p’s (Brinkman et al. 1974) provided many qualitative insights into the feedback mechanism for stabilizing the A phase through strong-coupling corrections, but its one adjustable parameter S cannot consistently fit the various experimental observations (see, for example, Halperin et al. 1976). Nevertheless, it remains popular because of its simplicity. In particular, Salomaa and Volovik (1985) have used it to investigate the pressure dependence of the vortex structure in the B phase. On this basis, they predict that the singly quantized v vortex remains the stable solution throughout the G L region, up to the polycritical point. Several recent calculations (Sauls and Serene 1981, Levin and Valls 1983, Bedell 1984) have yielded improved fits for the strong-coupling corrections to the p’s. Despite small differences among them, they are qualitatively similar and all differ significantly from the spin-fluctuation model. To investigate how these corrections affect the pressure dependence of the vortex core, Fetter and Theodorakis (1984) (see also Fetter 1985b) have used variational trial functions for the different components of the singly quantized axisymmetric vortices. With the spin-fluctuation model, they find, in agreement with Salomaa and Volovik (1989, that the v vortex is stable at all pressures. With the Sauls and Serene model, however, the four additional components that distinguish the v vortex from the o vortex diminish rapidly with increasing pressure and disappear at a critical pressure of about 10 bar. Above this value, their calculations predict that only the o vortex remains stable. As seen in section 4.3, Hakonen et al. (1983a, b) have experimental evidence for a first-order transition between two types of vortex cores in the B phase, but the data do not extend to the G L regime. In contrast, Salomaa and Volovik (1983b) find that any transition between the v and o vortices must be of second order. Thus it is not clear whether the suggested disappearance of the v vortex with increasing pressure is related to the observed core transition. Since a variational calculation cannot be considered definitive, additional

22

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exact solutions of the GL equations using the newer fits to the p’s would be desirable. This question remains open and intriguing. Several other distortions of the vortex structure might conceivably account for the observed vortex-core transition. The order parameter in eq. (4.12) assumes axisymmetry, but it is natural to consider more general nonaxisymmetric forms. If C in eq. (4.14) depends on both r and 4, it can be expanded in a Fourier series with different angular contributions exp(im4); axisymmetric solutions then become a special case with only m = 0. A small distortion with a single nonzero m leads to a linear problem, and the sign of the eigenvalue determines the stability. Alternatively, finite distortions can be studied with the full GL free energy, but the quartic couplings in eq. (4.16b) apparently preclude any closed finite set of m values. Preliminary calculations by Salomaa (1984) have found no evidence that such nonaxisymmetric vortices have lower free energy than axisymmetric ones. Another possibility is to allow helical distortions that break the translational invariance along the vortex axis (Salomaa and Volovik 1983a, 1985). More generally, the basis vectors for the spin and orbital coordinates in eq. (4.12) could differ by a rotation that depends on the distance from the vortex core and reduces to the unit matrix only at infinity. Such a rotation can qualitatively alter the centrifugal barriers near the vortex core, but it is not yet clear whether the potential reduction in the kinetic energy of circulating superflow can compensate for the increased bending energy associated with the gradients of the rotation matrix. Even more intriguing is the possibility of topologically distinct rotations joining the core and the asymptotic region, which could then produce the analog of inequivalent vacuum states in field theory. At present, there is no compelling argument for axisymmetric B-phase vortices (other than simplicity), and much work, both analytical and numerical, remains to be done. The preceding discussion of the structure of axisymmetric vortices in the B phase has omitted the possible effect of an applied magnetic field. This question must be considered, because NMR has been the principal means of detecting the vortices. It depends on the balance of several terms in the free-energy density, involving the dipole interaction between the nuclear spins, and the magnetic interaction of the spins with an external magnetic field [see eqs. (3.5) and (3.7)J. For the B phase, their detailed forms will be considered in sections 4.2 and 4.3, in connection with the it texture in a rotating container and the corresponding NMR. Here, it is sufficient to note that typical experimental magnetic fields ( ~ 3 0 G) 0 are too weak to affect the vortex-core structure significantly (Salomaa and Volovik 1985). This situation would be different for much higher fields, and such observations could be valuable in clarifying our picture of the vortices in superfluid jHe-B.

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23

Although the vortex structure is rigid, in the sense that moderate magnetic fields do not affect it, an applied magnetic field does lead to one very interesting and unusual effect, arising from the nonunitary character of the order parameter near the vortex core. In general, an order parameter in superfluid ’He is called unitary if the matrix representing the squared energy gap is proportional to the unit matrix. In essence, this condition means that spin-up and spin-down particles are equivalent, which indeed describes the bulk A and B phases in zero field. As expected, it fails in the presence of large magnetic fields, most notably in the formation of the A , phase, where Cooper pairing occurs in only one spin component. In the present context, the asymmetry of the order parameter CAPunder the interchange ( A , p ) * ( - A , - p ) [see, for example, eq. (4.18) and the explicit solution in fig. 21 indicates clearly the nonunitanty near the vortex core. It arises from the special coupling between the orbital and spin angular momentum [the broken spin-orbit symmetry introduced by Leggett (1973)l. The overall rotation of the container induces quantized vortices, whose preferred sense of circulating superflow acts roughly like an external magnetic field in distinguishing up and down spins. This effect had been predicted earlier (Sauls et al. 1982) for vortex lines in rotating neutron stars, where the Cooper pairing occurs in the ’P2 state. In the context of the B phase, its occurrence was suggested by Ohmi et al. (1983) and by Volovik and Mineev (1983), and detailed numerical work by Salomaa and Volovik (1983b) has provided a convincing fit to the data of Hakonen et al. (1983b). The order of magnitude of the effect can be estimated by noting that it arises from the asymmetry between particles and holes, since otherwise complete cancellation would occur. The net magnetization in the vortex core is of order nlr ( h ( A/ E ~ ) ’ = 4 X lo-’ erg/G cm’, where n is the atomic number density (“2 x cm-’), lylfr/2 is the magnetic moment of a ’He nucleus erg/G), and the two factors of = reflect the Cooper pairing and the particle-hole asymmetry. Multiplying by 47r times the “area of a vortex” 7rt2, where 6 is cm, gives a magnetic flux of order G cm’ per vortex line. Since the number of vortices in a container is unlikely to exceed lo4, the total magnetic flux is far smaller than a single flux quantum in a superconductor (of order 2 x lo-’ G cm’). Clearly, direct measurement of this effect would be difficult. Nevertheless, as seen in section 4.2, it exerts a significant influence on the ri texture in a rotating container and thus is observable in the shift of the NMR spectrum. The simplest way to quantify the effect is to recall the small term in the G L free-energy density (Ambegaokar and Mermin 1973) (4.19)

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that is linear in the magnetic field H. Here h = ( l y ( h H / k , T , ) is a dimensionless magnetic field, and 17 is a small dimensionless parameter, of order 0.01 (lsraelsson et al. 1984), that characterizes the particle-hole asymmetry. When evaluated for an order parameter of the form in eq. (4.12). the resulting integrated free energy becomes - m H,where the magnetic moment m per unit length of vortex line lies along the axis of the vortex (or, more generally, rotated by the local rotation matrix R ) and has a magnitude given by (4.20a)

(4.20b) a dimensionless integral of order 1.5 for the v vortex (Salomaa and Volovik 1985). Recent experiments (Israelsson et al. 1984) have provided detailed information on the pressure dependence of 17, but comparison with the observed NMR shifts (see section 4.3) is difficult because they are measured outside the GL regime. 4.2. TEXTURES IN A

ROTATING CONTAINER

In static equilibrium, the order parameter for the B phase has the form ” the given in eqs. (4.1) and (4.2). Since the angle 8 is fixed at ~ 1 0 4 by dipole coupling, the only remaining freedom is the spatial orientation of the unit vector ti. Although this direction is arbitrary in an unbounded isotropic system, the situation changes in the presence of perturbations that introduce one or more preferred spatial directions. Examples of interest here are boundaries of a container, an external magnetic field, or rotation and the associated vortex lines. In these cases, competing influences o n ri typically induce a spatially varying texture, obtained by minimizing the total free energy, subject to the boundary conditions. To understand this behavior in detail, it is first necessary to consider the various contributions to the free-energy density for a B-phase order parameter. One important term arises from the gradients of ti: substitution of eqs. (4.1) and (4.2) into eq. (3.3) yields the following (Smith ct al. 1977, Brinkman and Cross 1978)

+ g(ti .curl ti)’ + (ti X curl ti)’ - - F ( d i v r i ) ( t i .curl ti) + div[(ti .V)i ti div ti]}

f, = p,(h/2m,)’{ g(div ti)’

-

.

(4.21)

Ch. 1, 941


25

Here, p, is the superfluid density from eq. (4.5b), and the last term is a pure divergence, equivalent to a surface contribution. By itself, this gradient energy density tends to favor a uniform texture. The form of the remaining terms in the free-energy density depends crucially on the presence or absence of a magnetic field. Since all the Helsinki experiments on rotating 3He-B have used a strong field ( ~ 3 0G 0 9 Hd), only the latter situation will be considered here, but the generalization to the weak-field regime requires no new principles. Recall that the B-phase energy gap at the Fermi surface is isotropic in zero field, with a magnitude given in eq. (4.3). The addition of a strong magnetic field deforms the gap slightly into an ellipsoidal shape, with A, reduced along the direction of the field (Engelsberg et al. 1974, Fetter 1975, Leggett 1975). When combined with the dipole energy, this deformation leads to an orienting term in the bulk energy of the form (see, for example, Brinkman and Cross 1978) fH = - a H 2 ( i *

k)’ ,

(4.22a)

where

evidently reflects the combined role of the magnetic and dipole energies. Comparison of eqs. (4.21) and (4.22) suggests the introduction of yet another characteristic length [see eqs. (3.6), (3.8), (4.3) and (4.5)]

tH= ( p S / ~ ) ” 2 ( h / 2 m 3 Hz) ‘i Hd

5

H ’

(4.23)

that determines the scale of spatial variations of i in a large magnetic field. Note that tHvaries inversely with H and is therefore field-dependent (as well as temperature-dependent, through 5). Since the coefficient a in the orienting energy is very small (it involves the product of two perturbations), this length is macroscopic even in the present high-field limit ( tH= 0.05 cm for ~ 3 0 G). 0 For comparison, the containers used in the Helsinki experiments have radii about five times larger than t,, but this condition depends on the particular experiment. The combined effect of magnetic fields and vortices is slightly complicated, and it seems preferable to start with the simpler case of a fluid at rest in a stationary container, where the only additional contribution is the surface energy f,. In the strong-field limit, f, is proportional to -(If. R . i)’, where s^ is the normal to the surface (Smith et al. 1977.

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Brinkman and Cross 1978). For definiteness, consider a long cylindrical container, with the field parallel to the symmetry axis z^. A simple calculation then shows that the resulting surface energy is minimized by the values

where n, and ne are the radial and azimuthal components of h. In the typical Helsinki experiment (see the discussion by Jacobsen and Smith 1983), these values provide the boundary conditions that serve to determine the h texture in the cylinder, whose qualitative form now follows by comparing the energies f, and f,. The unit vector ii will be uniform and parallel to H (along the z-axis) in the interior of the cylinder to minimize eqs. (4.21) and (4.22a), but this texture cannot satisfy the boundary conditions in eq. (4.24). Thus h bends throughout a surface layer of thickness 5, from vertical to one of the values given in eq. (4.24), say n, = (1 /5)1'2 and n+ = (315)'". The resulting texture in a stationary cylinder is axisymmetric, with h exhibiting a spiral pattern near the boundary. To understand the NMR in both stationary and rotating containers filled with B-phase fluid, it will be important to consider this texture in greater detail. The configuration may be parametrized with a pair of angles a and /3, through the relations ti = sin

/3 ( i cos a +

sin a)+ i cos p .

(4.25)

Thus p is the polar angle of h , but a is not the conventional azimuthal angle, since eq. (4.25) involves cylindrical-polar unit vectors instead of Cartesian ones. This choice offers the important advantage that a and f3 depend only on the radial distance r from the center of the cylinder, and eq. (4.24) implies the corresponding boundary values

P ( R ) = arcsin d!, ~ 6 3 " .

a ( R )= 60",

sH),

(4.26)

Several studies of wide cylinders (radius R S both analytical (Hakonen and Volovik 1982, Maki and Nakahara 1983) and numerical (Jacobsen and Smith 1983) indicate that P ( r ) decreases rapidly within a surface region R - r < 5" from its boundary value of ~ 6 3 at " r = R toward its central value p ( 0 ) = 0. In the interior, it has a roughly parabolic form, with a curvature proportional to exp(-Rlf,). In contrast, the angle a varies smoothly, decreasing slowly to a central value determined by the ratio R / 5 , ; for R15, 1 a ( 0 )= 38". Figure 3 illustrates this behavior for the ~

Ch. 1, 541


27

.o Fig. 3. The C texture for nonrotating ’He-B in a cylinder of radius.R = 4 t H , with a and p defined as in eq. (4.25) (Jacobsen and Smith 1983, reprinted by permission of the authors and Plenum Press).

typical value R / [ , = 4 (Jacobsen and Smith 1983). As will be seen in section 4.3, the form of the angle P ( r ) , especially its deviation from a strict parabolic form, is important for the NMR spectrum for the B phase in a cylinder. It is evident that the ri texture in a stationary cylinder arises from the balance between the orientational energyf, in eq. (4.22a) and the gradient energy f, in eq. (4.21), supplemented by an appropriate set of boundary conditions. Furthermore, the small value off, implies that the corresponding characteristic length (, is macroscopically large. It also implies the crucial corollary that other small orientational energies will be equally significant in determining the overall texture. In the present context of rotating superfluid ’He-B in a strong magnetic field, two additional contributions are relevant. One, originally suggested by Gongadze et al. (1981), reflects the local alteration of the magnetic energy caused by the circulating superflow around the vortex and the resulting deformation of the order parameter near the core. This contribution f, is automatically quadratic in the magnetic field and thus even under a field reversal. The other, suggested by Ohmi et al. (1983) and Volovik and Mineev (1983), and studied in detail by Salomaa and Volovik (1983b), arises from the nonunitary order parameter near the vortex core and the associated magnetic moment [discussed in connection with eq. (4.20)]. This “gyromagnetic” contribution fern is linear in the magnetic field and is

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A.L. FETTER

(Ch. 1. $4

therefore odd under a field reversal. The distinct symmetries under field reversal are important in the experimental separation of the two effects. The original derivation (Gongadze et al. 1981) of the orienting energyf, relied on a formalism combining the effect of a uniform magnetic field and uniform hydrodynamic flow (see, for example, Brinkman and Cross 1978). The resulting (rather complicated) generalization of eq. (4.22) was then applied to the nonuniform flow about the vortex. Since the supcrfluid velocity diverges at the center of the vortex [see eq. (2.5) for the corresponding situation in ‘He], this procedure required a short-range cutoff to approximate the intrinsic deformation of the core. More recently, Salomaa and Volovik (1985) have introduced a simpler approach that relies explicitly on the localized character of the circulating supercurrent and core deformation. They start from the observation that the various vortices in zero field (discussed in section 4.1) are, in principle, exact solutions of the G L equations obtained from the bulk free-energy densities f,, f,. and f, [see eqs. (3.1) and (3.3)]; furthermore, the order parameter for each vortex automatically approaches the bulk form (4.4) far from the core. To include the effect of an applied field to leading order, it is sufficient to use first-order perturbation theory and evaluate the spatial integral of the magnetic energy densitiesf, [eq. (3.7)] a n d f , , [eq. (4.19)] for the known vortex order parameter in zero field. Although this procedure gives no orienting effect for the bulk uniform order parameter in eq. (4.2), it does properly include the orienting effect of the localized vortex-core structure; indeed, it is just the approach used to obtain the magnetic moment (4.20) of the vortex in section 4.1. The analysis of the order parameter near a vortex in section 4.1 must first be generalized to allow for a rotation matrix R that is locally constant over the intervortex separation of order a,. Thus eqs. (4.9a) and (4.12) arc rewritten as

where eq. (4.11) has been used to simplify the subsequent analysis. Substitution into eq. (3.7) then gives the associated magnetic energy density (4.28) whose average over the azimuthal angle vanishes unless K = A . A straightforward analysis yields an isotropic term, which will be ignored, and the relevant vortex-orientation contribution

ilc-gl2 I .

(4.29)

Ch. 1, 941


29

Averaging over the unit cell of area mug = n,' [see eq. (4.6)] then gives the corresponding vortex orientation energy density

(4.30) where the radial coordinate is scaled with write this contribution in the form

where

h denotes the

5. It has become conventional to

rotation axis and

is a dimensionless coupling constant, proportional to the vortex density n, and thus to the angular velocity. Although the integrand in eq. (4.31b) would vanish identically for a bulk B-phase order parameter, it is here nonzero because of the altered (non-B-phase) order parameter in and around the core. Indeed, the dimensionless integral diverges logarithmically for large r, which accounts for the finite upper limit, and has a value of order 25 (Salomaa and Volovik 1985). They ascribe the logarithmic contribution to the circulating superflow [it has the form originally suggested by Gongadze et al. (198l)l and the remaining constant to the "magnetic anisotropy inside the vortex core", but this separation is not unique. The other magnetic contribution to the textural free-energy density follows immediately from eq. (4.20). When eq. (4.27) is substituted into eq. (4.19), an average over the unit cell of the vortex lattice yields the gyromagnetic free-energy density fgm

= -(H-R . m ) n , ,

(4.32)

where the property RA X RB = R ( A X B ) has been used, and m is given in eq. (4.20a). It is conventional to rewrite this expression in a form analogous to eq. (4.31)

where

K

is another coupling constant, again proportional to the angular

A.L. FE’ITER

30

[Ch. 1, 84

velocity, but now with the dimension of a magnetic field K

=

-

:mn,/a.

(4.33b)

As noted previously, fv and fgm are even and odd, respectively, under a reversal of the relative directions fi and d. It is now possible to consider the equilibrium ri texture in a rotating container subject to a strong axial magnetic field. The total free energy in the rotating frame of reference is given by (4.34) where the last two terms comprise the effect of the vortex lines (and thus the rotation of the container). In principle, eq. (4.34) must be minimized subject to the boundary conditions in eq. (4.24), and it is again convenient to parametrize i with the two angles a and p introduced in eq. (4.25). To understand the qualitative effect of rotation on the texture, it is helpful to evaluate the quantity (H R z^) = ? H ( i R i) explicitly, where H = &Hi is assumed parallel to the rotation axis. Use of eq. (4.1) with cos 8 = gives the value +H[1 - sin’ p ] . Apart from an additive isotropic constant, it follows that the last three terms in the integrand of eq. (4.34) may be written as

-

- -

a H Zsin’@ [I - A ( l -

sin2p) 7 K / H ] ,

(4.35)

which characterizes the combined effect on the i texture of parallel (antiparallel) magnetic field and rotation. Since A and K are both positive, rotation parallel to H evidently acts to reduce a by an approximate factor 1 - A - K / H ,assuming that p 4 1. In particular, this reduction decreases the effective strength of the bulk orienting energy f, in eq. (4.22a); at the same time, it increases the effective value of the characteristic length in eq. (4.23). The qualitative effect of rotation ( h l l H ) on the ri texture in a cylinder is now clear: the reduced effective orienting energy implies a narrower central region where ri is parallel to the applied field (namely, where p = 0), and the increased tH implies a thicker surface region where P ( r ) decreases from its boundary value. Both of these effects act to increase the curvature of p ( r ) near the center, which will be important (section 4.3) in understanding the NMR. Reversing the magnetic field direction changes the sign of fgm but leaves f, unchanged. Experimental studies (Hakonen et al. 1983b) at -29 bar for TIT, 6 0.6 (well below the GL regime) yield the approximate values


Ch. 1. 841

31

A ~ 0 . and 8 K I H= 3 X lo-* for 0 = 1 rad/s and H = 300 G (see fig. 4). Thus the rotation has a significant effect on the overall texture, and numerical studies are necessary to obtain the detailed form of the angles a and p for various angular velocities and magnetic field strengths. The same formalism also applies to more general situations, for example, those with the magnetic field tipped relative to the rotation axis. In a finite container, this configuration is complicated because it breaks the

10

C

205

0

Lx10-Z

o,o,O (25', 155')

1.4 1.4 1.7 1.7 1.4

(LSo,135') P (L5',135') x (65',115') + (65',115') A

I C

3

\

#

2

I

-

-

I I I

I I I

1 1

0

05

d

8.r.,.

06

Atx

1

I

07

0.8

0.9

10

Fig. 4. Temperature dependence of the normalized parameters h l f l and K I O Hat 29.3 bar. The solid lines refer to a theoretical calculation (Hakonen et al. 1983b, reprinted by permission of the authors and the American Physical Society).

31

A . L . FETTER

[Ch. 1. 84

cylindrical symmetry; consequently, the only detailed studies of tipped fields have concentrated on the simpler question of the corresponding uniform ti texture, ignoring the finite-size effect of boundaries. The resulting minimization of the last three terms in eq. (4.34) produces a transcendental relation for the angle /3 that depends on the tipping angle and the two parameters A and K I H .When applied to the experimental data for the central region of a rotating cylinder (Hakonen et al. 1983b), these observations provide the most accurate values of the two parameters. Strictly speaking, the present description of the ri texture in a container applies only in the vicinity of T,, because the derivation of the gradient energy (4.21) and the three “coupling constants” a , A , and K in eq. (4.22), (4.31). and (4.33) relies o n the GL formalism. At lower temperatures, the various contributions are expected to have the same general form, but with altered coefficients that depend on temperature as well as pressure. In particular, Brinkman and Cross (1978) discuss the gradient energy, which involves Fermi-liquid corrections and several different coefficients for the distinct invariant terms in eq. (4.21). In addition, the parameter a in eq. (4.22) can be related to experimental values (Osheroff and Brinkman 1977) of N M R shifts in the bulk B phase. Alternatively, a, A, and K can be considered phenomenological parameters. to be determined by comparison with experimental data, using the theory to be discussed in section 4.3 (Hakonen et al. 1983b). As a result. the description of h textures in rotating containers should hold throughout the B phase. 4.3. N M R

IN A ROTATING CONTAINER

The preceding section provides a prescription for obtaining (in general, numerically) the equilibrium ti texture of ‘He-B in an arbitrary container, subject to a strong static magnetic field. The remaining question in determining the N M R spectrum is the linear response of the superfluid B phase to a weak external rf magnetic field. Soon after the original observation o f the NMR shifts in ’He by Osheroff et al. (1972), Leggett (1974) developed the basic formalism, which relies on an effective magnetic free-energy density

f,,,= y2S2J2xB- y S * H

+ fd + fk .

(4.36)

Here S(r) is the spin density, xu is the B-phase susceptibility, y is the “gyromagnetic ratio” for the - He nucleus, H is the total magnetic field,f, is the dipole energy density from eq. (3.5), and fk is the kinetic energy density associated with spatial variations in the order parameter, taken either from eq. (3.3) in the GL. regime or its generalization to lower

Ch. 1, 941


33

temperature. Althoughf, plays no role in a uniform system, it is crucial for understanding the NMR in a bounded geometry. The effective free energy density depends on the spin density and on the order parameter A,,. As shown by Leggett (1974), the orbital part of the order parameter remains fixed on the relatively short time scales associated with an rf magnetic field, so that only the spin part of the order parameter varies. Thus the theory involves two sets of dynamical variables, the spin density S and the spin part of A,, (the first index). In static equilibrium, the order parameter has the general form AZi given in eqs. (4.1) and (4.4), with the rotation angle arccos(- $ )and the rotation axis ri determined as described in section 4.2. This solution can also be shown to minimize the spatial integral of eq. (4.36) for the appropriate static magnetic field. Small deformations in the spin part of the order parameter can be characterized by an infinitesimal rotation 8, of magnitude 101 about the axis 8/181, leading to

where

For small-amplitude motion, the infinitesimal rotations themselves can be taken as the dynamical variables of the order parameter, and the canonically conjugate spin operators S (the generators of spin rotations) complete the set. They obey the usual commutation relations

If He,, denotes the spatial integral of the effective magnetic free-energy density feff, this quantity then acts as an effective Hamiltonian that governs the dynamical motion of these variables through the familiar relations (4.39)

To derive the explicit equations of motion, it is helpful to rewrite the dipole energy density in terms of the small rotations 8. Substitution of eq. (4.37) into eq. (3.5) eventually gives the result

A.L. FE'ITER

34

[Ch. 1. 44

(4.40) correct through second order in small rotations. Here, ti characterizes the unperturbed order parameter A:, , and a constant term that plays no role in the dynamical equations has been omitted. Consider, temporarily, a bulk uniform sample in a static magnetic field H O i , with G a spatial constant and a uniform static spin density S,, = x e H O / y . In the presence of an additional rf magnetic field H', the commutation relations in eq. (4.39) yield the pair of dynamical equations (4.41a) (4.41b)

15gdAin(n

where H is the total magnetic field. These equations are readily combined in a single equation that expresses the infinitesimal rotation in terms of the rf field H' d '8 _ _ -- w O $ at?

Y2 dH' xi-15-gg,A~h(G-8)-y XB

dt

'

(4.42)

where w,) = yH,, is the Larmor frequency. In the usual case that G is parallel to H,, [recall the coupling energy in eq. (4.22)], this equation predicts the familiar transverse response ( H ' perpendicular to i) at the Larmor frequency wo, and an additional longitudinal response ( H ' parallel to i) at the characterisic frequency (Leggett 1974) (4.43) which Wheatley (1978) tabulates for several pressures. Note that 0, is proportional to A , and thus vanishes as T-+ T , in the G L regime [see eq. (4.3)]. More generally, if ti differs from z^ for some reason, the bulk transverse and longitudinal responses are characterized by the approximate frequencies w: = wt 7

+~ ,

wz = f j t , n ; .

; ~ ( nl f- ) ,

(4.44a) (4.44b)

apart from corrections o f order (fl,/q,)', which are small in most cases of interest. These relations show that the ti texture can affect the NMR

Ch. 1, 141


35

response significantly, even in a bulk sample. In particular, the bulk transverse response shifts from the Larmor frequency whenever textural effects rotate ri away from the direction of the static field. To understand the actual NMR in a bounded container, it is essential to consider the bending contribution fk in eq. (4.36) (Smith et al. 1977, Brinkman and Cross 1978, Buchholtz 1978). Substitution of eq. (4.37) into eq. (3.3) eventually gives the approximate expression

where L, is the dipole length from eq. (3.6), k is a constant of order one [the ratio of the bending constants (K, + K3)/2K,, which equals 1 in the weak-coupling GL regime], and Tis a tensor that depends explicitly on the unperturbed i texture [compare eq. (4.1) for the angle arccos(-i)] T,, = ~(c?,, - 5ripriv + Y B E , , , ~ , ) .

(4.46)

Since the i texture varies on the scale ( ~ 0 . 0 cm), 5 whereas L, = cm sets the scale for spatial variations in 8, gradients of ri and T have been ignored in deriving eq. (4.45). This expression must be combined with the dipole energy density (reexpressed in terms of the frequency 0,) to give the total effective free-energy density (4.36). A generalization of the previous derivation then gives the dynamical equation for the oscillatory motion of the ith component of the small rotation angle 8 in the presence of a static magnetic field Hot and a weak rf field H' exp(-iot) oscillating at frequency o

L?i(Dij+ i i i j ) q- w2t$+ iowo(8 x t),= iwyHI,

(4.47)

where

D , = $ L i [ - S i j ( l + k)V'

+ kTildlTjmdm]

(4.48)

is a hermitian differential operator. This equation differs from eq. (4.42) only in the term involving the gradients of 8, which arises from the physical effect of the nonuniform i texture. The structure of eq. (4.47) resembles that of the Schrodinger equation, with an inhomogeneous driving term proportional to the rf field. To make the correspondence precise, it is convenient to project it onto the Cartesian spherical unit vectors qAfrom eq. (4.8), using the definitions

e,

=e

- 4, ,

e = e,+;

.

(4.49)

A.L. FETTER

36

[Ch. 1, $4

A straightforward analysis yields the equation for the spherical component

O’,(D,,, + tiAi;)fls - w(w

-

- Aw,)8,

= iwyH;,

(4.50)

-

where D,, = (ti, i , ) D l , ( i , 4;) is the projection of eq. (4.48) onto the new basis, and the summation convention is implied. Evidently D,, acts like the “kinetic energy” in the Schrodinger equation, and ti,;; acts like a (hermitian) potential V . In principle, all three components 8+, O,, and 8are coupled by these matrix operators, but this coupling becomes negligible in the usual case that (0,1w,)2 + 1 (Theodorakis and Fetter 1983). Thus eq. (4.50) separates into three inhomogeneous Schrodinger equations for the spherical components O,, with w ( w - Aw,)/O’, acting like the eigenvalue for the effective motion in the real potential V, = ltiA12, governed by the kinetic energy operator D, = D,,(here, no summation on A is implied)

O;(D,

+ V,)e, - W ( W - Aw,,)B, = i w y H ; .

(4.51)

These equations must be supplemented by appropriate boundary conditions. Since there is no flux of spin density into the walls, the normal derivative of S must vanish there (Smith et al. 1977, Brinkman and Cross 1978, Buchholtz 1978), which then gives the corresponding (Neumann) boundary condition on the small rotations

im, = 0 .

(4.52)

In dealing with an inhomogeneous equation of this form, it is convenient to introduce the eigenfunctions $; of the homogeneous equation, and the corresponding eigenvalues w : , defined by

(4.53) subject to the same boundary conditions in (4.52). If the spherical component 0, is expanded in this (assumed complete) set, eq. (4.51) readily yields the explicit solution

where the frequency w has been given a small positive imaginary part ie to ensure a causal response (see, for example, Fetter and Walecka 1980,

Ch. 1. 941


37

section 50). The structure of this equation is physically transparent, for the applied (uniform) rf field excites the ith normal mode with a strength proportional to its overlap integral with H;, and the response contains a causal resonant denominator that is large if the applied frequency w is close to the eigenfrequency w A. The physical quantity of interest is the mean power absorbed by the system dUldr = to

I

d’x Im H‘** M ( x ) .

(4.55)

Use of eq. (4.41a) and the relation M = yS leads to the expression (4.56) Equation (4.54), combined with the familiar relation ( x - iE)-’ = P x - ’ + i ?rS(x), where P denotes a principal-value integral, yields the approximate form

X

2u(2~ boo)-'

I

1’

d3x $;(I)

(4.57)

for the power absorbed from a weak external rf field with spherical component A . As expected, excitation of spin-wave normal modes occurs only if the applied frequency equals one of the resonant frequencies w : , and the strength of the absorption is proportional to the square of the overlap integral of the appropiate eigenfunction with the uniform rf field H;. In the present physically motivated picture, the study of the NMR spectrum in a bounded container reduces to solving an effective Schrodingcr equation (4.53) for the eigenfunctions and eigenvalues, subject to the (Neumann) boundary condition (4.52). Apart from the unusual boundary conditions, this procedure is familiar from quantum mechanics, and all the well-known techniques can be used to find the solutions. In particular, the Rayleigh-Ritz quotient provides an estimate for the lowest eigenvalue of a particular spherical projection

(4.58)

A.L. FEITER

38

[Ch. 1, 84

giving a variational principle for the shift in the NMR frequency from the bulk value (w,, for transverse excitation H i , @, for longitudinal excitation

H;).

This formalism applies directly to 3He-B in a long cylinder of radius R, where the equilibrium texture is axisymmetric and has the form discussed in section 4.2. The spherical components of Ii follow from eqs. (4.8), (4.25), and (4.49) as

if = (d)-’ sin p e”“’’)

, Ii =cos

p

(4.59)

with a and p functions of r only. As a result, the effective potential V, is axisymmetric; when compared with the texture for a bulk uniform sample with p = 0 , the transverse potential

V,

=

t sin’ p

(4.60)

is positive (“repulsive”) and raises the NMR above the (bulk) signal at the Larmor frequency wo, whereas the longitudinal potential

y, = cosz p

(4.61)

lies below that for a bulk uniform sample with p = 0 (“attractive”) and lowers the NMR below the (bulk) signal at 0,.Solutions in such axisymmetric potentials can be classified by their angular dependence [exp(imc$)], and eq. (4.57) shows that only the axisymmetric modes with m = 0 couple to the uniform rf field. It is convenient to deal first with the transverse modes, which have been studied in considerable detail (for theoretical work, see Hakonen and Volovik 1982, Theodorakis and Fetter 1983, Jacobsen and Smith 1983, Maki and Nakahara 1983; for experimental work, see Hakonen et al. 1983a, b). For axisymmetric modes, eq. (4.53) reduces t o a one-dimensiona1 equation

-

where T , , = G, T - i and k has been set equal to one. In a stationary container, the potential sin’ p follows the discussion below eq. (4.26) and from fig. 3; typically, it is quadratic near the center of the container, with exponentially small curvature, and rises within a surface layer of thickness &,(-GR) to its boundary value 0.8 [compare eq. (4.26)]. As a first approximation, the potential near the center of the cylinder can be

Ch. 1, §4]


39

replaced by a parabola ( plr)2,where p, is proportional to the curvature and has the dimension of an inverse length. In this way, eq. (4.62) describes an effective two-dimensional harmonic oscillator. The characteristic length (4.63) determines the spatial scale of the approximate eigenfunctions, and the resulting approximate eigenvalues are equally spaced, with transverse NMR absorption occurring at the approximate frequencies

where n = 0 , 1 , 2 , . . . . In this approximation, the intensity of the nth normal mode relative to a bulk sample of area ?rRZis given by

I,,lI,,,, = 4 r : / R 2 .

(4.65)

As expected from the role of r , in setting the scale of the wave functions, this ratio is of order ( r l / R ) 2 ;its independence of n arises from a cancellation between the increasing size of the harmonic-oscillator wavefunction and the reduced overlap integral, which does not hold for the exact eigenfunctions. The observations of transverse NMR in stationary cylinders (Hakonen et al. 1983a) agree qualitatively with this approximate theory; in particular, a set of nearly equally spaced absorption peaks is seen above the Larmor frequency, but the intensity falls off with increasing n. To incorporate the deviations from the harmonic-oscillator model, Jacobsen and Smith (1983) have integrated eq. (4.62) numerically, using the exact equilibrium ti texture for various stationary cylinders (recall fig. 3); fig. 5 shows a typical spin-wave eigenfunction found for the exact sin’ p and (for comparison) the corresponding approximate potential with equal curvature at the origin. In detail, the exact solutions differ significantly from the approximate ones, and the data clearly support the numerical studies. The same description also holds for a rotating cylinder, since (see section 4.2) the dominant effect of the induced vortex lines is to modify the equilibrium ti texture. As noted below eq. (4.35), rotation with d parallel to l&,increases the curvature p, of the polar angle P ( r ) near the center of the cylinder and hence increases the splitting of the transverse spin-wave modes relative to that in a stationary cylinder [see eq. (4.64)]. This qualitative effect [first suggested by Maki and Nakahara (1983) and analyzed in detail by Hakonen and Volovik (1982)J agrees well with the observa-

A.L. FETTER

ICh. 1 . 14

Fig. 5 . Effective potential sin' p (solid line) and a typical spin-wave eigenfunction (dotted line) for nonrotating 'He-B in a cylinder of radius R = 56,. For comparison. an approximate quadratic potential (dashed line) with equal curvature at the origin is also shown (Jacobsen and Smith 1983. reprinted by permission of the authors and Plenum Press).

tions of Hakonen et al. (1983a). In practice. extensive numerical studies are required to determine: (1) the equilibrium ti texture in a rotating cylinder for given values of H , R, R, 6 , . A and K , and (2) the position and intensity of the corresponding transverse NMR satellites above the Larmor frequency. In this way. Hakonen et al. (1983b) have determined the parameters A and K as a function of temperature at 29.3 bar, as shown in fig. 4 (this figure also includes data obtained for NMR experiments in tipped magnetic fields, which provides clearer separation of A and K ) . The most striking feature of the data is the discontinuity at TIT, = 0.6, which is interpreted as arising from a first-order phase transition in the vortex core (since, as discussed in sections 4.1 and 4.2, the core structure determines the parameters A and K ) . More recent NMR experiments (mentioned in Pekola et al. 1984) have mapped out this transition in the p T plane (see fig. 6); it appears to approach the normal-superfluid transition line near the polycritical point (about 20 bar), but the small value of the NMR shift in the GL regime makes quantitative studies difficult. At present, there is no general agreement on the character of this phase transition, and much additional effort will be required. This question remains one of the outstanding problems associated with rotating 'He-B.

Ch. 1, 541

VORTICES IN ROTATING SUPERFLUID ’He 10

I

1 ’He

“1

I

SOLID

2.0

1.5 T

41

25

3.0

(mlo

Fig. 6. Observed transitionsin’He-B. The Odenote the transition seen in fig. 4usingNMR; the + and denote that inferred with the gyroscopic techniques (Pekola et al. 1984, reprinted by permission of the authors and the American Physical Society).

Pekola et al. (1984) have studied the same phenomenon with a rotating powder-filled torus that performs torsional oscillations. As seen in fig. 6, the resulting phase transition differs somewhat from that determined with NMR. The gyroscopic measurements presumably involve vortex lines in small interstices, which need not be similar to those in the open geometry used for the NMR experiments. Thus the precise connection between the two techniques remains unclear. The same formalism also describes longitudinal NMR in rotating cylinders (Theodorakis and Fetter 1983), although the corresponding experiments have not been performed. The principal difference from eq. (4.62) for the transverse spin-value modes is the appearance of the longitudinal potential cos2 p, which is always negative relative to the bulk value for p = 0 and is most attractive near the boundary where cos2 p = 0.2. Thus the longitudinal spin-wave modes are concentrated at the outer edge of the cylinder and could provide a useful probe of boundary conditions. As in the case of the transverse modes, the effect of rotation occurs through the change in the overall equilibrium ri texture, which correspondingly alters the position and spacing of the spin-wave eigenvalues. Since the same ri texture determines both the transverse and longitudinal “potentials”, experimental detection of the longitudinal resonances would help confirm the present theoretical picture of the rotating B phase. In principle, other techniques could detect the vortex lines in rotating 3 He-B (or in turbulent B-phase counterflow). Two obvious possibilities

A.L. FETTER

42

[Ch. 1, $5

that have been suggested for the A phase (see section 5.4) are ions and zero sound. In each case, the intrinsic A-phase anisotropy of ionic mobility and zero-sound attenuation plays a crucial role, whereas these properties are effectively isotropic in the B phase. Thus ions in 'He-B are more likely to behave as in 4He, where trapping on the vortex cores is the central feature (see, for example, Fetter 1976). It must be noted that the ionic radius is of order 10-20 8, in both liquids; this value is large compared with the vortex core (a few A) in 4He, yet small compared with that in 3He-B (a few hundred A). Mineev and Salomaa (1984) predict that the attractive potential between the B-phase vortex and the ion is rather small, suggesting that vortices in rotating 'He-B will at best trap ions only weakly. Experiments at the Helsinki University of Technology plan to study this question in detail, and additional theoretical work will doubtless be required.

5. Vortices in superfluid 'He-A In the A phase, the order parameter is intrinsically more complicated than in the B phase, and properties such as the hydrodynamic variables and the free-energy density merit separate consideration (section 5.1). The resulting rich internal structure of the order parameter allows both singular and nonsingular vortices, with various quantum numbers for the circulation (section 5.2). The NMR spectrum for these vortices is studied in section 5.3, and other possible detection techniques are discussed in section 5.4.

5.1. GENERAL PROPERTIES OF 'HE-A In the A phase of superfluid 'He, the triplet p-wave order parameter A P j factorizes into a direct product of vectors, a real one for the spin component p. and a complex one for the orbital component j

As in eq. (4.2), the overall coefficient A ( T ) is a temperature-dependent amplitude, given in the GL regime by

A: = a / 4 P 2 , , .

(5.2)

In contrast, the remaining (intrinsically complex) structure of the order parameter for the stationary bulk A phase is very different from the real rotation matrix of the stationary bulk B phase. Specifically, the unit vector

Ch. 1, 551


43

d is perpendicular to the unit spin of the Cooper pairs, which will be important (section 5.3)in understanding the NMR. In addition, the orbital part of the order parameter consists of two orthogonal unit vectors A, and A,, combined in the complex form A, + id,. This structure incorporates a preferred sense of rotation, defined with the right-hand rule by a third orthogonal unit vector i= A, X A,; thus A,, A,, and i constitute an orthonormal triad. Apart from its overall magnitude A( T), complete specification of the A-phase order parameter involves the orientation of the unit vector 2 and the orthonormal triad A,, A,, and i, all of which can vary in space, depending on the boundaries and external perturbations. The unit vector i has the physical interpretation as the orbital angular momentum of the Cooper pair. Furthermore, the energy gap in the A phase is anisotropic, with uniaxial symmetry relative to the local direction of i and magnitude given by A( T) sin(&), where O,, is the angle between i and the direction i on the Fermi surface. Note that A( T) is the maximum energy gap, occurring at the equator of the Fermi sphere, with nodes at the poles (defined relative to i). These nodes lead to special and unexpected divergences in the low-temperature hydrodynamics (Cross 1975, Brinkman and Cross 1978), and it remains an interesting challenge to detect such be.havior experimentally (for example, by applying sufficiently large magnetic fields to stabilize the A phase for all temperatures below T,). To interpret the remaining orbital vectors A, and A,, it is first necessary to consider several matters of principle (see, for example, Mermin 1978): How is the supertluid velocity v, defined? What constitutes a vortex in the A phase? What contributes to the mass supercurrent j , and associated angular momentum density r X j,? For simplicity, consider the special case of a constant i vector (i= i), and a constant orbital order parameter f + i j . If this system then undergoes a uniform translation with velocity

v, = hk/2m,

(5.3)

[compare the discussion above eq. (2.1)], the resulting orbital vector in the order parameter merely acquires an additional phase factor

A,

+ i s , = e""(i + i j ) ,

(5.4a)

which follows either from the GL kinetic-energy density (3.3) and the corresponding expression for the mass current (see, for example, Fetter 1975), or from the general properties of Galilean transformations (Ho 1978a, Mermin 1978). A simple calculation shows that this expression can be rewritten as

A.L. FE’ITER

44

A,

+ iA2 = [icos(k

*

(Ch. 1, $5

r ) - y sin(k r ) ]

+ i [ i sin(k. r ) + j

-

cos(k r ) ] ,

(5.4b)

which represents a rotation of complex vector pair i + i j about the i-axis through a spatially varying angle - k r. Thus the orbital part of the order parameter acquires a helical form, analogous to the polarization vectors for circularly polarized light, with i as the symmetry axis. More generally, uniform flow with velocity usinduces a spatial rotation of the orbital part of the order parameter about the orthogonal direction i, with magnitude -(2m,u,lh) r. For many purposes, it is preferable to use the equivalent prescription

-

-

which defines the superfluid velocity even for nonuniform flow. Substitution of eqs. (5.1) and (5.4a) for constant 2 into the G L kinetic-energy density (3.3) for equal K,’s yields

Use of eq. (5.3) shows that the superfluid density tensor p, has the general form

This A-phase anisotropy, which holds outside the G L regime, is of relative order unity. The preceding treatment considered only the special case of a constant i vector, but i t is not difficult to allow for spatial variations. Equation (5.5) continues to define the superfluid velocity; in contrast, the supercurrent now contains additional terms (similar to the c curl M contribution to the electromagnetic current in a nonuniform ferromagnet) j , = p,u,

+ C curl 1 .

(5.8)

Here, the tcnsor C has a form similar to p, in eq. (5.7a)

c,,= c:q,

A

-

^

C0lll,,

(5.9a)

Ch. 1 , 851


45

where (in the GL regime)

C = 4(h/2m3)p,,, C, = 2C,

(5.9b)

and the generalization to lower temperatures merely renormalizes the various coefficients in eq. (5.8) (Cross 1975, Brinkman and Cross 1978). The same situation holds for the gradient contribution to the free-energy density (including spatial variations of 2). It can be expressed either in the GL form near T, [substituting eq. (5.1) into eq. ( 3 . 3 ) ] , or in the “hydrodynamic” form

which holds throughout the A-phase region of thep-T plane. Near T,, the various coefficients are readily expressed in terms of the three GL bending coefficients { K , } , and the generalization to lower temperatures (including Fermi-liquid corrections) can be found in Cross (1975), Brinkman and Cross (1978), Fetter (1979) and Fetter et al. (1983). Equations (5.8) and (5.10) together constitute the “hydrodynamic” theory that is valid wherever the order parameter retains its A-phase form in eq. (5.1). This approximate description requires that the various unit vectors d, Al,A’, and i vary slowly on the scale of the coherence length; it ceases to hold in singular regions, where eq. (5.1) fails. The hydrodynamic theory has the virtue of providing an explicit form for the free-energy density, but it involves a potential source of confusion. Specifically, usand i appear to be independent orbital variables (implying five distinct orbital parameters, since i is a unit vector), when, in fact, specification of the orbital part of the order parameter requires only three independent parameters (since it is equivalent to a rigid orthonormal triad). Thus the set us and i must obey additional constraints, which have the form (Mermin and Ho 1976) a;u,, - aju,; = ( h / 2 m 3 ) i (a;i . x dji).

(5.11)

This Mermin-Ho condition generalizes the irrotational condition ( 2 . 3 ) on usin superfluid 4He (and in the B phase). In the present case of 3He-A,eq. (5.11) shows that the superfluid vorticity no longer vanishes identically;

46

A.L. FETTER

[Ch. 1 , 05

instead, it is fully determined by the spatial variations of the 1^ vector. Although curl usindeed vanishes for a uniform i texture, the more general situation will be of great interest in connection with vortices in the rotating A phase. For superfluid 4He, the phase function @ characterizes the single degree of freedom associated with the order parameter at a given temperature T. Its relation to usis made explicit in eq. (2.1), which automatically enforces the irrotational condition in eq. (2.3). The same general relation applies to the B phase, for, apart from the deformation near the vortex core, the phase function @ again characterizes the single degree of freedom in the order parameter. However, the situation is more complicated for 3He-A because the orbital triad has three distinct degrees of freedom. For many purposes, it is helpful to introduce specific parametrizations for the orbital variables; this will be seen to enforce the Mermin-Ho condition automatically. Among the various ways to characterize the orientation of the orbital triad relative to some fixed reference set i ,y^, i,the Euler angles a,p, and y (see fig. 7, and Fetter and Walecka 1980, section 29) are especially convenient, but one must be aware of possible spurious singularities in 1y if sin p vanishes (Mermin 1978, section 111, Mermin 1979, p. 596). In the

I

/

I/

/

I I

I

I

Fig. 7 . Schematic diagram of the Euler angles a.Band y , used to define the orientation of the orbital triad for superfluid 'He-A.

Ch. 1, SS]


47

familiar quantum-mechanics convention (Edmonds 1957), the angles p and a are the usual spherical-polar coordinates of the vector i on the unit sphere, with p the polar angle relative to i, and a the azimuthal angle relative to i .The remaining angle y then represents an additional rotation about the direction i. To be precise, define the auxiliary orthonormal triad

i=

(i cos a

+j

sin a ) sin p

+ i cos p ,

rii = (icos a + j sin a ) cos p i = -i sin (Y

+j

- i sin

p,

(5.12)

cos (Y .

The full orbital part of the order parameter in eq. (5.1) then has the representation

A1 + iA, = e-'Y(i

+i i ) ,

(5.13)

so that spatial variations of the Eule! angles induce corresponding variations in the order parameter A, + id,. To relate the superfluid velocity to the Euler angles, recall that the superfluid velocity at some point r in 4He was defined in terms of the change in the order parameter on moving from r to some nearby point r + Sr [see eq. (2.2)]. A similar approach will apply for 3He-A, but the details are more complicated naturally. It is evident that the infinitesimal change in the rigid orbital triad must represent an infinitesimal rotation Sa proportional to Sr. Since an infinitesimal rotation acts like a vector (Fetter and Walecka 1980, section 7), S a can be decomposed by projecting it onto the original orthonormal triad. The part of Sn perpendicular to i corresponds to a change in the direction of i,correctly requiring two free parameters because iis a unit vector. The remaining part of S f l parallel to i represents a rotation about the local direction i, with magnitude S a i. The discussion below eq. (5.4b) relates this local rotation of the orbital triad about i to the superfluid velocity

-

(5.14a)

-

and elementary geometry shows that Sa i = Sy + cos p &a (see fig. 7). Comparison of these expressions yields the fundamental result (5.14b) which also follows from eqs. ( 5 . 5 ) and (5.13). This relation replaces the

A.L. FEITER

48

(Ch. 1, 55

simpler eqs. (2.1) for ‘He and (4.5a) for 3He-B; it has several striking features: ( 1 ) Whenever the spatial variations of the orbital triad involve a local rotation about the direction i. there is a corresponding local superfluid velocity us and vice versa. (2) The superfluid vorticity does not, in general, vanish. Instead, a direct calculation gives curl u, = sin p (Vp x Va),

(5.15)

so that nonzero curl u, requires simultaneous changes in both a and p. Apart from singularities in the third Euler angle y, the superfluid vorticity is wholly determined by i a n d its derivatives [see eq. (5.12)], which merely restates the Mermin-Ho (1976) condition from eq. (5.11). This distributed vorticity differs qualitatively from the line singularities of the B phase and superfluid 4He. (3) The angles a and y a r e “phase-like”, in that the single-valuedness of the orientation of the orbital triad requires them to return to their original values modiilo 2~ on traversing some closed path C in the fluid. Thus the superfluid circulation us d s in the A phase involves a quantized part [from the first term of eq. (5.14b)], and another (in general) nonquantized part because of the presence of the factor cos p in the second term of eq. (5.14b). This nonquantized circulation in ’He-A is a direct consequence of the distributed superfluid vorticity. Two other contributions to the free-energy density from section 3 must be rewritten in specific A-phase form. The first is the dipole-energy density f,. Direct substitution of the A-phase order parameter (5.1) into eq. (3.5) yields an irrelevant isotropic term and the following orienting term

I,. -

fd =

-2g,AZ,(d-

i)’ ,

(5.16a)

which explicitly exhibits the coupling between the spin and orbital degrees of freedom. More generally, eqs. (3.6) and (5.7) can reexpress fd in hydrodynamic form fd

=

-~Pa(ti/2rn,)’l,,2(d.i)l1

(5.16b)

which holds outside the GL regime. The dipole coupling acts to align d and l” in a bulk fluid, and deviations from this oriented configuration are expected only within small regions of characteristic size L, (=lo p n ) . The other contribution to the energy arises from the magnetic freeenergy density (3.7), which, for the A phase, becomes

Ch. 1. 851


f ,= 2g,A:(d

- H)’ .

49

(5.17a)

Evidently, the magnetic field tends to orient d perpendicular to its local direction. Using eq. (3.8) and the previous relations yields the equivalent (but more general) hydrodynamic form

f,

=

~ ~ l l ( f i 1 2 m , ) 2 ( H / H , L ,ri)’ ) 2 ( d. ~

(5.17b)

Comparison of eqs. (5.16b) and (5.17b) indicates that the orienting effect of the magnetic field predominates if H 9 Hd,in which case d tends to lie in the plane perpendicular to 8 ;the dipole coupling then orients i along d, apart from regions of characteristic size L,. In the hydrodynamic (or “London”) theory (Mermin 1978) of 3He-A, the spin vector d and the orbital triad serve as dynamical variables whose spatial orientations vary slowly on the scale of the GL coherence length (3.4). The corresponding total hydrodynamic free-energy density, given by the sum of eqs. (5.10), (5.16b) and (5.17b), holds throughout the A-phase region of the p - T plane. Since this description assumes that the order parameter retains its A-phase form, it can never provide a full description of singular regions, such as singular vortex cores, where the order parameter deviates significantly from eq. (5.1). In such cases (discussed in section 5 . 2 ) , the full G L theory from section 3 must be used to incorporate local regions with more general order parameters, as was done in section 4.1 for the (non-B-phase) core of a vortex in the B phase. Unfortunately, this latter approach is restricted to the vicinity of T , , and the construction of a uniform theory throughout the phase diagram remains for future investigations. Apart from a few special cases, the hydrodynamic theory suffices for a description of vortices in rotating 3He-A. Near the phase transition, it explicitly reduces to the G L form, but it otherwise includes Fermi-liquid corrections and the anisotropic energy gap (Cross 1975, Brinkman and Cross 1978). Near the melting pressure, the various hydrodynamic coefficients have been evaluated explicitly (Fetter 1979, Williams 1979), and Fetter et al. (1983) provide a brief table. The gradient energy (5.10) assumes that d and i are independent, each with its own set of bending coefficients. If they each vary slowly on the scale of L,, however, the dipole energy (5.16) locks them together. Equation (5.10) can then be simplified considerably by setting d = i; the resulting expression retains the same form (omitting the last two terms) but with renormalized dipole-locked bending coefficients KI or K L , also tabulated in Fetter et al. (1983).

A.L. FETTER

50

[Ch. 1. $5

The equilibrium configuration of 'He-A in a rotating container is determined by minimizing the total free energy in the rotating frame. The general transformation to a reference frame in uniform motion is slightly intricate. involving both Galilean and Legendre transformations (Ho 1978a. b, Mermin 1978). One unusual feature for 'He-A is that both the normal and superfluid densities are anisotropic [see eq. (5.7)]. Similar considerations apply to a frame in uniform rotation. Detailed calculations, analogous to those in eqs. (2.7)-(2.9) for 'He, verify that the normal component executes solid-body rotation (2.6) (Fujita et al. 1978, Williams and Fetter 1979); furthermore, the transformed free-energy densityfin the rotating frame follows from f f o r 0 = u, = 0 with a simple substitution. In the GL regime, the gradients V in eq. (3.3) are replaced by V i(2rn3lh)l2 X r, analogous to the minimal gauge substitution in the Schrodinger equation for a charged particle in a uniform magnetic field. In the hydrodynamic theory, the superfluid velocity us is replaced by the relative velocity u, - u,, and an additional additive term can be omitted, as in eq. (2.9). Although the normal-fluid corrections are generally small, they are important in comparing the free energy of different proposed vortex configurations.

5.2. SPECIFIC VORTICES

IN

'HE-A

Consider a long cylinder of radius R( % Ld) filled with liquid 3He in the normal phase. When cooled slowly into the superfluid A phase, the system is expected to attain the state of lowest free energy. If the container is at rest in zero magnetic field, the resulting texture, first studied in detail by Mermin and Ho (1976), has unit quantized circulation about the boundary, even though it remains nonsingular throughout the container. Suppose instead that the container rotates at fixed angular velocity about its symmetry axis. When cooled into the superfluid A phase at constant 0 (and again zero field), the Mermin-Ho (MH) texture persists essentially unchanged up to a (generally low) critical angular velocity = 10(h/2rn3R'), above which the texture changes to one with three units of circulation at the boundary, still remaining nonsingular everywhere (Williams and Fetter 1979). For somewhat larger angular velocities, the sequence of states is not known, but the situation eventually becomes simple when the total circulation about the boundary contains many quanta (0B @,). In this latter case, the boundary plays a negligible role, and the system can be treated as a bulk rotating fluid. For zero magnetic field and moderate rotation speeds, a square array of nonsingular textures, similar to that proposed by Mermin and H o (1976), is believed to have the lowest free energy. At still higher angular speeds (0>O.lh/2m3L; =

n,,

Ch. 1, 151


51

30 rad/s), a transition to a triangular lattice of singular vortices has been predicted (Fetter et al. 1983), and similar classes of singular states should persist up to the upper critical field OC2= h/8m3,f2(of order 10" rad/s), when the singular cores of radius 6 begin to overlap (Schopohl 1980). The situation is quite different in large magnetic fields (H %- H d ) parallel to the rotation axis, since (as seen below) the MH texture is strongly suppressed. In this case, the overall texture has b and i predominantly uniform and aligned in the plane perpendicular to H,but with isolated vortex cores of characteristic dimension L, = 10 km arranged on a (presumably triangular) lattice (Volovik and Hakonen 1981, Salomaa and Volovik 1983a). In principle, these cores can be either singular (and singly quantized) or nonsingular (and doubly quantized). Theoretically, the former seem to have lower free energy, but (see section 5.3) experiments (Seppala et al. 1984) provide strong evidence for the latter textures. This discrepancy remains unresolved. Since a magnetic field has a dramatic effect on the rotating A phase, it is helpful to consider separately the two limiting cases of H I H , G 1 and Bl. The following discussion will amplify the preceding brief summary, paying particular attention to the difference between singular and nonsingular textures. 5.2.1. Singular and nonsingular textures As noted in section 5.1, an orbital texture in 3He-A is specified by giving the orientation of the rigid orthonormal triad A,, A2, and i at every point of the fluid. A nonsingular texture is one for which the configuration is well defined everywhere, whereas a singular texture typically has one or more regions where the orientation has different values depending on the path used to reach the particular point in question. This idea is familiar in the case of supertluid 4He and 3He-B, where the phase variable CP is equivalent to the orientation of a unit vector in a plane. Figure 1 illustrates various singular configurations for circulation p = ? 1; each different integral circulation state represents a distinct type of singularity, implying an infinite set, one for each integer. The situation can be clarified by recognizing that the phase variable @ is topologically equivalent to the angular variable around the circumference of a circle or a cylinder. On traversing a single closed path in physical space, the single-valuedness requires that CP return either to its original value or to one that differs by 2 r p , where the integer p characterizes the number of times that the phase angle increases by 2 r . Thus a p-fold circulation in 4He is analogous to a rubber band wrapped p times around a cylinder, and states with different winding numbers p are topologically distinct, because they cannot be

A.L. FEITER

S2

[Ch. 1. 55

continuously deformed into each other (in other words, without cutting the rubber band). To appreciate the crucial role of the order parameter’s intrinsic structure, consider a Heisenberg ferromagnet. The effective order parameter is now a three-dimensional spin vector with fixed length, whose orientation is equivalent to the position on the surface of a unit sphere. This system differs profoundly from the previous one, for a rubber band around the equator of a sphere can be removed merely by pulling it to one pole. As a result, a ferromagnet has no stable line defects (Mermin 1979). For 3He-A, the remarkable new feature is that it has only a small number of distinct singularities or defects, and many apparently different textures are. in fact, topologically equivalent. In particular, if 2 and i are dipole-locked, a general theorem (Volovik and Mineev 1977, Mermin 1978, 1979) asserts that there are only two distinct defects, with quantum numbers zero and one; all other configurations can be continuously deformed into one or the other of these. Although explicit construction of such deformations is rarely easy, the general theory ensures their existence. The Euler angles (fig. 7) provide a useful way to think about this question. The angle /3 ranges from 0 to n-, but the angles a and y can increase without limit and are analogous to the phase angle Q, for superfluid ‘He. Many of the unusual properties of 3He-A arise from the presence of two such functions in specifying the orbital order parameter. In general, line singularities in a or y lead to vortices and singular textures, and several different possibilities exist. Suppose that a and /3 are constant (so that i is constant). The superfluid velocity then becomes [see eq. (5.14b)l u, = - ( h / 2 m 3 ) V y ,

(5.18a)

essentially the same as eqs. (2.1) and (4.5a) for ‘He and 3He-B. Thus any singularity in y produces a “phase” vortex, with quantized circulation. A simple example is to take 7 = -PdJ + Y”

7

(5.18b)

where 4 is the polar angle in cylindrical coordinates. The resulting circulating velocity p f i /2m,r represents a p-fold quantized vortex line, whose divergent behavior near the center forces the order parameter to vanish for r + 0. This “normal” core is analogous to that for the o vortex in ‘He-B [see the discussion below eq. (4.18)]. A second important case is to take y constant, so that U, =

- ( h /2rn,) cos p V a

( 5.19a)

Ch. 1 , 551


53

Generally, a line singularity in a produces both a singular superflow (unless cos P vanishes there) and [see eq. (5.12)] a singular i texture (unless sin /3 vanishes there). In any case, however, the free-energy density diverges at the singularity, which represents the vortex core. One simple example is to take a

= p 4 + 010

9

P = P(r)

7

(5.19b)

where P ( r ) increases smoothly from f T at the center to 7r far from the core. Thus i = -? asymptotically, and the corresponding superfluid velocity looks like a p-fold quantized vortex line. With decreasing r , however, i rotates to a horizontal orientation, so that u, in eq. (5.19a) is everywhere bounded. Nevertheless, the texture is still singular because i itself has a “disgyration” (shown in’fig. 1 for p = ?1). Muzikar (1978) has considered various possible core structures for p = 1 and finds that an A-phase radial disgyration changes continuously into the “polar” phase for r < 6. Since the polar phase is not found in bulk superfluid 3He, this behavior is similar to that of the v vortex in the B phase. The preceding discussion shows that singularities in either a or y typically generate singular textures, and the same situation usually holds if a and y both have singularities. In certain cases, however, the singularities cancel, leaving a nonsingular texture that varies smoothly throughout the fluid. This possibility arises because the rotations labeled by a and y are equivalent if P = 0, so that only the sum a + y is relevant. In this case, the superfluid velocity in eq. (5.14b) contains only V ( a + y ) , and opposite singularities in a and y can therefore yield a nonsingular us. One simple and important example is the MH texture (fig. 8a), in which the orbital triad lies along i ,y^, i at the center, with i upward in the vortex core. To define the global configuration, move outward in a radial direction, rotating the triad by 17r about the azimuthal direction; consequently, far from the center, i points radially outward. It is evident that the resulting texture is nonsingular everywhere, yct an examination of fig. 8a shows that the orbital triad executes a single negative rotation about the (radial) i direction on moving once around a large circle at infinity. Thus, by the discussion of eq. (5.14a), there is a net circulating superfluid velocity at infinity with unit quantum number ( p = 1). It is not difficult to see that this MH texture has the equivalent Euler-angle representation

where p ( 0 ) = 0 and P ( r ) = i7r for r + = . velocity is

The corresponding superfluid

[Ch. 1, 85

A.L. FETTER

54

E ?

*

--*

(a)

Fig. 8. Typical nonsingular singly quantized vortices in 'He-A. The orientation of the orbital triad at different locations shows the rotation of I from vertical at the center to horizontal far away, as well as the single negative asymptotic rotation about ion once encircling the core. Neither (a) the Mermin-Ho (1976) texture nor (b) the mixt-twist texture (Ho 1978a) can tile the plane by itself.

with distributed vorticity ( h/2rn,r) sinp (dp/dr)z^. Since p ( r ) approaches was r--* z.u, looks asymptotically like a singly quantized vortex; near the origin, however, P ( r ) tends linearly to 0, and u, vanishes smoothly. In addition, i i s uniform in the core region, so that the MH texture is indeed nonsingular. With the additional assumption of dipole locking (2 = i),the resulting texture is believed to represent the ground state of 3He-A in a long cylinder of radius R >> L, and zero magnetic field, because it

Ch. 1, 551


55

automatically satisfies the boundary condition that i be normal to the walls. Apart from nonunitary corrections that are important only in the currently unattainable regime R 6 506 (Passvogel et al. 1984), the order parameter retains its A-phase form everywhere. Thus the hydrodynamic theory is applicable, and detailed calculations show that P ( r ) increases nearly linearly from the center to the outer boundary (Buchholtz and Fetter 1977). The resulting orientational part of the free energy per unit length is independent of R, because the spatial derivatives (of order R - ), cancel the R Zfactor from the cross-sectional area. This texture is also interesting because it has a permanent ground-state orbital angular momentum =( p , / p ) h per particle. More generally, adding a constant a, to a in eq. (5.20a) rotates i uniformly about the z^ direction by angle a, [see fig. 1 for several different values of a,; Maki (1983) and Zotos and Maki (1984) have called the textures for 0 and T “radial”, and that for ;T “circular”]. Although such textures obviously violate the boundary conditions for a cylinder, they are of interest as candidates for vortices in the rotating A phase. In addition, they have different divergence and curl, and hence different free energies, even though us remains unaffected. Detailed calculations (Fetter et al. 1983) incorporating particle conservation show that a,, near + T minimizes the free energy, but the numerical differences are not large. Anderson and Toulouse (1977) have suggested another, closely related, nonsingular texture; it differs from eq. (5.20a) and fig. 8a only in the asymptotic boundary condition that P ( r ) = IT for r - w . Equation (5.20b) shows that the Anderson-Toulouse vortex is doubly quantized asymptotically, yet it again has distributed vorticity and remains nonsingular throughout the fluid. As will be seen below, these vortices can tile the plane, because their asymptotic i texture is uniform, in contrast to the MH vortices, whose asymptotic i texture involves a net rotation. One other nonsingular texture is important, in which i points down at the center, so that P ( 0 ) = T . The superfluid velocity in the core [eq. (5.14b)l now involves V ( a - y ) , and the two phase-like angles now must have the same singularities to produce a nonsingular texture. A particular example [called “mixt-twist” by Ho (1978a) and “hyperbolic” by Maki (1983) and by Zotos and Maki (1984)] arises from the choice

’

(5.2 1a) with boundary values P ( 0 ) = 7~ and p ( r ) = 4 T for r ing superfluid velocity

0s.

The correspond-

(5.21b)

A.L. FETTER

56

[Ch. 1, $5

again looks asymptotically like a singly quantized positive ( p = 1) vortex with distributed vorticity throughout the core. The mixt-twist (MT) i texture is analogous to the phase pattern in fig. Id, since the 1 vector at infinity rotates once in the negative sense on once encircling the origin (see fig. 8b). Although both the MH and MT textures have uniform and continuous cores, it is evident that they have very different asymptotic symmetries. These particular nonsingular textures with asymptotic circulation by no means exhaust the various possibilities, and other qualitatively different nonsingular configurations will be considered in connection with high magnetic fields. At present, however, these structures suffice for a description of rotating superfluid 'He-A in low magnetic fields.

5.2.2. Textures in low magneric fields The transformation to the free-energy density f for 'He-A in a frame rotating with constant angular velocity Oi has been considered at the end of section 5. I . As in the analogous case for 'He [see the discussion below eq. (2.11)]. the bulk rotating A phase is expected to have an array of vortices, arranged on a regular lattice. If each unit cell has a net superfluid circulation ph/2m3, then eq. (4.6a) shows that its area is A,

=

rph/2m3R.

(5.22)

The order parameter will depend only on the two-dimensional coordinate r in the xy-plane, and i and u, - u, both must have the periodicity of the lattice. In particular [see eq. (5.12)], the Euler angle a(.) should be periodic modulo 2 7 ~ .If, in addition, a ( r ) is continuous, the same arguments used in obtaining eq. (2.12) imply that Va - ( r q / A C ) ix r must be periodic, where the integer q is the net winding number of a on once encircling the unit cell (in general, q differs from the circulation quantum p mentioned above, since p involves the winding numbers of both a and y ). Since Va is strictly periodic, the second term violates this periodicity unless q = 0 (or a is not continuous, contradicting the initial assumption). For example. neither the MH nor the M T vortices can form lattices by themselves, because each has a net winding number in a, with q # 0. A texture with no net winding number of a has been proposed independently by H o (1978a) and by Fujita et al. (1978). in which a ( r ) has p = 1 singularities at the corners of a square lattice and p = - 1 singularities at the center of each square (fig. 9). The unit cell contains equal numbers of positive and negative singularities, so that the net circulation of a vanishes. In addition. t h e i texture flares upward at the

Ch. 1. 551


57

p = 1 singularities (0’s in fig. 9), and downward at the p = - 1 singularities ( X’S in fig. 9), with the polar angle p(r) having the symmetry of the lattice. This i texture (called “type-I” by Fujita et al. 1978) can be considered as a pair of interpenetrating square lattices (like two-dimensional NaCI), one with MH textures (the 0’s in fig. 9) and on with MT textures (the X ’ S in fig. 9). The preceding analysis of these separate structures shows that the overall i texture has finite elastic energy density everywhere; to ensure that us is also finite, it suffices merely to choose the remaining Euler angle y ( r ) to have p = -1 singularities at both sets of sites. In effect, a given square contains two singly quantized vortices, and [see eq. (5.22)] the nearestneighbor separation between like vortices is b = ( ~ /m,O)”*, h because p = 2 in a given square cell. To calculate the free energy of this nonsingular vortex lattice, recall that the excess (gradient) free energy of a MH texture in a cylinder is independent of the cylinder’s radius R, because the polar angle /3 varies smoothly on the same characteristic scale R [see the discussion below eq. (5.20b)l. In effect, R serves as the core radius of the nonsingular texture. A similar situation holds here as well: the polar angle p ( r ) varies smoothly (on the scale of the lattice spacing b) from 0 at the up-sites to T at the down-sites, with b also determining the characteristic core size. Thus spatial gradients are of order b-’, and the quadratic character of fk in eq. (5.10) means that the free energy per unit cell integrated over the area bZis

Fig. 9. Type-I lattice of nonsingular singly quantized vortices in rotating ’He-A at low magnetic fields (Fujita et al. 1978). The unit cell encloses two MH (0sites) and two MT ( X sites) vortex textures, shown individually in fig. 8 (Fetter et al. 1983, reprinted by permission of the authors and the American Physical Society).

58

A.L. FETTER

[Ch. 1, 15

a sum of the various bending constants with dimensionless coefficients, independent of the parameter 6 . This property arises directly from the nonsingular nature of the texture, since, otherwise, additional length scales introduce logarithmic factors (as seen below for singular textures or nonzero magnetic fields). Fujita et al. (1978) have evaluated the dipolelocked free energy of the type-I lattice per unit cell numerically, explicitly incorporating the four-fold lattice symmetry. A simpler but approximate approach is the Wigner-Seitz (WS) model (Fetter et al. 1983). in which the MH and MT vortices are treated separately, each confined to circular cells of area A c / 2[see eq. (5.22)] and radius a, = (fi/2rn3R)”’,as in eq. (4.6b). For the currently available rotation speeds R ( s l rad/s), the WS cell radius a, (=lo0 pm) far exceeds L, (=lo pm). so that deviations from dipole locking are insignificant. In this regime, both calculations (Fujita et al. 1978, Fetter et al. 1983) conclude that the type-I lattice has lower free energy per quantum of circulation than other candidates [such as a triangular lattice of nonsingular Anderson-Toulouse doubly quantized vortices (Volovik and Kopnin 1977). described in the previous section, or specific singular vortices, to be considered below; both these vortices can tile the xy-plane since i = -2 far from the core]. In contrast to the scale invariance of the nonsingular vortices, singular vortices involve additional length scales associated with the internal structure. One important example is that in eq. (5.19) (called the z vortex by Fetter et al. 1983), where i and 2 are dipole-locked along -2 far from the core, with kinetic energy arising from the unit circulating velocity. For r s L,, however, the texture can lower the kinetic energy by rotating 1 from vertical to radial. at the cost of introducing dipole energy (since d remains along -i). The resulting radial disgyration in i has a singular div i near the origin, forcing the order parameter to change character from A phase to polar phase inside an inner core of characteristic radius 5 (Muzikar 1978). In the WS approximation, the resulting free energy per quantum of circulation has additional logarithmic factors proportional to In(a,/L,) and In(L,/,$). The first of these decreases with increasing angular speed, and detailed calculations show that a (presumably triangular) lattice of these z vortices becomes favored at a temperature-dependent angular velocity that rises rapidly with decreasing temperature from about 30 rad/s for T/ T , = 0.99. Since this angular velocity considerably exceeds current experimental ones, rotating ‘He-A in zero magnetic field should quite generally contain a type-I nonsingular vortex lattice. 5.2.3. Textures in high magnetic @el&

For rotating ‘He-A in a high magnetic field (H + H,),the magnetic energy

Ch. 1, ‘351


59

density (5.17) forces the 2 vector to lie perpendicular to H (assumed, for simplicity, parallel to the rotation axis 2; the more general case is discussed briefly at the end of this section). Furthermore, the curvature energy density [the last two terms of eq. (5.10)] acts to keep auniform, which can be assumed constant (2 = i , say) throughout most of the sample. The dipole coupling energy in eq. (5.16) then ensures that i is also essentially uniform, minimizing the corresponding part of the curvature energy (5.10). As seen in comparing eqs. (5.10) and (5.16), however, 2 and ican become unlocked within regions of characteristic size L,, where i deviates from its asymptotic uniform texture; this region represents the core of the high-field vortex [Volovik and Hakonen (1981), called the “soft core” by Salomaa and Volovik (1983a)l. In principle, also can deviate from i within the smaller distance L , ( H , / H ) ; in most cases, however, this effect turns out to be unimportant, and 2 is frequently assumed to be uniform throughout the fluid. Seppala and Volovik (1983) have proposed two remarkable candidates for the nonaxisymmetric vortex textures in rotating 3He-A in a large magnetic field. Both have a dipole-locked asymptotic region far from the core ( r % Ld), where i and 2 are uniform and parallel (say, to i ) .One of these is singly quantized and singular; its outer region can be obtained from that of the MH texture by rotating the orbital triad in fig. 8a about -i through the polar angle 4, thus rotating i from radial to the i direction (see fig. 1Oa). It is obvious by inspection that this modified texture still involves a single negative rotation of Al + iA2 about i ( = i )on once completing a large circle around the core, so that it indeed represents a singly quantized vortex [see eq. (5.4b) and the following discussion]. To avoid a divergent superfluid velocity in the core, an additional rotation of 1.. about the direction -A2 on moving radially inward brings A, parallel to i in the core (see fig. 10a). Consequently, the infinitesimal rotations near the center involve only a local reorientation of the direction i, ensuring that us vanishes there [see the discussion above eq. (5.14a)l. It is also clear from inspection of fig. 10a that this texture remains singular at the center, where i has no unique direction. For a quantitative analysis of this texture, it is convenient to parametrize the orbital part 8, + i& of the order parameter by

i1= i cos 7) - ( j sin 4 + i cos 4) sin 7 , &=jcos$-isin+,

(5.23a)

where this modified notation will facilitate comparison with eq. (5.24a) below. Here the angle T ( r ) varies smoothly from 0 near the origin to 1 mfor r S= L,, and a possible additive constant in the angle 4 has been omitted.

A.L. FE'ITER

[Ch. 1, 95

n' ?

(b)

Fig. 10. Typical nonaxisymmetric vortices proposed by Seppala and Volovik (1983) for rotating 'He-A in high magnetic fields. (a) Singular singly quantized texture (5.23) shows the single negative asymptotic rotation about i o n once encircling the core; the core clearly has singular structure. (b) Nonsingular doubly quantized texture (5.24) shows the double negative asymptotic rotation about ion once encircling the core; the core clearly has a unique structure.

The corresponding i texture and u, become (5.23b) confirming that the circulating velocity field looks asymptotically like a singly quantized vortex yet vanishes smoothly inside the core [compare the discussion below eq. (5.19)]. The singularity in the i vector for r % L ,

Ch. 1, §5]


61

means that the order parameter ultimately must depart from the A-phase form as r - 0 , like Muzikar's (1978) proposed polar core for a radial disgyration. It is also evident from fig. 10a that i deviates from d ( = i ) inside the core, and numerical studies (Seppala and Volovik 1983, Vulovic et al. 1984) based on the WS circular-cell model with constant d indicate that this core region has a radius of order 2Ld. The second high-field vortex texture proposed by Seppala and Volovik (1983) is doubly quantized and nonsingular. It is obtained from eq. (5.23a) by imposing an additional negative rotation about the local i direction

A,+ i6, = e'"(t, + i & ) ,

(5.24a)

at the core to 1 7 for ~ i, eq. (5.23b) continues t o describe i, but with the extended range for q. Specifically (see fig. lob), inow rotates continuously from i in the outer region to -i at the center; it remains in the xy-plane along certain radial directions, whereas it becomes vertical along certain others. The additional phase factor in eq. (5.24a) ensures that this vortex has two quanta of circulation far from the core, yet us is nonsingular, given everywhere by and by extending the parameter 77 to run from -

T

r B L,. Since this additional rotation is purely about

(5.24b)

Furthermore, the order parameter near the center is also nonsingular, with the orbital components z^ + i j . This last property exemplifies the previous assertion at the start of section 5.2 that 3He-A has only two distinct line defects (with quantum numbers 0 and 1). Detailed numerical studies (Seppala and Volovik 1983, Vulovic et al. 1984, Seppala et al. 1984) using the WS circular-cell model for uniform d indicate that the core radius fo: the doubly quantized nonsingular vortex is =4Ld, within which distance 1 deviates significantly from d = i . It is interesting to compare the free energy of these two different textures. Although the doubly quantized vortex has more circulating superflow and therefore more hydrodynamic kinetic energy, its nonsingular core has less energy than that of the singular vortex. Seppala and Volovik (1983) conclude that their singly quantized singular vortex has slightly lower free energy per quantum of circulation for currently accessible rotation speeds ("1 radis). They predicted a transition to a lattice of doubly quantized nonsingular vortices at a critical angular velocity of order 3 r a d / s near T / T c = 0 . 8 that increases slowly with increasing temperature, and Vulovic et al. (1984) confirmed this conclusion. Nevertheless, NMR experiments (section 5.3) indicate unambiguous-

62

A.L. F E T E R

(Ch. 1, 05

ly (Seppala et al. 1984) that rotating 'He-A in high fields indeed contains nonsingular vortices, with no evidence at all for singular ones. This discrepancy between theory and experiment remains unresolved. Figure 10b illustrates an important alternative view of the high-field doubly quantized vortex, for, by inspection, the i vector lies along the rotation axis in two regions (one up and one down). Locally, these regions are similar to the MH and MT vortices (figs. 8a, b); thus the overall texture in the core of the nonsingular doubly quantized vortex can be considered a bound pair of nonsingular singly quantized vortices ( H o 1978a, Zotos and Maki 1984, Maki and Zotos 1985a), neither of which can, by themselves, form a lattice because of topological constraints. Zotos and Maki (1984) have used this picture to construct a trial function for the high-field doubly quantized nonsingular texture and confirm the (relatively large) characteristic core size. Unfortunately, their intrinsically two-dimensional variational parametrization requires a more intricate numerical analysis than eq. (5.24a). Furthermore, they did not consider the alternative possibility of singular high-field vortices [see eq. (5.23)], and their different form of the free-energy density precludes direct comparison with the calculations of Seppala and Volovik (1983). On the other hand, Maki and Zotos (198Sa) have made a detailed analysis of ultrasonic attenuation by these nonsingular vortices, which will be considered as a potential detection scheme in section 5.4. Ohmi (1984) has extended this picture of a triangular lattice of bound pairs to include the orientation of the pair axis relative to that of the lattice. At present, such orientational effects remain unobservable. Volovik and Hakonen (1981) and Maki and Zotos (198Sb) have also considered a tipped magnetic field, whose principal effect is to orient perpendicular to both H and R. lifting the rotational degeneracy of i i n the xy-plane. In particular, Maki and Zotos (1985b) find that the separation of the bound-pair constituents decreases with increasing tipping angle. The resulting altered NMR signal (section 5.3) is consistent with the observations of Seppala et al. (1984). The equilibrium texture of rotating 3He-A evidently depends strongly on the magnetic field, and it is natural to consider the transition between the two extremes of low and high fields. Specifically, the low-field texture involves nonsingular dipole-locked MH and MT vortices, arranged on an interpenetrating square lattice (fig. 9), with the central core region of each vortex having a radius comparable to the lattice spacing. An axial magnetic field forces 2 toward the xy-plane, and the dipole coupling then pulls i along with it. This magnetic deformation shrinks the core and flattens the texture, retaining the fundamental four-fold symmetry. At some point, however, the free energy of this deformed type-I lattice exceeds that of the

Ch. 1 , SS]


63

doubly quantized Seppala-Volovik vortices [as seen in section 5.3, experiments of Seppala et al. (1984) conclusively rule out singly quantized high-field vortices]. Fetter (1985a) and Zotos and Maki (1985) have compared the free energy of these two configurations and conclude that a first-order transition occurs at a critical field of order H , that depends weakly on temperature and on the rotation speed. It remains an open question whether the low-field square lattice can instead dimerize and deform through shear into the high-field array of bound pairs.

5.3. NMR

IN ROTATING

3HE-A

Given the high-field texture from the preceding section, the corresponding NMR follows as in section 4.3 for the rotating B phase (Volovik and Hakonen 1981, Fetter 1983, Seppala and Volovik 1983, Seppala et al. 1984, Vulovic et al. 1984, Zotos and Maki, 1984). The basic dynamical variables for the A phase are the spin density S and the spin part of the order parameter, now specified by the unit vector d. Thus the effective freeenergy density [compare eqs. (4.36), (5.10) and (5.16)] becomes

where the tensor x incorporates the anisotropy in the magnetic susceptibility, and K, is dimensionless, normalized relative to ~ ~ , ( h / 2 mSince , ) ~ .>acts like a vector under rotations of the spin coordinates, it obeys the usual commutation relations (5.26)

and a straightforward calculation analogous to eq. (4.39) gives the desired dynamical equations

(5.27)

Here, Sf/Su = d f / d u - d , [ d f / d ( d , u ) ] , where fdenotes the last three terms of eq. (5.25), and x, is the normal-state susceptibility. In static equilibrium, d lies in the xy-plane perpendicular to the uniform field H, = fig, and, for simplicity, d will be assumed to be uniform and along i (as in the preceding section). When subjected to a weak rf field H ’exp(-iwt), d acquires a small oscillating part d’ (perpendicular to the

A.L. FETTER

64

[Ch. I . $5

original static direction i ) ,with components d:y^+ d : i . Elimination of the associated spin density S ' then yields a pair of coupled inhomogeneous equations for the amplitudes d : and dj. As in the similar case for the B phase [see eq. (4.51)], these equations decouple. in the usual situation that the Larmor frequency w,, = y H , , is much bigger than the characteristic A-phase frequency fl, = ( y h /3m,L,)( p,,/,yn)"'. For transverse excitation, the resulting equation becomes

where

D

=

L i [- K,V'

-

( K , - k,)(V- i ) ( i * V ) ]

(5.28b)

is a scalar differential operator [compare eq. (4.48)],with the dipole length L,, setting the characteristic scale. Once again. eq. (5.28a) resembles an inhomogeneous Schrodinger equation. with an aftractive potential

v,= -(If + 2 1 3 .

(5.2%)

and the solution can be expressed in terms of the corresponding eigenfunctions 4, and eigenvalues wz of the homogeneous equation

(D + V { ) $ ,= R A 2 [ 0 f- ( w ; + n;)]& .

(5.29b)

Note that V, involves the components of i perpendicular to d and is therefore significant only in regions where i differs from the local direction of d (here i ) .Since V , is attractive, the lowest solutions represent bound states with squared resonant frequencies (eigenvalues) below the bulk transverse value w: + 12;. It has become conventional to characterize these eigenvalues with the parameter R,(< l ) , defined by the equation

so that the "eigenvalue" on the right-hand side of eq. (5.29b) is just RI - 1. To find the lowest bound state, it is necessary first to know the i texture explicitly, and then to solve for t h e ground state of eq. (5.29b). Variational techniques are particularly convenient for this second step, since they provide a variational principle for Rz - 1 [compare eq. (4.58)]

Ch. 1, 65)


65

(5.31) which determines the position of the NMR satellite through eq. (5.30). The corresponding power absorbed at the resonant frequency is easily found as in eqs. (4.55)-(4.57), and the relative intensity of the shifted NMR satellite signal relative to that of the bulk A phase is given by the squared overlap integral of the ground-state wavefunction with unity (becaues the rf field is effectively uniform on these scales) I,

= V-’lJ

dV $ I 2 .

(5.32)

This formalism applies directly to rotating 3He-A in a strong parallel magnetic field. As shown at the end of section 5.2, the corresponding i texture is uniform and parallel to 2 = i throughout most of the sample, so that V, vanishes in these regions. Near each vortex core, however, i deviates from 2, producing an array of attractive cylindrical potentials, aligned along the rotation axis, and with radius of order 4L, for nonsingular doubly quantized vortices and of order 2L, for singular singly quantized vortices (Seppala and Volovik 1983, Seppala et al. 1984, Vulovic et al. 1984). Since these cores are widely separated at currently available rotation speeds, it is reasonable to use a “tight-binding’’ approximation that treats only a single vortex. Thus the high-field transverse NMR for rotating 3 He-A reduces to the solution of the effective Schrodinger equation (5.29) with an attractive two-dimensional potential V,determined by the i texture of a single vortex. This picture is evidently quite different from that used for rotating ’He-B, where the ti texture was fixed by the combined effect of the boundaries and the weak orienting terms in eq. ( 4 . 3 9 , and varied only on the (very large) characteristic length tH.Here, in contrast, the orbital i texture varies with the dipole length L,, which also characterizes the vortex core size. It is clear from eq. (5.29a) that the potential V,has a depth of order unity and a range of order L,. Such two-dimensional potentials typically have only a small number of bound states (Morse and Feshbach 1953). Since L, also sets the scale for spatial variations in the eigenfunctions [see eq. (5.28b)], the binding energy is very sensitive to the range of the potential. For the doubly quantized nonsingular vortex in eq. (5.24), the relatively large core size (-4L,) produces a tightly bound state, with R, significantly less than 1 [see eq. (5.30)j. Furthermore, the binding energy determines the spatial extent of the wave function (Landau and Lifshitz 1959), so that

66

A.L. FETTER

[Ch. 1, gS

this tightly bound state is well localized within the core. Equation (5.32) then implies that the doubly quantized nonsingular high-field vortex should have a relatively weak NMR satellite signal at a “bound-state” frequency shifted well below the bulk value. In contrast, the situation is quite different for the singly quantized singular high-field vortex in eq. (5.23), whose smaller core size (=2L,) yields only a weakly bound state with a significantly bigger spatial extent. Thus this singular vortex would be expected to have a relatively strong NMR satellite signal [from eq. (5.32)] at a frequency only slightly below the bulk frequency [from eq. (5.31)]. In practice, the detailed calculations involve considerable numerical work. first to determine the equilibrium i texture of each vortex, and then to find the ground-state eigenvalue and eigenfunction of eq. (5.29). The problem is further complicated by the explicit angular dependence of i in eq. (5.23b), which means that the effective potential V, is nonaxisymmetric, precluding the usual separation into partial waves. Seppala and Volovik ( 1983) analyzed the transverse NMR for both high-field vortices near T , and found R f = 0.5 for the nonsingular texture. with relative intensity 0.047 at 1 rad/s. In contrast, the singular vortex gave values of 0.98 and 0.56 for the same parameters, in accordance with the preceding discussion of bound states in two dimensions. Vulovic et al. (1984) extended the calculations to lower temperatures, using the hydrodynamic parameters of Williams (1979). and concluded that R : and the relative intensity both decrease with decreasing temperature. Seppala et al. (1984) have observed the A-phase NMR satellite at 29.3 bar for a range of temperatures and for various orientations of the field relative to the rotation axis. They find that R : is independent of tipping angle but depends on temperature, falling nearly linearly from about 0.6 near T, to about 0.3 near TIT, ~ 0 . 7 The . strength of the satellite absorption peak increases linearly with the angular velocity (and hence with the number of vortices), having a value of about 0.06 at 1 rad/s. In contrast to the magnitude of the shift. however. the relative intensity increases with tipping angle, in agreement with the calculations of Maki and Zotos (1985b). In addition, the improved variational calculations on the i texture near T , (Seppala et al. 1984) confirm the earlier analyses. Comparison of the experimental and theoretical values shows unambiguously that rotating ’He-A in a high magnetic field contains doubly quantized nonsingular vortices. This conclusion remains somewhat puzzling, because theoretical estimates (Seppala and Volovik 1983. Vulovic et al. 1984) indicate that the singular singly quantized vortices have lower free energy per quantum of circulation at angular velocities of order 1 rad/s and should therefore be preferred, at least for an experiment that cools the liquid in a state of

Ch. 1, 551


67

rotation. Various types of metastability have been suggested, but none is entirely satisfactory; this question remains unresolved. A similar analysis (Vulovic et al. 1984) describes the longitudinal NMR (with the rf field parallel to the applied field H,Z^), and the principal difference is the substitution of V, = -21: - 1: for V, in the effective Schrodinger equation (5.29b), along with the elimination of the Larmor frequency on the right-hand side. Since the itexture in eq. (5.23b) for both high-field vortices is essentially symmetric in the y and z components, the resulting frequency shifts and relative intensities are very similar to those for the transverse case. At present, no such experiments have been reported, but they could provide valuable additional confirmation of the previous identification.

5.4. OTHER DETECTION

SCHEMES

In principle, when rotating liquid .‘He is cooled from the normal sate to the superfluid A phase, it passes first through the transition at a,, (compare the discussion at the start of section 5.2), because the GL coherence length always becomes comparable with the intervortex spacing ( p h / 2 r n 3 0 ) 1’2 [see eqs. (4.6b) and (5.22)] sufficiently close to T,. In low magnetic fields, this implies the formation of a lattice of singular vortices (Schopohl 1980, Fetter et al. 1983), analogous to those seen in type-I1 superconductors at Hc2.The situation in 3He-A is more complicated, however, because many different types of vortices are possible; this singular lattice is expected to be stable only in an extremely narrow temperature interval, below which the nonsingular type-I lattice (see fig. 9) has lower free energy and should be the equilibrium state. Conceivably, singular vortices might remain metastable at lower temperatures, leading to hysteresis; in this case, the alternative procedure of accelerating the superfluid A phase from rest to some final angular velocity (of order 1 rad/s) should indeed produce the nonsingular lattice. Experimental study of this nonsingular low-field lattice and investigation of possible hysteresis both require a detection scheme that is independent of the magnetic field. Two approaches have been suggested, each of which depends on the anisotropy of certain parameters in ‘He-A. The first is the anisotropy of ultrasonic attenuation a (Serene 1974, Wolfle 1978), which has the following explicit dependence on the angle 8 between the propagation vector of the sound and the local direction of i a

=

a,,C O S ~e

+ 2a, sin’ e cos’ e + a , sin4 6 .

(5.33)

68

A.L. FEITER

(Ch. I . $5

The strong anisotropy observed in a for "He-A (Paulson ct al. 1976) arises from the rapid variations of (Y, and a . with temperature and frequency. Nakahara et al. (1979) have proposed that this anisotropy might be a useful probe of the texture of the rotating A phase, especially in low fields when NMR is inapplicable. Specific calculations by Fetter et al. (1983) show that different possible low-field vortex lattices should produce quite different temperature dependences for the spatially averaged attenuation coefficient 6 , reflecting the overall differences in the itextures. In addition, Zotos and Maki (1985) note that ultrasonic attenuation would also provide a sensitive detector of the transition between the low-field nonsingular type-I lattice. where i is distributed roughly uniformly in the xy-plane, and the high-field doubly quantized nonsingular lattice, where i (and d ) have a preferred direction in the xy-plane. Since this transition occurs at a field of order H , (see the discussion at the end of section 5.2), direct use of NMR is likely to be difficult. They have also suggested (Maki and Zotos 1985a) that ultrasonic attenuation may bc more sensitive than NMR in studying the specific form and temperature dependence of the high-field texture. An experimental program to investigate these various possibilities would be most valuable. The other potential probe of rotating 'He-A relies on the anisotropy of ionic mobility (Roach et al. 1977). In an electric field E , the steadystate velocity u depends on the relative local orientation of i according to

where p,,Ip -0.1 is a measure of the anisotropy. For a given E , an ion moving perpendicular to i has a higher velocity than one moving parallel. Thus. each specific texture of the rotating A phase should produce a characteristic distribution of arrival times for an initial ion pulse (Williams and Fetter 197'3) moving along the rotation axis. Specific calculations (Fetter et al. 1983) for the low-field textures indicate significant differences between the nonsingular type-I and the singular lattices. The same technique should also be able to detect the field-induced distortion of the nonsingular type-I lattice (discussed at the end of section 5 . 2 ) , since the narrowing of the core region with increasing H forces i preferentially into the xy-plane. The experimental study of ions in superfluid 'He has provided much valuable information (Fetter 1976), and a similar study of 'He-A would be of great interest.

Ch. 1, 561


69

6. Discussion

This chapter has presented an account of the static properties of vortices in rotating superfluid 3He, emphasizing the similarities and differences between the A and B phases, as well as the more familiar superfluid 4He. A series of experiments at the Helsinki University of Technology have provided much detailed information and stimulated enormous theoretical efforts. Much is now known about the static textures, but several problems remain unresolved. There is clear evidence for some sort of transition in the vortex cores of 'He-B (figs. 4 and 6), but a theoretical description has proved elusive. A second question concerns the equilibrium texture of rotating 'He-A in high magnetic fields. Experiments (section 5.3) indicate the presence of doubly quantized nonsingular vortices, independent of the order of cooling and rotating, yet theoretical calculations [see discussion below eq. (5.24)] imply that singular singly quantized vortices have lower free energy per quantum of circulation for angular velocities less than =3 rad/s. This real discrepancy should not be ignored. A third challenge is the experimental detection of the low-field structure of rotating 'He-A, which is predicted to contain a square array of singly quantized nonsingular vortices (fig. 9). Since NMR is inapplicable, new techniques will be necessary (section 5.4). For the future, the vortex dynamics in 'He has scarcely been considered. Even in 'He, the creation and destruction of vorticity is not well understood in detail, and the problems are likely to be even more severe in the present case. Bunkov et al. (1983a) report some studies of time-dependent effects following spin up, but much additional work remains. Another question is the possibility of vortex waves in either the equilibrium A or B phases, and the specific effect of orbital viscosity in the A phase (Brinkman and Cross 1978). Hook and Hall (1979) have exhibited certain time-dependent textures for 'He-A in a magnetic field, but more realistic models would be desirable, especially in connection with rotation. Conceivably, heat flow along the rotation axis might induce either static or dynamic helical distortions. The question of superfluid turbulence in 'He and the role of vortices also remains unexplored. These questions will challenge the ingenuity of both theorists and experimenters for the foreseeable future. Note added in proof (3/10/86)

Two recent papers (Thuneberg 1986 and Salomaa and Volovik 1986) have reported that the B-phase v vortex (previously thought to be the preferred state) is unstable at low pressures with respect to a finite-amplitude quadrupole deformation of the core. In addition, Thuneberg uses the Sauls and Serene 1981 strong-coupling parameters to study the pressure dependence of both

70

A.L. FETTER

[Ch. 1

states. He has found a first-order transition to the v vortex a few bar below the A-B transition, in qualitative agreement with experimental observations. References: Thuneberg, E.V. 1986. Phys. Rev. Lett. 56. 359. Salomaa, M.M., and G . E . Volovik. 1986. Phys. Rev. Lett. 56, 363

References Ambegaokar, V . and N.D. Mermin, 1973, Phys. Rev. Lett. 30, 1981. Anderson. P.W.. 1967. in: Progress in Low Temperature Physics, Vol. 5, ed. C.J. Gorter (North-Holland. Amsterdam) p. 1. Anderson, P.W.. and W.F. Brinkman, 1978, in: The Physics of Liquid and Solid Helium, Part 11. eds K.H. Bennemann and J.B. Ketterson (Wiley, New York) p. 177. Anderson. P.W.. and G. Toulouse, 1977. Phys. Rev. Lett. 38,508. Andronikashvili. E.L., and Yu.G. Mamaladze, 1967. in: Progress in Low Temperature Physics, Vol. 5. ed. C.J. Gorter (North-Holland, Amsterdam) p. 79. Balian, R.. and N.R. Werthamer. 1963, Phys. Rev. 131, 1553. Bedell. K.S.. 1984. unpublished. Brinkman. W.F.. and M.C. Cross, 1978. in: Progress in Low Temperature Physics, Vol. 7A. ed. D.F. Brewer (North-Holland. Amsterdam) p. 105. Brinkman, W.F.. J.W. Serene and P.W. Anderson, 1974, Phys. Rev. A10,2386. Buchholtz. L.J.. 1978. Phys. Rev. B18, 1107. Buchholtz. L.J.. and A.L. Fetter, 1977. Phys. Rev. B15, 5225. Bunkov, Yu.M., P.J. Hakonen and M. Krusius, 1983, in: Quantum Fluids and Solids1983, eds E.D. A d a m and G . G . lhas (American Institute of Physics, New York) p. 194. Cross, M.C.. 1975, J. Low Temp. Phys. 21, 525 and 24, 261E. Edmonds. A . R . , 1957. Angular Momentum in Quantum Mechanics (Princeton University, Princeton) p. 6. Engelsberg, S.. W.F. Brinkman and P.W. Anderson, 1974, Phys. Rev. A9, 2592. Fetter. A.L.. 1975. in: Quantum Statistics and the Many-Body Problem, eds S.B. Trickey, W.P. I r k and J.W. Dufty (Plenum, New York) p. 127. Fetter, A.L.. 1976, in: The Physics of Liquid and Solid Helium, Part I . eds K.H. Bennemann and J.B. Ketterson (Wiley. New York) p. 207. Fetter, A.L.. 1979, Phys. Rev. 820.303. Fetter. A.L.. 1983. in: Quantum Fluids and Solids - 1983, eds E.D. A d a m and G.G. lhas (American Institute of Physics, New York) p. 229. Fetter, A.L 1985a, J. Low Temp. Phys. 58.545. Fetter, A.L.. 1985b. Phys. Rev. B31. 7012. Fetter, A.L.. and S. Theodorakis. 1984. Phys. Rev. Lett. 52, 2007. Fetter. A.L.. and J.D. Walecka. 1980. Theoretical Mechanics of Particles and Continua (McGraw-Hill. New York) sections 7, 29 and 50. Fetter, A.L.. J . A . Sauls and D.L. Stein. 1983, Phys. Rev. BZS, 5061. Feynrnan, R.P., 1955, in: Progress in Low Temperature Physics. Vol. 1. ed. C.J. Gorter (North-Holland. Amsterdam) p. 17. Fujita. T.. M. Nakahara. T Ohmi and T. Tsuneto. 1978. Progr. Theor. Phys. 60.671.

.

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Gongadze, A.D., G.E. Gurgenishvili and G.A. Kharadze, 1981. Fiz. Nizk. Temp. 7,821 [Sov. J. Low Temp. 7, 3971. Hakonen, P.J., and G.E. Volovik, 1982, J. Phys. CIS, L1277. Hakonen, P.J., O.T. Ikkala. S.T. Islander, O.V. Lounasmaa and G.E. Volovik, 1983a, J. Low Temp. Phys. 53,423. Hakonen, P.J., M. Krusius, M.M. Salomaa, J.T. Simola, Yu. M. Bunkov, V.P. Mineev and G.E. Volovik, 1983b, Phys. Rev. Lett. 51, 1362. Halperin, W.P., C.N. Archie, F.B. Rasmussen, R.A. Buhrman and R.C. Richardson, 1976, Phys. Rev. B13,2124. Ho, T.-L., 1978a, Ph.D. thesis (Cornell University, Ithaca, NY) unpublished. Ho, T.-L., 1978b, Phys. Rev. B18, 1144. Hook, J.R., and H.E. Hall, 1979, J. Phys. C12,783. Israelsson, U.E., B.C. Crooker, H.M. Bozler and C.M. Could, 1984, Phys. Rev. Lett. 53, 1943. Jacobsen, K.W., and H. Smith, 1983, J. Low Temp. Phys. 52,527. Landau, L.D., and E.M. Lifshitz, 1959, Quantum Mechanics (Pergamon, New York) Section 45 and pp. 156-157. Langer, J.S., and J.D. Reppy, 1970, in: Progress in Low Temperature Physics, Vol. 6, ed. C.J. Gorter (North-Holland, Amsterdam) p. 1. and R.C. Richardson, 1978, in: The Physics of Liquid and Solid Helium, Part Lee, D.M., 11, eds K.H. Bennemann and J.B. Ketterson (Wiley, New York) p. 287. Leggett, A.J., 1973, J. Phys. C6, 3187. Leggett, A.J., 1974, Ann. Phys. (USA) 85, 11. Leggett, A.J., 1975, Rev. Mod. Phys. 47, 331. Levin, K., and O.T. Valls, 1983, Phys. Rep. 98, 1. Maki, K., 1983, in: Quantum Fluids and Solids - 1983, eds E.D. A d a m and G.G. Ihas (American Institute of Physics, New York) p. 244. Maki, K., and M. Nakahara, 1983, Phys. Rev. B27,4181. Maki, K., and X. Zotos, 1985a, Phys. Rev. B31, 177. Maki, K., and X. Zotos, 1985b, Phys. Rev. 831,3116. Mermin, N.D., 1978, in: Quantum Liquids, eds J. Ruvalds and T. Regge (North-Holland, Amsterdam) p. 195. Mermin, N.D., 1979, Rev. Mod. Phys. 51,591. Mermin, N.D., and T.L. Ho, 1976, Phys. Rev. Lett. 36,594. Mineev, V.P., and M.M. Salomaa, 1984, J. Phys. C17, L181. Morse, P.M., and H. Feshbach, 1953, Methods of Theoretical Physics, Vol. I1 (McGrawHill, New York) p. 1654. Muzikar, P., 1978, J. Phys. (France) Colloq. 39, C6-53. Nakahara, M., T. Ohmi, T. Tsuneto and T. Fujita, 1979, Prog. Theor. Phys. 62,874. Ohmi, T., 1984, J. Low Temp. Phys. 56, 183. Ohmi, T . , T. Tsuneto and T. Fujita, 1983, Progr. Theor. Phys. 70,647. Onsager. L., 1949, Nuovo Cimento 6, Suppl. 2,249. Osheroff, D.D., and W.F. Brinkman, 1977, unpublished. Osheroff. D.D.. W.J. Gully, R.C. Richardson and D.M. Lee, 1972, Phys. Rev. Lett. 33,584. Passvogel, T., N. Schopohl, M. Warnke and L. Tewordt, 1982, J . Low Temp. Phys. 46,161. Passvogel, T., N. Schopohl and L. Tewordt, 1983, J. Low Temp. Phys. 50,509. Passvogel, T.. L. Tewordt and N. Schopohl, 1984, J . Low Temp. Phys. 56, 383. Paulson, D.N., M. Krusius and J.C. Wheatley, 1976, J. Low Temp. Phys. M,73. Pekola, J.P., J.T. Simola, P.J. Hakonen, M. Krusius, O.V. Lounasmaa, K.K. Nummila, G. Mamniashvili. R.E. Packard and G.E. Volovik, 1984, Phys. Rev. Lett. 53,584.

12

A.L. FETTER

Ch. I]

Peshkok. V.P.. 1964. in: Progress in Low Temperature Physics. Vol. 4. ed. C.J. Gorter (North-Holland. Amsterdam) p. 1. Roach. P D.. J.B. Ketterson and P.R. Roach. 1977. Phys. Rev. Lett. 39,626. Salomaa. M.M., 1984. unpublished. Salomaa. M.M.. and G . E . Volovik. 1983a. in: Quantum Fluids and Solids - 1983. eds E.D. Adams and G . G . lhas (American Institute of Physics, New York) p. 210. Salomaa, M.M.. and G . E . Volovik. 1983h. Phys. Rev. Lett. 51. 2040. Salomaa. M.M.. and G . E . Volovik. 1984. Phys. Rev. Lett. 52. 2008. Salomaa. M.M.. and G . E . Volovik. 1985. Phys. Rev. B31. 203. Sauls. J.A.. and J.W. Serene. 1981. Phys. Rev. B24. 183. Sauls. J.A.. D.L. Stein and J.W. Serene, 1982. Phys. Rev. D25. 967. Schopohl. N.. 1980. J. Low Temp. Phys. 41,40!2. Seppali. H.K., and G . E . Volovik. 1983. J . Low Temp. Phys. 51.279. Seppalii. H . K . , P.J. Hakonen. M. Krusius. T. Ohmi, M.M. Salomaa. J.T. Simola and G.E. Volovik. 19x4. Phys. Rev. Lett. 52. 1802. Serene, J.W.. 1974. Ph.D. Thesis (Cornell University. Ithaca. NY) unpublished. Smith, H.. W.F. Brinkman and S. Engelsherg. 1977. Phys. Rev. B15. 199. Stewart. G . R . . 1984. Rev. Mod. Phys. 56,755. Theodorakis, S . , and A.L. Fetter. 1983, J. Low Temp. Phys. 52, 559. Tkachenko. V.K.. 1%S. Zh. Eksp. & Teor. Fiz. 49, 1875 (1966. Sov. Phys.-JETP 22. 12821. Tough, J.T.. 1982. in: Progress in Low Temperature Physics, Vol. 8. ed. D.F. Brewer (North-Holland. Amsterdam) p. 133. Vinen, W.F., 1961. in: Progress in Low Temperature Physics. Vol. 3, ed. C.J. Gorter ( North-Holland. Amsterdam) p. 1. Volovik. G . E . . and P.J. Hakonen. 1981. J. Low Temp. Phys. 42, 503. Volovik. G . E . . and N.B. Kopnin. 1977. Pis’ma Zh. Eksp. & Teor. Fiz. 25, 26 [JETP Lett. 25, 221. Volovik. G.E.. and V.P. Mineev. 1977, Zh. Eksp. & Teor. Fiz. 72. 2256 [Sov. Phys.JETP 45, I186j. Volovik. G.E.. and V.P. Mineev. 1983. Pis‘ma Zh. Eksp. & Teor. Fiz. 37. 107 [JETP Lett. 37. 1271. Vulovic. V.Z.. D.L. Stein and A.L. Fetter, 1984, Phys. Rev. B29. 6090. Wheatley, J.C.. 1975. Rev. Mod. Phys. 47,415. Wheatley. J.C.. 1978. in: Progress in Low Temperature Physics. Vol. 7A, ed. D.F. Brewer (North-Hotland, Amsterdam) p. 1. Williams. M.R.. 1979, Ph.D. thesis (Stanford University. Stanford. CA) unpublished. Williams. M.R.. and A.L. Fetter. 1979. Phys. Rev. BZO. 169. YMe. P., 1978. in: Progress in Low Temperature Physics, Vol. 7A, ed. D.F. Brewer (North-Holland, Amsterdam) p. 283. Zotos, X.. and K. Maki. 1984. Phys. Rev. B30. 145. Zotos. X . , and K . Maki. 1985. Phys. Rev. B31, 7120.

CHAPTER 2

CHARGE MOTION IN SOLID HELIUM BY

A.J. DAHM Case Western Reserve University, Cleveland, Ohio 44106, USA

Progress in Low Temperature Physics, Volume X Edited by D.F. Brewer 0 Elsevier Science Publishers B.V., 1986

13

Contents .. 1. Introduction . . . , . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . 2. Background . . . . . , . . . . . . ........................ . . . . . unties in solid helium . . . . . . . . 2.1. Description of vacancies 2.2. Structure of ions . . . ............................

....................... . . . . . ............................ 2.2.1.2. Negative ions . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Ion structures in solid helium . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1. Positive ions . , . . . . . . . . . .................... 2.2.2.2. Negative ions . . . , . . , . . . 2.3. Experimental methods of measuring mob 2.3.1. Space-charge limited currents , . . . . . . . , . . , . . . . . , . . . . . . . . . . . . . . . .............. 2.3.2. Transient space-charge limited currents . . . . 2.3.3. Time of flight techni 2.3.4. Comparison of techni

measurements. . . . . . . . . . . . . . . . . . . . . 3. ionic mobilities , . . . . . . . . , . . . . ................... ................... 3.1. Early investigations . . . . . , . 3.2. Mobility measurements . . , . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Theory , . . . . . . . . . . . . . . ........................ 3.3.1. Negative ions . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1. Vacancy-assisted mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2. Surface adatom diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Positive ion ...................................... 3.3.2.1. Ho ....................................... 3.3.2.2. Vacancy diffusion. . 3.3.3. Species-independent mechanisms. . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . 3.3.3.1. Phonon-assisted ion motion . . 3.3.3.2. Motion by a plastic flow mechanism . . . , . . . . . . . . . . . . . . . . 3.3.3.3. Vacancy-wave scattering 3.3.3.4. Ions as quasiparticles. . . 3.3.3.5. Bound ion-vacancion complexes. . . . . . . . 3.4. Ion velocities in large electric fields . . . . . . . . . . . . . . . . . 3.5. Related phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Ion-dislocation ring complexes . . 3.5.2. Motion of charged grain boundaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Test for the collapse of the negative-ion cavity . . 3.5.4. Search for vacancy waves . . , . , . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5. Motion of charged droplets of liquid helium through the solid . . .

phase. . . . . . . . . . . . . . . . . .

........................... 74

76 77 77 79 79 79 80 81 82 82 84 85 86 89

9u Y1 91 92 101 101 101 104

1M 106 106 108 108 108 108 112 113 114 116 116 119 121 122

123

3.5.6. Theory for positive-ion mobility in a dilute 4He-3He solution. . . . . . . 3.5.7. Theory for negative ions in an rf electric field . . . . . 4.1. General comments . . . . .

123

. . . . . . . . . . . 125

......................

75

130

1. Introduction

The study of ions in liquid helium proved to be very rewarding, and developed into a subfield of superfluid physics. These probes provided information on quantized vorticity and turbulence and were used to study vortex lattices and vortex excitations; they were useful in studying the quasiparticle spectrum through scattering from phonons. rotons and 'He impurities and were used in roton creation; they were used to study the 'He-'He interface and the condensation of 'He atoms on surfaces and vortex lines in dilute solutions of 'He in 'He, and have been used to probe the anisotropic properties of superfluid 3He. In addition, the structure of these ions were of intrinsic interest. Much of this work was reviewed by Schwarz (1975) and Fetter (1976). In light of the success of these studies in liquid helium, it seemed reasonable to expect ions to be valuable in probing the solid phase as well. The work on charged particles in solid helium has been much less extensive. in large part due to the low mobility of these probes in the solid phase and the requirement of excellent crystal quality to probe the solid over a large temperature range. At this point the experimental work is quite thorough, and although there is a need for further efforts, particularly theoretical, it seems an appropriate time to review the results to date. Early reviews on ions in solid helium were included in Shikin (1977a) and in Spear and le Comber (1977). In the next section we give some background necessary for the understanding of the experimental and theoretical work. A short summary of the nature of vacancies and isotopic impurities in solid helium is given as background to theories of ion motion. The structure of ions in liquid helium is discussed since their structure in the crystal phase may be expected to be similar. We then examine the theory of similar structures in t h e solid phase and evidence for the cavity structure for t h e negative ion. The experimental techniques for probing ion motion are also described here. The experimental and theoretical work are presented in section 3. For pedagogical reasons, the results of each experimental effort are not compared with theory. Instead, the pioneering work and all low-field experimental work are presented first, and the results are compiled. Then theories of the mobility and predictions for ion velocities in large electric

CHARGE MOTION IN SOLID HELIUM

Ch. 2. $21

77

fields are given, followed by experimental measurements in large fields. Related experiments and predictions are also given in this section. The data are summarized and discussed in light of present theories in section 4 and a summary is given in section 5. The units used are cgs units with the following exceptions. Pressures are given in atm, since this is the convention used in all the work which is reviewed. Boltzmann’s constant is set equal to unity, and, except for section 2.2, energies are measured in Kelvin. Other units, used in figures, are specified.

2. Background OF 2.1. DESCRIPTION

VACANCIES A N D ISOTOPIC IMPURITIES I N SOLID HELIUM

Some theories for ionic motion in solid helium are based on concepts first developed for vacancies and isotopic impurities in helium. Furthermore, an understanding of the nature of vacancies is essential for understanding their role in ion motion. The energy of a vacancy or impurity in a lattice is periodic in space. If the transport of a vacancy from one site to an equivalent site occurs via tunnelling, a band of vacancy states will be formed analogous to electron states in metals. In solid helium, the large zero-point motion of atoms leads to a large tunneling rate via exchange of sites for both vacancies and isotopic impurities. These excitations are calculated to occupy a band of states of width @ (Hetherington 1968, Andreev and Lifshitz 1969, Guyer and Zane 1970). The tight-binding approximation is used to calculate the band states for these excitations in 4He. The dispersion curve in bcc ‘He is given by &

= E0

+ ;@[l+ cos(k,ald3)

cos(k,ald3) c o s ( k , a ~ ) ,]

(1)

where E() is the energy at the bottom of the band, k is the wavevector of the excitation, and a is shortest lattice vector. Vacancies and isotopic impurities are treated as quasiparticles with well-defined energy and momentum, and having an effective mass given by

m * = cr,h’/a% ,

(2)

where a, = 6 for the bcc phase. The values of @ are somewhat uncertain: 1-10K for vacancies (Guyer et al. 1971, Mineev 1972, Hetherington K for 3He atoms in solid 4He (Guyer et al. 1971, 1978) and (1-5) X

78

A.J. DAHM

(Ch. 2 , 02

Mikheev et al. 1977). Delocalized vacancies are called vacancy waves or vacancions and have been reviewed by Guyer (1972) and Andreev (1976, 1978). The impurity quasiparticles are called mass fluctuation waves or impuritons and are discussed by Andreev (1976, 1978). The diffusion constant for these excitations is D = uA 16, where A is the mean free path and the velocity u i s of the order of a@lh. The vacancy mean free path is limited by phonon, impurity or vacancion-vacancion scattering. In the phonon scattering regime, A, a T-”. where n = 9 (Andreev and Lifshitz 1969) or for T Q @, n = 7 (Kagan and Maksimov 1983). Estimates of the temperature for which A, 9 a are: T < $18 (Pushkarov 1970), i.e., at all temperatures in the solid phase, and T < 8 ( h w , / 8 ) ” 2=S 1 K (Landesman 1974). Here 8 is the Debye temperature and w, is the vacancy hopping frequency. The situation is different in solid ’He. Here vacancies and ‘He impurities have a definite spin associated with them. Suppose that the vacancion wavefunction in ‘He extends over N atomic sites. Then the probability of any of these sites being vacant is 1/N. In the case of ’He, suppose that an atom of spin f has been removed. Then only sites of spin t can be vacant. The random orientation of spins in ’He destroys the coherence of the vacancy wavefunction. and it cannot extend uniformly over N sites. The mean free path is A = a. A band of energy levels exists, and the tunnelling probability is still = @ / A . Thus, D ua = @a2/h. Vacancy and impurity diffusion have been studied by NMR techniques. See the reviews by Guyer et al. (1971) and Landesman (1975). These studies yield the diffusion coefficient, D,, and bandwidth, @,, for impuritons and the vacancy creation energy E,,. The diffusion coefficient for ’He impurities in ‘He is presented as a function of T - ’ in fig. 1. The diffusion coefficient decreases exponentially with T - ’ at high temperatures where impurity motion proceeds via vacancy hopping. In this regime, D , is proportional to the concentration of vacancies c0 = exp(-e,,/T). At lower temperatures the ’He atoms diffuse as quasiparticles with a temperatureindependent mean free path which is limited by ’He-’He collisions. For sufficiently dilute solutions and large molar volumes, the impuriton mean free path is limited by phonon scattering, and D, varies as T below 1.5 K (Mikheev et al. 1977, Allen and Richards 1978, Esel’son et al. 1978). In this regime, A, 9 a. Kagan and Maksimov (1983) examined the localization of quasiparticles when energy levels on neighboring sites are shifted with respect to each other by an amount greater than the bandwidth. Localization can occur due to fluctuations in the energy levels caused by thermal phonons. In this case the diffusion of quasiparticles decreases with increasing temperature. Quasiparticle interactions also cause static shifts in the energy levels, in

-

Ch. 2, $21


lo"

79

I-

-121

10

0.5

,

I

0.7

,

I

,

0.9

, 1.1

,

I

,

1.3

T-' (K-')

Fig. 1. Temperature dependence of the diffusion coefficient of impurities in a hcp crystal of 4He. The symbols represent: isotopic impurities, 0 0.75% 'He, 0 2.17% 'He, V,,, = 20.7 cm3; 0 positive ions, V, = 20.76 cm3. After Keshishev (1977).

which case fluctuations allow tunnelling to a neighboring site, and the diffusion coefficient increases with temperature. This occurs for large concentrations, 3 2 % 3He, and has been observed by Mikheev et al. (1982). For the case of static shifts, a maximum in the diffusion coefficient as a function of temperature is predicted. One final relevant note is the theoretical prediction of 'He impurityvacancy bound states by Locke and Young (1976). 2.2. STRUC~URE OF IONS 2.2.1. Structure of ions in liquid helium 2.2.1.1. Positive ions A model of the positive ion, which is supported by a huge body of experimental data, was proposed by Atkins (1959) to explain the large ion-roton cross section. He suggested that the density of liquid in the vicinity of the ion core is enhanced by electrostriction and is greater than the melting density for R > 6 ,&This model neglects any structure within the solid and treats the helium as a classical continuum. The center of the positive entity is a singly charged bare helium ion. This is assumed to be a He: ion with an internuclear separation of 1.08 A

(Ch. 2, 02

A.J. DAHM

80

(Pauling 1967). However, Patterson (1968) reported evidence for a stable He,' ion in gaseous helium below 200K with a dissociation energy of 0.17eV to He: + He. This conclusion is consistent with theoretical calculations (Pdshusta and Zetik 1968, Vauge and Whitten 1972) which give a linear molecule with adjacent nuclei separated by 1.2 A. The effect of the surrounding medium on the core is uncertain. Electrostriction in the surrounding medium results from the field gradient due to the bare ion. The force on an atom in the presence of an electric field gradient is

(3)

f = grad f a E ' ,

where a is the atomic polarizability. The pressure gradient in the helium near the charge is grad p = n grad :aE'

= grad(nae?/p'r")

- l(ae'/&'r')grad n .

(4)

Here n is the number density, &(r) is the dielectric constant and r is the distance from the charge. The second term on the right-hand side of eq. (4) is always small compared to the first term. The pressure in the liquid p ( r ) is approximately 7

7

.

p ( r ) - p, = +nae-/&-(r)r"

(5)

where p , is the external pressure applied to the liquid. Just inside the solid the pressurep,(r) isenhancedover the meltingpressurep,(m) by thesurface tension pressure pt

The exact value of p , depends on the structure of the solid. By combining these equations we obtain an approximate value for the radius R . of the positive ion R , = (nae'/2~;( p , ( s )

where

E,

+ p , - pe)]"'

,

(7)

is the dielectric constant at the melting pressure.

2.2.1.2. Negative ions The negative ion in liquid helium consists of an electron trapped in a cavity void of helium atoms. This structure was proposed by Ferrell (1957) to explain the long lifetime of ortho-positronium in liquid helium. Careri et al. (1959, 1960) suggested that this model might also apply to the structure

Ch. 2, $21


81

of negative ions. This model has been subjected to a number of interesting experimental tests, and is consistent with all experimental data. This cavity structure results from the short-range Pauli repulsive interaction between an electron and a helium atom. The requirement that the wavefunctions describing an excess electron and helium atomic-core electrons be orthogonalized leads to large curvatures in the excess-electron wavefunction when it is injected into a dense helium medium. The resultant large kinetic energy leads to a barrier for electron penetration of the liquid. This barrier is 6, = 1 eV at the vapor pressure. The formation of a cavity of radius R - around an excess electron in the liquid leads to a large reduction in the kinetic energy of the electron with a small cost in surface-tension energy, electrostatic energy (due to a decrease in the local dielectric constant) and enthalpy. The summation of these terms gives a free energy of

F = E , ( R ) + 4 ~ a R ’where u is the surface tension, and p is the ambient pressure. A good approximation for the zero-point kinetic energy E, is obtained by assuming a square density profile at the cavity surface and treating the electron as a particle of mass m in a square well-potential of height V,( p ) and radius R . For the descriptive nature of our discussions here, it is sufficient to assume V,, = =. Then E, = h2m2I2mR2. The equilibrium radius R - is obtained by minimizing F with respect to R . This yields the condition

Theoretical values of R - obtained from eq. (9), corrected for a finite value of V,, are: R - = 17 %, at p = 0, and R - = 12.5 A at p = 25 atm. Values of R - deduced from experiments carried out at the vapor pressure, range from 14-17A. More detailed discussions of these models and the theoretical and experimental radii are given in the reviews by Schwarz (1975), Fetter (1976) and Shikin (1977a).

2.2.2. Ion structures in solid helium The physical arguments leading to the solid cluster of atoms surrounding the positive ion and the negatively charged cavity in the liquid apply to the

A.J. DAHM

82

[Ch. 2, $2

solid phase as well. In the solid phase the lattice-strain energy contributes to the free energy of the structure.

2.2.2. I. Positive ions Unless the positive ion is a hole which propagates with a diffusive velocity of the order of the sound velocity, an idea which can be rejected on the basis of the small experimental ion mobilities, there must be an enhanced density surrounding the ion core due to electrostriction. The electrostrictive pressure, eq. ( 5 ) , in a continuum model is -300 atm at a distance of one atomic spacing from the bare ion. Far from the ion, the regular lattice structure exists. Thus a few A from the center of the ion a highly strained transition region to a regular lattice is expected. 2.2.2.2. Negative ions Cohen and Jortner (1969) examined the cavity model of the negative ion in solid helium. They estimated the ground-state energy of the quasifree (conduction band) state by the Wigner-Seitz method. This energy represents the barrier for electron penetration into the crystal and is given by the relation

where k,, is determined from the boundary conditions; tan k , ( r , - a , ) = kor, .

(11)

Here r, is the Wigner-Seitz radius and a, is the electron-helium atom scattering length. Their calculated value of V,, is 1.6 eV at 30 atm. The calculation of the ground-state energy and the radius of the cavity is identical to that for the liquid phase except for the inclusion of the lattice-strain energy. These authors treat the solid as an isotropic elastic medium and introduce a rigid cavity of radius R,) into the lattice. This may not represent the energetically stable configuration of the system, as elastic shear forces may change the cavity radius from R,, to R , > R,. The corresponding energy change is (Frenkel 1948)

where C is the elastic constant. This represents an upper limit since a nonrigid cavity would relax. This is the only term in the free energy which

Ch. 2. S2l


83

depends on R,;

A minimization of the free energy with respect to r, yields

R , = R,,

and

E,,(R,, R,)

=0

The above argument is intended to show that the elastic shear contribution is small and may be neglected. In any case, the expression for E,, is small, even if the cavity would relax and the values of R, - R,, were relatively large. The elastic constant is =5 X 10' erglcm3 (Greywall 1971). For R, = 10 8, and R , - R,, = 1 A, E,, = lo-' eV. With the neglect of Esh, the energetics of the cavity in the solid are identical to those of cavity in the fluid. For a pressure of 30 atm, Cohen and Jortner calculate a cavity radius of 9 8 , and a total cavity energy of 0.5eV- iv). Thus the stabilization criterion is satisfied. These authors neglect the small electrostrictive term in eq. (13), but use an estimate for the solid-vacuum surface tension of us"= 2.4 dynlcm at 30 atm. This value appears to be quite large. Amit and Gross (1966) calculate the surface tension for the liquid phase to be 0.7fins, where n is the number density and s is the sound velocity. This value of us"would give an -10% larger radius. A detailed presentation of the Wigner-Seitz method applied to an electron in helium and the energy of the cavity is given in Springett et al. (1967). Smith et al. (1977) discuss models for V,, the pressure dependence of the surface tension and the surface profile. Shikin (1971b) calculates the cavity radius at various pressures. He uses the Born approximation to calculate V, (Lenz 1929, Levine and Sanders 1967);

y, = 2.rrfi2na/m .

(14)

This approximation neglects correlations, and underestimates V,, perhaps by 50%. The result is a greater penetration of the electron outside of the cavity, and a smaller cavity energy and radius. He includes the electrostatic energy, but neglects the lattice-strain and surface-tension energies. The latter is important. The energy associated with the electron penetration into the surrounding medium is doubly counted in his calculation. Shikin calculates radii of 12 8, at 30 atm and 8.5 8, at 140 atm. He points out that

84

A.J. DAHM

[Ch. 2, 92

there is a slight density enhancement just outside the cavity due to electrostriction. The electrostrictive pressure at 10 8,from the center of the cavity is -6atm. The actual potential barrier step for the electron occurs at R - a, for a square helium-density profile. This will slightly increase the cavity radius. The value of 0%is 1 . 1 9 = ~ 0.64 ~ A, where a, is the Bohr radius (O’Malley 1963, LaBahn and Callaway 1964, Golden 1966). Two experiments have been carried out to test the cavity model for positronium in solid helium. Trifthauser et al. (1968) measured the angular correlation of gamma rays from the decay of para-positronium. This measurement yields the momentum distribution of the positronium atom. The data show a clean signature of the momentum components associated with the positronium atom in the cavity state. Hernandez and Choi (1969) analyzed these data and obtain a radius of 210 8, at 30 atm. Smith et al. (1977) measured the lifetime of ortho-positronium as a function of the pressure in liquid and solid helium. They obtain an excellent two-parameter fit to the cavity model with a radius of 9.6 8, at 30 atm. The electron cavity is expected to have a larger radius because of its larger zero-point energy. The ortho-positronium lifetime varies smoothly with pressure from p = 0 to 130 atm. The smooth variation through the melting curve verifies that the lattice-strain energy is unimportant in determining the cavity radius. The theoretical ortho-positronium-helium atom scattering length is 1.390, (Drachman and Houston 1970). The values, ( 1.0-1.5)0,, deduced from various experiments are discussed by Smith et al. (1977). These values are approximately equal to the electron-helium atom scattering length. The potential barrier V, for penetration into helium is inversely proportional to the mass. Thus, the barrier for positronium should be less than for electrons. The experimental evidence for a cavity structure for positronium provides strong support that this same structure is valid for an electron in solid helium. 2.3. EXPERIMENTAL METHODS

OF MEASURING MOBILITIES

The conventional method of measuring ionic mobilities in liquids and gases is to measure the time of flight across a drift space. Most experimental cells employ one or more grids. For the case of a solid sample, the crystal must be grown with the grids, in situ. Early attempts to use grid structures in solid helium resulted in difficulties with charge trapping on crystal defects (Keshishev et al. 1969a. Dionne et al. 1972). For this reason some

Ch. 2. 021


85

experimentalists have chosen to measure mobilities in a diode configuration, using either steady-state or transient space-charge limited currents. The formulas for the steady-state and transient currents for various ion velocity-electric field relations and a discussion of possible difficulties with this technique for small mobilities are given below. Other techniques are discussed briefly, A thorough discussion of diode techniques to measure mobilities including the effects of charge traps is given by Lampert and Mark (1970). 2.3.1. Space-charge limited currents One method of measuring mobilities in a diode is to plate one electrode with a tritium-alloy beta-emitter. An electric field is applied across the diode to extract charges of either sign, determined by the sign of the field, from the beta tracks. These tracks are ==lopm in condensed helium. The temporal and steady-state currents for a space-charge limited (SCL) diode are obtained by solving Poisson’s equation with charge density p = j / e u ( E )subject to the boundary condition that the field E vanish at the cathode. Here j = I/S is the ratio of the total current to the diode area, and u is the ion velocity. Various forms of u ( E ) have been used to fit experimental data. For small electric fields, the mobility p in solid helium is well defined: u = p E. For large electric fields, two other formulas for u ( E ) have been used to fit experimental data. Keshishev (1977) derived the relation u = x ( E + E,)3, where x and E, are constants, and Efimov and Mezhov-Deglin (1981) obtained an empirical fit of the data with the form u ( E ) = ( p i y ) sin y E . The steady-state SCL I-V characteristics in a diode plate with separation L are given for these cases. For a well-defined mobility (Thomson and Thomson 1928)

I = 9&psv’/32rL3. For the case u = x ( E

+ E0)3 (Keshishev 1977)

I = 625&x(V+ EoL)4S/4096.rrL5. For u = ( p / y ) sinh y E the relation is [ycosh-’y-(y’ - l ) l ” ] I ( y - l ) = y V ~ L . Here y = 1 + 4rILy2/p&S).

(15)

A.J. DAHM

86

(Ch. 2, 92

Gudenko and Tsymbalenko (1979) derived the following relations for a general form of u ( E ) ,

u ( E , ) = ( 4 7 r L / ~ A d) l l d E , ,

(18b)

where E, is the field at the collector. The velocity can be determined from a numerical differentiation of experimental curves. Low-field characteristics for a sample of 4He are shown in fig. 2. In principle, very precise values of the mobility can be obtained from the slopes of these curves. One problem with any technique is the possible presence of charges trapped on dislocations or grain boundaries. In a diode, the presence of trapped charge results in an intercept of the abscissa in an I ’ ‘-V plot at V = V, > 0. The amount of trapped charge per unit area is -V,14reeL, and V, increases with decreasing temperatures. Trapped charge has a negligible effect on these curves for V 9 V,, and, in addition, some of the charge is extracted from traps at large electric fields. An advantage of the SCL-current technique is that the I”2-V plot gives a relative measure of the crystal quality, and poor crystals can be immediately discarded. The density of dislocations in a good crystal can be estimated from fig. 2. For V,l L 1 statV/cm at lower temperatures and an estimate of 10’ charges trapped per cm on a dislocation (Marty and Williams 1973), an order-of-magnitude estimate yields a dislocation density of -lO3/cm2. A more serious difficulty was pointed out by Keshishev (1977). X-rays, created in the source by bremsstrahlung radiation or by electron excitation of atoms in the source or substrate can eject high-energy photo-electons from the collector. The number of ions created near the collector is a very small fraction of the number created at the source. However, at sufficiently low mobilities and consequently low currents, charges created near the collector where the applied field is relatively large, 3 V / 2 L , contribute a measurable current. This contribution results in an intercept of the ordinate on an I”’-V plot at I, > 0. The value of I, is small, as can be observed in fig. 2. No quantitative analysis has been carried out to determine errors which may be introduced by these effects. However, large deviations in the mobility, measured in a cell with a strong beta-source, have been attributed to this effect as discussed below in section 3.2.

-

2.3.2. Transient space-charge limited currents The time of Hight of ions can be determined from the transient response to

Ch. 2. 821


1.7 K t +/

v

87

1.6Kt

(Volts)

Fig. 2. I”’-Vplots for a SCL diode at various temperatures. V,,,= 20.5 cm3. 0 positive ions, 0 negative ions. Curves taken by Lau (1978).

a step increase in voltage applied to a SCL diode (Efimov and MezhovDeglin 1978b). The transient response has the following features: (1) An instantaneous increase in current induced in the collector by a sheet of current injected at the source to maintain the condition E ( 0 ) = 0. (2) A monotonic increase in current, as more charge is injected, up to a maximum at time t , , when the initial current sheet reaches the collector.

A.J. DAHM

X8

[Ch. 2. $2

(3) A decrease in current after the sheet is collected. (4) Small oscillations in t h e current as the charge distribution in the diode approaches steady state. The cusp at the maximum. which may be -20% greater than the steady-state current. is rounded by diffusion. The transit time is obtained by measuring t , . If a step voltage is applied from V = 0, the time 1 , for u = p E (Many et al. 1961, Helfrich and Mark 1962) is

and for u = ( p / y ) s i n h y E , t , = ( y L / p ) {z - [ 1

+ (z’

-

1) e-’]’’’},

z = coth(yVI2L) .

(20)

A small incremental voltage step yields a maximum at f , = ( 7 r / 3 ) i Oif v = p E (Baron et al. 1964). The expression for f , in the case of a small

incremental voltage step and u = ( p / y )sinh y E is much more complicated and is not given here. Samples of transient traces are shown in fig. 3. For high-quality crystals, steady state is attained quickly and a second maxima in the current is observed as steady state is approached. Trace C was taken on the same crystal as trace B, after a strain was intensionally introduced. There is a I

I

100

200

300

400

500

600

t (sec)

Fig. 3 . Space-charge limited current transients following a step change in voltage o n crystals of ‘He at V, = 20.5 cm’. The vertical lines indicate the time at which the electrometer was shorted and the voltage was changed. The thicker horizontal bars denote the steady-state current. The curves and voltage changes are : (A) 1.1 K , positive ions. +3OO t o +SO0 V. The maximum at -150s from the vertical line is due t o noise: (B) 1.3 K. negative ions before straining crystal. -12OOV to -14ooV. (C) 1.3 K. negative ions after strain. - 1 O V to -8OOV. Curves taken by Lau (1978).

Ch. 2. 92)


89

long decay to steady state as charge traps fill up in a characteristic trapping time T~ 9 1 , . The effect of trapped charge on transients and SCL steadystate currents is discussed by Many and Rakavy (1962), Lampert and Mark (1970) and Gudenko and Tsymbalenko (1979). The transient response allows a separate check on the steady-state SCL current mobility and gives a stringent test of crystal quality. The effect of a small current source, introduced at the collector by photo-electron ejection, on the transient response has not been analyzed.

2.3.3. Time of flight techniques Marty and Williams (1973) used a cylindrical photo-diode with a spacing of 1 0.1 mm. The cell was fabricated from beryllium and the outer electrode was gold-plated. A pulsed X-ray source ejected electrons preferentially from the gold electrode. These electrons ionized helium atoms to a depth of -10 Fm, and charges of either sign could be injected into the diode. This design conveniently eliminates grid structures while measuring the time of flight of a sheet of charge. A strong thermal coupling with a liquid-helium bath was required to extract the heat deposited in the gold-electrode by the X-rays. Triode cells are operated in one of two modes. Both use ionizing radiation from a beta-emitter as a source of ions. In one method, used by Keshishev (1977), a constant potential is applied across the drift space between the grid and the collector. An additional step voltage is then applied between the source and the grid, and the induced collector-current waveform is monitored. The collector current increases linearly in time from Z = 0, when the first ions pass the grid, to a steady-state value which is attained when the leading edge of the ion beam reaches the collector. The separation between these changes in the derivative of the waveform is the time of flight. A more accurate mode of operation for short flight times (-1 s) was introduced by Cunsolo (1961), and used by Golov et al. (1984b) in the bcc phase of 4He and by Keshishev et al. (1970). In this method, a fixed voltage is applied between the source and grid so that a source of ions is always present at the grid. A square-wave voltage of frequency f is applied between the grid and the collector, and the time-averaged collected current (Z) is monitored as a function of frequency. If the time of flight across the drift space I, is greater than 1/2f, the charges do not reach the collector and return to the grid during the alternate half cycle, (( Z) = 0 ) . For t , < 1/2f the current is (Z) = Z,( 1 - 2r, f ) , where I , is a constant. The drift time is determined from a plot of ( I ) versus frequency.

*

A.J. DAHM

00

[Ch. 2 , $2

2.3.4. Comparison of techniques, and the effect of crystal strain on mobility measurements

Gudenko and Tsymbalenko (1979) compared different methods of measuring the mobilities and tested the effects of crystal strain on these measurements. They measured a nonlinear u ( E ) relation, using eqs (18a) and (18b) and the steady-state SCL I-V characteristics. The agreement with time of flight measurements made in a triode cell by Keshishev (1977) were excellent, although there was more scatter with the steady-state SCL technique. The agreement between mobilities determined by SCL transients and steady-state SCL measurements was better than 15%, and both sets of measurements were within 20% of the triode cell measurements of Keshishev. The effect of crystal strain on these measurements was studied by applying a controlled strain, in situ. They attached the collector of their diode to a magneto-strictor, and the diode spacing was changed by the application of a magnetic field. The maximum crystal deformation, ed = A L / L , was 9 x For E~ < 3 x the current changed by 20cm3 in 'He, he finds that plots of

Ch. 2, 831


97

In p versus T - for positive ions have a positive curvature below 1 K. This curvature can be seen in fig. 1. He attributes this curvature to a temperature dependence in the pre-exponential mobility factor. Activation energies measured by Keshishev are displayed in fig. 6. Efimov and Mezhov-Deglin (1978a,b,c) measured the SCL diode transient response to determine the ion mobilities. They measured the thermal conductivity of the sample in the diode as a check on crystal quality. A small activation energy -6" was reported for negative ions in bcc 3He at molar volumes V,,, >22cm'. Activation energies from their data are included in fig. 7. A large increase in p- and a smaller decrease (-2) in p, on entering the 4He bcc phase had been reported by Marty and Williams and by Keshishev. This transition was studied in more detail by Golov et al. (1984b) who measured the activation energies in both phases. A plot of p versus T-I for one sample is shown in fii . 9. The measurements in the bcc

I

1

-

10' 0.5

0.6

T-'1

0.7

0.8

(K-')

Fig. 9. Mobility versus inverse temperature in the vicinity of the bcc-hcp phase transition in 'He. The symbols are: n positive ions and 0negative ions both in the same sample; positive ions in another sample. After Golov et al. (1984b).

w

[Ch. 2, $3

A.J. DAHM

phase were taken along the phase boundary, but the variation in molar volume, 21.1-20.8cm 3, is not expected to change the qualitative behavior. Upon cycling through the phase boundary, the mobilities in bcc 'He were reproducible to within +20%, while mobility values in the hcp phase varied by a factor of 2 and the activation energies were not as reproducible. The activation energies in the bcc phase were A + = 15 K and A- = 10 K. The average activation energies in the hcp phase were A + = 9 K and A =22.5K. Lau et al. (1978) noted the large range of positive-ion activation energies reported at a given molar volume in hcp 'He. They studied ion motion in a large number of crystals with nearly identical molar volumes. The positiveion mobilities are plotted in fig. 10 as a function of inverse temperature for five crystals with a molar volume, V,,, = 20.5 cm-3. Their results suggest an anisotropic mobility tensor characterized by different activation energies for motion along the c-axis and in the basal plane.

I

I

I

I

I

I

0.5

0.6

0.7

0.8

0.9

1.0

T-I ( K-I)

Fig. 10. Positive-ion mobility versus inverse temperature for a narrow range of pressures. The solid lines are given by eq. (24) with the parameters given in the text. The values of 0 for a best fit are given in the figure. After Lau et al. (1978).

Ch. 2, $31


99

They define the ratio of the velocity component along the field to the field as p E . For an hcp crystal 2

pE = V . E / [ E ( =

sin2 4

+ pc cos 2 4 ,

(23)

where pb and pc are, respectively, the mobilities in the basal plane and along the c-axis, and 4 is the angle between the field and the c-axis. The solid lines in fig. 10 are given by p = p1 e-’IlT cos2 e

+

,+ e-”/’sin2

0,

(24)

where A, = 10.2K and A, = 17.8K. The crystal orientation was not measured, and a value of 8 was assigned to each crystal. The value of 8 is assumed to be 4 or 7~ - 4. Five crystals grown in a second diode ‘at the same molar volume (Lau 1978) gave a fit to eq. (23) with A , = 12.1 K and A, = 19.8 K. Evidence for anisotropic behavior for negative ions was much less convincing, particularly in the light of the strong binding of negative ions to defects. The range of activation energies for the negative ions was 21-25 K in the first diode and 22-24 K in the second one. These crystals were strained, in situ, in an attempt to alter the measured activation energies. The current decreased, but the values of A were reproduced to within small errors (Lau 1978). Crystals which showed signs of substantial trapping were rejected. Lau was unable to explain these results in a model in which a large number of traps were assumed. A source was employed which was a factor of 15 weaker than the source used by Sai-Halasz and Dahm in order to reduce the number of ions created near the collector. Keshishev (1977) and Golov et al. (1984b) also observed that positive4 ion activation energies in hcp He were not entirely reproducible. Keshishev reproduced negative-ion activation energies to within 1 K on 13 crystals grown at the same molar volume. Marty and Williams (1973) measured mobilities in different regions of their cylindrical cell. They saw no evidence of anisotropy in the positive-ion mobility, although it is possible that all measurements were taken in the basal plane. They observed anisotropy in the negative-ion mobilities, but this observation may have been related to their problems with trapping. A very interesting feature of the temperature dependencies of ion mobilities for both species in 4He was observed by Efimov and MezhovDeglin (1978a,b,c). As the temperature is lowered in small fields, E < 2 x lo4V/cm, the mobilities decrease exponentially, reach a minimum, increase by a factor of order -2 to a maximum, and again decrease exponentially at lower T . This anomaly, which occurs in the region

4 A.J. DAHM

100 16’

l0*l0

CI

p loll

a v

m

(Ch. 2, $3

10”

ldO

41

10

lb’

10-12

0.2

~

lo1’

16l3 1.0

1.5

2.0

T-YK-~)

(a)

Fig. 11. Positive-ion current as a function of inverse temperature at constant voltages in a SCL diode with a gap of 1 mm. The voltage for each curve is given in kV. (a) ‘He, p = 32 atm; (b) ’He, p = 60 atm. After Efimov and Mezhov-Deglin (1978a).

0.7-0.9K, is shown for positive ions in fig. l l a . Plotted here is the SCL current in a 1 mm gap diode, at a series of fixed voltages, versus inverse temperature. The maxima decrease in amplitude and shift to slightly higher temprature as the field is increased. The infections disappear for E > 3 x 10 Vlcm. This behavior is peculiar to 4He. In 3He, the I versus T - ’ curves remain nearly exponential for positive ions above 70 atm and for negative ions at all pressures. Below 60 atm, there is a very large reduction in the slope of the In I versus T - ‘ curves for positive ions in the vicinity of 1 K. This is shown in fig. 1l b . The shapes of the curves are voltage dependent as in fig. lla. Measurements of the positive-ion velocities from SCL current transients in ‘He, which are less susceptible to error, exhibit the same behavior (Mezhov-Deglin et al. 1984, Golov et al. 1984a). These results are shown in fig. 12 for two crystals. The diode spacing was 0.3 mm. The height of the current and velocity maxima were specimen dependent.


Ch. 2. 831

200 10

101

lo00

100

.5

h

u)

Y

> 10

10

,

I

I

I

I

I

1.o

I

I

I

T-’ (K-’1

I

I

1.5

, \ , \ L

I

2.0

Fig. 12. Positive-ion velocity versus inverse temperature for two crystals of 4He. The symbols are: 0 p = 31.2 atm; 0 p = 31.0atm. The curves are labeled with the diode voltages. After Mezhov-Deglin et al. (1Y84).

3.3. THEORY 3.3.1. Negative ions

3.3.I . I . Vacancy-assisted mobility 3.3.1.I . 1. Classical vacancies. The first theoretical treatment of ion mobilities in solid helium was carried out by Shikin (1971a,b; see also 1977a) who calculated the vacancy contribution to the negatively charged cavity. In the presence of an electric field, the electron pressure on the cavity walls is asymmetric. This leads to a net flow of vacancies from high-pressure to low-pressure regions. Shikin solved for the change in the normal electron pressure Sp,(O) on the surface of the cavity in the presence of an electric field E , and related the ion velocity u to E through the following conditions: V2c(r)= 0 ,

(25)

I02

A.J. DAHM

[Ch. 2 , 93

Here c ( r ) is the local vacancy concentration, 6 c ( r ) is the change in concentration, c, is the concentration at the cavity surface in zero-field, 8 is measured from the direction of E and R is the volume of a vacancy. We labeled the vacancy diffusion coefficient 0:with a superscript a to indicate that D,should be evaluated with A, = a. This follows from the fact that an appreciable gradient in the vacancy concentration occurs only over atomic dimensions. Equation (25) states that there are no sources of vacancies in the bulk, and eq. (26) states that the local normal velocity of an element of the cavity surface, Un(8), is determined by the local vacancy current and that the surface profile remains fixed during ion transport. For small values of 6 p , , ( 8 ) ,i.e. small electric fields, eq. (27) is approximated as

The distortion of the cavity is expanded in Legendre polynomials P,(cos O), and the electronic wavefunction may be written as (Gross and Tung-Li 1968)

where j , ( k r ) are spherical Bessel functions. The electron pressure is obtained from spatial derivatives of $(r, 8) evaluated at r = R - . The total excess pressure, which is given by the deviation in the terms in eq. (9) from equilibrium, is expanded in odd Legendre polynomials, and 1 = 1 is the only nonvanishing term for steady-state motion of the cavity. An estimate of the excess pressure on the cavity surface is given by (Shikin 1977a) 6 p , ( $ ) = p;,eER- cos 81Ez ,

(31)

where p e0 l = 3,EZ/3R- is given by eq. (9) and R is the volume of the cavity. With dc(r)/drlR-= d6c(r, O)l, l a R - , and eqs. (26), (29) and (31), the following approximate expression for the mobility is obtained p,. = eL)~c,RlrrR'T .

Shikin's more exact expression yields

(32)

Ch. 2, 931


103

where A = 1.1 and is given in terms of spherical Bessel functions, and A, - A, = h27r212mR2is the separation of the ground and first-excited electronic level of the unperturbed cavity. The interaction between the negative charge and a vacancy consists of a repulsive term due to the polarization interaction (Shikin 1971b) and an attractive term due to the elastically strained medium (Shikin 1977a). The net interaction is nearly zero in an isotropic elastic medium, and c , is approximately equal to the bulk concentration c,. With approximate values for A and A, - A, the vacancy assisted mobility for the negative ion is

This formula agrees with the vacancy contribution to the diffusion coefficient of microscopic voids in classical solids (Kelly 1967)

Dvoid= D , c , a 3 3 / 2 7 r R ~ ,

(35)

where f - 1 is the correlation factor for vacancy motion.

3.3.1.1.2. Vacancy-wave assisted motion. At low temperatures in 4He, T S 1 K, vacancies are assumed to be delocalized. For this case, Shikin uses a classical Boltzmann distribution function f, to describe these quasiparticles. The cavity surface is assumed to be a perfect absorber for vacancies, and detailed balancing requires that the number of thermal vacancies emitted be equal to the number absorbed. In equilibrium, the flux of vacancies impinging on an element of the ion surface is

Here u is the vacancy velocity, uT is its thermal velocity, and the z-axis is along the normal to the surface element. In the presence of an electric field, the distribution function f(r, 8) has a small correction which depends on the angle 8. This is a consequence of the fact that the vacancies leave the cavity preferentially from the rear where the electron pressure is reduced. The change in the local vacancy current density from equilibrium becomes

6 j , ( R - ) = 6 c , u T / 2 f i = u cos 8 .

(37)

A.J. D A H M

104

[Ch. 2. 53

The mobility follows from eqs. (29), (31) and (37). It is

The thermal velocity is given by

- ( 2 T / m * ) ' - a ( T @ ) " ? / h , for T 6 @ , U T - u@!ii. for T * @ ,

uT

(39)

where the symbol @ without a subscript denotes the vacancy bandwidth. Phenomenologically, this result follows from writing the ion diffusion coefficient as D = 5'167. Here, the probability of a vacancy-cavity collision in the limit A " % R - is T - ' =(c,,/a')u,..rrR?, and, on the average. the ion moves a distance 6 = a'/fi.rrRy each time a vacancy is absorbed and another is emitted from a random location on the cavity surface. The result, eq. (38), should be compared with eq. (32), since the same approximations were used. The increase in mobility of a factor of R _ / a may be cancelled by a numerical factor. 3.3.1.1.3. Nonlinear velocity-field relation. Shikin (1971b) extended his calculations to larger electric fields, E > 10' V/cm. For large fields eq. (27) must be used instead of the approximation eq. (29). The excess vacancy concentration is written as

The expression for 17, eq. (28), is substituted into eq. (40), and only terms which contain cos 8 are kept, in order to satisfy the boundary condition, eq. (26). The first-order correction leads to

3.3.1.2. Surface adatom diffusion Sai-Halasz and Dahm ( 1972) compared their measured negative-ion activation energies and diffusion-coefficient prefactors with the corresponding quantities for 'He atomic impurities: A - = l S A , at Vn,= 20.6cm' and A _ is much more pressure dependent; D , / D %= 10'. Theoretical arguments given above yield Db /Di= L&/R' = 10- ' for a classical vacan-

Ch. 2, $31


105

cy diffusion mechanism and -a2/R! = lo-’ for vacancy-wave diffusion. Here 0,is the volume of a 3He impurity. These authors ruled out a vacancy assisted mechanism for negative-ion motion in hcp 4He on the basis of these comparisons. Sufficiently small microscopic voids in classical solids move by the diffusion of surface absorbed atoms as opposed to a vacancy mechanism (Kelly 1967). The void diffusion coefficient for adatom-assisted motion is Dvoid= (3a4/2.rrR!)D,, D,= pl‘w, exp(SG) . Here D, is the surface diffusion coefficient (Bonze1 and Gjostein 1969), 2 : o, is taken as the Debye frequency and SG is the Gibbs free energy required to create and move a diffusible adatom. Equation ( 4 2 ) is to be compared with the vacancyassisted diffusion coefficient given by eq. ( 3 5 ) . In classical solids the adatom diffusion-coefficient prefactor is smaller by a factor of all?, but the adatom activation energy is the energy required to create a surface vacancy plus a small barrier for surface motion, and is less than the activation energy associated with vacancy motion, E,, + E ~ where , E~ is the energy barrier for vacancy motion. Thus large voids move by vacancy diffusion and smaller voids move by surface-adatom diffusion. The situation in solid helium is different in two respects. If we consider the surface adatom to be nonlocalized, then the number of available surface states is -exp(SS) = 47rR21a2. In addition, the jump distance of the adatom is 1 R . For this case, D,-68az/h for adatom motion and is much larger than the coefficient associated with vacancy-assisted motion, which is determined by the vacancy tunnelling probability. On the other hand, the creation energy for an adatom is larger than because it includes the work performed against the electron pressure in the creation process. Sai-Halasz and Dahm suggest that negative ions move via an adatom mechanism with I = R . The expression for the mobility for this case is

0 1 , 1 is the mean free path of an adatom,

-

pa

-

(6e8a2/hT) exp(-Aa/T) .

(43)

Marty and Williams (1973) estimate the energy to inject an atom into the cavity as =60K. They treat the atom as uniformly distributed over the cavity. The energy required to inject a surface adatom is less than this estimate. Their estimated upper limit on the diffusion prefactor of 5 X lo-’ cm’ls for the negative ion does not include the possibility of a large adatom activation entropy.

I0 6

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(Ch. 2, $3

3.3.2. Positive ions 3.3.2.1. Hole hopping Shal'nikov suggested that a hole mechanism is responsible for the mobility of the positive ion. A hole hopping mechanism would require a transfer of an electron from a neighboring lattice site with a subsequent shift of the density hump to the new site. Sai-Halasz and Dahm argue that the rapid increase in activation energy with reduced molar volume (a factor of 2 in A+ for a 13% increase in density) is inconsistent with this model. A simple qualitative argument can be given. The binding energy of the positive ion to the density hump in an isotropic medium is IW,l

=

lox

(e2/2r2)(e(33)-.' - ~ ( r ) - 'd]r - w , ,

(44)

where w , is a small strain energy. The compressibility decreases with increasing pressure so that E(=) increases faster than E [ p ( r ) ] , where p ( r ) > p ( = ) , and I WBldecreases with pressure. The hopping barrier is the change in the integral in eq. (44),when the density hump is displaced by an This change and the displacement atomic spacing and is a fraction of 1 WBl. decrease with increasing pressure. 3.3.2.2. Vacancy diffusion Sai-Halasz and Dahm suggest vacancy diffusion as a mechanism for positive-ion motion. Their measured activation energy A + == 1.3A, was inconsistent with the penetration of a vacancy up to the ion core. On the basis of a suggestion by Guyer that the radius of the ion might be defined as the distance to which vacancies can penetrate the density hump, these authors suggest that the vacancies penetrate to within a couple of atomic spacings. Their measured diffusion constant, D, = D,, with a large quoted error for both values, is large even for a radius of 2a. Mineev (1973) calculated the vacancy-assisted mobility of a sphere of radius R in solid helium. The excess energy required to create a vacancy by removing a polarizable atom from a site located a distance r from a charge is ~ , ( r= ) ae2i2/2rJ.

(45)

Mineev writes the local vacancy concentration as c ( r ) = co exp( - V p /T ) .

(46)

He estimated R + . the distance to which vacancies could penetrate, by

Ch. 2, 831


107

setting V , ( R + ) I T= 1. This yields R+ = 10 A. His derivation of the mobility follows that of Shikin for the negative ion, except for the calculation of the normal pressure on the surface of the sphere. He applies a deltafunction force, eE6(0) to the positive charge and calculates the local displacement of the medium u(r, 8). The stress tensor is determined from the displacement, and the pressure tensor is expressed in terms of the stress tensor. The resultant normal pressure exerted by the field is 6Pn(r7 e > =

where v =

(2-

U)

eEcos8 4Tr2

7

(47)

is Poisson’s ratio. With eqs. (26), (29) and (47)

This expression is numerically equal to eq. (32) for R , = R - to within 25%. For the limit of a long vacancy mean free path, Mineev adapts the calculations of Shikin for the negative ion. He obtains

+

ec, R

(2-

” = 27-41 - V) R 2 , ( 2 ~ m * T ) ”‘ ~

(49)

His mobility pre-exponential factor is two orders of magnitude less than values reported by Sai-Halasz and Dahm. Shikin (1977a) points out that there is a second term in the vacancy-ion interaction which is attractive. This results from the attraction of a vacancy to the elastically stressed region surrounding the charge. He writes the vacancy-charge interaction as

where Oi, is the change in lattice volume and a;,is the stress tensor. For an isotropic elastic medium V, = - nuii,where the minus sign indicates that the volume is negative. Both terms in the interaction vary as Y4. Shikin writes the interaction as

[

v+=-,. 1-Rn (3 2 are 2 (1--5u)4 1 where n is the density of helium atoms.

IOS

A.J. DAHM

[Ch. 2 . $3

This reduction in V ; allows vacancies to penetrate closer to the core and reduces R , . Shikin suggests that a near cancellation of the two terms in V : may allow the vacancy to penetrate to the ion core. This would explain the experimental results of Keshishev and Shal'nikov (1975) who found 0 , = LI?.

3.3.3. Species-in depend ent nieciiun isms 3.3.1.1. Phonon-assisted ion niotiori As discussed above. Marty and Williams (1973) noted that the diffusion pre-exponential factors for both species of ions had an unusual and strong depcndence on A / & eq. ( 2 2 ) . This observation led the authors to suspect that a phonon-assisted mechanism might dominate ion motion. They speculated on a two-well model in which the ion is located in one of two neighboring symmetric potential wells and estimated the probability of a phonon-assisted transition to the neighboring well. They found that, as in other models. the prefactor is too small unless the mean jump distance is several lattice spacings. This would require a band of states.

3..1.3.,7. Morion by ri plastic pow mechanism Plasticity occurs in a solid when the difference between the radial and angular stress components is sufficiently large and positive. Efimov et al. (1975) point o u t that the sti'esses surrounding a charge in solid helium may favor a plastic zone of radius R , > R . . They quote a formula (Kachanov 1956) for R , ,

whew dp = p ( R ) - p ( x ) , a, is the flow stress of helium, and R , is taken to be =6 A . For the negative ion d p / 2 q , 2 10. They suggest that a plastic region would enhance t h e phonon-charge cross section and could possibly be detected as a charge-density dependent heat conductivity. An attempt t o detect phonon scattering by ions (Efimov and Mezhov-Deglin 197%) yielded the large upper limit for the phonon-ion cross section of lo-"' cm2. 3.3.3.3. Vacuncy-wave scatteririg of charged defects

3.3.3.3.I . Low-field niobihfy. Andreev and Meierovich ( 1974) pointed out a difficulty with t h e concept that the radius of a positive ion in solid helium be defined as the distance from the core to which a vacancy can penetrate. Unlike the negative-ion cavity, there is no surface. bounding the positive ion, which can serve as a source and sink for vacancies. Instead,

Ch. 2. 131


109

they point out that vacancies located near the bottom of the vacancy band have a wavelength which is larger than the size of the ion complex. On this ground they ignore the structure of the ions and calculate the inelastic scattering of vacancions from ions in a manner analogous to the scattering of vacancions from isotopic impurities subjected to a force field. This treatment is also discussed by Andreev (1975,1976). The vacancion is treated as a quasiparticle in a definite momentum state at a temperature T 4 @. If a vacancy appears at a nearest-neighbor lattice site to an ion, it may scatter elastically with the ion remaining at rest, or it may undergo an inelastic scattering event. In the latter case, the ion is transferred to the vacant site and the vacancion absorbs the potential energy loss of the ion, eE a, , and leaves in a different momentum state. Here a, is the lattice vector connecting the original ion and vacant site. The average ion drift velocity is given in terms of the inelastic ion-vacancion cross section a,(/?)

-

u=

c

a,

d3k ( 2 ~ ) - ~ a ( k ) u ( k ) [ DD ( ~( )E+ eE * a,)] ,

(53)

where u(k) is the velocity of a vacancion with wave vector k, D ( E )is the density of states for vacancions, and the sum is taken over nearest neighbors for which eE a, > 0. The displacements -a, are accounted for as inverse processes in eq. (53). Equation (53) is rewritten as u=

2 a,

I

dE ( 2 ~ h ) - ~ [ D -( & D )( E+ e E . a , ) ]

I

a,(/?) dS ,

(54)

where the last integration is over a constant-energy surface in momentum space. The authors use the fact that inelastic scattering cross sections for slow particles are inversely roportional to the velocity, and evaluate this integral as 4 7 r h a ( ~- E " ) 1 , where a For a Boltzmann distribution the velocity is

/P

u = ( a / 4 h 2 ) ( T / ~ ) exp(-qJT) "2

-

C a,[l n

-

exp(-eE.a,/T)].

(55)

This relatively simple and elegant approach leads to some interesting predictions contained in eq. (55). In a weak electric field eEu s T, the mobility is a tensor ui = B i k E k ,where

The magnitude of the mobility tensor differs from that for a charged point

A.J. DAHM

110

[Ch.2, 93

defect moving by localized vacancies by a factor of TI@. For strong electric fields, the velocity is approximately independent of the electric field

-

Keeping in mind that the sum is in the forward direction, E a, > 0, both the magnitude and direction of u change abruptly as the direction of E is varied. That is, the sum remains fixed until a plane passing through a lattice site and having a normal parallel to E , intercepts a set of atoms. The transition to a new magnitude has a width TleEa since exp(-eE a , / T ) = 1 for atoms lying near this plane. The final energy state of the vacancy must lie within the band: E - E,, < @. If Ihree-body processes are neglected, then, at large electric fields, only those lattices vectors for which eE a, < @ contribute to eq. (57). For large electric fields, ion motion is blocked except for directions of E nearly perpendicular to some vector a,. The velocity of the ion is nearly perpendicular to E for these directions of electric fields. Finally, the authors note that the sum C, a, depends on the sublattice in hcp helium and that in very large electric fields and certain orientations of E, the ions will be concentrated on one sublattice. The above calculations assume that a vacancion can penetrate to within one lattice spacing of the ion. If the interaction between a vacancion and an ion is sufficiently large and repulsive. V > @, then the inelasticcrosssection will besmall (Andreev 1976).

-

-

3.3.3.3.2. Largefields (eEa 9 @). Keshishev (1977) extended the vacancion scattering theory of Andreev and Meierovich to large fields eEa 9 @ in order to explain his empirical results IJ CK ( E + E,l)3. This extension, which includes vacancy scattering with the simultaneous emission of a phonon is based on similar ideas put forward by Andreev and Meierovich (1974) and Meierovich (1975a). The final state of the vacancion in a scattering process must lie within the vacancy band. Thus, the ion scattering process cannot proceed for eE - a > @, without the emission of a phonon (eEa 1 K at 2300 V/cm). For T -% eE a , the density of thermal phonons is small, and the emission is spontaneous with a probability ( f i ~ ) Energy ~. conservation requires

-

E,

+ e E - a = E~ + ho ,

where the subscripts i and f refer to the initial and final vacancion states.

Ch. 2. 831


111

Equation (53) is replaced by u a nc a,

I

where hw, = E yields

I

Y

d3k u ( k ) u ( k ) f ( k )

WI

w 3 dw

,

+ eE - a, - @ and ho, = E + eE - a , .

(59) The integral over w

*

where E a, is replaced by Ea and terms of order (@/eEa)' are neglected. For eEa @, the ion velocity can be written as u = x ( E + E,)3 with E , ( T ) given by

I,@'

@ ( E ) u ( E ) Dexp(-E/T)(E (E) - ;@) dE

eaE,( T ) =

(61) @,/"

~ ( E ) u ( E ) Dexp(-E/T) (E) dE

The cross section is assumed to be temperatures, T G @,

E,

= (3T-

cx u - ' ( E )

and independent of E. For low

@)/2ea.

(62)

The limit E,( T = 0) = @ / 2 e a reflects the fact that for T = 0, and large values of eEa, the average final state is in the center of the band. Keshishev and Meierovich (1977) have carried out a slightly more detailed calculation for T T, and there is very little control over the density. As a consequence of the high density, the ideas and predictions of the weakly interacting Bose gas are not applicable. Hydrogen, on the other hand, does not liquefy, and the density can always be controlled so that it behaves as a weakly interacting gas. The experimental realization of BEC is therefore one of the major goals of spin-polarized hydrogen research and its achievement holds the promise of exciting new studies of macroscopic quantum phenomena.

150

I.F. SILVERA A N D J.T.M. WALRAVEN

[Ch. 3 , $1

Currently, the maximum density of spin-polarized hydrogen that has been achieved is approximately 4.5 x 10"/cm3 (Hess et al. 1984), however, at a temperature T = 570mK which is much higher than the predicted critical temperature of 43 mK for this density. With current techniques it is conceivable that densities a factor 3 or 4 higher, corresponding to T , = 100mK, can be achieved, but with great difficulty. In particular the rapidly increasing recombination rate with increasing density will make it difficult to dissipate the heat generated by recombination, and thus difficult to attain temperatures of order T , at high densities. Limitation to lower densities for the study of degenerate properties is not a serious restriction since it is this regime of the weakly interacting boson gas that enables the most detailed comparison with theory. Some suggestions for achieving BEC in a very low-density, low-temperature sample of hydrogen shall be discussed in section 10 of this review. Hydrogen can be condensed on a liquid helium surface having a single, weakly bound state and two-dimensional translational motion (Miller and Nosanow 1978). The atoms d o not penetrate the helium surface (Miller 1978) and have an effective mass of order one. As such it represents an almost ideal two-dimensional gas, which among other properties is expected to display two-dimensional superfluidity (Edwards and Mantz 1980). The situation for deuterium is quite different. The maximum achieved density of order lOI4/cm3(Silvera and Walraven 1980a) can probably be increased by at least an order of magnitude, although no other experimenters have succeeded in collecting a measurable quantity since the first report. The Fermi temperature

corresponding to a density of l O I 5 cm-3 is 1.828 x K, with g = 1. However, the density requirements are even more severe, as in order to observe superfluidity due to fermion pairing, densities of order 10" cm-3 (T, = 1 K) are required (Leggett 1980). At lower densities, hydrogen and deuterium have already proven to be rich systems, and promise much more interesting physics in the future, both in fundamental and applied aspects. In the past few years our understanding and knowledge has been vastly increased in the area of recombination and relaxation; interactions between H atoms and with the He gas and surfaces have also been studied in detail. The interaction with the helium surface has turned out to be of fundamental importance and has presented a host of problems, both theoretical and experimental, ranging from the determination of the adsorption energy to the Kapitza thermal

Ch. 3 , 511

SPIN-POLARIZED ATOMIC HYDROGEN

15!

boundary resistance between a gas of the HJ. and a helium film. A hydrogen maser operating at low temperatures with helium-covered walls may have a frequency precision enhanced by a factor of 300 compared to a room-temperature maser (Berlinsky and Hardy 1981). The spin-polarized gas of hydrogen is actually a mixture of two hyperfine states, one a pure (Zeeman) state with electron and nuclear spin projections m, = - 1 , mi = - 1 and the other the ground state which is an admixture of electron and nuclear spin-up and down. A gas of hydrogen in the pure Zeeman state was first produced by Cline et al. (1981). This is called double-polarized hydrogen, or HJ.S , as both the electron and nuclear spins are polarized. In this pure state the recombination rate is greatly reduced and the decay of the gas is controlled by the much slower relaxation to the admixed ground state. Recently spin-waves have been observed in a low-density nondegenerate gas of H J t (Johnson et al. 1984). In high-energy physics experimental efforts are under way to use a low-temperature (high-density) gas of nuclear polarized hydrogen ( H i t ) as a polarized proton source or a scattering target at CERN and at Brookhaven (Niinikoski 1981, Niinikoski et al. 1984, Kleppner and Greytak 1983), while at Los Alamos an experiment is being developed with a source of atomic tritium to measure the rest mass of the neutrino. It is expected that by polarizing the nuclear spins in a fusion reactor, the fusion cross section and energy yield will be enhanced (Kulsrud et al. 1982). At this point in the introduction it is useful to address the question of why atomic hydrogen is the most abundant material in the universe and yet does not naturally exist on earth. Atomic hydrogen is a highly reactive gas and can always enjoy a much more stable state by forming a covalent bond with another H atom or other elements (H,O, etc.). Let us concentrate on recombination to form H,. Jones et al. (1958) calculated the probability for recombination with the emission of radiation and found it to be extremely small as the radiative transitions are electric-dipole forbidden for the homonuclear diatomic molecule. As a consequence a third body is required for conservation of energy and momentum. The rate equation for the decay of the density n of a gas of H at constant volume is (at this point we omit the hyperfine-state labels on the density) dn dt

- = - Kin'n, .

(1.3a)

Here n = N'/V where N' is the number of atoms in volume V , and n, is the density of third bodies. For a gas of pure H, K:+ K' and n,+ n and one obtains dn = - K V n 3 . dt

(1.3b)


152

[Ch. 3 , $1

The rate constant K l depends on the initial states of the gas, the temperature. magnetic field, etc.. A second place where atoms can recombine is on surfaces. For surface coverage u = N ' / A ( N s is the total number of atoms on the surface and A the surface area) the rate equation becomes (1.4) This process corresponds to a situation where hydrogen atoms are weakly bound to the surface and the collision of two atoms in the presence of the surface, which plays the role of the third body, can result in a recombination. Here we consider only the second-order process with the rate constant K : for H physisorbed on surfaces such as helium, although there are examples of first- and third-order surface decay of H. If the surface is helium. we shall drop the subscript x, so K : + K'. In thermodynamic equilibrium there is a relationship between the surface coverage u and the gas density n , and both n and u c a n be large so that recombination of atoms in both phases can be important. Since the total number of atoms in a system with area A and volume V is N = N' + N' = Vn + A u , combining (1.3b) and (1.4) we find AK"'.

and the decay of the atoms will depend critically on the details of the system. Let us first consider outer space, in the absence of surfaces or other gases. Here the lowest densities are estimated to be of order 1 atom/cm3. To characterize the lifetime of the gas we consider the time for the density to decrease by a factor of 2, or A n l n = From eq. (1.3b) we find

4.

fi,2=(:K'ni)

I ,

(1.6)

where n , , is the initial density. Using K' = 1.2 x cm' atom-'s-' for unpolarized hydrogen at room temperature (Mitchell and LeRoy 1977) we find, as a rough estimate, f,,, = 1.25 x s, a lifetime much longer than that of the universe. A gas of H at the density of the earth's atmosphcre at sea level would have a half-life of about lO-'s. In outer space H can condense on surfaces of interstellar grains and be catalyzed to H,. Likewise, on earth, surfaces can control the decay rate of an assembly of hydrogen atoms. Metallic surfaces are extremely active with first-order decay as the dominant process (Wise and Wood 1967) so that the lifetime is determined by the diffusion of the gas to the surface. For a

Ch. 3, $11


153

container with metallic walls and with a mean dimension of 1 cm and a mean atomic velocity of 5 x los cm/s, the lifetime is a few ks. On the other hand, hydrogen has a very low probabilit of sticking to a teflon surface Y and an atom can strike the surface 104-10. times before recombining (this implies a first-order process), leading to a lifetime of order 0.1 s for the same container. The properties of various other surfaces have been reviewed by Wise and Wood (1967). On a liquid-helium surface the adsorption energy of a hydrogen atom, E , , is quite small. As we shall see later in the low-density, high-temperature limit the adsorption isotherm takes a very simple form (T

=

nArhexp(s,/kT) ,

where Arh = ( 2 7 ~ h * / r n k T ) ”is~ the thermal de Broglie wavelength. For moderate temperatures and gas densities, the coverage will be quite low and the decay due to the surface term will be insignificant. We shall present numerical examples later. The first production of a long-lived gas of atomic hydrogen (Silvera and Walraven 1980a) employed l i q ~ i d - ~ Hcovered e surfaces. The gas was “stabilized” in a high magnetic field of order 7 T at low temperature, of order 300mK. The high field is of extreme importance in extending the lifetime of the gas. It maintains the hydrogen in a highly polarized state which reduces the recombination rate constant by 5 orders of magnitude for a field of 10 T. In the first experiment the gas ( n l O I 4 cm-’) was observed to have no measurable decay in a period of 10min. Later measurements with densities 2-4 orders of magnitude higher showed that the gas slowly decays. A gas of atomic hydrogen is never absolutely stable. The word stabilization is used to imply that the lifetime of the gas has been increased by several orders of magnitude. These lifetimes are sufficiently long to allow the gas to come into quasi-thermodynamic equilibrium and its properties can be studied by both static and dynamic techniques. To date, this has always been the observed experimental situation at the highest densities with the shortest lifetimes. With each stage of increased density, new barriers to high densities appear. Gases of H J t have recently been compressed to achieve densities in the 101*/cm’ region (Sprik et al. 1983, 1985, Hess et al. 1983, 1984). At these densities three-body volume and surface recombination becomes the dominant and limiting process.

-

1.2. BOSESTATISTICS The phenomenon of Bose-Einstein condensation for the ideal gas is found


I54

[Ch. 3, I1

by a careful examination of the Bose distribution function N; =

g exp[(E, - p ) / k T ]- 1

'

where N , is the average population of the ith state, g the spin degeneracy, p the chemical potential and E, the energy. The volume ( V ) density, n , is found by summing (1.8). For the noninteracting gas with E; = p f / 2 m , this is evaluated by converting the sum to an integral over p, after first separating out the p = 0 term. By the usual procedure (Huang 1963) one finds

where g,(z) = El'=, z'll" is a Bose integral with g3,z(l)= 2.612. N,,is the number of particles in the zero-momentum state, the condensate. The chemical potential is always negative and approaches zero as the temperature is reduced. Bose-Einstein condensation takes place at a finite temperature

(-)

h2 n T , = 3.31 mk g

2/3

(1.10)

when the argument of g,,, becomes 1, or p - 0 . &, which is microscopic for T > T , , becomes macroscopically populated, of order N, the total number of atoms in the system. Expression (1.10) remains valid for a weakly interacting gas. However, in this case p takes on a small positive value (the interaction energy) for T S T , . The pressure of the gas at BEC is

where g,,,( 1) = 1.342. For the weakly interacting gas, the critical pressure increases with the strength of the interaction (Huang 1963). For a two-dimensional surface of area A the same procedure is carried out. only we now extend the energy of E, = pf/2rn - E , , where E , (a positive number) is a single-particle surface adsorption energy and p, is a two-dimensional momentum. The integral can be evaluated in closed form to find the surface coverage

Ch. 3, $11


-(glh~,)ln{l-exp[(p,+~,g-')lkT]},

(+=

155

(1.12)

where p, is the chemical potential of the atoms on the surface. In this case the p = 0 term plays no special role and the expression diverges for ps + &,g-' = 0. Bose-Einstein condensation does not take place (however, the two-dimensional system can have a normal-superfluid phase transition). Let us now consider the case of a gas H i in contact with a surface. (For simplicity we shall take the degeneracy g = 1.) The total number of particles, N = nV + (+A,and the distribution of particles over V and A at equilibrium are found by setting p = ps,in eqs. (1.9) and (1.12); this yields the adsorption isotherm. For the ideal gas an unphysical result is found! When p, = the number of particles on the surface, eq. (1.12), diverges. The addition of further particles cannot increase p to zero, so BEC cannot occur in the gas phase. Silvera and Goldman (1980) and Edwards and Mantz (1980) resolved this problem by taking into account the interaction energy of the two-dimensional gas of H i ; we shall discuss this is section 9. Expression (1.7) for the adsorption isotherm is easily found by setting p = p, in eqs. (1.9) and (1.12) in the high-temperature, low-density limit. In the ensuing sections of this review, we shall first describe the single-particle properties and the interatomic interactions, before discussing the techniques of stabilization. We shall then deal with the theory of recombination and spin relaxation, thermodynamic properties, magnetic properties, and interactions with surfaces. This will be complemented with a presentation of the experimental determination of the physical properties in section 5. The emphasis will be placed on H with a number of complementary treatments of D; Twill receive relatively little attention. In section 10 we present some of the speculations and difficulties to be encountered in attempts to achieve BEC. An alternate means of stabilizing atomic hydrogen was studied some years ago: matrix isolation in a solid lattice (H in H,, for example). These so-called free-radical studies (Bass and Broida 1960, Hess 1971) have many interesting properties, but because the atoms are immobile and isolated from each other, the aspects of Bose and Fermi gases are suppressed. The subject will not be treated in this review. A general review of atomic hydrogen has appeared about three years ago (Silvera 1982); recent reviews by Walraven (1984) and Greytak and Kleppner (1984) are more up-to-date. Hardy et al. (1982) have reviewed resonance techniques as applied to low-density hydrogen in low magnetic fields; Nosanow (1980) has reviewed some of the thermodynamic properties of spin-polarized quantum gases.

I . F . SILVERA A N D J.T.M. WALRAVEN

I56

(Ch. 3. $2

2. Single-atom properties The atomic hydrogens ( H , D and T) are one-electron atoms with 'S,,, electronic ground states. Higher excited electronic states fall outside the region of interest in thischapter; we shall confine o u r attention to the ground state spin multiplet. The proton and triton have spin i = whereas the deuteron has spin i = 1 with associated magnetic moments for the electron, p , = -g,pHs and the nucleon, p N = g,,p,,i. Here p13and p,, are the Bohr magneton and the nuclear magneton, respectively; g, and g, are the corresponding g factors. Values of the constants are given in table 2.1. We note that p, is oriented antiparallel to s so that in a magnetic field the lowest energy state of the atoms will have the electronic magnetic moments parallel to the field but the electronic spins antiparallel. We also define the gyromagnetic ratios here: y, = g,p,/h and y,, = g,,p,,/h. Throughout this article we use p,, and pBfor nuclear and electronic Bohr magnetons. respectively; pN,pI, and p, refer to magnetic moments of the nucleus. H atoms and electrons, respectively. Similar notations are used for the g-factors or gyromagnetic ratios. Furthermore, we use lower case letters to denote spin or angular momentum of single particles and uppercase to distinguish that of a molecule or a pair of atoms. J is used for the rotational angular momentum of an H, molecule, whereas I (or L ) is used for the angular momentum of unbound (scattering) pairs of atoms.

2.1. HYPERFINE ENERGIES

A N D STATES

There are two single-atom interactions which lift the spin degeneracy: the Zeeman and the Fermi contact hyperfine interactions given by

H = H,+H,,,,=-(-g,p~s+g,p,i)~B+ai~s. Diagonalization of this Hamiltonian to find the hyperfine energies is straightforward. In zero field the total angular momenturn f = s + i is a good quantum number, whereas in high field ( p e B S u). the spin projections m ,and m , (thus, also m,) are good quantum numbers. The energies of the states for hydrogen and tritium are:

E., = - : u

-

!u[l + ( p + B / U ) y ,

(2.24 (2.2b)

E,

= -:a

+ !all + ( p + f ? / a ) y.

(2.2c)

Table 2.1 Single-atoms constants of H, D and T in the ground electronic state. Values for the electronic and nuclear Bohr magnetons are ps = 9.274 078(36) X J T - I and pn= 5.050 824(20) X lo-” J T I. The magnetic moment of the nucleon is pN= g.p.1. The ratio of the deuteron magnetic moment to the proton magnetic moment is 0.307 012 192( 15) [Wimmet 19531. The ratio of electron to proton magnetic moment is 658.2106880(66), ti= h/27r = 1.0545887(57)X 1 0 - ” J s . Quantity

Hydrogen 1

Electron spin gc RepB

fi?e

Nuclear spin R. g.p.

=fix

alh

2.002 319 313 4(70) 1.8569664(72) X 1 0 - ” J T

?

I

5.585 691 2(22) 2.821 234(1l) X IO-” 3 T - ’ 1420.405751 773( 1) MHz

References:

Tritium

Deuterium

t

I

2.002 319 313 4(70) 1.8569664(72)X 1 0 - ” J T

t

I

5.9577 3.0091 x 10 - ”J 7 -I 1516.701470 808 7(71) MHz

2.002 319313 4(70) 1.8569664(72) X 10 ” J T 1

0.8.57 419 2(64) 4.330673 x 10 ‘ b J T I 327.384 3.52 522 2( 17) MHz

Ref.

v,

-u

FE

N

rn

’

0

I11 (2-51

>

8

50 x < 0

72

[ 11 Ramsey (1956). [2] Petit et al. (1980). [3] Hellwig et al. (1970). [4] Mathur et al. (1967). [5] Wineland and Ramsey (1972).

0 0

m Z

158


E d = : a + ?' p - B ,

(Ch. 3, 62

(2.2d)

*

where p - = g,pg g,p,,. For deuterium we have 6 hyperfine states, with energies

E = - - : -A a / 2 p + = 58.5 G for D. Hydrogen, tritium and deuterium all have two pure Zeeman states, b and d (for H and T) and y and 5 (for D). All other states are mixed-spin states. The mixing parameters in the figures are

for H and T:

E

=sin 8 = { 1 + ( p + B / a+ [ 1 + ( p L + B / a ) 2 ] 1 / 2 ) 2;}

77 = cos 8

for D:

(2.4a)

,

E=

=sine,

=

77-

=COS~,

,

(1 + f ( Z '

+ R')2}-"2; (2.4b)

with Z f = ( p + B / a )? i . In the high-field limit (&8+ -0). E + al(2p'B) and ~ = - a l ( d & + B ) . For a magnetic field of -10T, E and E? are 2.54 x (for H) and 8.26 x (for D), respectively. The 77 and q 2 are defined here for later use.

Hydrogen ,Tritium

im mI> 8 - - r O , B d =

n

1 W M h)

I

\

3 3>-11-1> I-2 2

Fig. 2.1. The hyperfine energies and states of hydrogen, tritium and deuterium as a function of magnetic field B . Allowed magnetic-dipole transitions are indicated with vertical arrows. N , , and S,, are nuclear and electronic transitions, respectively; F,, are the longitudinal resonance transitions.

Q

P

160


[Ch. 3, $2

2.2. ELECTRON SPIN POLARIZATION In thermodynamic equilibrium at low temperatures and high fields the polarization of the electron spins is almost complete. From eqs. (2.1) we find that for a field B = 10T, the splitting between the lowest and the highest pairs of hyperfine states amounts to A E l k = 13 K. For a temperature of 200mK the thermal occupation of the Ic) and ld)-levels is suppressed by a factor 10”. Unfortunately, this does not mean that t h e electronic spin “up” component is also suppressed by this amount, because the lowest hyperfine state (state la) in fig. 2.1) has an admixture of spin “up” due to the hyperfine coupling between the nuclear and electronic spins. To be quantitative, it is useful t o introduce the concept of state polarization.

and electron spin polarization,

Here P,, is the probability that an atom is in state i or j , PI, = PI + P,, and one easily finds f,,= tanh( p B B I k T )= 1 - 2 e x p ( - p , B / k T ) for the highfield, low-temperature limit. For B = 10T, P,, is essentially temperatureindependent for a temperature below 1 K. To determine f,, we evaluate P i = C, P h l ( ~ l h ) 1 2and P t = C, P,l(Tlh)(’, where Ih) is a hyperfine state and P, the occupation probability. Let us examine the case in which P,, = 1, i.e., only the hyperfine states a and b are occupied. For hydrogen we easily find P, = 1 - F’ = 1 - ( a / 4 p R B ) ’for high fields. We see that the electron spin polarization depends on the degree of admixture, which is controlled by the magnetic field. As shown by eq. (2.4a) it cannot be rendered negligible with currently accessible fields. For a field of 10T. (a14pBR)’ 2: 6.4 x IFh, which is rather small, but by no means negligible. In sections5 and 6 . we shall see that for H J , this small admixture of reversed electronic spin provides an open channel for recombination to the ‘ 2 ; bound molecular H, state, with rate proportional to F ‘ ; similar considerations hold for D.1 and T.1. Finally we note that if only the hyperfine state Ib) is populated, then the sum has but one term and f, = 1 .

2.3. MAGNETIZATION For an assembly o f hydrogen atoms, each with magnetic moment p , , ,

Ch. 3. $21


161

the magnetization is

The effective magnetic moment of the hydrogen atom depends on the hyperfine state, Ih), and is given by

where E, is the energy of the state, eqs. ( 2 . 2 ) and (2.3), and V, is the gradient with respect to the magnetic field. The magnetic moments for the four hyperfine states of hydrogen are shown in fig. 2.2 (we ignore p N since p., h ) ; a similar diagram exists for D. For later purposes it is useful to consider the magnetization in high field, where only hyperfine states la) and Ib) are populated. Then from eq. ( 2 . 6 ) we find

*

Here na and nb are the number densities of atoms in states a and b. Since in the high-field regime a gas of a- and b-state atoms corresponds to electron spin-polarized hydrogen we see that in this case the magnetization is proportional to the atomic density n of the gas, to the approximation that the nuclear magnetization can be ignored.

4-

B-

Fig. 2.2. The effective magnetic moments for the four hyperfine states of hydrogen, shown in fig. 2.1, as a function of applied magnetic field. States a and c reach their high-field values for E %- 507 G (after Silvera and Walraven 1981a).

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1.F. SlLVERA A N D J.T.M. WALRAVEN

ICh. 3, $2

The hyperfine energy level diagrams can be studied by means of magnetic resonance. The problem is analyzed in terms of the Hamiltonian eq. (2.1) in which B = B,, + 2B, cos w f , B , being a small polarized rf perturbation field and B,, the applied static field. The transition rate from an initial state Ih) = Ism,, im,)to If) = Is'm:, i ' m : ) is given by 27T r,, = P , (I Ivl fl

) I 2 6 ( E ,- E , - h w ) ,

(2.9)

where P, is the probability that an atom is in hyperfine state Ih) and V is the perturbation for magnetic dipole transitions, V = -2p,,B,cos of.ESR transitions are allowed which only involve a change in the electron spin (Am, = 1, Am, = 0), or N M R transitions which only involve a change in the nuclear spin (Am, = 0, Am, = 1). Pure ESR and NMR transitions only occur in high magnetic fields, where the mixed nature of the hyperfine states la) and Ic) may be neglected. In addition, for low external fields, a longitudinal resonance ( B , I I B ) with If. m , ) + If t 1, m , ) is allowed. The field dependence of the transition frequencies for hydrogen is shown in fig. 2.3. We shall make a few general remarks about the line strengths, which are

Fig. 2.3. Magnctic-field dependence of the hyperfine transitions in the hydrogen atom. The notation is given in fig. 2.1. Fa, is the transition used in the hydrogen maser; Nah3 Sadand S,, have also been observed experimentally in H i and Hit.

Ch. 3, $31


163

proportional to the square of the magnetic transition moment. ESR transitions are of order ( g,..+Jg,pn)2 = 4.4 X 10' stronger than NMR transitions. We note that the a- b and c--* d transitions, which are pure NMR transitions in the high-field limit, are greatly enhanced at low field due to the spin state mixing caused by the hyperfine interaction. For longitudinal resonance the transition rate r,,a sin 28 (sin 8 is given in eq. 2.4a). This rapidly weakens with increasing field, i.e.

rhf(B#O)lrhf(B=0)=sin28=a/pL'B=2~.

(2.10)

The hyperfine states have very long lifetimes and as a consequence the natural linewidths are extremely small. The observed linewidths are determined by radiation damping or inhomogeneities in the external field B. The latter is generally the most severe limitation. There are two field regimes where this problem can be minimized: for the a + c transition at approximately zero field and for the (a-b) or (c+d) transition at B = 0.648 T (at 765.48 MHz and 654.92 MHz response, respectively). For both fields, the derivative of the transition energy with respect to field is zero (see fig. 2.3). Due to this, field broadening is a second-order effect and less important. The first resonance experiments on gaseous atomic hydrogen at low temperature were designed to optimize sensitivity for low-density gases, exploiting the considerations discussed above. Crampton et al. (1979) studied the a + c transition in a field of approximately lo-' T , whereas Hardy et al. (1979) studied the a + b transition at B = 0.65 T. At both fields the homogeneity requirements are quite modest and a field of 0.65 T is still low enough to assure a considerable hyperfine enhanced transition probability. Hardy et al. (1980a,b), Morrow et al. (1981) and Jochemsen et al. (1981) used zero-field resonance techniques to study H below 1 K. At the high magnetic fields (B 3 5 T) required to study the spin-polarized gas, the homogeneity requirements are severe. Yurke et al. (1983) studied the build-up of nuclear polarization using NMR. van Yperen et al. (1983) succeeded in using ESR on the a + d and b + c transitions to study the time evolution of the hyperfine occupations n, and nb in a stabilized gas of hydrogen; Statt et al. (1985) have also used ESR to study properties of H in high fields. Recently, Johnson et al. (1984) have observed nuclear spin-waves in a low-density gas of hydrogen in high fields. 3. Interatomic interactions

One of the most attractive aspects of the hydrogen system is the high degree of precision to which the intraction between pairs and with external

I.F. SILVERA AND J.T.M. WALRAVEN

164

[Ch. 3 , 93

fields is known. Unlike most larger atomic systems, these ab initio potentials have been confirmed by experiment. In addition, the pair approximation. i.e. the approximation that the interaction energy of the many-body system is the sum of the pair interactions, is a very good approximation for most properties of interest. The Hamiltonian of a gas of atomic hydrogen can be expressed as

Here the kinetic energy is

the Zeeman energy is

and the hyperfine interaction is

H,,

=u

C i, -

S,

.

(3.4)

I

Terms (3.3) and (3.4) have already been discussed in the previous section and are responsible for the mixing of the spin states (fig. 2.1). In addition, during a collision there is a hyperfine interaction between an electron on one atom and a nucleus on another, which we shall ignore. The most important pair interaction, which determines the binding of atoms into a stable molecular state, is H , + Hexuhdue to the Coulomb interaction. This interatomic potential has been calculated from first principles by Kolos and Wolniewicz (1965, 1974, 1975) and is the most accurately known pair potential of all atomic systems. In the ground electronic state. ignoring nuclear spin (i.e., the hyperfine interaction), there are two potential curves: the singlet, S = 0, and the triplet, S = 1, state, where S = s 1 + s2. In fig. 3.1 we see the singlet potential ‘2.; = V, which has a well minimum E , / k = 55 100 K at r,,, = 0.74 A. This potential supports bound states. Dabrowski (1984) has extensively analyzed the spectrum of HZand made comparisons between theory and experiment. The ground vibrational-rotational state of H z has an experimental dissociation energy of D,(H,) = 36 118.3(3) cm- I = 51 967 K (Herzberg 1970), D,(D,) = 36 748.88(30) cm-I = 52 874 K (LeRoy and Barwell 1975). The best value available for T, is probably the theoretical value o f

Ch. 3, 93)

SPIN-POLARIZED ATOMIC HYDROGEN I

-1 3 22 -E

t6

- L

h

L

I

165 I

I

I

@-

-

t

0 1

$ 2 a w 2 w o

a

W Z W O

-I

Q 5-2

5-1 W c 0 a -2

W

t-

0

L-L

NUCLEAR

SEPARATION

IAI

-

Fig. 3.1. (a) The singlet and triplet interatomic potentials of hydrogen according to calculations of Kolos and Wolniewicz (1965, 1968, 1974, 1975). (b) The potentials of (a) in a high magnetic field on a magnified vertical scale.

Kolos and Wolniewicz (1968): D,(T,) = 37 028.89 cm-I = 53 277 K. LeRoy (1971) has made the most extensive study of the number of bound and quasi-bound states of the ‘ 2 ; potential of H, using the KolosWolniewicz potential and finds 301 bound levels and 47 quasi-bound states. The quasi-bound states are long-lived states with an energy in the continuum. These are resonance states of the effective potential, i.e., the singlet or triplet potential plus the (repulsive) rotational barriers. A few of these potentialsare shown in fig. 3.2 for both the singlet and triplet potentials, and high-lying bound states near the continuum are shown for H in fig. 3.3. Potential parameters and energy levels will differ for D, and T, due to the dependence of zero-point energy on mass (table 7.1). Due to the indistinguishability of the protons, the wavefunction of the H, molecule must be antisymmetric under proton exchange. This imposes a restriction on the molecular quantum numbers. Under nuclear exchange the rotational part of the wavefunction JI, is symmetric for even rotational quantum number J and antisymmetric for odd J. The allowed values of the nuclear spin states are I = 1, where I = i, + i,. The three I = 1 states correspond to symmetric wavefunctions; I = 0 to antisymmetric. As a consequence of this, molecules with even J have I = 0 and odd-] ones have I = 1. The former is called para-H,, the latter ortho-H,. The same combinations and notation exist for T,, whereas D, must have a symmetric

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30

I

- 20 Y

I

1

P

I-

z

w

I-

F 10 W

:5 0

w

u. h 0 -5 I

I

I

I

L

5

R

I

6

~AI

Fig. 3.2. The effective triplet interatomic potential of hydrogen, including the centripetal barrier, for a few of the lower rotational states. The angular momentum I is designated L in the text (after Walraven 1984).

wavefunction under nuclear exchange as the spin of the deuteron is 1. The designations and states are summed up in table 3.1. In contrast to the singlet state, we see in fig. 3.la that the triplet potential 3 .+ 2 = V, is essentially a repulsive interaction on the scale shown. Since the long-range Van der Waals interaction is always attractive and the repulsive electronic overlap forces are short-range, the atoms must have a negative potential well. The well can be seen on the expanded scale in fig. 3.lb (curve labeled M , = 0). Its well depth of E,/k = 6.46 K at r,,, = 4.16 A is insufficient to support a bound diatomic-molecular state. In the same figure we show the effect of an applied magnetic field. Taking into consideration only the electronic magnetic moment, we see that the singlet potential is field-independent, whereas the degeneracy of the triplet potential is lifted by the field.


Ch. 3, 13)

,

I

I

I

167 I

I

50

-_______.

J3:

to

1

2

V=lL J=4

v

IK1 50

V=lL J =3

i i

-100

V=14

J.2

-15t

V = l 4 J =1

B=lOT

-20(

I

I

0

1

2

3

R(A)

-

1

5

6

Fig. 3.3. High-lying bound states near the continuum of the singlet potential of hydrogen. (After Walraven 1984.)

It is noteworthy that the attractive part of the H-H triplet potential is the weakest of all interatomic potentials. For comparison, in fig. 3.4 we show a number of other potentials for weakly interacting species. It is believed that two 4He atoms with a well depth of Elk = 10.5 K will just form a bound state with dissociation energy D l k 4 1K, although the bound dimer has never been observed.

Table 3 . 1 Allowed comhinlttion of n u c l c a r - q ~ n \tiites and rotational 5t;itch f o r hydrogcn. dcutcrium, and tritium. and thc ortho-para designations. Antisymmetric is abbreviated by AS and symmetric by S: is thc total molecular nuclciir spin and L the rotational quantum number. P ~

Hydrogen. Tritium:

Ih =.

I?

0

Even

Symmetry

AS I S

S

AS

Odd AS

AS

State

Symmetry Deuterium; I, = 1

para

St'ltc

State

Symmetry State Symmetry

1 AS

0.2 S

Odd AS Even S

z 0

ortho

S

3

S

h

para

>

ortho

m

r

a

$

z

Ch. 3, 831


LENNARD-JONES

POTENTIALS

169

4

Fig. 3.4. Lennard-Jones potentials of some weakly interacting pairs (after Silvera 1984).

Later in this section, after a complete discussion of interactions, we shall return to discuss how spin polarization can be used to stabilize atomic hydrogen. We now express the direct and exchange term of eq. (3.1) in terms of V, and V,:

where V, = a(V, + 3V,) and J = V, - V,. A rather accurate fit to the tabulated potential values of Kolos and Wolniewicz (1974), useful for calculations, has been given by Silvera (1980):

V, = exp(0.096 78 - 1.101 73r - 0.039 45r2)

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1.F. SILVERA AND J.T.M. WALRAVEN

[Ch. 3. 13

with the cut-off function

r > 1.28r, mln

=1,

,

(3.7)

where r r m l nis the minimum in the triplet potential curve. In eq. (3.6) atomic units are used (1 Hartree = 219 474.6 cm-', 1 Bohr = 0.529 177 A). The fit is based on approximating the potential with an exponential repulsive part (first term) plus the long-range attractive part (second term) which is exponentially attenuated at short distances as proposed by Ahlrichs et al. (1976). The exchange energy can be fit over a range from 1 to 12Bohr by J ( r ) = exp(-0.288

-

0.275r - 0.176r'

+ 0.0068r') .

(3.8)

V, can be obtained by combining eqs. (3.6) and (3.8), however, this is restricted to r > 1 Bohr since Kolos and Wolniewicz did not tabulate the potential for smaller values of r . For calculations with V, in the region of its well, the tabulated values are recommended, with interpolation for intermediate points. A word of caution is in order for using the Kolos and Wolniewicz (KW) potential curves for calculational purposes. The "best" tabulated values for the singlet- (1975) and triplet-potential (1974) energy curves (KW 1975 and KW 1974, respectively) are obtained within the Born-Oppenheimer approximation. To reproduce the experimental values for the vibrational and rotational levels this potential is not sufficient and one also has to apply adiabatic, nonadiabatic and relativistic corrections (for phenomena not included in eq. 3.1), which are incompatible with the simple potential concept (see Kolos and Wolniewicz 1975, Wolniewicz 1983). Bishop and Shih (1976) have proposed an effective Schrodinger equation to calculate nonadiabatic rovibronic energies, yielding improved agreement with experiment. For some properties (like spin exchange) at very low temperatures these small corrections may have a rather dramatic effect, depending on the exact location of the Hz(u = 14, J = 4) vibrationalrotational level, which is close to the dissociation limit (see Dabrowski 1984). The final set of important interactions for the hydrogens are the magnetic dipole-dipole interactions, which can involve electron-electron, electron-nucleon and nucleon-nucleon magnetic dipole interactions. All are of the same form and can be written as

Ch. 3, 13)


171

where

f ( s i , i,) = [ s i * i, - 3(sj i j , ) ( i j. i j i ).]

(3.9b)

Here rji is the distance between spin si and ij and i, is a unit vector. As we shall see later, the term Hddis the dominant cause of relaxation between the hyperfine levels in high magnetic fields and is responsible for threebody recombination of highly polarized atomic hydrogen at high densities. In general, the term f ( i i , i,) is negligible. Until now we have only discussed the electronic singlet and triplet potentials of the H-H interaction. When nuclear spin is also taken into account, due to the spin multiplicity (4 x 4 = 16 spin states), the number of interaction potentials is far greater, but it usually suffices just to consider the singlet and triplet. Nevertheless, it is useful to consider all the possible potential curves. First, consider the interaction potentials between a pair of H atoms in zero magnetic field. In the absence of the hyperfine and dipolar interactions, we have the ground-state singlet and degenerate triplet potentials. Harriman et al. (1967) have analyzed the H-H hyperfine curves and Milleur et al. (1968) D-D curves; Uang et al. (1981) have treated H-D, D-T and H-T in an applied magnetic field. The H-H potentials are shown in fig. 3.5. There are 16 hypefine states, but due to degeneracy, only 11 distinct potential curves exist for H-H (36 with 22 distinct potentials for D-D). H , , Hexch,and Hdddepend strongly on the nuclear separation, whereas the hyperfine constant, a , is independent of range, except for very close approach. Comparing the magnitudes to a , we find a = ( p00/47r)(p f I r 3 )for r = 3.3 8, (6.2 au); the hyperfine splitting is equal to the singlet-triplet splitting [ a = J(r)] for r = 5.8 8, (11 au). For the long-range region, r 2 6.75 A, the exchange can be ignored ( J l a < 0.1) a n d x = si + ii, mfj are good quantum numbers. For separated atoms (J, p f l r 3 - + O ) in zero applied field, there are three curves with energies - $ u , - $ a ,and i a with 1-,6- and 9-fold degeneracies, respectively. These correspond to both atoms singlet, singlet-triplet, and triplet-triplet, respectively. For intermediate ranges, 5 < r < 6.75 A, all terms in eq. (3.1) is dominant and must be considered. For short ranges, r < 5 A, Hexch S = s1 + s2, M , and Z = i, + i,, M, are all good quantum numbers. For longer ranges, where I is no longer a good quantum number, the

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R (a,) Fig. 3.5. Spin independent part of the 16 (S are degenerate) interatomic potentials of H for the hyperfine states as a function of separation R given in Bohrs. For large separation there are three curves with degeneracies 1. 6 and 9 and energies - 3 a / 2 . - a / 2 , and a12 depending only o n the single atom hyperfine interactions (after Harriman et al. 1967, their notation is used).

ortho-para classification made for short ranges in terms of the quantum number I can still be made in terms of J . A simplified argument can be made by considering large distances, so that we are only concerned with atomic exchange. Since the Hamiltonian, eq. (3.1), is symmetric under permutation (P) of two atoms (see table 6.4a), the spin states can be separated into 10 symmetric (P = + 1 ) and 6 antisymmetric (P = -1) states. Under atomic exchange the wavefunction must be symmetric for the composite particles. Thus, the six odd-spin states must couple with the odd-orbital (ortho) states, and the other 10 spin states must have even rotational quantum numbers (para). The restriction to atom exchange is

Ch. 3 , 931

173


easily lifted, but the argument is longer. For deuterium one finds 15 symmetric and 21 antisymmetric spin states (see table 6.4b). It is now clear that the spin character of a state depends on the range. In order to characterize a potential curve as triplet or singlet, it is useful to examine the fractional triplet character of a pair of interacting atoms which is given by t(S’)

=

3 + (s, ‘S?).

(3.10)

This has the value 1 for a pure triplet state and 0 for a pure singlet, and varies with separation for other states. Similarly (Z’) gives the fractional triplet character of the nuclear spin. Consider now the effect of an applied magnetic field. In the infinite-field limit ms,and m,, are good quantum numbers for individual atoms. For pairs of atoms the electronic singlet has M, = ms,+ ms2= 0 and the triplet has M, = 1 , 0, -1 as shown in fig. 3.1. Inclusion of nuclear spin increases the number of states to 4 and 12 for the singlet and triplet, respectively; these additional states and splittings are not shown in fig. 3.1. For a high, but finite, field the triplet character of the states can be determined by evaluating eq. (3.10). Consider two hydrogen atoms at large separation in the Ib) state (fig. 2.1), then (S’) = 1. However for two la)-state atoms the amount of triplet character is + a(2sin’ 6 1)*= (1 - sin2 6) for high fields; the atoms have a small but finite singlet character. For short range the exchange will dominate and S becomes a good quantum number; the interaction is then either singlet or triplet in nature. Let us now discuss how spin polarization stabilizes a gas of atomic hydrogen. Since a pair of atoms interacting on the triplet potential cannot form a bound state, clearly, if an assembly of atoms could be maintained in the spin-polarized state such that all pairs mutually interact on the triplet potential, the gas will be stable against recombination to H,. An assembly of atoms in electron spin-down states (high-field hyperfine states la) and ( b ) of fig. 2.1) is called spin-polarized hydrogen and symbolized by H i ; an assembly in the electron spin-up states Ic) and Id) is represented by HT. If the assembly of atoms are all in the Ib) state, this is called doubly-polarized hydrogen, H i t . From the discussion of the previous paragraph we see that for H i t , all pair interactions are pure triplet and we expect this to be more stable than HJ. The wavefunction of a pair with at least one atom in the a-state will, in high field, have a small yet finite degree of singlet character, enabling recombination to the molecular state. Analogous definitions are used for deuterium (D) and tritium (T). Finally, we note that if an atom in the la) or Ib) state interacts with one in the Ic) or Id) state, the interaction

a

174


[Ch. 3, P4

contains a strong singlet component and recombination during a collision has a much greater probability. From fig. 3.1 we see that in zero-field the singlet state is lower in energy than the triplet state for all separations. At T = O K the singlet is the equilibrium state to which a triplet state will relax, followed by recombination into the bound molecular state. At elevated temperatures ( T B 2 h B / k ) then the thermal occupation of singlet to triplet state is 1:3 (for H, D and T). If equilibrium is maintained, the gas will continue to rapidly recombine until no atoms remain. With an applied magnetic field, the triplet M,< = -1 state is lower in energy than the singlet. Outside of a certain range and for low temperatures ( T G 2 p B B l k )where H i is the equilibrium state, a great degree of stability is offered to the system. Again at high temperatures, population of singlet states enables recombination to rapidly proceed. However, even at low temperature, we note that since a gas of H i is always in motion the region with V, < V, is accessible and at best we can speak of metastability for magnetic fields that are available in the laboratory.

4. Single-atom interactions with helium surfaces

I n the study of the lifetime enhancement of a gas of atomic hydrogen, the surface appears to play a dominant and controlling role for all densities, at low enough temperatures. As already mentioned in the introduction, at elevated temperatures the interaction with most surfaces, in particular metals. leads to very rapid recombination. Two types of recombination processes can be identified on surfaces with attractive potentials for atoms. In one. deep traps are rapidly filled with hydrogen atoms; mobile atoms recombine at these sites to form H, and the sites arc then again rapidly filled by another atom. This leads to a first-order decay process of the surface number density u, i.e. u = -u. In the second type of process, mobile atoms o n the surface interact with each other to recombine; this can be a second- or third-order process. Early experiments by Crampton et al. (1979), Hardy et al. (1979) and Silvera and Walraven (1979) demonstrated that H could be observed for T 3 5 K. although lifetimes were quite limited. In the initial stabilization experiments of H i (Silvera and Walraven 1980a) at low temperatures ( T < 1 K ) and high fields, it was found that on ordinary surfaces H J rapidly recombined and no detectable sample could be produced. However, when the surfaces were covered with a thin film o f ‘He, H i could be collected for long periods of time. The reason for this is that H has a very small adsorption energy, E . , , on ‘He, and according to eq. (1.7), u =

Ch. 3 , $41


175

nh,, exp(s,lkT), the surface coverage should be small. On ‘He, only second- and third-order decay processes are significant, and these become negligible for T 5 s,/k due to the small value of u. Because of its importance, we shall concentrate our attention on this surface. The properties of free-helium surfaces have been reviewed by Edwards and Saam (1978) and more recently by Edwards (1982). The currently accepted picture is that ‘He presents a flat, translationally invariant impenetrable surface to hydrogen, with an adsorption energy, E,lk 2: 1 K. The adsorption potential is believed to be about 5 K deep with a single bound state; E , is the energy required to remove an H atom from this surface state. The expectation value for the distance of the atom above the surface is about 68,. The atom moves on the surface as an almost free-particle in two-dimensional momentum states, with effective mass m 2:m H . The helium surface itself is flat (but not rigid, as elementary excitations such as ripplons exist) and there is no evidence that the adsorbed hydrogen causes a significant puckering or dimpling of the underlying surface. The helium density does not reduce abruptly from its bulk value to zero at the surface, but has a more gradual fall-off similar to a Fermi function with a width of a few 8, (Edwards and Fatouros 1978). Recent calculations (Pandharipande et al. 1983) indicate that this width may be as large as 7 A. The interesting question of whether hydrogen and its isotopes penetrate into bulk helium has been addressed by Miller (1978, 1980), Guyer and Miller (1979), and more recently, by Kurten and Ristig (1985). The latter, more rigorous calculation confirms the general trends of the earlier results. The quantity calculated is the change of the chemical potential, p , when a He atom from the bulk is replaced with a foreign atom. For positive p , the atom does not penetrate into the bulk (at T = 0 K) unless it has kinetic energy p. At the saturation density of 4He, Kiirten and Ristig obtained values of p 2:75 K, 40 K, and 27 K for H , D and T , respectively, so that none of these atoms penetrate the surface (uncertainties of order 5-10 K exist due to uncertainties in the He-H pair potential). p is strongly density dependent and in going over to 3He surfaces with a molar volume of 36.8 cm3/mol compared to 27.6 cm3/mol for ‘He, these values could dro by more than a factor of two. In fact T on ’He might be similar to 3He on ?He with the possibility of dissolving into the bulk. Another problem of significance for the stability of H i is the question of “what happens to the H, produced by recombination?”. If it accumulates on the He surface it could seriously affect the environment of the H i gas. Silvera (1984) has investigated this problem experimentally. While spraying a cold molecular beam of H, on the surface of superfluid liquid ‘He, he observed the formation of macroscopic clusters of H, within the bulk liquid

176

I.F. SILVERA A N D J.T.M. WALKAVEN

(Ch. 3. $4

‘He. This provided evidence that H, penetrates the “He surface either in monomolecular or clustered form. Kiirten and Ristig have also analyzed this system and found p = -23 K for H, with lower values for D, and T,, implying that all of these molecules penetrate into the bulk liquid as single molecules. 4.1. THESURFACE

ADSORPTION POTENTIAL

An accurate calculation of the adsorption potential presented to a hydrogen atom by a He surface is a formidable task. At large distances, z, from the surface it is easily shown that the potential varies as z-’. Howevcr. near the surface the calculation requires knowledge of the surface density profile. the two-particle (He-He) distribution function and the kinetic-energy density, none of which are precisely known. Calculations have been carried o u t by Mantz and Edwards (1979), Guyer and Miller (1979) and De Simorle and Maraviglia (1979). Stwalley (1982) has provided a scaling model which yields nice predictions for the isotopes. Of the ab initio type calculations. only the Manz-Edwards result, which gives a lower bound for F , of 0.6 K. is reasonably close to the value which was cxperimentally determined afterwards (see section 5). Mantz and Edwards used an extension of the Feynman-Lekner variational principle. The Hamiltonian for the N particle “He liquid with a He atom replaced by an H atom at the surface was written down. The expectation value of the energy was minimized and the resulting Euler-Lagrange equation could be written in t h e form of a single-particle Schrodinger equation of the H atom in the effective potential V,,, of the helium. The motion of the particle in the plane (s.y ) was described by free-particle momentum states, whereas the distribution in the z direction was found from

Solutions of this equation give the bound states of hydrogen on ‘He, and eigenvalues

where the translational energy in the plane is included, K being a two-dimensional wave vector. H was found to have a single bound state, F : = - E , with F , / k 3 0 . 6 3 K . Due to the larger masses, and thus smaller as well as a very weakly zero-point energy, D and T had larger values of bound second state. The probability density, + ? ( z ) ,and V , , , ( z ) are shown

Ch. 3, 441


177

in fig. 4.1. The peak of c#’(z) is several 8, from the surface and the distribution is quite broad, extending to -20 8,for H. Surface deformation due to electric polarization effects which increase E , have been estimated by Papoular et al. (1984). The effective potential for 3He on ‘He is also shown in fig. 4.1. By contrast to H, the 3He resides right at the surface (z = 0). This potential has the interesting property that the energy is lower in the interior than in the vacuum so that atoms “desorb” into the bulk. Stwalley (1982) has used a potential of the form - c / z 3 , cut-off at a distance zu from the surface by an infinite repulsive wall and solved for the bound states using a semi-classical solution. His model apparently has nice predictive values for the isotopes (see table 5.1, with a comparison to experiment). 0.3

P(Z )/Po

0.2 I

&Z)

(a+)

0.I

0

-5

0

10

20 1

Fig. 4.1. V,,, for H on 4He and ’He on ‘He. The upper graph shows the probability density for H, D, T and ‘He as well as the surface profile of 4He (after Mantz and Edwards 1979).

4.2.

INELASTIC SURFACE COLLISION

In thermodynamic equilibrium of hydrogen in a chamber with liquidhelium walls there is a rapid and continual exchange of atoms between bulk (gas) states and surface states. In this section we examine some of the

178


(Ch. 3. 44

processes for establishing equilibrium. Let us first provide some numerical estimates. In the low-density, high-temperature limit, u = hthnx exp(e,/kT). The thermal de Broglie wavelength A,,,= 39.0 A, and u = 5.7 x 10” atoms/cm’ for n = 10Ihatoms/cm3, e,lk = 1.0 K and T = 0.2 K. This is a seemingly small, but very important surface coverage. Under these conditions. the atomic collision rate per unit surface area, & = I . ,nu, - is 1.63 x 1019atomscm--2 s-l [V = ( U k T / ~ r n ) = ’ ’6482cm/s ~ is the average atomic velocity]. Alternatively, the surface coverage is given by

where I., is the average adsorption time and a, is the sticking probability per collision. Using both expressions for u yields I, = [ 4 h , , e x p ( ~ , l k T ) ] l u a ,.

(4.4)

Evaluating for our standard conditions, we find t, = 1 X 10-’ s, where we have used the experimental value a, = 0.035(4) measured by Jochemsen et al. (1981) using resonance techniques. They determined this from the ratio of the time between wall collisions to the time between stickings. The former was computed. the latter extracted from either relaxation time T 2 or the linewidth. For H on ’He surfaces. they found a, = 0.016(5). In all experiments carried out to date. t h e gas and the He surface are close to being in thermodynamic equilibrium. Evidently, equilibrium is rapidly established and requires only a few surface collisions. For a sample cell with a typical wall-to-wall dimension of I cm, the equilibrium time is of order or less than 1 ms. This assumes that there is a reasonable transfer of energy per collision. Salonen et al. (1982) measured a fractional energy loss per collision of a =0.2(1), using ballistic pulses of HJ. This experiment used the planar heater and bolometer detector shown in fig. 4.2. During a heat pulse of a few ps. gas atoms striking the heater pick up energy. This group of atoms propagates towards the bolomoter, depositing energy which is detected by monitoring the heating of the bolometer. From the time-of-flight spectra, the energy transfer coefficient a could be determined. In view of the small value of the sticking coefficient, this large value of 0.2 is surprisingly large. Recently, Salonen et al. (1984) have confirmed this result by a static measurement and find the temperature dependence shown in fig. 4.3. In this experiment a thin-film carbon bolometer is suspended in the gas and ohmically heated to temperatures above the cell temperature. Introduction of H J gas cools the bolometer by energy transport to the cooler walls. By monitoring the power required to

Ch. 3 , 94)


179

Mixinp Chamber

D eater etector rigger Bolome! t e r

Fig. 4.2. The cell used for studying ballistic heat transfer of H to He walls. All surfaces are covered with He and the cell is filled with HJ. HJ atoms traverse from the heater to the bolometer (after Salonen et al. 1982).

maintain a constant temperature as a function of gas density, a was determined. The important point to note in fig. 4.3 is that in this temperature range a decreases with increasing temperature, rather than increases. On theoretical grounds one expects a to go to zero as T-* 0 K. There are three types of surface collisions: elastic, inelastic nonsticking and sticking collisions which are, of course, inelastic. In the following we shall discuss some of the mechanisms for energy transfer. Until now we have considered the surface to be flat and static with a uniform massdensity profile normal to the surface (2). In order to accommodate energy, the surface must have dynamical modes, and the height of the surface, h(r), must be able to vary with position. Phonon excitations of the bulk liquid helium are important for higher temperatures ( T 3 1 K). The intrinsic elementary excitations of the 4He surface are ripplons (Atkins 1957) which are related to film height fluctuations by 1 h(r) = p h, eiq’r, 4

(4.5)

1.F. SIIXERA A N D J.T.M. WALRAVEN

I110

T, = LLO rnK

A

[Ch. 3 . $3

T O = 329 r n K

378 mK LO3 r n K

t OIL 0

2

L

6

8

nH [ 10'5 cm-31 Fig. 4 . 3 . The energy transfer coefficient as a function of HJ density. T , is the bolometer lcnipcrature and T,, is the gas temperature (after Salonen et R I . 1984).

where

Here r ,

is a ripplon creation operator for wavevector q and and p are the surface tension and the density of the liquid. and in this case g represents the Van der Waals force per unit mass exerted by the substrate on the helium. When an H atom strikes the surface. the surface is distorted by dynamic or static creation of ripplons. In the former case, an energy hw, is transferred. In the latter. the surface is statically deformed (dimpled) in the vicinity o f the H atom; the distortion moves with the atom, and the atom, along with its distortion, is called a polaron. Kumar (1981) and Guyer et al. (1981, 1982) introduced the idea of a polaronic distortion, and it appeared to be an important effect with a large effective mass for H and H-H coupling via ripplons. This now is recognized as invalid, mainly due to an unphysical choice of the H-ripplon coupling (Zimmerman 1982, Wilson and Kumar 3983). The calculated effective mass of H is only a few percent larger than the bare mass and the static surface distortion is minor. This implies that two-dimensional free-particle states are a good representation f o r translational motion.

w f = gy + ( y / p , , ) q 3where . y

Ch. 3. $41


181

4.3. STICKING PROBABILITY The sticking probability of H on 4He has been calculated by Zimmerman and Berlinsky (1983) and Kagan and Shlyapnikov (1983). Both articles used essentially the same model; we shall describe Zimmerman and Berlinsky’s results. They expanded the film thickness dependent H-He interaction in ripplon coordinates and used time-dependent perturbation theory to calculate the transition rate of free-atoms into bound surface states. The dominant contribution arose from single-ripplon creation. Normalizing to the incident flux of atoms gave the sticking probability as a function of energy and angle of incidence. This was then averaged over the angular coordinates and averaged with a Boltzmann distribution to yield a,(T), which for low Tvaried as T”*. In order to obtain numerical results with tractable integrals they modeled the Mantz-Edwards adsorption potential with a Morse potential. The calculated result, shown in fig. 4.4, is about 50% higher than the experimental value of 3.5 X lo-*. A serious criticism of this calculation can be presented. The approxim-

bt 0

I “0

0.1

0.2 0.3 Temperature ( K l

0.L

05

Fig. 4.4. Thermally averaged values of the sticking probability of H on ‘He as a function of temperature (solid line). The dashed line is an approximation using a 9-independent coupling. The vertical lines represent experimental values. The circle, square and plus are values found by varying E, with the circle representing the lowest value (0.89 K ) (after Zimmerman and Berlinsky 1983).

I R?


(Ch. 3, 84

ation o f replacing the long-range ( f 3surface ) potential with a short-range Morse potential was not investigated and does not appear to be good. Brenig (19130) has shown that the matrix elements for the two potentials have different q dependencies which implies different temperature dependencies. 4.4. KAPITZA THERMAL

RESISTANCE AT THE GAS-LIQUID

INTERFACE

Due to the release of recombination energy, D,, = 4.6 eV, there is a large source of internal heating in a gas of H i . The rate of heating (Q) increases with density. since Q = ~VD,(K'"n' + K(')n') where V is the volume and the K"s represent recombination-rate constants. For the gas to remain cold, this heat must be transferred to the He film and from the film to the cell (copper) which is connected to a thermal bath (refrigerator). If the paths of thermal conduction are not well-designed, large thermal gadients can develop in the system. In the following discussion the finite thermal conductivity of the gas, the helium and the copper will be ignored, as we shall concentrate on the Kapitza resistance, R,. R, exists at heat path interfaces connecting materials in different phases or different materials. This results in a temperature step, AT, at the interface. It is defined in terms of the heat flow per unit time Q

=

ATIR,.

(4.7)

R , is inversely proportional to the contact area, A , and in general is temperature dependcnt, with RK'-+ 0 as T-0 K. Due to the thermal step there can he three temperatures: the gas temperature, T, the helium temperature, T,,cand the cell wall temperature, T,. The thermal steps can be substantial, with ( T - T u ) /T = 0.25 or greater, as the cell temperature is lowered. In fact, this can be the most significant barrier for achieving BEC in HL as either high densities or low temperatures are required, a situation which creates a profound conflict. Unfortunately at this time the gas-liquid Kapitza resistance is not well understood. The helium-cell wall (or helium-solid) thermal resistance is much better understood. or at least a great deal of experimental data exists. In this case R , can be greatly reduced by increasing the interface area, using silver or copper sintcr. On the other hand, very little is known about the gas-liquid R , at prcsent and in order to design an experiment to achieve BEC, its magnitude and temperature dependence must be known. It appears as if increasing the surface area will be of little help since the surface recombination process is expected to be dominant, so that Q = A . The remaining alternatives are to select a critical temperature for BEC at which R , is

Ch. 3, 841


183

not a serious problem, or find a strategy for reducing the heating. We shall concentrate further discussion on the gas-liquid Kapitza resistance. The theoretical problem was first discussed by Castaing and Papoular (1983) who used a semi-classical model and confined their attention to nonsticking processes. Recently, Kagan et al. (1985) have made a fully quantum-mechanical calculation, including the adsorption process which turns out to be the dominant energy-transfer process at low temperature. They calculate the energy flux J from the gas to the liquid:

where

Here n q , nk and N , are occupation numbers for ripplons, atoms in the bulk and adsorbed phase, respectively. E~ is the bulk kinetic energy and E, that on the surface; U ( Z ) is the static adsorption potential and (01 represents a bound state of H. Kagan et al. use a Morse potential with a longrange Y 3fall-off to model the Mantz-Edwards potential and from eqs. (4.7) and (4.8) (with J = Q / A ) find

R -K' = R -Ka'

+

R -K's

(4.10)

7

where RKais the contribution for sticking (adsorption) which is scaled to its value at THe= 1 K

RKa(THe) = RKa(l)'THe

7

(4.1la)

and RKs is the term for inelastic scattering

Amongst other quantities RK1 a n. As a result we see that at low temperature the Kapitza conductance RKI

I84

I.F. SILVERA A N D J.T.M.WALRAVEN

[Ch. 3 , P4

must be dominated by the sticking term, with the weaker temperature dependence, even though the sticking probability is only of order 3 X at 100 mK (see fig. 4 . 4 ) . We also note that this picture must be incomplete as it predicts a monotonically increasing value of R K (T H e )with decreasing temperature. whereas experiment indicates an increasing dependence in the 300mK region (see fig. 4 . 3 ) . Boheim et al. (1982) have examined sticking coefficients to second order in perturbation theory and find that they can exhibit increasing (nonmonotonic) behavior with decreasing incident energy for T > 0 K. In the same approximation Kagan et al. (1985) have also considered the effect of higher surface coverages on R,, in which E , is reduced in value, going to zero at the critical density nc for BEC (see section 9 ) . They find that the Kapitza resistance is still dominated by a term of the form RKa. Below T , only the normal (noncondensate particles) of density n, can adsorb (since the condensate particles are in the k = 0 state and behave like a gas at T = 0 K) and an additional temperature dependence enters since R K x l/n,, and n n = ( T / T , ) " ' . It is useful to make some numerical estimates of the Kapitza resistance and compare to experiment. For the adsorption process Kagan et al. find Q l ( A i 3 T )= S n k ; T / 4 h k * .

-

(4.12)

Here k , is Boltzmann's constant, k* 4 x 10' cm-I represents the squared matrix element (see eq. 4 . 9 ) , among other constants, and T is the gas temperature. Evaluating for T = 0.3 K and n = 10"/cm3, we find Q / ( A A T ) = 1.68 mW cm K-.'. To compare this to experiment, we note from section 4.2 that Salonen et al. (1982) measured the energy loss per collision with the surface, (Y = 0.2 at T-0.3 K. We relate this to the heat transport by

-'

Q = : n C A a ( T ) A E = f n ( 8 k , T / ~ m ) 1 i Z A ~ ( T ) ( 2 k , i 3 T ) ,(4.13a) where the first factor is the particle flux and AE is the energy transfer, from kinetic theory. This can be rewritten as Ql(AAT)= ~ n ( 8 k ~ / ~ r n ) " ' ~ ( T ) T 1 ~ ~ .

(4.13b)

Evaluating for the same conditions used for eq. ( 4 . 1 2 ) , we find Q / (Ad T ) = 1 1 mW cm ' K I. This means that at T = 0.3 K, for a fixed Q l A , the experimental temperature drop is 6.5 X smaller than the theoretical prediction. More important, as already pointed out, the predicted tcmperature dependence has a slope opposite to what is experimentally found so that the discrepancy will become larger at lower temperatures.

Ch. 3.

W]


185

Kagan and Shlyapnikov (1983) point out that to achieve BEC, one may take advantage of the fact that the sticking probability goes to zero for low T. The idea is to reduce the heating by allowing the surface to get out of thermal equilibrium so that (+ < vsatfor T < T, (see section 9 for a discussion of u s a tIn ) . equilibrium, as the atoms on the surface recombine they are replenished by atoms from the bulk, to maintain the adsorption isotherm. At a sufficiently low temperature, due to the low sticking probability and reduced flux (- aniA) the surface coverage becomes smaller than its equilibrium value and heating is reduced. This effect is enhanced below T,, since condensate atoms will not stick. Unfortunately the estimate of the temperature at which the system comes out of equilibrium is T* = 5 mK. The heating rates for a saturated surface are far too high to cool a sample with (+ = vsatto 5 mK. One experimental strategy might be to start at a temperature T 6 T* and slowly fill the cell with hydrogen so that the cell is never submitted to a severe thermal load. Even so, the experimental cooling powers available at these low temperatures appear to be incompatible with the recombination power that is dissipated and consequently this would be an extremely difficult experiment. 4.5. THEHYPERFINE

FREQUENCY SHIFT OF

H

In section 3 we have introduced the single-atom hyperfine constant a (eq. 3.4) which, for a many-particle system of hydrogen atoms, can be considered to be a constant since the average H-H separation is quite large. On the other hand, when hydrogen is adsorbed on a He surface or in a high-density buffer gas, the average H-He separation is small or the collision frequency is high, giving rise to a change in a. This is usually observed by measuring the zero-field shift in the frequency of the a to c hyperfine transition of fig. 2.1. In appropriate units, this has the frequency v = ( a ) l h , where ( ) represents an average over the paths or motion of the atoms. This frequency can be measured with great precision and one usually measures the fractional hyperfine shift (HFS), A ( a ) / a ( m ) = [ ( a ( R ) )- a(a)] la(m). The hyperfine interaction strength is proportional to the density of electronic spin at the nucleus. Calculation of the HFS is a formidable task which involves computing the electronic spin density at the hydrogen nucleus, which will depend on the separation R between H and He

where ri is the position of electron i with spin s, and $(I?) is the complete

186

1.F. SILVERA A N D J.T.M. WALRAVEN

[Ch. 3 , 84

Fig. 4.5. The fractional hyperfine frequency shift calculated by Davison and Liew (1972) and Ray (1975). The solid line is the average of the two (after Jochemsen and Berlinsky 1982).

wavefunction for the He-H system. At long-range, the mutual polarization of the atoms causes a spreading of the electronic wavefunction and thus a lower spin density at the nucleus, resulting in a negative frequency shift. At short-range, the shift is positive resulting from two effects: deformation of the orbitals due to the additional interaction of the He and distortion of the wavefunction due to the Pauli exclusion principle. A detailed calculation requires a determination of the wavefunction, I(r(R), using extensive multiconfigurational self-consistent field techniques. Results of calculations by Davison and Liew (1972) and Ray (1975) are shown in fig. 4.5, from the article by Jochemsen and Berlinsky (1982), who have performed quantum statistical averages of the HFS for a He buffer gas. From fig. 4.5 we see the general trend of the HFS is to be negative for large R and positive for small R . very similar to a pair potential. For H in a He buffer gas at room temperature, a positive HFS is found since collisions favor the short range part of Aa. However, at liquid-helium temperatures the interatomic potential is such that the atoms mainly

Ch. 3. $41


187

sample the negative part of the HFS curve of fig. 4.5, and the shift is negative. When adsorbed on He, the larger average separation also results in a negative shift. Zero-field measurements of the HFS of H on 4He and 3 He as a function of temperature have been exploited to determine E, (see the review by Hardy et al. 1982). The motionally averaged frequency shift depends on the percentage of time that an atom spends on the surface, which is proportional to aln = hthexp(s,lkT), so that ( A a ( R ) ) 0: hrhexp(s,lkT). A measurement of ( A a ( R ) ) enables extraction of E , . In fig. 4.6 we show the HFS as measured by Morrow et al. (1981) as a function of temperature. At high temperature the 4He vapor pressure is high and the pressure-shift dominates the HFS, whereas at low temperature as the pressure drops the H atoms populate the surface, and the wall-shift becomes dominant. The experiment is described in greater detail in the following section.

T (Kl Fig. 4.6. The hyperfine shift versus temperature. The solid line is a fit to the data, the dashed line shows the gas-phase pressure shift contribution (after Morrow et al. 1981).

188

I . F . SILVERA A N D J.T.M. W A L R A V E N

[Ch. 3. $5

5. Experimental developments

I n a long review such as this i t is difficult to treat theory and experiment separately. We have attempted throughout to support and amplify theoretical discussions with experimental results. In this section we shall concentrate on experiment, referring to theory in previous and subsequent sections where necessary. We shall cover the principal developments of this field rather than give a complete coverage of all experiments performed. The handling of material will be in an essentially historical order. focusing on efforts to understand decay mechanisms and achieve high densities. The earliest published suggestion that H J might be an interesting system to produce and study was by Hecht (1959), with subsequent theoretical articles on the subject by Etters (1973) and Stwalley and Nosanow (1976), who were unaware of Hecht's work. Experimental work was started in the early 70's at the University of Amsterdam. Walraven et al. (1978) attempted t o stabilize spin-polarized hydrogen (Hf ) on a cold surface, using atomic-beam techniques. This was unsuccessful due to rapid spinrelaxation and recombination on the surface. At MIT, Crampton et al. (1975)) observed zero-field hyperfine transitions of H in contact with H,-walls at liquid-helium temperatures; Hardy et al. (1979) at UBC studied H in the presence of H,-covered walls and He vapor using low-field NMR; Silvera and Walraven (1979) used an atomic beam to show that H could easily be transported through teflon tubes to a region o f H,-covered liquid-helium cooled walls, and that the H thermalized with the walls. The conclusion drawn from t h e three latter experiments was that a reasonable flux (or density- l o i 7 to 10" cm-') of H atoms could be cooled to low temperatures in a steady-state flow experiment. However, when the H discharges were turned off the H density rapidly disappeared. indicating a short lifetime in the presence of H,-walls at low temperature. Spin-polarized hydrogen was first stabilized in a long-lived state in 1979 (Silvera and Walraven 1980a) and D.1 was stabilized shortly thereafter (Silvera and Walraven 1980b). Low-density gases of H J were found to have unmeasureably long lifetimes (on a scale of hours), whereas DJ was limited to minutes. Limitations of the lifetime and density of D.1 were found to be due to surface recombination, and the first measurement of the adsorption energy on 'He was made in the latter paper. The adsorption energy of H J on 'He was subsequently measured (Morrow et al. 1981. Matthey et al. 1981). Matthey et al. also showed that the surface recombination rate constant depended inversely on the square of the magnetic field. The next big step was the demonstration by Cline et al. (1981) of nuclear spin polarization in H i , resulting in significantly lower recombination rates and opening the way to higher densities. A series of

Ch. 3, $51


189

interesting measurements on the nuclear spin relaxation rate ensued. Until this point, densities of order 3 X 10i7/cm3 had been achieved. Using compression techniques Sprik et al. (1983, 1985) and Hess et al. (1983, 1984) were able to increase the density by an order of magnitude; a new density limitation was encountered: three-body recombination. Recently, in a research unrelated to decay of the density, Johnson et al. (1984) have observed nuclear spin-waves in a low-density gas of H i t .

5.1. STABILIZATION OF H i In order to stabilize and measure atomic hydrogen several essential requirements have to be met: (1) atoms must be made (by dissociating H,), (2) the electron spins must be polarized so that atoms interact via the 3 + 2 potential, (3) the spin polarization must be maintained, (4) the atomic hydrogen must be confined to a cell, and the atomic density of the cell walls must be kept low to suppress wallrecombination, and (5) a detector of H must be available to establish its existence. In the following, we discuss these various requirements and show in detail how they were implemented in the apparatus of Silvera and Walraven (1980a) shown in figs. 5.1 and 5.2.

5.1.1. Dissociation of H, A room temperature microwave discharge was used to dissociate H?. This is described in detail by Walraven and Silvera (1982). (rf fields can also be used.) The high-frequency electromagnetic field accelerates free electrons in a gas of H, and inelastic collisions create neutral hydrogen atoms through a multi-step process. This discharge can take place at room or lower temperatures. Cline et al. (1980a, b) used a liquid-nitrogen temperature rf discharge. The higher temperature source requires in the order of 5-20W of electromagnetic power to sustain a discharge. In a good discharge, 90-95% of the gas is atomic. Hardy et al. (1980) used a helium temperature low average power pulsed discharge to produce H. 5.1.2. Polarization At this point we discuss polarization of the samples with the aid of figs. 5.1 and 5.2. Atoms from the discharge flow up a teflon lined tube to a low-temperature sample cell in the center of a solenoidal magnetic field.

LIQUID NITROGEN

' I

CURRENT LEAD

I

LlPUlD HELIUM

I

I 2.5 TESLA

MAGNET

, , LlPUlD NITROGEN

5

M ovc

/

HYDROGEN STABILIZATION

I 11 TESLA

MAGNET

, ,'H BATH

,

, , 77 KELVIN I

RADIATION SHIELD

Fig. 5.1. The apparatus used to stabilize HJ and D1. (IVC and OVC: inner and outer vacuum chambers, respectively.) The 2.5 T magnet was not used. The 'He baths used to cool the cell and the copper braid connection are not shown. 190

Ch. 3. SS]


191

Fig. 5.2. A more detailed view of the cryostat of Silvera and Walraven. The cell (HSC) and HEVAC were cooled with 'He evaporation refrigeration which could achieve no load temperatures of about 250 mK. The cell was also provided with sintered copper for cooling. This was later removed as it apparently was not necessary, and a well-defined surface area was desired.

To enter the field the atoms must pass through a field gradient and they undergo a force, -V( pH . B). The electron spin-up atoms are repelled by the gradient and do not enter the cell, whereas the spin-down atoms are attracted into the cell. Spin-up atoms on the wall can either recombine or relax to the spin-down state and then enter the cell. 5.1.3. Sustenance of polarization In order to maintain the electron spins polarized, the sample must be at

I . F . SlLVEKA AND J.T.M. WALRAVEN

191

[Ch. 3. $5

low temperature and in a high magnetic field so that the spin-down state is thermally favored. As an example, in equilibrium, the ratio of down-spins to up-spins is nl / n f = exp( g c p B B / k T )= 2.7 x 1OIy when we evaluate for R = 10 T. T = 0 . 3 K.

5.1.4. C'onfinement 5.1.4. I . Magnetic compression The cell in fig. 5.1 is open at the bottom, and the atoms are confined to the cell by the magnetic gradient. It is useful to analyze this in some detail. For a simple solenoidal magnet. the field can be approximated by

B,

=0

(5.1)

.

Here. z is the axial direction, z,,= 51 mm for the magnet of fig. 5.1, pis the transverse coordinate and R,, = 10 T is the field at r = 0. In what follows we neglect R for simplicity. Walraven and Silvera (1980) analyzed the spatial density dependence of the gas. In the low-density high-temperature limit, neglecting interactions, we can use eq. (1.9) with N,,= 0 and g = 1 and approximate g , ?(z) by the first term in the summation,

or n

= exp[( p

+ p H B ) / k T ] A , i ,'

(5.2b)

with g = 1. Since the chemical potential p must be uniform, we evaluate it at B = 0 to find n ( R = 0) 3 n,, or

n( R ) = n,, exp( p , B / k T ) .

(5.2c)

We note from cq. (2.8) that since the gas is electron spin-polariz.ed the density and magnetization M = (MI are related by a constant

Here

I![,

is the local atomic density in hyperfine state h and we take

Ch. 3. $51


193

i n . The nuclear contribution can be ignored, as the splitting ( E , - E,)/k= 55 mK at B = 10 T. The axial and radial density distribution, n ( z ) and n ( p ) , can be evaluated for the field of eq. (5.1) n, = n , =

where

Thus, field gradients give rise to enormous density gradients or magnetization gradients with a Gaussian axial shape of half-width Az = ~ ~ ( k ? . / p ~ B , At , ) "low ~ . temperatures, HJ will be highly localized to the center of the field and pressed against the walls of a tubular confinement cell. As a numerical example, let us assume that we have a density of 10")/cm3 in zero field: then in a field of 1 0 T at 300 mK, n(B,) = 5 x 10'Y/cm3.A gas of HJ in a low-field region will be compressed into a high-field region. It is useful to introduce the compression ratio

c,+,(B,- B)= n(B,)/n(B)= exp[pB(B,,- B ) / k T ].

(5.5)

The density n(B,]), eq. (5.4c), is the maximum or saturation density to which a cell will fill, given a zero-field density, no. Another consequence of eq. ( 5 . 4 ~ is ) the following: if a cell is filled with a central density n(B,) and the zero-field density is reduced to zero, n must also go to zero. The important question here is "What is the time scale for this thermal escape of out of the cell?". To analyze this, let us first consider B = 0 and assume that a cell with volume V is connected to a (filling) tube of cross section A with a Clausing flow conductance factor K . From kinetic theory, assuming free molecular flow, the flow of atoms out of the end of the tube is found to be d N / d t = KnUA, where n = N / V is the density in the volume V and U is the average atomic velocity. This gives an exponential decay of N with time constant

a

re,

(6.61)

.

The projection amplitudes ( S ,M , , ~ m , , l S M , S , )are summarized in table 6.11. In this new representation the matrix elements of Vpd are readily

Table 6.11 Decomposition of the total election spin representation ISM,%S,)with respect to the partially s , . P = \/i and y = fi. symmetrized spin states / S , M s , m s , ) DHere m,,=i S

M,

S,

0 0

1

1

m , ,= 0

1

1-1

0 0

1

1

1 0

1-1

I 1

-0

Y

B

-Y 1

P

Y

P

Y 1

Ch. 3. 961

SPIN-POLARIZED ATOMIC HYDKOGEN

287

evaluated using table 6.8. We obtain for the single spin-flip process (-klVydl - + ) = - ~ ( 1 - ~ ~ h ~ / 4 7 r ) y ~ ( 3 Y 7 ;~’/(5i ;))’/ ”r ;(.) (6.62a) Here ii = &, in the notation of fig. 6.2 and the quantization axis is the direction of the applied magnetic field. In the transition, angular momentum is transferred from the spin system to the orbital system. Similarly for the double spin-flip process

( ilVydl -

i ) = ~ ( 1 - ~ , , h * / 4 7 r ) y ~ ( 6 7 rY/ 5; *) ”( i~; )( / r ; .)

(6.62b)

Thus, as far as the spin part is concerned, the transition amplitude for the double spin-flip process is twice as large as that for the single spin-flip process. This leads to a factor 4 difference in the transition rate. As in the case of exchange recombination, KVS evaluated their theory in the T = 0 K limit, where the unperturbed orbital wavefunction is invariant under permutation of the atoms [IpI,q l ) = l p z , q 2 ) = Ip3. p 3 ) , and the states are meant to be undistorted] and the sum T E + T , + T , is obtained by simply adding the spin parts. Squaring the matrix element and summing over all final spin states, the transition rate may be compactly expressed (note the absence of the thermal average which arises from the wellbehaved T = 0 K limit of the initial-state orbital wavefunction, see section (6.2)

T / V = - bn3e1

c

a(v/27rh2)’( /4,h2/47r)*(7r/5)

(6.63) where el =

c

hihzhj

c

ili2i3 I( [MI

-

;lIM,llE

+ A + B(hlh,h,)12.

An explicit expression for el in terms of a^ and 6.10.

(6.64a)

6 is obtained using table

In writing eq. (6.63) and eq. (6.64), we have lost the information on the final hyperfine state of the third body. To retrieve this information one has to compare the probabilities 1(001M,, hi1 kM,OIM,1)l2for the various


288

[Ch. 3, $6

processes. as may be seen from eq. (6.60). Using table 6.7, we have calculated these probabilities for both single ( M , = - ;) and double = + i ) spin-flip process in the high-field, T = 0 K limit. The results are listed in table 6.12. The l a n d M,quantum numbers to be used are obtained for any set h,h,h, from the projection on the total spin representation (table 6.10). A point of great practical importance in view of the stability of HJf at high density is the magnetic field dependence of dipolar recombination. Comparing with eq. (6.15) or eq. (6.50) we note that in eq. (6.63) an explicit field dependence is absent, yet the calculated rate for dipolar recombination between three H-atoms in the b-state shows a pronounced field dependence as illustrated in fig. 6.4. The curves shown were obtained by de Goey and Verhaar (1984) and represent the single and double spin-flip contributions for recombination to the u = 14, L = 3 molecular final state. In contrast to Van der Waals or exchange recombination, this field dependence does not arise from the spin part of the matrix elements but originates in the momentum mismatch and density of final states considerations discussed in section 6.1.2.1. We illustrate this point using eq. (6.37) and fig. (6.5). The final-state relative wave vector is given by qhi= [ ( 2 v ) ’ ” / h J [ ~+~ E,

+

~ , E~~

-

-

E,,]‘”

(6.65)

,

where ph is the Zeeman energy of hyperfine state lh,). For the single spin-flip process (E’ in fig. 6.5) q h ;= [(2v)1’2/n][p,,I - 2PBB]I .

This implies a cut-off field B

=

(6.66a)

IEVLI/(2pLg)beyond which thermal actiTable 6.12

Relative probabilities of various dipolar recombination channels. Both the single and double spin-flip cases are presented. The table is valid for the high-field, T = 0 K limit. The hyperfine states of the incoming atoms are given on the left. The molecular final states are indicated by the nuclear spin projection M, (only ortho final states are allowed at T = 0 K). The final hyperfine state of the third body is given in parentheses. Double spin-flip h,h,h,

M , =1

M,=0

Single spin-flip

M,=-1

M,=l

M,=O

M,=-1


Ch. 3, 861

I

"

"

'

"

"

"

"

I

"

"

1

5-

jl

4:

1

4

I

2 3

/

0

E

u"

0

10

spin flip process

, double

\ \\

i

i3

20

289

final state ti2(1k,3)

.

,

single spin flip process

B (TESLA)

30

40

Fig. 6.4. Theoretical results for the Kagan process in the volume. The single and double spin-flip contributions to the volume rate K,,, are shown separately.

Fig. 6.5. Diagram showing the energy available for the relative motion of the third atom with respect to the molecule for both the single (E') and double (E") spin-flip process.

290


[Ch. 3 , 16

vation is required for recombination to the H,(u, L ) final state. For the double spin-flip process (E" in fig. 6 . 5 ) qhi = [ ( 2 ~ ) ''l h ] [ l E , , I - 4 4 3 1 '

',

(6.66b)

corresponding to a cut-off field which is a factor 2 lower. For the H,( 14,3) molecular final state. the cut-off fields are B = 48 T (single spin-flip) and B = 24 T (double spin-flip), as may be seen in fig. 6.4. The maxima in this figure are due to two competing effects. With growing field the initial- and final-state momenta are better matched while the density of final states (proportional to ~7,,~,see eq. 6.8) is decreasing. becoming zero at the cut-off field. Note that the maximum in the double spin-flip curve is four times higher than the maximum for the single spin-flip case. This difference is due to the difference in spin matrix elements for the two channels (eq. 6.62) and appears explicitly in eq. (6.63). A plot similar to fig. 6.4 could be made f o r the H,(14.1) molecular final state. For this channel cut-offs would be found at B = 68 T and B 2: 136 T. The maxima would be lower than for the H,(14,3) channel in view of the difference in rotational degeneracy of the final state (2L + 1 = 3 versus 2 L + 1 = 7). We note the latest experimental results for the vibrational-rotational levels (Dabrowsky 1984) imply a cut-off field of 26.85 T for recombination to the H,( 14,3) molecular state via the double spin-flip process. The results quoted in the previous paragraph were obtained for bbbrecombination in the T = 0 K limit. It requires little argument that the qualitative aspects presented remain valid for ?' > 0. It is therefore unlikely that the field dependence would depend strongly o n temperature. However, if some a-state is present in the gas, recombination to para-H? final states becomes possible at higher temperatures. The appearance of this new channel will affect the overall field dependence in a qualitative manner. 6.1.4.3. The KVS mechanism on rhe surface

It was first pointed out by KVS that three H-atoms are required on a surface for direct recombination of H J f . These authors estimated the surface rate using a scaling argument in which the pair-correlation between the atoms on the surface is assumed to be identical to that in the bulk. Such a quasi-bulk theory in a good ansatz, in particular if the bound-state wavefunction o n the surface has a wide extent as is the case for H on 'He (see fig. 4.1 ). The quasi-bulk model is discussed in some detail by Kagan et al. (1982) and implies the conservation of essentially the complete phenomenology discussed in the previous paragraphs when changing from bulk- to surface-dipolar recombination. Thus, the dominant molecular

Ch. 3, 861


291

final state is H,(14,3) and the matrix elements for the double spin-flip process dominate by a factor four over the single spin-flip process. A more detailed analysis of the KVS dipole mechanism for bbbrecombination on the surface of 4He was made by de Goey et al. (1984) who were interested in the limitations of the quasi-bulk model. O n the surface one deals with a very anisotropic situation. Only the component of the momentum parallel to the surface is conserved; similarly, only the normal components of the angular momentum vector. As a result, in a partial wave analysis of surface scattering, the surface normal is the natural choice for the quantization axis. In contrast, the spin system is quantized with respect to the direction of the applied magnetic field. As the dipolar recombination mechanism irivolves the transfer of angular momentum from the spin system to the orbital system the above considerations lead to an intrinsic anisotropy of surface dipolar recombination with respect to the angle between magnetic field and surface normal. Such an anisotropy is lost when using the scaling argument. De Goey et al. (1984) applied a “2~-dimensional”(2;-dim) model, used earlier by Ahn et al. (1982) for a calculation of the surface dipolar relaxation rate (see section 6.2), to describe the relative motion of the H-atoms on the He surface. The model was found to agree on a 30% level with a full three-dimensional description of the surface relaxation. In the 2t-dim model, the b-b relative wavefunction + i ( p ) consists of a 2dimensional plane-wave exp(ik p ) , distorted near the scattering center by the triplet interaction averaged over the z-motion of the atoms. The full initial state is written as

-

where the 40(z) represent single-atom bound-state wavefunctions of the type shown in fig. 4.1, thep, and 5, are defined as in fig. 6.2, be it that the symbols now represent two-dimensional vectors in the plane of the surface. As for bulk bbb-recombination the two-dimensional vectors q3 and r), represent either p , and 6, or p 2 and g2depending on the dipolar interaction under consideration (Vyd or V i d , respectively). This extremely useful approximation was first introduced by KVS in relation to bulk exchange and dipolar recombination, and amounts to the neglect of the distortion of the wavefunction due to the third atom when considering the interaction of the remaining pair. For the final state the same wavefunction is used as the bulk bbbrecombination (eq. 6.56b). This choice may be unexpected at first sight. We therefore deviate a bit and compare the role of the adsorption potential in dipolar and Van der Waals recombination mechanisms on the surface. In

202


[Ch.3, 16

the latter case the potential not only serves to confine the initial state wavefunction to the surface, but also enables the momentum transfer required for recombination. Hence the molecule has to desorb. On the contrary, in the surface KVS mechanism, in principle, the potential only serves the confinement role. Since the interaction with the third H-atom may also account for the momentum transfer, the 3BCM can remain on the surface and the final state particles do not necessarily have to desorb. The approximation that the 3BCM remains on the surface is reasonable, in particular for low magnetic fields where up to 73 K (see fig. 6.1) is available for the relative H-H2 motion so that one of the final-state particles can enter the bulk of the He liquid, while the other particle is desorbing. In fact, this approximation, adopted both by Kagan et al. and de Goey et al., amounts to a complete neglect of the He surface in the final state, and the bulk wavefunction may be used. In high magnetic fields, the complete neglect of the He surface is much less realistic. This triggered de Goey et al. to investigate the case where the wave associated with the relative H-H, motion, expanding around the 3BCM, reflects elastically from the surface. However, this approach did not lead to a significantly different qualitative behavior, but only to modest quantitative differences of order 10% in the overall rates. The effects of the anisotropy may be very pronounced. Lagendijk (1982) pointed out that in the case of surface dipolar relaxation the relaxation rate can be made to vanish by orientating the magnetic field along the surface normal. This will be discussed in some detail in section 6.2 of this chapter. For surface dipolar recombination the effects are less pronounced. This is due to the desorption of the final-state particles. To calculate the bbb-recombination rate on the surface, an expression of the type given in eq. (6.36) may be used. For the quasi-bulk model used by Kagan et al. (1982), explicit thermal averaging may be avoided by working in the T = 0 K limit. This leads to an expression quite similar to eq. (6.63). In the 2 -dimensional model such a simplification is not feasible due to the logarithmic nature of the relative wavefunction $:( p ) . We return to this 2-dimensional pathology in section 6.2. Hence, de Goey et al. were forced to do the thermal average over the initial-state wavefunction and found the rate to increase roughly by a factor of 2 over the temperature range 0.2-0.6 K. It may be useful to point out that this does not imply that the full temperature dependence was extracted. In fact, the calculation carries a strong T = 0 K signature since only s-wave scattering was considered and the (unperturbed) incoming orbital wavefunction was taken to be invariant under permutation of the atoms as was done in deriving eq. (6.63). The theoretical results of de Goey et al. may be expressed as

Ch. 3, 561


1 Kbbb

=-

C C

MS n=0.2.4

A , , ~ ~ ( B ) P n ( c o8)s

7

293

(6.68)

where P , is a Legendre polynomial and 8 is the angle between B and the surface normal. The coefficients A, are shown in fig. 6.6, and display a behavior which is quite similar to that of fig. 6.4. For 8 = 0 the calculated rate is K,,, = 6.5 x cm4 s-' for B = 7.6 T. The experimental value of Hess et al. (1984) is an order of magnitude larger, K,,, =5.3 x cm4 s-', moreover the rate was found to drop slightly with growing field. Hence, it is quite possible that yet something else is going on at these very high surface densities. A possible explanation for the discrepancy (Haftel 1985, Verhaar 1985) could be a combined process of three-body dipolar relaxation and Van der Waals recombination, which we briefly compare with the KVS process to conclude this section. In the KVS process both the singlet admixture and the momentum transfer associated with the recombination is enabled by the presence of a third H-atom. In the above mentioned combined process

Fig. 6.6. The rate constant for surface dipolar recombination as a function of magnetic field. For the definition of the A , see the text.


194

[Ch. 3 . 56

the singlet character is admixed by the passage of a third H-atom, but the momentum transfer is realized by the molecule-surface interaction as in the case of surface Van der Waals recombination. To our knowledge no other suggestions exist in thc current literature that may explain this discrepancy. 6. I .5. Relarionship to phenonzenological experimental rates

In the coming section we wish to bridge the gap between the rather formal theoretical discussion of the sections 6.1.3 and 6.1.4 and the rate equations which are used for analyzing the experiments. The procedure is more involved than that of section 6.1.2.3, where we gave some examples for writing down rate equations for second-order processes. For third-order processes we not only have to account for the loss of two atoms per recombination event, but also keep track of what happens with the spins on the third body. We first consider the example of aaa-recombination via the exchange process in the volume for the high field, T = O K limit, an example considered before in some detail at the end of section 6.1.3.1. The overall recombination rate is written down in terms of using eq. (6.36),

c,,

N f V = -21'iV=

-

:a3c,aa.

(6.69a)

In terms of a simple rate equation we write NiV= -2Kaac,a3.

(6.69b)

Comparing we find

K,,, =

LC,:,,, .

(6.69~)

Along the same lines of reasoning this result may be readily generalized to

We next look at the recombination probabilities for the various aaachannels:

Ch. 3 , 861

2%


+ a,

aaa-, p-H,

2 I‘

aaa-, o-H,(M, = 0) + a, aaa-, o-H,(M, = 1)

+ b,

2 :9 E

9

&=EE &=i

para ‘orbital-1

2

7

ortho

rorbi,al-2 ,

~

orbital-3

.

~

r

~

r,

here r,, and are the partial contributions of the processes considered (see eq. 6.36) and the r,!?,:i:ta,.l, etc., are the factors due to for the rate the overlap of the orbital parts alone. Assuming the volume, V , to be time independent, we obtain

c,,

3

u = -(2+ ();caaa- ,

6 = +;(r,,,a”

(6.71)

where

represents the fraction of recombination events in which the third body changes hyperfine state. Thus aaa-recombination gives rise to a source term in the rate equation for the b-state, with the result a = -(2

+ ()Kaaaa3, 6 = +(Kaaaa3.

(6.72)

We estimate ( = 3 , using the KVS result that r ~ ~< ~ :‘pal. ~ t Any al three-body process may lead to source terms of the kind that appear in eq. (6.72). The importance of these source terms depends on the details of the process considered, hence, general rules cannot be given here. Later in this section we shall return to these source terms when discussing the example of dipolar recombination. First, however, we compare the K,,, for exchange recombination with the rate constant KaaHefor Van der Waals recombination to obtain information about the difference in recombination efficiency when one replaces the helium atom by a third H-atom. For this purpose we somewhat crudely assume = rorbital-2 = and add the rates for the three channels for aaa-recombination

This result is compared with a similar expression for KaaHe,derived by using eq. (6.27) and eq. (6.14), KaaHe 2: Z1 E 2‘orbital

’

(6.74)

Hence K,,, = 3Kaak,,and we conclude that aaa-collisions are -9x more

~

~

I.F. SILVERA AND J.T.M.WALRAVEN

2%

[Ch. 3 , 16

efficient than aaHe-collisions. The additional factor 3 arises because for a = He there are 3x more aaHe triples in a H-He mixture than in a pure a-state gas (note that for the aaa-process. the total gas density is a, whereas for aaHe it is a + r ~ , , ~The ) . K,,, efficiency may deviate significantly from the above result, as determined by the details of the orbital integrals. To conclude this section we present the rate equations associated with the dipolar recombination terms included in (6.64b). The equations were derived using table 6.12 a

=

-(2

+()R,.

b

=

-(2

+ OR,.

ci = + & r i a , c = +(R,,

(6.75a)

where

The rate of recombination events is given by R , + R,; 6 is the fraction of events that proceed via the double spin-flip process. From eq. (6.63) one concludes that K,,,, = Kaah= Kahh= K,,,. In the limit that the a-state density vanishes eq. (6.75a) reduces to b = -(2

+ ( ) K h h h b 3,

C

= (K,,,b3 .

(6.75b)

When the c-state atoms have a larger probability to recombine than to relax back to the b-state the total loss of particles is given by ri = -2( 1 + ()K,,,,n'(

- Ln')

.

(n =b) .

The symbol L has been used to characterize the literature. From the above discussion experimental information concerning the obtained if the state of polarization and relaxation channels are well-known.

the rate of the KVS process in it will be clear that accurate KVS process may only be the efficiency of the various

6.2. RELAXATION The wealth of recombination processes discussed in section 6.1 is by itself not sufficient to estimate the stability of a gas of hydrogen atoms at low temperature. A key ingredient is missing as long as the occupation o f the various hyperfine levels is unknown. To obtain this information the various

Ch. 3, 161


297

spin-relaxation mechanisms have to be included in the rate equations. Fortunately, from the abundance of relaxation channels at our disposal a rather restricted set appears to be relevant to our problem. In low field, spin-exchange is dominant, whereas in high magnetic fields dipolar processes govern the physics of relaxation. Moreover, at least for the currently accessible density regime, only two-body relaxation mechanisms come into play. This has led us to divide section 6.2 of this chapter into three subsections. First we give an introduction, comparing the particularities of three-dimensional and two-dimensional scattering as far as is relevant to the relaxation problem. The next two subsections deal with spin-exchange and dipolar relaxation, and we also make the link with the phenomenological rate equations. 6.2.1. Introduction-Volume and surface processes

For the relaxation problem we are interested in the transition rates between the various hyperfine levels. The total rate of relaxation events is given by eq. ( 6 . 3 ) , where li) and If) now represent symmetrized two-body initial and final states, respectively. Piis the probability that an initial pair-state is occupied and (fl Tli) is the two-body T-matrix. Momentum conservation is satisfied implicitly by expressing li) and If) in two-body center of mass (2BCM) system li)

=

Ih,h,; k) ,

If) = 1h;h;; k’) .

(6.76)

The initial state li) represents a pair of H-atoms in hyperfine states h, and h,, moving with relative momentum k. The final state If) is defined analogously. The probability Pi is normalized to the total number of H-H pairs, analogous to eq. ( 6 . 5 ) ,

(6.77) with

298


[Ch. 3 , 86

P,,,(k) represents a Boltzmann momentum distribution (eq. 6.6) and p = : m H is the reduced mass of the pair. For some remarks concerning the

use of symmetrized versus nonsymmetrized states, the reader is referred to the discussion of eq. (6.5). With the usual continuum transition C,- [ V / ( 2 7 ~ I) ~dk) and integrating over all final k ( = / k l ) states, the energy S-function disappears and eq. (6.3) becomes (6.78a) with

(6.78b) Here, n is the bulk density of the H gas, k‘ denotes the unit vector corresponding to k’ and we evaluated the three-dimensional density of final states p , ( E ) = phk/(27r)’ for the momentum k = k‘. The factor in eq. (6.78b) serves to assure that the symmetrized final states contribute just once in the summation. Energy conservation is satisfied if

4

h’k‘’l(2p) = Ei - Ehi- Ehi E ‘ a O .

For k < k ’ , the process is thresholdless; for k > k ’ it is (thermally) activated (see also the discussion below eq. 6.9). The transition rate is related to an energy-dependent cross section a ( E ) via

-__ - Umh,hz-h;hi

7

(6.78~)

The V and 6h,h,-h;hi represent where E = f i ’ k 2 / ( 2 p )and U = (8kT/7rp)liZ. thermally averaged quantities. A E is the activation threshold ( A E a 0). The cross section is defined as (6.78d)

Ch. 3 , $61


29Y

Analogously, one derives the total transition rate on the surface in a two-dimensional treatment

with

(6.79b) where a is the surface density of H and k and k’ refer to two-dimensional vectors in the plane of the surface. The two-dimensional density of momentum states is a constant, p , ( ~ ‘ = ) p./27rh2. The transition rate is related to an energy-dependent cross length via

- -

= VAhlhz-h;hj

where E = h Z k 2 / 2 p V;

3

= (7rkT/2p)’/*. The

(6.79~) cross length is defined as (6.79d)

The various two-body relaxation processes may be described with eq. (6.78) or eq. (6.79) as they only differ in the choice of the T-operator. Here we do not enter in the details of the evaluation of the matrix elements, which is left for the coming sections, but rather discuss some fundamental differences between two-dimensional and three-dimensional pair wavefunctions when the distortion due to the interatomic potential is taken into account. Such wavefunctions are needed for calculations within the distorted wave Born approximation (DWBA), which is often required to obtain accurate results. We first consider scattering between two H-atoms in free space, interacting via the central potential V ( r ) , which may be the singlet or triplet potential. Since V ( r )is short-ranged and well-behaved, the solution


300

[Ch. 3. $6

to the three-dimensional Schriidinger equation for the relative motion of the two atoms may be expressed in terms of a partial wave expansion (Messiah 1970)

(6.80a) Here k and r are the relative wave-vector and position, 1 and m = rn, are the quantum numbers of the orbital angular momentum of the pair, Y'y * ( k ) and Y y ( r )are spherical harmonics describing the angular dependence of the wavefunction in terms of the directions of k and r with respect to an appropriate quantization axis, and F,(k, r ) is related to the distorted radial wavefunction (1 / r ) y , ( k , r ) via y , ( k , r ) = (21 + 1)i' e"'"F,(k, r ) / k .

(6.80b)

The phase shift 77, is defined by the asymptotic behavior of Fi(k, r ) . F,(k,r)-sin(kr-

$IT++,),

k cotan q o ( k )=

a1 + 51 r,k'

r+x,

(6.80~) (6.80d)

Here re is the effective range of the potential and a the s-wave scattering length [for H-atoms interacting via the triplet potential a = 1.33 a, and re = 323 a, (Friend and Etters 1980, Uang and Stwalley 1980a)l. For vanishing potential the phase shift vanishes ( q - + O ) and eq. (6.80a) changes continuously into the partial wave expansion of a plane wave, while F,(k, r ) / k r approaches the Bessel function j / ( k r ) , as may be seen from the radial Schrodinger equation [d21dr' - t(l + l ) / r ' - U ( r ) + k ' ] y , ( k , r ) = 0 ,

(6.81)

where V ( r )= ( 2 p / A Z ) V ( r and ) k Z = ( 2 p / A 2 ) E . The second term in eq. (6.81) is the effective centrifugal barrier which is illustrated in fig. 3.2. Also the I = 0. k-0 limit of eq. (6.80a) is well-behaved in three dimensions. If we substitute lim,,,+O = ka into eq. (6.80~)we find y , ( k , r ) (1 /k)sin k ( r + a) ( r + a ) for r > re and y , ( k , r ) becomes k independent:

-

1 S ; ( r ) = -r y J r )

-

(for small values of k) .

Here yo(') is the I = 0, k

=0

(6.82)

solution of eq. (6.81). This is particularly

Ch. 3, $61


30 1

useful in the context of this section as it enables rapid estimates of processes by taking the T+O limit for the relative motion, thus avoiding elaborate thermal averaging. This approach has been followed quite impressively by Kagan et al. (KVS 1981) in their exploratory study of many decay processes in H. Scattering of two H-atoms bound to a surface of liquid helium is quite different. Although for a single H-atom on a perfectly flat surface the motions parallel and normal to the surface are decoupled, such a simplifying feature is absent, at least in principle, for interacting atoms where both degrees of freedom are coupled via the interatomic potential V ( r ) . Hence, in principle, scattering on a surface is a highly anisotropic three-dimensional problem. Fortunately, in practice, a three-dimensional analysis is not required since the characteristic times for the events in which we are interested are much longer than the oscillation time of the bound-state so that it is sufficient to work with a quasi-two-dimensional approach and average quantities of interest over the bound-state wavefunction. This last approximation is called the 2 1 -dimensional approach by van den Eijnde et al. (1983), who compared the threedimensional approach with the 2;-dimensional approach for the case of nuclear dipolar relaxation on a He-surface. To derive the radial Schrodinger equation for the 2 ;-dimensional case, the three-dimensional wavefunction for the surface problem F,( p, zl, z 2 ) is approximated by ~o(zl)~o(z2)ym( p ) exp(irn4), where ~ $ ~ ( is z ) the bound-state wavefunction, real and normalized to unity (see fig. 4.1), and p is the relative position vector in the plane of the surface. Then the three-dimensional radial equation reduces to

U, is related t o the surface adsorption energy through Ua = ( 2 p / h 2 ) ~ , . Multiplying on the left by ~o(zl)~o(z2) and integrating over z1 and z2 one arrives at the 2 1 -dimensional radial equation [ (d2/d p ')

+ a ( 1 - 4m2)/ p

-

U( p ) + k 2 ]y , ( p ) = 0 .

(6.84a)

Note that the zero of energy has been shifted for convenience; further, (6.84b) The 2;-dimensional model was first applied to the H-H problem on the

I.F. SILVEKA AND J.T.M. WALRAVEN

301

[Ch. 3. 86

He surface by Edwards and Mantz (1980)in their mean-field estimate of E , as a function of the H coverage (see section 9). Further simplifying eq. (6.84a) by replacing U ( p ) by U2-,( p ) = U ( p, 0, 0), one obtains the pure two-dimensional limit for the radial equation. In contrast to the three-dimensional case, the 2;-dimensional (or two dimensional) case has not such a nice limiting behavior for T+ 0 K. This is discussed in detail by Verhaar et al. (1984,1985). First of all notice that for rn = 0 the centrifugal "barrier" is attractive, thus increasing the binding forces between the atoms. For U ( p ) = 0, eq. (6.84a) is perfectly wellbehaved and the solution consists of cylindrical Bessel functions. However, for U ( p ) f 0 the phase-shift is given by 1

i

1

+ log f ku) + rfk' , 27T

cotan q , , ( k )= - ( y 7T

(6.85)

where a and re are the two-dimensional scattering length and effective range. respectively. and y = 0.577 215 665 . . . is Euler's constant [for H-atoms interacting via the triplet potential a = 2.3 a , and re = 14.3 a,, (Verhaar et al. 1984, 1985)l. Equation (6.85)is plotted in fig. 6.7 along with approximate expressions for qt,.Although it is encouraging that the effective range theory may be carried over to two dimensions, its use is more involved in view of the logarithmic dependence on k (and a) of the phase shift for k - 0 . In practice, this implies that a vanishingly weak

Oer

O7I

06

,

b(k'

Elkg

(M

O q t

1

0

. . . . ..'.

1

1

01

I

I

02

I

I

03

I

.I-. 0 .L

__.

. u 0.5

Fig. 6.7. The triplet phase shift as a function of the relative energy of two H-atoms for the two-dimensional scattering model.

Ch. 3 , $61


303

potential may lead to large phase shifts and that in two dimensions thermal averaging cannot be avoided even for s-wave scattering in the T-+ 0 K limit.

6.2.2. Spin-exchange relaxation In zero magnetic field spin-exchange is known to be the dominant relaxation channel. It has been extensively studied in relation to experiments with the atomic hydrogen MASER. The classic reference for spin-exchange scattering is the paper by Balling et al. (1964). Indistinguishability effects have been discussed in detail by Pinard and Laloe (1980). Berlinsky and Shizgal (1980) did the first low-temperature calculation of the spin-exchange cross sections and phase shifts. KVS (1981) studied exchange depolarization as a function of magnetic field for a few initial and final state energies. Morrow and Berlinsky (1983) did an improved low-temperature calculation, using the best currently available potential energy curves and also calculated spin-exchange between atoms in the adsorbed state. Statt (1984) analyzed the magnetic field dependence. Following the line of Pinard and Laloe (1980), Bouchaud and Lhuillier (1985) recently reanalyzed spin-exchange in relation to a study of novel spin-wave modes in H i . To our knowledge, in all, except the KVS calculation, spin-exchange was trcated as an elastic process. In this section we aim to review the essentials of spin-exchange, including the field dependence. We shall discuss why the process, while dominant for B = 0, is completely negligible in high fields, unless the gas is brought out of thermal equilibrium with respect to the population of c- and d-hyperfine levels, for instance by absorption of microwave radiation. We also choose to discuss spin-exchange within the golden-rule approach used throughout this section. This implies explicit incoherent summations over initial and final states, which is correct as long as the phase of the wavefunction may be treated as a random property such as in a system in thermal equilibrium. Usually (see for instance Balling et al. 1964). spin-exchange is treated within the density-matrix formalism, which implicitly keeps track of the occupation of all states. This is of course extremely convenient for calculational purposes but has the disadvantage that the contributions of the various spin-exchange channels do not appear in a transparent manner. Another feature of the density matrix is that it enables the user to keep track of the phase of the wavefunction. This is indispensable if one is interested in coherently excited systems such as spin-exchange in a hydrogen maser but is merely ballast for our present purpose where we are interested in T,-like relaxation phenomena in a (quasi-)thermal system.

304


(Ch. 3 , $6

We start with the matrix element in the expression for the transition rate (eq. 6.78). For spin-exchange the two-body T-operator is given by eq. (6.10a) where H = H , + V and H , = K + H , + H,, (analogous to eq. 3.1). The {li)} and {If)} are eigenstates of H,. The interaction V = V, + VeXch enables the transition and describes the interaction between the two H-atoms. The triplet potential is obtained for V acting on a pair state with total electronic spin S = 1, and the singlet for S = O . Using the C+ operator, we rewrite the transition amplitude, (6.86)

(flTli) = lim i,(flVG+li) . 7-0

The G, operator induces the proper distortion of the initial state due to the interaction between the atoms. As in section 6.1, we use the sudden approximation for the spin part of the incoming wavefunction (Wigner spin rule), neglecting the time dependence of ( S M , I M , ) , arising from the off-diagonal elements of H , , (see table 6.1). References to the limitations of thesc approximations which may break down at very low temperatures are given by Pinard and Laloe (1980). To evaluate the T-matrix we project the hyperfine states h , and h2 onto the total spin basis { ( S M , I M , ) } 11) =

=

Ih,h,; k ) = f l [ l h , h , ; k ) + I W , ; -k)l

fi

c

[ISMsIM,;k)(SM.SIM,Ihlh,)

.SM,IMI

+ (SM,IM,; -k)(

SM,IM,Ih,h,)]

.

(6.87)

A similar expression holds for the final pair state If). Here, as is commonly done, we treat the H-atoms as simple compositc bosons. The states Ik) represent plane waves. Pinard and Laloe also analyzed the internal structure of the atoms. The spin-projection amplitudes ( SM,yZM,\h,h,) are summarized in table 6.4a. Substituting eq. (6.87) into (6.86) and taking advantage of the diagonality of V within the total spin representation, we find for the T-m at ri x

Ch. 3. 46)


305

With the substitution

Ik), 3 Ik) + (-1)q-k).

(6.88~)

Expanding eq. (6.88b) into partial waves, one is left (for h , , h, #hi, hi, which we assume from now on) at most with one term. For 1 = even this is the loo00) term; for 1 = odd this is one of the three 1001M,) terms. Since both the singlet and triplet potential are isotropic, the orbital angular momentum is conserved during the collision and the angular dependence may be factored out of the T-matrix. Squaring the T-matrix and integrating over the angles d and k ‘ expressions may be derived, using eq. (6.78d), for cross sections m + and cr- corresponding to even and odd partial waves respectively:

where at most one term in the sum contributes, as may be seen from table 6.4a and (6.89b) The { l + }and {I-} represent even and odd integers, respectively. Further,

j,0(kr) 3 krj,(kr) ,

(6.89~)

3%


[Ch. 3. 86

where F;(kr) and Fl(kr) are radial wavefunctions distorted by singlet and triplet potential, respectively (see eq. 6.80) and j,(kr) is a Bessel function. For thermally activated processes u *( k ) vanishes below the threshold value. Treating the scattering as elastic (k = k’), the cross sections are usually given in terms of the triplet and singlet phase shifts (6.89d) Equation (6.89a) shows very concisely which spin-exchange transitions are allowed within the sudden approximation (Wigner spin rule), at the same time providing the field dependence. For a precise determination of the cross sections, both the singlet and the triplet phase shift should be known to high accuracy. In particular, the singlet phase shift is extremely sensitive for small variations in the potential and likely to he not accurately known. In fig. 6.8 we show the results of Berlinsky and Shizgal (1980). Note the important contribution ( a 5 )of the H z ( 14.5) resonance to the u cross section. In view of the recent experimental observation by Dabrowski (1984) that H,( 14.5) is not quasi-bound, this contribution is likely to be much smaller. In fig. 6.9 we show the thermally averaged spin-exchange cross sections calculated by Berlinsky and Shizgal (1980). Only the s-wave contribution is seen to be important for typical experimental conditions, T < 1 K. These results were obtained with the (1965) results of Kolos and Wolniewicz for the singlet and triplet potentials. With better potentials (Kolos and Wolniewicz 1974, 1975), which are refinements to the 1965 results, the cross sections are reduced by approximately a factor 2 or 3, witnessing the strong dependence of spin-exchange on the detailed shape

40-

20 -

0

2

4

6

E (meV)

8

K!

Fig. 6.8. The spin-exchange cross section D ’ and v as a function o f the relative energy. Only the (T ’ does not vanish for , 5 4 0 . v,is the contribution o f the u = 14. L = 5 level to L T - (after Berlinsky and Shizgal 1980).

Ch. 3, $61


ZF

cE

307

f

2.0

1.o

0

2

4

6

8

10

T(K) Fig. 6.9. The thermally averaged spin exchange cross sections 6’ and 6 -as calculated by Berlinsky and Shizgal (1980). Note that at low temperature s-wave scattering ( 5 ’ ) is dominant.

of the potential (Morrow and Berlinsky 1983, Bouchaud and Lhuillier 1985). From fig. 6.9 one notes that 0’ is of order 1 A’ which is much smaller than the room temperature value ~7’= 23 A’ (Allison 1972) and therefore attractive for those interested in a cryogenic hydrogen maser (Berlinsky and Hardy 1981). Nevertheless, as a relaxation cross section this value is extremely large. Comparing eq. (6.89,) with a diffusion cross section estimated from the diffusion constant of Lhuillier (1983), u,,= 16 A’ at T = 0.5 K, one findsthat oneout of 10-20collisionsleadstospin-exchange.In high field all but the bd $ ac spin-exchange cross sections are suppressed by a factor E’ = (a/(4h)]’( = lop5for B = lOT). We consider two examples in detail; first the high-field, low-temperature limit, where only the a- and b-states are populated in thermal equilibrium. We use eq. (6.89b) and table 6.4a to analyze the field dependence of the various cross sections for aa, ab and bb collisions and the exchange depolarization rate as introduced by KVS. (a) aa-collisions. In aa-collisions only even partial waves contribute as may be seen from table 6.4, where the total spin states corresponding to even partial waves (I + S, even) are labelled with an asterisk. As a consequence all u - cross sections vanish identically


308

Here k,, is the momentum threshold value for the a-c Likewise :

[Ch. 3 , 96

transition.

(b) ab-collisions.

(c) bb-collisions. Here, the exchange of the electronic spins does not affect the hyperfine states of the individual atoms which trivially excludes any spin relaxation. This example shows that the cross section decreases as 1 / B 2 with growing magnetic field. In fact the rate decreases even faster. Only atoms from the high energy tail of the momentum distribution carry with them enough energy to enable the transition. As this fraction of atoms drops off exponentially with magnetic field, spin-exchange depolarization is entirely negligible in the high-field/low-temperature limit. Note further that spin-exchange does not lead to b-a relaxation in H i t . These features cause nuclear magnetic b + a relaxation to be dominant over spinexchange in high field (see section 6.2.3). As a second example we treat bd-, ac spin-exchange. From table 6.4a it may be seen that this is the only spin-exchange channel which is not suppressed in high magnetic fields: ( ~ t ~ - ~=~81(ac(0000))’ ( k ) [(00001bd)’a’ (k) =

f(+

& 1 )-a ’-+

A compilation of all spin-exchange cross sections derived using eq. (6.89a)

is given in table 6.13.


Ch. 3, 861

309

Table 6.13 Compilation of all spin-exchange cross sections 6 , , h 2 h i h i . The Ahi columns contain the number of h,-atoms lost (-) or gained (+) per event. The weight reflects the relative contribution to the sum over all initial and final states (see eq. 6.78b). Note that all but the ac+bd spin-exchange contributions vanish as E’ B in high field. Moreover, many of the terms are strongly suppressed in high field due to the inelastic nature of the process.

- ’

h,h,+h;h:,

Aa

Ab

Ac

Ad

Weight

aa+cc cc-aa aa-ac ac+ aa aa-, bd bd+ aa ac+ bd bd+ ac ac+ bd bd +ac ac-cc cc-, ac bd +cc cc+ bd ab- bc bc+ ab ad+ cd cd+ad

-2 +2 -1

0 0 0 0 +I -1

+2 -2 +I -1 0 0 -1 +I -1 +1 +I -1 +2 -2 +I -1 +1 -1

0 0 0 0 +I

it2

+1

-2 +2 -1

+1 -1

+1

+I

-1 +I

-1 0 0 -1 +I 0 0 0 0

-1

+1 0 0 -1

+1 -1

+I

(Ih1h2-hihi

(T*(aa+ ~ ) 8 & ~ q ‘ it’ (J+(CC--,~C)~E‘V‘ 46’ a+(aa+ca)t&*q2(q2- E2)* 482 c~+(ac+aa)2~’if(q’ E ’ ) ~ Ji’ 0+(aa+ bd)2&’qZ 4 6d 6* (bd +aa)2&’q2 it c+(ac+bci)i(q2 - E ’ ) ~ 62 a+(ixi-?ac)i($ - E ~ ) ’ it (T (ac-+,bd)t a-(bd+ac)t 466 ‘Tt(aC--*CC)2EZ~’(~’ - E’)’ ? t z 5 ’ ( c i + a c ) 2 ~ ’ q ~ ( q-~E ~ 468 a ‘(bd-r C c ) 2 E 2 $ 4 t 2 a * (cc+ bd)2E2q2 66 ~~(ab-+bc)2~’q’ 6t Cr- (bc-+ab)2E2q’ id 0 (ad+cd)2&’q2 td &(cd+ ad)2~’q’

-

-1

+1 -1

+I -1

0 0 -1 +1

0 0 0

0

)

6.2.2.1. Relation to the rate equations With the overall rate of spin-exchange events given by eq. (6.78) and the cross sections given in table 6.13, it is straightforward to derive the rate equations for spin-exchange. The result is given somewhat schematically intable 6.14, for zero field and T % 68 mK.To be specific we consider a c

*

Table 6.14 Summary of all zero-field contributions to the spin-exchange transition rate for T % 68 mK where inelastic effects are negligible. The h, columns contain the number of h,-atoms lost (-) or gained ( + ) per event. This number appears as a prefactor to the rate constant G,(h,h,-h;h;) in the rate equations. The weight reflects the density dependence of the various terms. h,h,t*h;h; aa-cc aa c,bd ac-bd bd-cc abwbc ad c)cd

u

-2 -2 -1 0 -1 -1

c

d

0

+2

+1

0 -1 +2 +1

0 +I +1

b

+I -1 0 0

+1

-1

0 0

GL(hlh2 h;hl)

Qvo’

:ucr‘ 4 60 I&?+ f 60 jvo

Weight ( a z - c’) (a‘ - bd) (nc - bd)

(bd - c’) (06 - bc) (ad - c d )

~

I.F. SILVERA A N D J . T . M . WALRAVEN

310

1Ch. 3. 86

relaxation in detail. For this we sum the d and c columns with the appropriate weights, accounting explicitly for the number of a- and c-state atoms gained or lost per event. Subtracting the results we find d

? - - +

- ( u - c) = --n-L'u

dt

(fi

+ c:)(n^

-

c*)

-- n'urr

(6 t a)(; - 6 ) (6.93a)

a

For t h e special ease that a^ = d = c^ = = f , this leads to the well-known result (see for instance Berlinsky and Shizgal 1980) that 1 =

T,

(6.92b)

IVn(U+ -e 0 - ) . -

Another interesting example is the high-field limit for T 3 68 mK. From table 6.13, one finds only the a c s bd spin-exchange channel is left. Hence. guided by eq. (6.78) one find5 d = - hG(6.

b=+

+ n - )(ac -- b d ) ,

;v(a' + 6 - )(ac - b d ) , (6.93a)

r = - $ ( 6 ++ G - ) ( a c - h d ) , ci =

+ f L ; ( c r * t 0 -)(ac - b d )

Comparing with the notation of section 5 (eq. 5.15) one finds for the rate constant for spin-exchange Gt-= { U ( 6 * + G ) .

(6.93b)

In particular d dr

- ( U C - h d ) = i i +~ UC - bd =

-+ - !fi([T

--

hd

+ G - ) ( a c- h d ) .

(6.03C)

Hence, in high field, spin-exchange drives the system to a state in which ( u l b ) = ( d / c ) .The relaxation time T , is also as given by eq. (6.92b). This effect plays an important role in ESR experiments in high field where it may prevent the experimental modification of t h e a l b ratio by selectively pumping the a 3 d or b+ c ESR transitions.

Ch. 3. 661

SPIN-POLAKIZED ATOMIC HYDKOGEN

31 1

6.2.3. Dipolar relaxation In high magnetic fields only the weak interatomic dipolar forces are effective in establishing thermal equilibrium between the various hyperfine levels. We distinguish nuclear-spin relaxation between a and b levels or c and d levels, and electron-spin relaxation, between the lower and upper pair of hyperfine levels. For convenience these names are used over the full range of magnetic fields although the simple distinction is only correct in a strict sense when hyperfine admixtures are absent or negligible. Apart from these intrinsic processes, dipolar interactions with impurity spins on the substrate surface may also lead to relaxation. The nuclear relaxation process was first studied by Statt and Berlinsky (1980), predicting the double-polarized state ( H J f ) , and also by Siggia and Ruckenstein (1981) in relation to collective phenomena in the Bose condensed gas. Factor-of-two inaccuracies were resolved in a careful study by Ahn et al. (1982,1983) who also accounted for the inelastic (k # k ’ ) nature of the relaxation process. Lagendijk et al. (1984) showed the process to be particularly suited to look for exchange effects of complete hydrogen atoms and used this to demonstrate the Bose nature of the H-atom. The surface process was first analyzed by Lagendijk (1982) in a two-dimensional picture, predicting an anisotropy depending on the direction of the magnetic field with respect to the surface normal. Improved calculations were done by Ruckenstein and Siggia (1982), Statt (1982) and Ahn et al. (1982), all within the 2:-dimensional model. The last authors also discuss a reduction of the anisotropy due to surface roughness of the helium film substrate. A full three-dimensional calculation of the surface process was done by van den Eijnde et al. (1983). The electronic relaxation process was studied by KVS (1981) to estimate the dipolar depolarization rate in high field. The process was also considered by Lagendijk et al. (1984). In the coming sections we shall discuss why the dipolar interaction although much weaker than the direct and exchange forces, plays an important role in spin relaxation. First, we present general features of the dipolar interaction, showing how it may be decomposed into terms transforming like the spherical harmonics Y y ( k ) . Subsequently, we study the transition rate from the double-polarized state HJf. Comparing with surface Van der Waals recombination, we discuss the result of Statt and Berlinsky (1980) that at low temperatures the nuclear dipolar relaxation process is limiting the overall decay of the sample. Then we turn to nuclear relaxation on a helium surface. The section is concluded with a discussion of the electronic relaxation process, showing that in high magnetic fields it dominates over spin-exchange.


312

(Ch. 3 , I6

6.2.3. I . Dipolar relaxntion-general In this section we discuss both the nuclear and electronic depolarization of pure H i t , i.e., a gas of only b-state atoms. To calculate these transition rates we pick up the treatment of section 6.2.1 at eq. (6.78). The T-operator is given by eq. (6.10a) where H = H,, + V,, and H,, = K + H , + H,, V , + Vexch(analogous to eq. 3.1). The {li)} and {If)} are eigenstates of H,,. which are written as a linear combination of waves distorted by the singlet and the triplet interactions. Here the collision is treated within the sudden approximation with regard to the hyperfine interaction ( Wigner spin rule). This approximation was checked against a close-coupling treatment by Ahn et al. (1983). The dipolar interaction between the atoms is given by Vdd.

+

where the operator expressions f(i,, iz) (eq. 3.Yb) are defined in terms of second-rank tensor operators by -2

C

f ( s , . i2) = I,,

T;Y;'(~),

-- - ?

Here the s , and S _ are raising and lowering operators for the spin angular momentum. s, is the component of s along the quantization axis. Similar definitions hold for i * , i . and i,. The matrix elements of f(sl, s z ) and - { f ( s , . i 2 )+ f ( i , . s:)} in the total spin representation. { \ S M . s I M , ) }are , given in tables 6.8 and 6.9, respectively. The matrix elements of f ( i , , i 2 ) arc identical to those of f(s,. s z ) . The dipolar interaction is very weak in comparison to the direct and exchange interactions. At the zero-crossing of the triplet potential ( r = 3.66 A ) the electron-electron contribution represents only 51 mK, which is more than two orders of magnitude weaker than the triplet interaction around its minimum. Hence, the motion of the particles is

Ch. 3, 161


313

negligibly distorted by the dipolar interaction, enabling an accurate perturbative calculation within the distorted-wave Born approximation (DWBA). The electron-proton term is even weaker by a factor yJy, = 658. The proton-proton term is negligible for all practical purposes. The range of the dipolar interaction is large in comparison to the range of the triplet potential, so that even a simple plane-wave Born approximation (PWBA) yields accurate results. This feature makes dipolar relaxation, in contrast to spin-exchange, into a property which is very insensitive for the exact shape of the potential and as such ideal for studying particleexchange effects in H J f , as shown by Lagendijk et al. (1984). In H& the initial-state pair wavefunction involves a bb-pair and is distorted purely by the triplet potential. In principle, the final state may contain both triplet and singlet character. Within the DWBA the transition amplitude is given by (flTli) = (flVddli)= +(fl[l + P]v,,[I

+~]li)

= V2(flVddli) .

(6.95a)

Here we have treated the H atoms as simple composite bosons and used the invariance of V,, under permutation of total atoms. Note that only one side of the T-matrix needs to be symmetrized. A complete treatment, also addressing the internal structure of the atoms, was given by Ahn et a]. (1983). To evaluate the T-matrix within the sudden approximation, we reexpress the initial state in the total spin basis { l S M s Z M I ) } :

li) = Ih,h,; k) = fi[lh,h,; k) + Ih,h,; -k)l

(6.95b) Here we have used the relation (SM,IM,Ih,h,) = (-l)'fS(SM,ZM,(h,h,) and the shorthand notation of eq. (6.88~)(see table 6.4). States with S = 1 (S = 0 ) are implicitly assumed to be distorted by the triplet (singlet) potential. Analogously, for the final state we have If) = b,h,; k ) =

c

SM>IM,

(SM.dM,; k)(SM,lM,(h,h,) .

(6.9%)


(Ch. 3. 56

Table 6.15 Summary of the transition matrix elements for electronic and nuclear relaxation starting from bb. a b and ac initial pair states. Only the terms contributing to highest order are included. The symbols c (single spin-Rip) and d (double spin-flip) are defined in table 6.8. Scattering is assumed to proceed exclusively via s-waves in the initial state. All terms have a common factor. ( p,,h2/47r)( 1 / r ) 3 ( 6 ~ / 5 ) " ' . ac

bh

Substituting eqs. (6.95b) and ( 6 . 9 5 ~ into ) (6.95a) we find for the 7-matrix (flV"(i) =

c

(h;hi[S'M,.t'M,.)(SM,tM,lh,h:)

5V51M, 5 M\ I ' U ,

x ( S ' M , t ' M l t ;k'~Vdd~SM,IMI; k),,, . (6.95d) In H J , the only available initial pair states are of the type Ibb; k ) , lab; k ) and (aa: k). Equation (6.95) and eq. (6.94). in combination with table 6.8 and table 6.9. enable us to calculate t h e contributions of all final states allowed by V,d. The results are listed in table 6.15. From table 6. IS, one finds that the dominant nuclear relaxation process is the bb-. ah channel. For electronic relaxation, two spin matrix elements are of the same order. These correspond to bb-. bc and bb-, cc channels. The latter may be excluded as it is a double (electron) spin-flip process. These are negligible in comparison to single spin-flip processes due to the lack of high momentum states in the translational bath which are required for energy conservation in the transition. 6.13.3.2. Nuclear spin relaxation in the bulk gas For the bb- ah relaxation channel, the transition matrix (eq. 6.95d) reduces to

Ch. 3 . 661

(f{V,,li)


315

= (1-~oh’/4‘rr)y,y,(3r/lO)’’~(l + &ye/y,)

x [ ( k ’ / y ; * ( i ) / r 3 /+k () k ’ / y ; * ( i ) / r 3-Jk)] .

(6.96)

The ab+aa channel leads to identically the same result. The factor (1 ~ y , / y , ) = (1 + 16.68/B), with B in Tesla, arises since both the yeyn and the 7 %terms of V,, contribute to the rate. The former represents the process where the nuclear spin is flipped due to the passage of another atom carrying a Bohr magneton. The latter term corresponds to an electronic spin-flip, allowed energetically due to the presence of the spin-up admixture in the a-state, vanishing in high field as 1 / B with the &-admixtureof spin-up. For typical magnetic fields used in the experiments E 2 : yn/y, and the two processes contribute by approximately the same amount. Using eq. (6.78d) to relate the T-matrix to the scattering cross section, expanding Ik) and (k’)into partial waves (see eq. 6.80a) and integrating over all angles k and k‘ we find, after a tedious calculation,

+

a,,,,,(k)

=

Q,,l/(k’k’)[l

+ (16.68/8)]’

(6.97a) where Q,, = ( 2 4 ~ / 5 ) [p0p/4‘rr)y,yn]2 ( = 2.34 x

I

T /.,, ( k , k ’ ) = d r Fi.(k’,r)( 1 / r 3 ) F ) ( k I, ) .

cm2 ,

(6.97b) (6.97~)

In the relaxation process, angular momentum is transferred from the spin system to the orbital system, with the component of the angular momentum along the quantization axis being conserved. At low temperature, the 1, m I= 0, O-+ I, m I= 2, + 1 transition is dominant. In fig. 6.10 the cross sections for the bb-ab and ab+bb are compared. The difference between both cross sections is due to the difference in the overlap integral (eq. 6.97~).At low relative energies, the elastic-scattering approximation breaks down. Note the cut-off at E / k = 5 O m K for the ab+bb cross section. Below this threshold there is not sufficient energy in the translational motion to flip the nuclear spin. One also observes that the PWBA yields a very accurate result. From fig. 6.10 one notes that the cross section


316

----

0L

03

l

I

0.2

I

1

03

I

ICh. 3. 96

DWBA PWBA

I

OL

I

I

05

I

L 0.6

Fig. 6.10. Spin-relaxation cross sections, chh . r h and u,,, .hh as functions of energy. Also shown is the elastic scattering approximation (HTL). Each function has been calculated in J3WB.A (drawn curves) and in PWBA (dashed curves). The plot is for R = 8 T (after van den Eijnde 1984).

for b-+ a dipolar relaxation is of order lo-' A' for T 9 50 mK. The reason that this very weak process is the dominant decay mechanism in high magnetic fields is that hb-t ab or ab-, aa spin-exchange channels do not exist (hence absent from table 6.13) and all allowed spin-exchange channels for ab initial states are strongly suppressed as they scale with 1l B 2 and involve transitions to the c- and d-states. The b-+a transition rate T(b-+a) is found using eq. (6.78)

= (Tah+aa. Rate Here we have used t h e result (see eq. 6.96) that equations of the kind used in section 5 are obtained if we identify

- (Jaa-ah. Treating the system as where we have used the relation an effective two-level system, considering only relaxation between the a and b levels and keeping track of the gain or loss of atoms per event, the rate equations become

(6.98~)

Ch. 3. $61

SPIN-POLAKIZED ATOMIC HYDROGEN

317

The approximate expressions hold for temperatures much higher than the level splitting ( T S 50 mK). The relationship between the relaxation rate constants Ciaand G:,, and the relaxation time T, is found by adding u and b in eq. (6.89~)and comparing with d dt

- (b

-a)= -

r,1 ( b - a )

(6.98d)

One finds 1

- = (Cia+ G : , ) n = 2G:,n .

(6.98e)

TI

An important general relation between G:a and G:, is derived by assuming the system to be in thermal equilibrium ( u = b = 0) (G:,,/Gi,)

= b / a = exp[ - ( E , -

E,)/kT].

(6.98f)

To first order the b + a relaxation rate will become independent of temperature for T+ 0 K, being dominated by s-wave scattering. Equation (6.89f) implies that G;, vanishes exponentially at low temperature (T450mK). Both theoretical and experimental values for G:a are given in fig. 5.15. Note the weak temperature dependence of the rate constant. This figure, taken from Lagendijk et al. (1984), also shows the dramatic importance of symmetrization. Symmetrizing H atoms as composite fermions, i.e., summing over odd 1 and I’ in eq. (6.97a), leads to a rate constant which deviates way beyond experimental error from that with proper Bose symmetrization. As such, this result provides a clear demonstration of the Bose character of H J t . An accurate expression for G:, is given by van den Eijnde (1984) Gia= (6.329T”’

+ 7.572A,,T-1’2)(1 + 16.68/B)2 x

l a A,, = - - [l + 2 y n h B / a ] (for B 2 1 T) . 2 k

cm3 s - ’ (6.99a) (6.99b)

6.2.3.3. Nuclear spin relaxation on the surface For bb-, ab relaxation on the surface, eq. (6.96) remains valid; however, we must interpret k as a two-dimensional vector in the plane of the surface. To obtain the cross length, Ik) and Ik’)are expanded in two-dimensional

l . F SILVERA A N D J.T.M. WAI.RAVEN

31s

(Ch. 3 , $6

partial waves. Before we present an expression for the cross length, we give some general considerations concerning the two-dimensional relaxation process. One o f t h e most fascinating features of surface dipolar relaxation is the anisotropy o f the rate with respect to the angle between magnetic field B and surface normal n. It was first pointed out by Lagendijk (1982) that the surface rate can be made to vanish by orienting B parallel to h. This effect may be seen from cq. (6.96). If B / / h the direction i o f the interatomic vector is perpcndicular to h , i.e. in polar coordinates 0 = f r ,which is a condition for which Y i * ( t )vanishes. Another interesting point is that the selection rules are affected by the presence o f the surface. As discussed in section 6.1.2.2. only the component of the angular momentum along the surface normal is conscrved. This implies that spin transitions can occur without a change in motional angular momentum. Evaluating eq. (6.7Yd). one finds for the cross length, within the pure two-dimensional model.

A,,;,),.h,,,:

=

( 1 5 / 3 3 r ) Q , , [ ll(k'k')][ 1

+ (16.68/B)]'

where tl is the angle between B and h , and (6.100b)

t COS'

e).

where in the elastic approximation G,,, and G,,' are given by G < , [T, () = (0.96 -- 0.827' + 0.717")[1

+ (16.68/8)]' x

G \..,( T ) = (0.038 + 0.278T)[1 + (16.68/B)]' x 10

~

''

10- I' m'' s

I

,

m's-' .

The expressions arc from the paper by Ahn ct al. (1982). These authors also give an angular average accounting for surface roughness: ( G , (T ) ) = (0.69 - 0.45T + O . 5 2 7 ' ) [ 1 + ( 16.68/R)I2 x lo-'' m2 s

I

.

Ch. 3, $61

Here zero.


8, the

319

angle between the field and macroscopic surface normal, is

6.2.3.4. Electronic spin relaxation As far as the theory is concerned, there is little difference between nuclear and electronic dipolar relaxation in HJf. The main aspect to point out is that under typical experimental conditions in a high field we are dealing with the extreme inelastic limit. For the bb+ bc relaxation channel, the transition matrix, eq. (6.95d) reduces to

(f1Vdd(i)= ( ~~2/4.rr)yf(3.rr/10)1’2(kf(Y:*(i)/r3(k),ven . (6.101) The cross section becomes, analogous to eq. (6.97a), ubb-bc(k)

= Qee

1 T[,[,(k, kf)J2(21+ 1)(21’ + 1) k’k3 / . / ‘2 =even (6.102a)

where T[+/,(k,k’) is defined as in eq. (6.97) and Qee

= (24.rr/5)[(~,,,~/4.rr)yf]’ = 1.02 x lo-’’ cm’ .

(6.102b)

One may analyze b-, c relaxation as a two-level system, analogously to eq. (6.98), while defining Gic= $ 6 ~ and 7 Glb ~ =~fUacb-b,,. ~ ~ The ~ strong inelastic nature, in which only the atoms in the fast tail of the Boltzmann distribution contribute to the b-, c relaxation, gives rise to an exponential B l T dependence as found both by KVS (1981) and Lagendijk et al. (1984)

Gr, = e x p [ 2 ~ ~ B / k T ] G ;,,

(6.103)

Comparing with the theoretical result GI,, = 9.7 x 10-” em's-' of Lagendijk et al. (1984), for B - 7 T and T-0.71 K , we calculate an effective cross section for the c+ b relaxation of 1.1 X 10 - 3 A’, which is three orders of magnitude smaller than the zero-field spin-exchange cross sections (-1 A’, see section 6.2.2). This in turn is at least two orders of magnitude larger than all but the ac+ bd spin-exchange cross sections in high field which are suppressed by at least a factor E* (see table 6.13; E* -- 10.’ for B = 10 T). KVS (1981) were the first to compare the exchange and dipolar relaxation mechanisms between the lower and upper pairs of hyperfine levels as a function of magnetic field. In fig. 6.11 we have redrawn their results using the notation G r = GJ exp(-2pBB/kT)] analogous to eq. (6.103). We have no qualitative explanation for the origin of the minimum in the exchange contribution around 10 T.

3 20


[Ch. 3 . 87

r

spin exchange l\spin

dipolar

10'

10 10 B (Tesla)

lo2

Fie. 6.11. 'Theoretical results of Kagan et at. (1981). comparing exchange and dipolar depolarization mechanisms. See text for further details.

7. Thermodynamic properties

In the beginning of the 1970s, theoretical evidence of the gaseous nature of H J at T = 0 K became available. In this section we review the theoretical activities in which the nature of H J , DJ and TJ was established. This involves the calculation of a restricted set of thermodynamic properties. To fully characterize the gas, many thermodynamic properties are of interest. We mention specific heat, compressibilities, pressure, etc., all as a function of temperature, in particular near the critical density for BEC. These properties are conveniently compiled in the recent review by Greytak and Kleppner (1984) to which the reader is referred. 7.1.

QUANTUM THEORY OF CORRESPONDINGSTATES

An elegant unifying description of the thermodynamic properties of H i , DJ and TJ can be given within the framework of the quantum theory of corresponding states, originally proposed by de Boer (1948) and extended by Nosanow and co-workers to enable a comparison of the various quantum fluids (Nosanow et al. 1975, Miller et al. 1975, 1977, Nosanow 1977b). The theory has been reviewed by de Boer and Bird (1954), and its application to macroscopic quantum systems has been reviewed by Nosanow (1980). Thc first application of the theory to the spin-polarized hydrogens is due to Hecht (1959), who calculated critical temperatures. He obtained negative values for both HJ and DJ implying the absence of a liquid state,

Ch. 3, $71


32 1

whereas for TJ. he obtained a value T, = 0.95 K. In 1977 Miller and Nosanow applied the extended version of the corresponding states theory to the hydrogens, using new values for the potential parameters. According to their analysis, DJ. could have a liquid state at T = 0 K, possibly under applied pressure. However, H i would remain gaseous at all temperatures, including T = 0 K, up to pressures of 54 atm (at T = 0 K) where it would go directly into the solid state. TJ would have a liquid ground state. Recently, more accurate calculations by Panoff et al. (1982), to be discussed in the next section, provided theoretical evidence for the existence of a liquid ground state for the deuterium system. The quantum theory of corresponding states (QTCS) applies to each class of systems, where the potential energy u may be characterized by two parameters, one which sets the energy scale ( E ) and the other the length scale ( a ) .Nosanow and co-workers assumed the potential to be pairwise additive and of Lennard-Jones form, yielding a Hamiltonian given by

where

Here E is the well-depth and u is the hard-core diameter, r, represents the position of atom i and rij = Ir, - rjl. V; is the Laplacian with respect to r , . Writing r: = r , / u , eq. (7.1) may be reduced to

where V: is the Laplacian with respect to the reduced coordinates r r and

is a measure of the relative importance of the kinetic energy term in the dimensionless Hamiltonian H* and A * is the parameter introduced by de Boer. In table 7.1 we reproduce values for m , E , a, q, A* and other useful quantities for various systems of interest (Miller and Nosanow 1977). For the spin-polarized hydrogens they are based on accurate variational calculations of the ‘2 potential (Kolos and Wolniewicz 1974; see section 6.1.3).

I.F. SILVERA A N D J.T.M. WALKAVEN

322

[Ch. 3. 87

Table 7.1 Quantum parameters q for various substances. Also given are the masses ,4*, coupling constants F . "core diameters" u. e I d , and N , v ' (from Miller and Nosanow 1977). m

Elkh

Substance

(amu)'

(K)

Hi

'He H: D, Ne

1.008 2.014 3.016 3.016 4.m3 2.016 4.028 20.18

6.46 6.46 10.22 6.46 10.22 37.0 37.0 3.5.6

Ar

39.95

D1 'He

T1

1m.o

"1 amu = 1.66024 x 10. IT kg. bk = 1.38054 x 1W'' J K I.

e I(,'

(i) (atm) 3.69 3.69 2.556 3.69 2.556 2.92 2.92 2.74 3.41

17.5 17.5 83.39 17.5 83.39

202.5 202.5 235.8 412.3

N(,d (cm3/mol)'

9''

30.2 30.2 10.06 30.2 in.06

0.547 0.274 0.2409

4.65

n.ws

0.58

15.0 15.0

12.4 23.9

A'

3.29 3.08 0.183 2.69 0 . 1 ~ 1 5 2.68 0.0763 1.74 0.0382 1.23 0.00088 0.19

' N t ,= 6.02252 x 10" particles/mol. ' h = 1.05430 x 10 - " J s.

According to the QTCS. the free energy F of a one-component system may be reduced to a form ( F * = FINE; N is the total number of particles in the system) which only depends on the reduced variables for the temperature, T * = kTIE, n * = (r3NIV= 1 / V * as well as on q and the statistics that apply to the system:

F* = F * ( T * ,V * , 71).

(7.5)

It is also useful to define the reduced pressure p * = p a 3 / c . Nosanow and co-workers extended the de Boer theory by interpreting q as an additional thermodynamic variable (besides p * and T * ) . This enabled them to construct phase diagrams in a space spanned by p * , T * and 77 (Nosanow 1977b). To express a change in the free energy F * , we need to introduce a variable (4 * ) which is the thermodynamic conjugate of q (Nosanow et al. 1975) dF*= -S*dT*-p*dV*+4*dq,

(7.7) and the reduced entropy is S * = SINk. Using the powerful framework of statistical mechanics and thermodynamics, the phase diagram in p*-T*-q-space can be mapped out. Clearly, 71 only has physical significance for certain discrete values

Ch. 3, 871


323

(corresponding to real physical systems), but the concept of using r] as a continuous thermodynamic variable is useful in obtaining insight into the various phases which are accessible to macroscopic quantum systems. Using Lennard-Jones potentials, the ground-state properties of manybody Bose and Fermi systems as a function of r] have been calculated by Nosanow and co-workers and published in a series of papers dealing with the liquid-solid phase transition (Nosanow et a]. 1975), liquid-gas phase transition (Miller et al. 1975, 1977) and generalized phase diagrams and critical behavior (Nosanow 1977a,b). Two-dimensional systems were studied by Miller and Nosanow (1978). Some features of the ground state are immediately clear from eq. (7.3). For small r] the system behaves “classically”, i.e., the potential energy is dominant and the atoms are located at the equilibrium sites of a crystal lattice. With growing r] the kinetic energy becomes increasingly important and beyond a critical value the system melts. If we continue to increase r] we reach the point where the system no longer can sustain a many-body bound state and it becomes a quantum gas. However, other features such as the dependence of the results on the statistics of the system are more subtle. For the Fermi-case, due to the Pauli principle the ground-state energy depends markedly on the nuclear spin degeneracy. For DJ this implies a dependence on the occupancy of the various hyperfine states. To distinguish the various cases Miller and Nosanow (1977) introduced the where the index v refers to the ground-state degeneracy or notation DJY, number of allowed nuclear spin states. In the limit v + m , keeping n constant, the difference between Fermi and Bose behavior vanishes (Bose limit). Miller et al. (1975,1977) calculated the location of the critical point as a function of r ] , using a variational method to determine the ground-state energy (for more detail on the method see the next section) and found that beyond r], = 0.33 (for v = 2 fermions) and qc = 0.46 (for bosons) (see fig. 7.1) the systems are gases at T * = 0. The v = 1 fermion case was studied by Miller and Nosanow (1977) who found qC= 0.35. Comparing the values of r] for H i , DJ. and TJ (table 7.1) with the relevant values for r],, we note that at all temperatures HJ. should behave as a fluid above its critical point, i.e., it remains a gas down to T = 0 K and up to the solidification pressure. TJ. is expected to have a liquid ground state. The results for DJ are not unambiguous. DJ. could be a liquid or exhibit two coexisting fluid phases. The uncertainty is due to the approximate nature of the theory. Recent Monte Carlo calculations predict D.1 to be a liquid at T = 0 K (Panoff et al. 1982). Such calculations are not available at present for D.1, and DJ 2 . The remarkable phenomenon of two coexisting fluid phases in a one-component system at T = 0 K is illustrated in fig. 7.2, reproduced from Miller et al. (1977). Focusing on the curve for r] = 0.31, one notes that at

REDUCED DENSITY o.

REDUCED DENSITY p.

Fig. 7.1. Reduced energy E' (a) and the corresponding reduced pressurep* (b) as a function of the reduced density p* (designated n * in the text) for systems obeying Bose-Einstein statistics. The arrows locate the positions of the energy minimum for each density (from Miller et al. 1977). 324

w q = 0.27

0.05

I

0.0

I

1

0.05

0.10

I

I

1

0.15

0.20

0.25

I 0.30

d

I

REDUCED DENSITY p*

1

I

I

I

I

1

B

i

-

-

0.0

0.05

0.10

0.15

0.20

0.25

I 0.30

0.35

REDUCED DENSITY D*

Fig. 7.2. Reduced energy E* (a) and the corresponding reduced pressure p' (b) as a function of the reduced density p' (designated 'n in the text) for a system obeying Fermi-Dirac statistics. It is evident that at low densities the statistical repulsion between fermions dominates the total energy (after Miller et al. 1977). 325

326


[Ch. 3. 97

low densities the energy of the system is just proportional to the Fermi energy

where v is the spin degeneracy. At higher densities the curve starts to bend over as interactions become important and shows a minimum at n* = 0.15. Beyond a critical density it becomes energetically favorable for the system to phase separate into a low- and a high-density fluid phase. In both phases the pressure and chemical potential are required to be equal. Clearly the phenomenon of coexisting fluid phases does not exist in a Bose system, where both statistics and interactions tend to bring the particles closer together (see fig. 7.1). Although the QTCS is not decisive as far as the nature of the D.1 ground state is concerned, it leaves little doubt that D.1 may be pressurized into a liquid state. With the aid of fig. 7.3, one finds for the critical point T c = 1 . 2 9 K for DJ,, T c = 1 . 6 8 K for D J , , and T c = 2 . 5 K for TJ. In a recent paper, Hecht (1981) has redone his original calculation (Hecht 1959) of the critical points, based on more recent values for the LennardJones potential parameters and an extrapolation scheme that conserves the

QUANTUM PARAMETER 7

Fig. 7.3. Reduced temperature T' versus quantum parameter 17 based o n a Lennard-Jones potential. The experimentally available points are indicated by the open circles. The dark circles represent the T = 0 K theoretical estimate of qc for the gas-liquid phase transition. The estimated values for D.1 , , D1 and T1 as obtained by interpolation between experimental and theoretical points are indicated by the arrows.

Ch. 3, 87)


327

critical ratio pZVZITZ = 0.3 as suggested by theory (Nilsen and Hemmer 1969). The results are T, = 1.56K for DL, and T, =3.28K for TJ.. Statistical corrections were neglected, with reference to a paper by Lieb (1967). To conclude this section on the corresponding-states theory we present estimates of the solidification pressures ( p 5 ) .These were obtained using the procedure of Nosanow (1980), based on results of Nosanow et al. (1975) and Nosanow (1977) for the liquid-solid phase transition. In view of the rather speculative nature of these figures we only give the results: p, = 54 atm for H i , p , = 12 atrn for D.1, and p, = 5 atm for TJ..

7.2. GROUND-STATE CALCULATIONS After the more general discussion of the spin-polarized hydrogens within the framework of the QTCS, in this section we discuss a number of calculations of the ground-state energy for H i and its isotopes. These calculations were intended to yield the “best available” estimate for the nature of the ground state. The first calculations to provide convincing evidence of the gaseous nature of HL down to T = 0 K were by Etters and co-workers (Etters 1973, Dugan and Etters 1973, Etters et al. 1975, Danilowicz et al. 1976), who applied the Monte Carlo technique to obtain the energy, pressure and compressibility of the ground state of HJ, D.1, and TJ. for a variety of densities, ranging between 40 and 200cm3/mol. In 1978, this work was extended to lower densities of order 2 x lo4 cm3/mol (Etters et al. 1978). Also the solid, and solid-fluid phase transition were studied by these authors (Danilowin et al. 1976), but will not be discussed in this review. Accurate approximate results for the ground-state properties were also obtained by Miller and Nosanow (1977), who applied two different cluster-expansion methods. Apart from improving upon the accuracy of existing theory, this work showed that for D.1 the calculations were not sufficiently accurate to decide upon the nature of the ground state at T = 0 K. Relatively clear-cut results were obtained for H i (gaseous) and T i (liquid). The required high accuracies for DJ. challenged a number of theorists to apply the most advanced methods of Fermi-fluid theory to decide upon the nature of the DJ. ground state. Clark et al. (1980) and Krotscheck et al. (1981) used the method of correlated basis functions and a Fermi-HNC approximation to calculate the ground-state energy, but this work also lacked the required accuracy. The first results that point convincingly towards the existence of a liquid ground state at zero pressure came from a variational Fermi-Monte Carlo calculation for D.1 3 . Within statistical


328

[Ch. 3. 67

error a negative upper bound was obtained for the energy for a range o f densities (Panoff et al. 1982). It is not our aim to give a detailed account of the various calculational techniques employed. For this purpose the reader is referred to reviews by Feenberg (1969), Zabolitzky (15)77), Clark (1979) and Clark et al. (1980). In discussing the various results we shall reference the original literature where useful. The Hamiltonian of a N-body system of atoms interacting pairwise via the triplct potential V, is given by

H =

-

fi?

7 -ni

2, Of + 2V , ( r l , ) , 1'

(7.9)

I

where r, represents the position of atom i and r,, = ( r ,- r , ( .To obtain the ground-state energy per particle one has to calculate the expectation value of the Hamiltonian with respect to the ground state 14)

(7.10) 1.2.1. The hoson case

Both Ettcrs et al. (1975) and Miller and Nosanow (1977) chose a variational wave function of the Jastrow type to approximate the ground state t,hH of the Bosc systems H J and TJ (7.11)

This ground state is built up as a product of N ( N - I ) functions f(r,]) describing the correlations o f the pair (i. j ) . Then the ground-state energy per particle for the boson case, ( E )B, follows from repetitive use of eq. (7.11). the relation

G,E'= F

2 O,In j ( r , , ) ,

(7.12)

I T 1

and integration by parts (scc McMillan 1965) (7.13) where d~ is used to indicate integration over the spatial coordinates of all N particles. Introducing the (Bose) pair-correlation function (7.14)

Ch. 3, 971


329

one obtains (7.15) We first discuss the approach of Etters et al. (1975), who used a Monte Carlo method on a collection of 32 particles that had proven to be very successful in calculating the properties of the Bose fluid 4He (McMillan 1965). The density of this model system was varied by changing the volume of a cubic box; periodic boundary conditions were used to simulate an infinite system. Originally (Dugan and Etters 1973, Etters et al. 1975), a Morse potential fitted to the '2; potential energy curve of Kolos and Wolniewicz (1965) (see section 3) was used V,(r) = E[exp 2 4 1 - r / r , ) - 2 exp c(1- r / r , ) ] ,

(7.16)

where r = r,,, Elk = 6.19 K, r m = 4.1527 8, is the position of the potential minimum, and c = 6.0458 is a dimensionless constant. Equation (7.16) provides a good fit for both the well-region and the short-range part. However, it was pointed out by Stwalley and Nosanow (1976) that for long-range, the Morse potential is incorrect and therefore is not suited for an analysis of the low-density properties. In a later paper, Etters et al. (1978) used the analytic form (3.6), which represents a much better fit to the '2; potential of Kolos and Wolniewicz (1974). Etters et al. (1975, 1978) used a biased random walk procedure to calculate the ground-state energy. This involved the numerical evaluation and averaging of eq. (7.13) for lo5configurations. The form of the Jastrow function chosen was

which represents a WKB solution to the Morse potential, eq. (7.16), at short-range. The Monte Carlo procedure implies random generation of the configurations, which are accepted or rejected by a biasing procedure that conserves the probability distribution lF(r,, . . . , r N ) ) * .To exploit the variational nature of eq. (7.17) Etters et al. (1975,1978) minimized the energy for each density. The pressure and compressibility were obtained by taking the appropriate derivatives of the energy with respect to volume. The results of Etters et al. (1978) for H J are reproduced in table 7.2 and shown in fig. 7.4 along with the results of Miller and Nosanow (1977) and the results of the hard-sphere model (Friend and Etters 1980). The monotonic increase of the energy with density points convincingly to the gaseous nature of HJ.

I.F. SlLVERA AND J.T.M. WALRAVEN

330

[Ch. 3, 87

Table 7.2 The Monte Carlo results for the energy E and pressure P versus density. a’/u is the hard-sphere model expansion parameter. where a = 6.5 au and u is the volume per atom. N J N is the fractional number of atoms in the condensate.

l0’n

(A - ’)

E (K) 1.902 2 0.081 1.105 * 0 . W 0.681 A0.027 0.543 ?0.031 0.326 20.024 0.192 20.014 0.101 5 0.012 0.037 k0.007 0.0165 2 0.004 0.0125 2 0.003 0.0064 2 0.002

3.010 2.007

1.505 1.204 11.803 0.607 0.301 0.120 0.060 0.045 0.030

lO’P (kg/cm*)

lO2a’/u

N,,IN

108O.0 431.0 217.0 129.0 46.0 72.0 4.5 0.72 0.28 0.065 0.018

-

-

-

-

-

-

2.46 1.23 0.49 0.25 0.16 0.12

0.764 0.833 0.895 0.926 0.939 0.047

-

16c

2 g

14-

1210-

U

Z08-

r

W

0604-

04

08

12

1.6

2.0

2.4

2.8

3.2

Fig. 7.4. The ground-state energy per H-atom versus density. The dots represent the Monte Carlo results of Etters e t al. (1978). T h e squares are the results of Miller and Nosanow ( 1977). The dashed line corresponds to the hard-sphere results of Friend and Etters (1980).

Ch. 3 , 97)


331

In their calculation Miller and Nosanow (1977) tried two potentials, the “exact” ’2 potential (Kolos and Wolniewicz 1974) and the LennardJones fit to this potential given in table 7.1. The “exact” potential was supplemented for the long-range regime using a polarization expansion determined by Bell (1966)

-

V ( r ) -c6r - 6 - c8r- 8 - clflr-lfl,

(7.18)

where c6 = 4.506 X lo4 K A6; c, = 2.415 X lo5 K A8 and clfl= 1.786 X 10‘ K A”. The treatment of Miller and Nosanow (1977) was also variational in nature, using a trial wave function of the type eq. (7.11), however, the McMillan form was chosen for f(r): (7.19) where 6 is a variational constant. Etters et al. (1975) also attempted this form for f(r) in conjunction with a Lennard-Jones potential, but obtained lower energies with eq. (7.17) and the Morse potential. To calculate the ground-state energy Miller and Nosanow (1977) used two different cluster-expansion methods, which appear to provide a good approximation, in particular in the low-density regime. The methods used are the “BBGKY-KSA” due to Born, Bogoliubov, Green, Kirkwood and Yvon, which includes the Kirkwood superposition approximation (KSA), and the hypernetted chain (HNC) approximation. In fig. 7.4 we only show the BBGKY results, which were found to be slightly lower than the results obtained with the HNC method. The results obtained with the LennardJones potential are also included to provide some perspective of the accuracy of the predictions based on the quantum theory of corresponding states.

7.2.2. The fermion case For the fermion case the wavefunction has to be antisymmetrical, suggesting a Slater-Jastrow-type trial wavefunction for the ground state

Here F is defined by eq. (7.1 l ) , 4 is a Slater determinant of free fermions


331

(Ch. 3, 87

with spin cr,. and (7.21)

cb = det{exp(ik, .r,)X,(s,),>) ,

where ,y,(cr,) is the spin state of the j t h particle. Based on eq. (7.20) various equivalent expressions may be derived for ( E ) F . the ground-state energy per particle for the fermion case. These result from different partial integration schemes for eq. (7.10) (Zabolitzky 1977). We give an expression obtained by use of the Jackson-Feenberg (1961) identity

I

1

IJ*V’G d7 = 3

i

{[+*V’+

-

(V+*)(V+)]

+ [cc]} d 7 .

(7.22)

Substituting q . (7.20) and using eq. (7.12) one finds

1

+ ~ ‘ [ 4 * V ; d , -Vz1412]d 7 .

(7.23)

Introducing the Fermi pair-correlation function, (7.24) one arrives at

( E ) , = :(h2/2rn)kf +

+N

1

F’

2

gF(r)[-(h’/2rn)V’ In f(r)

(h2/2rn)V’l41’d7/~ l$,:l’dT.

+ V ( r ) ]d r (7.25)

The first term is t h e free-fermion kinetic energy, dominant for n +0. with k , the Fermi momentum. The second term is closely analogous to eq. (7.15) but contains. in addition to the dynamical correlations accounted for by gR(r),also the statistical correlations implicit in 141’. The last term of eq. (7.25) requires evaluation of V f l 4 l ’ and represents a kinetic energy correction. There arc various methods to evaluate ( € ) F . A well-known procedure, due t o Wu and Fcenberg (1962) is to treat the fermions as bosons and t o account for the statistical correlations in an approximate way by means of a

Ch. 3 . 371


333

cluster expansion. Various methods to generate such a statistical cluster expansion are discussed in the book by Feenberg (1969). This method is expected to work particularly well if the dynamical correlations keep the atoms sufficiently far apart for the statistical repulsion to be small. The Wu-Feenberg approach was taken by Etters et al. (1975) to study D.1 in analogy to Schiff and Verlet’s (1967) application of the method to liquid 3He. The associated D.1-Bose problem was solved using the same Monte Carlo program that served for the HL and T.1 calculations (see foregoing section). Miller and Nosanow (1977) and Miller et al. (1975, 1977) also used a statistical cluster expansion to correct for the statistics, but chose for the BBGKY-KSA and HNC integral equation to approximate gB(r).In addition to DL,, Miller and Nosanow (1977) also studied 0.1,. More advanced approximate solutions for eq. (7.25) may be obtained by using the Fermi-hypernetted-chain (FHNC) method in which both dynamical and statistical correlations are treated on equal footing. For a detailed discussion of the method in terms of cluster diagrams the reader is referred to the reviews by Zabolitsky (1977) and Clark (1979). Krotscheck et al. (1981) applied the FHNC-method to D.1 D.1 and D.1 3 , using an “optimized” Jastrow function, obtained by minimizing the energy functional with respect to lnf. The results are shown in fig. 7.5 along with the D.1 results of Miller and Nosanow (1977) obtained with the McMillantype Jastrow function given in eq. (7.19) with chosen b(n) to minimize the

,,

I

”

0

1

2 3 4 DENSITY (10’k3

5

6

I 1

I

Fig. 7.5. Energy per particle for D J D1, and DJ as obtained by Krotscheck et al. (1981) using correlated basis functions for D.1 and D.1 3 . The result of Miller and Nosanow (1977) is given as a full line; the results of the variational Monte Carlo calculation of Panoff et al. (1982) for DJ, are indicated by the dots. j,

I.F. SILVERA AND J.T.M.WALRAVEN

334

(Ch. 3. 98

energy for each density studied. The results of Etters et al. (1975) were the first available for D.1 and stimulated theoretical and experimental work, but are not included in fig. 7.5 since they are no longer considered up-to-date. Focusing on the FHNC results, one notes that the minimum in the DJ energy versus density curve lies below the minima for the D J 2 and the DJ results, as one expects intuitively. For detailed considerations and error estimates of the FHNC-method the reader is referred to the original literature (Krotscheck et al. 1981). The FHNC results substantiated the conclusion, already drawn by Miller and Nosanow (1977), that for D.1 the kinetic and potential energies almost cancel each other. Therefore, very accurate calculations are required to decide upon the nature of the ground state. Although the theory presented was not sufficiently accurate to be decisive in this respect, in particular D.1 is likely to have a liquid ground state. If not, only a very slight applied pressure should suffice to liquify any of the D J , modifications. The first convincing result that a liquid ground state of D.1, exists was obtained by Panoff et al. (1982), using a full Fermi-Monte Carlo evaluation of ( E ) Ffor up to 99 particles. The application of this method was made possible by recent advances in high-speed computing facilities. The calculation used the complete antisymmetrized wavefunction (7.21) with an “optimal” choice off(r) taken from the paper by Krotscheck et al. (1981). The same f(r) was used for all densities. The results are also included in fig. 7.5 and show. within statistical error, a negative minimum, required for a liquid ground state. In view of the “exact” nature of the Monte Carlo evaluation of ( E ) these results represent a rigorous upper bound for the energy. Moreover, the interaction potential is also known to high accuracy giving confidence that D.1 liquifies at sufficiently low temperatures.

,

8. Many-body static and dynamic magnetic properties

Many of the very new aspects of H , D, and T are due to the magnetic moments associated with the electronic and nuclear spins. The magnetization of the gas has already been generally discussed in section 2.3 and the inhomogeneous density or magnetization due to a magnetic field gradient has been briefly handled in section 5.1.4.1. In this section we consider in greater detail the interaction with a magnetic field, in particular under conditions in which quantum degeneracy is important. We first discuss static properties which display unusual behavior due to the statistics, then dynamical properties o r magnetic excitations.

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33s

8.1. STATIC MAGNETIC PROPERTIES

8.1.1. Noninteracting gases An intriguing question concerning HJ is "what happens to the density distribution if the gas Bose condenses?". Since the condensate represents a macroscopic population of the lowest energy state of the gas, one might expect the condensate to accumulate in the highest field regions of an inhomogeneous magnetic field. In order to get some insight into this behavior, Walraven and Silvera (1980) analyzed the problem for a field which varied quadratically with displacement from the field center, such as for a simple solenoid. They made a further (nonphysical) approximation that the field gradient only existed in the z direction so that eqs. (5.1) become

with zo = 51 mm, which represents an actual experimental solenoid. In this case the Schrodinger equation for noninteracting atoms in the m, = - $ state becomes

where the energy is measured with respect to the static energy (-pBB)in the field center. This Hamiltonian corresponds to harmonic oscillator motion in the z direction and plane wave motion in the x-y plane. The wavefunction can be factored: @(rl,r2,. . . , r N )= @ ( r l ) J / ( r* z ) I)(.,). The single-particle wavefunctions are of the form

where 4nzis a harmonic oscillator (HO) wavefunction with oscillator energy splitting

ho,

2 1/2

= h(2hB"/rnZ")

.

(8.3b)

Here, oo2:6.8 x lo3 rad/s for a field of 10 T with zo = 51 mm. Thus in this field the HO states are very closely spaced and of order lo8 states are populated for T = 0.1 K. Populations are governed by the Bose distribution function, eq. ( I .8),

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1.F. SILVEKA AND J.T.M. WALRAVEN

N, = { e x p [ ( ~-, p ) / k T ]- 1 )

'

(Ch. 3, 98

and the density distribution is calculated,

using

Here $ , ( r ) denotes a single-particle eigenstate of eq. (8.2). The density distribution for T 9 7 , shown in fig. 8 . l a is Boltzmann-like; a classical approximation is used for ($,I2. Below the critical temperature for BEC the k , = 0. ground oscillator state is macroscopically occupied. The width o f the HO ground state is approximately 7 pm (fig. 8.la). The axial density profile for this noninteracting case below T, is a broad (nonBoltzmann) normal component with a spatially localized, sharp. distinctive peak at the field maximum. representing the condensate fraction. It is interesting to contrast the Bose and Fermi gases (Silvera and Walraven 1981a), the tatter represented by D.1. At T = 0 K only the ground oscillator state of the Bose gas is occupied, yielding a sharp density peak around z = 0: however, the Fermi gas has a very broad density distribution shown in fig. 8.lb. This is also calculated using eq. (8.4),but with the

'I I

Bosons

c3 Fer m I ons

T= 0 I

2lL-

L~

Distance from center

Fig. 8.1. (a) The axial density distribution for noninteracting bosons in an inhomogcneous solenoidal magnetic field above and below 7 n,, the critical density. Interactions, represented by the parameter a' in the figure, result in an increase of p at n,, but the curve and its derivative (compressibility) are still quite characteristic of BEC. It is clear to any reader who has read through this review that techniques are available to measure p, n and T (see section 5.12), and thus the EOS. The problem, however, is to achieve a sufficiently high density or low temperature. The highest density yet achieved (Hess et al. 1984) of 4.5 x 10" was at T = 570 mK ( T , = 43.5 mK for this density). The lowest temperatures of study, T 60-70 mK, have been achieved with bulk densities probably in the range of 10'5-10'6/cm3 (corresponding to T, of 159-740 kK). The source of these problems is also clear. At both high and low densities, recombination heats the sample and cell or is sufficiently rapid that a high density cannot be built up. The heating problem can be of two sorts:

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1.F. SILVEKA AND J . T . M . WALRAVEN

3%

[Ch. 3. $10

( 1 ) the heat load is so high that the temperature of the refrigerator rises,

or

(2) the gas sample itself heats up due to thermal gradients and Kapitza resistance so that the gas is not in equilibrium with the cell walls. 10.2.1. Compression o f bubbles

The first problem rnentioncd above can be resolved by using small samples (the bubbles mentioned in section 5.12) so that the total heat load is reduced. Let us analyze the heating problem in a bubble at BEC (Silvera 1985). We discuss and encounter some of the experimental problems by way of a numerical example. For the sake of clarity we do not parameterize the problem to find an optimum configuration, but rather use a typical situation to demonstratc some of the difficulties that arise. Assume that we seek to attain BEC with a critical temperature, T, = 100 mK, corresponding to 12, = 1.57 x 10"//cm3. A bubble of H J t with radius r is created in a vessel of liquid 'He (see section 5.12) at this temperature with n < n,. As the pressure is increased at constant T , t h e density increases and sweeps through n,. Thus, we can trace p versus n at constant T to determine the EOS. The HJf sample is under a pressure p = phyd+ PSIwhere phydis the hydrostatic pressure due to a column of liquid helium and pl, = 2 a I r

(10.1) I

is the pressure due to surface tension, Q: = 3.78 X lo-' N m . For bubbles somewhat smaller than one millimeter in diameter, pSI alone can be sufficient to pressurize HJS into BEC. Due to recombination in the bubble, rshrinks andp\, increases. From eqs. ( 1 . 1 1 ) and (10.1) we see that

the critical pressure is proportional to T5" and the critical radius is rr = 67.9( 1001T,)5 with T , in mK and r, in pm. For the conditions TL= 100 mK and n, = 1.57 x IO"/cm', r c = 67.9 p n and 2 X l O I 7 atoms are in the gas phase of the spherical bubble. We establish a bubble having r > rc and monitor p and n during its decay. We first determine the decay rate. From an analysis of bubble decay (Sprik et al. 1985), one finds that the inverse time constant of the bubble of volume V is

',

(10.2)

All of thcse terms are known and T - 1 = (1.79 + 6.84) = 8.63 s ', or T = 116 ms, where we have taken usa, = 5 x 10"/cm2. The power dissipated is

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359

- N D 0 / 2 ( D ois the dissociation energy) or - N = -riV= 1.8 x 1014s-’ and = 66 pW. We make a few remarks at this time. 66 pW at 100 mK is a heat load which a moderate sized 3He-4He dilution refrigerator can handle. A decay time of 116ms (time for volume decrease of -l/e) is rapid, but a measurement time of 10ms/point seems possible. We note from the calculation of 7 - l that the decay is dominated by the surface term. This is quite unfortunate since otherwise one could use smaller bubbles to reduce this contribution to the heating, as A = 47rr’. Larger bubbles with lower T , have greater heat dissipation due to the larger area (of course we could compress these with phyd to reduce the area, but as already stated our object is to trace through one example). Now the most important question is “can the temperature of the bubble be maintained at 100mK in the presence of the released recombination energy?”. First assume that all of the recombination energy is dissipated uniformly in the H i t gas. There are two heating problems: (1) the thermal gradient across the bubble due to the finite thermal conductivity of the gas, and ( 2 ) a thermal step at the boundary due to the Kapitza resistance. We first calculate gradients due to thermal conductivity. For a density of 1.57 x l O I 9 ~ r n - ~using , data from Lhuillier (1983), the collision mean free-path hmfp= 3 ~ / ( n C , u )= 0.7 pm, where K is the thermal conductivity and C, the specific heat. The thermal gradient is A T = (1 /15)( O/V)r’/K W/K m, we find AT > 1 K. (Sprik et al. 1985). Using K = 3.5 x Next we consider the Kapitza resistance. Using expression (4.13a) with a=O.3 we find Q = 8 . 6 X 1 0 - 5 A T W K - ’ , or A T z 7 5 0 m K for T = 100 mK. With this type of heating the bubble will explode due to the thermal instability observed by Sprik et al. (1983) and Tommila et al. (1984), and studied by Kagan et al. (1984). The problem is thus the following: the gas develops a large thermal gradient and the heat cannot be conducted out through the He walls fast enough. What can be done to help? Certainly the example used is not an optimum, but the problem is severe enough that major improvements are required. One would be to use 3He surfaces which reduces osat by a factor of about0.35, orthe heatingdue tosurfacerecombination(proportiona1to(+:a,) by a factor of 23 (we ignore changes in bubble size for T, = 100 mK due to the reduced value of the surface tension for 3He). This alone would be a major gain. However, the use of 3He-4He mixtures, to create 3 He surfaces, introduces another problem since the thermal conductivity of 3He-4He mixtures is a few orders of magnitude lower than that of pure 4He (Rosenbaum et al. 1974) and the 3He-4He itself may develop thermal gradients.

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I.F. SILVEKA A N D J.T.M. WALKAVEN

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Thc situation for the bubbles, as stated, may be pessimistic. For example. not all of the recombination energy goes into the gas. As pointed out by Silvera (1984), it may be possible for molecular H2 formed by recombination to go directly into the liquid and relax there without transferring energy to the gas. In section 6 it was shown that the initial recombination step divides -70 K of energy (of the 51 967 K available) between the excited molecule (Hr) and the spectator atom. The average number of collisons to relax from one state to another of HT may be of order lo2 (the splittings between levels of HS is large compared to the kinetic energy so that the integrals representing relaxation cross sections will be small due to the oscillations in the wavefunctions). Thus, it might be advantageous to use very small bubbles, even though the density and T, will be higher. The purpose of this discussion on bubbles, which ends here, was not to be definitive, but to demonstrate with numerical examples some of the problems facing the experimentalists. 10.2.2. Traps for low-field seekers Recently Hess (1985) has proposed a scheme for magnetic trapping of hydrogen at 20 mK and cooling to temperatures below 10 pK by “evaporation” to obtain BEC. Although the information available in his abstract is limited, we shall discuss what we believe to be possible advantages and problems. The great advantage of a trap is that one rids the system of helium walls which catalyze recombination. Unfortunately, it is not possible to create a static magnetic field in charge- and current-free space which has an absolute field maximum. Any maximum of a component is always a saddle-point (Wing 1984). In a standard solenoidal geometry (fig. 5.2) the atoms are compressed in along the axis to the field center; they are forced radially outwards resulting in higher densities at the confining side walls. Maxwell’s equations do allow static local magnetic-field minima, with IBI not necessarily zero at the minimum. Since states a and b of fig. 2.1 are high-field seekers, it is not possible to trap H J with a static magnetic field. On the other hand, states c and d are low-field seekers so that it is possible to trap Hf in a field minimum. Pritchard (1983) has discussed magnetic trapping of neutral atoms in such fields. Hf is a perfectly acceptable two-component thermodynamic system which can Bose condense. The important questions are “can a trap be filled, can the trapped atoms be cooled, what are the lifetimes for decay to HJ. or H z , and how are the trapped atoms detected?”. Several methods can be used to produce H f . c and d-state atoms can be

Ch. 3, 0101


36 1

collected by passing an H atomic beam through a hexapole magnet, which focuses c- and d-state atoms and defocuses a- and b-state atoms. Alternately, a 180”ESR pulse on HJ in a high magnetic field will convert HJ, to HT, which is then ejected to low field. Still another technique is to use a field gradient at low temperature which drives HJ to high fields and HT to low fields. Whatever the technique, Hess proposes to thermalize the HT on 20mK walls. The field would then be activated to trap and isolate the atoms from the He walls. The trapped atoms can then be cooled by reducing the trapping gradient so that hot atoms “boil” out of the trap and stick to the He walls. The remaining atoms thermalize by collision and cool to temperatures less than 10 pK, according to Hess. The density required for BEC of a (uniform density) one-component gas at 1 0 k K is 1 . 6 ~ 10”/cm’. The gas could possibly be studied by NMR or ESR. The greatest problem in this proposal seems to be the thermalization of the atoms without their sticking to the wall before being trapped. Let us estimate the equilibrium density, n. If the walls have a ‘He surface with E,/k = 0.34 K, then from eq. (1.7), with T = 20 mK, n = u/30with n in cm-3 and u in cm-’. The heating rate is Q = Ksu2(D,,/2)A.Using a value of K s (table 5.1) corresponding to B = 1 T and A = 10cm2, we find u 2 x 1010/cm2for 0 = 10 pW, which corresponds to the cooling power of a moderate refrigerator operating at 20 mK. The result is n = a130 = 6.7 x lo8cm-’ near the thermalizing walls surrounding the trap. Thus the trap cannot be loaded from a fully thermalized gas. This means that the atoms must have a moderate accommodation coefficient and a very low sticking coefficient, a,.Jochemsen et al. (1981) measured a,= 0.016(5) for 3 He at T = 100 mK. Zimmerman and Berlinsky (1983) calculated a,a T l’*, so that at 20 mK, a,= 0.007, or an average of 140 bounces before sticking. The scaling of the sticking coefficient is somewhat questionable, as has been mentioned in section 4, Zimmerman and Berlinsky did not use a long-range potential in their calculation, which may be very important. Measurements by Salonen et al. (1984) indicate that the Kapitza conductance is increasing with decreasing temperature. Since this is an energy moment of as,it means that a,probably also increases with decreasing T. In any event, filling of the trap may be a critical problem. The final consideration is the lifetime of the sample. By assumption the losses due to escape are acceptable and advantageous in the cooling of the gas. The other mechanisms are: (1) three-body recombination, (2) spin-exchange decay, and (3) electron spin relaxation from c, d to a- and b-states, which are then ejected from the trap by the field gradients. The first process is negligible. If the field B < 500 G (0.05 T) then c-state

-

362


[Ch. 3, §I0

atoms will escape from the trap as their magnetic moment decreases in low field (see fig. 2.1), leaving pure d-state, which trivially has no spinexchange problem. There are no accurate calculations of magneticrelaxation rates at low fields, and new decay channels open up, compared to high fields (Kagan et al. 1981). A rough estimate indicates lifetimes of order 1-10 s for electronic relaxation with density 5 x 10"/cm3. All told this seems to be a promising proposal although it is replete with uncertainty, the largest being the filling of the trap and the question of the cooling time of atoms in the trap which must be much shorter than the decay time.

10.2.3. Traps for high-field seekers Although it is not possible to make a static trap for the high field seeking aand b-state atoms, it is possible to make an ac or dynamic trap. This problem has recently been analyzed by Lovelace et al. (1985). The dynamic magnetic trap takes advantage of a principle used in strongfocussing particle accelerators. Consider a solenoidal field B,(r) of the form of eq. (5.1). The static forces on a magnetic moment are (ignoring radial components of the field)

F,, = p,,B,pIz~,

F, = -2p.,,B,zI~, 2 .

(10.3)

The force in the z-direction is inward and leads to confinement of the aand b-state atoms, whereas the force in the radial direction is outwards and leads to expulsion. The natural harmonic frequency wo for confined particles is given by eq. (8.3b), oo= ( 2 ~ B 0 / m z2 , 1)1 2. Clearly, as discussed earlier, this static field is nonconfining. Now, a dynamic trap can be achieved if B,, Is superimposed on a large static field in the z-direction and is modulated at a frequency w . On alternate phases, the forces in eq. (10.3) are positive and negative since orientation of the magnetic moment in space (or spin state) does not change. It can be shown (Landau and Lifshitz 1976) that at a sufficiently high frequency (in practice, of order kHz or greater), the particles undergo a slow motion corresponding to the natural motion (without the ac field) and a fast ac motion. As a result a particle will oscillate with amplitude 5 at frequency w about the slowly varying coordinate po(f) such that the particle has larger displacements from the origin, \p,,l + 161, during the phase when the force is inward and \pol - 15) when the force is outward. Since the force is proportional to p, there is a net inward force. This holds for both F,, and F, so that the particle is confined. Lovelace et al. (1985) have simulated such traps and find that fields of

Ch. 3 , $10)


363

order of 1T at a few kHz are required to trap a gas at a temperature of a few mK. The gas can be cooled by reducing the amplitude of the confining field so that hot atoms can escape and stick to cold walls. Internal viscous heating due to the interaction of the atoms with the ac field puts some limitation on the cooling. There are some experimental difficulties with the dynamic trap. A very careful design must be made to avoid eddy current heating of the refrigeration system. The most serious problem, again as discussed for the static trap, is to fill the trap with a gas at a temperature of a few mK. The advantage over the static trap is that the atoms are in the two lowest hyperfine states so that the lifetime will not be limited by electronic spin relaxation.

10.2.4. Two-dimensional superfluidity The final subject that we shall discuss is two-dimensional superfluidity. At this time the most sensitive means of detecting this phenomenon seems to be by a study of third-sound in a pill box shaped resonator such as has been used for studying ‘He (Ellis and Hallock 1983). In this case H & twould be condensed on a 4He film in such a resonator and the shift of the ‘He third-sound mode or new modes due to H3.f would be sought (see section 9). The signals are estimated to be very small but within the sensitivity of measuring techniques. A large uncertainty is in the stability of the 4He film thickness. Godfried et al. (1985) observed large variations in 4He film thickness during the H & filling phase of the hydrogen cell. Since the frequency of the ‘He third-sound depends on the film thickness, the thickness should be stabilized to parts in 10’ to lo6. A second and perhaps more serious problem for observing two-dimensional BEC is that it is probably not possible to fill the pill box with a sufficient density of H i t to achieve the required coverages shown in fig. 9.1. In the geometry of Ellis and Hallock the pill box resonator has a hole at the center for filling with the condensable gases. If the hole is too large, the Q of the cavity will be attenuated. Estimates indicate that the steady-state density in the cavity, with flow-in through the hole of limited diameter as a source, and surface recombination as a sink for atoms, is too low to achieve the densities required for two-dimensional superfluidity. Nevertheless, such an experiment at lower coverages may give insight into surface decay processes and allow one to study a large region of the adsorption isotherms of fig. 9.1. Although the pathway to the study of degenerate phenomena in these quantum systems is filled with uncertainty, the rapid developments and many new ideas which have emerged in the past several years encourages experiments and theorists alike to meet the challenge of nature.

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I . } . SILVERA AND J.T.M. WAL.RAVEN

[Ch. 3

Acknowledgement

We acknowledge and thank many colleagues and students for stimulating discussions which have contributed to this review. but in particular: V.V. Goldman. E. Eliel, H. Godfried, A . Lagendijk, R. Sprik and B. Verhaar. J . Gillaspy and R. van Rooijen helped with the programming of some functions. Hans van Zwol and Erik Salornons aided with the calculation of some of the tables in section 6. IFS acknowledges support from DOE contract number DE FG02-85-ER-45190: JTMW from the Stichting FOM.

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CHAPTER 4

PRINCIPLES OF AB INITIO CALCULATIONS OF SUPERCONDUCTING TRANSITION TEMPERATURES BY

Dierk RAINER Physikalisches Institut der Universitat Bayreuth 0-8580 Bayreuth, Fed. Rep. Germany

Progress in Low Temperature Physics, Volume X Edited by D.F. Brewer 0Elsevier Science Publishers B.V., 1986 371

Contents 1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Many-body aspects diagram analysis ......................... 2.1. Technical preliminaries, notation ................. 2.2. Classification and calculation of self-energy diagrams . . . . . . . . . . . . . . . . . . . 3 . The low-energy equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Bloch-function representation . . . . . . . . . . . . . . . . . . . . . . 4 . Band-structure theory and the electron-phonon interaction . . . . . . . . . . . . . . . . . .

.

.................

373 371

383

401 402

....................... ........................

411

5.2. Calculation of T, from Eliasberg’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . Conclusion . . . . . . . . . ........................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 419 421

372

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1. Introduction

By an ab initio theory we understand here a theory which starts from the fundamental Hamiltonian of solid state physics, and calculates directly measurable properties of superconducting materials. These properties include, among many others, the critical temperature T , , critical fields, the tunneling conductance and the electromagnetic absorption spectrum. The fundamental Hamiltonian is that of electrons and nuclei coupled by electromagnetic forces. The advantage of ah initio theories is that only a few universal and precisely known input parameters are needed: in our case the fundamental constants h , e , m , M , k , and the chemical composition of the metal. The disadvantage is the complexity of the problem, which has not yet been fully mastered by theoretical physicists. Recent progress towards a practicable ab initio theory of superconducting materials, however, leads to some optimism that this ambitious goal will be reached in the near future. The progress has mainly been made possible by the increase in computer capacities. One can nowadays attack problems which were thought unsolvable some decades ago, when the foundations of the theory of superconductors were established. The essential concepts and strategies for an ab initio theory were already developed in the early years of the modern theory of superconductivity (Scalapino et al. 1966, Scalapino 1969). It is remarkable that most present calculations still follow these old concepts. I recapitulate in this review the standard concepts of an ah initio theory of superconducting materials, in a formulation which is probably the most efficient one for numerical calculations. Of course, this article does not cover all aspects of the theory of T,. I refer to excellent recent textbooks and reviews by Allen (1980), Grimvall (1981), Vonsovsky et al. (1982), Dolgov and Maksimov (1982), Allen and Mitrovid (1982), and Ginzburg and Kirzhnits (1982) for supplementary details. I emphasize more than is usual the rigorous basis of the theory, namely, the asymptotic expansion in small parameters like kT,IE,, h ~ ~ , , < and , ~ ll lEk ,~l * . All three are very good

' E,. stands for a typical electronic energy in metals (Fermi energy, plasmon energy. bandwidth, etc.), k ; ' is a characteristic electronic wavelength. mphona typical lattice frequency, and I the mean free path of conduction electrons. 373

374

D.RAINEK

(Ch. 4, 81

expansion parameters for conventional superconducting metals*. They are typically of order lo-’ or smaller; this explains the outstanding accuracy of the microscopic theory of superconductivity, and is basically responsible for the fact that superconductivity is one of the best understood many-body effects. The intention of this review is to clearly distinguish approximations which are asymptotically correct in the above sense from other approximations which need different justifications or are made simply in order to bypass presently insurmountable difficulties. It might be necessary. even in the future, to use at some stages unjustified approximations in order to set up a tractable calculation. Nevertheless, one should keep an eye on the conceptual weak spots; in case of failure of a first-principles calculation these are the most probable error sources. The standard strategy for setting up an a b initio calculation of superconducting parameters is guided by the existence of two well separated energy scales in a conventional metal. We have the ’high-energy range’ of typical electronic energies (characterised by E, ), and the ‘low-energy range’ where we find the phonon energies. k’7;, the energy gap, and other typical energies of the superconducting state. The following calculational procedure is correct to leading order in the ratio of low and high energies. One first keeps the phonons frozen in, omits all superconducting correlations, and solves the ‘high-energy problem’ of Coulomb-interacting electrons in the fixed potential of the nuclear charges. This part of the ab initio theory needs to be mastered by many-body and bandstructure theorists. They have t o calculate the stable state of the crystal. the phonon modes and frequencies. and furthermore the low-lying electronic excitations in the frozen in lattice (bare conduction electrons**), their residual interactions, and their coupling to phonons. This knowledge is needed for the next step, which is to solve the ‘fow-energy problem’, i.e., the dressing of bare conduction electrons with phonons, and the eventual formation of the superconducting state, i.e., the condensation of conduction electrons into Cooper pairs. The low-energy problem is solved by the strong-coupling theory of superconductivity. It finally calculates measurable properties of superconductors. Clearly, a complete first-principles theory of superconducting materials needs the cooperation of many-body theory. bandstructure theory, and the theory of superconductivity. In sections 2 to 5 I will specify and describe in more detail the The assumption o f a small hmphc,n/ E , is not fulfilled for ‘heavy fermion’ superconductors (Stewart 1984). Hence. thc conventional strong-coupling theory of superconductivity does not apply t o these systems since it is based o n a small f ~ w , , ~ ~ , ~The / E , weak-coupling :. theory , still work. which needs a small kT,i E , ;ind k T , / f i w , , ~ , should * * ‘bare‘ stands for ‘undreswd by phonons’.

Ch. 4, $11

AB INlTlO CALCULATIONS OF T,

3 75

various parts of a complete ab initio theory. I will concentrate on a single material parameter, the transition temperature T,. This quantity is the most important and also the most extensively studied. Furthermore, a first-principles calculation of T , is quite representative of any other first-principles calculation of superconducting parameters. The methods, the solved and unsolved problems, and the amount of computational work, are essentially the same, whatever the quantity of interest is. I will first discuss, in sections 2 and 3, the many-body aspects of an ab initio theory of T,. Section 4 reviews what is needed from bandstructure theory for calculating T,. Finally, section 5 specifies the role of the strong-coupling theory of superconductivity in an ab initio calculation of T,. This last part of the ab initio theory is already settled to a large degree and is under reasonable control. The most significant progress in recent years has been achieved in the band-structure part of the theory of T,. A few questions are still open at present but will probably be settled in the near future. Many-body problems in the theory of T, are mostly unsolved. We have no comprehensive theory of the effect of electron-electron correlations on T,, although there are many studies of particular correlation effects, like plasmon effects (Ruvalds 1981) and paramagnon effects (Berk and Schrieffer 1966). There are also no first-principles calculations of the interaction parameter p * which could replace presently used ad-hoc rules (Bennemann and Garland 1972). Nevertheless, many-body theory has made very fundamental and indispensable contributions to the present scheme for a first-principles theory of superconducting materials. This will be discussed at length in section 2. A flow chart of the scheme is shown in fig. 1. The idea is to divide the full calculation into several steps with well-defined input parameters and well-defined quantities on the output side which are needed as input for the next step. In this way it is possible to work simultaneously on various parts of the chain of problems. For each part one has to develop a machinery to calculate the output from any physically sensible input. When the parts can finally be put together we expect to have a useful ab initio theory of T,. One might then think of using the theory for a computer assisted search for materials with higher Tc's. It is well known that T, is strongly linked to many other material parameters (Beasley and Geballe 1984). It depends on phonon properties (soft phonons are good for a high T , ) , properties of conduction electrons (a high density of states is favourable, a stong mutual repulsion is unfavourable) and the electron-phonon coupling (the stronger the better). Unfortunately, it turns out that the rules for achieving a high T,, listed in brackets, were more or less useless in the search for new high- T , materials (Matthias 1969). The reason is that fundamental properties like the

D. KAINER

376

fundamental

data

many-body

theory

[Ch. 4, $ 1

prescriptions for constructing band- s t r u c t u r e band-structure

potentials

theory

I

(=F strong-coupling

theory

Fig. 1. Flow chart for an ab initio theory of Tt

phonon frequencies, the electron density of states, etc., cannot be varied itidependenily by any practical means, i.e., by changing the chemical composition or the structure of compounds and alloys. For example an increase in the density of states (favourable!) is often accompanied by a detrimental decrease in the electron-phonon coupling (Hopfield 1969) or by an increased effect or the mutual repulsion of electrons (Berk and Schrieffer 1966). Because of our present failure to understand such interdependences we have no insight in how to achieve properties favourable for a high T, without introducing unwanted countereffects. Obviously, the problem of detrimental interdependences cannot be solved by the theory o f superconductivity; this theory takes the phonon frequencies, the electron-phonon coupling, etc., as given quantities, and calculates T, from these input parameters. Relations among the input parameters can only be studied by the deeper theories dealing with band-structure and many-body effects. Because of the barely understood interdependences it is dangerous to draw conclusions about T , from a mechanism which is singled out. The failure of all theoretical suggestions for high-7, mechanisms is probably due to an unjustified neglect of countereffects. It seems that a theoretical understanding of T , needs a complete ab initio theory. The hope is that future computer codes which are based on a first-principles theory will help in finding the relevant mechanisms for a high T , , what spoils them, and how to tailor superconducting materials with an optimized T,.

Ch. 4, 521

AB INlTIO CALCULATIONS OF TL

377

2. Many-body aspects, diagram analysis Many-body theory is of central importance in a first-principles approach to superconductivity, for three reasons: - It explains why one-particle theories, such as band-structure theory o r the weak- and strong-coupling theories of superconductivityt, work so well for the correlated many-particle system of conduction electrons in a metal. - Many-body theory has developed a calculational scheme, the Green’s function technique, which is the most powerful tool in the microscopic theory of superconductivity. - Many-body theory is called on to calculate important input material parameters of the theory of superconductivity, such as the Coulomb pseudopotential p * , and other conduction electron parameters to be discussed below. There is strong evidence that the standard theory of superconductivity works equally successfully for nearly free-electron metals (Al, Pb, etc.) with presumably little electron-electron correlations, and for metals with strongly correlated conduction electrons (V, Rh, and probably many other superconducting transition metals and their compounds). In fact, there is no example of a clear failure of the theory of superconductivity. The present understanding of this broad range of applicability of the microscopic theory of superconductivity is based on a sophisticated many-body interpretation of seemingly trivial one-particle terms like ‘conduction electron’, ‘Bloch state’, ‘electron-phonon interaction’, etc. Such terms were introduced originally in the model of non-interacting (or weaklyinteracting) electrons. Their generalized interpretation, which also applies to a system of strongly interacting electrons, is due to Landau. He developed the concept of ‘quasiparticles’, and explained the evident success of one-particle theories in metal physics by their superficial similarity to better justified one-quasiparticle theories. What are simply called ‘conduction electrons’ in metal physics are, in Landau’s interpretation, low-lying quasiparticle excitations of the correlated electron liquid which have the signature of single electrons but are, in general, quite complicated many-body objects. A quasiparticle can be viewed as a moving electron plus its ‘dressing’, i.e., the accompanying distortion of its local environment. In this scheme, the commonly used term ‘electronphonon interaction’ stands for the interaction of quasiparticles with phonons. This comprises the interaction with phonons of both the electron t The standard theories of superconductivity study single particles which are coupled to selfconsistently determined “mean fields”. Such mean field theories are categorized. in o u r notation, as one-particle theories.

37x

D. RAINER

[Ch. 4. 52

and its dressing. Similarly, the Coulomb pseudopotential p * is determined by the interaction of two quasiparticles, which differs substantially from the Coulomb interaction of two electrons. The largely unexplored internal structure of the dressing makes the ah initio calculation of conduction electron properties, in general, an involved many-body problem. The recipe which bypasses unsolved many-body problems in non-first-principles theories is to use quasiparticle parameters which are taken directly from experiments. This was done very successfully in a multitude of weakand strong-coupling calculations for superconductors in the last decades. In the weak-coupling theory, the quasiparticle density of states N * ( E , ) , the transition temperature T , , normal and spin-flip lifetimes of quasiparticles, and other quasiparticle properties, are traditionally adjusted to experiments. This approach does not demand a careful distinction between particles and quasiparticles. It is more or less immaterial if the adjusted parameters are interpreted as belonging to bare electrons or to quasiparticles. A very careful distinction is, in fact, vital only in a first-principles theory of superconductivity. The microscopic derivation of the quasiparticle concept was achieved by Green’s function techniques. I will repeat this derivation here in some detail for the following reasons: - It gives us a constructive definition of electronic quasiparticles (which we alternatively call ‘conduction electrons’). This includes the introduction of a rigorous crystal potential which comprises all many-body effects, and allows a simple calculation of quasiparticle wave functions from a oneparticle Schrodinger equation. - It gives us a precise and constructive definition of interaction parameters such as the matrix elements of the electron-phonon interaction, and the Coulomb pseudopotential p * . - It separates clearly the calculation of phonon and conduction electron properties. These calculations can be done in parallel. The separation is correct to leading order in the adiabatic parameter hophon/ E,. - It demonstrates that the quasiparticle concept in the theory of superconductors is correct to leading order in the fundamental expansion parametcrs kT,lE,,, h w p h o n l E Fand 11kJ. - It shows that the superconducting condensation has only a negligible effect of order (kTcIEF)’on the internal structure of a quasiparticle and on the (relevant) phonons. Hence, quasiparticle and phonon parameters can be conveniently calculated in the normal state. Subsection 2.1 is a brief presentation of some notation, definitions and the central equations of the Green’s function technique. Subsection 2.2 contains a discussion of the asymptotic expansion in h ~ ~ ~ ~ and , , l kT,l E , E, which forms the rigorous basis of the standard theory of superconductivity.

Ch. 4, 821

AB INITIO CALCULATIONS OF 7,

379

This expansion leads quite naturally to the concept of quasiparticles and allows us to precisely define the meaning of the terms ‘conduction electron’, ‘conduction electron interaction’, ‘phonon’ and ‘electronphonon coupling’. Sections 2 and 3 contain quite formal many-body arguments. They are supplemented at the end of section 3 by a very brief review of concrete model calculations of many-body effects in conduction electron properties of superconductors. 2.1. TECHNICAL PRELIMINARIES, NOTATIONS

The central quantity in our ab initio scheme for calculating T, is the electron Green’s function G(x, x‘; E ” ) which is a 4 x 4 Nambu matrix defined by 1/T

d7 eiEnT( TT(+,(x,-iT)$;.(x’, 0))) .

(1)

The 4-dimensional Nambu indices 1, I‘ describe spin- and particle- hole degrees of freedom,

The G’s are formally the Fourier coefficients of a Matsubara Green’s function with time ordered field operators at purely imaginary times. Thermal equilibrium is inferred by the KMS boundary condition for the time ordered G, or equivalently, by using the Fourier series expansion with frequencies E,, = ( 2 n + l ) r k T , n = 0, -tl, . . . (Matsubara frequencies for fermions). The statistical averages and the time dependence of the field operators are governed by the fundamental Hamiltonian of a metal,

H= T

+ V ( r i ,R n ) - p N .

(3)

T is the kinetic energy of electrons and ions ( = nuclei + core electrons), and V contains the Coulomb interaction between electrons, ions, and between electrons and ions. The zero of energy is shifted, for convenience, by p N , the chemical potential times the number of electrons outside the cores. ri and R , denote the coordinates of electrons and ions (with mass M , and charge -2,e). It is convenient to replace R , by the deviation from a fixed position R: (in most cases the equilibrium position), R,

= R”,+R,,

(4)

I>. RAINER

3x0

[Ch. 4. $2

and to expand V in powers o f SR,,:

V = V ' " ' ( r ,R. i : )+ V " ' ( r , , R : : ,S R , , ) + . . . .

(5)

v'"'contains

one-body potentials u::), which describe the Coulomb coupling of electrons to the lattice of ionic charges at their fixed positions, and two-body interactions, namely the Coulomb repulsion of electrons, (0' u,, and the c-number Coulomb energy of the fixed ions u:,"':

.

The most important part of V " ' is the interaction of electrons with small lattice vibrations, us;'. I call this term the 'fundamental electron-phonon interaction'*. It is simply given by the coupling of electrons to little electric dipoles of dipole moment -Z,eSR,, at the lattice points Rlt:

The Green's function technique uses the second-quantized equivalents of ( 3 ) . (6) and ( 7 ) . which leads to a convenient graphical representation of V in terms of fundamental interaction vertices, as shown in fig. 2. The electron Green's function needs to be supplemented by the phonon ' One ha\ to distinguish carefully among the different meanings of the term 'interaction' at the three ,tage5 of an ah initio theory. I use the adjective 'fundamental' for all interactions (couplings) that occur in the fundamental Hamiltonian (3). 'Interaction' (or 'coupling') \\ithout ;t special addition. like 'electron-phonon interaction'. 'conduction-electron interaction' refers t o the residual interactions which remain after forming (hare) quasiparticles. I h e w interactions enter. e g., the strong-coupling theory o f superconductivity. A third meaning o f the tern1 'interaction' is that used in the weak-coupling theory of superconductivit y . I t rcfcn to tlie inter;iction of physical quasiparticles which are dressed with phonons in addition to the "correlation ifrcssing' already huilt into 'hare quasiparticles'. This last kind o f interaction i\. i n general. the most important one hut will barely he used in this chapter. I will reserve no special n a m e for these interactions.

AB INITIO CALCULATIONS OF T ,

Ch. 4, $21

381

-

electron

phonon

Green’s

Green’s

functions

function

A

A

(bl

...

A (el t I

inter cti

vertices

Fig. 2. Diagrammatic symbols for conventional Green’s functions and interaction vertices. The arrows on the electron Green’s functions distinguish diagonal (arrows point in the same direction) and off-diagonal (arrows point in opposite direction) Green‘s functions in the particle-hole index. Whenever the arrows are omitted in a diagram, they must be attached in all possible ways compatible with particle conservation. The interaction vertices describe the various terms in an expansion of the Coulomb interaction in powers of 6 R , . They represent:

Green’s function.

The w, are Matsubara frequencies for bosons, w, = 2mmkT, v and p ( = x , y , z) are Cartesian indices. The graphs for G and D are displayed in fig. 2 .

The central equations of the Green’s function method are Dyson’s equations for G and D :

382

D. RAINER

[Ch. 4. $2

All effects of the interaction V are contained in the self-energies 2 and r . Dyson’s equations formally have the structure of single-particle equations, with 2 and T acting as effective potentials and force constants, respectively. The self-energies have a diagrammatic expansion in terms of fundamental interaction vertices and the full Green’s function G and D.Some typical self-energy diagrams are shown in fig. 3. I will alternately use two representations for the Green’s functions, the space and the momentum representation. The momentum representation takes into account the periodicity of the crystal, and is useful whenever conservation of crystal momentum is essential for the argumentation. G depends, in the momentum representation, on the crystal momentum (short: ‘momentum’) p , which lies in the first Brillouin zone (BZ), and on two reciprocal lattice vectors Q,: Q 2 .The position space Green’s function G(x, x’; E,,) is given in terms of G( p , Q 1 , Q 2 ;E,,) by:

Fig. 3. Representative diagrams for the electron self-energy (a) and the phonon self-energy (b).

Ch. 4, 821


383

The standard variables of the phonon Green’s function D(q , I , 1 ‘ ; o m )in the momentum representation are the ‘momentum’ q and two ‘branch indices’ 1, I’ which run from 1 to 3 N 0 . No is the number of atoms in a unit cell. In a simple lattice the branch and Cartesian indices can be chosen identical, and one finds:

The Green’s functions contain a substantial amount of information of physical properties of the metal. I refer to standard text books on many-body theory (Fetter and Walecka 1971, Landau and Lifschitz 1980) for more details on the Green’s function method, especially on how to extract the physical information from the Green’s functions. The information on the superconducting transition temperature is contained in the off-diagonal (in the particle-hole index) elements of the electron Green’s function. These elements vanish in the normal state, and T , can be calculated by determining the highest temperature with non-vanishing off-diagonal parts of the Nambu matrix G.

2.2. CLASSIFICATION AND CALCULATION

OF SELF-ENERGY DIAGRAMS

Dyson’s equations map the many-body problem of calculating the Green’s functions onto one-body equations with specifically designed ‘potentials’, the self-energies (Fetter and Walecka 1971, Landau and Lifschitz 1980). Such a transformation of the many-body problem is useful if the selfenergies have a simpler mathematical structure than the Green’s functions themselves. This happens to be the case for metals at excitation energies low compared to E,, which covers the relevant energy range for conventional superconductivity. The following formal calculation of the low energy part of the self-energies by diagram techniques uses the method of asymptotic expansion in kT,lE, and hwphon/EF. To leading order in the expansion parameters one gets tractable expressions for the self-energies in terms of low-energy Green’s functions and certain material parameters such as the electron-phonon coupling parameters, p *, or Landau’s parameters for the conduction-electron interaction. The diagram technique allows a precise definition of these ‘quasiparticle parameters’ in terms of the fundamental interactions of the electron-ion system. I adopt the notation of Serene and Rainer (1983) and introduce the quantity (small) which stands for the order of magnitude of kT,lE, or ho,,,,lE,t. For reasons to be explained below one needs the electron t For convenience, I assign the same order of magnitude to both ratios, although kT,lE, is in most cases much smaller than ~ W , , , ~ ~ / E ~ .

D. KAINER

3x4

[Ch. 4. 92

self-energy 2 to order (small)'' and (small)', and the phonon self-energy 7~ to order (small)'! In order to get these leading order self-energies one has t o convert the standard diagram expansion, which is an expansion in powers of bare coupling constants, into an asymptotic expansion in the expansion parameter (small). The procedure is to first estimate the order in (small) for each diagram. and then sum up all diagrams which contribute to the leading order self-energies. I start by estimating the order in (small) of various important quantities in an ab initio theory of superconductivity. I use atomic units ( e = h = m = k = 1) such that any quantity constructed from e, h , m ,k alone is of order 1 = (small)'! In these units many basic parameters of a typical metal, like the lattice constant, the Fermi energy and momenta, the conduction electron density, t h e plasmon frequency, etc., are of order 1. The only quantity in the fundamental Hamiltonian (3) of nontrivial order in (small) is the ion mass* M which is of order (E,lhwphon)' = (small)-'. All fundamental interaction vertices u::), u : : ' , etc., are independent of the ion mass. Consequently, they are of order 1 (at typical interparticle distances which are of the order of the lattice constant). A more careful order of magnitude analysis is necessary for the Green's functions because they depend on arguments with no fixed order in (small). the momenta p, q and the Matsubara frequencies E , , w,. The order of magnitude of a Green's function will. in general, depend on the values of its arguments. The order of magnitude analysis is easy for the unperturbed (2 = 5~ = 0) Green's functions G " ) , D'"'which are given in the momentum representation by:

.

G IS large, of order (small)-', if both E , and ( p + Q)'/2 - p are of order (small). This happens in p-space in a small shell around the Fermi surface. 6""is of order 1 or smaller if either p lies outside this shell or E,, is of order 1 or larger. The phonon Green's function, on the other hand is very small (of order (small)') for w, of order one, and is of order 1 for small w, irrespective of q . ('I)

* To avoid unessential notational ballast. the order of magnitude considerations of this section are done for a Bravais lattice without a basis. and with a single ion mass M.

Ch. 4. $2)

AB INITIO CALCULATIONS OF T,

385

A basic assumption on the theory of conventional metals is that these order of magnitude estimates are still valid for the full Green's functions G and D. The following analysis will demonstrate the consistency of this assumption. A very useful book-keeping device for our order of magnitude considerations is to introduce a 'high-energy' and 'low-energy' range in momentumfrequency space. By definition, an electron Green's function is in its low-energy range if its momentum p lies in a shell of width Ap around the correct Fermi surface and if its frequency is smaller than a 'cut-off energy' w,, i.e., I E " ~ < w,. A p and wc separate the order (small) from order 1: (small) G Ap G 1, (small) G w, G 1. Ap and w, are purely auxiliary quantities, and must drop out of any physical result. The low-energy range of a phonon Green's function includes all phonon momenta q and the frequency region Iw,I < w,. The momentum-frequency space outside the lowenergy range will be called the 'high-energy range'. The 'volume' of the low-energy range for electrons has a very small size of order (small)2 [one order of (small) comes from the frequency window, one from the size of the momentum shell]. The low-energy range for phonons has a 'volume' of order (small). Another useful book-keeping device is to decompose the Green's functions into 'low-' and 'high-energy Green's functions':

G ( p , . . . ; E n ) = G'loW(p,. . . ; E n ) + GhiBh(p,.. . ; E n ) ,

(154

D ( q, . . . ; w , ) = D ' " " ( q , . . . ; w , ) + D h i g h ( q , . . . ; w , ) ,

(15b)

where (?'low, D'O" are identical to G, D if the momentum-frequency arguments are in the low-energy range, and are zero otherwise. It is convenient t o introduce low- and high-energy Green's functions in a diagram expansion. To do this, one best starts from conventional diagrams in terms of unperturbed Green's functions. Such a diagram with, say, n conventional Green's function lines is decomposed into 2" diagrams, with low- and high-energy lines replacing in all possible ways the conventional lines. One then assembles diagrams which differ only by self-energy insertions on low-energy lines into skeleton diagrams with full low-energy Green's functions and unperturbed high-energy Green's functions. The graphs for low- and high-energy Green's functions are displayed in fig. 4. The non-symmetric treatment of low- and high-energy lines is useful. The advantage of these new diagrams (with four instead of the conventional two types of Green's function lines) is that one can estimate quite easily their order in (small). One has to count the number of low-energy electron lines and high-energy phonon lines. Each contributes a factor (small)-' or (small)2,

386

[Ch. 4. $2

D. RAINER

----

(a)

( C )

low- energy

electron

(b) ,-\-,-s.,'-~-

p h on o n

(d) high-energy

Fig. 4. Diagrammatic symbols for the low- and high-energy Green's functions. T h e low-energy lines represent full Green's functions in the low-energy range.

respectively. All other parts of such a diagram. namely interaction vertices, high-energy electron lines and low-energy phonon lines, are of order 1. The powers in (small) from the Green's function lines have to besupplemented by further powers which come from the momentum integrations and frequency summations. These 'phase-space factors' take care of the vanishing of low-energy lines outside their low-energy range. They are obtained by estimating the reduction of the 'volume' of the domain of integration (summation) by theconstraint that allargumentsof the low-energy lines fall in the low-energy range. The phase-space factors from momentum integrations depend critically on the dimensionality of the metal. This chapter deals exclusively with 3-dimensional metals. Hence, all the following rules and examples for order of magnitude estimates are restricted to the 3-dimensional case. The determination o f the phase-space factors is the only nontrivial aspect of the order of magnitude estimates. These factors depend on the number of low-energy lines and on the topology of a diagram. Several instructive examples of self-energy diagrams and their orders in (small) are shown in figs. 5 and 6. Figure 5 displays an example of Migdal's (1958) theorem; a diagram for the electron self-energy with two or more intersecring phonon lines is of order (small)' or smaller. The two diagrams in fig. 5 have the same number of lines and vertices but are topologically different. Their different order in (small) comes from different phase-space factors for the momentum integrals*. Figure 6 shows some typical * In both diagrams the momenta pI and p . have t o be constrained to the low-energy range. This contributes a factor (small)'. No further constraints on the momentum integrals' are needed for diagram (a) since the constraint o n p , puts p , ( = p , ) automatically in the low-energy range. Diagram ( b ) needs an additional constraint. since having p. p I .p 2 in the low-energy range near the Fermi surface does not imply a p , ( = p + p? - p , ) near the Fermi surface. The necessary constraint on p , - p : to keep p , in the low-energy range leads t o a further factor (small).


Ch. 4, §2]

387

Fig. 5. Examples for Migdal’s theorem. The two diagrams have the same number of lines and vertices but are of different order in (small). Momentum conservation implies p 3 = pI for diagram (a), and p , = p + p2 - pI for diagram (b). Note that the left diagram is not a skeleton diagram; it is used here only to demonstrate how an order of magnitude analysis works.

o+o+o

o+o+o

o+o+o

=O

=O

=O

8

Q

+- +--y +\

-2+1+2 =1

._

. d

#

*

o+o+o o + o + o -1+1+1 -1+1+1 -1+1+1 =O

=O

0+1+0 =1

-1+1+1

-3 +2 +3 =2

0+1-0 =1

=I

=I

=l

=I

-1+1+1 =I

-1+2+1 -3+2+3 -2+2+2 -1+2+1 =2 =2 =2 =2

-5+3+5 =3

+2+0+0 =2

+2+1+0 =3

Fig. 6. Typical self-energy diagrams and their order in (small). The total order is decomposed into the contributions from Green’s function lines + constraints on frequency summations + constraints on momentum integrations.

388

D. RAINER

(Ch. 4. 82

diagrams for the phonon and electron self-energies together with their order in (small). For clarity I have decomposed the powers in (small) into the contributions from Green's function lines, and phase-space restrictions from frequency summations and from momentum integrations. It is important to note that the order of magnitude estimates of fig. 6 hold only for external variables p , E , in the low-energy range (in the case of electron self-energies). and for external momenta q of order one* and w, small (in the case of phonon self-energies). This is the relevant range of external variables for our problem. A presentation of the complete theory of order of magnitude estimates is certainly outside the scope of this chapter. In the following I will list the four most relevant results of the order of magnitude analysis. The above discussions and the representative examples in fig. 6 should enable the interested reader to work out the proofs of these results. (a) We collect into a 'high-energy blob' all high-energy elements of a diagram (high-energy lines and fundamental interaction vertices) which are interconnected by high-energy lines or vertices. Diagrams which are equivalent in the sense that they differ only by the internal structure of their high-energy blobs are of the same order in (small). Hence, it is sensible to replace the sum of all equivalent diagrams by a skeleton diagram, which consists of low-energy lines and 'high-energy vertices' (see fig. 7). A high-energy vertex is fully characterised by its external lowenergy legs and the variables attached to them. A basic assumption of the theory of conventional metals is that the scale for the dependences of a high-energy vertex on its external momenta and frequencies is set by the high-energy scales (kf.., E F ) .This means a weak momentum and frequency dependence in the low-energy range. In some sense this assumption defines a 'conventional' metal. To leading order in (small) a high-energy vertex is temperature independent, and unaffected by the superconducting condensation (this condensation does affect the low-energy parts!). N o phonon lines appear in a high-energy vertex. Therefore. all high-energy vertices can be calculated from the adiabatic Hamiltonian (infinite ion masses), i.e., for fixed ion positions. (b) The leading terms in the phonon self-energy are of order 1 = (small)". This verifies. via Dyson's equation (lo), the order 1 for the full phonon Green's function in the low-energy range. The leading phonon self-energy is the two-phonon high-energy vertex (see fig. 7d) which consists of the fundamental two-phonon vertex and a purely electronic part. The above-mentioned properties of high-energy vertices imply that the phonon self-energy can be taken at zero temperature and zero external * The estimates break down in the long-wavelength limit. i.e.. for q / k , of order 'small'.

AB INITIO CALCULATIONS O F T ,

Ch. 4, $21

I .

\

++ ++

I

/*\ +--++ --/

;*; +

+

+--++

389

...

Fig. 7. Examples for the construction of high-energy vertices (cross-hatched symbols) from fundamental vertices (dots) and high-energy electron Green’s functions (dashed lines).

frequency. In this limit the phonon self-energy ~ ( n vm, p ; 0) describes the change in the ground-state energy of the adiabatic Hamiltonian ( M = 30) to second order in static displacements 6 R , of the ions*: T(nv, m p ; 0) = 6 2 ~ o ~ ~ ~. , , ~ ~ , ,

(16)

(c) The leading terms in the electron self-energy are of zeroth and first order in (small). The zeroth order term is a two-point high-energy vertex ~ o ( xx’; , E , ) (see fig. 7a). Our assumptions about the weak frequency dependences of high-energy vertices at small E, imply that $o can be expanded in powers of E,:

* For our purpose, R should be correct to order (small)”. Electrons in their low-energy range first contribute to R in order (small)’. Hence these low-energy processes can (but need not) be taken into account in calculating S2E,ISR,,SRm,.

D. RAINER

3!Jo

[Ch. 4 . 92

The term linear in P, is of order (small)'. All further terms in the Taylor expansion can be neglected. They are of higher than first order in (small). There are only two relevant skeleton diagrams for 2 which are of first order in (small)*. The first one, which I denote by 2:' (see fig. 8a) contains one low-energy line connected to a four-point high-energy vertex (electron-electron interaction). The second diagram, denoted by $Yp in fig. 8b, describes the coupling of low-energy electrons to phonons. 2;' and $yp are both influenced strongly by the superconducting condensation. vanishes in the unperturbed normal state but becomes finite below T,. $fP, but not 2;'. shows a strong frequency dependence in the low-energy range which mainly reflects the strong frequency dependence of the phonon Green's function. 2;'' and its frequency derivative are of the same order as E, of eq. (17). Both contribute in leading order to normal the term metal properties. 2;'.however. changes in the superconducting state, whereas is unaffected. (d) In our context, the Fermi surface is given by those momentap in the first Brillouin zone for which the free electron kinetic energy f corrected by the zeroth order electron self-energy $,(. . . , F , = 0) equals the chemical potential**. This definition and our estimates for the self-energy imply that the operator acting on G in Dyson's equation (9) is of order (small)' for small E , and p near the Fermi surface [because leading terms of order (small)" cancel!]. Hence, G in this range is of order (small)-', which agrees with our initial estimate. and demonstrates the consistency of our order of magnitude arguments.

ere

(a)

(0

(b)

Fig. S. The threc electron self-cnergy diagrams of order (small)'.

The only other diagram o f order (small)' is shown in tig. 8c. It has a weak frequency dependence a@ IS unaffected by the superconducting condensation. Hence. it has the same properties as I,, but is of higher order in (small). and can be neglected as compared to Z,,. * * More precisely. this means that T + 2,,- p has a zero energy eigenvalue: ( - ~ : : 2 - p ) $(r) + j ~ ~ . i , ~ ~ , ( x . x$ '( x: o' ) )= o . for a Blotch function I)(x) which satisfies $(I

+ R:)

= exp(ipRy) ~ ( x )


Ch. 4, 831

391

A

2 =

+

+

Fig. 9. The phonon and electron self-energies in leading order in (small). The diagrammatic notation is explained in figs. 4 and 7.

We are finally left, as a result of our order of magnitude analysis, with one phonon and three electron self-energy diagrams which represent the leading terms in an expansion in 'small'. These diagrams are shown in fig. 9. In sections 3 and 4 I discuss methods for evaluating these diagrams, which are the basic diagrams for an ab initio theory of superconductivity.

3. The low-energy equations

The diagram analysis of section 2 has led to a closed theory for the low-energy Green's functions, and hence for T , , provided one knows the four high-energy vertices that enter the leading self-energies (see fig. 9). Insertion of the self-energies, displayed diagrammatically in fig. 9, into Dyson's equations (9) and (10) yields a coupled system of equations for the low-energy parts of G and D.Equation (10) for D does not contain G l o w . Its solution is a perfectly settled problem of the theory of lattice dynamics (for a recent review see Dederichs and Zeller 1980). Therefore, 1 concentrate on the equations for GI"", assuming D to be given. The following formal manipulationson Dyson'sequationfor GI"" are necessary forestablishingthe connections to band-structure theory and to the strong-coupling theory of superconductivity. The final goal is a complete formulation of the theory of superconductivity in terms of functions on the Fermi surface. This reduces a 3D problem to a 2D problem, which is very useful for practical calculations. I start from Dyson's equation with all leading order self-energies taken

[Ch. 4. 53

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into account:

where a.-' and H,, collect the terms in Dyson's operator which are unaffected by the superconducting condensation:

H , , ( x ,x ' ) = (-V1/2.- p ) S(x - x ' ) + &(x.

1';0)

.

(20)

Equation (19) defines a - 1 , the inverse* of the 'quasiparticle renormalizfor noninteracting ation factor' a ( x , 1').n ( x , x ' ) is equal to 6(x electrons or in the Hartree-Fock approximation. Hence, a substantial deviatkn of a ( x , x ' ) from 6(x - x ' ) is an indication of strong correlation effects in the electron system. Now eliminate a(x, x') formally from Dyson's equation by absorbing it into the high-energy vertices and by a redefinition of G . This procedure leads t o a new Dyson's equation which resembles that of uncorrelated particles, and can be interpreted as Dyson's equation for quasiparticles. The new high-energy vertices in this equation describe quasiparticle interactions and the quasiparticle-phonon coupling. The trick is to define the following renormalized quantitiest: XI)**

* The inverse is defined by d'x" a '(x. I")n(x". I')= fi(x - 1'). * * This corresponds t o thc familiar result. a = 1. in the momentum representation for translationally invariant systems. In these systems (J( p F ) gives the jump in the momentum distribution function at p F .

?The factors i ,are an inessential convention.

Ch. 4, 931


393

The use of these renormalized functions turns Dyson’s equation (18) into the desired form:

One can call h , the ‘ideal band-structure Hamiltonian’. Band-structure data obtained from this Hamiltonian are exactly those needed in the theory of superconductivity*. It is common to write a band-structure Hamiltonian as a sum of the one-particle kinetic energy and a crystal potential: h,(x, x’)

2

= (-V,/2

- p ) S(X

-

x‘)

+ uo(x, x‘) .

(25)

Equation (25) defines u,(x, x’), the exact crystal potential (excitation potential) for calculating conduction electron properties. I emphasize that this crystal potential is guaranteed to be physically meaningful only in the small momentum shell around the Fermi surface. There is an intimate relation between the excitation potential of eq. (25) and the electron-phonon vertex in the self-energy 6.“’. In order to derive this relation (eq. 29) we first consider the high-energy electron-phonon vertex, defined in fig. 7c. I denote it, in position representation, by reP(x,x’lnv). rep is given by the derivative of the two-point vertex 2, (shown in fig. 7a) with respect to the ion position R::

rep(& x’lnv) = 62,(x,

x’; O)/SRH”.

(26)

This relation follows directly from the diagrammatic definitions of rep and 2,. The renormalized electron-phonon vertex y e pdiffers from rep by two quasiparticle renormalization factors: yep(x,X’ 1 nv)

=I

d3y d3y’~ ‘ ’ ~ (yx) rep( , y, y’ 1 nv) ~ ” ~ ( yx’) ‘ , . (27)

This definition of yepis sensible since it allows us to write the renormalized self-energy bepin terms of 8 in standard form:

* The same Hamiltonian also governs low-frequency [of order (small)] transport properties of a metal.

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[Ch. 4 . $3

Equation (28) can be represented by the same diagram (fig. 8b) as thc original self-energy Xip. One just has to interpret the electron Green’s function line as a renormalized ( g ) line, and the cross-hatched vertices as renormalized ( y e p )vertices; the phonon Green’s function is identical in both interpretations. Equation (26) and definitions (20), (22), ( 2 5 ) , and (27) imply the following useful relation*:

I

y y x , x ‘ nv) = 6u,,(x, x ’ ) / G R ; , .

(29)

This is an important result. The coupling of conduction electrons to phonons is determined by the change in the excirarion potenrial when the ions are statically displaced from their equilibrium positions. Hence, a band-structure calculation, if it is based on the correct potential, includes all many-body effects and gives the exact electron-phonon coupling parameters needed for calculating T,, except for small corrections of order h ~ ~ , ,E, ~ ,or , l kT,l E,.

3.1. BLOCH-FUNCTION REPRESENTATION The band-structure Hamiltonian h,, is defined in eq. (22). Taken for the equilibrium R:’s, it determines the energy spectrum and the wave functions of the conduction electrons. These one-quasiparticle data can be calculated from Schrodinger’s equation for a single particle in a periodic (in general nonlocal) potential:

i is a band index, i,b(xlp, i) are properly normalized Bloch functions = I). and & ( p ) is the energy of the conduction electron measured from the Fermi energy. Equation (30) is meaningful only in the small low-energy shell around the Fermi surface (FS) which is given by:

(Ice,,d’xj$J’

FS = { p , I3i:t,(p,.) = O} .

(31)

‘The derivation of eq. (29) from eq. (26) takes a few steps:

The last two terms contribute only in order (small) to matrix elements with conduction electron states near the Fermi surface [/I,,$ = (small). $1. and can be neglected.

Ch. 4, $31

A B INITIO CALCULATIONS OF T ,

395

A typical Fermi surface and its low-energy shell are sketched in fig. 10 as an illustrative example. The band-structure Hamiltonian is diagonal in the Bloch-function representation, which in turn leads to an easy solution of the low-energy equations. I will transform in the following Dyson's equation for the low-energy Green's functions into the Bloch-function representation, and solve them formally. Of course, the practical solution needs first solving explicitly Schrodinger's equation (30) as well as the calculation of matrix elements with the eigenfunctions of h,, (see section 4). It is very convenient, particularly in the theory of superconductivity, to replace the conventional momentum-band variables (p, i) by the set (s, 0, where 6 is the energy ( 6 = (,( p)) and s a two-dimensional Fermi surface variable. The lines of constant s are normals to the Fermi surface (see fig. 10). Phase space integrals in the low-energy range (LE) are most easily evaluated in terms of s-6 variables;

The cut-off in momentum space( t&) is small on the scale of band energies but commonly chosen to be large compared to the frequency cut-off (w, 4 6, 4 l), for practical reasons to be explained later. N ( 0 ) n(s) is the Jacobian of the transformation from (p, i) to (s, 6 ) * . N ( 0 ) is the bandstructure density of states at the Fermi energy ( 6 = 0),

Fig. 10. Two-dimensional sketch of the Fermi surface and its surrounding low-energy shell. The dashed lines are surfaces of constant (( = ?&); the thin solid lines are lines of constants.

* The Jacobian is, in general, (-dependent. This dependence is on the high-energy scale, and can be neglected in the low-energy range to leading order in (small).

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and n ( s )d's is the fractional contribution to N ( 0 ) of states within d's of the Fermi surface point s. The s-[ representation will be used extensively in sections 3 and 3. We will need the Green's function and the electronphonon vertex in the s-[ representation: d7.rd3X' $*(XIS,

[ ) g ( ~x ,' ; E , ) $ ( x ' ~ s 5, )

(34)

Here and in the following, N denotes the number of unit cells in the crystal. The [-dependence of I"' has been omitted. It can be neglected in the low-energy range. It is not necessary to know the full [-dependence of g. The theory of superconductivity can be formulated completely in terms of the so-called [-integrated Green's function i ( s ; E , ) (Eilenberger 1968, Larkin and Ovchinnikov 1968), defined by

The advantage of using the (-integrated functions is substantial for practical calculations. One has eliminated one variable, and is left with a theory which involves only functions at the Fermi surface. The solution of Dyson's equation (24) is trivial in the s-5 representation since c?~', C?." and g are diagonal in this representation. One finds: R(s; F , , ) = E,--* lim

J

I

d t iE,l?, - 5 - &"(s) - GCp(s;E , ) c,

'

(37)

where

aCP(s: = T 2 E , ~ )

d's' n ( s ' ) h ( s , s'; wn,)i ( s ' ;

W,,

with A($. s'; w ) defined by

E,

- w,,,) ,

(38)

Ch. 4, $31


397

The last quantity to be discussed in this section is the self-energy &", which describes 'molecular fields' acting on the conduction electrons. &ee is the renormalized version of (see eq. 23) which is shown in diagrammatic notation in fig. 8a. i'f"jnvo1ves the four-point high-energy vertex f ", defined in fig. 7b. It is sensible to renormalize this vertex by attaching quasiparticle renormalization factors to each of its external legs. This defines the renormalized vertex y e e :

eye

a, = f , .1 denotes thespinvariablesofelectrons.Two1imitsofthisvertexoccur in the standard theory of superconductivity, namely the particle-hole vertex and the particle-particle vertex (see fig. 11).Only the particle-particle vertex enters T,, and shall be discussed here*. One needs the following particular matrix elements of this vertex with Bloch functions at the Ferrni surface:

Fig. 11. The two limits of the high-energy electron vertex which appear in the theory of superconductivity. The particle-hole vertex (a) has zero total momcntum in the particle-hole channel. It enters thediagonalself-energy asshowninfig. (a). the particle-particlevertex (b)has zero total momentum in the particle-particle channel. It enters the off-diagonal self-energy ,as shown in (b).

* The particle-hole vertex is intimately related to Landau's parameters, which determine Fermi liquid effects in the conduction-electron system.

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[Ch. 4 . $3

Again. the [-dependence of the matrix elements can be neglected in the low-energy range. Here, - s denotes the state time reversed with respect to s. The matrix elements. made dimensionless by a factor N ( O ) , are traditionally called the 'Coulomb pseudopotential' ( p * ) in the theory of superconductivity. Spin-rotational invariance allows us to decompose p * into singlet and triplet components:

For ordinary singlet superconductivity 7,depends on ps alone, and one can drop the index S. The self-energy 6'' is given in terms of p* ( s, s ' ) by: Ci"(s)

=

I

d's' n ( s ' ) p * ( s . s ' ) 7

c'

f(s':

F,,)

,

(43)

t-,,

where f a r e the components of g which are off-diagonal in the particle-hole index. The frequency sun1 must be cut off at our low-energy cut-off w,. This is indicated by the prime on the sum. The cut-off dependence of the sum is compensated by an intrinsic cut-off dependence of p * , such that T , and all other physical quantities are independent of w,. The cut-off dependence of p * will be discussed in more detail in section 5. I have defined in section 2 and 3 the central quantities of an ab initio theory of superconductivity. The definitions were justified by arguments based on the smallness of typical energies of the superconducting state (low energies) as compared to typical normal state electronic energies (high energies). All relevant high-energy effects were assembled into three high-energy quantities: ( 1 ) The phonon self-energy T ,which determines the complete lattice dynamics [ r is defined diagrammatically in fig. 7d and in eq. (16)]; (2) the crystal potential (excitation potential) u,,, which determines the energy spectrum and the wavefunctions of conduction electrons, as well as the electron-phonon coupling yep[uo and -yep are defined by eqs. (25) and (29), and figs. 7a,c]; (3) the Coulomb-pseudopotential p * [defined by the diagrams in fig. 7b and eq. (41)].

Ch. 4. $31

AB INITIO CALCULATIONS OF 7,

399

These quantities enter some rather simple equations of the one-particle type, eqs. ( l o ) , (30), (37), (38) and (43), which govern the low-energy properties of a metal, and allow for the calculation of T,. Methods for solving the low-energy equations are discussed in sections 4 and 5. The prospects for their complete and controlled solution on a computer are excellent. However, there is very little knowledge of how to calculate the needed high-energy quantities, which are all ground-state properties of the electron liquid immersed in the rigid background of the ionic charges. I will close this section with a few comments on high-energy quantities. An important part of the crystal potential is the bare ion potential screened by the electrons (of density p ( x ) ) : u ( x ) = u:l‘’(x)

’I

+ e-

d3x’ p ( x ’ ) / ( x - x’( .

(44)

The rest, which comprises all subtle many-body effects, is conventionally called exchange-correlation potential (u,,): U()(X, w‘) =

u ( x ) S(X

-

x‘)

+ UJX.

x’) .

(45)

Exchange-correlation (xc) potentials have been discussed extensively in the context of the density-functional scheme (Hohenberg and Kohn 1964, Kohn and Sham 1965, Sham and Kohn 1966; for a review see Lundqvist and March 1983, and Dahl and Avery 1983). This method collects all many-electron effects into a ‘ground-state potential’ whose xc-part is usually denoted by p,,t. The ground-state potential is a universal functional of the electron density. By the density-functional method one can calculate selfconsistently the electron density and the total energy of the electrons at any (frozen in) position of the ions. Hence, the ground-state potential can be used to calculate the equilibrium crystal structure, the phonon self-energy (from eq. 16), and the change in electron density when the ions are displaced from their equilibrium positions. Such calculations are part of a complete ab initio theory of T,. Conduction electron properties and the electron-phonon coupling can, in general, not be obtained from the ground-state potential but are connected with the ‘excifution potential’ u,(x, x’). uo is also a universal functional of the density. The ground-state and the excitation potentials differ, in principle. However, it is not known if this difference is important for practical purposes (Overhauser 1974). The excitation potential has been studied much less intensively than the t p , , is a local potential; p,, = p x , ( x )6(x - x ’ )

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(Ch. 4, 93

ground-state potential. The review by Hedin and Lundqvist (1969) on many-body calculations of the xc-potentials covers the developments in the field up to 1969. All tirst-principles calculations of the xc-potential for excitations employ the GW-model (von Barth and Hedin 1974), which approximates the self-energy processes by the emission and reabsorption of collective modes, such as density fluctuations or plasmons (Lundqvist 1968, von Barth and Hedin 1974, Rietschel and Sham 1983, for an elaborate calculation see MacDonald et al. 1980), and spin fluctuations (Doniach and Engelsberg 1966, Berk and Schrieffer 1966, Lundqvist and Wilkins 1973, Rietschel et al. 1980; for a more involved calculation see MacDonald 1981). The characteristic GW-diagram is shown in fig. 12a. The GW-model also provides a scheme for calculating p * (see fig. 12b). The few first-principles calculations of p *. or of conceptually related quantities, are based on this model. For the mode W they use either the spin-fluctuation propagator (Berk 1966, Levine and Valls 1978, Rietschel and Winter 1979) or the dynamically screened Coulomb interaction (in RPA) Masharov 1981, Rietschel and Sham 1983, Grabowski and Sham 1984, Iordatti et al. 1984). GW-calculations are about at the limit of present capabilities of manybody calculations in the traditional Green’s function scheme. It should be noted that there is no deep justification for the GW-models. No firstprinciples arguments have yet been found for neglecting vertex corrections or the multiple exchange of collective modes (see fig. 12c). It is conceivable that the Green’s function technique is not the most practical scheme for many-body calculations of u,, and p * , and that a solution of these problem will be achieved by alternative many-body techniques (Fantoni 1981. Emrich and Zabolitzky 1984, Hirsch and Scalapino 1986). Many-body theory is challenged to develop new ideas for solving this part of the ah initio theory of T,.

A G

IC) Fig. 12. Electron self-energy (a) and (irreducible) particle-particle interaction (h) in the GW-approximation. (c) shows typical diagrams neglected in the GW-approximation.

Ch. 4, 541


40 1

4. Band-structure theory and the electron-phonon interaction Coherent multiple scattering of electrons by the (periodic or non-periodic) crystal potential strongly affects the energy spectrum and the wave functions of the electrons. These ‘band-structure effects’ are of primary importance for understanding the dependence of T, on crystal structure and chemical composition. Band-structure effects control the phonon and conduction electron properties, which both influence T,. Consequently, a band-structure calculation is an indispensable constituent of an ab initio theory of T,. Band-structure theory is often misunderstood as a one-electron theory which ignores electron-electron correlations. The idea behind modern band-structure theory is to include all relevant many-electron effects by means of a sophisticated choice of the crystal potential. The choice of a potential might depend on the physical quantity of interest. A calculation of the ground-state energy, and of properties derived from it, can be based on the ground-state potential of the density functional theory; an ideal calculation of conduction electron properties and the electron-phonon coupling requires the ‘excitation potential’ (see section 3). Several approximate potentials, or better, construction schemes for potentials are presently in use. This includes empirical pseudopotentials (Harrison 1966; for recent applications in the theory of T , see Whitmore 1981, Jaffe and Ashcroft 1981, Perlov and Fong 1984), xa-potentials (Slater 1979; for an application in the theory of T , see Harmon and Sinha 1977), and self-consistent local density potentials (see e.g., Koelling 1981). None of these approximations has a full first-principles justification. Hence, an important mission for band-structure theory is to check competing potential recipes by calculating a large variety of experimentally accessible data. Knowing the capacities and, more important, the limitations of various recipes leads to a step by step improvement, and finally to a universal and useful construction scheme for crystal potentials. Superconducting data are especially well suited for this purpose. The ab initio theory of T , needs from band-structure theory a calculation of: (1) the equilibrium structure of the crystal; (2) the dynamical matrix for phonons; (3) matrix elements of the electron-phonon potential with conduction electron states at the Fermi surface, and certain Fermi surface averages. Points (1) and (2) refer to ground-state properties of electrons in the rigid lattice. These problems can be treated successfully by means of the density-functional theory. I refer to several reviews (Lundqvist and March 1983, Dahl and Avery 1983; for a recent review of first-principles lattice

402

D. RAINER

[Ch. 4, 04

dynamics see Weber 1984) for an extensive discussion of this field. The present status of first-principles calculations of crystal structures and phonons is quite satisfactory, especially in view of the insensitivity of T , to fine details of the phonon spectra (Allen and Dynes 1975). The art of calculating electron-phonon coupling parameters is less advanced. The resuits obtained by different methods do not yet agree satisfactorily (an instructive comparison was given by Perlov and Fong 1984). I will concentrate in the following on the electron-phonon interaction and its band-structurc aspects. This review does not cover the many details of modern band-structure techniques, like the advantages and disadvantages of various sets o f basis functions, and the specific conceptual and numerical tricks which speed up the solution of Schrodinger's equations and the calculation of matrix elements ( I refer to recent reviews on band-structure theory by Mackintosh and Andersen 1980. Koelling 1981, Williams and von Barth 1983, and contributions by O.K. Andersen. M.L. Cohen, J.C. Phillips, and M. Schliiter to 'Highlights of Condensed Matter Theory' (Bassani et al. 1985)). I will simply repeat the definitions of various electron-phonon parameters which are of interest in the theory of T , , and discuss the most frequently used approximations for calculating them. 4.1. THEELECTRON-PHONON

COUPLING PARAMETERS

The starting point of this section is a recipe for calculating the ground-state potential and the excitation potential* from the electronic density p ( x ) . This is the basis for calculating all electron-phonon coupling data of interest in the theory of T,. The most intricate parts of the recipe are the exchange-correlation potentials for the ground-state ( p x c )and the lowlying excitations ( u x c ) :

For a given potential recipe it is a routine job for band-structure theory to compute self-consistently the density p ( x ) (from the ground-state potential), and the energy bands and Bloch functions (from the excitation potential) for a periodic crystal. A somewhat more involved problem is the

1 continue distinguishing the ground-state and cxcitation potcntials although they are considered identical in all practical calculations.

Ch. 4, 541


403

self-consistent calculation of the change in density 6 p ( x ) when an ion is slightly displaced from its equilibrium position by an amount SR,. Both calculations are, in principle, necessary for computing the electronphonon coupling parameters. 6 p ( x ) determines the screening of the bare electron-phonon potential (7), and the exchange-correlation contributions to the electron-phonon potential via the functional dependence of u,, on the density. The total electron-phonon potential yep(x,x’; nv) whose matrix elements determine the Eliashberg function a 2 F ( o ) is the sum of the screened dipole potential y zyp, and the exchange-correlation potential y which simulates all many-body effects in the electron-phonon coupling:

:,”

x 6(x

-

x’) ,

(48)

yzyp can also be expressed in terms of the inverse dielectric function E-’(x, and the fundamental electron-phonon interaction (7) (see, e.g., Hanke 1978): XI)

The xc-part in yep is neglected in any RPA-type approach to the electron-phonon coupling (Cohen and Anderson 1972); the consequences of this approximation are not yet known [a discussion of exchange effects has been given by Kim (1978)l. A fully self-consistent calculation of the electron-phonon potential is probably within the reach of present computer capacities, but has not yet been done. A very often used approximation which avoids the costly calculation of Sp(x) is the rigid-ion approximation. This method is reviewed in detail in chapter 4.7 of Grimvall’s (1981) book. The rigid-ion approximation assumes that the total crystal potential can be decomposed into a sum of terms associated with each ion*, * For simplicity, I usc local ion potentials: u,(x. x’) = u,(x) 6(x - x’), ycp(x,x’ I nv) = y‘P(xI

n v ) b ( x - 1’).

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and that the potential is carried rigidly with the displaced ion:

Specific versions of the rigid-ion scheme are the rigid muffin tin (RMTA) or the rigid atomic sphere (RASA) approximations. One assumes that the potentials (and the charge density) follow rigidly the displacement of an ion in its Wigner-Seitz cell, and are unchanged outside this cell (perfect screening) : -av,,(x)/dx,

x in cell n ,

elsewhere.

(53)

The rigid-ion approximation is at least a useful starting point for a full calculation of yep(Winter 1981); it seems to be a good approximation €or transition metals (Klein and Pickett 1982). However, a systematic numerical study of the accuracy and the limitations of rigid-ion approximations is still required in order to put this popular approximation on a firm basis, or to improve it whenever necessary. The calculations by Winter (1982) show that such studies are feasible, and will probably be completed in the near future. The next step, after having determined the electron-phonon potential y '", is the numerical computation of the matrix elements of y e pwith Bloch wave functions at the Fermi surface, and subsequently the calculation of the Eliashberg function a * F ( w ) . The function a'F(w) is the spectral function of the averaged electron-phonon interaction function A( w m ) , A(w,,,) = 2

IOx

dw a'F(w)

,

w-

w

+ w;

(54)

A comparison of eq. (54) with eqs. (91) and (39) leads to a formula for a * F ( w )in terms of local quantities, such as the coupling matrix elements to displaced ions at R: and R L , and the propagator of a lattice distortion from

RZ to R t , : a ? F ( w )= N ( 0 )

I

FS

d's d*s' n(s) n ( s ' )

Ch. 4, 841

AB INITIO CALCULATIONS O F T,

405

The matrix element Zep(s, s ’ l n v ) , which is defined in eq. (359, describes the coupling to a displacement of ion n in direction v(v = x, y, z). B is the spectral function of the phonon Green’s function D [see eq. (12)],

D(nv, m p ; w i ) = 2

w

dw B(nv, m p ; o) 0

+wi

.

Alternatively, after diagonalizing B , a’F(w) can be written in terms of coupling matrix elements g( p, p ‘ ; j ) to a phonon mode with wavevector Q = p’ - p, branch index j , polarization vectors E,,(Q, j ) and frequency w(Q, j ) : t

wherett

fl=

I

x yep(X,x‘; nv) @(XI I

Nu

,i) .VV,”(X-

R:)

Here +(x I p) are Bloch wave functions at the Fermi surface+*,normalized to 1 in a unit cell. The first and second lines of eq. (58) are the exact result for 1 gl’, whereas the last lines present the rigid-ion approximation only. t For convenience, I use several different notations for the same Fermi-surface integral, namely

t t T h e sum C?, is over all atoms in the unit cell. The summation can be dropped for a simple lattice without a basis. t’ I drop, as usual, the band indices which are redundant at the Fermi surface since p already characterizes the state completely.

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[Ch. 4, 94

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Commonly, a numerical calculation of a’F(w) is based on eqs. (57) and (58). The two essential steps are the computation of the coupling matrix elements (58).and the double Fermi-surface averaging (57). A calculation of the matrix elements can be done by several different methods, depending on which band-structure technique has been used to generate the wave functions. For instance, in pseudo-potential schemes the matrix elements are given directly in terms of pseudo-potential form factors, whereas tight-binding schemes connect I gl to a few matrix elements with localized orbitals. Pseudo-potential calculations of the full spectral function a ’ F ( w ) have been published by Carbotte and Dynes (1968), Tomlinson and Carbotte (1976), Tomlinson and Swihart ( 1979), Jain and Kachhava (1980). Whitmore (1981), Jaffe and Ashcroft (1981), Ashraf and Swihart (1982), MacDonald and Leavens (1982). Perlov and Fong (1984); tight-binding calculations for a * F ( w ) have been published by Peter et al. (1977) and Simons et al. (1981). The most powerful scheme for calculating (g1’ was devised by Gaspari and Gyorffy (1972) (see also John et al. 1980, and Pickett 1982). Their method is presently used in all APW (Harmon and Sinha 1977, Veprev and Shirokovskii 1980), KKR (Butler et al. 1977. Pinski et al. 1978, Butler et al. 1979, Pinski et al. 1981). and LMTO (G16tzel et al. 1979. Glatzel et al. 1981) calculations of a ’ F ( w ) . A nonlocal version of the Gaspari-Gyorffy scheme (Rietschel 1978) has been employed by Winter et al. (1978), Rietschel et al. (1980), Schell et al. (1982). The Gaspari-Gyorffy analysis starts from the rigidion approximation for spherical symmetric muffin-tin potentials, and uses the following representation of a Bloch wave function inside the muffin-tin spheres:

4(Xl P) =

c c B , (P.Rj:)p ; ( r ) ’I

YL(i)

I

3

(59)

where L represents the angular momentum quantum numbers (I, m). Y, ( i )is a spherical (or cubic, etc.) harmonic, and p y ( r ) the radial part of the (nonsingular) solution of Schrodinger’s equation in the muffin-tin sphere around Rj:; r = Ix - RRI, i = (w - R R ) / r . p ; vanishes outside its muffin-tin sphere. Representation (59) allows us to write lgl’ in terms of radial matrix elements 111 , standard angular integrals (Y, (ilY, . ) =

I

d’i Y t ( F ) Y , , , ( i ) i,

and the expansion coefficients B,: , .VM

Ch. 4, 141


407

The angular integrals lead to the well known selection rule 1 - I' = ? l . Gaspari and Gyorffy have shown that the radial matrix element

can be expressed in terms of the radial wave functions u / ( T )= r p ; ( r ) (or alternatively by the logarithmic derivative D,= rp;, / p y ) at the muffin-tin radius rsn (of atom n):

or equivalently

z:'+'= sin(',

- a,+ I ) .

(63)

to the phase shifts of the muffin-tin Equation (63), which relates '":Z potential at atom n, holds for the special normalization

(Evans et al. 1973). The Gaspari-Gyorffy method leads to significant technical simplifications [the costly integral in (58) is replaced by a sum over a few terms], and allows for a physical interpretation of trends in the electron-phonon coupling strength (Pettifor 1977). Thorough discussions on the limitations of this method have been published by Pettifor (1977), and Zdetis et al. (1981). Clearly, the Gaspari-Gyorffy method (and also the pseudo-potential or tight-binding methods) permits a very fast calculation of the coupling matrix elements. The much more time consuming part of a calculation of a 2 F ( w ) is the remaining double Fermi-surface integral in eq. (57). As I will outline below, explicit calculations of this four-dimensional integral by several groups have demonstrated that the numerical integration can be done with controlled accuracy, and within reasonable computation time. Although various different strategies and numerical techniques have been used to solve this problem, a generally accepted 'best strategy' has not yet emerged. There is, therefore, still room for new ideas and substantial technical improvements. Most calculations start from formula (57) and split the full integration

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into two conveniently chosen steps. One method (Butler et al. (1977), Pinski et al. 1978) is to calculate first the function

r(Qli) = 2 4 ,

3

~ ( Z ( P ) ~) ( Z ( P+ Q ) ) I d P , P + Q ; ; ) I 2

4Q, i)

I

(64) which amounts to doing a one-dimensional integral over all pairs of Fermi surface points with distance Q. y ( Q , j ) is the phonon line width due to electron-phonon interactions. The function a ‘ F ( w ) can then be obtained from y ( Q , j ) by the 3D integral

This scheme for calculating a 2 F ( o ) and, as an intermediate step, the phonon lifetimes has been introduced by Allen (1972). Another method, which also has a physically relevant by-product, is to calculate first

and then

Calculations of a’F(w) via a ’ F ( p , w ) have been performed by Tomlinson and Carbotte (1976), Harmon and Sinha (1977), Veprev and Shirokovski (1980). a21;(p,w ) determines the local mass enhancement at various points (or orbits) on the Fermi surface (Crabtree et al. 1979; for a recent experiment see Joss and Crabtree 1984), and allows the authors to discuss the relative contributions of different parts of the Fermi surface to a 2 F ( o ) . A third. somewhat different scheme was presented by Glotzel et al. (1979). The authors start from the ‘local’ formula ( 5 9 , and decompose a‘F(w) into contributions from pairs of atoms (at R : , RY,):

Ch. 4, 541


409

where q(nv, m p ) = N(O) J F S d2s d's' n(s) n ( s ' ) Zep(s,s' I nv) Zep(s',s I m p ) (69)

is a generalization of the single-atom parameters q, of McMillan (1968) and Hopfield (1969) to pairs of atoms. The numerical results show that T , is rather insensitive to pairs at a larger than nearest-neighbor distance. The most ambitious and complete calculations of the band-structure input into Eliashberg's equations are by Peter, Ashkenazi, and Dacorogna (1976) (see also Peter et al. (1982)), and by Butler (1980). Peter et al. partition the Fermi surface by dividing it into pieces (22 in the irreducible zone), calculate the piece to piece coupling functions, and solve the anisotropic Eliashberg equations. They obtain information on local mass enhancements, the clean and dirty limit T,, and the gap anisotropy. The work by Butler also contains an ab initio calculation of the full anisotropic a 2 F function, and, based on this result, a very detailed strong-coupling analysis of the anisotropic critical fields of Nb. My brief review of the various approaches to calculate a 2 F ( w ) from band-structure data does not cover the many conceptual and technical details which are vital to a successful calculation. I refer to the original publications for these details. The exact theory of T , needs a full a 2 F ( w ) calculation. However, the work of McMillan (1968), and Allen and Dynes (1975) (see also Kahn and Allen 1980) indicates that T, can already be estimated from a few moments of the phonon spectral function F ( w ) , and the McMillan-Hopfield constants q,, q, = N ( 0 )

2 Id's

d2s' n(s) n(s')lZeP(s, s'lnv)12 .

Y

The parameter 77, describes the averaged squared coupling matrix element to displacements of the ion n. One finds from eq. ( 5 5 ) , and the sum rule 2

I I

dw wB(nv, m p ; w ) = Sn,mSv3plMn ,

the following relation between the first moment of a 2 F ( w )and the 7,'s: 2

dw w a 2 F ( w )=

From the definition

No

2 q,lM, n-1

.

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one obtains McMillan’s exact (by definition) relation between the electron-phonon mass enhancement A = 2 I d w a ‘F( w ) / w and the McMillanHopfield constants.

The short-cut calculation of A (or T,) assumes that ( w ’ ) can be computed approximately by replacing a Z F ( w )by the phonon spectrum. In this way, one can bypass the costly full calculation of a ’ F ( w ) . The reduction in computation time is appreciable, since q,,can be obtained directly from standard output of band-structure routines by using a formula of Gaspari and Gyorffy (1972) for v,, within t h e RMTA:

Here, 6, are the phase shifts at the Fermi energy of the muffin-tin potential at lattice position R:. The factors v, = N,(O)/Ni” describe the partial crystalline density of states (in cell n ) normalized by the corresponding single scatterer density of states. Alternative formulations of eq. (74) have been discussed by Pickett (1982). Most of the band-structure work in the theory of T , follows the short-cut approach to T , (the most elaborate study is by Papaconstantopoulos et al. (1977), who calculate T , for 32 metals with 2 S 49). I refer to an excellent review by Klein and Pickett (1982) for a thorough discussion of the RMTA. a comparison of the results by various groups, and a complete list of references up to 1982. The short-cut version of an ab initio theory of the electron-phonon coupling permits computer studies of essentially all compounds of interest. Typical recent publications are on A15-compounds (Jarlborg et al. 1983), metal hydrides (Gupta 1984), technetium (Chatterjee 1983), lanthanum and actinium under high pressure (Dakshinamoorty and Iyakutti 1984), C15-compounds (Klein et al. 1983), and LaS (Vlasov et al. 1984). The short-cut calculations of T, work well if the shape of the a Z F ( w ) function resembles F ( w ) ; in other words, if the coupling function a ’ ( w ) (=a’F(o) / F ( w ) ) has little structure. It is conceivable, however, that high- T , superconductivity is produced by atypical situations, e.g. by an abnormally strong coupling to favourable (low-frequency) phonon modes (Hanke et al. 1976). The consequences for T , of shifting the coupling

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AB INITIO CALCULATIONS O F T,

411

, .2

.4

.6 .8

QL/Q Fig. 13. Dependence of T, on the shape of the a ’ F ( o ) spectrum at fixed first moment. The curve was calculated for a fixed phonon-spectrum F ( w ) and McMillan-Hopfield constant q. The model spectrum F(w) is shown in the insert. The amount of coupling to longitudinal phonons (qL) and transverse phonons (qr) is changed gradually from pure transverse coupling (qL./q= 0) to pure longitudinal coupling (ql./q = 1). The sum ql, + qT = q is kept constant.

strength from (low-frequency) transverse modes to (high-frequency) longitudinal modes is shown in fig. 13. One can see that this shift of coupling strength at fixed phonon spectrum and McMillan-Hopfield constant produces sizable (a factor of 4 in our example) changes in T,. Hence, an estimate of T , from the phonon spectrum and the 77,’s alone is, in general, insufficient. In order to gain insight in the mechanisms of high- T, superconductivity it is probably necessary in the future to perform full size a ’F(w) calculations. The techniques for such calculations need further improvement before a full a’F(w) calculation can be done routinely, and one can start a systematic computer study of high-T, superconductivity.

5. Strong-coupling theory of the transition temperature

The strong-coupling theory of superconductivity (Eliashberg theory) solves the low-energy problem of conduction electrons coupled to phonons (Eliashberg 1960, 1961). All high-energy effects are absorbed into effective coupling constants and conduction electron parameters*. This theory, * Recent claims that some high-energy correlations, e.g. spin-fluctuation effects, must still be added to the conventional Eliashberg theory are, in my opinion, incorrect.

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[Ch. 4. 65

which is exact in leading order in the small parameters fiwphonlEFand kT,l E,. . does not need kT, to be small compared to the typical phonon energy. Hence the strong-coupling theory is appropriate for studying hypothetical high-temperature superconductivity (as long as kT, is small on typical electronic scales), as well as conventional superconducting materials. The critical temperatures of presently known superconducting materials are small on the scale hwphon.All these materials are quite accurately described by the mathematically simpler weak-coupling theory (BCStheory. Bardeen et al. 1957) which is exact for small kT,lE, and kT,lAw,,,,, . Strong-coupling corrections to measureable properties of such conventional materials are typically of order 10%. and are often outside the experimental reach. such that a costly strong-coupling calculation is rarely justified. This general statement is no longer true if one is interested in the critical temperature. The weak-coupling result for T , ,

T, = l.13~,,exp(-l~gB,,),

(75)

is cxact and well known, but quite useless in an ab initio theory of T,. The problem of calculating Tc is simply shifted by this formula to calculating the BCS-coupling constant g,, . A calculation of gBCsfrom microscopic parameters requires, however, the solving of the strong-coupling equations, which takes the same efforts as calculating T , directly from these equations. Hence. it is best to skip the weak-coupling detour, and to base a calculation of T,, even for a weak-coupling metal, on the strong-coupling theory of superconductivity. A calculation of T, from Eliashberg's strong-coupling equations is a standard numerical problem which can be solved easily on a small size computer. I will describe in subsection 5.2 a numerical scheme for solving Eliashberg's equations for T , . This scheme, or other, equivalent ones (Bergmann and Rainer 1973, Leavens 1974, Allen and Dynes 1975, Peter et al. 1977, Louie and Cohen 1977). can be readily transformed into a computer code by anyone who needs to calculate T,*. Subsection 5.1 presents some useful notations and formulas of the strong-coupling theory of 7 c . and links this theory to the results of sections 2-4. 5.1. THELINEARIZED

ELIASHBERG EQUATIONS

The self-energy equations (38), (43). and eq. (37) for the conduction * I d o not recommend using baroque empirical formulas which have been obtained hy fitting a set of exact computer results to an exponential function (several formulas for Tc are reviewed in chapter 6 . 7 of Grimvall's (1981) hook).

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electron Green's function g form a complete set of coupled equations which determine g, &", and in the normal and superconducting states. The superconducting state is characterized by non-vanishing anomalous (off-diagonal in the particle-hole index) matrix components of g, Gep,or &ee. These components vanish continuously when T, is approached, and one can linearize the equations for g, Sep,and be' at T , with respect to the anomalous components. T , is found as the highest temperature at which the linearized equations have a solution with non-vanishing anomalous components. This method for calculating T, will be used here. There is an alternative method which starts from the Bethe-Salpeter equation for the two-particle Green's function, and determines T , as the temperature at which the pair susceptibility diverges (Gorkov and Pitaevski 1962, Luttinger 1966, Appel and Kohn 1971). This method calculates T , by looking for an instability of the normal state. Both methods lead to exactly the same linear equations. At this stage it is necessary to leave the compact Nambu-matrix notation, and to work with the matrix components of & and g. The 16 matrix components of g(s; E , ) can be written (I use the notation of Alexander et al. 1985) in terms of two scalar functionsg(s; E,,) andf(s; E , , ) , and two spin-vector functions g(s; E , ) and f(s; E , ) : *

eee

I will omit the vector components in the following. g can be dropped at T , , and f describes triplet pairing which will not be discussed in detail here. E,,) + &"(s) are The scalar parts of the total self-energy &(s; E , ) = eCp(s; conventionally expressed in terms of two functions En($) and 4,(s):

From eq. (37) one finds through first order in 4: g(s;

E,)

=

-in sign E, ,

(78)

* Remember that we need to know the components of g only at the Fermi surface. The variable s refers to a coordinate system on the Fermi surface, --s is time reversed point to s. and E, are the Matsubara frequencies.

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Inserting (78) and (79) into the self-energy equations (38), (43) finally gives the linearized Eliashberg equations: E,(s) =

F,

(for

E,

+ TT,

=

I

d%' n ( s ' ) h ( s , s'; E, -

f",

(2i + l)7rTc,

E,)

sign

F,

w, = 2m7rTc).

(81)

For completeness, 1 present the analogous equations for triplet pairing. Equation (80) is unchanged, and eq. (81) turns into

4,,(s)

=

nT,

c'Id's' n ( s f )(A(s, s';

E,

-

E " ~ )-

&(s,

s'))

Fnl

* 4m(wlE,(s')l

(82)

7

where &(s) is the triplet component of the anomalous self-energy. c$,(s) is an odd function of s whereas 4,(s)is even. Equations (80) and (81) are of different complexity. A calculation of the normal self-energy from (80) amounts to doing a Fermi surface integral and a finite sum over Matsubara frequencies. The more involved eq. (81) presents a coupled set of linear homogeneous integral equations for the anomalous self-energy &,(s). T , is the highest temperature at which eq. (81) has a nontrivial solution, or, tantamount thereto, at which the kernel K ( s n ; s ' n ' ) = T T , ~ z ( s ' ) ( A (5'; s,

has an eigenvalue 1:

E,

-

E,:) -

p * ( s ,~ ' ) ) / l E , ' z ( . ~ ' ) l

(83)

AR INIT10 CALCULATIONS OF Tc

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415

One can easily show that all eigenvalues of K are real*, and that the largest eigenvalue of K becomes 1 at T , (Rainer 1968). Hence, a calculation of T , from Eliashberg’s equations is a standard eigenvalue problem which can be solved by standard numerical methods (see also subsection 5.2). The dimension of the eigenvalue problem can be reduced substantially in the dirty limit where all anisotropies in +,(s) are smeared out by impurity scattering, and 4 becomes independent of the Fermi surface variable s. This ‘mean free path effect’ has been studied in detail by the theory of superconductivity (Markowitz and Kadanoff 1963, Allen and Mitrovic 1982). Most ab initio calculations of T , are done in the dirty limit**. Such an approximation is reasonable, because the mean free path effect on T, is usually much smaller than the uncertainties in the calculated T,’s. Because of its importance, I will briefly discuss a derivation of the dirty limit of the linearized Eliashberg equations. The generalization of eq. (79) to an impure metal is (I use the notation of Alexander et al. 1985) [E,,(s)

+ 1/2T(S)]f(S; E , )

= 7r sign

E,

-

$

I

d2srn(s‘)W(s,s‘) f(s‘;

E,)

4, (s) .

(85)

For an impure metal the solution f(s’; E ~ of) eq. (85) replaces the term &(s’) l E , ( s ’ ) on the right-hand side of Eliashberg’s equation (81). W(s,s’) is the transition rate from state s to s’. The lifetime of state s is T(S)9

I

d’s‘ n ( s ’ ) W(s,s ‘ ) .

1/ T ( s ) =

One can check that the impurity terms drop out of eq. (85) for an isotropic (s-independent) f. The solution of eq. (85) in the dirty limit [ ~ / T ( s ) + w ,W(s, s‘)+m] is: 7r

f(s;

En)

=

sign E,,

I

d’s’ n(s’) +,(sf)

4 n + O ( T ). + O ( T )= 7r sign E, T En

d’s’ n ( s ’ ) E , , ( s ’ )

(87) * K can be transformed into a hermitian kernel

1q-

K(si; s ’ j ) =

K by

K ( s i ;s ’ j ) ( q n ( s ’ ) i l . , ( s ’ ) l ) - ’ .

** A calculation including the full anisotropy has been performed by Peter et al. (1977). and utler (1980).

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[Ch. 4. 55

Hence, the dirty limit of f(s; E,,) term is isotropic, and determined by the Fermi surface averages of &,(s), i,,(s). The weighting in the averaging procedure is given by the density of states function n ( s ) . En

=

I

d’s n ( s ) E,(s) ,

4,[

=I

d’s n(s) &(s) .

(88)

By averaging eqs. (80) and (81) we obtain our final Eliashberg equations for the averaged self-energies E n , cb,, :

h(w,,) and p * are the averaged electron-phonon kernel and Coulomb pseudo-potential: A(wn,)=

I

d’s d’s’ n(s) n(s‘) A(s, s’; w,,,) ,

p * = / d b d’s’ n ( s ) n ( s ‘ ) p * ( s , s‘) .

(91)

(92)

Equations (89) and (90) are the simplest and most frequently used equations for calculating T, from Eliashberg’s theory. We can again interpret (90) as an eigenvalue equation. However, instead of an integral equation the dirty limit theory leads to a simple matrix-eigenvalue problem,

with the matrix K given by

Equation (94) has been studied numerically as well as analytically by several groups (a complete list of references is given by Allen and Mitrovic (1982) in chapter IV). Its formal mathematical properties and its physical consequences are largely understood. Some of these aspects are briefly discussed in subsection 5.2.

AB INITIO CALCULATIONS OF 7;

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5.2. CALCULATION OF T ,

FROM

417

ELIASHBERG’S EQUATIONS

In the dirty-limit Eliashberg theory, T , is a functional of the Eliashberg function a - F ( w ) and j~ *. The theory also contains a cut-off wcwhich can be chosen quite freely. p * depends on w c i , while T , is, of course, cut-off independent. An @,-dependence of T , from the summation cut-off in (93), and the cut-off dependence via p * ( w c )cancel exactly. The freedom in choosing w, can be used profitably in numerical calculations. A numerical routine for calculating T , needs the following three components: - a preparatory part to set up the kernel (94), i.e., an integration routine for calculating A(w,) from a * F ( w )(eq. 54), and the subsequent calculation of En (eq. 89); - a method to calculate the largest eigenvalue of the kernel; - a strategy to determine the temperature at which the largest eigenvalue becomes 1. Some typical eigenvalue spectra of kernels of Eliashberg’s equation are shown in fig. 14. A significant feature is that the physical (largest) eigenvalue is well separated from the unphysical ones, which accumulate at zero. Hence, for most practical problems, the iteration m

converges fast to the physical eigenvector 4n, and yields the physical eigenvalue. This method needs to be generalized only in exceptional cases, for instance, for a large p * where an unphysical negative eigenvalue becomes dominant. A typical eigenvector 4,, is displayed in fig. 15. Its characteristic features are a smooth structure which extends up to about five times the phonon energies, and a region of asymptotically constant #n

-.5

.5

0 I

1.

I

1

I

1

(6) Fig. 14. Eigenvalue spectra of Eliashberg’s equations at T, for an Einstein model ( a z F ( w )= j A o , 8 ( w - w , ) ) with A = 1.6;j~*(50,,) = 0.2. The two spectra are calculated for a cut-off of 20 wD (A), and 10 w , (H). The physical eigenvalue (=1) is cut-off independent. t Papers where authors calculate be treated carefully.

j ~ *without

stating their particular choice of w, should

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0

SW, ~

~

~0

1

I

0

-

[Ch. 4. 85

IOU,

E" .

0

.

.

.

0

0

.

0

0

.

0

0

l

j

-

Fig. IS. Frequency dependence of the solution of Eliashberg's equation at T'. The closed circles and crosses are. respectively, the off-diagonal and diagonal self-energy functions 6" and F,/e, at the Matsubara points E , = (2n + l ) n T , . The results were obtained numerically for an Einstein spectrum with A = 1.6. p * ( S w , , ) = 0 . 2 . They are representative for any other n?F(w).

at high energies. The cut-off in Eliashberg's theory must be put in the region of constant b,,. It is a simple exercise to show that the formula for resealing p* from a cut-off at wcl t o another at wc2is, for o , , ,S~ wphun, given by:

Traditionally. p * is listed for a cut-off of five times the maximal phononenergy. If, forconvenience, adifferentcut-off isused in acalculation, p * must be adjusted to the actual cut-off by eq. (96). p * diverges at a cut-off OC,

R, = w, exp( 1/ p * ( w , ) ) .

(97)

The energy 0, is a material parameter which characterizes p * ( w c ) = 1/ln(l&/w,) for arbitrary w,, without making reference to an artificially fixed standard cut-off. w, is restricted to w, < f2, in Eliashberg's theory. In the Matsubara technique, the temperature can not be varied continuously at fixed cut-off. The allowed temperature points are determined by the condition that n, = w C / 2 x Tis integer. This leads to a temperature mesh of width d T = Z . r r T f / ~ ,I.n general, T, will not be on the mesh. A possible strategy for determining T , at fixed cut-off is the following. One first finds the temperature points on the mesh near which the largest eigenvalue E m , , passes through 1. One then interpolates a few of these

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419

T-points to get E,,,(T) near T,, and calculates T, from E m a x ( T c=) 1. Other equivalent methods are described elsewhere (Bergmann and Rainer 1973, Leavens 1974, Allen 1974). A larger wc leads to a finer temperature mesh but increases the dimension n , of the eigenvalue problem, and hence the computation time. A standard choice of w, is 5-10 times the maximum phonon frequency. The above described scheme allows an easy and precise numerical calculation of T , from a 2 F ( o ) and p * . Recent applications of such T,-calculations have been worked out by Mitrovic et al. (1984), Leavens and Fenton (1981), Lie and Carbotte (1980), Rainer and Culetto (1979), (see also Carbotte 1981). The scheme has proven to be more efficient than older methods (McMillan 1968), which work with Eliashberg’s equations at real frequencies. The problem of calculating T, from Eliashberg’s theory seems to be settled satisfactorily. If required, the calculation can be done routinely.

6. Conclusion To set up a tractable ab initio theory of T, is a very complex project which needs the cooperation of experts from quite diverse fields of solid state theory. T, is controlled by phonon properties, band-structure effects, the electron-phonon coupling, and many-body correlations in the electron system. It seems that all parts are equally important for understanding even global trends in T,, and especially the conditions which lead to high-T, superconductivity. The actually observed T, is the result of various counteracting mechanisms. Hence, it is relatively easy, by picking out particular mechanisms, to find explanations for a measured high or low T,, but it is difficult to make a reliable prognosis for new materials. Of course, the ultimate goal of an ab initio theory of T, is the ability to predict Tc’s reliably. In my opinion, we have not yet fully reached this goal (see also Rainer 1982), and more cooperation, critical thinking, and creativity are needed in order to complete the theory of T,. I see a good chance that the open problems will be solved by some joint efforts in the near future. The only serious problems exist in the many-body part of the theory. In particular, there is evidence that the presently used excitation potentials begin to fail if correlation effects become important. Striking examples are VN (Rietschel et al. 1980) and V,Au (Jarlborg et al. 1983), for which the calculated Tc’s (32.3K and 10.2 K, respectively) deviate significantly from the observed Tc’s(8.6 K and 2.8 K ) , whereas the calculated and observed Tc’sagree reasonably well for compounds with less electronic correlations in the same classes of super-

420

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[Ch. 4. 16

conducting materials. The failure of present ab initio theories to describe correlated electrons is. of course, detrimental to their credibility in predicting high- 7, materials. I do not expect that the correlation problem in thc theory of T , will be solved by many-body theory alone. It is more likely that a ‘practical solution’ will be found by a trial and error strategy in a similar way to that in which the correlation problem has already been ‘solved’ in the density functional method. This strategy requires one to check available exchange-correlation potentials and p *’s systematically for their capacities in reproducing the observed data, to explore their limitations, and toimprove them in collaboration withmany-bodytheoryuntilone hasfounda useful universalconstruction recipe for potentials and p *. A systematiccheck would consist of: - A quite precise and fully controlled calculation of the electron-phonon coupling parameters (from a given potential-recipe), which does not rely o n empirical rules or ad-hoc approximations. - A critical comparison of the results with all experimental data which are connected with the excitation potential. such as Fermi-surface data, mass enhancements. transport data. and the superconducting data T,, Hc2 and tunneling spectra. This comparison must cover a wide range of materials including simple metals. transition metals. and compounds. Unfortunately. we are not yet in a position to perform the systematic check. This present situation is well demonstrated by the case of Nb. The calculation of two groups (Simons et al. 1981, Perlov and Fong 1984) reproduce very well experimental results for A (and T c ) , whereas other groups (Papaconstantopoulos et al. 1977, Peter et al. 1977, Harmon and Sinha 1977, Butler et al. 1979, Gliitzel et al. 1979, John et al. 1980. Peter et al. 1982) obtain an electron-phonon coupling which is much too large. Since each group employs different approximations, it is not yet clear if cither the present potential recipes work well for Nb, with some approximations spoiling the otherwise correct result. or if present recipes do not incorporate all correlation effects, with some calculations profiting from a compensation of errors. The intention of this review was to outline the general aspects of an ab initio theory of T , , to pin down the various partial problem (whosc solutions need the specific know-how of several fields in solid-state theory). and to describe how all these partial results are put together to form a complete theory of T,. A particular aim of this review was to summarize the arguments supporting the belief that an ab initio theory for the practical calculation of the superconducting T, is conceptually feasible. Various groups have already implemented much of the required machinery, and we anticipate that future research will complete this task, leading to powerful a b initio routines for calculating T , with the capacity of predicting new superconducting materials.

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42 1

Acknowledgements

I wish to thank Dr. Chris Kennedy, Dr. Alan Singsaas, and especially Dr. Paul Muzikar for reading the manuscript and for many valuable critical comments, and Ingrid Raps for typing the manuscript.

References Alexander, J.A.X., T.P. Orlando, D. Rainer and P.M. Tedrow, 1985, Phys. Rev. B31, 5811. Allen, P.B., 1972, Phys. Rev. B6,2577. Allen, P.B., 1974, Technical Report No. 7, TMC/4/1974 (Cavendish Laboratory, Cambridge). Allen, P.B., 1980, in: Dynamical Properties of Solids, Vol. 3, eds G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam) ch. 2. Allen, P.B., and R.C. Dynes, 1975, Phys. Rev. B12, 905. Allen, P.B., and B. MitroviC, 1982, in: Solid State Physics, eds F. Seitz, D. Turnbull and H. Ehrenreich ( Academic, New York). Appel, J., and W. Kohn, 1971, Phys. Rev. B4,2162. Ashraf, M., and J.C. Swihart, 1982, Phys. Rev. B25,2094. Bardeen, J., L. Cooper and J.R. Schrieffer, 1957, Phys. Rev. 108, 1175. Bassani, F., F. Fumi and M.P. Tosi, eds, 1985, Highlights of Condensed-Matter Theory, Proc. School of Physics “Enrico Fermi”, Course LXXXIX (North-Holland, Amsterdam). Beasley, M.R., and T.H. Geballe, 1984, Phys. Today 37 (lo), 60. Benneman, K.H., and J.W. Garland, 1972, in: Superconductivity in d- and f-Band Metals, ed. D.H. Douglass (American Institute of Physics, New York) p. 103. Bergmann, G., and D. Rainer, 1973, Z. Phys. 263,59. Berk, F.M., 1966, thesis (University of Pennsylvania) unpublished. Berk, N.F., and J.R. Schrieffer, 1966, Phys. Rev. Lett. 17,433. Butler, W.H., 1980, Phys. Rev. Lett. 44,1516. Butler, W.H., H.G. Smith and N. Wakabayashi, 1977, Phys. Rev. Lett. 39, 1004. Butler, W.H., F.J. Pinski and P.B. Allen, 1979, Phys. Rev. B19, 3708. Carbotte, J.P., 1981, in: Recent Progress in Many-Body Theories, eds J.G. Zabolitzky, M. dellano, M. Fortes and J.W. Clark, Lecture Notes in Physics, Vol. 142 (Springer, Berlin) ‘p. 135. Carbotte, J.P., and R.C. Dynes, 1968, Phys. Rev. 172,476. Chatterjee, P., 1983, Phys. Rev. B27,4722. Cohen, M.L., and P.W. Anderson, 1972, in: Superconductivity in d- and f-Band Metals, ed. D.H. Douglass (American Institute of Physics, New York) p. 17. Crabtree, G.W., D.H. Dye, D.P. Karim, D.D. Koelling and J.B. Ketterson, 1979, Phys. Rev. Lett. 42. 390. Dahl, J.P., and J. Avery, eds, 1983, Local Density Approximation in Quantum Chemistry and Solid State Theory (Plenum, New York). Dakshinamoorty, M., and K. Iyakutti, 1984, J . Phys. F14, L121. Dederichs. P.H.. and R. Zeller, 1980, in: Springer Tracts in Modern Physics. Vol. 87 (Springer, Berlin) p. 1. Dolgov. O.V., and E.G. Maksimov, 1982, Sov. Phys.-Usp. 25 (9), 688.

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D. RAINER

[Ch. 4

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AUTHOR INDEX Agnolet, G., see McQueeney, D. 353,368 Ahlrichs, R. 170,364 Ahn, R.M.C. 291,311,312,313,318,364 Alexander, J.A.X. 413,415,421 Allen, A.B., see Khan, F.S. 409,422 Allen, A.R. 78,94, I34 Allen, D.W., see Hellwig, H. 157,366 Allen, P.B. 373,402,408,409,412,415, 416,419,421 Allen, P.B., see Butler, W.H. 406,420,421 Allen, P.B., see Pinski, F.J. 406,408,423 Allison, A.C. 307,364 Alt, E.O. 253,364 Ambegaokar, V. 23,70 Amdur, I. 239,364 Amit, D. 83, I34 Andersen, O.K., see Mackintosh, A.R. 402, 423 Anderson, P.W. 3, 11,55, 70 Anderson, P.W., see Brinkman, W.F. 21, 70 Anderson, P.W., see Cohen, M.L. 403,421 Anderson, P.W., see Engelsberg, S. 25, 70 Andreev,A.F. 77,78, 108, 109, 110, 112, 114,352,134,364 Andronikashvili, E.L. 3, 70 Antonov, V.N., see John, W. 406,420,422 Appel, J. 413,421 Araki, K., see Iwasa, J. 90,135 Archie, C.N., see Halperin, W.P. 21, 71 Ashcroft, N.W., see JafTe, J.E. 401,406, 422 Ashkenazi, J., see Peter, M. 406,412,415, 420,423 Ashraf, M. 406,421 Atkins, K.R. 79, 179,352,353,134,364 Audoin, C., see Petit, P. 157,368 Avery, J., see Dahl, J.P. 399,401,421 Balian, R. 15, 70

Balling, L.C. 303,364 Bardeen, J. 412,421 Baron, R. 88,134 Banvell, M.G., see LeRoy, R.J. 164,367 Bashkin, E.P. 226,341,364 Bass, A.M. 155,364 Bassani, F. 402,421 Bear, D.R.,see Heald, S.M. 95, 127, 131, 135 Beasley, M.R. 375,421 Bedell, K.S. 21, 70 Bell, D.A. 220,223,233,235,364 Bell, D.A., see Hess, H.F. 148, 153, 188, 206,207,220,229,233,234,235,293,357, 366 Bell, R.J. 331,364 Benneman. K.H. 375,421 Bergmann, G. 412,419,421 Berk, F.M. 400,421 Berk, N.F. 375,376,400,421 Berkhout, J., see van Yperen, G.H. 163, 214,216, 227,228,370 Berlinsky, A.J. 151,222,225,303, 306, 307, 310,340,356,364,365 Berlinsky, A.J., see Greben, J.M. 239,241, 258,366 Berlinsky, A.J., see Hardy, W.N. 155, 163, 174, 188, 189, 211,212,224, 225, 241, 259, 366 Berlinsky, A.J., see Jochemsen, R. 163, 178, 186,207,208,209,212,361,367 Berlinsky, A.J., see Morrow, M. 163, 187, 188,205,206,207,209,210,212,303,307, 368 Berlinsky, A.J., see Reynolds, M.W. 206, 207,220,221,229,369 Berlinsky, A.J., see Statt, B.W. 163, 203, 214,218,221,229,258,311,345,369 Berlinsky, A.J., see Zimmerman, D.S. I8 I , 361,370 425

426

AUTHOR INDEX

Bernstein, R.B., see Roberts, R.E. 239,369 Bigelow, N., see Johnson, B.R. 15 1, 163, 189,226,227,345.367 Bigelow, N., see Yurke, B. 163,206,207. 212,213,217.218,219,222,225,226,370 Bilz, H., see Hanke, W. 410,422 Bird, R., see de Boer, J. 320,365 Bishop, D.M. 170,365 Bodensohn, J. 133, 134 Bodensohn, J., see Savignac, D. 123, 137 Kheim, J. 184,365 Bonzel, H.P. 105.134 Bouchaud, J.P. 303,307,346.365 Boyer, L.L., see Klein, B.M. 410,422 Boyer, L.L., see Papaconstantopoulos, D.A. 410,420,423 Bozler, H.M.. see Israelsson, U.E. 24, 71 Bredohl. H. 239,240,365 Brenig, W. 182,365 Brenig, W., see Bijheim, J. 184,365 Bridges. F., see Reynolds, M.W. 206,207, 220,221,229,369 Brinkman, W.F. 12, 13,21,24,25,26,28, 32, 36.43.45,49. 69, 70 Brinkman, W.F.. see Anderson, P.W. 1 I , 70 Brinkman. W.F.. see Engelsberg, S. 25, 70 Brinkman, W.F.. see Osheroff, D.D. 32, 71 Brinkman, W.F., see Smith, H. 24.25.36, 72 Briscoe, C.V., see Triftshauser. W. 84, 137 Brisson, J.G.. see Godfried, H.P. 206,207, 222,363,366 Brisson. J.G.. see van Yperen, G.H. 163, 214,216,227,228,370 Broida, H.P., see Bass, A.M. 155,364 Bruschi, L. 117, 118,134 Buchholtz, L.J. 36, 55.70 Buhrman, R.A., see Halperin, W.P. 21, 71 Bunkov. Yu.M. 69,70 Bunkov, Yu.M., see Hakonen, P.J. 23, 30. 31, 32, 38, 71 Butler, W.H. 406,408,409.41 5,420, 421 Butler, W.H., see Pinski, F.J. 406,408,423 Callaway, J., see LaBahn, R.W. 84, 136 Carbotte, J., see Mitrovic, B. 419,423

Carbotte, J.P. 406,419,421 Carbotte, J.P., see Lie, S.G. 419,423 Carbotte, J.P., see Tomlinson, P.G. 406, 408.424 Careri, G. 80,91, 117,134 Castaing, B. 183,365 Chattejee, P. 410,421 Chester, G.V., see Panoff, R.M. 149, 321, 323,328, 333,334,368 Choi, S.-l.,see Hernandez, J.P. 84, 135 Clark, J.W. 327,328,333,365 Clark, J.W., see Krotscheck, E. 327, 333, 334,367 Clark, J.W., see Panoff. R.M. 149, 32 1, 323,328,333,334,368 Cline, R.W. 151, 188, 189, 197, 198,206, 207,212,213,215,216,218,219,222,365 Cline, R.W., see Hess, H.F. 188, 206,229, 233,234.366 Cohen. M.H. 82, 121, I34 Cohen, M.H., seespringett, B.E. 83, 121, 137 Cohen, M.L. 403,421 Cohen, M.L., see Louie, S.G. 412,423 Collaudin, B., see P a p u l a r , M. 177,368 Condat, C.A. 337,365 Cooper, L., see Bardeen, J. 412.421 Cotts, R.M., see Miyoshi, D.S. 94, 136 Crabtree, G.W. 408,421 Crabtree, G.W., see Joss, W. 408.422 Crampt0n.S.B. 163, 174, 188, 195,365 Crampton, S.B., see Mathur, B.S. 157,368 Crank, J. 122,134 Crooker, B.C. 356,365 Crooker, B.C., see Israelsson, U.E. 24. 71 Cross, M.C. 43,45,49, 70 Cross, M.C.,see Brinkman, W.F. 12, 13, 24.25, 26,28, 32, 36,43,45.49,69, 70 Culetto, F.J., see Rainer, D. 419,423 Cunsolo, S. 89, 134 Cunsolo, S., see Careri, G . 117, 134 Curtiss, C.F., see Roberts, R.E. 239,369 Curtiss, L.A., see Milleur, M.B. 171,368 Dabrowski, 1. 164, 170.239,240.290,306, 365 Dacorogna, M., see Peter, M. 406,409,

AUTHOR INDEX 412,415,420,423 Dahl, J.P. 399,401,421 Dahm, A.J. 92, 122,134 Dahm, A.J., see Guenin, B.M. 1 19, 120, 131,135 Dahm, A.J., see Lau, S.C. 94,98, 119, 125, 126,136 Dahm, A.J., see Sai-Halasz, G.A. 92, 93, 94, 104,122,125,136.137 Dahm, A.J., see Smith Jr, J.B. 83,84, 129, 137 Dakshinamoorty, M. 410,421 Danilowicz, R.L. 149, 327,365 Danilowicz, R.L., see Etters, R.D. 327, 329,330,366 Davison, W.D. 186,365 de Boer, J. 320,365 de Goey, L.P.H. 220,234,288,29 I, 365 de Goey, L.P.H., see Verhaar, B.J. 302,370 De Simone, C. 176,365 Dederichs, P.H. 391,421 Denker, J.S., see Johnson, B.R. 151, 163, 189,226,227,345,367 Denker, J.S.,see Yurke, B. 163,206,207, 212,213,217,218,219,222,225,226,370 Desaintfuscien, M., see Petit, P. 157,368 Deville, G., see Sullivan, N. 95, 126, 137 Dharma-Wardana, M.W.C., see MacDonald, A.H. 400,423 Dionne, V.E. 84, 121, 134 Dolgov, O.V. 373,421 Doniach, S. 400,422 Drachman, R.J. 84,134 Driesen, J.P.J., see de Goey, L.P.H. 220, 234,291,365 Dugan, J.V. 327,329,365 Dugan, J.V., see Danilowicz, R.L. 149, 327,365 Dugan, J.V., see Etters, R.D. 327, 328, 329, 33 I , 333,334,365 Dupre, F., see Careri, G. 117, 134 Dye, D.H., see Crabtree, G.W. 408,421 Dynes, R.C., see Allen, P.B. 402,409,412, 421 Dynes, R.C., see Carbotte, J.P. 406,421 Eckardt. J.R. 352,365

427

Eckstein, Y., see Rosenbaum, R.L. 359, 369 Economou, E.N., see Zdetis, A.D. 407,424 Edmonds, A.R. 47, 70 Edwards, D.O. 150, 155, 175,302,347, 352,365 Edwards, D.O., see Eckardt, J.R. 352,365 Edwards, D.O., see Mantz, I.B. 176, 177, 368 E h o v , V.B. 85,87,90,94,95,97,99, 100, 108,115, 125, 126, 129, 131,134,135 Efimov, V.B., see Golov, A.I. 89, 94, 97, 98, 100, 118, 125,135 Efimov, V.B., see Mezhov-Deglin, L.P. 100, 101,136 Ehrenfest, P. 149, 365 Eilenberger, G. 396,422 Eliashberg, G.M. 41 I , 422 E M , E.R., see Godfried, H.P. 206,207, 222,363,366 Eliel, E.R., see Walraven, J.T.M. 188,370 Ellis, F.M. 353,363,365 Emrich, K. 400,422 Engelsberg, S. 25, 70 Engelsberg, S., see Doniach, S. 400,422 Engelsberg, S., see Smith, H. 24, 25, 36, 72 Esel’son, B.N. 78, 135 Esel’son, B.N., see Grigor’ev, V.N. 94, 135 Esel’son, B.N., see Mikheev, V.A. 78,136 Etters, R.D. 188,327,328, 329, 330, 331, 333,334,365,366 Etters, R.D., see Berlinsky, A.J. 340, 365 Etters, R.D., see Danilowicz, R.L. 149. 327,365 Etters, R.D., see Dugan, J.V. 327,329,365 Etters, R.D., see Friend, D.G. 300,329, 330,337,366 Evans, R. 407,422 Fantoni, S. 400,422 Farberovich, O.V., see Vlasov, S.V. 410, 424 Fasoli, V., see Careri, G. 80,134 Fatouros, P.P., see Eckardt, J.R. 352,365 Fatouros, P.P., see Edwards, D.O. 175,365 Feenberg, E. 328,333,366 Feenberg, E., see Jackson, H.W. 332.367

428

AUTHOR INDEX

Feenberg, E., see Wu. F.Y. 332.370 Feng. D.H.. see Pan. J. 341.368 Fenton, E.W., see Leavens, C.R. 419,423 Ferrante. R.F.. see Uang, Y.H. 171,370 Ferrell, R.A. 80.135 Feshbach, H., see Morse, P.M. 65, 71 Fetter, A.L. 6.21. 25, 36,42.45,46,47,49, 51, 55, 57. 58,63,67.68, 76. 81, 383, 70. 135,422 Fetter, A.L.. see Buchholtz, L.J. 55, 70 Fetter. A.L.. seeTheodorakis, S. 36, 38. 41, 72 Fetter, A.L., see Vulovic, V.Z. 61.63.65, 66.67. 72 Fetter. A.L., see Williams, M.R. 50, 68. 72 Feynman, R.P. 3, 70 Fong, C.Y.. see Perlov, C.M. 401,402,406, 420,423 Freed, J.H. 149,341.342. 345,366 Freed, J.H.. see Johnson, B.R. 151, 163, 189,226.227.345.367 Freed, J.H., see Yurke, B. 163,206,207, 212,213,217,218, 219,222,225,226,370 Frenkel, J. 82, I35 Friend, D.G. 300,329,330,337,366 Fujita, T. 50, 56. 57, 58, 70 Fujita, T., see Nakahara, P1. 68, 71 Fujita, T., see Ohmi. T. 16, 23, 27, 71 Fumi, F.,see Bassani. F. 402,421 Furth, H.P., see Kulsrud, R.M. 151, 356, 36 7 Gaeta, F.S., see Careri, G. 80, 134 Garland, J.W .. see Benneman. K.H. 375. 421 Gaspari, G.D. 406,410,422 Gaspari, G.D., see Evans, R. 407,422 Gasparini, F.M., see Eckardt. J.R. 352,365 Geballe, T.H., see Beasley, M.R. 375,421 Geldart, D.J.W., see MacDonald, A.H. 400.423 Gillaspy, J.D., see Godfried, H.P. 206. 207, 222. 363,366 Ginzburg, V.L. 373,422 Gjostein, N.A.. see Bon;ral, H.P. 105, 134 Glaze, J., see Hellwig, H. 157,366 Glcicke. W. 253,366

Glotzel, D. 406,408,420,422 Glukhov, N.A., see Kagan. Yu. 183, 184, 290,292. 351.367 Godfried, H.P. 206,207,222,363,366 Godfried, H.P., see Silvera, I.F. 196.369 Golden, D.E. 84, 135 Goldenberg, H.M., see Kleppner, D. 148. 367 Goldhaber, M., see Kulsrud, R.M. 151, 356,367 Goldman, V.V. 337,339,349,366 Goldman, V.V., see Berlinsky, A.J. 340, 365 Goldman, V.V., see Silvera, I.F. 155. 347, 369 Golov, A.I. 89,94.97,98, 100, 118, 125, I35 Golov, A.I., see Mezhov-Deglin, L.P. 100, 101,136 Gompf, F., see Schell, G. 406,423 Gongadze, A.D. 27.28.29, 71 Gorkov, L.P. 413,422 Could, C.M., see Israelsson, U.E. 24. 71 Grabowski, M. 400,422 Grassberger, P., see Alt, E.O. 253,364 Greben, J.M. 239,241,258,366 Greenberg, A S , see Miyoshi, D.S. 94, 136 Greytak, T.J. 155,231,320,356. 357.366 Greytak. T.J., see Bell, D.A. 220, 223,233. 235.364 Greytak, T.J..seeCline, R.W. 151, 188, 189, 197, 198, 206,207, 212.213, 215, 216. 218.219, 222,365 Greytak, T.J.,seeCrampton, S.B. 163, 174. 188,365 Greytak, T.J.. see Hess, H.F. 148, 153. 188, 206,207,220,229, 233,234,235,293,357, 366 Greytak, T.J., see Kleppner, D. 151, 253, 367 Greywall, D.S. 83, I35 Grigor'ev, V.N. 94, 135 Grigor'ev, V.N., see Esel'son, B.N. 78, 135 Grigor'ev. V.N.. see Mikheev. V.A. 78. I36 Grimvall, G. 373,403,412,422 Gross, E.P. 102,338,135.366 Gross, E.P., see Amit, D. 83, 134

AUTHOR INDEX Gudenk0,A.V. 86,89,90,94,135 Guenin, B.M. 119, 120, 131,135 Gully, W.J., see Osheroff, D.D. 3, 11, 32, 71 Gupta, M. 410,422 Gurgenishvili, G.E., see Gongadze, A.D. 27,28,29, 71 Gurney, R.W., see Mott, N.F. 128,136 Guyer, R.A. 77.78, 175,176,180,353,354, 355,135.366 Guyer, R.A., see Condat, C.A. 337,365 Guyer, R.A.. see Ellis, F.M. 353,365 GyorfTy, B.L.,see Evans, R. 407,422 GyoBy, B.L., see Gaspari, G.D. 406,4 10, 422 Hafner, J., see Hanke, W. 410,422 Haftel, M. 282,293, 366 Hakonen, P.J. 21,23,26,30,31,32,38,39, 40,71 Hakonen, P.J., see Bunkov, Yu.M. 69, 70 Hakonen, P.J., see Pekola, J.P. 40,41, 71 Hakonen, P.J., see SeppdP, H.K. 51.61, 62,63,65,66, 72 Hakoncn, P.J., see Volovik, G.E. 51, 59, 62.63, 72 Hall, H.E., see Hook, J.R. 69, 71 Hallock. R.B., see Ellis, F.M. 353,363,365 Halperin. W.P. 21, 71 Hanke, W. 403,410,422 Hanson, R.J., see Balling, L.C. 303,364 Hardy, W.N. 155, 163, 174,187,188,189, 209,21 I, 212,224,225,241,259,366 Hardy, W.N., see Berlinsky, A.J. 151,225, 307,356,364 Hardy, W.N., see Jochernsen, R. 163, 178, 207,208,209,212,361,367 Hardy, W.N., see Morrow, M. 163, 187, 188,205,206,207,209,210,212,368 Hardy, W.N., see Reynolds, M.W. 206, 207,220,221,229,369 Hardy, W.N.,seeStatt, B.W. 163,203,218, 221,227,229,258,345,369 Harmon, B.N. 401,406,408,420,422 Hamman, J.E. 171,172,366 Harrison,W.A. 401,422 Heald, S.M. 95, 127, 131, 135

429

Hebral, B., see Crooker, B.C. 356,365 Hecht, C.E. 188,320,326,366 Hedin, L. 400,422 Hedin, L., see Von Barth, U. 400,424 Helfrich, W. 88, 135 Hellwig, H. 157,366 Helmbrecht, U., see Pandharipande, V.R. 175,368 Hemmer, P.C.,see Nilsen, T.S. 327,368 Hernandez, J.P. 84,135 Herzberg, G. 164,240,366 Herzberg, G., see Bredohl, H. 239,240,365 Hess, H. 360,366 Hess, H.F. 148, 153,188,206,207,220, 229,233,234,235,293,357,366 Hess, H.F., see Bell, D.A. 233,235,364 Hess, R. 155,366 Hetherington, J.H. 77,135 Hirai, A., see Mizusaki, T. 94, 131,136 Hirayoshi, Y., see Mizusaki, T. 94, 131, 136 Hirsch, J.E. 400,422 Hirschfelder, J.O., see Harriman, J.E. 171, 172,366 Hirschfelder, J.O., see Milleur, M .B. I7 I , 368 Ho, T.-L. 43,50, 54, 56.62, 71 Ho, T.L., see Mermin. N.D. 45, 50, 54, 71 Hohenberg, P. 399,422 Hohenberg, P.C. 351,366 Hook, J.R. 69, 71 Hopfield, J.J. 376,409,422 Houston, S.K., see Drachman, R.J. 84, I34 Howe, L.L., see Herzberg, G. 240,366 Huang, K. 154,366 Huse, D.A. 337,367 Ifft, E.L. 91.94.95.135 Ikkala, O.T., see Hakonen, P.J. 21.40, 71 Iordatti, V.P. 400,422 Islander, S.T.,see Hakonen, P.J. 21,40,71 Israelsson, U.E. 24, 71 Iwasa, J. 90,135 Iyakutti, K., see Dakshinamoorty, M. 410, 42 1 Izyumov, Y u.A., see Vonsovsky, S. V. 373, 424

430

AUTHOR INDEX

Jaakkola, S., see Salonen, K. 178, 180,361, 369 Jaakkola, S., see Tommila, T. 1%. 223, 229,232,359,370 Jackson, H.W. 332,367 Jawbsen, K.W. 26,27.38,39,40, 71 Jaffe, J.E. 401,406,422 Jain. S.C. 406,422 Janak, J.F., see Papawnstantopoulos, D.A. 410,420,423 Jarlborg, T. 410,419.422 Jarlborg, T.. see Peter, M. 409,420,423 Jeffers Jr, W.A., see Lau, S.C. 94,98, 119. 125, 126,136 Jochemsen. R. 163, 178, 186,207,208,209, 212,361,367 Jochemsen, R.,see Hardy, W.N. 155, 163, 189,211,212,224,225,241,259,366 Jochemsen, R.,see Morrow,M. 163, 187, 188,205,206,207,209,210,212,368 John, W. 406,420,422 Johnson, B.R. 151, 163, 189,226,227,341, 345,367 Johnson, B.R.. see Yurke. B. 163,206,207. 212,213,217,218,219,222,225.226,370 Johnson,M.H., see Jones Jr, J.T. 151,236, 36 7 Jones Jr, J.T. 151,236,367 Jortner, J., see Cohen, M.H. 82, 121. I34 Jortner, J., see Springett, B.E. 83, 121,137 Joss, W. 408.422 Junod. A,, see Jarlborg, T. 410,419,422 Kachanov, L.M. 108, 135 Kachhava, C.M., see Jain. S.C. 406,422 Kadanoff, L.P., see Markowitz, D. 41 5. 423 Kagan,Yu. 78, 131, 181, 183, 184, 185, 220,223,231.232,233,234,240,241,259. 280,290,292, 301,303.3 I 1 , 3 19.320.340, 351,359,362, 135.367 Kalos, M.H., see Panoff, R.M. 149,321, 323,328, 333,334,368 Karhunen, M., see Salonen, K. 178, 180, 361,369 Karim, D.P., see Crabtree, G.W. 408,421

Katz, S., sey Jones Jr, J.T. 151. 236,367 Kelly, R. 103, 105, 135 Keshishev. K.O. 79,84,85,86,89.90,92, 93,94,96,97,99,108,110, 111,114,115, 116, 125, 131, 135.136 Ketterson, J.B., see Crabtree, G.W. 408, 421 Ketterson, J.B., see Roach, P.D. 68, 72 Ketterson, J.B.,see Roach, P.R. 124, I36 Khan, F.S. 409,422 Kharadze, G.A., see Gongadze, A.D. 27, 28.29, 71 Kim, D.J. 403,422 Kirzhnits, D.A., see Ginzburg, V.L. 373, 422 Klein, B.M. 404,410,422 Klein, B.M., see Papaconstantopoulos, D.A. 410,420,423 Kleppner, D. 148, 151,253.367 Kleppner, D., see Bell, D.A. 220,223,233, 235,364 Kleppner, D.. see Cline, R.W. 15I , 188, 189,197, 198,206,207,212,213,215,216, 218,219,222,365 Kleppner, D., see Crampton, S.B. 163, 174, 188,365 Kleppner, D.. see Greytak, T.J. 155,320, 356,357,366 Kleppner. D., seeHess, H.F. 148, 153, 188. 206,207,220,229,233,234,235, 293. 357, 366 Kleppner, D., see Mathur, B.S. 157,368 Kochanski, G.P., see Bell, D.A. 220. 223, 233,235,364 Kochanski, G.P., see Hess, H.F. 148, 153, 188,206,207,220,229,233,234.235,293, 357.366 Koelling, D.D. 401,402,422 Koelling, D.D., see Crabtree, G.W. 408, 421 Kohn, W. 399,422 Kohn, W., see Appel, J. 413,421 Kohn, W., see Hohenberg, P. 399,422 Kohn, W., see Sham, L.J. 399,423 Kolos. W. 164, 165, 169, 170,239,240,306, 321,329,331,367 Kompaneets, D.A., see Andreev, A.F. 352,

AUTHOR INDEX 364 Kon, L.Z., see Iordatti, V.P. 400,422 . Kopnin, N.B., see Volovik, G.E. 58.72 Kosterlitz, J.M. 352,367 Kosterlitz, J.M., see Nelson, D.R. 352, 368 Kovdrya, Yu.Z., see Keshishev, K.O. 84, 92,135 Krotscheck, E. 327,333,334,367 Krotscheck, E., see Clark, J.W. 327,328, 365 Krupczak, J.J., see Crampton, S.B. 195, 365 Krusius, M., see Bunkov, Yu.M. 69, 70 Krusius, M., see Hakonen, P.J. 23, 30, 31, 32, 38, 71 Krusius, M., see Paulson, D.N. 68, 71 Krusius, M., see Pekola, J.P. 40,41, 71 Krusius, M., see Seppala, H.K. 5 I , 6 I , 62, 63,65,66, 72 Krusius, M., see Tommila, T. 196,223, 229,232,359,370 Kubrik, P.R., seeHardy, W.N. 163, 189, 21 1,212,241,259,366 Kulsrud, R.M. 151,356,367 Kumar, P. 180,367 Kumar, P., see Wilson, B.J. 180,370 Kurmaev, E.Z., see Vonsovsky, S.V. 373, 424 Kiirten, K.E. 175,367 LaBahn, R.W. 84,136 Lagendijk,A. 219,222,223,292,311,313, 317,318,319,367 La&, F., see Lhuillier, C. 226,341, 346, 368 Lal&, F., see Pinard, M. 246,303,304,368 LaloE, F., see Tastevin, G. 345,370 Lamb. W.E. 148,367 Lampert, M.A. 85,89,136 Landau, J., see Rosenbaum, R.L. 359,369 Landau, L.D. 65,362,383,71,367,422 Landesman, A. 78,136 Landesman, A., see Hardy, W.N. 163, 189, 21 1,212,241,259,366 Landesman, A., see Sullivan, N. 95, 126, 137 Langer, J.S. 3, 71

43 I

Lantto, L.J. 349,351,367 Larkin, A.I. 396,422 Lau, S.C. 87,94,95,98,99, 114, 115, 119, 125, 126,136 le Comber, P.G., see Spear, W.E. 76,137 Leavens, C.R. 412,419,423 Leavens, C.R., see MacDonald, A.H. 406, 423 Leduc, M., see Tastevin, G. 345,370 Lee,D.M. 11, 71 Lee, D.M., see Johnson, B.R. 151. 163, 189,226,227,345,367 Lee, D.M., see Lovelace, R.V.E. 362,363, 368 Lee, D.M., see Osheroff, D.D. 3, 1 1, 32, 71 Lee, D.M., see Yurke, B. 163,206,207, 212,213,217,218, 219,222,225,226,370 Lee, M.A., see Panoff, R.M. 149,321,323, 328,333,334,368 Legg, J., see Triftshauser, W. 84, 137 Leggett, A.J. 11, 12, 23, 25, 32,33, 34, 150, 341,343, 71.367 Leggett, A.J., see Goldman, V.V. 337, 339, 366 Leiderer, P., see Bodensohn, J. 133,134 Leiderer, P., see Savignac, D. 123,137 Lenz Jr, W. 83,136 LeRoy, R.J. 164, 165.367 LeRoy, R.J., see Mitchell, D.N. 152,368 Levchenko, A.A. 119,136 Levin, K. 21.71 Levine, J. 83,136 Levine, K. 400,423 Levine, M., see Hellwig, H. 157,366 Levinson, J., see Many, A. 88,146 Lkvy, L.P. 227,344,345,346,367 Lkvy, L.P.,see Johnson, B.R. 151, 163, 189,226,227,345,367 Gvy, L.P., see Yurke, B. 163,206,207, 212,213,217,218,219,222,225,226,370 Lhuillier, C. 226,307, 341, 342, 343, 344, 346,359,367,368 Lhuillier, C., see Bouchaud, J.P. 303,307, 346,365 Lie, S.G. 419,423 Lieb, E.H. 327,368 Liew, Y.C., see Davison, W.D. 186,365

432

AUTHOR INDEX

Lifshitz, E M . , see Landau, L.D. 65. 362, 383, 71,367,422 Lifshitz, I.M., see Andreev. A.F. 77, 78. I34 Locke. D.P. 79. 136 London, F. 149,368 Louie, S.G. 412,423 Lounasmaa, O.V.. see Hakonen. P.J. 21, 40, 71 Lounasmaa, O.V., see Pekola, J.P. 40,41, 71 Lovelace, R.V.E. 362,363,368 Lundqvist, B.I. 400,423 Lundqvist, S. 399,401,423 Lundqvist, S., see Hedin, L. 400,422 Luttinger, J.M. 413,423 MacDonald, A.H. 400,406,423 Mackintosh, A.R. 402,423 Maekawa. S.,see Mizusaki, T. 94, 131,136 Maidanov, V.A., see Mikheev, V.A. 79,94. 127. 131,136 Maki, K. 26. 38, 39, 55. 62.66.68, 71 Maki, K.,see Zotos, X. 55,62,63,68,72 Maksimov, E.G., see Dolgov, O.V. 373, 421 Maksimov, L.A.. see Kagan, Yu. 78, 13I , 135 Mallardeau, C., see Godfried, H.P. 206, 207,222,363,366 Mmaladze, Yu.G.. see Andronikashvili, E.L. 3, 70 Mamniashvili, G.. see Pekola, J.P. 40,41, 71

Mantz, I.B. 176. 177,368 Mantz, I.B., see Edwards, D.O. 150, 155, 302,347,352,365 Many. A. 88.89. 136.146 Maraviglia, B.. see De Simone, C. 176.365 March, N.H., see Lundqvist, S. 399.401. 423 Mark. P., see Helfrich, W. 88. 135 Mark. P., see Lampert. M.A. 85,89, 136 Markowitz, D. 415,423 Marsolais, R.M., see Hardy, W.N. 163, 189,211,212,241,259,366 Marty, D. 86, 89.94,95,96,99, 105. 108,

121, 125, 126, 130,136 Masharov, N.F. 400,423 Mathur, B.S. 157,368 Matthey, A.P.M. 188, 199,205,206,207, 209,211,212,213.214,255,368

Matthey, A.P.M., see van Yperen, G.H. 207,208,214,220,370 Matthey, A.P.M.,see Walraven, J.T.M. 197, 198,370 Matthias, B.T. 375,423 Mayer, R. 204,227,368 Mazzoldi, P., see Bruschi, L. 117,118.134 Mazzoldi, P., see Careri, G. 117, 134 McGervey, J.D., see Smith Jr, J.B. 83. 84, 129, 137 McMillan, W.L. 328,329,409,419,368, 423 McQueeney, D. 353,368 Mehanian, C., see Lovelace, R.V.E. 362, 363,368 Meierovich, A.E. 110, 113,136 Meierovich, A.E., see Andreev, A.F. 108, 110, 112, 114,134 Meierovich, A.E., see Keshishev, K.O. 111, 114, 115,135 Mello, E.V.L. 351,368 Mennin, N.D. 3,43,45,46,49,50, 52, 54, 71 Mermin, N.D., see Ambegaokar, V. 23, 70 Messiah, A. 274,300,368 Mester, J.C., see Godfried, H.P. 206, 207, 222,363,366 Meyer, H.L., see Jones Jr, J.T. 151,236, 367 Mezhov-Deglin, L.P. 100, 101, 133,136 Mezhov-Deglin, L.P., see Efimov, V.B. 85, 87,90,94,95,97,99, 100,108, 115, 125, 126, 129, 131,134. I35 Mezhov-DegIin, L.P., see Golov, A.I. 89, 94,97,98,100, 118, 125, I35 Mezhov-Deglin, L.P., see Ifft, E.L. 91.94. 95,135 Mezhov-Deglin, L.P., see Keshishev, K.O. 84. 89.92.93, 135. 136 Mezhov-Deglin, L.P., see Levchenko, A.A. 119,136 Migdal, A.B. 386,423

AUTHOR INDEX Mikheev, V.A. 78,79,94, 127, 131,136 Mikheev, V.A., see Esel’son, B.N. 78, 13.5 Mikheev, V.A.. see Grigor’ev, V.N. 94,135 Mikhin, N.P., see Esel’son, B.N. 78, 13.5 Mikhin, N.P., see Mikheev, V.A. 78, 79, 94, 127, 131, 136 Miller, M.D. 150, 175, 320, 321, 322, 323, 324, 325, 327, 328, 329, 330, 331, 333, 334, 349,368 Miller, M.D., see Ellis, F.M. 353, 36.5 Miller. M.D., see Guyer, R.A. 175, 176, 180,353,366 Milleur, M.B. 171,368 Milleur, M.B., seeHarriman, J.E. 171, 172, 366 Mineev, V.P. 42,77, 106, 71.136 Mineev, V.P., see Hakonen, P.J. 23,30, 31, 32,38, 71 Mineev, V.P., see Volovik, G.E. 23,27,52, 72 Misha, W.A. 259,368 Mitchell, D.N. 152,368 Mitrovic, B. 419,423 MitroviC, B., see Allen, P.B. 373,415,416, 421 Miyoshi, D.S. 94, 136 Mizusaki, T. 94, 131, 136 Morrow, M. 163, 187, 188,205,206,207, 209,210,212,303,307,368 Morrow, M., see Hardy, W.N. 155, 163, 189,211,212,224,225,241,259,366 Morrow, M., see Jochemsen, R. 163, 178, 207,208,209,212,361,367 Morse, P.M. 65. 71 Moruui, V.L., see Papaconstantopoulos, D.A. 410,420,423 M o w , E., see Peter, M. 409,420,423 Mott, N.F. 128,136 Muzikar, P. 53, 58.61, 71 Nacher, P.J., see Tastevin, G. 345,370 Nakahara, M. 68, 71 Nakahara, M., see Fujita, T. 50, 56, 57, 58, 70 Nakahara, M., see Maki, K. 26,38, 39, 71 Nelson, D.R. 352,368 Nemoshkalenko, V.V., see John, W. 406,

433

420,422 Nicolet, M.A., see Baron, R. 88, 134 Nieminen, R.M., see Lantto, L.J. 349, 351, 367 Niinikoski, T.O. 151,368 Nilsen, T.S. 327,368 Nizhnikova, G.P., see Vlasov, S.V. 410, 424 Nosanow, L.H. 116, 155,320,322,323, 327,136,368 Nosanow, L.H.. see Miller, M.D. 150,320, 321, 322,323, 324, 325, 327,328, 329,330, 331,333,334,349,368 Nosanow, L.H., see Stwalley, W.C. 188, 329,369 Nummila, K.K., see Pekola, J.P. 40,41, 71 Ohmi, T. 16,23,27,62, 71 Ohmi, T., see Fujita, T. 50, 56, 57, 58, 70 Ohmi, T., see Nakahara, M. 68, 71 Ohmi, T., see SeppalH, H.K. 51, 61,62,63, 65-66, 72 OMalley, T.F. 84,136 Onsager, L. 3, 71 Oppenheimer, J.R., see Ehrenfest, P. 149, 36.5 Orlando, T.P., see Alexander, J.A.X. 4 13, 415,421 Osheroff, D.D. 3, 1 I , 32, 71 Ovchinnikov, Yu.N., see Larkin, A.I. 396, 422 Overhauser, A.W. 399,423 Packard, R.E., see Pekola, J.P. 40.41, 71 Padmore, T.C. 1 17,136 Palmer, R.W., see Etters, R.D. 327, 328, 329,330,331,333,334,365,366 Pan, J. 341,368 Pandharipande, V.R. 175,368 Panoff,R.M. 149,321,323,328,333,334. 368 Panoff, R.M., see Clark, J.W. 327,328,36.5 Panoff, R.M., see Krotscheck, E. 327,333, 334,367 Papaconstantopoulos, D.A. 410,420,423 Papaconstantopoulos, D.A., see Klein, B.M. 410,422

434

AUTHOR INDEX

Papaconstantopoulos, D.A., see Zdetis, A.D. 407,424 Papoular, M. 177,208,368 Papoular, M..see Castaing, B. 183,365 Parish, L.J., see Miller, M.D. 320. 323, 324, 325,333,368 Parish, L.J., see Nosanow, L.H. 320, 322. 323,321,368 Parker, E.R., see Washburn, J. 119.137 Passvogel, T. 13, 19,55, 71 Patterson, P.L. 80. 136 Pauling. L. 80,136 Paulson, D.N. 68. 71 Pekola, J.P. 40,41, 71 Penco, R.. see Ahlrichs, R. 170,364 Pentilg, S., see Niinikoski, T.O. 151,368 Perlov, C.M. 401,402,406,420,423 Peshkov, V.P. 3, 72 Peter, M. 406,409.4 12,415,420,423 Peter, M..see Jarlborg, T. 410.419.422 Peters, H.E., see Hellwig. H . 157.366 Petit. P. 157,368 Pettifor, D.G. 407,423 Phillips, W.D., see Crampton, S.B. 163, 174, 188,365 Pickett. W.E. 406,410,423 Pickett, W.E., see Klein, B.M. 404,410, 422 Pieper, S.C.,see Pandharipande, V.R. 175. 368 Pinard, M. 246,303, 304.368 Pinski, F.J. 406, 408, 423 Pinski, F.J., see Butler, W.H. 406.420.421 Pinsky, F.J., see Nosanow, L.H. 320, 322, 323,327,368 Pipkin. F.M., see Balling, L.C. 303,364 Pitaevski, L.P., see Gorkov, L.P. 413.422 Pollack, L.. see Bell, D.A. 233. 235,364 Poshuta, R.D. 80, 136 Pritchard, D.E. 360.369 Pushkarov. D.I. 78, 136 Rainer. D. 415,419,423 Rainer, D., see Alexander, J.A.X. 4 I3,4 15. 421 Rainer, D., see Bergmann, G . 412,419,421 Rainer, D., see GIBtzel, D. 406,408,420,

422 Rainer, D., see Serene, J.W. 383, 423 Rakavy, G., see Many, A. 89,136 Ramsey, N.F. 157,369 Ramsey, N.F., see Kleppner, D. 148,367 Ramsey, N.F., see Mathur, B.S. 157,368 Ramsey, N.F., see Wineland. D.J. 157,370 Rasmussen, F.B., see Halperin. W.P. 21, 71 Ray, S. 186,369 Rayfield, G.W. 116,136 Rehr, J.J., see Mello, E.V.L. 351.368 Reich, H.A. 95, 122, 126, 136 Reichardt, W., see Rietschel. H. 400.406. 419,423 Reichardt. W., see Winter, H. 406,424 Reif, F., see Rayfield, G.W. 116, 136 Reppy, J.D., see Crooker, B.C. 356.365 Reppy, J.D., see Langer, J.S. 3, 71 Reppy, J.D., see McQueeney, D. 353,368 Retherford, R.C., see Lamb, W.E. 148,367 Reuver, C.J.. see Ahn, R.M .C. 29 1.3 1 I , 318,364 Reuver, C.J., see van den Eijnde, J.P.H.W. 220,301,3\1,317,370 Reynolds, M.W. 206,207,220,221,229, 369 Rice, M.J.. see Leggett, A.J. 341,367 Richards, M.G.. see Allen, A.R. 78,94,134 Richardson, R.C., see Guyer. R.A. 77, 78. 135 Richardson, R.C., see Halperin, W.P. 2 I , 71 Richardson, R.C., see Lee, D.M. 1 I. 71 Richardson, R.C., see Miyoshi, D.S. 94, 136 Richardson, R.C., see Osheroff, D.D. 3, 1 I , 32, 71 Ridner, A., see Mayer, R. 227,368 R i a , G., see Winter, H. 406,424 Rietschel, H. 400,406,419,423 Rietxhel, H.. see Schell, G. 406.423 Rietxhel, 11..see Winter, H. 406,424 Rieubland, J.M.. see Niinikoski. T.O. 151. 368 Rijllart, A., see Niinikoski, T.O. 151.368 Ristig, M.L., see Kiirten, K.E. 175,367 Roach, P.D. 68, 72

AUTHOR INDEX Roach, P.D., see Roach, P.R. 124,136 Roach, P.R. 124,136 Roach, P.R., see Roach, P.D. 68,72 Robert, J.B., see Papoular, M. 177,368 Roberts, R.E. 239,369 Robinson, A.L., see Amdur, I. 239,364 Rodrique, V., see Baron, R. 88,134 Rosenbaum, R.L. 359,369 Ruckenstein, A.E. 31 1,369 Ruckenstein, A.E., see Gvy, L.P. 227, 344, 345,346,367 Ruckenstein, A.E., see Siggia, E.D. 3 1 1, 340,369 Rudnick, I., see Atkins, K.R. 352,353,364 Ruvalds, J. 375,423 Saam, W.F. 352,369 Saam, W.F.,see Edwards, D.O. 175,365 Sai-Halii~~,G.A.92,93,94, 104, 122,125. 136,137 Salomaa, M.M. 16, 19, 20.2 I , 22, 23, 24, 27,28,29, 51,59,69, 70, 72 Salomaa, M.M., see Hakonen, P.J. 23, 30. 31.32, 38, 71 Salomaa, M.M., see Mineev, V.P. 42, 71 Salomaa, M.M., see Sepplla, H.K. 51,61, 62,63,65,66, 72 Salonen, K. 178,180,361,369 Salonen, K., see Tommila, T. 196,223,229, 232, 359,370 Salonen, K.T. 178, 179,184,369 Sample, H.H. 95,137 Sanders Jr, T.M., see Levine, J. 83, I36 Sandhas, W., see Alt, E.O. 253,364 Santini, M., see Bruschi, L. 117, 1 18,134 Santini, M., see Careri, G. 117,134 Saslow, W., see Guyer, R.A. 354,355,366 Sauls, J.A. 21,23, 72 Sauls, I.A., see Fetter, A.L. 6,45,49, 51, 55, 57, 58.67.68, 70 Savignac, D. 123,137 Savignac, D., see Bodensohn, J. 133, I34 Scalapino, D.J. 373,423 Scalapino, D.J., see Hirsch, J.E. 400,422 Scaramuzzi, F., see Careri, G. 80,91,134 Schaffrath, M.M.J., see Verhaar, B.J. 302, 370

435

Schell, G. 406,423 Schiff, D. 333,369 Schmid, E.W. 253,369 Schmidt, K.E., see Panoff, R.M. 149,321, 323,328,333,334,368 Schober, H.R., see GIBtzei. D. 406,408, 420,422 Schopohl, N. 51,67, 72 Schopohl, N., see Passvogel, T. 13, 19, 55, 71 Schrieffer, J.R., see Bardeen, J. 412,421 Schrieffer, J.R., see Berk, N.F. 375,376, 400,421 Schrieffer, J.R., see Scalapino, D.J. 373, 423 Schwarz, K.M. 76,81,137 Scoles, G., see Ahlrichs, R. 170, 364 Seidel, G., see Mayer, R. 204,227,368 Seitz, F. 128,137 Sepplla. H.K. 51, 59,60,61,62,63,65, 66, 72 Serene, J.W. 67, 383, 72,423 Serene, J.W., see Brinkman, W.F. 21,70 Serene, J.W., see Sauls, J.A. 21.23, 72 Shal’nikov, A.I. 91,92, 137 Shal’nikov, A.I.,see Ifft, E.L. 91,94,95, 135 Shal’nikov, A.I., see Keshishev, K.O. 84, 89,92,93,96, 108,135, 136 Sham, L.J. 399,423 Sham, L.J., see Grabowski, M. 400,422 Sham, L.J., see Kohn, W. 399,422 Sham, L.J., see Rietschel, H. 400,423 Shen, S.Y.,see Eckardt, J.R. 352,365 Shevchenko, S.I. 352,369 Shewmon, P.G. 122,137 Shih, S., see Bishop, D.M. 170,365 Shikin, V.B. 76,81,83,91, 101, 102, 103, 104, 107, 123, 124, 129,137 Shikin, V.B., see Efimov, V.B. 108, 135 Shinkoda, I., see Reynolds, M.W. 206, 207, 220,221,229,369 Shirokovskii, V.P., see Veprev, A.G. 406, 408,424 Shizgal, B., see Berlinsky, A.J. 303, 306, 310,365 Shlyapnikov, G.V., see Kagan, Yu. 181,

436

AUTHOR INDEX

183, 184. 185.220, 223, 231, 232, 233, 234, 240,241.259,280.290.292,301,303,311, 319,320,340,351,359, 362.367 Siggia, E.D. 31 I, 340.369 Siggia, E.D., see Huse, D.A. 337,367 Siggia, E.D., see Ruckenstein, A.E. 31 I. 369 Silvera. I.F. 148. 150. 153, 155, 161, 169, 174, 175, 188, 189, 194, 196, 197,204,205, 207,209.253.254,256,336,340, 347,358. 360,369 Silvera, I.F.,seeAhn, R.M.C. 291,311, 318,364 Silvera, I.F., see Berlinsky. A.J. 340,365 Silvera, I.F.. see Godfried. H.P. 206, 207, 222.363.366 Silvera, I.F., seeGoldman. V.V. 337, 339, 349.366 Silvera, I.F.. see Matthey, A.P.M. 188, 199, 205,206,207.209,211, 212,213,214,255, 368 Silvera. I.F., see Salonen. K.T. 178, 179, 184.369 Silvera, I.F.. see Sprik, R. 153. 189, 196, 203,206,207.216.217,218,219,220,221, 223.224. 229. 230. 231, 232, 235,258,358, 359.369 Silvera, I.F., see van Yperen, G.H. 163, 207.208,214,216,220, 227,228,345,370 Silvera, I.F., see Walraven, J.T.M. 188, 189, 192. 197, 198,335, 337,370 Simhony. M., see Many. A. 88, 146 Simmons, R.O., see Heald, S.M. 95. 127, 131,135 Simola, J.T., see Hakonen. P.J. 23, 30, 31, 32. 38, 71 Simola, J.T., see Pekola, J.P. 40.41, 71 Simola, J.T., see SeppiIa, H.K. 51,61,62, 63.65.66. 72 Simons. A.L. 406,420,423 Simpson. R.W., see GIBtzel, D 406,422 Sinha. S.K., see Harmon, B.N. 401,406, 408,420.422 Slater, J.C. 401.424 Smith, D.A., see Cline, R.W. 189, 197, 198, 365 Smith, D.A.,seeCrampton,S.B. 163, 174.

188,365 Smith, E.N., see Crooker, B.C. 356,365 Smith, H. 24, 25,36, 72 Smith, H., see Jacobsen, K.W. 26,27,38, 39,40. 71 Smith, H.G., see Butler. W.H. 406,408. 42 I Smith Jr, J.B. 83, 84, 129, 137 Smith, R.A., see Krotscheck, E. 327, 333, 334.367 Souza, S.P., see Crampton, S.B. 195,365 Spear, W.E. 76,137 Sprik, R. 153, 189, 196,203,206,207,216, 217, 218, 219,220, 221, 223,224, 229, 230. 231,232,235,258,358, 359,369 Springett, B.E. 83, 121, 137 Statt, B.W. 163,203, 206, 207. 214, 218. 221,227,229.253,258,303,3 11, 345,369 Statt, B.W., see Hardy, W.N. 163. 189.21 I . 212,241,259,366 Statt, B.W.. see Reynolds. M.W. 206. 207. 220, 22 1,229,369 Stein, D.L.. see Fetter, A.L. 6, 45,49, 51. 55, 57, 58,67,68, 70 Stein, D.L., see Sauls, J.A. 23. 72 Stein, D.L., see Vulovic, V.Z. 61, 63,65. 66.67, 72 Stewart, A.T., see Triftshauser, W. 84. 137 Stewart, G.R. 10. 374, 72,424 Stutzki, J., see Boheim, J. 184, 365 Stwalley. W.C. 176, 177, 188, 207.239.240. 329.369 Stwalley, W.C., see Uang, Y.H. 171. 300, 370 Sullivan, N. 95, 126, 137 Swenson, C.A., see Sample, H.H. 95,137 Swihart, J.C., see Ashraf, M. 406,421 Swihart, J.C., see Tomlinson, P.G. 406,424 Tabakar, V.P.. see lordatti, V.P. 400,422 Takano, Y., see Crooker, B.C. 356,365 Tastevin, G. 345,370 Tedrow, P.M., see Alexander, J.A.X. 41 3, 415.421 Tewordt, L., see Passvogel. T. 13, 19.55, 71 Theodorakis, S. 36, 38.41. 72

AUTHOR INDEX Theodorakis, S., see Fetter, A.L. 21, 70 Thomas, A.W. 253,370 Thomas, A.W., see Greben, J.M. 239,241, 258,366 Thomson, G.P., see Thornson,J.J. 85, I37 Thomson, J.J. 85, I37 Thomson, J.O., see Careri, G. 80.91, I34 Thouless,D.J., see Kosterlitz, J.M. 352, 367 Thuneberg, E.V. 69, 70 Titus, W.J., see Nosanow, L.H. 116, I36 Tjukanov, E., see Salonen, K. 178, 180, 361,369 Tjukanov, E., see Tommila, T. 196,223, 229,232,359,370 Tkachenko, V.K. 9,72 Tomizuka, C.T., see Dionne, V.E. 84, 121, I34 Tomlinson, P.G. 406,408,424 Tommila, T. I%, 223,229,232,359,370 Tommila, T.,see Salonen, K. 178, 180, 361, 369 Tommila, T.J., see Lovelace, R.V.E. 362, 363,368 Tosi, M.P., see Bassani, F. 402,421 Tough,J.T. 3, 72 Toulouse, G., see Anderson, P.W. 55, 70 Tnftshauser, W. 84, I37 Tsuneto, T., see Fujita, T. 50, 56, 57, 58, 70 Tsuneto, T., see Nakahara, M. 68, 71 Tsuneto, T., see Ohmi, T. 16,23,27, 71 Tsymbalenko, V.L., see Gudenko, A.V. 86, 89,90,94, I35 Tung-Li, H., see Gross, E.P. 102, I35 Twerdochlib, M., see Haniman, J.E. 171, 172,366 Twerdochlib, M.,see Milleur, M.B. 171, 368 Uang, Y.H.

171,300,370

Valeo, E.J., see Kulsrud, R.M. 151, 356, 367 Valls, O.T., see Levin, K. 21, 71 Valls, O.T., see Levine, K. 400,423 van den Eijnde, J.P.H.W. 220,301,311, 316,317,370

437

van den Eijnde, J.P.H.W., see Ahn, R.M.C. 291,311,312,313,318,364 van den Eijnde, J.P.H.W., see Verhaar, B.J. 302,370 van Yperen, G.H. 163,207,208,214,216, 220,227,228,345,370 van Yperen, G.H., see Lagendijk, A. 222, 223,311,313,317,319,367 van Yperen, G.H., see Salonen, K.T. 178, 179,184,369 van Yperen, G.H., see Sprik, R. 203,206, 207,216,217,218,219,220,221,235,258, 369 Varma, C.M., see Simons, A.L. 406,420, 423 Vartanyantz, LA., see Kagan, Yu. 220, 223,231,232,233,234,240,241,259,280, 290,292,301,303,311,319,320,351,359, 362,367 Vauge, C. 80, I37 Veprev, A.G. 406,408,424 Verhaar, B.J. 203,232,282,293,302,370 Verhaar, B.J., see Ahn, R.M.C. 29 1.3 1 1, 312,313,318,364 Verhaar, B. J., see de Goey, L.P.H. 220, 234,288,291,365 Verhaar, B.J., see van den Eijnde, J.P.H.W. 220,301,311,317,370 Verlet, L., see Schiff, D. 333,369 Vessot, R.F.C., see Hellwig, H. 157,366 Vilches, O.E.,see Mello, E.V.L. 351,368 Vinen, W.F. 3, 72 Vlasov, S.V. 410,424 Voermans, M.A.J., see Verhaar, B.J. 302, 370 Volovik, G.E. 23,27, 51,52,58,59,62,63, 72 Volovik, G.E., see Hakonem, P.J. 21,23, 26,30, 31, 32, 38. 39,40, 71 Volovik, G.E., see Pekola, J.P. 40.41, 71 Volovik, G.E., see Salomaa, M.M. 16. 19, 20, 21, 22, 23, 24, 27,28, 29, 51, 59, 69, 70, 72 Volovik, G.E., see Sepplll, H.K. 51, 59, 60,61,62,63,65,66, 72 Von Barth, U. 400,424 Von Barth, U., see Williams, A.R. 402,424

438

AUTHOR INDEX

Vonsovsky, S.V. 373,424 Vredenbregt. E.J.D., see Verhaar, B.J. 302. 370 Vulovic, V.Z. 61. 63,65, 66,67, 72 Wakabayashi, N.. see Butler, W.H. 406. 408,421 Walecka, J.D.. see Fetter, A.L. 36.46,47, 383,70,422 Walraven, J.T.M. 155, 166. 167, 188. 189, 192, 197, 198,254,335,337,370 Walraven. J.T.M., see de Goey. L.P.H. 220,234,291,365 Walraven, J.T.M., see Lagendijk, A. 222, 223,311,313,317,319,367 Walraven, J.T.M., see Matthey, A.P.M. 188, 199,205,206,207,209,211,212,213, 214,255,368 Walraven, J.T.M., see Salonen, K.T. 178, 179. 184,369 Walraven, J.T.M.. see Silvera, I.F. 148, 150, 153, 161, 174, 188. 189, 194, 196, 197, 204,205,207.209,253,254.256,336.340. 369 Walraven, J.T.M., seeSprik, R. 153, 189, 1%,203.206,207,216,217,7-18,219,220, 221,223. 224,229, 230. 231,232.235, 258, 358,359.369 Walraven, J.T.M.. see van Yperen, G.H. 163, 207, 208. 2 14. 2 16, 220, 227. 228, 345, 370 Warnke, M ,see Passvogel. T. 13. 19, 71 Washburn, J. 119. I37 Weber. W. 402.424 Weber, W., see Simons, A.L. 406,420.423 Weinrib, A.. see Crampton, S.B. 163. 174, 188.365 Weisz, S.Z.. see Many, A. 88, 146 Werthamer, N.R.. see Balian, R. 15, 70 Wheatley. J.C 1 1. 34. 72 Wheatley. J.C.. see Paulson, D.N. 68. 71 Whitehead, L.A.. see Hardy, W.N. 174. 187. 188,209,224,366 Whitmore. M.D. 401,406,424 Whitten, J.L.. see Vauge, C. 80, I37 Wilkins. J.W.. see Lundqvist, B.I. 400,423 Wilkins, J.W., see Scalapino, D.J. 373,423 Williams. A.R. 402,424

Williams, A.R., see Papaconstantopoulos. D.A. 410,420,423 Williams, F.I.B., see Marty, D. 86, 89,94, 95,96,99, 105, 108, 121, 125. 126, 130, I36 Williams, G.A. 353,370 Williams, M.R. 49, M,66,68, 72 Wilson, B.J. 180,370 Wimmett, T.F. 157.370 Wineland, D.J. 157,370 Wing, W.H. 360,370 Winter, H. 404,406,424 Winter, H., see Rietschel, H. 400,406,419, 423 Winter, H., see Schell, G. 406.423 Wiringa, R.B.,see Pandharipande, V.R. 175,368 Wise, H. 152, 153,370 Wolfle, P. 67, 72 Wolniewicz, L. 170,239,370 Wolniewicz, L., see Kolos, W. 164, 165, 169. 170,239,240,306,321.329,331.367 Wood, B.J., see Wise, H. 152. 153.370 Wright, R.S., see Jones Jr, J.T. 151. 236, 367 Wu, C., see Pan, J. 341,368 Wu, F.Y. 332,370 Wu, T.T. 338,370 Yaple, J., see Guyer. R.A. 180. 366 Young, R.A.,see Dionne, V.E. 84, 121. I34 Young, R.A.,see Locke, D.P. 79. 136 Yuan, J., see Pan, J. 341,368 Yurke. 8 . 163,206, 207,212. 213, 217, 218. 219.222.225,226,370 Zabolitsky, J.G. 328, 332, 333,370 Zabolitzky, J.G., see Emrich, K. 400,422 Zabolitzky, J.G., see Pandharipande, V.R. 175,368 Zane, L.I., see Guyer, R.A. 77.78.135 Zarate, H.G., see Mitrovic, B. 419,423 Zdetis, A.D. 407, 424 Zeller. R., see Dederichs, P.H. 391,42/ Zetik, D.F., see Poshuta, R.D. 80,136 Ziegelman. H.. see Schmid, E.W. 253,369 Zimmerman, D.S. 180, 181,361,370 Zitzewitz, P.W., see Hellwig, H. 157,366 Zotos. X. 55,62,63,68, 72 Zotos. X., see Maki, K. 62,66,68, 71

SUBJECT INDEX Alpha= 23 A-phase radial disgyration 53 Accommodator 194 Activation energy 91,93,94,95,97,98,99, 105, 106, 114,115, 125, 127, 129,130, 131, 132 Adatom 104, 105, 128, 129, 130, 133 Adsorption energy 154,174, 195,205,207 Adsorption isotherm 153,155,201,347, 349,350 Adsorption potential 175, 176 Adsorption time 178 Anderson-Toulouse vortex -doubly quantized 55 Angular momentum 156 Anisotropic energy gap 49 Anisotropy 43,63 -in3He-A 67 -of ionic mobility 68 APW 406 Areal density 15 Axisymmetricmodes 38 Axisymmetric structure 18 Axisymmetric vortices 22 Axisymmetry 17 B-phase energy gap 25 Ephase superfluid density 15 Band-structure Hamiltonian 393 Band-structure theory 401 Barrier 82 Barrier V, 121 bcc 'He 91, 119,127, 129 bcc4He 94,98 bcc-hcp phase. transition 92,95,96,97,130 BCS coherence length 12 BCS theory of superconductivity 10 Bending energy 12 Bloch-function representation 394 Bloch functions 394

Bohr magneton 156 Bolometric detection 196 Bose condensation 4 Bose statistics 149, 153, 185, 349,356 Bound dimer 167 Bound ion-vacancy state 132 Boundpair 62 Bound states 64 Bound states in two dimensions 66 Boundary conditions 24,36 Broken spin-orbit symmetry 23 Bubble decay 224 Bulk order parameter 12, 14 Bulk phases; 'He A and B 1 1 Bulk quadratic energy 12 C tensor 44 Canonically conjugate operators 33 Cartesian basis 16 Causal response 36 Centrifugal barriers 18,22 Characteristic length 25,27,30,48 Charge trapping 9 1 Charged droplets 123 Chemical potential 154 Circular I5 Circulating supertlow 29 Circulation 50,51 Classical theory 130 Clawing factor 193 Coexisting fluid phases 323 Coherence length 6,lO. 45 Commutation relations 33, 34, 35,63 Complex orbital component 42 Composite boson 149 Composite fermion 149 Compression experiments 229,233 Compression of bubbles 358 Compression ratio 193 Condensate amplitude 10

439

440

SUBJECT INDEX

Condensate fraction 336 Confinement 192 Constants 157 Constraints 45 Continuous cores 56 Cooper pairing in JP2state 23 Cooper pairs 10,43 Core 9.61 Core radius 61 Coulomb interaction 164 Coulomb pseudopotential 398,416 Coupling constant 29. 32 Critical angular velocity 8, 50 Critical pressure 154 Critical temperature 154 Crystal strain 90.99 Crystal structures 402 Current transients 118 Cut-off 398.417,418 Cut-off function 170 Cylindric potentials 65 de Boer parameter 321 Decay 203 Defects 3,7 Density-functional method 399 Density of states 395 Density profiles 336, 339 Depolarization 320 Deuterium 158.209 Diffusion coefficient 78,79,93,95, 102, 103. 104, 105. 106, 112, 122, 125, 126, 127, 128, 130, 131, 132, 133 Dilute ‘He-’He solution 123 Dipole coupling 24,48,49 Dipole coupling constant 13 Dipole energy 25, 33.58, 59 Dipole energy density 14,35,48 Dipole interaction 22 Dipole length 13.64,65 Dipole-locked bending coefficients 49 Dipole locking 52.58 Dirty limit 415 Disgyration 53 Dislocationloops 118, 119, 133 Dislocation ring 116 Dislocations 86, 118, 119, 120, 121, 122,

133 Dissociation 189 Dissociation energy 164 Distinct defects 52 Distorted wave 245 Distorted-wave Born approximation 3 I3 Distortions of the vortex structure 22 Distributed vorticity 48, 54, 55, 56 Double polarization 214 Double-polarized hydrogen 15 I, 203 Doubly quantized nonsingular lattice 68 Doubly quantized nonsingular texture 61 Doubly quantized nonsingular vortices 65, 66 Doubly quantized vortices 20,62 DWBA 313 Dynamical instability 340 Dyson’s equations 38 1,39 1,392 Effective free energy density 35,63 Effective Hamiltonian 33 Effective magnetic free-energy density 32, 33 Effective magnetic moment 161 Effective mass 14 Effective range potential 337 Effective rate constants 200 Eigenfunctions 36,64 Eigenstates of S, and L, 16 Eigenvalues 36.64 Elastically strained medium 103 Elastically stressed region 107 Electron cavity 121 Electron Green’s function 379 Electron self-energy 389,390 Electron-phonon interaction 380, 394,402, 403 Electron-phonon interaction function 404 Electron-phonon vertex 393,396 Electrostriction 79, 80.82, 83, 84 Eliashberg theory 411,412,414,415,416, 417 Energygap 43 Equation of state 357 ESR 162,216,227, 310,345 Euler angles 46,47,48, 52.53,57 Exchangexorrelation potential 399.402

SUBJECT INDEX Exchange interaction 169 Exchange recombination 262 Excitation potential 393,399,401 Explosive recombination 232 External rfmagnetic field 32 Faddeev equations 253 Feedback mechanism 21 Fermi contact interaction 156 Fermi-liquid corrections 32,45,49 Fermi surface variable 395 Field gradient 191 First-order transition between two types of vortexcores 21.40 Free-energy density 13 Free energy in rotating frame 50 Fusion 151,356 Gaspari-GyorKy analysis 406 Gauge substitution 50 Generators of spin rotations 33 GL bending coefficients 45 GL coherence length 12, 19,67 GL equations 19 GL formalism 13, 32 GL free-energy density 23 GL regime 14,42,48 GLtheory 49 Golden Rule 235 Gradient energy 49 Gradients of i 15 Grain boundaries 119, 120, 121 Ground-state energy 327,330 Ground state in cylinder 54 Grouptheoretic classification I9 GTB 239 GW-model 400 “Gyromagnetic” contribution 27 Gyromagnetic freeenergy density 29,30 Gyromagnetic ratios 156 Gyroscopic measurements 41 Harmonic-oscillator modes 340 hcp ’He 96,127 hcp4He 91,94,98, 105, 119, 121, 127, 129, 133 ’He atomic impurities 96, 104

’He impurities 127 IHein‘He 122 ‘Hebccphase 97 ‘He3He liquid solutions 124 Healing length 338 Heat piping 195 Heisenberg ferromagnet 52 Helical distortions 22 Helical form 44 Helium surface 150, 174 HEVAC 195 High-energy range 385 High-field limit 158, 163 High-field vortex 59 Higly strained transition region 82 Hole hopping 106 Hydrodynamic coefficients 49 “Hydrodynamic” form 45 Hydrodynamic modes 352 Hydrodynamic theory 45.55 Hydrogen maser 148,151,356 Hyperbolic texture 55 Hyperline energies and states 156,159 Hyperline frequency 185 Hyperline interaction 164 -hydrogen 248 - deuterium 249 Hyperfine states 158 Hysteresis 67 Identical-spin rotation effect 341 Impurities 130,216 Impuritons 78 Impurity relaxation 221 Indistinguishability 165 Inequivalent vacuum states 22 Interatomic interactions 163 Internal degrees of freedom 11 Intervortex spacing 8, IS, 28.67 Ion cavity structum in He 81,82,83, 84, 101, 102, 103, 122, 129 Ion drift velocity 109 Ion mobility 82.97, 101, 121 Ionpulse 68 Ion structure 81, 113, 131 Ion velocity 85, 101, 110, 11 1, 112, 113, 114, 115, 120,131

441

442

SUBJECT INDEX

Ions as quasiparticles 1 12, 132 Ion-vacancion complexes 1 13 Ion-vacancy bound states I14 Irrotational condition 46 Irrotational condition on vs 45 Irrotational frame 7 Isotopic impurities 76,77,93, 109, 125, 130, 131, 133 I-Vcharacteristics 85.90.91, 92 Jastrow wavefunction 328 Kapitza resistance 150, 151, 182, 359 KKR 406 Kolos-Wolniewicz potential I64 Kosterlitz-Thouless transition 349, 352 KVS 240

i texture

60,64,65,66 Lambshift 148 Large magnetic field 59 Lamior frequency 34,35,39,64,67. 344 Lamor’s theorem 10 Lattice strain 82, 83, 84 Lennard-Jones potential parameters 326 Linedefects 52 Line singularities 48 Line strength 162 Liquid barrier 81 Liquid ground state 334 LMTO 406 Local density potentials 401 Localization 121 Localized vacancies 110 Longitudinal NMR 67 Longitudinal NMR in rotating cylinders 41 Longitudinal response 34,35 Lower critical angular velocity 15 Magnetic anisotropy inside the vortex core 29 Magnetic compression 192 Magnetic confinement 194 Magnetic dipole field 13 Magneticdipole interaction 12, 25. 170, 23 1 Magnetic excitations 334, 340

Magnetic field 22, 24, 25 -tipped 31 Magnetic freeenergy density 48 Magnetic impurities 216, 221 Magnetic interaction 22 Magnetic moment 24,27, 156, 160 Magnetic susceptibility 63 Magnetic traps 360 Magnetization 160, 192 Mass enhancement 408,410 Matrix isolation 155 Matsubara frequencies 379. 381,414 Maximum density 214,235 McMillan-Hopfield constants 409 Mean power absorbed 37 Mezhanism 131 Mermin-Ho condition 45.48 Mermin-Ho texture 50 Metallic surfaces 152 Metastability 67 MH texture 53, 54,56, 57 M H vortices 55,58,62 Migdal’s theorem 386 Mixing parameters I58 Mixt-twist texture 55, 56 Mobility 84.85, 86,90,91,92,94, 95,96, 98.99, 104, 107, 108, 121, 122, 123, 124, 128, 133 Mobility tensor 98, 109. 112, 126, 130, 133 Monte Carlo method 329 Morse potential 329 MTtexture 57 M T vortices 58.62 n texture

14,23, 27,30, 39,41, 65 ri texture in cylinder 26 Nambu matrix 379 Nearest-neighbor separation 57 Negative ion 76, 80, 81, 82.92, 93,94,95, 99, 100, 101. 103, 105, 107, 108, 114, 116, 121, 123, 124,128, 129, 130, 131, 132 Negative-ion activation energies 104 Negative-ion cavity 108, 132 Negative-ion mobility 95 Negative-ion radius 122 Negatively charged cavity 81, 131 New ultra-low-temperature phases of 3He

SUBJECT INDEX 10

NMR 13.22, 162,217,224,225,345 NMR in a rotating container 32 NMR in rotating 'He-A 63 NMR of the rotating B phase 14 NMR satellite 65, 66 NMR shifts in bulk B phase 32 NMR spectrum 23.27 Node in the order parameter 7 Nonaxisymmetric potential 66 Nonaxisymmetric vortices 60 Nonaxisymmetry vortex textures 59 Noninertial frame 7 Nonquantized circulation 48 Nonsingular Anderson-Toulouse doubly quantized vortices 58 Nonsingular cores 5 1 Nonsingular texture 50, 51, 55, 56, 58 Nonsingular vortices 57, 58.62 Nonunitary character of order parameter 23 Nonunitary corrections 55 Nonunitary order parameter 27 Normalcore 52 Normal fluid 4 Notation 201 Nuclear magneton 156 ovortex 19,52 Orbital angular momentum of Cooper pair 43 Orbital part of the order parameter 33,45, 47 Orbital viscosity 69 Order parameter 3,4,5, 10, 13, 16, 17, 28, 33,61 - single-valued 5 - spatial rotation of orbital part 44 Orienting effect 28 Orienting effect of magnetic field 49 Orienting energy 30 Orienting term 48 Ortho-Hz 254 Ortho hydrogen 166 Ortho-positronium 80,84, 129 Ortho-positronium cavity 129 Orthonormal triad 47

443

Ortho-para classification I72 Overlap integral 37,65 pwave pairing 10 Pairarrelation function 328 Para-H2 254 Para hydrogen 166 Particlehole asymmetry 23 Permanent ground-state orbital angular momentum 55 Phaseangle 52 Phase diagram 12 Phase factor 43 Phase shift 302,410 Phasevortex 52 Phenomenological parameters 32 Phonon-assisted ion motion 108 Phonon Green's function 380, 381,405 Phonon line width 408 Phonon self-energy 388,389 Plane-wave Born approximation 3 I3 Plastic flow 90,108, 127 - phonon-assisted I28 Polar core 61 Polar phase 53,58 Polarization 189 Polycritical point 21 Positive ion 79, 81,82, 91, 92.93, 94, 95, 98,99, 100, 106, 108, 114, 115, 119, 121, 122,123, 124, 127, 130, 131,132 Positive-ion diffusion coefficient 96 Positive-ion drift velocity 1 14 Positive-ion mobility 129, 130 Positive-ion velocity 100, I18 Potential barrier V,, 84 Pressure gauge 199 Principal-value integral 37 Pseudo-potential 401,406 PWBA 313 QTCS 321 Quadratic contribution 11 Quantization of magnetic flux 3.10 Quantization of superfiuid circulation 3, 5, 10, 15, 16, 52 Quantized vortices 6,23,52, 53 Quantum theory of corresponding states

444 320 Quartic contributions 1 I Quasiparticle renormalization factor

SUBJECT INDEX

392

Radial disgyration 58, 61 Radius R, of positive ion 80 Rate constant 152, 236,257 Rate equation 151. 200, 309 Rayleigh-Ritz quotient 37 Recombination 151, 207, 21 I . 236 -dipolar 280 -. field dependence 258.288 - rate constants 294 -- surface 255, 290 -theory 235 Rectilinear vortices 8 Relaxation -. dipolar 31 I -- electronic 223 - electronic spin 3 I9 -- impurity 220 -intrinsic 218 --spin 296 - spin exchange 303 -surface 317 -theory 235 Resonance 162 Resonance recombination 239 Resonant frequencies 37 rf field 63.67, 124 Rigid-ion approximation 403.404.406 Rigid muffin tin 404 Ripplons 179, 352 Rotating cylinder 39 Rotating frame 8 , 9 Rotating neutron stars 23 Rotating supertluid ‘He 4 s-wave scattering length 337 Schriidinger equation 35.36. 37, 50, 64, 65. 67 SCL current 86.90, 100, 114 SCL current mobility 89 SCL technique 127 Screening 403, 404 Second-order recombination 238,241 Second-sound surface mode 354

Selection rule 237 Self-energies 382 Single-atom properties 156 Single-valuedness 5 I Singlet state 10, 164 Singly quantized singular M H texture 59 Singly quantized vortices 8.20, 54, 56 Singular cores 5 1 Singular i texture 53 Singular singly quantized vortices 5 1, 58, 62, 66 Singular superflow 53 Singular textures 5 1. 53 Softcore 59 Solid-body rotation 7, 8, 50 Space-charge limited current transients 88 Specific-heat anomaly 11 Spectator atom 232 Spectator index 261 S p t r u m in bounded container 37 Spin current 342 Spin degeneracy 326 Spin density 32, 33, 63 Spin diffusion coefficient 342 Spin exchange 303,361 Spin-fluctation model 21 Spin-fluctuation effects 41 1 Spin part of the order parameter 63 Spin polarization 160, 173. 343 Spin-polarized deuterium (Dl) 148 Spin-polarized hydrogen (HJ) 148, 151, 355 Spin-polarized tritium (TI) 148 Spin waves 39, 151, 226, 340 Spin waves in a ferromagnet 5 Square lattice 56,62 Slabilkition 153, 173, 188 Stabilization of DJ 204 Stabilization of H i 189 State polarization 160 Sticking probability 178, 181 Strain field 132, 133 Strained crystals 91 Strong-coupling theory 12,21,41 I Structure of cavity 129 Structure of ions 76, 109 Superconductors 10 - type-I1 10

SUBJECT INDEX Supercurrent 44 Superfluid density tensor 44 Superfluid turbulence 69 Superfluid velocity 5,43,47,48, 52, 55 Superfluid velocity for nonuniform flow 44 Superfluid velocity in the B phase 15 Superfluid vorticity 48 Surface chemical potential 347 Surface collisions 179 Surface coverage 154, 178 Surface energy 25 Surface layer 26,38 Surface recombination 188 Surface recombination rate constant 208, 213 Surface region 30 Surface tension 23 1,358 Surfaces 346 Symmetrized states 241 T-operator 246 Texture of the B phase 13 Textures in rotating container 24,68 - topologically equivalent 52 Theory of positive-ion mobility 123 Thermalescape 203 Thermodynamic conjugate 322 Third-order decay constant 231 Third-order recombination 238,259 Third-sound waves 353 Three-body center of mass 241 Three-body recombination 361 Tight-binding approximation 65,406 Time of flight 86,89,90,94 Tipped magnetic field 62, 66 Topological constraints 62 Topologically distinct rotations 22 Torsional oscillations 41 Transient response 87,89,90,97, 11 5 Transient spacecharge limited (SCL)currents 85,86 Transit time 88,9 I Transition in vortex cores 69 Transverse excitation 64 Transverse modes 38 Transverse NMR 39,65,66 Transverse NMR satellites 40

445

Transverse response 34, 35 Trapped charge 86,89,91,94,99, 127 Triangular lattice 5 I , 58 Triplet 164 Triplet p-wave order parameter 42 Triplet pairing 1 I , 413,414 Triplet potential 173 Tunnelling 77, 78, 79, 105, 112, 113, 127, 132, 133 Two-dimensional harmonic oscillator 39 Two-dimensional superfluidity 150,351, 363 Two-dimensional surface 154 Two-particle Green’s function 413 Two quanta of circulation 61 Type-I lattice 58, 62, 67, 68 Type-I texture 57 Type-I1 superconductors 67 Ultrasonic attenuation 62.67 Uniform field 63 Uniform translation 43 Unitarity 23 Unsymmetrizd states 242 Upper critical field 5 1 v vortex 19,24,53 Vacancies 76,77, 101, 103, 104. 105, 106, 107,108, 109, 110,114, 120, 125, 127, 128, 130, 131, 132, 133 Vacancion 78,109, 110, 111, 112,113,122, 128, 131, 132 Vacancy-assisted mechanism 105, 122, 123, 129, 130 Vacancy-assisted mobility 103, 106 Vacancy-assisted motion 128, 132, 133 Vacancy mechanism 133 Vacancy waves 122 van der Waals interaction 166 van der Waals recombination 241,246,255 Vapor compression 194 Vibration-rotation levels 240 Virial expansion 351 Void radius 129 Vortexcore 3,7, 16,23,27,42,49, 53 Vortexcore radius 15 Vortex core structure 18

446 Vortex density 8 Vortex dynamics 69 Vortex in the A phase 43 Vortex lattice 9 Vortexlines 30 Vortex nucleation 4 Vortex-orientation contribution 28 Vortex orientation energy 29 Vortex ring I 16, 1 17. I18 Vortex waves 69 Vortices in superconductors 10

SUBJECT INDEX Weak-coupling limit 12 Weakcoupling parameters 19 Weakly interacting Bose gas 337 Wigner spin rule 306 Wigner-Seitz cell 9 Wigner-Seitz model 58 WSmvdel 61 zvortex 58 Zerosound 42 f?-decay 356

Wall coverage 1V4 Wall recombination 194 Weakcoupling GL regime 35

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