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PROGRESS IN LOW TEMPERATURE PHYSICS XIV

This Page Intentionally Left Blank

PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

W.P. HALPERIN Chairperson, Department of Physics and Astronomy Northwestern University, Evanston, IL, USA

VOLUME XlV

1995 ELSEVIER AMSTERDAM, LAUSANNE, NEW YORK" OXFORD, SHANNON"TOKYO

9 Elsevier Science B.V., 1995. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA O1923, USA. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the copyright owner, unless otherwise specified. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions of ideas contained in the material herein. ISBN: 0 444 82233 X

PUBLISHED BY: ELSEVIER SCIENCE B.V. P.O. BOX 211 1000 AE AMSTERDAM THE NETHERLANDS

Printed on acid-free paper PRINTED IN THE NETHERLANDS

PREFACE The fourteenth volume of Progress in Low Temperature Physics marks 40 years of achievement that have appeared in this book series, charting developments in low temperature physics in what has become a highly diversified and increasingly important subject area. As our literature becomes progressively larger and more interdisciplinary many of us have serious concerns that we are not able to keep ourselves au courant. And now the growing number of conference proceedings, preprints, periodicals, books, and popular journal articles have been joined by various electronic forms of dissemination of our research. In this environment, the book series Progress in Low Temperature Physics assumes a particular responsibility to continue the strong tradition of excellent reviews, guiding our reading of the literature and providing direction for future research possibilities. In the present volume of this series you find the main theme to be research on superfluid and adsorbed phases of helium. In chapter 1, Peter McClintock and Roger Bowley review one of the essential characteristics of superfluid 4He, "The Landau Critical Velocity". Landau showed that the critical velocity is determined by elementary excitations, rotons in the case of superfluid 4He. However, it was soon discovered that vortex nucleation, rather than the creation of elementary excitations, dominated dissipation in most experiments. Progress came from measurements of negative ion transport in superfluid helium at low temperatures and modest pressures from which our understanding of critical velocity is now consistent with Landau's theory. Still, there remain challenging problems such as why rotons appear to be created in pairs. Yuriy Bunkov reviews the amazing properties of coherent spin dynamics in superfluid 3He in chapter 2, "Spin Supercurrents and Novel Properties of NMR in 3He". Many of the experiments discussed here were performed at the Low Temperature Laboratory of the Kapitza Institute for Physical Problems. One of the consequences of triplet state superfluidity in 3He includes formation of homogeneous precessing domains in a magnetic field gradient, Josephson spin current phenomena, and vortices of spin supercurrents. New research directions are suggested which make use of these spin supercurrents to investigate rotational states of 3He and the dynamics of the transition between superfluid A- and B-phases. In superfluid 3He one finds a unique situation with a number of thermodynamic transitions between different superfluid states. However, it is a puzzle to

vi understand the mechanism for their nucleation. This fascinating low temperature problem was identified some years ago by Tony Leggett and is not yet understood. In chapter 3, Peter Schiffer, Doug Osheroff, and Tony Leggett describe the current experimental and theoretical situation. They have discovered that ionizing radiation can serve to nucleate the A- to B-phase transition, and that this process is consistent with the theoretical interpretation referred to as the baked Alaska model. The Low Temperature group at Stanford discovered a means to maintain the superfluid A-phase in a metastable condition, supercooled at low field and low temperatures. The technique is a key feature of their experimental work on the use of ionizing radiation to study nucleation and also has broad potential application to research on the low temperature properties of superfluid 3He-A. However, it remains for future work to determine the precise role of surfaces and textures in the nucleation process. Properties of phases of 3He adsorbed on graphite are discussed by Henri Godfrin and Hans Lauter in their chapter, "Experimental Properties of 3He Adsorbed on Graphite". This work emphasizes the structural aspects of adsorbed phases with considerable input from neutron scattering. The importance of understanding structure is the key to the related work on two-dimensional magnetism of the solid layers and superfluidity of helium films on this important substrate material. Since the role of the substrate is central to the phenomena observed in adsorbed helium phases, its understanding paves the way for further research on two-dimensional helium systems. In a complementary chapter Bob Hallock reviews "The Properties of Multilayer 3He-aHe Mixture Films". These two-dimensional quantum fluids can be understood in terms of the energetics of a 3He impurity in 4He. Both the free surface and the role of the substrate are crucial. Predictions for multiple 3He surface states and possible 3He superfluid phases in thin mixture films are pointed out. The recently discovered wetting transition of helium films on alkali metals is discussed including sensitivity to the isotopic mixture ratio. This chapter will serve as an important base for future work on two-dimensional superfluids. It is particularly my pleasure to acknowledge Douglas Brewer's many contributions as an advocate of low temperature physics while he served as editor of this series. Douglas played this key role for over 15 years following C.J. Gorter who founded the series and edited the first 6 volumes. Douglas Brewer's contributions also include the early stages of organization of the present volume. Under their stewardship, we have come to expect that Progress in Low Temperature Physics will bring into focus some of the important current themes of research at low temperature and I hope that this rich tradition can continue.

Bill Halperin Evanston, October 1995

CONTENTS VOLUME XIV Preface ..........................................................................................................

v

Contents ........................................................................................................

vii

Contents o f previous volumes .......................................................................

xi

Ch. I. The Landau critical velocity, P.V.E. M c C l i n w c k a n d R.M. Bowley ................................................ 1. Introduction ......................................................................................................................... 2. Quest for the Landau critical velocity ................................................................................. 2.1. The dispersion curve and excitation creation in He-II ................................................. 2.2. Critical velocity measurements in He-II ...................................................................... 2.3. Field emission in liquid helium .................................................................................... 2.4. Measurement of ionic drift velocities .......................................................................... 2.5. Observation of the Landau critical velocity ................................................................. 3. Theory of roton creation in He-ll ........................................................................................ 3.1. Early theories of supercritical dissipation .................................................................... 3.2. Roton creation by a light object ................................................................................... 3.3. Theory of single-roton creation .................................................................................... 3.4. Theory of roton pair creation ....................................................................................... 3.5. Comparison of the theory with experiment .................................................................. 3.6. A regime of negative resistance? ................................................................................. 3.7. Roton creation in extremely weak electric fields ......................................................... 4. Measurement of the Landau critical velocity ...................................................................... 4.1. Experimental details ..................................................................................................... 4.2. Velocity measurements in weak electric fields ............................................................ 4.3. The critical velocity ..................................................................................................... 4.4. The matrix element for roton pair creation .................................................................. 5. Roton creation at extreme supercritical velocities ............................................................... 5.1. Velocity measurements in high electric fields ............................................................. 5.2. Comparison with theory ............................................................................................... 6. Roton creation by "fast" ions ............................................................................................... 7. Conclusion ........................................................................................................................... References ................................................................................................................................

vii

3 5 5 8 11 13 15 18 18 19 23 28 33 35 38 40 40 46 50 53 54 54 55 61 65 66

viii

CONTENTS

Ch. 2. Spin supercurrent and novel properties of NMR in 3He, Yu.M. Bunkov ....................................................................................

69

1. Introduction ......................................................................................................................... 2. Basic properties ................................................................................................................... 2.1. Spatially uniform NMR ............................................................................................... 2.2. Spin supercurrent ......................................................................................................... 3. Experimental methods ......................................................................................................... 4. NMR and spin supercurrent in 3He-B ................................................................................ 4.1. Pulsed NMR ................................................................................................................. 4.2. CW NMR ..................................................................................................................... 4.3. Processes of magnetic relaxation ................................................................................. 4.3.1. Spin diffusion and intrinsic relaxation .............................................................. 4.3.2. Surface relaxation .............................................................................................. 4.3.3. Catastrophic relaxation ...................................................................................... 4.4. HPD oscillations .......................................................................................................... 5. Steady spin supercurrent ...................................................................................................... 5.1. Spin supercurrent in a channel ..................................................................................... 5.2. Phase slippage .............................................................................................................. 5.3. Josephson phenomena .................................................................................................. 5.4. Spin supercurrent vortex .............................................................................................. 6. Spin supercurrent in 3 H e - A ................................................................................................ 6.1. Instability of homogeneous precession ........................................................................ 7. Spin supercurrent at propagating A - B boundary ................................................................ 8. Conclusion ........................................................................................................................... Acknowledgments .................................................................................................................... References ................................................................................................................................

71 75 79 85 93 98 98 103 107 107 112 114 119 124 124 128 132 134 138 139 146 152 154 154

Ch. 3. Nucleation of the AB transition in superfluid 3He: experimental and theoretical considerations, P. Schiffer, D.D. Osheroff and A.J. Leggett ......................................

159

1. 2. 3. 4.

Introduction .......................................................................................................................... Background of the B phase nucleation problem ................................................................... Experimental history of the B phase nucleation problem ..................................................... The recent experiments at Stanford ...................................................................................... 4.1. Experimental design ..................................................................................................... 4.2. Initial B phase nucleation observations ........................................................................ 4.3. B phase nucleation by irradiation ................................................................................. 4.3.1. Data acquisition ................................................................................................. 4.3.2. Dependence on radiation type ............................................................................ 4.3.3. Dependence on temperature and magnetic field ................................................ 4.4. Monte Carlo simulations .............................................................................................. 5. The baked Alaska model: theoretical considerations ........................................................... 6. Conclusions .......................................................................................................................... Acknowledgments .................................................................................................................... Appendix A: Probability of depositing energy E in a radius much less than the "typical" radius R(E) ...........................................................................................................................

161 163 167 170 170 174 177 177 179 181 184 190 200 203 204

CONTENTS

ix

Appendix B: Relaxation of the magnetization by flow ............................................................ Appendix C: Analytical model of the thermodynamics of superfluid 3He .............................. References ................................................................................................................................

206 206 210

Ch. 4. Experimental properties of 3He adsorbed on graphite, H. Godfrin and H.-J. Lauter .............................................................

213

1. Introduction .......................................................................................................................... 2. Graphite substrates ............................................................................................................... 2.1. Exfoliated graphite ....................................................................................................... 2.2. Physical properties of exfoliated graphite .................................................................... 2.2.1. General properties of different exfoliated graphites ........................................... 2.2.2. Chemical impurities ........................................................................................... 2.2.3. Structural properties ........................................................................................... 2.2.4. Specific area ....................................................................................................... 2.2.5. Electronic properties .......................................................................................... 2.2.6. Specific heat ....................................................................................................... 2.2.7. Electrical conductivity ....................................................................................... 2.2.8. Thermal conductivity ......................................................................................... 2.2.9. Magnetic susceptibility ...................................................................................... 3. Physical adsorption of 3He on graphite ................................................................................ 3.1. Adsorption potentials .................................................................................................... 3.2. Interaction potential and zero point energy in adsorbed layers .................................... 3.3. Layering ........................................................................................................................ 3.4. Coverage scales ............................................................................................................ 4. Experimental techniques of surface Physics at low temperatures ........................................ 4.1. Experimental details ..................................................................................................... 4. I. I. Experimental cells .............................................................................................. 4.1.2. Preparation of the adsorbed 3He sample ............................................................ 4.2 Adsorption isotherms ..................................................................................................... 4.3. Heat capacity ................................................................................................................ 4.3.1. Guide to the literature ........................................................................................ 4.3.2. Techniques ......................................................................................................... 4.4. Nuclear magnetic resonance ......................................................................................... 4.4.1. Guide to the literature ........................................................................................ 4.4.2. Techniques ......................................................................................................... 4.5. Neutron scattering ......................................................................................................... 4.5.1. Guide to the literature ....................................................................................... 4.5.2. Techniques ........................................................................................................ 4.6. Other techniques .......................................................................................................... 5. Structure and phase diagram of the adsorbed films ............................................................. 5.1. Submonolayer coverages ............................................................................................. 5. I.I. Very low coverages ........................................................................................... 5.1.2. The first layer fluid phase ................................................................................. 5.1.3. The commensurate phase ................................................................................... 5.1.4. The intermediate coverage region ...................................................................... 5.1.5. The incommensurate phase ................................................................................ 5.2. Second layer ................................................................................................................. 5.2.1. The second layer fluid phase ..............................................................................

215 215 216 217 217 217 218 219 219 220 221 226 228 229 230 233 235 237 240 241 241 245 247 248 248 252 253 253 256 261 261 262 269 270 270 270 272 279 285 288 292 292

x

CONTENTS

5.2.2. Second layer solidification ................................................................................. 5.2.3. The second layer commensurate phase R2a ....................................................... 5.2.4. Remarks about the second layer density ............................................................ 5.2.5. The second layer intermediate region (0.178 ,~-2 to 0.26 ,~-2) ......................... 5.2.6. The second layer incommensurate phase above n = 0.26 ]~-2 ........................... 5.3. Multilayer films ............................................................................................................ 6. Conclusions .......................................................................................................................... References ................................................................................................................................

296 297 300 301 306 308 312 314

Ch. 5. The properties o f multilayer 3He-4He mixture films, R.B. Hallock ......................................................................................

321

1. Introduction ......................................................................................................................... 2. Bulk interfaces ...................................................................................................................... 2.1. The bulk free surface .................................................................................................... 2.2. The bulk-wall interface ................................................................................................. 2.3. Other surfaces ............................................................................................................... 3. H e l i u m films ......................................................................................................................... 3.1. Theoretical overview .................................................................................................... 3.2. Thickness scales ........................................................................................................... 3.3. Energetics experiments ................................................................................................. 3.3.1. Heat capacity experiments ................................................................................. 3.3.2. Nuclear magnetic resonance experiments .......................................................... 3.4. Other experiments ......................................................................................................... 3.4.1. Third sound experiments .................................................................................... 3.4.2. Oscillator measurements .................................................................................... 3.4.3. Selected other experiments ................................................................................ 3.5. Future directions ........................................................................................................... 4. S u m m a r y .............................................................................................................................. A c k n o w l e d g m e n t s .................................................................................................................... References ................................................................................................................................

323 324 324 329 333 334 334 344 345 345 355 387 387 416 425 433 435 435 436

A u t h o r Index .................................................................................................

445

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

463

CONTENTS OF PREVIOUS VOLUMES

Volumes I - V I, edited by C.J Gorter Volume I (1955) I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. XVIII.

The two fluid model for superconductors and helium II, C.J. Goner ............................................................................ Application of quantum mechanics to liquid helium, R.P. Feynman ............................................................................... Rayleigh disks in liquid helium II, J.R. Pellam .................... Oscillating disks and rotating cylinders in liquid helium II, A.C. Hollis Hallett ................................................................ The low temperature properties of helium three, E.F. Hammel ................................................................................ Liquid mixtures of helium three and four, J.M. Beenakker and K.W. Taconis ................................................................. The magnetic threshold curve of superconductors, B. Serin ..................................................................................... The effect of pressure and of stress on superconductivity, C.F. Squire ............................................................................ Kinetics of the phase transition in superconductors, T.E. Faber and A.B. Pippard ........................................................ Heat conduction in superconductors, K. Mendelssohn ........ The electronic specific heat in metals, J.G. Daunt ................ Paramagnetic crystals in use for low temperature research, A.H. Cooke ........................................................................... Antiferromagnetic crystals, N.J. Poulis and CJ. Gorter ........ Adiabatic demagnetization, D. de Klerk and M.J. Steenland ....................................................................................... Theoretical remarks on ferromagnetism at low temperatures, L. N6el ........................................................................ Experimental research on ferromagnetism at very low temperatures, L. Weil ........................................................... Velocity and absorption of sound in condensed gases, A. van Itterbeek ......................................................................... Transport phenomena in gases at low temperatures, J. de Boer ......................................................................................

1-16 17-53 54-63 64-77 78-107 108-137 138-150 151-158 159-183 184-201 202-223 224-244 245-272 272-335 336-344 345-354 355-380 3 81-406

xii

CONTENTS OF PREVIOUS VOLUMES

Volume H (195 7)

II. III. W~ V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV

Quantum effects and exchange effects on the thermodynamic properties of liquid helium, J. de Boer ................... Liquid helium below 1~ H.C. Kramers ............................. Transport phenomena of liquid helium II in slits and capillaries, P. Winkel and D.H.N. Wansink ......................... Helium films, K.R. Atkins .................................................... Superconductivity in the periodic system, B.T. Matthias ..... Electron transport phenomena in metals, E.H. Sondheimer. Semiconductors at low temperatures, VA. Johnson and K. Lark-Horovitz .................................................................. The de Haas-van Alphen effect, D. Shoenberg .................... Paramagnetic relaxation, C.J. Gorter .................................... Orientation of atomic nuclei at low temperatures, M.J. Steenland and H.A. Tolhoek ................................................ Solid helium, C. Domb and J.S. Dugdale ............................. Some physical properties of the rare earth metals, F.H. Spedding, S. Legvold, A.H. Daane and L.D. Jennings ........ The representation of specific heat and thermal expansion data of simple solids, D. Bijl ................................................ 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) I. II. \III. W~ V. VI. VII. VIII.

IX. X. XI.

Vortex lines in liquid helium II, W.F. Vinen ........................ Helium ions in liquid helium II, G. Careri ........................... The nature of the ;t-transition in liquid helium, M.J. Buckingham and W.M. Fairbank ......................................... Liquid and solid 3He, E.R. Grilly and E.F. Hammel ............ 3He cryostats, K.W. Taconis ................................................. Recent developments in superconductivity, J. Bardeen and J.R. Schrieffer ....................................................................... Electron resonances in metals, M.Ya. Azbel' and I.M. Lifshitz .................................................................................. Orientation of atomic nuclei at low temperatures II, W.J. Huiskamp and H.A. Tolhoek ................................................ Solid state masers, N. Bloembergen ..................................... 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

1-57 58-79 80-112 113-152 153-169 170-287 288-332 333-395 396--429 430--453 454-480

CONTENTS OF PREVIOUS VOLUMES

xiii

Volume IV (1964)

II.

III. IV. V. VI. VII.

VIII. IX. X.

Critical velocities and vortices in superfluid helium, V.P. Peshkov ................................................................................ Equilibrium properties of liquid and solid mixtures of helium three and four, K.W. Taconis and R. de Bruyn Ouboter ................................................................................. The superconducting energy gap, D.H. Douglass Jr and L.M. Falicov ......................................................................... Anomalies in dilute metallic solutions of transition elements, G.J. van den Berg ................................................. Magnetic structures of heavy rare-earth metals, Kei Yosida Magnetic transitions, C. Domb and A.R. Miedema .............. The rare earth garnets, L. N6el, R. Pauthenet and B. Dreyfus ................................................................................. Dynamic polarization of nuclear targets, A. Abragam and M. Borghini .......................................................................... Thermal expansion of solids, J.G. Collins and G.K. White.. The 1962 3He scale of temperatures, T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde ....................

1-37

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

Volume V (1967)

II.

III. IV.

V. VI.

VII.

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 II, E.L. Andronikashvili and Yu.G. Mamaladze ........................................................................... Study of the superconductive mixed state by neutrondiffraction, D. Gribier, B. Jacrot, L. Madhav Rao and B. Farnoux ................................................................................. Radiofrequency size effects in metals, V.F. Gantmakher ..... 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 ........................................................

1--43

44-78 79-160

161-180 181-234 235-286 287-322

xiv

CONTENTS OF PREVIOUS VOLUMES

Volume VI (1970)

II. III.

IV. V~

VI.

VII. VIII. IX. X~

Intrinsic critical velocities in superfluid helium, J.S. Langer and J.D. Reppy ..................................................................... Third sound, K.R. Atkins and I. Rudnick ............................. Experimental properties of pure He 3 and dilute solutions of He 3 in superfluid He 4 at very low temperatures. Application to dilution refrigeration, J.C. Wheatley ......................... Pressure effects in superconductors, R.I. Boughton, J.L. Olsen and C. Palmy .............................................................. Superconductivity in semiconductors and semi-metals, J.K. Hulm, M. Ashkin, D.W. Deis and C.K. Jones ...................... Superconducting point contacts weakly connecting two superconductors, R. de Bruyn Ouboter and A.Th.A.M. de Waele .................................................................................... Superconductivity above the transition temperature, R.E. Glover III .............................................................................. Critical behaviour in magnetic crystals, R.F. Wielinga ........ Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities, G.R. Khutsishvili ........ The international practical temperature scale of 1968, M. Durieux .................................................................................

1-35 37-76

77-161 163-203 205-242

243-290 291-332 333-373 375-404 405-425

CONTENTS OF PREVIOUS VOLUMES

XV

Volumes VII-XIII, edited by D.E. Brewer Volume VII (1978) Further experimental properties of superfluid 3He, J.C. Wheatley ............................................................................... Spin and orbital dynamics of superfluid 3He, W.E. Brinkman and M.C. Cross ............................................................. Sound propagation and kinetic coefficients in superfluid 3He, P. W61fle ....................................................................... The free surface of liquid helium, D.O. Edwards and W.F. Saam ..................................................................................... Two-dimensional physics, J.M. Kosterlitz and D.J. Thouless ........................................................................................ First and second order phase transitions of moderately small superconductors in a magnetic field, H.J. Fink, D.S. McLachlan and B. Rothberg Bibby ...................................... Properties of the A-15 compounds and one-dimensionality, L.P. Gor'kov ......................................................................... Low temperature properties of Kondo alloys, G. Griiner and A. Zawadowski .............................................................. Application of low temperature nuclear orientation to metals with magnetic impurities, J. Flouquet .......................

.

~

,,

Q

,

,

,

,

1-103 105-190 191-281 283-369 371-433

435-516 517-589 591-647 649-746

Volume VIII (1982) ~

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 IX (1985)

,

Structure, distributions and dynamics of vortices in helium II, W.I. Glaberson and R.J. Donnelly ................................... The hydrodynamics of superfluid 3He, H.E. Hall and J.R. Hook ..................................................................................... Thermal and elastic anomalies in glasses at low temperatures, S. Hunklinger and A.K. Raychaudhuri ..........

1-142 143-264 265-344

xvi

CONTENTS OF PREVIOUS VOLUMES

Volume X (1986) Vortices in rotating superfluid 3He, A.L. Fetter ................... Charge motion in solid helium, A.J. Dahm .......................... Spin-polarized atomic hydrogen, I.F. Silvera and J.T.M. Walraven .............................................................................. Principles of ab initio calculations of superconducting transition temperatures, D. Rainer ........................................

~

2. 3. .

1-72 73-137 139-370 371-424

Volume XI (1987) Spin-polarized 3He-aHe solutions, A.E. Meyerovich ........... Long mean free paths in quantum fluids, H. Smith .............. The surface of helium crystals, S.G. Lipson and E. Polturak Neutron scattering by 4He and 3He, E.C. Svensson and VF. Sears ..................................................................................... Characteristic features of heavy-electron materials, H.R.

~

2. 3. 4. ~

O

t

t

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

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,-"

1-73 75-125 127-188 189-214 215-289

Volume XII (I 989) High-temperature superconductivity: some remarks, V.L. Ginzburg ............................................................................... Properties of strongly spin-polarized 3He gas, D.S. Betts, F. Lalofi and M. Leduc .............................................................. Kapitza thermal boundary resistance and interactions of helium quasiparticles with surfaces, T. Nakayama ............... Current oscillations and interference effects in driven charge density wave condensates, G. Griiner ....................... Multi-SQUID devices and their applications, R. Ilmoniemi and J. Knuutila ......................................................................

~

~

~

.

1-44 45-114 115-194 195-296 271-339

Volume XIH (1992)

~

,

~

.

Critical behavior and scaling of confined 4He, F.M. Gasparini and I. Rhee .......................................................... Ultrasonic spectroscopy of the order parameter collective modes of superfluid 3He, E.R. Dobbs and J. Saunders ......... Thermodynamics and hydrodynamics of 3He-4He mixtures, A.Th.A.M. de Waele and J.G.M. Kuerten .................. Quantum phenomena in circuits at low temperatures, T.P. Spiller, T.D. Clark, R.J. Prance and A. Widom .................... The specific heat of high-Tc superconductors, N.E. Phillips, R.A. Fisher and J.E. Gordon ................................................

1-90 91-165 167-218 219-265 267-357

CHAFFER 1

T I ~ LANDAU CRITICAL VELOCITY BY

P.V.E. McCLINTOCK School of Physics and Chemistry, Lancaster University, Lancaster, LA1 4YB, UK

and R.M. BOWLEY Department of Physics, The University, Nottingham, NG7 2RD, UK

Progress in Low Temperature Physics, Volume XIV Edited by W.P. Halperin 9 Elsevier Science B.V., 1995. All rights reserved

Contents 1. Introduction ......................................................................................................................... 2. Quest for the Landau critical velocity ................................................................................. 2.1. The dispersion curve and excitation creation in He-ll ................................................. 2.2. Critical velocity measurements in He-ll ...................................................................... 2.3. Field emission in liquid helium .................................................................................... 2.4. Measurement of ionic drift velocities .......................................................................... 2.5. Observation of the Landau critical velocity ................................................................. 3. Theory of roton creation in He-ll ........................................................................................ 3.1. Early theories of supercritical dissipation .................................................................... 3.2. Roton creation by a light object ................................................................................... 3.3. Theory of single-roton creation .................................................................................... 3.4. Theory of roton pair creation ....................................................................................... 3.5. Comparison of the theory with experiment .................................................................. 3.6. A regime of negative resistance? ................................................................................. 3.7. Roton creation in extremely weak electric fields ......................................................... 4. Measurement of the Landau critical velocity ...................................................................... 4.1. Experimental details ..................................................................................................... 4.2. Velocity measurements in weak electric fields ............................................................ 4.3. The critical velocity ..................................................................................................... 4.4. The matrix element for roton pair creation .................................................................. 5. Roton creation at extreme supercritical velocities ............................................................... 5.1. Velocity measurements in high electric fields ............................................................. 5.2. Comparison with theory ............................................................................................... 6. Roton creation by "fast" ions ............................................................................................... 7. Conclusion ........................................................................................................................... References ................................................................................................................................

3 5 5 8 11 13 15 18 18 19 23 28 33 35 38 40 40 46 5O 53 54 54 55 61 65 66

1. Introduction The Landau critical velocity for roton creation, 1)L, representing the minimum velocity at which a moving object can create elementary excitations in superfluid 4He, is one of the fundamental parameters of the liquid. Originally predicted by Landau (1941, 1947) as part of his celebrated explanation of superfluidity, it subsequently proved to be surprisingly difficult to observe (on account of complications associated with quantized vortices; see below). Experimental evidence for the reality of the Landau critical velocity did not start to emerge until the work of Meyer and Reif (1961), Rayfield (1966, 1968), Doake and Gribbon (1969) and Phillips and McClintock (1974), based on the use of negative ions; the magnitude of VL was eventually measured by Ellis et al. (1980b) and, more accurately and over a wider range of pressures, by Ellis and McClintock (1985). It should be noted at the outset that a finite value of VL is a necessary, but not sufficient, condition for superfluidity. It is not sufficient because, in addition to elementary excitations, there may also be a possibility of converting kinetic energy into other, metastable, states of the liquid, such as vortices. In practice, the most appropriate set of criteria for superfluidity will usually depend on the type of problem being considered. It is helpful, in this context, to recall Vinen's (1983) identification of the two distinct traditions or threads of development in research on superfluid 4He, dating from around the time of the original discovery (Kapitza 1938; Allen and Jones 1938; Keesom and Macwood 1938; Daunt and Mendelssohn 1938) of superfluidity. The first thread originated in London's (1938, 1954) suggestion that liquid 4He should be regarded as a Bose-condensed system, which can therefore be described by a single macroscopic wavefunction. The second one started from Landau's (1941, 1947) picture of the liquid as an inert background containing (for finite temperature) a gas of excitations. These two seemingly very different perceptions of He-II were effectively unified by the work of, especially, Bogoliubov (1947) and Feynman (1955), and are now understood to represent different aspects of the same underlying physical reality. Nonetheless, it remains true that either one or other of the two pictures will usually be found more apposite to any given type of problem. In considering rotation or annular flow, for example, London's macroscopic wave function normally provides the more revealing and fruitful approach (Leggett 1991) and, because VL is SO enormous (typically -50 ms-1, depending on pressure; see below) compared to other relevant critical velocities, it can often safely

4

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w1

be ignored. In dealing with the movement of a small object through the superfluid, on the other hand, as in the present chapter, it is the Landau picture that is usually the more helpful. In this chapter, we review the saga of the Landau critical velocity, describing how it was measured and discussing in some detail the process of roton creation that sets in for velocities above VL. In doing so, we will make frequent reference to six Lancaster/Nottingham papers in Philosophical Transactions of the Royal Society in which many of the original ideas were developed. For convenience, we will cite them as follows, using Roman numerals: I Allum et al. (1977) II Ellis et al. (1980a) III Bowley et al. (1982) IV Nancolas et al. (1985a) V Ellis and McClintock (1985) VI Hendry et al. (1990) Of these, papers I, II and V address the roton creation problem explicitly; III, IV and VI, devoted to vortex creation are also relevant, partly because vortex creation by a moving object can be treated successfully in terms of a generalized Landau argument (see below), but mainly because, as we shall see, the major obstacle to be overcome in order to be able to measure VL was the uncontrolled conversion of bare ions to charged vortex rings. All of the investigations to be discussed relate to the regime below 1 K in which, according to Landau's (1941, 1947) excitation model, liquid 4He is best viewed as an inert "background" fluid containing a dilute gas of thermal excitations. The excitations carry the whole entropy of the liquid; the background fluid has zero entropy, and it displays superfluid properties because of the relative difficulty of converting the kinetic energy of a moving object, or of a macroscopic flow, into excitations. For most of the work to be described, the presence of the excitation gas can be ignored. It merely provides a very weak, usually negligible, additional drag force tending to slow the moving probe (a negative ion) that is the subject of the investigations. The topic of prime interest is the much larger drag force arising from direct excitation creation by the probe. In section 2 we discuss the relationship of VL tO the excitation spectrum in He-II, and we review briefly the experimental techniques available for the investigation of roton creation together with the main results obtained in liquid helium of the natural isotopic ratio. The theory of roton emission from a moving object is outlined in section 3. Experiments on roton emission in isotopically pure 4He in very weak electric fields, leading to a precise determination of v L, are described and discussed in section 4. In section 5, we describe an investigation of roton emission in the extreme supercritical limit of very strong electric fields, providing a rigorous test of the Bowley and Sheard (1977) theory of ro-

Ch. 1, w1

THE LANDAU CRITICAL VELOCITY

5

ton creation. Experiments on roton emission from the enigmatic "fast" ions are discussed in section 6. Finally, section 7 summarises the principal results and remaining puzzles, and draws conclusions. Note that, in reviewing experimental results from a wide variety of sources, the authors have not felt it appropriate to re-label original (sometimes historic) figures in order to enforce a consistent set of physical units. Values of pressure appear, for example, in bars, atmospheres, N m -2 and Pascals, and the reader should accordingly bear in mind that 1 bar = 0.987 atm = 105 N m -2 --- 105 Pa (other equivalents, such as between electric fields in V cm -1 or V m -l are more obvious).

2. Quest for the Landau critical velocity 2.1. The dispersion curve and excitation creation in He-H

Landau's argument (1941, 1947), in essence, was that dissipation in liquid 4He must occur through the conversion of kinetic energy, e.g. of a macroscopic moving object or of a hydrodynamic flow, into elementary (thermal) excitations. It is a simple matter (see e.g. I) to demonstrate that, if energy and momentum are to be conserved in such processes, the initial velocity of the moving object must exceed a critical value v' = (e/hk + h k / 2 m ) ~ n ,

(2.1)

where m is the mass of the object and e, hk are, respectively, the energy of the created excitation and the magnitude of its momentum. For a massive object (but not for the ions used in the work to be described below), the second term is negligible and the Landau critical velocity is V L =

(e/hk)n~n.

(2.2)

The peculiar shape of the dispersion curve for the elementary excitations in HeII, shown in fig. 1, ensures that v L is non-zero and hence the possibility that the liquid will have superfluid properties. If the vicinity of the roton minimum in the dispersion curve is assumed to be parabolic, of form e(k) = A + h 2 ( k - ko)2/2mr,

(2.3)

where the roton parameters A, k0, mr specify the energy, wavenumber and effective mass of a roton at the minimum, it is straightforward to show that (2.2) leads to

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY

I

!

15

lO

5 / s J

s I s s s s

"

0

I

I

10

20

30

klnm-1 Fig. 1. The dispersion curve for excitations in superfluid 4He at a temperature of 1.1 K and under a pressure of 25.3 atmospheres (after Henshaw and Woods, 1961): the energy e of an excitation is plotted against the magnitude of its wavevector k. Excitations near the local minimum are known as rotons. Equation (2.2) is satisfied by rotons at the point where a straight line drawn from the origin makes a tangent with the curve, and the gradient of this line therefore represents the Landau critical velocity for roton creation, v L.

/3 L "-

[(2Amr+/:12 k02)1/2 _ hko]/mr.

(2.4)

Because (2Am,/h2ko 2) << 1, this result can be well approximated (to within 1%) by VL = A/hko,

(2.5)

which is the conventional expression used for the Landau critical velocity. Insertion into (2.5) of accepted values of the Landau roton parameters (see e.g. Brooks and Donnelly 1977) shows immediately that, in the low temperature limit, VL is .-60 ms -1 under the saturated vapour pressure; it decreases monotonically with increasing pressure, falling to --45 ms -l at 25 bar just below the solidification pressure. The physical significance of VL is that it represents the minimum velocity at which dissipation can occur through creation of elementary excitations. This is, of course, the most basic and direct form of dissipation; the liquid would not be

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

7

a superfluid if/)L = 0. On the other hand, it should be noted that there is nothing in Landau's argument to preclude other possible forms of dissipation, for example through the production of non-elementary (metastable) excitations, perhaps at velocities lower than v L. Ignoring such complications, however, and assuming that the temperature is low enough for the drag caused by excitation scattering to be negligible, the simple Landau picture suggests that the drag on an object moving through He-II should vary with its velocity as indicated in fig. 2. The drag remains zero until the critical velocity u L is reached, above which dissipation sets in very abruptly; the theory makes no prediction about how the drag varies with velocity above rE, however, and the curves (a), (b) and (c) would all be equally consistent with Landau's picture. Similar arguments can, of course, be applied to the case of He-II flowing through a channel showing that, for velocities less than v L, there should be no viscous resistance to flow. The temperature dependence of the roton parameters of He-II (Brooks and Donnelly 1977) implies that 1)L must also be temperature dependent. Below 1 K, however, the dependence is extremely weak and all the investigations to be described below refer, in effect, to u L in its T--> 0 limit. This is, of course, usually the most interesting regime because the drag on a moving object due to the normal fluid component (the excitation gas) is then very small and the onset of dissipation at/)L i s correspondingly dramatic. For the opposite situation, however, where the superfluid component moves and the walls (and normal fluid component) are stationary, the onset of dissipation at 1)L can be well defined even at temperatures near that of the lambda transition, Ta. In particular, Andrei and Glaberson (1980) were able to find evidence for a finite/)L through the investigation of fourth sound resonances in a highly packed powder that effectively clamped the normal fluid component. As T---> T;t, A --> 0, but/Co remains drag

b

vL

C

velocity

Fig. 2. The drag on a moving object due to excitation creation as a function of its velocity, according to Landau. Takken (1970) predicted that, for experimentally feasible measurements of negative ion characteristics, curve A would be followed; but curves such as b or c would be equally consistent with Landau's theory, which merely predicts the absence of drag for velocities b e l o w v L (Allure et al. 1977).

8

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

finite and so one expects that VL --->0 from eq. (2.5). What Andrei and Glaberson observed was that, for 1.2 < T < T~t, the velocity of fourth sound was markedly shifted by the presence of a superflow (produced by steady rotation of their cell). The velocity shifts became more pronounced as Ta was approached. The authors were able to interpret their results on the assumption that, close to powder grains, VL was being exceeded, so that the resultant local breakdown of superfluidity effectively reduced the porosity of the system. The reduction in porosity increased the refractive index of the medium for fourth sound and correspondingly reduced its velocity. Although the explicit temperature dependence of VL could not be extracted reliably, the fact that good agreement was obtained between experiment and theory on this basis can be taken as evidence for the validity of the concept of the Landau critical velocity. It should be noted that, although the Landau criterion (2.5) was introduced specifically in relation to rotons in He-II, the arguments are readily generalized to encompass other excitations in He-II, and other fluid systems with welldefined excitations characterized by dispersion curves for which the minimum value of energy/momentum is non-zero. Vortex ring creation by ions in He-II, for example, is readily interpretable on the basis of a generalized Landau argument (see III, IV and VI). Evidence for a Landau critical velocity Ac/pF corresponding to Cooper pair-breaking in superfluid 3He has been obtained from experiments on ions (Ahonen et al. 1978) and vibrating wires (Fisher et al. 1991). Here Ao is the energy gap and PF is the Fermi momentum. Critical current densities in superconductors can be related to pair-breaking above a Landau critical velocity (Tilley and Tilley, 1990) in a very similar way. Such phenomena are important and extremely interesting, but they lie beyond the scope of the present chapter, which is devoted to the problem of roton creation in He-II as originally formulated by Landau.

2.2. Critical velocity measurements in He-H A very large number of experiments on the flowing superfluid which test the predictions of Landau's theory have been carried out, and are reviewed in the standard texts on superfluid helium, e.g. Wilks (1967), Keller (1969), Wilks and Betts (1987), Tilley and Tilley (1990) and Donnelly (1991). Critical velocities have indeed been observed for He-II in a wide range of geometries including orifices, capillary tubes, adsorbed films and tightly packed powders. In every case, however, the experimental value of the critical velocity has turned out to be much smaller than rE, often being mm s-~. The reason for these low critical velocities is now understood to be associated with quantized vortices (Donnelly, 1991). Drag due to the expansion of vortices pre-existent in the liquid (which appears to be the universal situation, regardless of the liquid's history; see

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

9

Awschalom and Schwarz, 1984) sets in at a relatively low velocity and effectively masks the onset of roton creation at v L. An excellent review of recent experiments on orifice flow, in which discrete dissipative events are observed, has been given by Varoquaux et al. (1991). The events in question occur for flow velocities that are relatively high (several ms-l), but are still much smaller than VL. They are associated either with vortex depinning/repinning or, more probably, with vortex nucleation ab initio (cf. III, IV, VI). The other possible experimental approach is, of course, to move an object through stationary superfluid. Negative and positive ions constitute particularly convenient objects for this purpose. They can readily be injected into the liquid by a variety of different techniques, they can be moved through the liquid by application of electric fields, and their arrival at an electrode can be observed as a pulse of current. The so-called ions which can exist in liquid helium are, in fact semi-macroscopic objects with radii of ~1 nm and effective masses of ~100m4, where m4 is the mass of a 4He atom. Numerous investigations of ion motion in liquid 4He have been carried out. The early work has been reviewed, with extensive bibliographies, by Fetter (1976) and by Schwarz (1975); a modern discussion and critical analysis will be found in Donnelly (1991); see also IVI. In such experiments it has been found that, as the electric field is increased from zero, the drift velocity of the ion, which is limited by the scattering of thermal excitations, also at first increases; but in almost every case, at a critical velocity of ca. 30 ms -1, the bare ion undergoes a transition and, thereafter, its velocity falls with increasing electric field in precisely the manner expected of a charged vortex ring (Rayfield and Reif 1964). Such experiments have been extremely rewarding and have led, for example, to accurate measurements of the quantum of circulation. Because the bare-ion to charged-vortex-ring transition can usually be characterized by a critical velocity which is less than VL the experiments have not, however, enabled any satisfactory test of Landau's roton emission theory to be carried out. The only exceptions seem to be in the particular cases (a) of normal negative ions moving through liquid helium under pressure and (b) the so-called fast negative ion, to which we return in section 6. Meyer and Reif (1961) discovered that the behaviour of negative ions in pressurized He-II was quite different from that of positive ions, or of negative ions at lower pressures, in that for temperatures near 0.6 K, it was possible to accelerate them to what seemed to be plateau velocities of 50--60 ms -l as shown in fig. 3. This was later confirmed by Rayfield (1966, 1968). He found that for P > 12 bar (1 bar = 105 Pa) it was possible to accelerate negative ions to velocities approximating to VL; he deduced that in his highest electric fields of 7 kV m -1 the ions were approaching a limiting velocity; and he found (see fig. 4) that this apparent limiting velocity rose as the pressure was reduced. The latter was precisely the behaviour expected of VL; AJhk0 increases with a decrease in pressure as indicated by the dashed curve in fig. 4, owing to changes in the shape of

10

P.V.E. McCLINTOCK and R.M. BOWLEY

-

T'.505~

60-

-

9

20

I0

O 0

e

9

9

-

.,.

P=

50,-

=30

9

Ch. 1, w

"

"

-

1

!, 9

_ I I0

I 20.

I 30

I ... ! 40 50 -F..,Volts/cm

f 60

! 70

I 80

Fig. 3. Drift velocities, here called U, of negative ions in pressurized He-ll measured as a function of electric field, here called r at two different pressures (Meyer and Rief 1961). There is no sign of the decrease of velocity with increasing field seen at lower pressures, corresponding to the creation of charged vortex rings. the excitation spectrum. Unfortunately, Rayfield's experiment was at a temperature, 0.6 K, where drag on the ions owing to excitation scattering was considerable and it was not, therefore, possible to be entirely sure that true critical velocity behaviour was being observed, or to measure the component of the drag arising from excitation creation as opposed to that arising from scattering. Attempts by Neeper (1968) and by Neeper and Meyer (1969) to repeat Rayfield's experiments at lower temperatures, where excitation scattering could be ignored, resulted in failure. As their experimental chamber was cooled, the vortex ring nucleation rate apparently increased until at 0.3 K, only charged vortex rings could be detected at the collecting electrode, and no bare ions. It seemed, therefore, that the existence of the critical velocity predicted by Landau was not going to be accessible to a direct experimental investigation. No reason to doubt this conclusion emerged until Phillips and McClintock (1973) observed some apparently anomalous current-pressure characteristics in a field emission cell, which they attributed to the presence of bare ions, even at temperatures as low as0.3 K.

Ch. 1,w

II

THE LANDAU CRITICAL VELOCITY "1-,~

"9- . . . ,.,

I

!

" " "" " ,....

, , ,.. , . -- . . . . , . m

Necessory

VelocIty For

Roton

Creotion

~ 40

0 ,,,J t,aJ

I

.._ PRESSURE

,

, IN

i

i

I

20

I

I

ATMOSPHERES

Fig. 4. Measurements of maximum velocities of negative ions in He-II for weak electric fields at -0.6 K, plotted as a function of pressure, demonstrating that the Landau critical velocity (dashed line) can be attained at high pressures (Rayfield 1968). 2.3. Field emission in liquid helium Field emission and field ionization enable comparatively large currents to be injected into liquid helium, and current sources based on the phenomena have a number of advantages over the radioactive sources which were almost universally employed in the early ion experiments. If a negative potential of a few kilovolts is applied to a sharp metal tip immersed in liquid helium, then, just as in a vacuum, electrons are able to tunnel from the tip and proceed towards a collecting electrode. The presence of the liquid, however, introduces a number of complications. In particular, gaseous charge multiplication processes can take place close to the emitter; and the velocity of the ions through the liquid is drastically reduced. The latter feature of the phenomenon leads to spacechargelimited emission at comparatively low currents of ca. 10-9 A; conversely, the magnitude of the emission current for a fixed emitter potential can provide information about the way in which the ions move through the liquid, often enabling the ionic mobility to be deduced. Field emission and ionization in liquid 4He have been studied in detail by Phillips and McClintock (1975).

12

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY 200 --

I

I

I'"

I

I

I

i

(b) O.D..O,,O..D~

(a)

0

~

~176

0

I t

P = 25x105Pa

9100 <

_

50

10SPa

-o-...

--.o...

_

30 0

I

I

I

10 IO-sP/Pa

1

20

I

"7--O.

I 0.3

I 0.4

I 0.5

,,.! 0.6

7/K

Fig. 5. Field emission characteristics in superfluid 4He under pressure. In (a) the current i from an emitter at 3.0 kV is plotted as a function of pressure at a fixed temperature, and is seen to increase rapidly above 10 x 105 Pa. In (b) the currents from an emitter at 2.0 kV are plotted against temperature T for two pressures P; the current at 25 x 105 Pa is considerably larger than at 105 Pa and is temperature independent below 0.5 K (Allum et al. 1977).

The observation (Phillips and McClintock, 1973) which is of special relevance to the present discussion relates to the field emission characteristics under pressure below 0.6 K; typical examples are shown in fig. 5. We note from (a) that the current rises rapidly with pressure above 10 x 105 Pa; and from (b) that the current at 25 x 105 Pa is almost temperature independent below 0.5 K. Under spacecharge-limited conditions, an increase in current implies an increase in the average velocity of the carriers. The results of fig. 5 therefore imply that an increasing proportion of the current consisted of bare ions travelling at velocities ~v L, rather than charged vortex rings, as the pressure was increased beyond 10 x 105 Pa. Of particular note was the temperature independence of the current near 0.3 K, suggesting that the proportion of bare ions did not in fact decrease with temperature. The strong implication was, notwithstanding the failure of Neeper's (1968) and Neeper and Meyer's (1969) experiments, that it would, after all, probably be feasible to propagate bare ions through liquid 4He at 0.3 K at velocities up to VL, and thus to test Landau's (1941, 1947) theory of the breakdown of superfluidity. The reason that elevated pressures are necessary to support currents of bare ions travelling at a velocity ~VL, limited by roton emission, can now be under-

Ch. 1, {}2

THE LANDAU CRITICAL VELOCITY

13

velocity

Vc

VL

0

I i i i i i I

10

!

20

P (bar)

Fig. 6. Sketch to show the dependences on pressure P of the Landau critical velocityV L, and of the critical velocity ve for the creation of a charged vortex ring by a negative ion. Roton creation experiments are possible for pressures above ~10 bar where the two curves cross over. stood in terms of the pressure dependences of VL and of the critical velocity vc for vortex ring creation (see III, VI). They are of opposite sign, as sketched schematically in fig. 6. For pressures below ~ 10 bar, where the vc(P) and the VL(P) curves cross over, an accelerating ion is likely to nucleate a vortex and to undergo the transition to a charged vortex ring before attaining the critical velocity VL for roton creation. Above 10 bar, on the other hand, the situation is reversed. The ion will usually emit a roton and decelerate before it can reach Vc. In practice, the distribution of ionic velocities is described by a function (Bowley and Sheard 1977) which has a small high velocity tail extending beyond vc. Thus there is a finite probability of vortex creation, even when P > 10 bar, which naturally decreases as the pressure rises and the separation of v c and v L increases (see III; Hendry et al. 1988; and VI). The strong temperature dependence of Neeper and Meyer's bare ion signal, and its unexpected disappearance at --0.3 K, appear to be inconsistent with the picture, but are now understood to have been caused by 3He isotopic impurities (Bowley et al. 1980, 1984; and IV) in the sample of natural helium used for their experiments: such effects are large when electric fields are weak (as was the case); they would have been much less pronounced in the field emission data discussed above, for which the fields were relatively strong.

2.4. Measurement of ionic drift velocities A wide variety of techniques has been used for the measurement of ionic drift velocities in liquid helium (see e.g. Fetter 1976). Of these, the single-pulse

14

Ch. I,w

P.V.E. McCLINTOCK and R.M. BOWLEY

I I I I I I I I I

(a)

I I I I I I I I I

I I o I I I I I o

G3C

Gl G~_

(b) V~

~

i i o !

'

I ! ! i o i i

! ! I

o I i i

1

,,

I

,, I ! ! i i

!

,(

X

1

!

I o ! ! ,

i

I(

tl

t2

|

o o

!

o

t

,

i

t

I i I I !

:i Ol

II

t3 t( t 5 t s

Fig. 7. The single-pulse time-of-flight technique. The main components of the electrode structure are shown diagrammatically in (a). In (b) are sketched, as functions of time t, the transient negative potentials Vs (=1.5 kV) and VG1 (=30 V) applied respectively to the field emission source s and to the grid G 1; and the resultant negative current i c induced in the collecting electrode C by the arrival of the disk of ions (Allum et al. 1977).

time-of-flight technique introduced by Schwarz (1972) is overwhelmingly the best and most straightforward, and the least likely to lead to ambiguities or yield artefacts. It was used, in modified form, for all of the experiments described below. The method, slightly modified for use with a field emission ion source, is illustrated diagrammatically in fig. 7. The ionic velocity is determined by measurement of the transit time of a disk of ions across a region of uniform electric field, between G2 and G 3, whose length is known. A retarding electric field of a few kilovolts per metre is usually maintained between the gate grids Gl and G2, thus preventing the ions emitted from the field emission source S from entering the drift space G2-G 3. By applying a negative pulse of a few tens of volts to Gl the gate can, however, be opened momentarily, thus admitting a group of ions to the drift space. These then propagate across G2-G 3 at a characteristic speed which will depend on the electric field and on the temperature, pressure, and purity of the liquid helium. Their arrival at the collecting electrode C can be

Ch. 1, [}2

THE LANDAU CRITICAL VELOCITY

15

observed as a pulse of current whose transit time can be used to compute the velocity. The Frisch grid G 3 is required to screen the collector from the influence of the approaching charge. In practice, to minimize heating effects, it was necessary to pulse the field emission source, rather than running it continuously; and, because the resultant switching transients saturated the signal processing system, it was necessary to delay the gate pulse relative to the start of the tip pulse as illustrated in fig. 7(b). The current Ic seen at the collector is shown, in somewhat idealized form, in the lowest diagram; the transients at X and Y arise from the emitter switching on and off; and those at t I and t2 are caused by the gate pulse. The rise in ic at t 3 indicates the first appearance of ions on the collector side of G 3, and t4 represents their arrival at C. The passage of the back end of the pulse through G 3 occurs at t5, and the last ions reach the collector at t6. The periods (t4- t3) and (t6- t5) represent the transit time of an ion across the space between G 3 and C. The transit time rD between G2 and G3, which we wish to measure, is (t5 - t2). The signal at the collector is usually comparable with, or smaller than, the intrinsic electrical noise of the system, so that some method of signal enhancement is an essential feature of the single-pulse technique; which is why it could not be used effectively until digital signal averagers became available in the early 1970s. In the experiments to be described, Nicolet 1080 and 1280 dataprocessors were used for averaging the collector signal. The roton creation experiments were carried out in three main stages. First, using natural helium (before the effect of 3He isotopic impurities had been appreciated) in a 10 mm cell mounted in a 3He cryostat, the Landau velocity was observed unambiguously for the first time. The results of these experiments are presented in the next sub-section, below; fuller details are given in I. Secondly, using a very large (100 mm drift space) precision made cell mounted in a dilution refrigerator, and with a sample of isotopically purified 4He, roton creation was investigated in extremely weak electric fields in order to make a precise determination of VL; this experiment is discussed in section 4. Thirdly, as discussed in section 5, roton creation was studied in the extreme supercritical regime of high electric fields, using a very short (1 mm drift space) cell. (Historically, the project on the extreme supercritical regime was completed before the determination of v L) Additional velocity data were obtained from a quite separate series of experiments (see III) on vortex nucleation, based on analyses of electric induction signals rather than time-of-flight measurements.

2.5. Observation of the Landau critical velocity Some typical time-of-flight signals obtained from the 10 mm cell for a pressure of 25 bar, using the technique described in the previous section, are shown in

16

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

Fig. 8. Typical data recorded for a temperature of 0.3 K and a pressure of 25 x 105 Pa: the current ic appearing at the collector is plotted (with arbitrary origin) as a function of time t, measured from the moment at which the gate was opened. The transient negative potential VS applied to the emitter was 1.5 kV, and that VG, applied to the first grid was 30 V; the stopping potential, when the gate was shut, was 10 V. The emitter switching-off transient (Y of fig. 5b) occurred near t = 500/~s, and so is not visible in these results. The parameter settings for each signal are listed in Table 1 (Allum et al. 1977).

fig. 8. T h e ionic drift velocity was d e t e r m i n e d by m e a s u r e m e n t o f the interval t 5 - t2 (see fig. 7). K n o w i n g the length L s e p a r a t i n g grids G2 and G 3, the drift v e l o c i t y U = U ( t 5 - t2) c o u l d be f o u n d i m m e d i a t e l y . T h e m a i n s y s t e m a t i c error

Ch. 1,w

17

THE L A N D A U C R I T I C A L V E L O C I T Y TABLE 1 Values of the electric field E, the number n of repetitions in each average, and the vertical calibration factor, for each of the signals illustrated in fig. 8 (Allum et al. 1977).

Signal

E (kV m-l)

N

Ic/div (nA)

a b c d E f

50 50 65 65 100 200

1 2048 1 128 64 32

0.625 0.625 2.50 2.50 12.5 25.0

lay in the d e t e r m i n a t i o n o f L, and a m o u n t e d to --5%. Full details o f the c r y o g e n ics, e l e c t r o n i c s and e x p e r i m e n t a l p r o c e d u r e s are g i v e n in I.

i

51

I

1

i

I

0 0 0 0 0

4 tl-

.4.0 K

O

--

0 0 0 0 0

2; 3

o

o

_

o

r~

0.35 K

X

o

o

O 0

,

2

O

_

o

o O o o o -

o

-

o

o

', o ', ~ I

0

I

I

._.

20

I

40

I,~J)Zl

60

~/(m s-~) Fig. 9. The drag on an ion moving through superfluid 4He at 0.35 K, as a function of the average ionic velocity "v. For comparison, the equivalent plot for an ion moving through normal (nonsuperfluid) 4He at 4.0 K is also shown, emphasizing the qualitative difference which exists between the two cases. It is clear that drag in the superfluid sets in abruptly at a critical velocity which is very close to the critical velocity for roton creation, v L, predicted by Landau (Allum et al. 1977).

18

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

It was found that the signal became extremely weak at the 3He cryostat's base temperature of 0.29 K, especially for low electric fields; the lowest temperature at which good data could be recorded was 0.35 K. Some results are shown in fig. 9, plotted in the form of the average drag force (= eE) as a function of the average velocity U of the ion through the superfluid for more convenient comparison with the predictions of fig. 2, to which they bear a remarkably close correspondence. For comparison, the corresponding curve for a negative ion in normal (non-superfluid) liquid 4He, where it can be characterized by a constant mobility, is also plotted. The behaviour is seen to be qualitatively different in the two cases. The 0.35 K data of fig. 9 clearly indicate that the drag on the moving ion approaches zero near the value of VL calculated from accepted values of the roton parameters (as shown by the dashed line). They can thus be regarded as providing a satisfying and convincing vindication of Landau's excitation model and of his explanation of superfluidity in He-II. It is also evident that the ionic velocity readily exceeds v L in electric fields that are not particularly strong (3 x 105 V m -I at maximum in fig. 9). In order to measure VL from data of this kind it is clearly essential to construct a theory of roton emission that will enable the measurements to be extrapolated back to zero drag, i.e. to zero electric field, in order to identify the velocity at which the onset of dissipation first occurs.

3. Theory of roton creation in He-ll

3.1. Early theories of supercritical dissipation The first detailed discussions of supercritical dissipation in a Bose superfluid appear to be those of Iordanskii (1968) and Volovik (1970). When Bowley and Sheard (1975) sought to explain the first experimental results (Allum et al. 1975) several years later, they were unaware of these earlier calculations. They found, however, that they were able to account for the data in considerable detail with a kinetic theory based on of the roton pair emission hypothesis to be described below in section 3.4. We return to consider Iordanskii's and Volovik's contributions in more detail in the context of the actual experiment, in section 3.7. The only other theory of supercritical dissipation in He-II prior to the experiments was apparently that of Takken (1970), who treated roton creation on the basis of a classical wave radiation model, in analogy with Cerenkov radiation. He assumed therefore that the object would, as it travelled through the superfluid, be pushing at a point of fixed phase in a roton wave pattern. He concluded that an upper bound on the velocity of a negative ion would be

Ch. 1,w l)u.b. -" I ) L ( 1

THE LANDAU CRITICAL VELOCITY + 10-16E2),

19 (3.1)

where Vu.b. and VL are in ms -l and the electric field E is in V m -1, and therefore that, for realizable electric fields, it should be almost impossible to observe any increase in the ionic velocity beyond VL. This remarkable conclusion appeared at the time to be in accord with Rayfield's (1966, 1968) experimental data, but it is clearly at variance with the results of fig. 9. Furthermore, the shape of the measured U (E) curve, which is similar to (c) of fig. 2, is quite different from Takken's prediction, which would be similar to (b). It seems clear, therefore, that Takken's theory is not applicable to the motion of negative ions through pressurized He-II below 0.5 K. The reason is probably connected with the relatively small mass of the ion. The assumption that roton creation is a coherent process is in fact of doubtful validity unless the mass of the moving object is very large, because conservation of momentum dictates that, in emitting a roton, the forward momentum of the object be reduced by an amount Av 1 = hko/m. Unless m is very large, these events will have a tendency t o destroy any incipient phase coherence in roton creation. It is evident that a theory to describe the data of fig. 9 must make no assumption of phase coherence, must take explicit account of the recoil of the (relatively lightweight) ion each time it creates a roton, and must be valid in the weak-coupling limit in which the ion can exceed the threshold velocity by a substantial margin.

3.2. Roton creation by a light object In the case of a light object, the second term in eq. (2.1) cannot safely be ignored, and the critical initial velocity v~ for the emission of a roton may become substantially greater than VL. The ionic velocity ~" which we measure experimentally is, however, a time-averaged value over many roton creation events: we shall see that ~ need not be significantly larger than VL, even though v'I may be. As discussed in more detail in section 3.3, we will assume that the influence on U of the fluctuating thermal velocity of the ion can, to a good approximation, be ignored. We consider first an object travelling through superfluid 4He at a very low temperature such that the drag arising from the scattering of thermal excitations can be neglected, and under the influence of a force which is sufficiently weak that roton emission will occur at a velocity negligibly larger than v'l . Under these conditions the angle 0 between the direction in which the object is travelling and that in which the momentum of the roton is directed, is zero, and the problem is essentially one-dimensional. We assume that the emitted roton has

20

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

e n e r g y e'l, m o m e n t u m h x ' l, and that the final velocity of the object, after emission is v'[. T h e n eq. (2.1) b e c o m e s (3.2)

v' 1 - e ' l / h k ~ + h k ' l / 2 m .

In order to c o n s e r v e m o m e n t u m m y ' l = mv'~ + h k ' l,

whence, using eq. (3.2), pt

v 1 -

t

p

(3.3)

el/hk 1 -hk'l/2m.

F r o m eqs. (3.2) and (3.3), the a v e r a g e velocity for the onset of dissipation which we m e a s u r e experimentally, c o r r e s p o n d i n g to the time taken to travel a given distance (typically several tens of m m ) divided by that distance, (3.4) and the instantaneous velocity of the object as a function of time must be de-

U~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

U~

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

OL

.

U~ '

_.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

t

Fig. 10. Critical dissipation for an object being drawn by a constant weak force through superfluid 4He: its instantaneous velocity v is plotted as a function of time t. (a) Whenever the object reaches a critical velocity v i given by eq. (3.2), it emits a roton, dropping to the lower velocity v i' given by eq. (3.3) before accelerating once again to repeat the process. (b) If, on the other hand, dissipation occurred through two-roton emission events, then the amplitude and period of the waveform would be approximately doubled: simultaneous emission of two rotons would occur every time the object reached critical velocity v~ given by eq. (3.32), with the object decelerating discontinuously to a final velocity v~ given by eq. (3.33), before accelerating again (Allum et al. 1977).

Ch. 1, {}3

THE LANDAU CRITICAL VELOCITY

21

picted in fig. 10(a). To compute numerical values of v'l , v'~ and Vcl for an object of any given mass we need to find the value of k which makes the right hand side of eq. (2.1) a minimum. Differentiating it with respect to k, and setting equal to zero to find k',

1 de

k'i = hkl

-'h

(3.5)

e~.---k---hkl 2m-'

which, on using eq. (2.3) to substitute for e'1, becomes

h(~'~ -~o) = ~ -a~ mr

hk' l

h2(~ I -~o) 2 2mrhk' 1

hkl

~

.

(3.6)

2m

Solving this equation numerically for k~, assuming the values of the roton parameters for 25 bar, for various masses m of the moving object, yields the results shown in fig. 11. It is clear that although the recoil of the object, as indicated by the difference between v~ and v'~, is significant even for effective masses as large as a few hundred times m 4, the critical time-averaged velocity ~cl hardly deviates from VL provided that m > 30m4. The negative ion, which may be regarded as a non-localized electron trapped within a spherical void in the liquid, has an effective mass which is almost entirely hydrodynamic in nature and which therefore depends only on the ionic radius r i and the density p of the liquid. [Note that there is some evidence (Ellis et al. 1983) for a slight distortion of the ion at high velocities, perhaps caused by the Bernoulli pressure, leading to a very small p4 term in the ionic dispersion relation. In what follows, we shall ignore such effects.] Information concerning ionic radii at different pressures has been derived from a number of sources including, particularly, measurements of the mobility and of the trapping lifetime on superfluid vortices; and the data are found to be in satisfactory agreement with the simple bubble model first described by Kuper (1961) and subsequently developed by Springett et al. (1967). The available experimental information has been correlated, on the basis of this model, by Schwarz (1975) who has concluded that the ionic radius at 25 x 105 Pa is between 1.08 and 1.12 nm, depending on the precise value taken for the surface tension under pressure. Assuming a liquid density of 172 kg m -3, the corresponding hydrodynamics mass (= ZTr= 2ytr/3 p 1 3 ) lies between 68 and 76m4. We shall therefore assume that, for 25 bar, m i = 72m4, a value that is consistent with direct measurements of the inertial mass (Ellis and McClintock 1982; Ellis et al. 1983). From fig. 11 it can be seen that, for an ion of this mass, although (v'l - v " I ) is about 10% of Vcl, ~cl ---VL tO an excellent approximation, and we may therefore write eqs (3.2)-(3.4) in the forms

22

Ch. 1,w

P.V.E. M c C L I N T O C K and R.M. B O W L E Y

,

i

,

i l lll

I

i

i

i

i Ill,

I

i

i

i

i w lll

I -

(a} 100

V2

v

/,.

50 . . . . . . . . .

QI

vl' o

"~ --30 1 47

10 I'

I

I II

100

IIl

I

I

I I Ii

il i

1000 IIII I

I

I

I I

I

,

, , , ,,,I

{b)

46

_~

z

?' Pt.

45

0

l

, ' , ,',,I

10

I

I

l I IIIII.

100

1000

Fig. 11. Critical dissipation: influence of the mass m of the moving object, determined by solving eqs. (3.6) and (3.31), where m4 is the 4He atomic mass (Allum et al. 1977). (a) Plots of the critical instantaneous velocities v~ required respectively for single-roton and two-roton emission events, and of the consequent final velocities v i' and v'~. All four characteristic velocities tend to the Landau critical velocity v L in the limit of large mass. (b) The resultant average velocities v , and Vc2 which would be measured experimentally for critical dissipation through single-rotoCn and two-roton emission processes, respectively. Except for very light objects, the average velocity remains closely equal to v L.

Ch. 1,w

THE LANDAU CRITICAL VELOCITY

23

v' 1 = v L + h k o l 2 m i ,

(3.7)

v" 1 = v L - h k o l 2 m i ,

(3.8)

Fel =VL,

(3.9)

and

with negligible error. Thus, although by eq. (3.7) the ion has to reach a critical instantaneous velocity v'1 for single-roton creation which is 2.5 ms -1 larger than the Landau critical velocity VL appropriate to an object of infinite mass, the drift velocity F at which dissipation is observed to commence experimentally should nevertheless be very closely equal to VL.

3.3. Theory o f single-roton creation As already mentioned, it is found experimentally that quite modest electric fields are sufficient to propel the ions at drift velocities which are several metres per second larger than VL. Since e~/hk~ is closely equal to VL we conclude that, if the dissipative process involves the emission of single rotons, then the instantaneous velocity as a function of time must be something like the graph sketched in fig. 12(a); the ion must, on average, continue to accelerate for a finite time r~ after the conservation laws for roton emission can be satisfied, before the roton is emitted; on average, it is travelling at a velocity re1 (>v]) immediately before emitting the roton, and at a velocity vfl immediately afterwards, having had its velocity reduced by Avl =Vel-vfl. A convenient way of approaching the problem theoretically is by perturbation theory, postulating that there exists a matrix element which determines the strength of the roton emission process, and then calculating the form of ~"(E) which would then be expected, leaving the matrix element as an unknown constant to be determined by experiment. This procedure was carried out by Bowley and Sheard (1975, 1977) and we outline below, in a somewhat simplified form, the salient features of their calculation. Before we do so, however, it is prudent to consider in more detail the influence of the dilute excitation gas, consisting mainly of phonons and 3He isotopic impurities but also including a few rotons, through which the ion is moving. At low temperatures in pure 4He, roton emission will tend to be the principal mechanism limiting the drift velocity of the ion, but there will of course also be a contribution to the net drag on the ion arising from the scattering of excitations. Furthermore, because of the relatively small ionic mass, individual scattering events will cause significant changes in the instantaneous velocity of the ion. To look at the situation in another way, the ion will tend to have a superimposed

24

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY

/)el

-~-~1

I i

i

I I

T1 1'4--" II

L'fl

.....I.....t iiiiiiiii

Vr

v;

VL

17h,

or

t 'j -

--t,....

Ca)

(b)

Fig. 12. Supercritical dissipation through single-roton emission. The ionic instantaneous velocity v is plotted as a function of time t: (a) for relatively weak electric fields E, such that the average final ionic velocity Vfl is less than the critical instantaneous velocity v i required for single-roton creation; (b) for stronger electric fields such that Vfl > Vi . It is assumed that, for any given value of E, the ion continues to accelerate for an average time r ! after v i has been exceeded, before the roton is emitted. In (b) it is always the case that v > v i , and so r I starts being measured from the moment at which the previous roton was created. As discussed in the text, the amplitude t/'2AVl of the sawtooth may to a good approximation be regarded as independent of E (Allum et al. 1977). r a n d o m t h e r m a l velocity a m o u n t i n g to about 8% of its average velocity in the direction of the electric field. Roton emission occurs, h o w e v e r , with a f r e q u e n c y of a b o u t eE/hk o, this being the inverse of the time taken by the ion to increase its velocity by hko/ml. E x c e p t for the case of very small values of E, this f r e q u e n c y is m u c h larger than the rate of excitation scattering events, and we m a y therefore c o n c l u d e that the influence on U of thermal fluctuations in the ionic velocity parallel to the field m a y be ignored. T h e effect on U of thermal velocity fluctuations p e r p e n d i c u l a r to the field is m o r e subtle. T h e quantity which we m e a s u r e e x p e r i m e n t a l l y is the timea v e r a g e d velocity c o m p o n e n t parallel to the field; but the t i m e - a v e r a g e d total

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

25

velocity will be larger because of the transverse thermal velocity of ca_ 4 ms -1. The measured value of U __=50 ms -l at 0.35 K will thus be smaller by about 0.4% than the average total velocity, which implies that the Landau critical velocity as deduced from the data will be smaller by about 0.2 ms -l than its true value. The magnitude of the difference will, of course, be temperature dependent, so we may anticipate that the apparent value of VL will increase as the temperature is reduced. These effects will, however, be smaller than other uncertainties in the measurements shown in fig. 9. We conclude that, for the range of electric fields and temperatures used in that experiment, the net effect on ~ of thermal fluctuations in the ionic velocity is, to a good approximation, zero. For an ion travelling precisely at v' there is only one state into which a roton can be emitted, while satisfying the conservation laws, and so the transition rate is negligibly small. As the ion accelerates beyond v', however, the rate R l ( v ) rises rapidly because of the increase in the number of possible final roton states. As suggested by Reif and Meyer (1960), it is convenient to treat this problem using Fermi's golden rule

2~

R l(v) = ~

k

IV1,12 6

e +--~-2mi

hk .v ,

(3.10)

where Vk is the unknown matrix element, e and h k are the energy and momentum of the emitted roton, m i is the mass of the ion and v its velocity immediately prior to emission, and the 6-function ensures conservation of energy. Evaluating the sum over all possible final states and assuming Vk is constant (= Vk0) over the range of interest, it may be shown (see Appendix B of I) that R l (v) = a ( v - v'j ) 1/2,

(3.11 )

where I Vk012 (2m r )1/2 k 3/2 a =

ath S/2 v

.

(3.12)

Although v appears in the denominator of the expression for a, we assume that its variation with E is so very much slower than that of v - v ] , that we may regard a as being constant within the velocity range of interest. We will also assume that the momentum lost by the ion in creating a roton is always hko; in fact, of course, there will be a spread of momenta centred approximately on ko; but the shape of the dispersion curve implies that, within the momentum range of interest, this spread is usually (cf. section 5) small compared with ko. Thus, in

26

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

a plot such as that of fig. 12, we assume that the effect of increasing E is to raise the sawtooth waveform to higher velocities and to reduce its period, but without causing any significant change in its amplitude; the velocity lost by the ion as a result of emission, to a very good approximation, is (3.13)

A V 1 = h k 0 / m i.

We will define the average time r l which elapses with v > v~ before roton creation occurs as R 1 (1)) dt = 1.

(3.14) !

Assuming that t = 0 at v = v'l , which can usually be true provided that/)fl < /)1 (a condition which, in terms of measurable variables, implies U l < VL + Avl, where ~1 is the drift velocity of the ion when limited by single-roton emission events), (3.15)

v = v' l + ( e E / m i )t.

If we use eq. (3.11) to substitute for Rl v, and (3.15) for v, (3.14) becomes rfoCt(eE I m i )1/2 tl/2 dt = 1,

(3.16)

whence r I = (312a)2/3(m i leE)l/3.

(3.17)

From fig. 12 it is clear that -61 = v L + ( e E / m i ) r I ,

(3.18)

so that we obtain -Vl = OL + ( 3 e / 2 a m i

)2/3 E2/3,

(3.19)

a result that was first derived by Iordanskii (1968). We still need to consider the high velocity situation where vfl > v~l 9In this case t = 0 at v = vfl and instead of eq. (3.15) we obtain /1 -- Vf I + ( e E I m i ) t .

(3.20)

Ch. 1,w

THE LANDAU CRITICAL VELOCITY

27

Inserting this, with eq. (3.11), into eq. (3.14) we find ~~t~(vfl +(eE/mi)t-v~)l/2 d t = 1,

(3.2~)

which yields (Vfl +(eEIm i )r I - v~ )3/2 -(Vfl _ vl )3/2 = 3eEI2mia.

(3.22)

Now, from fig. 12(b), !

vl--VL =Vfl +eE'rl / m i - v l ,

(3.23)

so that eq. (3.21) becomes (Vl - VL)3/2 --(vi -- VL -- AVl )3/2 = 3eE / 2mia,

(3.24)

where we have also noted that eErl/m i = Av 1. Finally, we observe that eqs. (3.19) and (3.24) can conveniently be combined to give an equation which holds true for U n both above and below v I 9 (Vl - VL)3/2 -(Vl - VL -- AVl )3/2 0(~ l - v L - Av I ) = 3eE / 2mia,

(3.25)

where 0 is the unit step function. By solving this equation we can determine l(E), appropriate to energy dissipation through single-roton emission processes, over a wide range of E, for comparison with our experimental results. We note that the more rigorous theoretical analysis by Bowley and Sheard (1975, 1977) in which they eschewed the concept of an average pre-emission time rl but, instead, set up and solved the appropriate Boltzmann equation, resulted merely in an additional factor of e x p ( - y 3/2 ) dy = 0.903, multiplying the right-hand side of eq. (3.25). Before comparing the equation with our data, it is of interest to consider its limiting behaviour. For ~'1< VL + AVl one naturally re-obtains eq. (3.19). For ~'~ slightly greater than VL + A v~ there is, of course, no simple analytic form of U l(E). For U 1 - V L >> A Vl, however, which is equivalent to the assumption of m ~ oo so that Av I then becomes negligible, we can usefully expand ( U l - V L - Avl) 3r2 using the binomial theorem, whereby eq. (3.25) becomes

28

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1,w

('Vl -- VL )3/2 --('Vl -- VL ) 3 / 2 [ ( 1 -- 3Avl ) / 2(vi - VL )+"" ] = 3eE / 2mia,

or, using eq. (3.13), ~1 = VL +(e / a h k o )2 E2,

(3.26)

which is independent of mass and of the same form as the expression (3.1) which Takken (1970) derived using the implicit assumption of a very heavy ion. If we assume, that m i = 72m4, then we find Av~ =4.5 ms -1. Thus, for ~'1 < 51 ms -l (at 25 bar), eq. (3.19) should be applicable and we would therefore expect a plot of ~ against E ~ to yield a straight line which would be extrapolated back to VL; while, for higher velocities, we would anticipate deviations such that d2~"/d(E~) 2 becomes positive and the data fall above the line. Unfortunately, if we plot the data in this way, we find immediately (see I) that these expectations are not fulfilled" it is impossible to draw a plausible straight line through the data and, furthermore d2U/d(E2/3) 2 remains negative throughout the whole experimental range. It is clear that eq. (3.25) is unable to describe the form of U (E) measured in the experiments. It would appear, therefore, that the ions do not dissipate the energy acquired from the electric field through the creation of single rotons (but see also section 3.7). Note, however, Brundobler's (1994) demonstration that the application of Fermi's golden rule to single-roton emission will not be selfconsistent in the limit E ~ 0. He argues, in essence, that there is a quantum mechanical timescale "t'qm characterizing the continuous rise of the single-roton amplitude, and a classical timescale rd characterizing the decay of the amplitude of the no-roton state" the golden rule approach will clearly fall if rcl << rqm, as is bound to occur in the E ~ 0 limit. Because, as already noted, the single-roton theory fails to describe the experimental results over any range of E, it is impossible to infer from the measurements of a value of the relevant coupling constant or to identify the value of E below which the golden rule calculation would become unreliable. In what follows, we will assume that the finite values of E used in the experiments are larger than this critical value.

3.4. Theory of roton pair creation If we suppose that, for some unknown reason, the ion cannot emit a single roton, then it will continue to accelerate, eventually reaching the critical velocity v' 2 necessary for the simultaneous emission of two rotons. Following arguments similar to those used in deriving eq. (2.1), and assuming that both rotons have the same wavevector, we find

THE LANDAU CRITICAL VELOCITY

Ch. ~,w

29 (3.27)

v' 2 = (e / h k + h k / m i )min"

Assuming that k'2 is the value of k which minimizes the right-hand side of eq. (3.27), we can use conservation of momentum to show that if emission takes place at the critical velocity, the velocities of the ion immediately before and after are V ,2

II

= F,2, / h k 2

(3.28)

+ hk 2/m i

I

(3.29)

v 2 = e21hk2-hk2/mi,

so that the average ionic velocity, ~c2 = e / h k '2,

(3.30)

for critical dissipation. To compute numerical values of U c2, v2 and v 2 we follow a similar procedure to that used above for single roton emission. The equation, analogous to eq. (3.6), which determines k' is I

ff

h ( k ' 2 - k o ) / m r = A / h k 2 + h ( k 2 - k o )2 / 2 m r k ' -

h k '2 / m i ,

(3.31)

numerical solutions of which have been used in calculating the values of ~c2, v~ and v~' displayed in fig. 11. We see that, for an ion of mass about 72m4, it is an excellent approximation, as in the case of single-roton emission, to assume that ~ c2 = VL; and thus that, in eqs. (3.28) and (3.29) we can replace the e~ and k 2 by A and k0, respectively. Hence, in analogy to eqs. (3.7)-(3.9) for singleroton emission, we now obtain v 2' =VL + h k o / m I1

v 2 = VL

i,

-- h k o / mi '

(3.32) (3.33)

and re2 = VL.

(3.34)

The v ( t ) behaviour of an ion which dissipates energy through two-roton emission should therefore be as depicted in fig. 10(b)" for any given electric field, the amplitude and period of the sawtooth will be twice as large as those in the single-roton case.

30

P.V.E. McCLINTOCK and R.M. BOWLEY

~e2

.....! ..... !

Ch. 1,w

,,

T2 [,,6...-!,

II .... J ....

t

! !

vf2

3-

"I- ~ --

"/t~ -~v~ I I

'

Ur eEr~rni

6

1

!

V2

" "i v; -I---i'll

VL

/

Vf2

v;

(.)

(b)

Fig. 13. Supercritical dissipation through two-roton emission. The ionic instantaneous velocity v is plotted as a function of time t: (a) for relatively weak electric fields E, such that the average final ionic velocity vf2 is less than the critical instantaneous velocity v~ required for creation of a pair of rotons; (b) for stronger electric fields such that vf2 > v~. It is assumed that, for any given value of E, the ion continues to accelerate for an average time lr2 after v ~ has been exceeded, before the rotons are emitted. In (b) it is always the case that v > v ~, and so r 2 starts being measured from the moment at which the previous pair of rotons were created (Allum et al. 1977).

We treat two-roton supercritical dissipation by employing a similar approach (fig. 13) to that used for the single-roton emission case above. Again, following Bowley and Sheard (1975, 1977), we can write down an expression for the emission rate

g 2 (1))

=

"~

k

q

~"~2

t~ e k + e

q

-h(k +q)'v+ h(k - ~+ q) j.l

(3.35)

Here, Vk,q is the unknown matrix element which permits the ion to create a pair of rotons with momenta hk and hq, and the other symbols have the same sig-

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

31

nificance as before. Brundobler (1994) has considered the self-consistency of treating two-roton emission in this way using Fermi's golden rule and, in the light of the matrix elements deduced from the experimental data (see below), has concluded that it may be expected to give accurate results because "t'qm < < rql- Summing over all possible final roton states it can be shown (see Appendix C of I) that

t

R 2 (/3) = f l ( v - 1)2 )2,

(3.36)

where

=

k41Vko,ko12 m r 2~2h3v2

,

(3.37)

and, again, we suppose that v 2 varies so much more slowly than ( v - v 2)2 that we can regard fl as remaining constant within the velocity range of interest. The average increment of ionic velocity Av2 =/3e2 -/3f2 lost as a result of each emission event will be twice as large as before, so that Av 2 = 2 h k o / mi,

(3.38)

assuming once more that all the emitted rotons have momenta approximately equal to hko. t We define the average time z2 which elapses with v > v 2 before emission occurs by

~2

R 2 (v) dt = 1.

(3.39)

Provided that vf2
(3.40)

and so, by using eq. (3.36), eq. (3.39) becomes ~~~ fl(eEm i )2 t2 dt = 1,

(3.41)

whence r 2 = (3 / fl)l/3 (m i / eE)2/3.

(3.42)

32

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

It is clear (fig. 13a) that (3.43)

~2 = VL + (eE / m i )r 2, which, on using eq. (3.42), becomes

(3.44)

v2 = 13L + (3e / tim i )1/3 Ell3

It is interesting to note that this is similar in form to the analogous equation (3.19) for single-roton emission, except that the 2/3 power law has now been replaced by a 1/3 law. We must also consider the case (fig. 13b) of vf2 > v 2, for which eq. (3.40) is replaced by 13= vf2 +(eE / m i )t.

(3.45)

Thus instead of eq. (3.41), we obtain [,~2 fl[13f2 + ( e E / m i ) t - v

(3.46)

~ ]2 d t = l ,

yielding f

P

(vf +eEr 2 / m i - v 2 ) 3 - ( v f 2 - v 2 ) 3 = 3 e E / f l m i.

(3.47)

Now (fig. 13b), p

(3.48)

132--VL -'13f2 + eEr2 / mi -132,

so eq. (3.47) becomes (~2 -- 13L )3 -- (~2 -- VL -- mY2 )3 --- 3eE

/ tim i ,

(3.49)

where we have also noted that eEr2/mi = 2hkolmi, and used eq. (3.38). This equation can be solved for ~ 2 but, first, it is convenient to combine it with eq. (3.44) to give a general expression which holds true for ~ 2 both above and below VL + Av2: ( ~ - - V L ) 3 - - ( v 2 - - V L - Av2)3 0 ( ~ 2 - - V L - A v 2 ) = 3eElflmi"

(3.50)

This equation, which is analogous to eq. (3.25) for the single-roton case, de-

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

33

scribes ~ 2(E) for dissipation through two-roton emission processes over a wide range of E. Again, Bowley and Sheard (1975, 1977) have shown that a more rigorous theoretical analysis, based on the solution of a Boltzmann equation rather than relying on the concept of an average pre-emission period r2, results in only a small numerical modification of the equation. In this case, the right hand side of eq. (3.50) should be multiplied by ~-Y 3 dy = 0.893,

to obtain a more exact version of the equation. Returning now to eq. (3.49) we obtain, after a little manipulation, ('~2 --/)L )2 -- AV2 ('~2 - VL ) " h I A v 2 -- eE / ~ m i A v 2 -" O,

which is quadratic in U 2 - VL)" Solving this, v2 = VL +--~Av2[1 +(4eE / flmiAv ~ --~),/2 ].

(3.51)

In the high electric field limit the first term under the square root will dominate, and therefore, by using eq. (3.38) ~2 = 1)L + (e / 2flhk o )1/2 E l l 2

(3.52)

determines the limiting behaviour of U 2(E) for two-roton emission. Assuming again that m i = 72m4, we find that Av2 = 2Avl = 8.9 ms -1. Thus for U < 55 ms -1 (at 25 bar) we might expect our data to be described by eq. (3.44) whereas, for higher velocities, it will be necessary to use eq. (3.51). Note, however, that there are also other complications to be considered for high velocities; these are discussed in section 5.

3.5. Comparison of the theory with experiment Some of the experimental results (Allum et al. 1975; see also I) obtained at 25 bar are compared with eq. (3.44) by plotting U against E ira in fig. 14. It is evident that straight lines may be drawn through most of the data, although there appear to be deviations in the limits of high and low electric fields. The full lines represent least-squares fits of the data in the figures to the equation -~ = v L + AE1/3,

(3.53)

34

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

within the ranges indicated, treating both VL and A as adjustable parameters. The values of VL obtained in this way, of 46.3 ms -l from the 0.35 K data and 46.1 ms -1 from the 0.45 K data, are in excellent agreement with the value of 45.6 ms -1 computed from the Landau parameters by using eq. (2.5), considering that the systematic experimental uncertainty is __.3% in U.

'

'

"

I

'~

!

I

' I

"

"i

_

~

55~

50 P = P_.a5xlO' Pa

:

oS 415

~

~"" 40

I

~o

I

55-

'

!

I

,

.41o

l

"

..,!

~o

-

I

I

_.o,

50--

-

~

-

0.45 K 25xlOSPa

f 45

[

6' ~9 6~ 0

4O 0

I

I

20

I..

E89

I

40 m-t) 89

!

!

60

Fig. 14. Comparison of some of the experimental data with eq. (3.44), plotting the measured ionic drift velocity v" against the (electric field, E)1/3, for two temperatures. The straight lines represent least-squares fits of the data to the equation, treating v L as though it were an adjustable parameter, for data within the range 5 < E < 200 kV-1 (Allum et al. 1977).

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

35

The discrepancy at low electric fields is attributable to the additional drag caused by scattering of excitations and 3He isotopic impurities from the ion; not surprisingly, it becomes more marked when the temperature is increased (fig. 14b). The small deviations from eq. (3.53) above 200 kV m -1 are independent of temperature and may be attributed to vf2 becoming larger than v'2, so that eq. (3.51) is the relevant equation rather than eq. (3.44). In fact, the pair-creation theory outlined above provides a good fit to the data over a very wide range of fields (Allum et al. 1976); we discuss the high field behaviour in more detail in section 5. Values of the matrix element Vk0,k0 can, of course, be deduced from the gradients of the data in fig. 14; better values can, however, be obtained from the more accurate, lower field, experiment described in section 4.

3.6. A regime of negative resistance ? The conclusion that rotons are created (at least predominantly) in pairs within the electric field range of the experiments appears to be inescapable. There remains, however, the possibility that single-roton emission might become important in sufficiently weak electric fields: the length of time over which the ionic velocity exceeds the threshold v'1 for single-roton creation, but remains below the value v 2 for roton pair creation, then becomes larger. It is under these conditions, if at all, that single-roton emission would be likely to manifest itself (but see also section 3.7). Thus, if each of the two emission processes is allowed to occur in principle, but the matrix element characterizing single-roton emission is very small compared with that for pair emission, then there is bound to be a characteristic electric field at which the dominance of one process gives way to that of the other. The resultant deviations from eq. (3.44), which may be expected to be quite complicated, are of considerable intrinsic interest in their own right and have been discussed in detail by Sheard and Bowley (1978). Here we will summarize the underlying physics of the phenomenon they predicted, and the possible implications for the precision measurement of VL discussed in section 4. We will assume that, although weak, E is always large enough that single-roton emission can properly be described in terms of the Fermi golden rule approximation (Brundobler, 1994). In figs. 15(a) and (b) we sketch the instantaneous velocity of the ion as a function of time close to the critical field where, within the approximation of an average pre-emission time, single-roton emission gives way to roton-pair emission. In fig. 15(a), the ion emits a single roton just before it reaches the pairemission threshold velocity v 2 (suffering a velocity decrement of hko/mi in the process) and then accelerates again; its drift velocity will be slightly less than v'1 . In fig. 15(b) after a marginal increase in the electric field, the whole trajectory has risen slightly so that, on average, the ionic velocity has exceeded

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY

36 (a)

~L

(c)

(b)

/

/ I

time

Et

Fig. 15. A simple explanation of the effect on the ionic drift velocity of the transition from singleroton to roton-pair emission. The ionic drift velocity (indicated by the heavy broken line) decreases at the transition as sketched in (c). A detailed calculation, based on the use of a Boltzmann equation, implies a possible region of negative resistance (fig. 16) rather than the discontinuous drop in velocity suggested by this simple model (Ellis and McClintock 1985).

P

before the ion has emitted a roton. Because the roton-pair emission process is so strongly favoured over single-roton emission, the ion emits a pair of rotons almost immediately after attaining v 2 and consequently suffers a velocity decrement twice as large as that in (a); its drift velocity will thus have decreased and will be only marginally in excess of v L. If, therefore, we plot the expected drift velocity as a function of E 3/2 over a range of weak electric fields, we might expect to see the type of behaviour depicted in fig. 15(c). To begin with, v(E 3/2) follows a straight line in accordance with eq. (3.19). At the onset field for pair emission, however, there will be a sudden discontinuous decrease in ~" as illustrated. In reality, as already mentioned there will always be a certain spread in the instantaneous ionic velocities at which roton emission occurs, so that the concept of an average pre-emission time represents something of an oversimplification. A more accurate description of the phenomenon requires a solution of the appropriate Boltzmann transport equation and this leads, not unexpectedly, to a somewhat less dramatic manifestation of negative resistance than that sketched in fig. 15(c). The result of Sheard and Bowley's detailed calculation is shown in fig. 16. The several different curves correspond, not to different ratios of the single- and pair-emission matrix elements (which have different 1) 2

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

37

dimensions and therefore cannot be compared directly), but to different ratios of the rate constants K~ and K2. These are defined in terms of the transition rates R I and R2 for the two processes as R 1(v) = K 1[ ( v - v'1) //)L] 1/2,

(3.54)

R21 (v) = K 2 [ ( v - v;) ] VL]2 ,

(3.55)

and

and their relations to the matrix elements are given by eqs. (3.11), (3.12), (3.36) and (3.37). Although the predicted behaviour is rather less striking that that implied by fig. 15(c), there is still a region of negative resistance for small enough values of Kl/K 2. Also shown in fig. 16 are some of the experimental velocity data of I. These are subject to considerable scatter, mainly because of the relative shortness (10 m m ) of the experimental cell; more serious is the droop at low electric

0.08 -

0.06 -

o o o/-/

L_

0.04 I

0.02

o

L

10_3 .0//~0

3

-

3x10-

/

- , - - o o

10-5

o

10-~ 1

0

5

10

15

20

25

E~/(V m-')l Fig. 16. Deviations from eq. (3.44) resulting from the onset of single-roton emission at very low electric fields, after Sheard and Bowley (1978). The calculated fractional difference between the ionic drift velocity ~ and the Landau critical velocity v L is plotted as a function of E1/3 for several ratios of the single-roton and roton-pair emission rate constants k I and k2, respectively, as indicated by the number adjacent to each curve. The circled points represent velocity data from I.

38

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

fields, arising from excitation and isotopic scattering as discussed above. These effects would clearly have precluded observation of the interesting non-linear phenomena illustrated in fig. 16. In experiments at lower temperatures and electric fields, however, deviations from eq. (3.44) might be expected and could clearly constitute a complication in the extrapolation of ~ (E 1t3) data to E = 0 in order to determine VL. The signature for the onset of the phenomenon, for a small KI/K2 ratio, is evidently that the data will start to deviate a b o v e the straight line U (E it3) behaviour followed at higher fields.

3.7. R o t o n c r e a t i o n in e x t r e m e l y w e a k electric f i e l d s The kinetic model considered above incorporates implicitly the assumption of weak coupling between the ion and the roton. That is, we have tacitly assumed that the dispersion curve of the ion is a simple free-particle parabola, even in the region where the instantaneous velocity of the accelerating ion passes across the critical velocity for roton creation. The theoretical approach introduced by Iordanskii (1968), and subsequently extended to cover the case of non-parallel emitted rotons by Volovik (1970) does not involve the kinetic equation. Their point of view is the following (see also Brundobler 1994). The interaction of the ion with rotons, close to the threshold for emission, leads to a renormalization of the spectrum of the particle. The energy of the particle can be written (Lifshitz and Pitaevski 1980; see section 35) as [(hto - e c ) + a ( p - Pc )] + ct[v c (P - Pc ) - (hto - e e

)]1/2 =

0,

(3.56)

where Pc is the critical momentum of the ion, and ec is the critical energy for roton creation. Here, A and a are constants, the latter being proportional to the square of the coupling constant between ion and roton; note that the critical ionic velocity v e is the same as our v'1 above. The square root term gives the decay of the ion for large to, and is generally smaller than the first term. But in the vicinity of the threshold the square root term is very important. There is some distance in momentum and energy from the threshold values below which the square root term is essential. Suppose the acceleration time in the electric field needed before the ion leaves this region is large compared to the time needed to emit excitations in this region. In this case the threshold region dominates; this is the case for sufficiently small electric fields. The opposite limit pertains for high electric fields: the ion is rapidly accelerated out of the threshold region before it can emit any roton, and its behaviour is governed by the first term. A kinetic description is then appropriate.

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

39

The average velocity of the ion is different in the two cases: for high fields, the single roton emission process leads to an E ~ dependence; for low fields Iordanskii finds an E ira dependence. The simplest argument to obtain this latter result is that of Volovik (1970), based on the uncertainty principle. The time need to reach the critical momentum is 6 t = ( p - Pc ) / eE,

(3.57)

and so the corresponding energy uncertainty is beE

dE ~ ~ . (P-Pc)

(3.58)

Emission of excitations is possible when the energy uncertainty is of the order of the difference in energy of the ion from that of an ion-plus-roton. This energy difference is AE = M i n [ e ( p - k)+ hto(k)]- e(p),

(3.59)

where e(p) is the energy of the ion and hto(k) is the energy of the roton. In the critical region one can show A E = f l ( p - Pc )2,

(3.60)

where fl is a constant, i.e. that AE is proportional to the square of the distance from the critical momentum. By equating dE to AE we find P - Pc ~ Ell3.

(3.61)

The increase in the speed of the ion above the Landau velocity is proportional to P - Pc, so that we obtain finally v = o L + ),Ell3,

(3.62)

where y is a constant. This result is only valid in the threshold region. Thus, quite remarkably considering the very different models being considered, the Iordanskii-Volovik (IoVo) approach predicts a result of the s a m e form as was obtained (eq. 3.44) from the kinetic model on the assumption of roton pair emission. Thus, both theories are consistent with the observed form of the experimental results. The conundrum that needs to be addressed, therefore, is: which theory - kinetic model or Green's function formalism - is correct? Neither approach makes

40

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

any prediction of the absolute magnitudes of the relevant coupling constants, which are simply left as adjustable parameters. At the time the IoVo theory was introduced, the limited accuracy of the sparse experimental data then available meant that they were certainly consistent with the strong coupling picture, in that it was unclear whether or not ions could be accelerated to drift velocities significantly beyond v L. Generally, it seems to have been assumed that they could not; and it was of course in this same context that Takken (1970) developed his classical wave radiation model (see above) of roton creation. The experimental data (see fig. 9) clearly show, however, that the strong coupling picture is not applicable for the typical values of electric field used in practice: it seems to be rather easy, in reality, to accelerate the ions to speeds 20-30% higher than VL. This is the limit in which the kinetic model of sections 3.2-3.4 will be applicable. The question remains, however, as to how weak the electric field must become for the kinetic model to fail and for the IoVo theory to come into its own. Although it is evident that this must occur for sufficiently weak electric fields, it is not yet clear whether the cross-over regime will ever be accessible experimentally. Another, related, question concerns exactly how the cross-over will take place. Depending on the magnitudes of the relevant coupling constants, it is conceivable that the (U - V L ) e~ El/3 region currently being observed would give way to an E ~ law for weak fields (see section 3.6), which would in turn give way to an E I/3 law again in the limit of extremely weak fields. Some very complicated ~ (E) behaviour is therefore to be anticipated.

4. Measurement of the Landau critical velocity

4.1. Experimental details The technique used to measure v L was similar to that described in section 2.3, but with some important differences. First, the cell (see below) was very much larger in order to improve the precision of the velocity measurements and to reduce the charge density in the ion cloud for any given collector current. The latter feature was important because it enabled measurements to be performed in lower electric fields without problems of spacecharge spreading of the travelling cloud of ions. Secondly, the sample of He-II was isotopically purified (McClintock, 1978; Hendry and McClintock, 1987) in order to minimize vortex ring creation (see IV). Thirdly, the cell was cooled to lower temperatures in a simple dilution refrigerator, rather than in a 3He cryostat. Finally, there were important differences of operating procedure, and of data-processing, in order to optimize the production and analysis of very weak signals; these are discussed below. The requirement for absolute velocity measurements of high accuracy at

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

41

Fig. 17. Sketch (not a section) showing the principal components of the experimental cell used for measurement of the Landau critical velocity (Ellis and McClintock 1985). low temperature was quite a challenging and interesting one, so a fairly detailed description is given below. Further information will be found in V. The principal components of the experimental cell are sketched in fig. 17. The cylindrical outer wall a and its top b were of stainless steel; the lid c and bottom d of the cell were of copper. The ions were injected into the sample E of He-II from a symmetrically placed array of seven tungsten field-emission tips f, each of which was spot-welded to a nickel shank that was held by a grub-screw (not shown) in a nylon holder g. The four grid-carrying electrodes h, i, j and k were all made of copper. Ions that passed through the gate formed by grids G l, G2 and G 3 entered the drift space between G 3 and the screening grid G 4 where a uniform electric field was maintained by suitable potentials applied to the copper field-homogenizing electrodes L. Finally, passing through G 4, they induced a signal in the collecting electrode m and were detected. The entire electrode structure was fixed to the lid c of the cell, which also carried the metal-glass seals (not shown) to admit electrical connections and the bushes for the samplefilling tubes (also not shown). An indium O-ring was used to provide the necessary seal between the top ring b of the chamber body and the lid c. The lid was

42

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

Fig. 18. Radial section through outer rim of the assembled structure. The labelling of components a-n is the same as in fig. 17. The structure is designed so that its axial thermal contraction will be

very closely equal to that of the stainless steel mounting rod, n (Ellis and McClintock 1985). secured by a ring of 30 M6 high-tensile stainless steel bolts, capable of withstanding the force of ca. 3 tonnes acting on it when the cell was at its maximum working pressure of 25 bar. The most important dimension of the cell, the length, L, of the drift space specified by the separation of G 3 and G 4 w a s determined by four stainless steel mounting rods, n, onto which electrodes were threaded as sketched in fig. 18. The electrodes are insulated from each other and from the stainless steel rods n by nylon washers, but in such a way that the nylon plays almost no role in the determination of the G 3 G 4 separation. One end of each rod screws into the grid carrier k that holds G 4. The grid carrier j holding G 3 is held firmly against a shoulder in the rod and is separated from it only by a steel washer p and a thin nylon insulating washer q. The precise spacings of the electrodes below j are, of course, much less critical and are determined by a series of nylon insulators r. A nut t acting via a spring washer s holds the whole electrode assembly tightly together on the rod at room temperature and, in particular, ensures that compo-

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

43

nents p, q and j are held firmly against the shoulder of n. When the cell is cooled, all its components contract, with the nylon ones contracting by a proportion that is about ten times larger than for the metal ones. Thus, the nylon spacers o tighten onto the rod n, but they have no reason to move along it, and the homogenizing electrodes I therefore remain in essentially the same positions on the rod. Because the nylon washer q is of negligible thickness compared with the length of the rod, its contraction can be ignored and we may assume that, to an excellent approximation, the thermal contraction in the length of the electrode structure is exactly the same as it would be if the whole assembly were constructed of the same type of stainless steel as the rods n: that is, about 0.3% between room temperature and the operating temperature. On these assumptions, the cold length of the drift space was L = (100.3 _ 0.2) mm. In operation, the voltage pulses applied to the electrodes were as sketched in fig. 19. The profile of the emitter pulse (fig. 19a) was designed to minimize heating of the cell, by keeping the high voltage part as short as possible; this also minimized vortex ring creation in the source region which would have occurred (see III) while the electric field between f and G I was high. The emitters were then held weakly negative for long enough for the emitted negative ions to pass through G I. Between pulse sequences the tips were held positive so as to collapse any charged vortex rings that had been created during the field emission part of the cycle. Two different types of pulse were applied alternatively to the gate grid G2, as indicated in fig. 19(b) and (c). The action of the gate pulse in (b) is to "burn a hole" in the ion cloud, as shown in (d); the wider pulse in (c) effectively blanks out the ion signal completely. By alternately adding and subtracting input signals to a memory block, it was thus possible to remove coherent noise from the signal, as well as averaging away the usual random noise. The analysis of the collector signal to obtain precise times of flight is discussed in detail in V, but the principles were as follows. A (somewhat idealized) collector signal, after averaging, is sketched in fig. 20(a). The two spikes at the beginning of the sweep represent pick-up by the collector circuit of the gate shutting and opening transitions. The resultant ion transit times between various electrodes, from each of which U can (in principle) be calculated, are shown as rs~, rs2, r01 and 302. The lengths in the cell corresponding to these times are readily appreciated by reference to the diagram of the electrode structure in fig. 17. The time at which the signal first starts to rise is unrelated to the gate pulse, and corresponds to a velocity that is rather poorly defined because of the nonuniform electric field around the emission tips. The interval rsl, on the other hand, is precisely equal to the ionic transit time between G 3 and G4. The time taken for the signal to fall to zero represents the time taken to cross from G4 to the collector m, so that rs2 corresponds to the G3-to-collector transit time. When the gate opens again, the situation is a little more complicated. Provided that the

44

Ch. 1, w

P.V.E. M c C L I N T O C K and R.M. B O W L E Y

1000

-

(a)

500

v,

0 --500 -- 1000

go

(b)

gG

(c)

,

I !

i

i |

t i

(a) 0

1

2

3

//ms Fig. 19. Sketches of voltage pulses applied to the cell and of signals induced in the collector, plotted in each case as a function of time t. (a) Typical form of the high-voltage pulse Vt applied to the field-emission tips. When resting, between emission events, the tips are maintained at a positive potential with respect to G ! (see fig. 17). A brief (ca. 100/~s) negative pulse is applied to induce field-emission of ions into the liquid; this is followed by a period of ca. 400/~s when the emitters are kept at a small negative potential to assist bare ions in penetrating G 1 and finally, the tips return to their positive resting potential. (b) A positive voltage pulse of magnitude VG = 5 V and duration ca. 40/xs applied to G 2 can be used to close the gate formed by G 2 and G 3. (c) A wider positive pulse, appropriately timed, can be used to ensure that none of the ions are able to reach the collector. In the acquisition technique normally employed (see text) narrow and wide gate pulses were used alternatively to eliminate coherent noise on the signals. (d) Sketch to show the form of the signal expected at the collector. The collector current ic is plotted as a function of time. If gate pulses are being applied, the centre is removed in a precisely defined manner to yield a twinpeaked signal as shown by the full line. The broken curve indicates the typical shape of the signal in the absence of a gating pulse (Ellis and McClintock 1985).

gate remained closed long enough for all the ions that were caught between G2 and G 3 to be drawn back to G2, then to, and to2 will represent the G2-to-G 4 and G2-to-collector transit times, respectively. If the gate pulse were shorter than this, however, the significance of rol and to2 would be poorly defined. The

Ch. 1, w

TIlE LANDAU CRITICAL VELOCITY

45

moment at which the signal finally falls to zero represents the transit time from f to m, with part of the passage through a region of non-uniform electric field and with no relation to the gate pulse. The most satisfactory absolute determination of ~" is obtained from measurements of rsl. This analysis procedure works well provided the signal/noise ratio is favourable. It is much less satisfactory when signal/noise is adverse, which was always the case for weak electric fields, especially for the lower pressures. For these reasons a quite different approach was then employed, based on digital crosscorrelation techniques, as follows. First, a reference signal at a value of electric field where the absolute velocity had already been measured, as described above, was recorded. The drift field was then changed and another ion signal averaged. Figure 20(b) sketches two such signals superimposed. The important quantity is the displacement t d of the data signal from the reference signal. Once this quantity is known, it is simple to find the new velocity. The relative velocity-analysis routine performs a cross-correlation between these two signals. The resulting cross-correlation function will have a definite maximum. If there is a zero time-difference between the reference and the data signal, then the correla(a)

(b)

'-------rs~------'-i

,

!

'i

'

I

''I

-I

.

i I ~

'

:=

,

i

-

roz

!

:-,

Fig. 20. Sketches to illustrate the two chief techniques used for measurement of the ionic drift velocity. In each case, the collector current i c is plotted as a function of time t and the portion of the abscissa corresponding to the transit of the ions between G 3 and G 4 (see fig. 17) has mostly been omitted. (a) For absolute velocity determination, measurements were made of the times rSl that elapsed between the gate-shutting transient and the corresponding point on the signal, as described in the text. (b) For relative velocity measurements, a cross-correlation technique was used to measure the delay r d between a reference signal (full curve) of known drift velocity and the signal (broken curve) whose velocity was to be determined (Ellis and McClintock 1985).

46

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

tion function is centrally placed. Any finite r d will result in a similar but shifted correlation function, the displacement from the mid-position being a direct measurement of rd. The advantage in using a cross-correlation analysis is that the whole of the signal is used in calculating the cross-correlation function and this can give a substantial improvement in precision. Even with quite noisy data, the crosscorrelation function is relatively smooth. This can be appreciated from the fact that white noise cross-correlated with white noise gives a zero result. Experimentally, it was found possible to degrade the signal/noise ratio appreciably, without altering the cross-correlation result. This meant that it was possible to use considerably smaller numbers of sweeps in many data averages, a few hundred often being quite sufficient. In the data analysed, the gating of the signal gives rise to additional structure in the cross-correlation function, with two smaller peaks appearing, one on either side of the principal maximum. Because the gating position is very stable, this additional sharpness in the cross-correlation function gives improved accuracy. The actual signal shape can, of course, be of any form, so long as it remains the same for the signal and reference signal. This meant that signaldistortion was quite unimportant and the gain ~p of the current amplifier could therefore be set to optimize the signal/noise ratio, notwithstanding the resultant extension of the risetime of the input circuit. The observed attenuation of the signals in very weak fields remains rather mysterious: it is not associated with spacecharge effects; and it is not related to events occurring at the grids G 3, G 4. Rather it appears to be a phenomenon that occurs in bulk liquid, far from any electrodes. It is discussed in more detail in V.

4.2. Velocity measurements in weak electric fields Figure 21(a), plotting the collector current ic, as a function of time t, shows a typical (averaged) signal in its entirety, with E = 2.0 x 103 V m -l and ~p = 108 V A -1. The initial transients in ic arise from the combined effects of the three-level field emitter pulse, the gate pulse, and the blanking pulse applied to alternate signal sweeps. There follows a region of flat baseline and then ion signal itself on the right-hand side of the figure, looking very similar in form to that expected (figs. 19, 20). Figures 21 (b)-(e) show the ion signals enlarged, with transients and most of the baseline omitted, under a variety of conditions. In (b) is shown an example of a signal recorded with the amplifier gain set to 107 V A -l. The risetime of the collector circuit is relatively fast at this setting, but the signal is correspondingly rather noisy before averaging. The response is fast enough for short regions of flat baseline to appear between the two peaks. Signals of this type have been

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

47

Fig. 21. Examples of ion signals recorded at a pressure of 24 bar. In each case, the collector current ic (in arbitrary units) is plotted as a function of time t. (a) Complete digitized signal sweep after averaging, showing the tip pulse and gating transients on the left-hand side and the twin-peaked ion pulse of the right, for an electric field E = 2.0 x 103 V m -l, an amplifier gain of ~ = 108 V A -1 and with n = 1000 sweeps on average. (b) Ion signal, enlarged, with E = 2.0 x 103 V m -l, = 107V m -1 and n = 4000, suitable for an absolute measurement of the drift velocity U . (c) Signal with E = 2.0 x 103 V m -1 as in (b), except that ~ = 108 V A -1 and n = 1000, suitable for measurement of changes in v- by use of the cross-correlation method. (d) Signal with ~ = 108 V A -1 as in (c) except that E = 1.1 • V m -1 and n = 500. (e) Superposition of the two signals shown in (c) and (d), to demonstrate the shift in arrival time (Ellis and McClintock 1985).

48

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, {}4

I

I

-256

-171

-85

0

85

171

256

r/~s Fig. 22. Cross-correlogram computed between signals (c) and (d) of fig. 21. The cross-correlation function fc(r) is plotted as a function of the offset time r between the two signals. The displacement of the central maximum from zero offset time (broken line) gives a direct measure of the extent to which the signal in (d) is delayed relative to that in (c) (Ellis and McClintock 1985).

used for making absolute measurements of the ionic drift velocity as described. The signal in fig. 21(c) was recorded under identical conditions to that in (b), except that the amplifier gain was set to 108 V A -l. At this setting, the response is slower, the peaks are broadened, and the baseline between the peaks has disappeared. The signal/noise ratio is significantly larger, however, despite the reduced number of sweeps included in the average. Signals of this type are well suited for relative velocity measurements based on the cross-correlation technique. The signal in fig. 21 (d) was recorded for an electric field of 1.1 x 103 V m -l, compared with 2.0 x 10 3 V m -1 in (c), but all other conditions were left unchanged. As a result, the signal in (d) is delayed slightly with respect to that in (c), but it retains almost exactly the same shape; the relative magnitudes of the two peaks, however, are different for the two signals. The delay of signal (d) relative to (c) is demonstrated more clearly in fig. 21(e), where the two signals have been superimposed on each other. Measurement of this delay is effected by computation of the cross-correlation function of the two signals, which is plotted in fig. 22. The displacement of the central maximum from the (broken) zerotime axis is equal to the delay and is readily measured digitally with the Nicolet data-processor.

Ch. 1, {}4

THE LANDAU CRITICAL VELOCITY

49

The signals of fig. 21 were all recorded with P = 24 bar, where vortex nucleation occurs at a minimal rate (see III) and the signals are consequently strong. At lower pressures, the signals obtained are much weaker and the signal/noise ratio is poor, even after averaging. Nonetheless, the cross-correlograms computed between such signals are quite smooth (see V), with welldefined maxima, so that it is possible to measure relative velocities to high precision. The determination of the absolute velocity for the reference point is inevitably slow, but can be achieved by prolonged averaging. Velocities measured in this way for pressures down to 13 bar are shown in fig. 23. They are subject to a possible systematic error of up to +_0.4%, plus a smaller random error. In each case, the filled circle represents the absolute measurement of U that provided the reference point for the other (relative velocity) measurements at the same pressure.

5~r 13 bar 14 15

53

16 17 18 51 I

19 20

cn

21 22

49

23 24 25 47

/ 451

o

I

I 6

I

I 12

I

I 18

Et/(v m-t)t Fig. 23. Measurements of U as a function of E 1/3 for several pressures in the range 13 < P < 25 bar. The solid straight lines represent least-squares fits of eq. (3.44) to the data, values of the fitting parameters being tabulated in Table 2 (Ellis and McClintock 1985).

50

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

TABLE 2 Experimental values of the Landau critical velocity v L, and of the modulus of the matrix element for roton pair creation by a moving negative ion IVk0,k01,as functions of pressure p (Ellis and McClintock 1985). P (bar)

v L (ms -1 )

IVk0,k01 ( 10-52 J m 3)

13 14 15 16 17 18 19 20 21 22 23 24 25

50.75 50.37 49.83 49.40 49.02 48.53 48.08 47.70 47.22 46.87 46.47 45.97 45.62

1.56 1.79 1.88 2.07 2.24 2.39 2.55 2.70 2.81 3.08 3.22 3.36 3.5

As the pressure was reduced towards 13 bar the signals became extremely weak, notwithstanding the use of the tailored-tip pulse technique described above. Bare-ion signals, were, however, still detectable at pressures below 13 bar. With very careful optimization of all the operating parameters and prolonged signal averaging, the arrival of bare ions could be detected down to a minimum pressure of ca. 10.5 bar, but the signals were far too small to be of use in making high-precision velocity measurements. For the range of electric fields above ca. 500 V m -l, the data as plotted in fig. 23 were linear in E It3 in accordance with eq. (3.44). For lower electric fields, however, significant deviations from eq. (3.44) were observed (Ellis and McClintock 1981; see also V), with the data points falling below the straight line drawn through the higher field data. This effect is not yet understood; it is not due to excitation or isotopic scattering, it differs markedly from the form expected to result from the onset of single-roton emission (fig. 16), and it is not associated with spacecharge spreading of the ion cloud. It could possibly be related to an increasing importance in weak electric fields of the scattering of ions from vortices present in the cell. The phenomenon is discussed in more detail in V.

4.3. The critical velocity On the basis of eq. (3.44), which evidently fits the experimental ~ (E) data accu-

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

51

rately for E > 500 V m -~, v L is equal to the ordinate intercept on plots such as fig. 23. It clearly depends on pressure, as expected. Fitting straight lines to the ~" (E it3) data for each pressure by the method of least squares then yields the lines on fig. 23 and the corresponding values of VL that are tabulated in Table 2 and plotted with an expanded vertical scale in fig. 24. The statistical errors involved in the least-squares fitting procedures, amounting typically to _10 -4 ms -l in the intercept, are found to be negligible in comparison to the systematic uncertainty of +_0.4% in its absolute value. We conclude therefore that, subject to the various caveats already mentioned, our experimental values of VL should be accurate to within +_0.4%. The solid curves of fig. 24 are plots of eq. (2.4) based on three different sets of published roton parameters; those of Donnelly (1972), of Maynard (1976) and of Brooks and Donnelly (1977); indicated respectively by the D, M and BD adjacent to the curves. It is immediately evident that the agreement of the Don-

51

50

49 v

48 M 47

I

BD

lT.

46 I

I

I

I

P/bar Fig. 24. The Landau critical velocity for roton creation v L a s a function of pressure P. The circled points are experimental values obtained by fitting eq. (1.1) to the data of fig. 23 and the solid curves represent theoretical predictions as discussed in the text (Ellis and McClintock 1985).

52

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

nelly and the Brooks and Donnelly predictions with our data is remarkably good. The Donnelly curve is in almost perfect agreement with the data obtained near the solidification pressure, whereas the Brooks and Donnelly curve appears to be slightly better near 13 bar; in no case does the difference between the data and either of these two curves exceed 1.5%. The Maynard curve, on the other hand, appears to be significantly less satisfactory. It is interesting to note that our experimental values of VL(P) show a stronger pressure dependence than expected on the basis of any of the sets of roton parameters; a conclusion which would, of course, still be valid even is we had somehow mis-estimated the systematic error in our measurements. If, for example, our cell were about 1% longer than we believe to be the case, all our data points in fig. 24 should then be scaled up by the same factor, which would clearly lead to excellent agreement with the Brooks and Donnelly curve at 17 bar, but to significant discrepancies at 13 and 25 bar. We have investigated the effect of attempting to make explicit allowance for the slight droop in U (E lr3) seen at lower fields (see above and V), because it could reasonably be argued that the incipient droop could perhaps give rise to an additional source of systematic error. In doing so, we re-fitted the data on the assumption that eq. (3.44) should be replaced by an empirical equation of the form ~ = v L + AEl/3 +CE-I,

(4.1)

where C is an additional constant. This relation provides a good fit to the ~" (E) data over a wide range of E, including the non-linear low field region where the droop occurs. The net effect of applying this fitting procedure for E above 500 V m -l was to increase the scatter in the fitted values of VL but without moving them significantly either up or down. The values of C turned out to be both positive and negative. We decided, therefore, that all of the data in fig. 23 should be regarded as being within the linear regions of their respective ~" (E 1t3) characteristics and that no advantage was to be gained from fitting them to eq. (4.1). We may conclude that the quality of the agreement between the predicted and experimental values of VL(P) should be regarded as highly gratifying. The small discrepancies are almost certainly not from any unforeseen deficiency of the Landau excitation model but, rather, from the relatively large uncertainty in the values of the roton parameters under pressure. Donnelly and Roberts (1977) have remarked that reliable inelastic neutron scattering data are lacking in this region of the helium phase diagram and that some accurate values, based on the improved neutron scattering techniques now available, are much to be desired.

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

53

4.4. The matrix element for roton pair creation It is immediately evident from inspection of fig. 23 that the straight lines on which the data fall are becoming considerably steeper as the pressure is reduced. This implies that the strength of the roton-pair emission process is decreasing or, equivalently, that the modulus of the relevant matrix element becomes smaller with decreasing pressure. In fact, the gradients, A, of the fitted lines are found to decrease by some 70% between 13 and 25 bar. It is straightforward to extract values of the square of the matrix element by use of eqs. (3.57) and (3.44) which, when the numerical factor derived by Bowley and Sheard (1977) is also included, yield (4.2)

I Vko,kol-- (52.88h3v3e / k 4mrmiA3 ).

In evaluating IVko,koI by use of eq. (4.2) and our experimental measurement of A, we have used the Brooks and Donnelly (1977) roton parameters together with measured values (Ellis et al. 1983) of the ionic effective mass m i. Values of

O O

0 0

E

0 0 0

I v

0

0

g

o

;

1'o

1

,5

2'0

....

P/bar Fig. 25. Experimental values of the matrix element for roton-pair creation IVko,koI derived from the data of fig. 23 by use of eqs. (1.1) and (4.4) and plotted as a function of pressure, P (Ellis and McClintock 1985).

54

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, {}4

IVko,k01deduced in this way are listed in Table 2 and plotted as a function of pressure in fig. 25. The pressure dependence of the matrix element is quite pronounced; indeed, it falls so rapidly with decreasing pressure that a linear extrapolation would imply a complete cessation of roton-pair emission below ~3 bar. The most significant consequence of decreasing the pressure is probably the increase in the radius of the ion (Springett and Donnelly 1966), so that one possible interpretation of fig. 25 is that the unknown mechanism by which pairs of rotons are created is entirely suppressed if the ion becomes too large. An alternative (or perhaps equivalent) explanation, equally consistent with the data, is that Vko,kopasses through zero, changing sign at --3 bar; perhaps because of some type of interference process occurring between the emitted rotons. If this were the case, IVko,koI would be expected to start increasing again at lower pressures. Of course, there is no physical basis for a linear extrapolation per se in fig. 25 and a shallow curve intersecting the abscissa axis close to the origin would look equally plausible. Further progress will require the development of an explicit physical model of the creation mechanism. It is unfortunate that the crossing of VL(P) with the critical velocity vc(P) for vortex creation (see fig. 4 and III, VI) effectively precludes any attempt to follow the roton-pair emission matrix element experimentally down to lower pressures. If, for some reason, pair emission does not occur at low pressures, it remains possible that roton emission in larger groups or, indeed, singly could take over as the dominant dissipative mechanism and could, in principle, be identified from the electric-field dependence of U.

5. Roton creation at extreme supercritical velocities

5.1. Velocity measurements in high electric fields Roton creation by negative ions in high electric fields is of some importance as a test of the Bowley and Sheard (1977) pair creation theory which, as discussed in section 3, predicts significant departures from the ( U - VL)~: E 1/3 law under these conditions. By use of a very short drift space, the U (E) measurements were extended to a field of 6 MV m -l. The experiment is described in detail in II. Here, we just outline the modified technique and review the principal results. The high-field cell differed from those already described mainly in respect of its very short drift space of a nominal 1 mm. As a result, particular care was necessary to ensure accurate alignment of the various electrodes. The grids were of nickel mesh, mounted on mild steel carriers; the small mis-match in their thermal expansivities ensured that the grids would, if anything, become tighter and flatter on cooling to cryogenic temperatures. Although the intended working

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

55

field of---5 x 106 V m -1 was about an order of magnitude below the breakdown electric field in liquid helium, as measured between smooth hemispherical electrodes (Gerhold 1972), considerable difficulty was experienced with electrical breakdown inside the cell, often occurring at about 2 x 106 V m -1. This was apparently a consequence of local field enhancement at the grid wires; the effect was substantially reduced by resorting to grids of a lower geometrical transparency. The grids themselves were of a nominal 2000 wires per inch, 2.5 l~m thick and of 25% geometrical transparency. The hole size (6.4/tm) was very much smaller than either the gate (nominal 0.2 mm), screen grid-collector (nominal 0.1 mm) or drift space lengths and thus ensured that the electric fields in these regions could be uniform, as well as preventing any serious "leakage" of the relatively high drift space field into the gate. In analysing the signals recorded from the short chamber, explicit allowance was made for the finite rise time of the collector circuit and also for the grid movement that occurred when the electric field was large. The latter effect required a correction of up to 10% in U and was thus of considerable importance; fortunately, the grid movement could be determined reliably from the changes in the gate and screening grid to collector transit times. The measurements of U, corrected to allow a number of complications (see II) including those mentioned above, are plotted as a function of E 1/3, for more convenient comparison with the Bowley and Sheard (1977) theory, in fig. 26. The possible systematic error is estimated at ___6%.The random error is indicated by the scatter of the data; it becomes larger at the strongest electric fields where the signals are at their weakest and the effect of noise is relatively serious.

5.2. Comparison with theory Comparison of the experimental data with the Bowley and Sheard (1977) roton pair emission theory (see section 3.4; yielding curve a in fig. 26) gives excellent agreement up to U = 65 ms -l. At higher velocities deviations appear and, at --70 ms -l , the data and the theoretical curve diverge quite abruptly. At the highest values of U, the drag experienced by the ion is --100% larger than the theoretical prediction. We conclude that the theory, in its original form, fails for large values of U. There are several reasons why this might have been anticipated, including: departures from parabolicity of the roton region of the dispersion curve; the momentum dependence of the pole strength for high momentum excitations; changes in the average momentum of the emitted excitations; a momentum dependence of the matrix element Vq,k; and the onset of other forms of dissipation. We now discuss each of these in turn. In the theory, it was assumed that the relevant part of the dispersion curve was parabolic, as described by eq. (2.3). However, for large velocities, such as

56

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY 80

"

'

'

'

'

I

'

'

'

~

I

'

'

'

'

all ,:/b

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oo

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~ff b d:)

70-

~/(ms-9 60-

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,

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50

,

i

,

,

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100

i

,

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i

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150

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2q

E~/CVm-9 Fig. 26. Measurements of the drift velocity U of negative ions, plotted as a function of (electric field, E) 1/3, for P = 25 bar, T = 0.34 K. The dashed curve a represents a fit of the theory of Bowley and Sheard (1977) to the data. The full curve b represents an extended form of the theory, as discussed in the text (Ellis et al. 1980).

are being considered here, it is energetically possible to create excitations far from the roton region, where eq. (2.3) will not be applicable. In fig. 27 we compare the curve corresponding to the values of the roton parameters (Donnelly 1972) that were used in the theory, with some neutron scattering measurements of the dispersion curve (Smith et al. 1977) at a pressure of 24.3 bar, very close to the 25.0 bar used in our experiments. Also included in the figure are (dashed) straight lines, drawn from origin, at gradients corresponding to the indicated velocities. Since an ion can, in principle, create excitations at all points on the dispersion curve lying below the line corresponding to its average velocity, it is clearly quite essential that departures from parabolicity should be taken fully into account when U > 65 ms -~. We also note from fig. 27 that the value of k0 is, in reality, slightly larger than that deduced from Donnelly's (1972) equations; in the calculations that follow, we have therefore taken ko = 20.5 nm -l, which is consistent with the results of Smith et al. (1977).

Ch. 1,w

57

THE LANDAU CRITICAL VELOCITY I

I

|

I

i

=,

I

I

I

t

I

i

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I

/

g

I,o,oop_ -

15 - 84 0

-

,,

o\

,, /o

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,,"60

-

.

(e/k,)/K ,L.. ,." IJ,

10

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s

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s,

s

s

s

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.,

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s SS s

,,t, t 7-'

s

js

i

,

LO

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20

!

i

i

i

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30

i

'

k/rim -1 Fig. 27. The high momentum part of the dispersion curve: the excitation energy ~ is plotted as a function of its wavevector k. The full curve is the parabola prescribed by eq. (2.3) with roton parameters (Donnelly 1971) at 25 bar, and the points represent the neutron scattering data of Smith et al. (1977) at 24.3 bar. The pressure difference of 0.7 bar is not expected to be of great significance. The dashed lines are drawn from the origin, at gradients corresponding to the velocities given (in metres per second) by the adjacent figures in each instance: on average, only the part of the dispersion curve that lies to the fight of the line is accessible to the dissipation mechanism (Ellis et al. 1980).

T h e detailed shape of the dispersion curve is important for two reasons. First, it affects the average m o m e n t u m of the emitted excitations; this point we discuss in m o r e detail below. Secondly, it is clear from fig. 27 that the density of states for excitations with k = 30 nm -1 will in reality be much larger than was tacitly a s s u m e d in the theory, which calculated the roton pair emission rate by m e a n s of

k2 dk q2 dq

R(v)=

'6(ek +-~q-hkvlt-hqco,u').

(5.1)

(2jr)3 h O n e m i g h t therefore expect that, if proper account were taken of the nonparabolic shape of the real dispersion curve where k is far from ko, the excitation e m i s s i o n rate w o u l d b e c o m e larger, with a c o n s e q u e n t increase in the drag experienced by the ion and a c o r r e s p o n d i n g decrease in U b e l o w the value pre-

58

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

dicted by the theory. In fact, however, it is essential that the momentum dependence of the pole strength of the high k excitation also be considered. As discussed below, it turns out that an additional correction must be applied to the theory which, by de-emphasizing the influence of the high k part of the dispersion curve, greatly reduces the importance of the correction for non-parabolicity. The best model available for describing excitations with momenta --30 nm -I is that given by Pitaevskii (1959); see also Zawadowski (1978) for a review. He assumes that the interaction of a quasiparticle of large momentum with two rotons gives the main contribution to the self energy of the excitation. The net effect is to alter the wave function and shift the energy of the excitation. A consequence is that the probability of creating one of these high momentum excitations is reduced by a factor f(q), the pole strength. This quantity can be measured in neutron scattering experiments. It is found that, as the momentum increases and the flat part of the dispersion curve is approached, the pole strength vanishes. If the energy of an excitation is written e = e ~ + ~ (q, e),

(5.2)

where Z(q,e) is the self energy, the pole strength is then f ( q ) = ( 1 - O ~ ' ( q ' e ) ) -~ Oe

(5.3)

and the group velocity, which enters the formula for the density of states, is

Oe Oq

= ( Oe~ O~ (q,e) )(l _ O~"(q'e) ) -~ Oq

Oq

Oe

(5.4)

To determine the corrected transition rate, we have weighted the integrals by the factorf(q)f(k) and have also included the real (non-parabolic) excitation energies as measured by Smith et al. (1977) at 24.3 bar. We ignore the difference of 0.7 bar between their experiment and ours; it changes VL by about 0.5%, which is less than the scatter in the data. The integrations over q and k were computed by numerical methods, and those over/t and/t' were evaluated analytically. We have to take values of the pole strength from experiments under the saturated vapour pressure (Cowley and Woods 1971) since measurements do not seem to have been reported for higher pressures, but the resultant correction to the transition rate turns out to be relatively insensitive to the exact values used.

Ch. 1,w

THE LANDAU CRITICAL VELOCITY

1.10-,

0 u

i

'

' .

.

.

.

I

.

.

.

.

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59

"'

'

1.05 /

I

~

0 u ikl I-i

o 1.00

0.95

50

60 v/(ms -1)

70

80

Fig. 28. Corrections to the emission rate R(v) used in Bowley and Sheard's (1977) theory, as a function of ionic velocity v. Curve a represents the correction factor that must be applied to take account both of the non-parabolic nature of the real dispersion curve at high momenta (fig. 24) and also of the momentum dependence of the pole strength of the excitations far from k0. Curve b also includes an additional weighting factor, to make explicit allowance for the fact that the average momentum of the emitted excitations increases slightly above hk0 for large velocities of the ion (Ellis et al. 1980a). In fig. 28 we plot the ratio of the corrected emission rate to that calculated by Bowley and Sheard (curve a). It will be noted that the net change in R(v), introduced by taking account simultaneously of the pole strength and of the real shape of the dispersion curve, is remarkably small. The reason for this is that the singular term (1-OX(q,elOe) in the density of states cancels with the identical term in equation for the pole strength. The calculations could be in error, but the significant conclusion is just how small a change (ca. 10%) in the predicted R(v) follows from these two corrections. As pointed out above, a change tenfold larger than this would have been required to account for the measured values of at high electric fields. In evaluating the drag on the moving ion, the theory assumed that the average momentum of excitations emitted from the ion was hko. Inspection of fig. 27 suggests that, although this will be an excellent approximation for drift velocities only slightly in excess of v L, it becomes progressively less accurate as increases and is no longer tenable for our highest velocities, where the whole of the flat region of the dispersion curve near 30 nm -1 has become accessible to the dissipation mechanism. Here again, however, the pole strength factor de-

60

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

emphasizes the influence of the high momentum part of the dispersion curve. A detailed calculation shows that the increase, above hko in the average momentum is 4%, at most. Inclusion of this additional weighting factor gives curve b of fig. 28. This slight increase in the rate of momentum loss can be used to construct a corrected theoretical curve. In principle, of course, we ought to repeat the full calculation, including the new R(v). In practice, however, the corrections are so small that it suffices to multiply the values of drag given by Bowley and Sheard by the factor given by curve b of fig. 28. The result is plotted as curve b of fig. 26, which clearly fits the data closely over a wide range of electric fields. The only remaining deviations are for U > 70 ms -l. These are relatively large, however, and are clearly still in need of an explanation. One possibility relates to the assumption that the matrix element Vq~,, for the emission of a pair of excitations is constant, irrespective of the magnitudes and directions of q and k. The original justification was that the emitted pair of rotons had roughly equal momenta, both lying almost parallel to the direction of motion of the ion. In the present instance, however, where we consider roton creation at very much higher ionic velocities, it is possible (indeed probable) that the matrix element varies significantly over the relevant range of momenta. To some extent, of course, we have already taken explicit account of this variation of Vqj, through our inclusion of the pole strength factor f(q)f(k). We note, however, that any smooth variation of Vv~ with q and k will be likely to lead to a smoothly varying U (E) curve. Indeed, the same would be true even if there was a discontinuous change in Vqj,, since the integrations over q and k tend to smooth out the variation of R(v) with v. It seems unlikely, therefore, that the relatively sharp break at U -- 70 ms -1 between the data and the theoretical curve can be ascribed to a momentum dependence of the matrix element. It is, of course, possible that the ion dissipates energy via a new mechanism for velocities above about 70 ms -l , this being related to the critical velocity for the process in question. For example, the simultaneous emission of more than two rotons might become important for U > 70 ms -l, and so also might the emission of phonons together with rotons. Dissipation involving the creation of vortex rings (see III) could be responsible. The production of charged vortex rings is quenched (Nancolas and McClintock 1982) in very high electric fields, but processes involving ring creation without capture of the ion may well still occur. It is known (Nancolas et al. 1985b) that an additional dissipation mechanism comes into play at low pressures where roton creation is apparently attenuated (fig. 25). It seems quite likely that the same mechanism is the cause of the extra drag seen at the highest fields in fig. 26. It is believed to involve the quasi-continuous creation and immediate shedding of vorticity. We conclude, therefore, that the relatively small deviation (fig. 26) of U (E) from theory for velocities in the range 65 ms -~ < U < 70 ms -~ are attributable to

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

61

the combined effect of non-parabolicity of the real dispersion curve, momentum dependence of the pole strength, and departures from hk 0 of the average momentum of the excitations emitted by the ion. When explicit allowance is made for these effects, the resultant theoretical curve is in excellent agreement with the data up to 70 ms -t, thus lending further strong support to the roton pairemission hypothesis.

6. Roton creation by "fast" ions All of the results discussed above, and the vast majority of the papers in the huge literature on negative ions in liquid helium, relate to the normal negative ion. There also exists, however, a so-called "fast" negative ion; it, too, can exhibit a roton-creation-limited drift velocity under appropriate conditions. The fast ion remains a rather mysterious entity. Its behaviour has not yet been investigated in any detail, probably because of the difficulty of creating it in sufficient fluxes and in a reliable and reproducible way. Its physical nature is still subject to considerable uncertainty. Nonetheless, there is no doubt as to its reality and it is potentially of considerable importance for future roton creation studies, not least because it is the only probe known to exceed v L in He-II under the saturated vapour pressure (in striking contrast to the normal ion which, as already discussed, is converted almost immediately to a charged vortex ring unless the pressure is above --10 bar). In this section we review what little is known about the fast ion and its propensity for roton creation. The fast ion was discovered as the result of a serendipitous observation by Doake and Gribbon (1969). While using a chopped-dc Cunsolo (1961) ion cell for the measurement of ionic mobilities, they became aware that there were two separate species of negative ion present in the cell, with very different low electric field mobilities, and strikingly different behaviour in strong electric fields, as shown in fig. 29. The lower curve and data correspond to the normal negative ion - the only negative species known up to that time - and the upper curve and data represent the fast ion. It is evident that the drift velocity of the fast ion does not undergo the "giant fall" in high fields displayed by the normal ion, corresponding to the formation of charged vortex rings (Rayfield and Reif 1964). Rather, it seems to level off at a velocity very close to VL = 59 ms -1 under the saturated vapour pressure. Unfortunately, Doake and Gribbon were unable to repeat the experiment to confirm this important result, despite considerable effort, and were therefore unable to carry out a more detailed investigation. Shortly afterwards, however, Ihas and Sanders (1971) discovered that an essential ingredient in the recipe for producing fast ions was that there must be an electrical discharge in the vapour above the liquid surface. (It may be inferred that such a discharge was probably also present in the Doake and Gribbon ex-

62

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1, w

6O ffl

E v U

o >

L.

~ i 0

I

I 300

I

EGC (V//cm)

I

600

Fig. 29. First observation of the fast negative ion in He-ll (Doake and Gribbon 1969). Its measured drift velocity is plotted (a) as a function of electric field, here called Ec_~, at 0.92 K. At large fields, the fast ion appears to reach a plateau velocity close to the Landau critical velocity (59 ms -1 under the saturated vapour pressure). In contrast, the normal negative ion (b) undergoes a giant fall in velocity at a critical electric field, corresponding to the creation of charged vortex rings.

periment on the (unique) occasion when fast ions were observed, but without the experimenters being aware of it.) Ihas and Sanders made a further, equally startling, discovery: there was not just one additional species present, but at least a dozen that could be characterized by their different low field mobilities. In fact, the precise behaviour they observed depended on the experimental conditions, as shown in fig. 30. When the level of the liquid in their cell was close to the lower of the two electrodes between which the discharge was struck, large fluxes of fast ions were produced as indicated in (a). Here, the upper trace G corresponds to a gate-opening pulse admitting ions to a drift space some distance below the surface. The lower trace s shows the signal in the collector, which is obviously double: the fast ions (1) arrive first followed, some considerable time later, by the normal ions (N). For a lower liquid level, however, the behaviour was entirely different as shown in fig. 30(b). Here, the gate pulse (G) has had to be made extremely short in order to be able to resolve the large number of discrete peaks lying between those due to the fast ion (1) and the normal ion (N). The temperature dependences of the low field mobilities for several species were measured precisely near 1 K; but the attainable electric field was insufficient for studies of roton creation. Eden and McClintock (1983) subsequently confirmed the original Doake and Gribbon results, using an ion source of the type developed by Ihas and Sanders (1971), and were able to show that the drift velocity of the fast ion remained almost constant up to a field of at least 5 x 105 V m -l. They also discovered (Eden and McClintock 1984) that at least three of the intermediate mobility

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

63

a}

~ b) GV

....

S~ 6 ~ 897 1112 N

Fig. 30. A profusion of different negative ion species in He-ll, discovered by lhas and Sanders (1971). The plots (lhas and Sanders 1974) show, as a function of time, the gate-opening pulse (G) and the signal at the collector (S) under different conditions: (a) with the liquid level at the lower source electrode; (b) with the level well below the electrode. In (a) the fast ion (1) and normal ion (N) signals are well resolved, and no other species are visible. In (b), using a much narrower gate pulse, the fast ion (1) and normal ion (N) signals are still visible, but there are also a large number of intermediate mobility species (2-12). Also, in comparison with (a), the fast ion signal has become very weak and the normal ion signal has become very strong. species apparently form charged vortex rings, and so cannot be used for roton creation studies. In fig. 31, the upper curve and crosses represent the fast ion; the lower curve and squares represent the normal ion; and the behaviour of three of the intermediates is shown up to the point of their giant falls. In practice the (much larger) normal ion signal tends to obscure the signals from intermediates within a certain electric field range and, although a number of intermediate ion signals could be seen in the range where their velocities decreased with increasing field on the right hand side of fig. 31 (only one is shown), there was considerable ambiguity about which of the low field species each of them corresponded to. Williams et al. (1989) subsequently undertook an experiment to try to establish whether or not roton emission from the fast ion occurred through a paircreation process, with the results shown by the data points in fig. 32. The plot is of U / v L against E 1r3, which produces a straight line as illustrated for the normal negative ion in accordance with eq. (3.44), as discussed above. It is immediately evident, first, that the matrix element for roton creation by the fast ion is much larger than that for the normal one and, secondly, that the data are too scattered,

64

Ch. 1,w

P.V.E. McCLINTOCK and R.M. BOWLEY

(ms -1) 6O

200

0

1

2

3 E (10 5 V m-1)

Fig. 31. Ionic drift velocities measured in strong electric fields (Eden and McClintock 1984). The earlier results (see fig. 29(a)) of Doake and Gribbon (1969) are confirmed: the fast ion data (crosses) start to flatten off towards a value near the Landau critical velocity, in sharp contrast to the normal ion data (squares). Three of the intermediate mobility species (data with diamonds, circles and point-down triangles) are seen to create charged vortex rings.

1-2

.- - o ' ~ ~

>-, 1-0 0.8

O~

02 9

0%

'

20

.

9

.

,

J

i0

-

80

Fig. 32. Ionic velocity ~" divided by the Landau critical velocity V L a s a function of (electric field, E) 1/3 for the fast negative ion (data points) at T = 1.03 K (Williams et al. 1989) compared with the normal ion data of Ellis and McClintock (1985).

Ch. 1, w

THE LANDAU CRITICAL VELOCITY

65

and cover too small a range of E, for one to be able to reach any conclusion about the form of the dependence of U on E. Note that the comparison in fig. 32 must be viewed with caution because the fast ion data are (necessarily) at the saturated vapour pressure, whereas the normal ion line is (necessarily) for pressurized He-II (actually at 25 x 105 Pa). Further work on the fast and exotic ions in all their aspects is much to be desired. Their physical nature has yet to be resolved. Sanders and Ihas (1987) suggested that the fast and intermediate ions were multi-electron bubbles; but there are also some counter-indications (Williams et al. 1988) that appear to weigh against this ingenious idea. A quite different suggestion by Watkins et al. (1984) was that the fast and intermediate ions might relate to excited states of the (He2*)- molecular ion. Although, as already mentioned, at least three of the exotic ion species apparently create charged vortex rings (Eden and McClintock 1984) in strong electric fields, and thus would be unusable for roton creation studies, there is no evidence, at least in experiments near 1 K, that the fast ion ever creates vortex rings. Thus, regardless of its structure, it would appear to offer enormous potential for future investigations of roton creation. In particular, the fast ion offers the possibility of a precise measurement of VL under the saturated vapour pressure, complementary to the measurements for pressures at 13 x 105 Pa and above reported in section 4.

7. Conclusion The investigations of roton creation by negative ions reviewed in this chapter have confirmed Landau's original (1941, 1947) explanation of superfluidity in liquid 4He. They have yielded a precise experimental value of the Landau critical velocity VL which is in good agreement with the theoretical prediction, the latter being based on the dispersion curve as determined (mainly) by inelastic neutron scattering. Roton creation has been shown (at least for the normal negative ion) to be a relatively weak dissipative process, in the sense that quite modest electric fields are sufficient to propel the ions to drift velocities considerably in excess of VL. Measurements of ~ (E) over a very wide range of electric fields are consistent with the hypothesis that the rotons are created by the moving ion in pairs, rather than singly. There remain a number of important unresolved questions that can only be settled by further work, both experimental and theoretical. Notably, what is the physical reason that the rotons are created in pairs? Does single-roton emission ever occur? Are the rotons emitted from the enigmatic fast negative ion also created in pairs? Why does the matrix element for roton pair creation by the normal negative ion fall so rapidly with decreasing pressure?

66

P.V.E. McCLINTOCK and R.M. BOWLEY

Ch. 1

Note added in proof In a very recent paper, Lenosky and Elser report (1995, Phys. Rev. B 51, 12857) that a theory has been proposed (Elser, unpublished; Basile and Elser, unpublished) which "naturally predicts two-roton emission and provides a means to calculate its rate". The theory is based on a continuous collapse formulation of quantum dynamics. Initial calculations based on it predict roton pair emission rates that are smaller by four orders of magnitude than those observed in the experiments; Lenosky and Elder discuss possible reasons for this discrepancy.

Acknowledgements It is a pleasure to acknowledge numerous stimulating discussions with, and a great deal of help from, our several present and former colleagues who collaborated in the Lancaster/Nottingham joint research programme on the breakdown of superfluidity in liquid 4He. These include, especially, David Allum, Van Eden, Terry Ellis, Walter Fairbairn, Philip Hendry, Chris Jewell, Stuart Lawson, Frank Moss, Graham Nancolas, Alan Phillips, Fred Sheard, Philip Stamp, Musa Wahab and Charles Williams. We must also acknowledge valuable discussions and correspondence with Mark Dykman, Lev Pitaevskii and S. Iordanskii, and especially their help in clarifying some of the issues raised and discussed in section 3.7. We are much indebted to Jack Allen, Russell Donnelly and Tony Leggett for critical readings of, and constructive comments on, an early draft of the manuscript; but we obviously retain full responsibility for any errors that remain. The experimental side of the programme relied heavily on the technical expertise of Norman Bewley, David Bidle, Ian Miller and Andy Muirhead. We are very grateful to Heather Coates for her rapid and accurate typing of this chapter from an awkward manuscript. The research was supported by the late Science and Engineering Research Council (UK).

References Allen, J.F. and H. Jones, 1938, Nature 141, 243. Ahonen, A.I., J. Kokko, M.A. Paalanen, R.C. Richardson, W. Schoepe and Y. Takano, 1978, J. Low Temp. Phys. 30, 205. AUum, D.R., P.V.E. McClintock and A. Phillips, 1975, in Proc. 14th Int. Conf. on Low Temperature Physics, Vol. 1, eds M. Krusius and M. Vuorio (North-Holland, Amsterdam) p. 248. Allum, D.R., R.M. Bowley and P.V.E. McClintock, 1976, Phys. Rev. Lett. 36, 1313. Allum, D.R., P.V.E. McClintock, A. Phillips and R.M. Bowley, 1977, Phil. Trans. R. Soc. London A 284, 179 (referred to in text as I). Andrei, E.Y. and W.I. Glaberson, 1980, Phys. Lett. A 79, 431. Awschalom, D.D. and K.W. Schwarz, 1984, Phys. Rev. Lett. 52, 49. Bogoliubov, N., 1947, J. Phys. Moscow 11, 23 (translated in Galasiewicz, Z.M., 1971, Helium 4, Pergamon, Oxford, p. 247).

Ch. 1

THE L A N D A U CRITICAL VELOCITY

67

Bowley, R.M. and F.W. Sheard, 1975, in Proc. 14th. Int. Conf. on Low Temperature Physics, Vol. 1, eds M. Krusius and M. Vuorio (North-Holland, Amsterdam) p. 165. Bowley, R.M. and F.W. Sheard, 1977, Phys. Rev. B 16, 244. Bowley, R.M., P.V.E. McClintock, F.E. Moss and P.C.E Stamp, 1980, Phys. Rev. Lett. 44, 161. Bowley, R.M., P.V.E. McClintock, F.E. Moss, G.G. Nancolas and P.C.E. Stamp, 1982, Phil. Trans. R. Soc. London A 307, 201 (referred to in text as III). Bowley, R.M., G.G. Nancolas and P.V.E. McClintock, 1984, Phys. Rev. Lett. 52, 659. Brooks, J.S. and R.J. Donnelly, 1977, J. Phys. Chem. Ref. Data 6, 51. Brundobler, S., 1994, Critical Dynamics of Electron Bubbles in Superfluid Helium, Ph.D. thesis, Comell University, Ithaca, NY. Cowley, R.A. and A.D.B. Woods, 1971, Can. J. Phys. 49, 177. Cunsolo, S., 1961, Nuovo Cimento 21, 76. Daunt, J.G. and K. Mendelssohn, 1938, Nature 141, 911. Doake, C.S.M. and P.W.F. Gribbon, 1969, Phys. Lett. A 30, 251. Donnelly, R.J., 1972, Phys. Lett. A 39, 221. Donnelly, R.J. and P.H. Roberts, 1977, J. Phys. C 10, L683. Donnelly, R.J., 1991, Quantized Vortices in Helium II (Cambridge University Press, Cambridge). Eden, V.L. and P.V.E. McClintock, 1983, in 75th Jubilee Conference on Helium-4, ed J.G.M. Armitage (World Scientific, Singapore) p. 194. Eden, V.L. and P.V.E. McClintock, 1984, Phys. Lett. A 102, 197. Ellis, T. and P.V.E. McClintock, 1981, Physica B 107, 569. Ellis, T. and P.V.E. McClintock, 1982, Phys. Rev. Lett. 48, 1834. Ellis, T. and P.V.E. McClintock, 1985, Phil. Trans. R. Soc. London A 315, 259 (referred to in the text as V). Ellis, T., P.V.E. McClintock, R.M. Bowley and D.R. Allum, 1980a, Phil. Trans. R. Soc. London A 296, 581 (referred to in the text as II). Ellis, T., C.I. Jewell and P.V.E. McClintock, 1980b, Phys. Lett. A 78, 358. Ellis, T., P.V.E. McClintock and R.M. Bowley, 1983, J. Phys. C 16, L485. Fetter, A.L., 1976, in The Physics of Liquid and Solid Helium, eds K.H. Benneman and J.B. Ketterson (Wiley, New York) part 1, ch. 3. Feynman, R.P., 1955, in Progress in LOw Temperature Physics, Vol. 1, ed C.J. Gorter (NorthHolland, Amsterdam) p. 17. Fisher, S.N., A.M. Gufnault, C.J. Kennedy and G.R. Pickett, 1991, Phys. Rev. Lett. 67, 1270. Gerhold, J., 1972, Cryogenics 12, 370. Hendry, P.C. and P.V.E. McClintock, 1987, Cryogenics 27, 131. Hendry, P.C., N.S. Lawson, P.V.E. McClintock, C.D.H. Williams and R.M. Bowley, 1988, Phys. Rev. Lett. 60, 604. Hendry, P.C., N.S. Lawson, P.V.E. McClintock, C.D.H. Williams and R.M. Bowley, 1990, Phil. Trans. R. Soc. London A 332, 387 (referred to in the text as VI). lhas, G.G. and T.M. Sanders, 1971, Phys. Rev. Lett. 27, 383. lhas, G.G. and T.M. Sanders, 1974, in Low Temperature Physics - LT13, Vol. 1, eds K.D. Timmerhaus, W.J. O'Sullivan and E.F. Hammel, (Plenum, New York) p. 477. lordanskii, S.V., 1968, Soviet Phys. JETP 27, 793. Kapitza, P., 1938, Nature, 141, 74. Keesom, W.H. and G.E. Macwood, 1938, Physica 5, 737. Keller, W.E., 1969, Helium-three and Helium-four (Plenum, New York). Kuper, C.G., 1961, Phys. Rev. 122, 1007. Landau, L.D., 1941, J. Phys. Moscow 5, 71 (reprinted in Khalatnikov, I.M., 1965, Introduction to the Theory of Superfluidity (Benjamin, New York) p. 185).

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P.V.E. M c C L I N T O C K and R.M. B O W L E Y

Ch. 1

Landau, L.D., 1947, J. Phys. Moscow 11, 91 (reprinted in Khalamikov, I.M., 1965, Introduction to the Theory of Superfluidity (Benjamin, New York) p. 205). Leggett, A.J., 199 l, Proceedings of the Blydepoort 1991 Summer School (Low Temperature Physics, ed M.J.R. Hoch and R.H. Lemmer, Springer-Verlag, Berlin: Springer Lecture Notes in Physics, Vol. 394); see especially lecture 4. Lifshitz, E.M. and L.P. Pitaevskii, 1980, Statistical Physics Part II, Landau and Lifshitz Course on Theoretical Physics, Vol. 9 (Pergamon, Oxford). London, F., 1938, Nature 141,643. London, F., 1954, Superfluids, Vols. 1 and 2 (reprinted 1964, Dover, New York). Maynard, J., 1976, Phys. Rev. B 14, 3868. P.V.E. McClintock, 1978, Cryogenics 18, 201. Meyer, L. and F. Reif, 1961, Phys. Rev. 123, 727. Nancolas, G.G. and P.V.E. McClintock, 1982, Phys. Rev. Lett. 48, 1190. Nancolas, G.G., T. Ellis, P.V.E. McClintock and R.M. Bowley, 1985a, Phil. Trans. R. Soc. London A 313, 537 (referred to in the text as IV). Nancolas, G.G., T. Ellis, P.V.E. McClintock and R.M. Bowley, 1985b, Nature 316, 797. Neeper, D.A., 1968, Phys. Rev. Lett. 21,274. Neeper, D.A. and L. Meyer, 1969 Phys. Rev. 182, 223. Phillips, A. and P.V.E. McClintock, 1973, Phys. Lett. A 46, 109. Phillips, A. and P.V.E. McClintock, 1974, Phys. Rev. Lett. 33, 1468. Phillips, A. and P.V.E. McClintock, 1975, Phil. Trans. R. Soc. London A 278, 271. Pitaevskii, L.P., 1959, Soviet Phys JETP 9, 830. Rayfield, G.W. and F. Reif, 1964, Phys. Rev. 136, 1194. Rayfield, G.W., 1966, Phys. Rev. Lett. 16, 934. Rayfield, G.W., 1968, Phys. Rev. 168, 222. Reif, F. and L. Meyer, 1960 Phys. Rev. 119, 1164. Sanders, T.M. and G.G. lhas, 1987, Phys. Rev. Lett. 59, 1722. Schwarz, K.W., 1972 Phys. Rev. A 6, 837. Schwarz, K.W., 1975 Adv. Chem. Phys. 33, I. Sheard, F.W. and R.M. Bowley, 1978, Phys. Rev. B 17, 201. Smith, A.J., R.A. Cowley, A.D.B. Woods and P. Martel, 1977, J. Phys. C 10, 543. Springett, B.E. and R.J. Donnelly, 1966, Phys. Rev. Lett. 17, 364. Springett, B.E., M.H. Cohen and J. Jortner, 1967, Phys. Rev. 159, 183. Takken, E.H., 1970, Phys. Rev. A 1, 1220. Tilley, D.R. and J. Tilley, 1990, Superfluidity and Superconductivity, 3rd edn (Hilger, Bristol). Varoquaux, E., W. Zimmerman, Jr. and O. Avenel, 1991, in Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, eds A.F.G. Wyatt and H.J. Lauter (Plenum, New York) p. 343. Vinen, W.F., 1983, in 75th Jubilee Conference on Helium-4, ed J.G.M. Armitage (World Scientific, Singapore) p. 2. Volovik, G.E., 1970, Soviet Phys. JETP 31, 1106. Watkins, J.L., J.S. Zmuidzinas and G.A. Williams, 1984, in Proc. 17th Int. Conf. on Low Temperature Physics, eds U. Eckern, A. Schmid, W. Weber and H. Wiihl (Elsevier, Amsterdam) p. 1197. Wilks, J., 1967, The Properties of Liquid and Solid Helium (Clarendon, Oxford). Wilks, J. and D.S. Betts, 1987, An Introduction to Liquid Helium, 2nd edn (Clarendon, Oxford). Williams, C.D.H., P.C. Hendry and P.V.E. McClintock, 1988, Phys. Rev. Lett. 60, 865. Williams, C.D.H, P.V.E. McClintock and P.C. Hendry, 1989, in Elementary Excitations in Quantum Fluids, eds K. Ohbayashi and M. Watabe (Springer, Berlin) p. 192. Zawadowski, A., 1978, in Proc. Int. Conf. on Quantum Fluids (Erice), eds J. Ruvalds and T. Regge (North-Holland, Amsterdam) p. 293.

CHAPTER 2

SPIN SUPERCURRENT AND NOVEL PROPERTIES OF NMR IN 3HE BY

YU.M. B U N K O V P.L. Kapitza Institute for Physical Problems, Moscow, Russia

and CNRS-CRTBT, associ~ gt l' Universit~ J. Fourier de Grenoble, France

Progress in Low Temperature Physics, Volume XIV Edited by W.P. Halperin 9 Elsevier Science B.V., 1995. All rights reserved

69

Contents 1. Introduction ......................................................................................................................... 2. Basic properties ................................................................................................................... 2.1. Spatially uniform NMR ............................................................................................... 2.2. Spin supercurrent ......................................................................................................... 3. Experimental methods ......................................................................................................... 4. NMR and spin supercurrent in 3He-B ................................................................................ 4.1. Pulsed NMR ................................................................................................................. 4.2. CW NMR ..................................................................................................................... 4.3. Processes of magnetic relaxation ................................................................................. 4.3.1. Spin diffusion and intrinsic relaxation ................................................................... 4.3.2. Surface relaxation ................................................................................................... 4.3.3. Catastrophic relaxation ........................................................................................... 4.4. HPD oscillations .......................................................................................................... 5. Steady spin supercurrent ...................................................................................................... 5.1. Spin supercurrent in a channel ..................................................................................... 5.2. Phase slippage .............................................................................................................. 5.3. Josephson phenomena .................................................................................................. 5.4. Spin supercurrent vortex .............................................................................................. 6. Spin supercurrent in 3 H e - A ................................................................................................ 6.1. Instability of homogeneous precession ........................................................................ 7. Spin supercurrent at propagating A - B boundary ................................................................ 8. Conclusion ........................................................................................................................... Acknowledgments .................................................................................................................... References ................................................................................................................................

70

71 75 79 85 93 98 98 103 107 107 112 114 119 124 124 128 132 134 138 139 146 152 154 154

I. Introduction

For the last 20 years, the superfluid phases of 3He, discovered experimentally by Osheroff et al. (1972), have been among the most interesting subjects of research in the field of condensed matter physics. 3He belongs to the class of liquids which undergo, at sufficiently low temperatures, transitions to coherent quantum states related to Bose-Einstein condensation. For these liquids quantum behavior determines the macroscopic properties. One such property is the phenomenon of superfluidity, discovered in liquid 4He by Kapitza (1938). The fundamental property of superfluidity is its ability to flow without friction. However, we use the term "superfluidity" to refer to numerous other properties which manifest themselves when the liquid becomes superfluid. Another macroscopic quantum phenomenon of this kind is superconductivity which occurs in a number of metals and is characterized in particular by their ability to conduct electric current without resistance. Since the conduction electrons have halfinteger spin and consequently are subject to Fermi statistics, direct Bose condensation is forbidden. However, at sufficiently low temperatures, Cooper pairs of electrons with zero spin form, so that the Bose condensation process may occur. The explanation of the phenomenon of superconductivity, resulting from the superfluidity of the Cooper pairs, was given by Bardeen et al. (1957). The superfluidity of liquid SHe, which has half-integer spin, is also the result of Cooper pair formation. As was shown by Pitaevsky (1959), the Van der Waals interaction between 3He atoms leads to the formation of Cooper pairs with nonzero orbital moment and, if the orbital moment is odd, owing to Pauli's exclusion principle, the pairs should form in the triplet state (S = 1). Hence, the Cooper pairs in superfluid 3He have both an orbital and spin moment. The Cooper pairs of 3He form a single coherent state, and consequently the superfluid state also exhibits quantum liquid crystal behavior and quantum magnetically ordered behavior. The abundance of quantum properties of 3He makes it one of the most interesting topics in the physics of condensed matter. To describe the properties of superfluid liquids it is convenient to introduce an order parameter which has the same symmetry as the wave function of the condensate. For 4He the order parameter can be written in the very simple form e i(1)(r,t),

(1.1)

where It/~2 = Ps is the density of the superfluid component and q~ is the phase of 71

72

YU.M. BUNKOV

Ch. 2, w

the wave function. The existence of a phase 9 is a manifestation of broken gauge symmetry. In the case that the order parameter is not uniform and described by a spatial gradient of phase 4, the corresponding quantum state involves motion of the superfluid with the momentum density J given by j

h =--psVtP. m

(1.2)

As a result, there is mass transport with unique quantum properties. In the case of the triplet pairing in 3He, several symmetries are broken simultaneously, i.e. gauge, spin and orbit symmetries. Therefore in addition to a global phase, the Cooper pair wave function is also described by the phases of rotations about axes in orbital space and spin space. Spatial gradients of the phase of rotation in orbit space give an additional term to the mass superfluidity, at least for 3He-A, while the spatial gradient of the phase of rotation in spin space provides a new transport process, the spatial transport of magnetization (spin supercurrent). Following Leggett (1975), we introduce the vector in spin space, d(k), which describes the broken spin symmetry and whose dependence on k describes the breaking of relative spin-orbit symmetry. Since the pairs have L = 1, d is a linear combination of spherical harmonics di = A',aka, where A/a is in general a complex matrix which characterizes the order parameter of superfluid 3He. As was shown by Barton and Moore (1974) for Cooper pairs with S = 1 and L = 1, there are 11 different possible forms of the order parameter. Two phases, known as the A and B phases, are stable in the bulk. Under the influence of an external magnetic field a third phase, A l, may appear. The order parameters corresponding to these phases were discussed theoretically by Anderson and Morel (1961) and by Balian and Werthamer (1963). Anticipating events, we note that the magnetic properties and, especially, the NMR properties of the phases with these order parameters, are extremely unusual, and indeed it was the NMR properties which enabled the identification of the A and B phases. Recently the question of correct identification of the A phase has been discussed. We consider it in section 7 with a description of specific properties of 3He-A. The quantum state for triplet Cooper pairing of 3He can be described by the sum of three spin projection states: (1.3) where W1"1",WJ,$ and Wr are the amplitudes associated with the spin substates I "1"!'>, I ,I,,1,> and I1",1,) respectively. It is useful to represent the relation between these substates by the vector d.

SPIN SUPERCURRENT AND NMR IN 3He

Ch. 2, w

W(k) = [ tlJt$ (k)

qJ**(k)} =

dz(k)

dx(k)+idy(k)"

73 (1.4)

Let us begin by examining the consequences of a spatially inhomogeneous rotation of d. For example, let us consider the state with a rotation gradient in the form:

d x + idy = Id•

(1.5)

where d• is the component of d perpendicular to the quantization axis and tr is the gradient in direction of R. We may consider the spin "up" Cooper pairs ~1"1" and the spin "down" Cooper pairs ~$$ as two separate superfluids. Therefore, for the function ~$$, we have a phase gradient /r directed along R, while for ~1"1" we have the opposite gradient (-R) and consequently a counterflow of these two superfluids. This counterflow transports magnetization without mass transport and is called the spin supercurrent. (It can be combined with mass transport at large magnetic field, when the difference in the density of the two components must be taken into account). Generally speaking, the existence of a spin supercurrent in superfluid 3He has been expected for a long time. The most general expression for the spin supercurrent is the following: h J ict = ~

P ijotfl~-~jfl ,

(1.6)

where Pijctfl is the spin superfluid density tensor and g-2j#are the phase gradients of the order parameter. The explicit form of these phase gradients will be given later (see eq. (2.23)). The spin transport equations (1.6) play an essential role in the spin dynamics of 3He. First, the solution of the spin dynamics equations with the spin transport equations gives the spin-wave spectrum (Leggett 1975). We will be interested here in solutions which correspond to long distance transport of magnetization by a spin supercurrent. The first attempt to describe this phenomenon was made by Vuorio (1976). He tried to explain the fast magnetic relaxation in 3He-A observed by Corruccini and Osheroff (1975a), as a transport of magnetization by spin supercurrent out of the sensitive region of the pick-up coils. In fact the fast relaxation in 3He-A has another explanation which is considered in section 7. Furthermore, the spin supercurrent was treated by Vuorio in direct analogy with superfluid mass current. It should be pointed out, however, that the analogy between the spin supercurrent and the mass supercurrent is limited. The magnetization is a vector quantity. Consequently there is no conservation law for its

74

YU.M. BUNKOV

Ch. 2, w

components and the spin supercurrent should be considered in relation to very complex spin dynamics, as was first done by Fomin (1984b). To create a long range magnetization transport by spin supercurrent, a long range spatial structure of d is necessary. It can be static (the texture), or dynamic, if d rotates together with the magnetization in NMR. In the first case a persistent spin supercurrent state should exist, as is obvious from the theoretical point of view, but has not yet been demonstrated experimentally. In the case of NMR, d is coupled with the precessing magnetization. The spatial structure of d is therefore determined by the spatial structure of the magnetization. There are two different modes of NMR in superfluid 3He, longitudinal NMR and transverse NMR. In the first case the value of the magnetization oscillates, in the second case the magnetization is deflected from the external magnetic field and precesses. In 1984 the spatial transfer of magnetization by spin supercurrent in the case of transverse mode NMR was observed experimentally in Moscow at the Institute for Physical Problems (Borovik-Romanov et al. 1984). Simultaneously Fomin (1984b) constructed a theory of spin supercurrent in 3He-B, describing the actual conditions of the NMR experiments. He showed that for the transverse mode of NMR, the spin supercurrent of the longitudinal magnetization Jp can be described as a function of gradients of the angle of deflection fl and the angle of precession a of the magnetization Jr, = Fa (/~)Va + F~ (~)Vfl.

(1.7)

Consequently, to create the long range spin supercurrent in 3He-B, it is necessary to induce a spatial gradient in the phase of the precession. As a result, spin supercurrent occurs for both pulsed or CW NMR in any inhomogeneous field. In practice, it is difficult to set up the experimental conditions for transverse NMR without excitation of spin supercurrents. For example many of the puzzling features of NMR in 3He-B (see for example the review of Lee and Richardson 1978) can now be explained on the basis of spin supercurrent transport. This chapter reviews the results of intensive investigations of spin supercurrent phenomena both in 3He-B and in 3He-A. There are many related phenomena that are included in this review which deal with experimentally observed phenomena and their practical usage for related studies. The main part of the present results have not yet been included in reviews of superfluid 3He physics. For general reference we refer readers to the recently published book of Vollhardt and W61fle (1990). An outline of the basic theoretical aspects of the spin supercurrent is contained in the review of Fomin (1990). Some early experimental results can be found in the reviews by Bunkov (1985, 1987) and BorovikRomanov and Bunkov (1990). In section 2, we consider the main properties of NMR in superfluid phases of 3He, we represent results of the theoretical and experimental investigation and

Ch. 2, w1

SPIN SUPERCURRENT AND NMR IN 3He

75

provide, for the convenience of the readers, the theoretical background to spin supercurrent phenomena. In section 3, we review the NMR methods for investigating the superfluid phases of 3He and, in particular, describe experimental conditions under which the investigations, reviewed in the following sections, were carried out. In section 4, we consider the process of magnetization transfer by spin supercurrent in 3He-B, which leads to the formation of a domain with homogeneously precessing magnetization (HPD). We discuss the effect of the spin supercurrent on the NMR properties of 3He-B. The measurements of magnetic relaxation due to the spin diffusion of the normal component of 3He, the spin-orbit interaction, the surface interaction, and a crossover relaxation due to crossing with a new mode of NMR (the NMR in molecular Fermi liquid field) are reviewed. Concluding this section, we review the experimental results of NMR in 3He-B, obtained previously, which can be explained by the spin supercurrent phenomena. In section 5, we review the experiments where the steady state with spin supercurrent was investigated. These include the states with magnetization transport along a channel and the spin current vortex state. In the first case the observation of Josephson phenomena, phase slippage and the crossing of a channel by spin current vortices are described. The spin supercurrent transport in 3He-A leadsto the instability of uniform precession and its decay in a spatially nonuniform structure. The influence of this process on the magnetic relaxation of magnetization in 3He-A is considered in section 6. The results of unsuccessful attempts to apply the technique of parametric excitation of 3He-A NMR are described. This negative result shows that the NMR properties of 3He-A do not correspond precisely to the existing theory. In the last section we review the results of the Los-Alamos A-B boundary propagation experiments and describe the observed magnetic signals as evidence of a new mode of spin supercurrent related to longitudinal NMR.

2. Basic properties Liquid 3He is first of all a Fermi liquid and its properties below the Fermi temperature ~ 1 K can be interpreted in terms of the phenomenological theory of Landau (1957). According to this theory, liquid 3He can be described as a gas of elementary excitations or "quasiparticles" and "quasiholes" with energy equal to zero at the Fermi surface and the dispersion curve determined by the Fermi liquid correction coefficients. At the transition to the superfluid state the excitation spectrum changes. The Cooper pair formation is accompanied by the appearance of the energy gap (A(k)) in the excitation spectrum. Excitations of this

76

YU.M. BUNKOV

Ch. 2, w

kind are usually named Bogoliubov quasiparticles (Bogoliubov 1958). In 3He-A the energy gap is anisotropic. There are two poles on the Fermi surface, where the energy gap is equal to zero. As a result 3He-A has very anisotropic features. In contrast, the 3He-B phase has an isotropic energy gap, which can be deformed by an external influence such as a magnetic field, boundary conditions or a counterflow. The magnetic properties of 3He are modified by the Fermi liquid conditions. According to the phenomenological Fermi liquid theory of Landau, the Fermi liquid correction of magnetic properties can be visualized first as a correction to the magnetic susceptibility.

Zn ~

X,,o ~ , {l+Fo~}

(2.1)

with F0 a being a coefficient corresponding to the first order antisymmetric correction of the Fermi liquid, Xn0 is the susceptibility of 3He with only effective mass correction and Z, is the experimentally measured susceptibility of 3He in the normal phase. Consequently the magnetization can be cast in the form M = j~n H = Xno(H+ HL),

(2.2)

From this equation one can see that susceptibility correction can be represented by the introduction of the effective molecular field M

H,~ = - F 0 a ~ .

~2.3)

Xn0 These two different representations of Fermi liquid correction lead to the same experimental results for static and dynamic experiments. The situation is drastically changed for the superfluid 3He owing the coexistence of two subsystems: the superfluid and normal components of liquid. In this case a new mode of NMR has been observed recently by Bunkov et al. (1992a). This mode corresponds to the relative precession of both components around the molecular field HL and confirms the existence of the molecular field in the Fermi liquid. We shall named this field the Landau field. Interestingly, the new mode of NMR can be considered as the magnetic analog of second sound. The susceptibility of the Cooper pairs of 3He depends on the orientation of the vector d. For d _L H it is the same as in normal 3He, while for d II H it falls with temperature, decreasing as Xn Y(T), where Y(T) is the familiar Yoshida

Ch. 2, {}2

SPIN SUPERCURRENT AND NMR IN 3He

77

function from the BCS theory of superconductivity which changes from 1 to 0 as the temperature falls from Tc to 0. In superfluid 3He-A the vector d is oriented perpendicular to the magnetic field. As a result the susceptibility of 3He-A is the same as for normal 3He. The susceptibility of 3He-B, taking into account the higher Fermi liquid correction term /72a, was calculated by Serene and Rainer (1977), and is equal to

2 +(l +3/ SFr ;tB =;in0 3 + Ft~ ( 2 + Y ) + l / 5 F~ ( l + ( 2 + 3 Ft~ ) y )

(2.4)

for the limit of small magnetic fields, when the magnetic distortion of the energy gap can be neglected. The parameter F0 a changes from-0.68 to-0.75 as pressure changes from 0 to 29 bar. It is interesting to point out that the Landau field, which corresponds to relation (2.3) (see Schopohl 1982), can be equal to the external field at some temperature. This leads to a new phenomenon, when the mode of ordinary NMR precession interacts with the mode of the relative precession of the normal and superfluid components of the magnetization around the Landau field. The cross relaxation of these two NMR modes was observed in Lancaster by Bunkov et al. (1992a) and is discussed in section 4.3.3. The superfluid phases of 3He have very complex magnetic properties, which are the result of magnetic ordering of the nuclear spin system. However, in contrast to ordinary magnetically ordered materials, where we have spatial ordering that can be described by the language of sublattices, here we have a mixing of quantum substates. The relation between these substates can be described by the vector d which is actually the axis of quantization of the Cooper pair state. The projection of the spin of the Cooper pair on the direction of d is equal to zero. So in some sense it is similar to the antiferromagnetic vector I in antiferromagnets. The different phases of 3He have different orientations of d. For 3He-A it is a single vector for all Cooper pairs and its dz component is equal to zero. This means that the substate ~% has zero density and 3He-A is a mixture of only two quantum substates. In 3He-B the vector d is a function of the Cooper pair orbital momentum k and is described by the equation: d = R(O,n)k with R(O,n) being a rotation matrix about the axis n through the angle 0. Therefore, 3He-B is a unique magnetically ordered substance with a very isotropic state, instead of the broken relative symmetry between spin and orbit spaces, which is described by the vector n. The spatial orientation of the order parameter (so-called texture) is determined by the balance of different energies: the dipole-dipole spin-orbit interaction energy, the magnetic orientation energy, the surface orientation energy, the banding (gradient) energy and so on. The dipole-dipole energy is given by

78

YU.M. BUNKOV

Ch. 2, w

F~ = - 3 Gd (T)(d .l)2 +const,

(2.5)

F B = 4G~(r)(cos o + 88

(2.6)

for 3He-A and 3He-B correspondingly, where Gd is the dipole coupling strength. The dipole-dipole energy and the Zeeman energy are the main factors determining the basic properties of magnetic resonance. The dipole energy couples the motion of the spin to the orbital degree of freedom, which allows one to use the NMR method to study the nonmagnetic properties of superfluid 3He including textures, currents and vortices. The different external fields deform the order parameter and consequently give an orientation torque. The magnetic field orientation energy is given by FHA = a A ( H . d ) 2 ,

(2.7)

YB = - a B (n. H ) 2 .

(2.8)

Hence, in a fairly intense magnetic field, in 3He-A the d and I are directed parallel to one another and perpendicular to the external magnetic field. In 3He-B, n is parallel to H, while the dipole-dipole energy fixes the angle 0 = cos-l(-l/4) = 104 ~ The wall of any experimental cell disturbs this orientation by additional surface energies. There is the surface dipole orientation energy,

Fs~ = -b((s . n ) 2 - 5 ( s

.n) 4 ),

(2.9)

and the surface field orientation energy, FsB = -d(s]~/-/) 2 .

(2.10)

The mass counterflow v defines an anisotropy axis in orbit space. If both magnetic field and counterflow are present, the energy term appears, which in leading order in H and v has the form FI~Bv= _ f ( , ~ / ) 2

= _16,0( v .l)2.

(2.11)

This energy can be treated as the difference of the kinetic energy of superflow along and transverse to the axis l =/~S which is the result of superfluid density anisotropy 6p induced by the magnetic field.

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

79

The gradient energy of the order parameter can be considered as the kinetic energy of superflow. The gradient energy, related to the gradient of the common phase of the order parameter corresponds to the kinetic energy of mass superflow, while the gradient energy of orientation of the magnetic part of the order parameter corresponds to the kinetic energy of the spin supercurrent. In other words, it corresponds to the kinetic energy of the counterflow of mass supercurrent components with opposite spin. As it is the main subject of this review article, we consider this energy in section 2.2.

2.1. Spatially uniform N M R

The equations of motion of the magnetization in the superfluid phases of 3He were derived by Leggett (1975) starting from the phenomenological Hamiltonian M 2 F = ---

2X

(2.12)

M . H + Fd

and have the following form: dM - -

dt

= ~,[M x H] +

-~-t=y

Rd, (2.13)

x H-

,

where y/2~ =-3.2435 kHz/Oe is the gyromagnetic ratio for 3He and the dipole torque

R d =6Gd[d• for 3He-A and R d = -~ G d sin 0(1 + 4 cos O)n,

for 3He-B. The solution of these equations leads to the presence of two branches in the NMR spectrum. One corresponds to the precession of the magnetization around the magnetic field, and the other to oscillations of the value of the magnetization (the longitudinal NMR).

80

YU.M. BUNKOV

Ch. 2, w

A more detailed consideration of the spin dynamics of the superfluid phases of 3He can be conveniently carried out using a dynamic theory in which the entire evolution of the order parameter reduces to rotation in three-dimensional spin space. A discussion of this type of theory, which is well known in magnetism, can be found in the review by Andreev and Marchenko (1980). The dynamic behavior of the system is described by three pairs of canonically conjugate moment and angle variables.

OMi Ot O (~ i

Ot

OF Of 6qbi O(6F/ 6Mi) t~F 6M i

(2.14)

Of O(6F / &Pi )

where F is a free energy and f is a dissipative function. We shall consider NMR as a solid body rotation of all spin space vectors. It is useful to introduce a moving system of coordinates (~',r/,~), rigidly coupled to

iI

v

Y ~

lVk

• Fig. 1. The Euler angles a, fl, ~pdetermine the orientation of the coordinate system ~, ~, r/moving with solid body rotation of the spin part of the order parameter.

Ch. 2, {}2

SPIN SUPERCURRENT AND NMR IN 3He

81

the system of vectors M and d. Let the vector ~" be parallel to M. It is useful to characterize the orientation of the spin part of the order parameter by the Euler angles: (see fig. 1) the angle of precession a, the angle of deflection fl and the angle of internal precession 9o. The last one is the result of broken spin symmetry and describes the rotation of d around M. If the magnetic field energy is retained in the free energy and the dissipation is neglected, the motion of the order parameter in spin space reduces to the superposition of two rotations: the rotation around M at the angular velocity to s = ~,MIZ and the rotation of M around the z axis at the Larmor frequency tOh = -~,H. Since the magnetic energy is independent of the angles a and tp, the moments conjugated to them, M~ and Mz, are integrals of the motion. On the other hand, the angle fl and the moment M,1 can execute oscillatory motion corresponding to nutation. As we are interested in the motion without nutation we shall not consider the equations for M,1 Following Fomin (1978) let us introduce the angle ~p = a + 90 instead of qg. Then the rapidly changing variable ~o can be excluded and the system of equations (2.14), without the dissipation term, become ~3F

= -r 37' 6F

-yH +

/ Z, (2.15)

M~ a

=

54~ = i, =

6F -~, 6a

t~F

In the case where we only consider Zeeman energy, then the right sides of the first and third equations are equal to zero and the quantities M~ and P = M~-Mz conserved. Let us now consider the influence of the dipole-dipole interaction (eqs. (2.5) and (2.6)) on the NMR properties. As we are not interested in nutation, we average this energy over the rapidly changing variable a. Then for 3He-A and 3He-B we have (Fomin 1978)

VA = _

~-'~ 2 8Y 2A [ ( l + c o s f l 2

) + ~ ( 1 +COS fl) 2 COS2@],

(2.16)

82

YU.M. BUNKOV

Ch. 2, w

Fig. 2. The free energy of the dipole-dipole interaction in 3He-A averaged over the fast variable, as a function of phase ~o and cos ft.

VB _ 15)'2a [2COS f l - 1 + 2(1 +COS fl) COSr 2

(2.17)

The shape of the energy surface V as a function of the angles fl and (p is illustrated in figs. 2 and 3. The equations of motion (2.15) now have the form OV

e=0, (2.18) (p=y

g

-H

,

&=)' OP These equations describe the specific features of NMR in superfluid 3He. The transverse NMR mode corresponds to the point of the potential V with the minimum of energy at constant ft. For 3He-A it corresponds to the condition q~ = 0 _ n~ and for 3He-B it corresponds to ~ = 0 _+2n~ for fl > 104 ~ and a more complex relation (see eq. (2.30)) for smaller angles of deflection, shown by the solid line in fig. 3. The frequency of transverse NMR is determined by the

Ch. 2, {}2

SPIN SUPERCURRENT AND NMR IN 3He

83

Fig. 3. The free energy of the dipole-dipole interaction in 3He-B averaged over the fast variable, as a function of phase 9 and cos/5. The solid line shows the minimum of energy at constant angle ft.

equation for &. Consequently for 3He-A it gives the dependence of the NMR frequency on the angle fl, 2 co = - y H _

_1 g2A_(1 + 3 cos/3). 8 yH

(2.19)

For 3He-B, for/5 > cosq(-l/4) --- 104 ~ it gives

a) = - y H

+ - - - - - - - (1 + 4 cos fl). 15 yH

(2.20)

For smaller angles of the magnetization deflection in 3He-B the minimum of the potential V does not depend on [3 and consequently the magnetization precesses at the Larmor frequency. It is very useful to visualize the trajectory of the spin part of the order parameter of 3He in the potential V after the RF pulse. Let us suppose that after the excitation the system is moved to the points (1) in figs. 2 and 3. (for 3He-A and 3He-B, respectively). The positions of these points are determined by the state of the texture before excitation and the RF pulse parameters. Owing to the internal processes of relaxation, the system moves to the periodic precession solutions (points (2)) for a short time (Fomin 1980). These solutions correspond to the minimum of V. After this the behavior of 3He-A and 3He-B is completely

84

YU.M. BUNKOV

Ch. 2, w

different. For 3He-B this solution is stable and is not accompanied by the internal magnetic relaxation process. So this solution should be persistent in the absence of external influences which otherwise lead to additional processes of magnetic relaxation, causing the system to slowly move along the valley to the region/5/= 0, and finally to the stationary state. The 3He-A behavior is different, owing to the curvature of the minimum of the potential VA (see fig. 2). As was shown by Fomin (1979, 1984a), the homogeneous precession of magnetization in 3He-A is unstable. It can be demonstrated as follows. If we assume the possibility of spatial splitting of 3He-A into two subsystems, which move in opposite directions along the minimum of the potential VA (tO points 3+ and 3- in fig. 2) with conservation of the total longitudinal magnetic moment, then the total dipole-dipole energy decreases. As a result the spatial splitting is energetically favorable and spatial perturbations grow exponentially. This process of perturbation growth requires spatial redistribution of the magnetization, which takes place by spin supercurrent transport. Finally the formation of spatially inhomogeneous structure leads to very effective spin diffusion relaxation observed experimentally. This phenomenon is discussed in detail in section 6. The longitudinal NMR corresponds to a coupled motion of Me and ~ at/~ = 0 and has the frequency 0) 2 _ 0

2 V

2

2

O$ 2 y2 H =Q2 ff2B,

(2.21)

which corresponds to the curvature of the potential V for undeflected magnetization. The high amplitude longitudinal NMR also has very nontrivial behavior. After high enough excitation the system can move from one minimum of V I#___0 to another. This kind of behavior is mathematically equivalent to a pendulum, which is able to rotate through its upside-down position. In this case the periodical rotation of the phase q~ is possible. The first observation of this kind behavior was made by Webb et al. (1975a). In these experiments the value of the magnetic field was sharply changed and the longitudinal NMR was excited. In the case of a large enough jump of the field, larger than the dipole field, the phase q~ begins to rotate and the ringing of longitudinal magnetization appears. It is interesting to point out that the same rotation of the phase ~ can be excited by the propagation of the A-B transition boundary through the chamber or by the fast melting of solid 3He. As a result the spin supercurrent can be excited and play a significant role in these experiments. These phenomena are discussed in section 7. In addition to the modes of NMR, described above, there is another mode, the so-called wall pinned mode (WP). This is the combined precession of the magnetization and the order parameter under conditions of relatively small ex-

Ch. 2, {}2

SPIN SUPERCURRENT AND NMR IN 3He

85

ternal field. Webb et al. (1977) observed spatial transport of magnetic excitation by this mode, but we do not consider it further in this review. At the end of this section we need to consider the influence of orientation energies on NMR. The main influence of walls and counterflow energies leads to a deformation of the shape of the potential V at fl = 0. Consequently there are large changes in the frequencies of longitudinal NMR and transverse NMR at small ft. This was studied by many NMR experiments in textures determined by restricted geometry, counterflow, etc. This subject has not yet been reviewed in comprehensive form. However, we should point out one very important circumstance. For a large angle of deviation of the magnetization, the Zeeman energy plays a relatively important role, so the other orientation energies do not have much influence. It is possible to say that pulsed NMR erases the stationery texture of the order parameter. For 3He-B, this process was studied experimentally by Borovik-Romanov et al. (1983), Bunkov et al. (1984) and Ishikawa et al. (1989), theoretically by Golo et al. (1983) and highlighted in reviews by Bunkov (1985) and Golo and Leman (1990).

2.2.

Spin supercurrent

The superfluid quantum states can be characterized by their gradient energy, which can be treated as the kinetic energy of the supercurrent. The magnetic gradient energy is well known and can be cast in the form h

Fv =-~mpU~,7~ffam,

(2.22)

where Pig~ is the superfluid spin density tensor, m is the mass of the helium atom and g2/~ are the spatial derivatives of the Euler angles (see Maki 1975, Fomin 1990),

oa

o,8

Ox~,

Ox~,

f21~ = - - - - - s i n f l c o s ~ o + ~ s i n

qg,

g22r = Oct sin fl sin 99+ Off cos 9 ,

f~ 3~ =

Ox~ Oa ~ cos Ox~

Ox~

(2.23)

c9q9 Oxr

fl + - - - -

To take into account the symmetry of the order parameter, we are able to write the magnetic gradient energy in the form (Fomin 1985)

86

YU.M. BUNKOV

Fr = ~2y2 (c~ij- didj )[ c2 (0 Sq -

Ch. 2, w

l$1,1) + cj~I$1,7]if2ir

j~

(2.24)

for the 3He-A phase, where c i and c, are phenomenological constants, which are the spin wave velocities for propagation transverse and parallel to the direction of l, respectively. For 3He-B the magnetic gradient energy has the form

F$ = [c,

2y 2

aua

, 7

-cl)(a; a jr/ "F"6/r/6j~

)]~"~ i~"~ jr/,

(2.25)

where c• and c, are phenomenological constants identified as spin wave velocities related to the direction of magnetic field. By incorporating the magnetic gradient energy in the equations of motion (2.18) and taking into account that

dF da

=

OF 0 OF Oa Ox~ OOx~ oa '

(2.26)

we obtain

Mr = y div( OF; ) av

OV -700 '

= 7(-H + Me I Z), (2.27)

Mz-lf/lz=[:'=Tdiv( OFv ) 0 Va ' d=Y

OP

The terms with divergence can be treated as spatial transport of the conserved quantities Mr and P = Me - Mz. give the spin currents

Orv OV~ orv

JM = - 7 ~

(2.28)

JP = - 7 ~ 0 Va

As we shall show, these currents Jp and JM have the nature of supercurrents and are determined by the phases of the order parameter.

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

87

Let us now consider the explicit expressions for the spin supercurrents in 3He-B. If we assume that all the variables depend only on one coordinate, z, then using eqs. (2.23), (2.25) we can write down the gradient energy in the form Fd = ~ z {q2[2(1-cos f l ) a ' ( a ' - $ ' ) + $ 2y 2 - 2 c,, - c l

'2 +

t0- cos

fl,2] (2.29)

I.

It is important to note that under experimental conditions the gradient energy is much smaller then the dipole-dipole energy. Therefore we can assume that the spin system remains at the minimum of the dipole-dipole energy Va (see fig. 3). For deflection angle 13 < cos-l(-1/4) this leads to the relation (2.30)

cos fl + cos q~+ cos fl cos ~p = 1 / 2,

which enables us to cancel out the phase of the order parameter ~p from the gradient energy. Then the spin supercurrent Je can be expressed as

Z(1-u)

{[uq 2 + (1-u)c~_ ] 2 a ' + ,f3(2c 2t - c , 2 )(1 + u ) - ' (1 + 4u) -'/2 u'},

(2.31) where u = cos ft. The sign of the term with u' depends on the branch of minimum dipole-dipole energy, or in other words on the direction of the vector n with respect to H. For the region of higher angles of deflection, the minimum of VB corresponds to the condition q~ = 0 and the term with u' in the spin supercurrent vanishes. The same treatment of the gradient energy for the case of gradients directed perpendicular to the magnetic field gives the expression F~ = ~ { q ~ [ 2 ( 1 - c o s 2), 2

fl)a'(a'-qb')+q~ '2 + 3 '2 ]

(2.32)

-(c, 2 - c ~ ) [ ( 1 - c o s / 3 2 ) a '2 +/3 ,2 ]}. Correspondingly, the spin supercurrent is described by the expression ~(1- u) { [(1- u)q~ +(1 +u)c 2 ] a r Y

3~/2 q2 (1 +u) -1 (1 +4u) -~/2 ur (2.33)

88

YU.M. BUNKOV

Ch. 2, w

for the angles of deflection 13 < cos-l(-1/4). For larger angles the second term is equal to zero. This treatment runs into difficulties if the deflection angle/3 = cos-l(-1/4). For this angle the system being at the minimum of the dipole-dipole energy, which corresponds to eq. (2.30), causes the gradient energy to be infinite. So in practice the system does not follow the relation (2.30) for this range of angle. One should remember that the dipole-dipole energy minimization, used above, is only an approximation for superfluid 3He. For the spin supercurrent JM, a similar consideration leads to the expressions J ~ = - ~ ( 2 c 2 -cH2 )q~', Y

J ~ = ---g c,~q~'. Y

(2.34)

Let us now turn back to eq. (2.27). One can see that the dipole-dipole interaction V leads to additional terms describing the nonconservation of Me. Consequently, spin supercurrent JM as a long scale phenomena can be observed only in the case of high level excitation of longitudinal NMR when additional sources or sinks of M~ can be averaged by fast rotation of phase ~p and neglected. That is exactly the case of the magnetization behavior at the propagating A-B phase boundary at low temperature. As a result the boundary propagation is accompanied by a spin supercurrent JM, which leads to formation of a magnetic precursor, as was seen experimentally by Boyd and Swift (1992) and demonstrated theoretically by Bunkov and Timofeevskaya (1992). We consider this phenomenon in section 7. We should point out that if the magnetization is not tilted from the direction of magnetic field, than u = 1 and Jp vanishes. Consequently this type of spin supercurrent exists only for transverse NMR. Similar to transverse NMR, longitudinal NMR is accompanied by the processes of magnetic relaxation which plays the role of a sink for the conserved quantities described above. The nature of magnetic relaxation in superfluid 3He is connected with the interaction of the magnetization of superfluid and normal components of the liquid. In the case of mass superfluidity and electric superconductivity the superfluid and normal components of the liquid do not interact with each other at low velocities. Hence, two-fluid hydrodynamics applies, and the persistent flow of the superfluid fraction is not degraded by indirect damping through interaction with the normal component. In the case of magnetic properties of superfluid 3He there are very effective collision mechanisms which tend to bring the spin polarization of the superfluid component into equilibrium with that of the normal component. Since the normal component magnetization relaxes on a time scale r of order the quasiparticle scattering time, rq, any model based on independent "magnetic fluids" can be valid only for the time scales shorter than rq. To observe phenomena like persistent spin supercurrents or per-

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

89

sistent magnetic precession, we need to work at the lowest temperatures, where the quasiparticle scattering time becomes very long and its density becomes very low, so the magnetization of normal and superfluid components decouples. The first of such phenomena, the persistent magnetization precession, was observed recently in Lancaster by Bunkov et al. (1992d) at a temperature 0.12Tc and confirmed experimentally in Moscow by Bunkov and Zakazov (1993b) at a temperature 0.14Te. But these experiments are outside the scope of this review and are considered only in the conclusion. For the conditions of the experiments described in this review magnetic relaxation is essential wherein the quantity P decreases in time. But, as we shall see, magnetization redistribution by spin supercurrents is usually much faster than magnetic relaxation. Consequently under conditions of pulsed NMR in 3He-B one can observe the transient process of formation of the dynamic texture called the homogeneous precession magnetic domain (HPD) and then its relaxation. Furthermore, under conditions of CW NMR the loss of quantity P and Zeeman energy by magnetic relaxation can be compensated for by radio frequency pumping, so the HPD can be maintained. This type of experiment is described in section 4. Under conditions when magnetic relaxation is compensated for by RF pumping, the steady spin supercurrent states can be revealed as shown in section 5. We consider the magnetic relaxation processes in detail below. However, it is worthwhile pointing out here that magnetic relaxation, which accompanies spin supercurrent transport is not a friction. Rather it can be considered as "evaporation" of the quantity P. Let us now consider other phenomena related to spin supercurrents. If the temperature is not extremely low, the direction of the magnetization of superfluid and normal components almost coincide with each other. Consequently the spatial gradients of the superfluid part of magnetization are equal to that of the normal component that accompanies the spin diffusion current. The spin diffusion current of the magnetization is described by Fick' s law: o OMi J~q = -Du~ 0 x~ '

(2.35)

where Dig,t is the tensor of spin diffusion coefficients. The existence of the Fermi liquid molecular field leads to nontrivial correction of spin diffusion that can be described as a complex number for the components of spin diffusion coefficients, transverse to the magnetization. In other words, the transverse gradient of the magnetization leads to the modification of the Landau field that causes a change in the frequency of precession. This phenomenon was discussed first by Silin (1958). For the normal phase of 3He the spin diffusion current of the transverse magnetization, for small angles of deflection, can be written as

90

YU.M. BUNKOV D

JM+

._

Dd-

VM + ,

Ch. 2, w (2.36)

1 -b i~W'E d

where the frequency independent diffusion coefficient Do• l/3VF2(1 + FOa)'t'd and r d is the characteristic time of order rq. The parameter 2 characterizes the influence of the molecular Landau field on the reactive part of the spin current and can be represented as =

El a / 3 - F~ . (1 § Foa )(1 + El a [ 3)

(2.37)

Having determined the spin supercurrent of the superfluid component and the spin diffusion current of normal component we can consider their influence on the CW NMR in an inhomogeneous magnetic field. In this case the equation of motion for the transverse component of magnetization M + = Mx + iMy in a reference frame rotating with the RF field hRF, has the form of the Schr6dinger equation dM+ dt

-(w(z)ORF)M

+

( cl2 w

D0-t-JtWrd ) d2M+ 1+ Jl,2(/) 2 V 2 dz 2 (2.38)

-i

D~ 1 +/120)2"/:2

d2M+

dz 2

+ i M Z),hRF,

where the first term describes the potential energy, the second one describes the kinetic energy, and the last terms describe relaxation and excitation. The term with c, in the kinetic energy is the spin supercurrent VJp at constant magnetization and small deflection angle (dM § M § z" MZ), while the second term in the kinetic energy is the result of the reactive part of the spin diffusion current. One can see that these currents have opposite sign, or in other words, opposite effective mass. In the region of field, where yH = WRF, the RF field deflects the magnetization and consequently reduces the value P. The spin supercurrent and the spin diffusion current compete with each other by transporting M § in opposite directions. In fig. 4 the spatial solutions of eq. (2.38) in a magnetic field gradient are shown for normal 3He and for superfluid 3He-B. If one restricts the propagation of spin currents by the wall the solutions in the form of stationary spin wave appears. This method was successfully used by Candela et al. (1987) for studies of the spin kinetics in 3He. They reported the existence of a stationary spin wave spectrum in the region of higher field for a cell of normal 3He and its formation in the region of lower field in 3He-B, when the spin supercurrent surpasses the spin diffusion current. The potential energy local mini-

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

(a)

~"

E

91

(b)

VR

-,---3,//

/

M•

j

My

N~

Fig. 4. The distribution of transverse magnetization in the region of resonance field and gradient of magnetic field, calculated for 1.1Tc (a) and 0.7T c (b) for 0 bar and to = 1 MHz.

mum can also be created by a texture of the order parameter. The standing spin waves spectrum, related to a similar minimum of the potential energy, was observed in experiments of the ROTA project (see Ikkala et al. 1982, Bunkov et al. 1983, Hakonen et al. 1983). For large angles of deflection the spin supercurrent of condensate and the spin diffusion current of the normal part of the Fermi liquid cannot be easily compared because they transport different projections of the magnetization; the diffusion current transports transverse magnetization while the spin supercurrent transports longitudinal magnetization. Nevertheless, recently at Cornell University a long-lived decay signal in the normal liquid 3He was found. This signal was explained as the formation of a two-domain structure after the magnetization is redistributed by the imaginary part of the spin diffusion current (Nunes et al. 1992). One very interesting question is the stability of the spin supercurrent and its critical velocity with respect to the critical velocity for creating excitations. (The analog of the Landau velocity.) This topic was a matter of controversy between Fomin (1988) and Sonin (1988). They have shown that the general analysis of eq. (2.27) for states with steady spin supercurrent can be obtained with an approach similar to the Ginzburg-Landau (1950) phenomenological model for the stability of superconducting currents. For the case of a transverse mode of NMR the value P has the role of the square of the order parameter, while the angles a and fl have the role of phases. Under these conditions the characteristic length (Ginzburg-Landau coherence length) is determined by the relation between

92

YU.M. BUNKOV

Ch. 2, w

gradient energy and dipole-dipole energy. This length characterizes the stability of the current against phase slippage. For 3He-B at angles of deflection below 104 ~ the dipole-dipole energy is constant and cannot provide a potential for stabilization of the spin supercurrent. Consequently the Ginzburg-Landau length is infinite and any current is unstable against phase slippage. For angles of deflection more than 104 ~ it is useful to express the dipole-dipole energy in terms of the frequency shift Ato from the Larmor frequency. In this case the Ginzburg-Landau coherence length is now equal to (Fomin 1987b) ~" GL =

c(o)Ato) -1/2 ,

(2.39)

where c is the spin wave velocity in the direction of the spin supercurrent. For 3He-A ~:CL is negative. This results in the exponential growth of any inhomogeneity in the precession. The Landau criterion for the critical velocity of spin supercurrents can be treated here as a criterion for spontaneous spin wave emission at the expense of the energy of the spin supercurrent. This question was discussed in detail by Fomin (1988) and Sonin (1988). Here we should pay attention to the fact that the spin supercurrent takes place in the nonequilibrium state and that it relaxes to an equilibrium state even in the absence of a current. Consequently the violation of the Landau criterion means that in addition to the spin diffusion relaxation, another relaxation mechanism has come into play. In experiments described in this review, it is two or three orders of magnitude less than the spin diffusion relaxation. So the experimental condition for the study of this phenomenon can be achieved only at very low temperatures, when all other relaxation processes vanish. At the end of this section we discuss briefly spin currents in magnetically ordered materials. To describe the spin supercurrent phenomena we have used the gradient energy, which can be considered as a result of magnetic ordering and exists for all magnetically ordered substances. Consequently the spin currents, described by the same equations, exist in ordinary magnetically ordered materials. (Actually, the explicit form of the exchange energy should be used instead of the dipole-dipole energy.) The difference between spin current in normal magnetically ordering materials and the spin supercurrent in 3He is based on the nature of the different magnetic ordering. The usual magnetic ordering is the result of exchange energy between neighboring magnetic moments and is characterized by a near diagonal form of the density matrix. The superfluid ordering is the result of quantum ordering in momentum space and characterized by a nondiagonal density matrix with correlation of the quantum states on a macroscopic distance. Consequently the spin current in magnetically ordered materials is analogous to solid body motion (or rotation), while the spin supercurrent in

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

93

3He is analogous to motion of the superfluid liquid. Aside from this physical difference there is also a quantitative difference, because the energy of direct exchange between 3He nuclei is much less then the gradient energy of the superfluid condensate. As was shown above, the magnetic part of the gradient energy is the result of counterflow of superfluid components with different spin states. This is the reason for us not to be in agreement with the spin supercurrent classification of Sonin (1988) based on his early theory of spin currents in magnetically ordered materials. Physically speaking the supercurrent given by P is the result of a counterflow, determined by the same spatial gradients of d(k) as the supercurrent St and, particularly for NMR of 3He-B, the gradients of the phases of the substates ~1"1"(k), ~$~(k) averaged over all k are exactly equal to Va (Timofeevskaya 1991).

3. Experimental methods Most of the experiments reviewed in this article were carried out on the nuclear demagnetization refrigerator, constructed in the Kapitza institute for Physical Problems, Moscow. This refrigerator is able to cool 3He down to 140 p K. Its construction and performance were reviewed by Bunkov (1985). The main design difference of this refrigerator compared with others is in the use of a pendulum like antivibration mounting. Also novel for the nuclear refrigeration stages and dilution refrigerator was the wide use of silver vacuum soldering and diffusion welding (see Bunkov 1989, Bunkov et al. 1990a). The ultralow temperature NMR experiments down to 120/~K were done at Lancaster University. This refrigerator was made using a nuclear stage of copper flakes totally enclosing the second nuclear stage containing the experiment. The original and most comprehensive description of this novel technology is given by Bradley et al. (1984). The experiments were carried out in removable experimental cells, which are presented in figs. 5, 6 and 7. They were made from Stycast 1266 epoxy resin. The thermal contact of 3He with the nuclear stage was made by silver sinter heat exchangers that were placed in a low part of the chamber, connected with the cell by a wide channel. For temperature measurements a platinum NMR thermometer was usually used, with its sensor placed in the lower part of the cell. For temperature measurements at very low temperature the vibrating wire viscometer was used (see Pickett 1988). The NMR coils were usually unconnected to the body of the cells and thermally anchored to the mixing chamber of the dilution refrigerator. The cell used in early experiments is shown in fig. 5a. This cell was designed for studying the instability of homogeneous precession in 3He-A (see section 6), so considerable attention was paid to the homogeneity of the stationary mag-

94

YU.M. B U N K O V

Ch. 2, w

Fig. 5. Sketches of the experimental cells for the experiments, reviewed in this article. 1, RF coils; 2, pick-up coils; 3, sensor for Pt NMR thermometer; 5, sintered silver heat exchangers.

Fig. 6. Sketches of the experimental cells for the experiments, reviewed in this article. 1, RF coils; 2, pick-up coils (m and n); 3, sensor for Pt NMR thermometer; 5, sintered silver heat exchangers; 6, copper shielding.

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

95

Fig. 7. Sketches of the experimental cells for the experiments, reviewed in this article. I, RF coils; 3, sensor for Pt NMR thermometer;4, vibrating wire thermometer; 5, sintered silver heat exchangers; 6, copper shielding; 7, inner cell; 8, outer cell. netic field and radiofrequency field within the experimental volume. To achieve this we used gradient coils to compensate for the field gradient in the cell. The volume of the experimental cell was connected to the remaining volume of the chamber by a relatively narrow (1 mm) channel. This enabled us to reduce considerably the amount of helium in the region of the spatially inhomogeneous radiofrequency field. The result of this "quasiclosed" condition of the cell allowed us to observe a high intensity long lived induction decay signal in 3HeB and to study its properties (see section 4). An additional cell in this chamber was used to study 3He in a restricted geometry. The second cell, presented in fig. 5b, was designed for an experimental test of the theory of Fomin (1984) for the formation of the domain with homogeneous precession of magnetization (HPD)

96

YU.M. BUNKOV

Ch. 2, w

in 3He-B. This theory explains the strange behavior of the signal in the previous chamber as a spatial splitting of the magnetization into two domains: a stationery one and a second with magnetization deflected by 104 ~ and precessing uniformly. So besides one pair of exiting coils, two pairs of pick-up coils with separate regions of sensitivity were used. With the help of this chamber, the spatial splitting of magnetization into two domains was demonstrated both for the case of pulsed and CW NMR. This splitting is the result of the spatial redistribution of the magnetization's precession by the spin supercurrent. The straightforward development of this study led us to the design of chambers where two cells with two HPD are connected by a channel. These chambers are shown in figs. 5c and 6. The first observation of a steady spin supercurrent and the phase slippage as well as many other studies of HPD properties were done in the chamber shown in fig. 5c, while the regular studies of steady spin supercurrent in a channel and Josephson phenomena were made in chambers shown in fig. 6 (see section 5). The main advantage of the last chambers is the screening of the RF coils by copper foils, so that there was no RF field in the channel and crossinteraction of the two RF coils was minimized. Two miniature pick-up coils were situated in the channel, giving an opportunity to observe directly the gradient of phase precession in the channel, as well the behavior of the magnetization during phase slippage. In the last version of this chamber, used for studying the Josephson phenomena, the main RF coils were surrounded by copper barrels, which improved the screening of the RF field, and an orifice was situated in the middle of the channel to study the Josephson phenomena. The next cells, illustrated in fig. 7, are quite different. The cell shown in fig. 7a was prepared in Lancaster and its construction is based on the Lancaster approach to the nuclear demagnetization refrigerator. It consists of an outside chamber made from Stycast 1266 and an inside chamber made from paper impregnated with stycast. The chambers are cooled by separate nuclear refrigeration stages, the first one by the flakes of copper, the second inner one by cooper plates covered by silver sinter. The outside chamber intercepts heat flow to the inner chamber, so that temperatures down to 100/~K can be achieved. The NMR of 3He was performed in a cell in the form of a finger, connected to the main chamber. The NMR in the internal Landau field, presented in section 4.3.3, was observed in this cell. The last cell is the current experimental cell, now in use in Moscow. Its construction is based on the idea of a double cell to prevent heat leak to the inner cell. The massive nuclear stage bundle of the nuclear demagnetization refrigerator was used to cool both cells. A final 3He temperature of 140/.t K was reached in this cell, while the ordinary cell reached only 180/~K under the same conditions. This difference of temperature seems to be not very large, but one should take into account that the Kapitza resistance between 3He and heat

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SPIN SUPERCURRENT AND NMR IN 3He

97

exchangers changes exponentially below 0.3Tc, R --exp(A/kT) (see Castelijns et al. 1985, Pickett 1988). The temperature of the cell is determined by an equilibrium between heat leak and Kapitza conductance. This means that in the case of the double cell, the heat leak to the inner cell was reduced at least by a factor of 10. Furthermore, our experience shows that the temperature of 0.14Tc is limited by the properties of the nuclear stage. At so low temperatures the physical properties of 3He are determined by quasiparticles density, which changes exponentially. New and very interesting features of NMR in superfluid 3He were discovered at such low temperatures, as will be considered in the conclusion. The experiments were performed by using both continuous (CW) and pulsed NMR. In pulsed NMR, a high power short radiofrequency pulse deflects the magnetization by a large angle. After this, an induction signal is observed, induced by the precession of the magnetization around the magnetic field. The receiving part of the NMR spectrometer consisted of a broad band preamplifier, heterodyning circuit and a broad band amplifier. After the amplifier, the induction signal was recorded in digital form in a Datalab-905 transient recorder. The stored signals were then processed by a computer to determine the timedependence of the frequency and the amplitude of the induction signal. Mathematical processing of the induction signals enabled us to use all the information contained in the radiofrequency radiation of the 3He spin system excited by the radiofrequency pulse. The pulsed NMR methods also include the spin-echo method, which consists of observing the spin induction signal which is rephased at a definite instant of time after two or more radiofrequency pulses are applied. In systems possessing nonlinear spin dynamics, such as the superfluid phases of 3He, or the nuclear spin system of magnetic materials under coupled nuclear-electron precession, there exists a whole series of new mechanisms by which the spin echo is formed (see Bunkov and Dumesh 1975, Bunkov 1976, Bunkov and Maksimchuk 1980). Hence, investigation of the nuclear spin system of magnetic materials using spin-echo methods have turned out to be extremely fruitful. With regard to the superfluid phases of 3He, despite the promising possibilities of spin echo methods (Eska et al. 1981), interpretable information is difficult to obtain. This is obviously due to the fact that the spin dynamics of the superfluid phases of 3He are not local. There are spin currents in 3He capable of transporting components of the magnetization over macroscopic distances. As a result, single-pulse NMR methods have turned out to be more informative. In the continuous NMR method, a small amplitude radiofrequency field is applied to the sample. As a result the magnetization deflects and precesses at the frequency of the RF field, inducing a signal in the RF coil. There are signals inphase and in-quadrature to the RF field, named absorption and dispersion respectively, that can be separated by a lock-in amplifier. The absorption signal is

98

YU.M. BUNKOV

Ch. 2, w

proportional to the energy transferred from the RF field to the sample, while the vector sum of signals is proportional to the amplitude of the transverse part of the precessing magnetization. For usual NMR systems the intensity of signals is proportional to the distribution of frequencies in the sample. Superfluid 3He differs from these by the fact that the NMR frequency depends on the amplitude of excitation and by spatial transport of magnetization by spin supercurrents. As a result nonlinear properties of CW NMR in superfluid 3He are very interesting.

4. NMR and spin supercurrent in 3He-B 4.1. Pulsed NMR

The spin supercurrent transport phenomena play a very significant role in the case of pulsed NMR in 3He-B. Owing to the spin supercurrents, the deflected magnetization not only relaxes but also leaks out of the region of spectrometer sensitivity, so that establishing experimental conditions of a nearly closed cell is better for obtaining results that admit theoretical interpretation. Our cell, illustrated in fig. 5a, satisfies these conditions. Initially, our interest was concentrated on the so-called long-lived induction decay signal (LLIDS). Corruccini and Osheroff (1978) and Giannetta et al. (1981) observed LLIDS as a weak signal, of longer duration by an order of magnitude than could be explained by the inhomogeneity of the magnetic field. In our experiments (Borovik-Romanov et al. 1984) LLIDS signal was up to 90% of the initial induction signal. We observed also the following strange phenomenon. In our experiments the LLIDS had the form of a slowly decaying sinusoidal signal with a slowly changing frequency. The range of frequency change was much larger than the inverse length of the signal and proportional to the magnetic field gradient on the cell. In other words the NMR signal does not correspond to the model of independent local oscillators, each with their own frequencies; in fact, we can demonstrate that this is a collective effect of magnetization precession at a common frequency, slowly changing in time. The change in the decay signal frequency with time served as the basis for explaining the nature of the LLIDS given by Fomin (1984). Here we would like to mention that one should be careful in using Fourier transform spectroscopy methods for describing the NMR signals of nonlocal systems. In the case of the long-lived induction signal the Fourier transformation looks like a set of stationary spin waves; this fact can be misleading. To describe the properties of pulsed NMR in a closed cell, let us consider the solutions of eq. (2.27) found by Fomin (1984). For a while we shall not take into account magnetic relaxation. This condition corresponds well to the experimental situation, because as is shown below, the spin supercurrent transport of magnetization is usually much faster than magnetic relaxation. We can imagine the

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

99

~L=="

tit

J A

B

C

D

Fig. 8. HPD formation after the RF pulse (A, B, C) and its relaxation (D).

behavior of magnetization deflected by an RF pulse as follows. Let us suppose that the initial deflection is spatially uniform. After a time t~T, due to the presence of a magnetic field gradient VH, a spatial dephasing of the precessing magnetization occurs (fig. 8a). The presence of the gradient in the phase of the precession leads to a spatial transfer of the magnetization by the spin supercurrent (2.30), so that the angles of deflection of the magnetization at higher magnetic fields are reduced and the angles of deflection of the magnetization at lower magnetic fields increased (fig. 8b). Hence, there is a gain in the value of the Zeeman energy with the total longitudinal magnetization of the specimen being conserved. This process has quite understandable limitations. In high fields, the angle of deviation of the magnetization from the magnetic field becomes zero, whereas in low magnetic fields this deviation of the magnetization is about 104 ~ when the sudden increase in the dipole-dipole energy becomes considerable and as a result a shift in the NMR frequency occurs. This frequency shift compensates for the spatial difference between the precession frequencies in 3He-B. The arguments given above can be put in a more quantitative form as follows. Let us consider the frequency of the precession of the magnetization as 16~ to z =)'Hz 0 +)'VH(z-zo)+-(cos fl(z)+ l/ 4), 15 ~,H

(4.1)

where we have taken as Zo the point at which cos fl =-1/4. Quite clearly, equilibrium between the second and third terms in eq. (4.1) is responsible for a state without generating Va and consequently without spin supercurrent. In other words, the spin supercurrent transports magnetization until a state is formed, in which the gradient of Larmor precession is compensated by the dipole-dipole frequency shift. Anticipating the results, we note that it has been possible to

100

YU.M. BUNKOV

Ch. 2, w

observe the establishment of this equilibrium experimentally. Thus, in a domain situated in lower magnetic fields, the magnetization precesses spatially uniformly, with a deviation of angle fl _> 104 ~ (a domain with homogeneous precession), while in another domain the magnetization is at rest (fig. 8c). Hence in 3He-B there is a solution corresponding to the minimum energy, provided the total longitudinal magnetization is conserved. This solution has the form of two domains, separated by a transition region. The domain wall of thickness 2-10-2 cm is situated at z = z0. The value of Zo is determined by the total longitudinal magnetic moment for the volume of the cell, i.e. by the initial angle of deviation of the magnetization. Over the whole volume of the domain of uniform precession, the magnetization precesses with the same frequency to = yHzo, i.e. with the local Larmor frequency at the point where the domain wall is situated. It is this uniform precession of the magnetization that produces the long-lived induction signal. If we take into account magnetic relaxation processes, which are slow in comparison with the spin supercurrent, the value of total longitudinal magnetic moment increases with time and consequently the position of the domain boundary changes. Consequently, the dimensions of the HPD decrease and its frequency of precession changes (fig. 8d). For a quantitative description of the process of HPD formation we shall solve the following problem. In the region corresponding to point q~ =0, fl = arccos(-l/4) the gradient energy goes to infinity, so that the minimization of dipole-dipole energy is not valid in this region, a fact we shall take into account. Following Fomin (1984) let us confine ourselves to the following model. We shall assume that the dipole-dipole energy is very large, so the relation (2.29) is valid. Then we shall consider only the region of deflection fl < arccos(-1/4). Under these conditions the role of the dipole-dipole frequency shift is played by the spin waves frequency shift dFv/dP. Then for the gradients directed along the magnetic field, we can put eq. (2.27) in the form

OJp

du

},2

dt

ooX Oz

da r 2 (OF v d-'t = - Y H ~ - v V H ( z - Zo ) + t~ \ 3 u

(4.2) Oz OVu '

where u = cos ft. This equation is the analog of the Josephson relation for supercurrent, with the phase a being an analog of the phase of the order parameter and the right-hand side of the second equation is the chemical potential/,. A stationary solution of this equation was derived by Fomin (1984). In this solution two domains with stationary magnetization and with homogeneous precessing magnetization are separated by the domain wall of thickness 2

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

101

(4.3)

,l LyHVH

As is expected for a stationary solution, the spin supercurrent is equal to zero; due to the homogeneous configuration in the domains indicated above and to the mutual compensation of the 7 a and Vfl terms in the domain wall owing to its shape. Numerical simulation of the transient process of HPD formation helps to understand this in detail. For this purpose we introduce the spin diffusion terms into eq. (4.2) to damp oscillations of any transient processes. du dt

da

~

dt

=... + D(u" +O~12 sin 132 + sin/~ -2Ut2 ), ~

,,,

d I-

s

.

3

1 [a,,2sinfl2+u,2sinfl_2],+a,,+a,fl,2cotfli"

l+4u l+u

(4.4) The numerical solution of HPD formation and transient process after the deflection of magnetization by 90 ~ in magnetic field gradient VH = 0.2 Oe/cm with cll = 800 crn/s, D = 0.1 cm/s 2 is illustrated in fig. 9. The verification of HPD formation after the RF pulse was performed in a chamber illustrated in fig. 5b. It follows from an analysis of the induction decay signal in the normal phase of 3He that the inhomogeneity of the magnetic field over the dimensions of the chamber is 10- 2 0 e in fields of order 100 Oe. The nonlinearity of the gradient, produced by the gradient coils, according to geometrical estimates, is not worse than 1%. The duration of the exciting radiofrequency pulse in these experiments was 10 periods at a frequency of 460 kHz, so

Fig. 9. Computer simulation of HPD formation after a 90~ pulse.

102

YU.M. BUNKOV

Ch. 2, w

that its spectral width was about 50 kHz. Consequently the radiofrequency pulse excited the spin system of3He practically uniformly in the magnetic-field gradients of up to 5 Oe/cm. The spin-induction signal in the normal phase of 3He, received by both up-side and down-side receiving coils, had the same characteristics. The situation changed drastically for the B phase. In fig. 10 we give an example of the envelopes of the induction decay signals obtained in 3He-B in a field gradient of 0.1 Oe/cm in two different pick-up coils for different directions of the magnetic field gradient. These signals demonstrate that considerable redistribution of the precessing magnetization occurs in the chamber. The LLIDS signal originates from the HPD, that forms in the region of lower magnetic field. It should be emphasized that in the applied magnetic field gradient the induction signal from the normal phase of 3He decays after 3 ms, which is almost two orders of magnitude faster than in 3He-B. The difference in the durations of the decay signals, shown in figs. 10a and b, is due to the fact that under the conditions shown in fig. 10a, the volume with the precessing magnetization of 3He-B is closed; the condition for which the theory has been developed. In the case of fig. 10b, this volume was connected by a long narrow channel with unexcited 3He in the main volume, so that a spin current should flow through this channel obviously decreasing the HPD. The precession frequency of HPD is determined by the local field in the domain boundary. Consequently, an analysis of the time-dependence of the precession frequency also provides information on the motion of the domain boundary. It should be noted, in this connection, that the frequencies of the inJ

~

'

il

'

'

'

'

a

0

100

200 t, m s

b

0

100

200 t, m s

Fig. 10. Signals from top and bottom pick-up coils of the cell, shown in Fig. 5b for the weak magnetic field gradient directed down (a) and up (b).

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

103

Fig. 11. Intensity and frequency of the induction decay signal from HPD. The smooth curves are theoretical fits (after Bunkovet al. 1986). duction signal obtained by both receiving coils are always identical; another fact that confirms the signal being formed by a single-phase region of homogeneous precession. In fig. 11 we show typical dependence of the frequency and intensity of the signal from HPD. The smooth curves are theoretical, corresponding to a fitting parameter, the rate of spin diffusion relaxation in 3He-B. The good agreement between the theoretical and experimental curves enables us, firstly, to convince ourselves of the correctness of the theoretical ideas of the two-domain structure of precessing magnetization and secondly, to calculate the dimensions of HPD at each instant of time and consequently to measure the magnetic relaxation in 3He-B.

4.2. CW NMR At low RF field excitation the shape and intensity of the NMR absorption and dispersion signals are governed by the texture of the order parameter and by inhomogeneity of the external magnetic field. Practically all who deal with CW NMR of 3He-B have seen that the signal loses symmetry with respect to the direction of field-scanning as the RF-field amplitude is increased. For large enough amplitude a very large absorption signal was observed by Corruccini and Osheroff (1975a) and by Webb (1977) on sweeping the magnetic field down. In general this phenomenon can be explained as a capture of the preces-

104

YU.M. BUNKOV

O.q

Ch. 2, w

F Absorption, Ix

~1,, . . . . . . . .

..

:

",.

/7.3

C g.2

0.1

Ho, Oe -l.'a

-~s "- Ho

~ ~ "

@

0~

b

a,

0e

a.3

ly,

o

<~~

Dispersion, ly

~

Mc'

@ [

rt - H0, Oe

~a _

-1.5

_

............. ....................................................'................

Fig. 12. The absorption and dispersion signals of CW NMR in an inhomogeneous magnetic field at high RF field excitation, and the phase portrait of the transverse magnetization, recalculated from the signals. The dashed line is the fitting of the magnetic relaxation terms, described in section 4.3.

sion frequency by the RF field, similar to the results of CW NMR in magnetically ordered systems which exhibit frequency pulling (see Borovik-Romanov et al. 1974). As was shown in our experiments (Borovik-Romanov et al. 1989) behavior of this kind for CW NMR in 3He is connected with spin supercurrent transport and HPD formation. These experiments were done in one cylinder of the cell shown in fig. 5c. The typical absorption and dispersion signals obtained in the case of a field gradient of 0.83 Oe/cm applied along the axis of the chamber and at sufficiently high RF field, are illustrated in fig. 12. The following aspects of the absorption signal (fig. 12a) obtained by reducing the field can be distinguished: a steep rise of the signal intensity in a narrow range of fields in the region at Ha, a continuous rise of the intensity from Ha to Hb, a steep fall of

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

105

the intensity in the region of Hb, a subsequent accelerated rise of the intensity, a jump of the intensity at He, and sudden disappearance of the signal at Hd. If the direction of scanning of the field was altered for any field from Ha to Hd, the field dependence of the signal was retained everywhere with the exception of the region Hc-H'c, where a hysteresis of the jump of the signal intensity was observed. In the case of decreasing the field from the region below Hd, the small signal can be seen only at fields above H b with a sharp increase of signal near H a. The field dependence of the intensity of the dispersion signal behaved similarly (fig. 12b). Taking into account that the absorption and dispersion signals are proportional to the two components of the deflected magnetization, along and transverse to the RF field, we can restore the amplitude and phase of the precessing magnetization. This is shown in fig. 12c, which illustrates the trajectory of the transverse magnetization while decreasing the magnetic field. From field H a to Hb it grows nearly linearly, then begins to rotate at near constant absolute value. This strange behavior of the absorption and dispersion signals is in good agreement with the hypothesis of formation of HPD. First let us remind ourselves of the influence of the spin supercurrent on the distribution of the transverse magnetization in the case of a weak RF field, illustrated in fig. 4 (see eq. (2.37)). Note that owing to the spin supercurrent, we find that the distribution of S• becomes asymmetric. In the direction of a weaker magnetic field it has the form of a helix, which is the result of the transport of magnetization by the supercurrent. This solution is possible for an infinite cell. If the cell wall is located at a distance of the order of the magnetization transport length, then standing spin waves can be observed. The increase in amplitude of the RF field gives rise to an increase in the deflection angle of magnetization, not only in the region of resonance, but also at a distance of order the magnetization transport length. At higher RF field strengths the situation is changed drastically. In the case of magnetization being deflected by an angle of 104 ~, the dipole-dipole frequency shift and HPD formation should be take into account. We numerically simulated the HPD formation after a switch on the RF field of 5 x 10-3 Oe intensity with the frequency in resonance with 3He situated at 0.6 mm from the wall of a chamber of length 2.5 mm. All the parameters for 3He were chosen the same as for the pulsed NMR computer simulation, given in fig. 9. This computer simulation is illustrated in fig. 13. After a transient period the solution with HPD is formed. The domain boundary of HPD is situated in the region where H = rORrdy. The transverse magnetization of the HPD is precessing with the same frequency as the RF field. Its phase is slightly behind the phase of the RF field 6~, so that energy: = ~ S_LhaF sin 6r d V

(4.5)

106

YU.M. BUNKOV

Ch. 2, w

Fig. 13. Computersimulationof the HPD formationafter a RF field has been switchedon. is absorbed from the RF field. This energy is compensated by the dissipation of Zeeman energy in the domain, so the solution is kept stationary in a frame rotating with the RF field. Stability of the solution can be understood because any deflection of the boundary position from a point where H = ~oRr./y changes the frequency of HPD precession and the phase &p. Consequently the amount of energy supplied from the RF field is changing which restores the location of the domain wall. If one slowly changes the magnetic field, the boundary follows the position where H = O~RF/Y. Now it is easy to explain the absorption and dispersion signals shown in fig. 12. At field Ha corresponding to the condition H = WRrJYon one side of cell, the HPD is formed. It grows as the field is decreased to the field Hb when the entire cell is filled by HPD. Consequently the absolute value of the transverse magnetization precessing homogeneously grows up to field Hb, and keeps constant at lower fields. There are different processes of relaxation, which are examined in the next section. One of this processes is connected with spin diffusion in the domain wall which vanishes at the field Hb, when the domain boundary disappears. On further decrease of the magnetic field, the Leggett-Takagi relaxation grows. At the field Hd the RF field cannot compensate the magnetic relaxation energy dissipation any more, so the HPD is destroyed. Sweeping the magnetic field in the opposite direction, the HPD formation appears only near the Ha field because it is easy to form the HPD for small dimensions. The model of HPD formation given above was tested under various conditions and corresponds well to all the experimental results. One case, which we do not understand very well, is a jump of absorption at Hc. Possibly it is due to the change of hydrodynamic solution for the spin supercurrent in a channel, connecting the cell with the rest volume of3He (see section 5). In conclusion we can say that HPD formation under conditions of pulsed and CW NMR has been observed and is a new macroscopic phenomena, giving a

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

107

very useful tool for study of the dynamic properties of the spin system as well as the magnetic relaxation processes in 3He-B.

4.3. Processes of magnetic relaxation Since superfluid 3He is the first superfluid liquid to carry magnetization, magnetic relaxation in a superfluid is a novel subject. The coherent quantum states of systems such as superfluid helium and the electron gas in superconductors exhibit the well-known persistent phenomena associated with superfluidity or superconductivity and which give the names to these states. In the case of superfluid 3He, where the Cooper pairs have a magnetic moment, the magnetic analogue to persistent currents is the persistent magnetization transport by spin supercurrent and persistent magnetization precession. In cases of mass superfluidity and electric superconductivity the superfluid and normal components of the liquid do not interact with each other. Therefore the persistent properties of the superfluid components are not destroyed by interaction with the normal component. This is not true for the magnetic properties of superfluid 3He, since there are very effective collision mechanisms which tend to bring the spin polarization of the superfluid component into equilibrium with that of the normal component on a time scale comparable with quasiparticle scattering time, rq. Consequently for the region of temperatures, when to rq ___ 1, where so-called hydrodynamic conditions occur, the magnetic relaxation processes should be incorporated into the superfluid component spin dynamics equations through the interaction with the normal component magnetization. For lower temperatures, when to rq _> 1, this relaxation process became weaker, but still exists, and only at very low temperatures, when to rq ~ ,,o, could one expect to observe the persistent properties of the superfluid magnetization. This kind of phenomena, observed recently in Lancaster by Bunkov et al. (1992d), is considered at the end of this review. Here we consider the magnetic relaxation properties of 3He-B, measured by HPD techniques. 4.3.1. Spin diffusion and intrinsic relaxation The two-fluid phenomenological model to be used here supposes that it is possible to distinguish a superfluid component Ms and a normal component (quasiparticles) Mq of magnetization. In the hydrodynamic regime, when to rq >__ 1, the collision mechanisms brings the spin polarization of the superfluid component into equilibrium with that of the normal component. Consequently, in contrast to mass superfluidity and superconductivity, we have to deal with a model of two strongly coupled fluids. It is useful to write the susceptibility of both components and the total susceptibility as

108

YU.M. BUNKOV M s = XsOHeff,

Mq = ,ZqOHeff,

M = ~'oHeff,

Ch. 2, w (4.6)

where Heff= H + H L (see eqs. (2.1) and (2.2)). Following Leggett and Takagi (1977) it is convenient to introduce the variable r/, which measures the extent to which the superfluid and normal components of magnetization are out of equilibrium. (4.7)

7'/= M s - Mseq = M s - 2(T)M,

where 2(T) describes the relative density of the superfluid component. The phenomenological treatment of coupling between two components can be expressed as

(4.8)

/1 = - - ~ / r L T (T),

where rLT is a phenomenological parameter. In terms of r/ the complete set of equations of motion now have the form dM ~ = ) , [ M x H ] + R d, dt

dt =)' d•

(4.9)

H

-~-t =), r/x H -

ZqO

+(l_~)Rd

r] ,'t."LT

This is the set of equations called the Leggett-Takagi equations, that describes the spin dynamics of superfluid 3He in hydrodynamic as well as nonhydrodynamic conditions. For the hydrodynamic case the phenomenological parameter rLT can be calculated from the quasiparticle scattering time and is given by XsO Xo

"L'LT = ~ ' t " q .

(4.10)

The magnetic relaxation process following the Leggett-Takagi equations is called intrinsic relaxation. We do not discuss here the general properties of this intrinsic relaxation (see the comprehensive reviews by Leggett and Takagi (1977) and Vollhardt and Wtilfle (1990)). The physical background of this relaxation can be described in the following way. The dipole-dipole torque R d acts

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

109

directly only on the superfluid component. This torque tends to break the equilibrium between the components and moves the instantaneous direction of equilibrium. The normal component of magnetization follows the instantaneous direction through quasiparticle scattering. Quasiparticle scattering conserves magnetization. However, the combined effect of scattering with the dipole-dipole interaction causes magnetic relaxation. The total energy dissipation by this intrinsic relaxation has the simple form dE

y2 =---rRFo dt Z

2,

(4.11)

where R D is the dipole-dipole torque averaged over the precession (see eq. (2.13)), and r R is an effective time of intrinsic relaxation q0 't" R :

~

l" L T .

(4.12)

ZOXsO One can see that the intrinsic relaxation takes place at nonzero dipole-dipole torque and consequently at conditions when the NMR frequency is shifted from the Larmor frequency. This relaxation mechanism is in good agreement with CW NMR line broadening (for small excitations) (Bozler et al. 1974, Gully et al. 1976), for the relaxation of the wall-pinned ringing mode (Webb et al. 1975b) and for the fast relaxation of the transverse mode of NMR in the case of the deflection of magnetization by more than the magic angle 104 ~ (Eska et al. 1982). But for many other conditions of NMR the processes of relaxation has been puzzling for a long time. Particularly in the case of pulsed NMR in 3He-A, relaxation was observed to be faster than the theoretical prediction by an order of magnitude. In 3He-B the linear dependence of the relaxation rate on the gradient of the magnetic field was observed (see Corruccini and Osheroff 1975a). The studies, described in this review, have solved the mysteries of magnetic relaxation in superfluid 3He. If the magnetization distribution is spatially nonuniform, in addition to intrinsic relaxation there exists also relaxation associated with the spin diffusion of the normal component of 3He. This magnetic relaxation can be cast in the form

dE dt

OM~ o g j -DiJ~q 0 x~ 0 xq

(4.13)

To study spin diffusion in superfluid 3He, an experiment was done by Eska et al. (1981). This experiment was based on the spin echo signal dephasing due to

110

YU.M. BUNKOV

Ch. 2, w

spin diffusion transport of magnetization and gives reliable results for temperatures near Tc. But for lower temperatures, where spin transport by spin supercurrent plays an important role, this method cannot be applied. The same can also be said about stationary spin wave measurements by Candela et al. (1986, 1987). We have considered these experiments in section 2. In the case of HPD measurements the spin diffusion relaxation term can easily be extracted for any region of temperature and magnetic field where HPD exists, as demonstrated below. The third mechanism of relaxation which should be taken into account was proposed by Ohmi et al. (1987) and is connected with the distortion of the bulk liquid precession mode near the walls (surface relaxation). Below we demonstrate that all these mechanisms of relaxation can be measured using the properties of the HPD. Magnetic relaxation leads to energy dissipation in the HPD and for the case of all gradients in the z direction, can be written as

dt = -

Dij33 ~~d'~'tROz OZ

g

"-d-~t+ ) ' [ H x S ]

dV+

I~'s cos/ads, (4.14)

where Ws is the rate of surface relaxation and/~ the angle between the magnetic field and the normal to the surface. For CW NMR the dissipation of magnetic energy in a cylindrical cell with axis parallel to gradient of magnetic field can be cast in the form -1

dE = xH2~R2 [ Do" 5 ~ + ~ ' R dt L '~F 16

(Va~)2 F ( L ) / + l,i,'s 2~RL,

(4.15)

1

where R is radius of the chamber and L = ((.ORF./~ -- Ha)[VH is the length of HPD. If the domain boundary is in the chamber, then F(L) = L3. When the HPD fills up the cell, the first relaxation term vanishes, the last one does not change, so we should change L to Lo, the length of cell, while the second one continues to change, and F(L) - L 3 - ( L - Lo)3. In fig. 12a these three terms give a reasonable fit to the CW NMR absorption signal. The first two terms can be extracted with high accuracy from the jumps at Ha and Hb and from the region of field lower than Hb. The measurement of the third term has some experimental difficulties. It is necessary to take into account the instrumental term connected with the changing of the spectrometer sensitivity at large dispersion signal. This term has the properties analogous to the original surface relaxation term. It can be diminished by using small Q value pick-up coils. In experiments described by

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

111

Borovik-Romanov et al. (1989c) separation of the surface relaxation term was accomplished using a cell with enlarged surface. In the case of pulsed NMR under conditions of slow relaxation, when the structure of the HPD can be considered as stationary, the dissipation of the Zeeman energy leads to decrease of the HPD dimensions and a change of its frequency, that can be expressed as dto dt

= -Vto V = -

4Vto

dE

5 z H 2 :t R 2 ---~t '

(4 16)

where V is the velocity of the domain boundary propagation. This relation is in good agreement with the properties of the induction decay signal, as shown in fig. 11. The relation for the amplitude of the decay signal is more complicated. One should take into account the spatial distribution of the pick-up coil sensitivity and the small twisting of HPD. The last effect occurs because the intrinsic relaxation happens in the region of lower magnetic field. It should be compensated by the magnetization transport from the region of the domain boundary. Therefore the spin supercurrent passes through the domain and consequently Va appears. The twist of a is given by the following solution (Bunkov et al. 1986):

or(z) = o~(0)-1-3-~-r R ),3 H(VH)2 c -2 (z 4 + 4 ( L - z ) L 3 ) .

(4.17)

The numerical result of these processes, without any additional fitting parameters except relaxation rates, is in good agreement with the amplitude of decay signal, shown in fig. 11. The good fitting of signals both for pulsed and CW NMR gives confirmation of the theory of intrinsic and spin diffusion relaxation and an opportunity to measure the characteristic time of the intrinsic relaxation rR and the spin diffusion coefficient D_L~ The results of these measurements, summarized by Bunkov et al. (1990c), are shown in fig. 14. In the hydrodynamic theory the spin diffusion grows and r R remains almost constant on cooling. The observed vanishing of both with cooling is determined by a transition to nonhydrodynamic spin transport, when tOrq > 1. Einzel has calculated the value of the spin diffusion coefficient taking into account this phenomena (see curves in fig. 14A). It is important that there is no fitting parameter, all parameters for calculations being chosen from non-NMR experiments. There is very good agreement between theory and experimental data. Possibly, the disagreement of experimental results for 0 bar is connected to a very large rate of relaxation, so that distortion of HPD and its boundary should be taken into account. Recent experiments in Lancaster (see Bunkov et al. 1992a), in which a higher magnetic field were used, are in excellent agreement with the theory for 0 bar, as shown in fig. 15.

112

Ch. 2, w

YU.M. BUNKOV I

I

i

I

I

o 0.4:

A

/

I

P(ba,r) -j/

o

11 0.1 0.0

0.4

0.5

0.6

r/Tc

0.8

0.20 ,

~R

(us)

--

-

__

A

,

,

,_"_.o_

zx

0.9

,.2,//

.y/l

1.0

13

_

0.10 0.05 0"% .4

0.5

0.6

T/Tc

0.8

0.9

1.0

Fig. 14. (A) Transverse spin-diffusion coefficient measured at a frequency of 460 kHz and four different pressures as a function of temperature. (B) The intrinsic spin-relaxation time at 920 kHz and 29 bar as a function of temperature. Solid lines, the theoretical treatment of the nonhydrodynamic solutions (after Bunkov et al. 1990).

Markelov proposed a nonhydrodynamic correction of rR with one fitting parameter. The experimental results, shown in fig. 14B, confirm his correction, but a more advanced theory of this correction is needed for quantitative comparison of theory and experiment. In concluding this section it is possible to say that there is now excellent agreement between theory and experimental data for intrinsic and spin diffusion relaxation in the hydrodynamic and nonhydrodynamic regions. 4.3.2. Surface relaxation As was mentioned in the previous section, the nature of this relaxation is the distortion of magnetization precession at the walls of a cell. According to the generally accepted approach, developed by Ohmi et al. (1987), the surface relaxation is a result of the magnetic susceptibility anisotropy near the walls. In

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He 100

I

"~

II

I

, ,

I

I

,

I

I

113

~1

--1

Suaac. \

".~

00000 ,o .~

--0.01

0

I

I

0.2

f

]

,! ="

0.4

..........

f

0.6

~I 0.8

Temperature, T/'i"c Fig. 15. Temperature dependence of the duration of the HPD signal (e), and persistent NMR signals, measured in Lancaster (O) and in Moscow (x). In the high temperature regime the experimental points are in good agreement with the signal duration (dashed curve) calculated according to the spin diffusion relaxation mechanism. The sudden shortening of the signal duration at 0.46Tc is the crossrelaxation with NMR in the Landau field. In the region of lowest temperatures the experimental points are in agreement with the signal duration (solid curve) calculated according to surface relaxation (after Bunkov et al. 1992d).

the case of NMR, when the vector n rotates around the magnetic field and the walls are not perpendicular to the magnetic field, Zeeman energy oscillations appear. This generates spin waves propagating out from the walls. Consequently this mechanism leads to the dephasing of homogeneous precession and to magnetic relaxation via the spin diffusion mechanism. The characteristic time for dephasing the homogeneous precession in the case of a slab geometry has been estimated by Ohmi et al. as Ts = ( ~ n - - ~ B ) ZB

2

cW

(4.18)

~ 2092 '

where Zn and )~B are the susceptibilities of the normal phase and B phase of 3He, respectively, ~ is the coherence length and W is the width of the slab. There have been three attempts to look for this surface relaxation. Pulsed NMR experiments in a slab geometry by Ishikawai et al. (1989) show that part of the observed relaxation is in qualitative agreement with the theory of surface relaxation. The surface relaxation was estimated in experiments of BorovikRomanov et al. (1989c) where the relaxation of HPD was measured in a chamber with an enhanced surface. The contribution of surface relaxation was not observed by Korhonen et al. (1990) in HPD measurements. In an experiment

114

YU.M. BUNKOV

Ch. 2, w

carried out recently in Lancaster (Bunkov et al. 1992c) the shortage of long lived decay (LLIDS) below 0.3Tc was observed. This shortage can be explained in terms of surface relaxation. The dependence of LLIDS duration on temperature is shown in fig. 15. For the conditions of the experiment, (~ = 5.72 x 10-6cm , c = 2700cm/s in the limit of zero temperature, L = 2 mm, and to = 2,rt x 106 1/s ), we have qualitative agreement of decay duration for temperatures below 0.3To with the theory of surface relaxation (solid curve in fig. 15). We should point out that in these conditions the surface relaxation appears to be the main process which is destroying the homogeneous precession. But it is possible that in the nonhydrodynamic regime, where spin diffusion is very small, this process does not lead to magnetic relaxation, but only to the dephasing of the precession of the transverse component of the magnetization. The evidence in favor of this conclusion is the formation of the persistent signal, which is considered later. 4.3.3. Catastrophic relaxation Bunkov et al. (1989) observed a crucial change in the NMR properties of 3He-B at temperatures near 0.4To. A new relaxation process accelerates the relaxation rate by more than 1000 times. This process, named "catastrophic relaxation", is seen as an abrupt shortening of the HPD induction decay signal. In the same temperature range, drastic growth in the dissipation of magnetic energy of HPD in CW NMR experiments was observed. In these experiments the HPD signal has not been observed for temperatures below those at which the catastrophic relaxation appears. Later, from NMR experiments in a homogeneous magnetic field, Bunkov et al. (1990) showed that the catastrophic relaxation is a function of the simple magnetic resonance processes, not related to any specific feature of the HPD, and that the acceleration of relaxation rate takes place after a time delay in pulsed NMR. The latter observation indicates that the catastrophic relaxation is due to the development of some kind of instability in the homogeneous precession. It was Fomin (see Bunkov et al. 1990) who drew attention to the fact that the catastrophic relaxation takes place in the region of temperature when the molecular Fermi liquid field (Landau field) and external magnetic fields become equivalent. He suggested that the catastrophic relaxation is in some way related to a sudden change in the dynamic response of the magnetization under these conditions. Figure 16 illustrates the value of the molecular Landau field scaled by an external field, calculated according to eqs. (2.9) and (2.10) with Fa~ correction taken from the book by Volhardt and Wolfle (1990). If the catastrophic relaxation is indeed a function of the crossing of the Landau and external fields, then it is of crucial importance that measurements should be made in the temperature region below the crossing temperature where the relaxation rate might be expected to decrease again and an HPD signal be re-established. This kind of

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

1.3

115

"

'ID m

1.2-

I,.

x

>, J:2

1.1-

"tO (,)

m 1.0

..

'lO r :3

m 0.9-

'lO C:: .,i

i

!

0.2

0.4

0.6

T/Tc

Fig. 16. The Landau field scaled by the external field, HLIHplotted against temperature for superfluid 3He-B at various pressures. From the melting pressure to around 10 bar the low temperature value of the Landau field is very similar in magnitude to the external field. At lower pressures the Landau field and external field cross in the region up to 0.5Tc. experiment was carried out in Lancaster by Bunkov et al. (1992a) using the cell shown in fig. 7a. Measurements were made of the duration of the HPD induction decay at several pressures from 0 bar to 11 bar over a range of temperature from 0.12Tc to 0.7Tc, i.e. scanning the region of catastrophic relaxation at around 0.4To. The results of these experiments are shown in fig. 17. They confirmed the hypothesis of the crossing of the value of the Landau and external fields and made clear the nature of the catastrophic relaxation. The usual mode of NMR is the total magnetization precessing around the external magnetic field, while the Landau field lies along the magnetization. The existence in superfluid 3He of two types of magnetization with different dynamic properties lifts the degeneracy with respect to the rotation of both components of magnetization around the Landau field. As a result a new mode of relative precession of superfluid and normal components appears. This mode of NMR, combined with the usual one is shown schematically in fig. 18. This precession is analogous to second sound, with respect to relative oscillations of the normal and superfluid densities. Generally speaking, it is possible to imagine the spin waves of this mode of NMR, which are the magnetic analog of second sound in superfluidity. The excitation of this new mode of NMR in a Fermi liquid is responsible for the catastrophic relaxation. The dynamic properties of the NMR mode discussed above are very interesting. Intuitively, its frequency is equal to yHL, while HE--Mq + Ms. For this

116

Ch. 2, w

YU.M. BUNKOV I

120

i

l

~ yP

I "= 10.7 bar

60

E

~ ~po~ ~,

0

.120

10 r

o

60

o qO c

|

L.

r

13.

I

I

I

"

o

-

0

_~~

~o

O0

o

~

o

oo

_

-

o.~~..

(~

~ _

0 100

,

,

\,

,

,

j

P = 3.2 bar 1

-r0 c o

~ed~oo

L.

r=

I

P = 5.6 bar-

oo8f~m

0

I

I

100

,

, \

OoO o

o~o 9

I

I

,

oOoX \

o

'

!

I

,

,

l

P = 0 bar

~ ~ ~

~

0.3

I

I

0.5

I

0.7

TIT e Fig. 17. Duration of the HPD signal measured as a function of temperature for four pressures. The sudden shortening of the signal duration shows onset of catastrophic relaxation at around 0.4Tc. This is readily apparent in all but the zero bar data. At lower temperatures the signal recovers except at pressures of 10 bar and above (after Bunkov et al. 1992a). mode of excitation Ms is deflected from Mq and consequently the frequency is reduced. Let us consider the experimental results for 5.6 and 3.2 bar, shown in fig. 17. On cooling the Landau field approaches the value of the external field and consequently the nonresonant excitation of NMR in the Landau field grows which has the effect of separating Ms and Mq. At some temperature a nonequilibrium conditions appears. The excitation of the NMR mode we are considering leads to a decrease of the Landau field and consequently to the higher excitation of this mode. This positive feedback process can be seen by the sharp change in the HPD signal duration. Under the conditions of HL < H the

Ch. 2, {}4

SPIN SUPERCURRENT AND NMR IN 3He

117

Fig. 18. Schematic representation of the Landau field and associated magnetizations. The total magnetization lies parallel to H L and precesses around the external field while the normal and superfluid components of M precess around HL. recovery of the decay time with decreasing H L has a smooth character, due to negative feedback. To explain the relation between the magnetic relaxation rate of the usual NMR mode and the level of excitation of the NMR mode in the Landau field, one should take into account that the mode of precession around the Landau field is accompanied by Leggett-Takagi relaxation which, as we know, depends on temperature. It is interesting that at 0 bar the crossing of the value of the Landau field and the external field take place at such a high temperature that internal precession is not strongly excited and a depression of the decay time is seen only for a short interval of temperature. It is interesting to point out that this NMR mode supplies the relation between parameters F0 a and F2a. This relation is nearly orthogonal to the relation for these parameters that can be calculated from the susceptibility data giving us the ability to calculate both parameters. Taking into account eq. (2.11) one can see that condition HL = H corresponds to the relation Foa = -

15+Fza +2F~Y(TL) , 20+(IO+6F~ )Y(TL)

(4.19)

118

Ch. 2, w

YU.M. BUNKOV

1.0

'LL. *"

sol"

0

-.5

so S~ 1 7 6 ss s s , ' sI " s s S S9S -

" ~' .

sss S~ ss ~176 ~oo ~ so S~ 9 o o~ o~ o~ so ~

-1.07~ -. , iO

.

.

-.735

.

.

.

9

~

~

.

-.720

.

.

.

.

.

.

.

-.705

.

-.690

Fig. 19. The relation between two Fermi-liquid parameters for 0 bar, follows from the crossing of Landau and external fields (solid line) and its possible error (dashed line). The relation follows from susceptibility measurements (0) and our analysis of possible error of susceptibility data (short dashed line), and relation from the spin wave measurements (0) (after Bunkov et al. 1992c). where TL is the temperature of the crossing of the Landau field and the external field. This temperature can be measured at 0 bar as a relaxation rate increase at T=0.48Tc (Y=0.144). The relation between Fermi liquid parameters, corresponding to this condition, is shown in fig. 19 by the solid line. The straight dashed lines correspond to possible error in temperature measurements _ 0.02Tc. The point (~I~) represents parameters, calculated from the susceptibility measurements by Hoyt et al. (1981). But it is important to notice that these susceptibility measurements, at least at lower field, can only provide a relation between F0 a and/72 a. Error analysis for fitting the experimental data by Scholz (1981) for 0 bar and 1 MHz frequency are plotted in fig. 19 by short dashed lines. The parameters calculated from Fermi liquid spin wave measurements by Candela et al. (1987) (shown by (o) in fig. 19) fall very near the susceptibility relation. It should be pointed out that the relation between Fermi-liquid parameters gained from spin waves is similar to that gained from susceptibility data. Following this analysis we can choose as a best compromise, the point corresponding to F0a = 0 . 7 1 3 _ 0.005 and F2a= + 0 . 4 _ 0.2. It is important to notice that from this analysis we have demonstrated that the parameter F2 a is positive. The parameters F0 a and F2 a alone determine the susceptibility and Landau field, so these two measurements should be sufficient to determine both. There are several acoustic measurements which have been used to derive F0 a and F2 a but in addition a number of other parameters must be simultaneously fitted. There is only one work by Movshovich et al. (1990) where F2 a can be estimated independently, the value found, F2a= 0.45 at 4.8 bar being in good agreement

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

119

with our estimate here. Further corrections can be considered, for example the influence of other types of Cooper pairing. Such corrections have been analyzed by Fishman and Sauls (1988) on the basis of the susceptibility and acoustic data. The inclusion of the information from the NMR in Landau field measurement may well put these analyses on a more solid footing. In concluding this section we should point out that the new theoretical work by Bunkov and Golo (1993) shows that in the region of catastrophic relaxation, the Leggett-Takagi equations exhibit chaotic behavior. The term "catastrophic", chosen on an in intuitive basis turns out to have some connection with reality in that it indicates a change from periodic behavior to chaos. Consequently the description above based on coupled precessions in the external and Landau fields can be viewed as a first approximation to more complex behavior which studies are currently in progress.

4.4. HPD oscillations The HPD formation corresponds to an energy minimum under the condition of conservation of the total longitudinal magnetization of the sample. There is a powerful feedback mechanism that returns the system to the homogeneous precession state after any disturbance. This is the excitation of spin supercurrent transport between nonhomogeneous states of the precessing magnetization. Therefore oscillations of the magnetization distribution near the equilibrium HPD state can take place. The frequencies of two modes of such oscillation were calculated first by Fomin (1986). There are torsional volume oscillations and surface oscillations. Under experimental conditions both volume and surface oscillation modes can be observed. A torsional oscillation, connected with the degeneracy of the states of the domain structure with respect to the phase of the precession a, is a lowfrequency volume mode of oscillation. This mode is named the twisting mode and is shown schematically in fig. 20b. It is formed by spatial oscillations of the phase of the magnetization precession inside HPD with spin supercurrent feedback response. The dispersion law for this mode of oscillation is obtained from the solution of eq. (4.2) provided that the parameters of the system change only along the z axis. The frequency of twisting oscillations is

~'~T =

Clk

2~22

(4.20)

+3y2H i '

where k is the wave vector and 4c12= 5Cl 2 - Cll2. There is no superfluid spin current along z on the walls of the chamber perpendicular to z. That is to say, a

120

Ch. 2, w

YU.M. BUNKOV

HPD f

v"l l" static domain

i

f

li.

~

!

f a

_

~

t b

?-C

9

c

..

t"--~ ~

d

Fig. 20. Schematic representation of the precessing magnetization in the rotating frame for an equilibrium HPD (a), twist oscillations (b) and surface oscillations (c, d).

node of the oscillations of P and an antinode of the oscillations of a with respect to their equilibrium values, are formed on the walls of the chamber. The opposite situation exists on the domain wall, since the flow of magnetization leads to a change in the form and position of the wall. Therefore, an antinode of the oscillations of P and a node of the oscillations of a are formed on the domain boundary. Consequently, (2n + 1)/4 wavelengths of these oscillations must fit into the dimensions of the domain. The fundamental mode is the mode with k = :g/2L, where L is the length of the domain. The surface oscillation at the domain boundary is analogous to gravitational surface waves in liquids. The Zeeman energy of the HPD plays the role of the gravitational potential energy of the liquid, and the gradient energy of the order parameter (which can be considered as the kinetic energy of the spin supercurrent) plays the role of the kinetic energy of the flow of the liquid. For a cylindrical cell we can visualize these oscillations as the surface waves of water in a glass. Two different kind waves can be excited: the axial symmetry waves and plane waves (see fig. 20c,d). According to Fomin (1986), the frequencies of the fundamental modes of the surface oscillations of HPD are

kLc2 3' ~2 _ ,f~VHH-I kClC2 tanh~ ff2c,

(4.21)

where k = Q/R, R is the radius of the cell, Q is the first nonzero root of the equation J'(x)= 0 (here J is the Bessel function J0 for axial symmetry waves and Ji for plain waves) and c22 = (5c, 2 + 3c• For L < R the value of tanh( ) is close to unity and we can make the approximation:

SPIN SUPERCURRENT AND NMR IN 3He

Ch. 2, w

121

(4.22)

~"~s2 = ACl c2 VH(HR) -!

with A = 2.6 for plain waves and A = 5.3 for axial symmetry waves. One can see that the twisting and surface modes are easy to distinguish. For instance, the frequency of the twisting oscillations does not depend on the value of the gradient of the external magnetic field, but it depends on the HPD length. The frequency of the surface mode is proportional to the square root of the gradient and practically independent of the domain length. Both modes of oscillation have been found in experiments with HPD. Let us first describe the experiments with the twisting oscillations by Bunkov et al. (1986). To excite these modes by pulsed NMR an additional weak radiofrequency pulse was applied during the induction decay after the main RF pulse. The phase of this pulse was synchronized with the phase of the induction signal. This additional pulse deflects the magnetization by an angle of order 5 ~ from its equilibrium position inside HPD. After this additional pulse the oscillations of amplitude and instantaneous frequency of the HPD signal were observed. Typical records of these oscillations are shown in the inset to fig. 21. The same figure also shows the period of these oscillations as a function of the dimensions of the HPD and for different values of the magnetic field gradient. These results

~'1~I 1-

0-3

2.1ff a

-

0

4

8

12 t. ms / ~

.=.

I

.

I

g

t

,,

!

4

!

,,,

I

6

,1

L, mm

Fig. 21. The period of twist oscillations as a function of the HPD length for 20 bar, 0.51 Tc and different magnetic field gradients. The insert shows an example of amplitude and frequency oscillations after a test RF pulse (after Bunkov et al. 1986).

122

YU.M. BUNKOV

Ch. 2, w

correspond well to the torsional oscillation mode. The ratio of the frequency modulation and amplitude modulation of the induction signal also corresponds to this mode. The amplitude modulation is shifted with respect to the frequency modulation by at/2, and is noticeable only for high magnetic-field gradients, i.e. under conditions when a spatial "twist" of angle a is essential (see eq. (4.12)). In order to excite a plain surface waves the disturbance of the magnetization should be inhomogeneous in the plane of the domain wall. These conditions are satisfied by the spatial inhomogeneity of the RF field in pulsed NMR experiments carried out in the Moscow and Lancaster experiments (see Bunkov 1987, Bunkov et al. 1992a). The most convenient method for studying these oscillations was developed for CW NMR (Bunkov et al. 1992b). In this method the frequency of the applied RF field is modulated at a low frequency. The amplitude of this modulation is 10-100 Hz, while the frequency of modulation is swept from 0 to 1000 Hz. Using this method it was observed that at some frequency of modulation, the HPD absorption signal increases as shown in the inset to fig. 22. Under the same conditions, the dispersion signal from the HPD also changes. All the experimental properties of this kind of resonance and particularly its dependence on VH, show that frequency modulation excites surface

6000

9

,

,

1

'

'

'

"

5000

4000

~

N

i. 3000 "-

0 2000

1ooo o

o.o

20

0.5

1 .o Gradient,

Oe/cm

4.0

1.5

60

80

2.0

Fig. 22. The square of the resonant frequency of the surface oscillations as a function of the gradient of the magnetic field for 20 bar, 0.62Tc (O) and 11 bar, 0.51Tc (o), measured by pulsed (O) and CW NMR (O). The insert shows the relative change of the HPD CW NMR absorption and dispersion signals as a function of frequency of phase modulation of the RF field (after Bunkov et al. 1992b).

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SPIN SUPERCURRENT AND NMR IN 3He

123

oscillations. Consequently the additional absorption can be connected to the increase of the domain boundary surface and additional spin diffusion losses, while changes in the dispersion signal can be explained as spatial dephasing of the HPD oscillation. The measurements of two different modes of oscillation enable us to calculate the two constants of the gradient energy in 3He-B (spin waves velocities). From twisting oscillations one can measure the value of cl 2, while from surface oscillations one can measure the value of ClC2. Following Fomin (1980), the spin waves velocities can be calculated for the region near Tc by the equation

4 2x,

/T

/

(4.23)

where VF is the Fermi velocity and x is a parameter of the order of unity depending on strong coupling effects. The experimental results of our measurements for 20 bar show that the temperature dependence of spin waves corresponds to

c,,=1230,r

and

c+=lo7o4xn/xx/1 T/Tr

for temperatures down to 0.4Te. Within the accuracy of our experiments (10%) the theoretical relation (c• 3/4) for the Ginzburg-Landau region (near Tc) seems to be valid for the entire temperature region of our experiment. From the relation (4.23) and our spin wave velocity results, one can estimate the strong coupling parameter. For 20 bar this gives x = 0.93 with 5% accuracy. This value is in good agreement with one estimated from a different kind of spin wave measurement by Osheroff (1977) at 35 bar (x = 0.92) The axial symmetry mode of surface oscillation has possibly been observed recently in Kosice in the cell with a post in its center. This post gives an inhomogeneity of the HPD excitation with axial symmetry and consequently a similar symmetry of surface wave mode excitation. The same mode may have been excited in pulsed NMR experiments in the region near catastrophic relaxation. It appears as a modulation of the HPD signal with a corresponding frequency. The original records with this modulation can be found in the article by Bunkov et al. (1989). In conclusion, we have demonstrated in our experiments the possibilities of HPD spectroscopy. We should mention that, in addition to the results discussed above, the use of HPD spectroscopy methods for studying mass counterflow and vortices in 3He-B. This type of experiment has been con-

124

YU.M. BUNKOV

Ch. 2, w

ducted in Helsinki as a part of the ROTA project. We do not review it here due to the specific nature of the subject. We believe that the methods of HPD spectroscopy can be used in the future for precise measurements of many parameters of 3He-B. Concluding this section we would like to note that there is no longer any puzzle in the behavior of large excitation NMR in 3He-B. All phenomena which were difficult to understand (see Lee and Richardson 1978) are due to long distance transport of magnetization by spin supercurrents, excited by gradients of the chemical potential of the precessing magnetization. Processes of this kind form a unique state with homogeneous precession of magnetization and consequently with homogeneous chemical potential. In this section we have mainly considered the transient processes caused by spin supercurrent. In the next section we describe studies of steady spin supercurrent between states with equivalent chemical potentials.

5. Steady spin supercurrent In the previous section we considered the state with an equilibrium distribution of chemical potential, i.e. the HPD, as well as transient processes for its formation by spin supercurrent magnetization transport arising from inhomogeneity of the chemical potential. In this section we consider states in which a steady spin supercurrent exists instead of a homogeneous distribution of chemical potential. This is the quantum spin vortex state, where a topological barrier keeps the current persistent. The steady spin supercurrent can be maintained also between two states with fixed difference in the phase of the wave functions. The latter was realized in experiments where two cells, each with an independent system of maintaining HPD, were connected by a long and thin channel.

5.1. Spin supercurrent in a channel

The idea of these experiments was very straightforward and based on the analogy of a superconducting bridge between two massive superconducting electrodes. Here we can consider two cells filled with HPD as such electrodes. The HPD also fills the channel between these cells. The role of the potential difference between the electrodes is equivalent to the difference of the HPD precession frequencies. This difference leads to an increase in the gradient of the phase of precession in the channel and consequently to the growth of the spin supercurrent. If one keeps the frequencies of HPD's precession the same, then the

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SPIN SUPERCURRENT AND NMR IN 3He

125

phase gradient in the channel remains constant and a steady state supercurrent has to pass through the channel. In the case of superconductivity this current is supplied by the leads of the normal metal, which have some resistance and consequently there is a voltage difference. In the case of the spin supercurrent the longitudinal magnetization is not conserved in the RF field. Therefore, the RF field can pump the longitudinal magnetization into one cell and pump it out in the other cell. The transport of the longitudinal magnetization along the channel in a magnetic field is accompanied by transport of Zeeman energy. This transport has been measured by the increase in one cell of the energy absorbed from the RF field with increasing phase difference, and its decrease in the other cell. In this way we were able to measure the current of longitudinal magnetization flowing out of one cell and into the other. Owing to the direct relation between the phase of magnetization precession and the phase of the order parameter, we were able to control the spin supercurrent through measurement of the phase gradient of the magnetization precession in the channel. This method has no analogue with superconductivity because there is no field that is sensitive to the phase of the wave function of the Cooper pairs in superconductors. Borovik-Romanov et al. (1987) made the first experimental observation of spin supercurrent in the channel. This experiment used the chambers shown in figs. 4c and 5. This consists of two cells in the form of a barrel with axes parallel or perpendicular to the magnetic field, connected by a channel perpendicular to field. The cells were surrounded by RF coils, and copper shielding prevented interaction between the coils. The channel was surrounded by additional shielding to prevent RF field penetration into the channel. The coils Nos. 1 and 2 were used to excite an HPD state in both cells and to control them. The frequency and phase of the precession of the domain with homogeneous precession in each of the volumes was determined by the frequency and phase of the radiofrequency field of the corresponding coil, supplied from separate highly stable generators. The cells were filled with HPD by sweeping down the magnetic field. When the domain boundary crossed the inlet to the channel, the HPD filled the channel. Miniature receiving radiofrequency coils (Nos. 3 and 4) were set up in the channel, and received a signal from the precessing magnetization in the channel. A small signal induced by the exciting coils was compensated by an electronic circuit. Most experiments were carried out in a magnetic field of 142 Oe with magnetic-field gradients of 0.40 and 0.75 Oe/cm. For HPD creation, equal frequency and phase of both RF generators was chosen, so we can assume that the difference of phase of precession in the channel is zero. Then the frequency of one of the RF generators was changed by ~to = 0.1 Hz. This causes the difference between phases of precession 6a to grow. A phase difference between the two HPDs is equivalent to a phase gradient along the channel. Figures 23a,b show what occurs when Va starts to rise. One can see in fig. 23a

126

YU.M. BUNKOV

~w,*

C'

i

i' -8~ -f-~..-/~.~

8.w,-

C'

~w,

Ch. 2, w

/.B

B

_.-.-J I.o' 1 /

i I/Ii :i ..//I .__. :8~ -'" / I: , ~ " ~

I-~=

Fig. 23. (a) The change in NMR absorption in cell K and (b) the precession phase difference between HPD in cell K and signals in pick-up coils m and n as a function of precession phase difference between two HPD. Dotted line represents a theoretical fit with corrections for spin-diffusion losses (after Borovik-Romanov et al. 1989).

the RF absorption signal in cell K rises and in cell L it diminishes. ( Due to the symmetry, the signal from cell K at negative A a corresponds to the signal from cell L at positive Aa). This process corresponds to a transfer of longitudinal magnetization, and consequently the Zeeman energy, from one chamber to the other. In fig. 23b one can see the phase of the HPD in the region of pick-up coils m and n in comparison with the phase of the HPD in cell K. The same behavior of current as a function of A a can be seen. All experimental curves in fig. 23 correspond to stationary solutions in the channel. To check this, we made the frequencies of the HPDs equal at a certain time (point D in fig. 23). Then the absorption signals from both HPD and gradient distribution in the channel did not change any m o r e - a steady state spin supercurrent continued to flow along the channel.

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SPIN SUPERCURRENT AND NMR IN 3He

127

With increasing Aa one can see that on reaching a critical phase difference Aac + at point B the absorption jumps to a smaller value (point C), then increases to the critical value again, etc. The jumps occur with period 2mr in Aa. Periods from 2a to 16z have been observed. Similar jumps can be seen in the phase of precession in the channel. It is clear that these jumps corresponds to 2mr phase slippage in the channel. Upon changing the sign of the frequency difference 6to at point D, the absorption goes down, reaches the initial value, and continues to decreases until it reaches the critical value in the opposite direction (point B'). If we change the sign of 6to at point D', we observe a hysteretic behavior (dashed line). To explain the behavior of spin supercurrent, represented in fig. 23, let us come back to the theory. The gradient of the phase of precession in the channel produces a spin supercurrent which, for the channel perpendicular to H, reads

Jp = -----~ (1-- COS/~)[(1- cos ~)c, 2 +(1 +cos ~)c 2 ]Va.

(5.1)

Y This supercurrent transports the longitudinal magnetization from cell L to cell K. The rise of the magnetization in cell K means a decrease of the angle ft. To maintain the resonance condition, the HPD in this cell begins to absorb more RF power (curve AB). The same supercurrent leads to an increase of angle fl in cell L. To prevent this the NMR absorption must fall in this cell (curve AB'). In other words the magnetic supercurrent transports some magnetic energy JE = -JpH from cell K to cell L. To compensate this energy flow the RF absorption rises in one cell by 6 W1 and falls in the other one by --6 W2. If the magnetization transported by the supercurrent were conserved, we would have 6Wl =-.-6W2. However, there are some relaxation processes caused by interaction between the magnetization of the normal and superfluid components. Spin diffusion of the normal component leads to a dissipation of magnetic energy in the channel, that grows with phase gradient. To maintain the resonance conditions for the HPD in the channel, the energy losses should be compensated by additional energy supply by spin supercurrent. So the spin current is greater at the inlet of the channel than at the outlet. The asymmetry of the experimental curve in fig. 23 about Aa = 0 is the result of magnetic relaxation within the channel. But this relaxation is not the result of friction, it can be treated as a relaxation of the eigenstate, which cannot be seen in the case of mass superflow or superconductivity due to the conservation of mass and charge. By taking this relaxation into account one can recalculate the distribution of Va along the channel: Va(x) =

exp A A a - 1 . A[ L + (exp AAa - 1)x]

(5.2)

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YU.M. BUNKOV

Ch. 2, w

Here L is the length of the channel and A=(64/145)DtORFC• -2, where D = (D, + D• is the effective spin diffusion coefficient along the channel as defined in the paper by Einzel (1981).

5.2. Phase slippage The spin supercurrent in a channel is limited by the instability of current against phase slippage. In this section we analyze the nature of phase slip centers for spin supercurrent. From a general point of view the phase slippage of spin supercurrent is analogous to that observed in superconducting wires (see review of Ivlev and Kopnin 1984) and mass superflow through a small hole, studied by Varoquaux, G. Ihas at al. (1992). We have learned from these superfluidity and superconductivity experiments that the superfluid density should be zero at the phase slip center. As a result the phase of the order parameter is not determined and the phase relation along the channel can have a discontinuity. The formation of phase slip is related to a change in some energy. If this energy is less, then the density of the kinetic (gradient) energy of the supercurrent, the phase slip appears. As a result of phase slippage, the phase difference along a channel will be decries on 2n.rt. Upon decreasing the kinetic (gradient) energy density the phase slip center becomes unstable and disappears. The main difference between the phase slip in superfluidity and superconductivity on the one hand and phase slip of spin supercurrent on the other is that in the latter case it is not necessary to destroy the superfluid state to create the phase slip. It is sufficient to destroy the spin supercurrent density which is proportional to (1 - c o s fl) (see eq. (5.1)) to maintain the spin supercurrent phase slip center. If fl = 0 in any part of the channel, the phases of precession of the HPD in the cells are no longer connected and the phase difference between the two HPDs can change by a multiple of 2~. The critical current for creation of the phase slip can be estimated by comparing the stiffness of the HPD state in a channel and kinetic (gradient) energy of a current. This corresponds to the phase gradient equal to the inverse value of the Ginzburg-Landau coherence length (eq. (2.39)) Va c = 1/~GL =tOL(tORF--tOL)/C• 2. As was shown by Fomin (1988) the local gradient energy is equal to the energy of HPD formation. In reality the situation is more complicated. One should take into account the spectroscopic correction to the gradient energy that leads to the frequency shift of precession f2v = OFvlOP: 5cl2 - c 2 Va2 ~v = ~ 9 4to

(5.3)

The value of the dipole-dipole frequency shift decreases with increasing current in order to compensate this gradient energy frequency shift and to keep the

Ch. 2, {}5

SPIN SUPERCURRENT AND NMR IN 3He

129

HPD in the channel in resonance. But when g2v surpasses the difference between the HPD frequency and the Larmor frequency in the channel, the HPD can no longer exist and the angle fl decreases. Therefore the density of P, proportional to (1 - c o s fl), decreases which makes the spin current solution unstable. Interestingly an analogous instability takes place in the case of mass supercurrent in 3He-B due to Fermi liquid corrections. As was shown by Vollhardt et al. (1979), the superfluid density in 3He-B decreases with increasing gradient of phase of the wave function (velocity). Consequently the critical supercurrent corresponds to a maximum value of current as a function of this gradient. By taking into account the circumstances given above, the critical spin supercurrent should correspond to the gradient:

Va =~ '40)L(('0RF--('OL) ~

(5.4)

-ci

In fig. 24 we show the experimental value of the critical phase difference between two HPD as function of tORF- to L. In order to compare these results with theory one should take into account the distribution of phase gradient in a channel, given by eq. (5.2). There is good agreement with the theory, particularly if we use the spin diffusion coefficient as a fitting parameter, that is D = 0.035 cm2/s (solid line in fig. 24). For the D• measured under the same conditions as the method described in section 4, we have D• = 0.058 cm2/s. This discrepancy is probably caused by spin-diffusion anisotropy (see Einzel 1981), demonstrated experimentally for the first time.

/

ZO~

/

/

/

/

J

10~

0

l

!

I

0.2.5

1

!

0.5

(~Rr

!

I

0.75

I

I

1.

I

I

1.25

I

I

l.S

- ~r.,) kMz

Fig. 24. Critical phase difference versus frequency shift in the channel, measured at 29.3 bar and 1.4 mK. Dashed curve represents the theory for nonrelaxing magnetization, while solid curve is the theoretical fit with spin-diffusion relaxation as a parameter (after Borovik-Romanovet al. 1989).

130

YU.M. BUNKOV

Ch. 2, w

Fig. 25. Phase portraits of the signal from pick-up coils in the channel. Solid curve shows the slowly changing phase and amplitude of the signal with positive (a) and negative (b) difference of HPD precession phases in the cells. The dotted line represents the fast digital record of the signal during phase slippage. The time interval between points is 50 kts. In contrast with mass superflow and superconductivity, the spin supercurrent gives a unique possibility to monitor phase slippage in a channel by receiving the local phase of precession by pick-up coils. In figs. 25 and 26 the phase portraits of signals from coil m (fig. 6a) are shown with Aa used as a parameter. Figure 25 show the same signals as were shown in fig. 23. The phase portrait of the signal is presented in the following way: the amplitude and phase of the signal for a time sequence of measurements are shown by points in a polar coordinate system. Consequently if we draw the vector from the coordinate center to a particular point, its length corresponds to the transverse component of magnetization in the channel averaged over the sensitivity region of the coil, while the phase corresponds to the phase of the precession measured with respect to the

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SPIN SUPERCURRENT AND NMR IN 3He

131

(a)

(6)

" ...:.:i-f-i.;.iI.i:....;...'....;:...-., Fig. 26. The same phase portraits as in Fig. 25, but for large frequency shift in the channel. phase of the RF field in cell K. The point A, A' corresponds to the time when changes in the phase difference between the HPDs was initiated. So at this point Aa = 0 and the transverse magnetization is a maximum. There is some phase shift between the transverse magnetization in the channel and the HPD due to magnetic relaxation in the channel. The solid line shows the change of the signal in the channel with slowly changing Aa. The phase portraits at two different directions of spin supercurrent are shown in fig. 25a,b. The decrease of the transverse magnetization with increasing IVal is connected with the averaging of the signal over the sensitivity region of the coil. When the spin supercurrent reaches its critical value (points B, B'), phase slip occurs and the system returns to the previous condition (points C, C') and continues to move to the points B (B'). In fig. 26a,b the phase portraits are shown for the same conditions as in fig. 25, but for the larger value of tORF- to L in a channel and consequently for shorter Ginzburg-Landau coherence length.

132

YU.M. BUNKOV

Ch. 2, {}5

If the time between phase slips is of the order minutes and determined by the pumping frequencies, the phase slip duration is tens of milliseconds. With a transient recorder, we were able to record the phase slip process. This is shown as points in figs. 25 and 26. The time between subsequent points is equal to 50/ts. The slips are accompanied by oscillations of the transverse magnetization, which correspond to a twist oscillation of the HPD. The behavior of the signals during the phase slip allows us to determine the position of the phase slip center (PSC) relative to the pick-up coil. If the PSC is situated between the coil and the reference HPD, then during the slip, the magnetization in the region of the coils continue to turn in the same direction as before, otherwise it is turned in the opposite direction. This is clearly seen in fig. 26a,b, where the phase portrait is plotted for precession at smaller values of the Ginzburg-Landau length, and consequently the dimensions of the PSC are shorter. As was discussed above, spin diffusion makes Va rise in the inlet of the channel. For this reason PSC is formed near the inlet, which is situated between the pick-up coil and reference HPD in fig. 26a, and on the opposite side of the channel for the opposite direction of the spin supercurrent in fig. 26b. The behavior of the magnetization and consequently the phase of the order parameter during the slip is very complex, but it is the first demonstration of phase slip dynamics of the supercurrent by monitoring the phase of the order parameter. A program for a PC computer can be obtained from the author, which display the experimentally recorded phase slips in real time.

5.3.

Josephsonphenomena

The Josephson effect is the relationship between current and phase of two weakly connected regions of coherent quantum states. It was described by Josephson (1962) for the case of two quantum states, separated by a potential barrier. This phenomenon is usually studied for the case of quantum states connected by a conducting bridge with the dimensions smaller than the coherence length. In this case the coherent state in the bridge cannot be established so there is no phase memory, which determines the direction of the phase gradient. As a result the supercurrent is determined only by the phase difference between the two states. J = J0 sin(A~).

(5.5)

As the dimensions of the conducting bridge increase, more complex dependence of the current on A ~ is observed. For bridge dimensions of the order of the coherence length, a transition to a hysteretic scenario with phase slippage appears. The Josephson effect was carefully studied in superconductors (see Lik-

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SPIN SUPERCURRENT AND NMR IN 3He

133

harev 1979), and for mass supercurrent in 4He and 3He by Avenel and Varoquaux (1985). In the case of mass and electronic supercurrents the coherence length is a function of the temperature. In the case of spin supercurrents, however, the GinzburgI-Landau coherence length (eq. (2.39)) is not only a function of temperature, but also a function of the difference between the HPD precession frequency and the local Larmor frequency. This value can be changed experimentally with a magnetic field gradient or position of the domain boundary. As a result we were able to change the coherence length in the region of the orifice in the channel and observe the changeover from simple Josephson phenomena to phase slip behavior. This experiment was done by Borovik-Romanov et al. (1988, 1989d) in a chamber, shown in fig. 6b. The orifice, of diameter 0.48 mm, was placed in the central part of the channel. The current-phase characteristics, observed in this experiment are represented in fig. 27 for different positions of the domain boundary related to the orifice. One can easily see that the current in fig. 27a corresponds to a pure Josephson relation, in fig. 27b to the nonlinear Josephson relation and in fig. 27c to a phase slip phenomenon. The first attempt to describe theoretically the spin supercurrent Josephson phenomenon was by Markelov (1988). In spite of some difficulties in presenting a simple mathematical model of the spin supercurrent in an orifice, his calculations have a qualitative agreement with the observed phenomena.

Fig. 27. Absorption in one HPD recorded both by increasing and decreasing the phase difference between HPDs for 0 bar, 0.47Tc, 71 Oe, ~GL ~ 1.3 mm (a), ~GL ~ 0.8 mm (b) and ~GL - 0.7 mm (c) (after Borovik-Romanovet al. 1989).

134

YU.M. BUNKOV

Ch. 2, w

5.4. Spin supercurrent vortex Since the original suggestion by Onsager (1949) and Feynmann (1955) that superfluid liquids under rotation should form quantized vortices, many very interesting investigations of quantum vortices have been performed. There have been many review articles published about vortices in 4He, 3He and superconductors, particularly in Progress in Low Temperature Physics (original paper by Feynman (1955), vortices in 4He by Vinen (1961 b), vortices in 3He by Fetter (1986) etc.). We do not review here the general properties of quantum vortices in superfluid liquids. Just for completeness the excellent review by Fetter is worth mentioning; the first observation of vortices in 3He-A was published by Hakonen et al. (1982) and in 3He-B by Ikkala et al. (1982). The quantization of circulation in superfluid 4He as well as quantization of magnetic flux in a superconductors follows from the requirement of singlevaluedness of an order parameter as a function of position. It should return to the same value for any closed path in the fluid. For the simplest type of order parameter such as for 4He and ordinary superconductors, this condition leads to the formation of linear singularities of the order parameter-vortex lines. In these lines, the density of superfluid liquid turns to zero and consequently the order parameter phase is undetermined. Around any path enclosing this line the phase of order parameter changes by 2nat, which corresponds to n quanta of circulation. Owing to the complex structure of the order parameter in superfluid 3He vortices with different internal structure have been observed, as well as in 3He-A, vortices without a singularity. There have been many exciting results of vortex studies in rotating superfluid 3He, observed recently, particularly by HPD spectroscopy methods. A theoretical consideration of this subject can be found in the book by Volovik (1992). One can expect that spin supercurrent vortices are a very rich subject due to the tensor form of spin supercurrent equations (eq. (1.6)). Up to now only the simplest kind of spin supercurrent vortices have been observed experimentally and considered theoretically with singularity of the current related to the transverse mode of NMR. Its formation does not require suppression of the superfluid state. Analogous to the case of phase slippage, the density of spin supercurrent in the region of the vortex core can be suppressed by maintaining the angle fl = 0. Consequently, for the magnetic part of the order parameter, phase is undetermined in the region of the core. One can imagine vortices directed along (Fig 28A) and transverse (Fig 28B) to the direction of the magnetic field and its gradient. Fomin (1988) has obtained the solution of the equations of motion of the magnetization in 3He-B which corresponds to the spin vortex directed along the magnetic field. In particular, he has shown that the spin vortex should have a core with the radius of the order of ~GL and fl changing from 0 at the vortex axis

Ch. 2, w

135

SPIN SUPERCURRENT AND NMR IN 3He

i

1 \

t

!

2

3

--

as Jp

o

1

Y~GL

(a)

.......,,.,,,, t "/ j

Vortex axis..

\ .../

J (b) Fig. 28. Schematic representation of the spin supercurrent vortex line (a) directed along and (b) transverse to the magnetic field, shown in the frame rotating at the NMR frequency. Also there is shown the spatial distribution of magnetization deflection angle and spin supercurrent density for the longitudinal vortex.

136

YU.M. BUNKOV

Ch. 2, w

to 104 ~ at distances much larger than ~GL"The spin current is circulating so that a is changing by 2xn along any loop around the vortex. This solution is shown also in fig. 28A. For the case of the transverse direction of the vortex axis, the solution is much more complicated due to the absence of axial symmetry and further theoretical investigations are necessary. There are two different experiments where spin supercurrent vortex line formation was observed. There is the observation of double phase slip in a channel and, in a specially designed experiment, the observation of formation of a state with one quantum of spin supercurrent circulation in the chamber. In the first case the double phase slip can be explain as a crossing of all the enclosed streamlines by a vortex. This kind of phase slip scenario, where a vortex is created on one wall of the channel, crosses it and then is destroyed on the other wall of the channel was suggested by Anderson (1966). In general, to study this kind of phase slip, it is necessary to make use of fast time resolution spectroscopy. We have seen a phenomenon that we can only explain as the formation of a stationary vortex line in the channel. Under certain experimental conditions we have observed a splitting of 2.rt phase slip in two jumps, as illustrated in fig. 29. This kind of phase slip has been observed in a chamber with an orifice, prepared for the Josephson experiments. It was observed at 500 kHz NMR frequency under conditions of short Ginzburg-Landau length. Both parts of the slip show hysteretic behavior. The state between jumps is persistent if we stop to change the phase difference between chambers. This splitting of 2~ slippage can be explained as the formation of a vortex on the lower part of the orifice and then its annihilation on the upper part of orifice. The stationary state of the vortex in the channel between these two events can be explained by two

i

t

--3 0

I

I I

I

AO~, R a d

I 2

Fig. 29. The form of HPD absorption signal jump at the critical spin supercurrent at 0 bar, 0.52Tc, 284 Oe and frequency shift in the channel 400 Hz (single jump), 300 Hz and 160 Hz (double jumps) (after Borovik-Romanov et al. 1989c).

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SPIN SUPERCURRENT AND NMR IN 3He

137

circumstances. First, the Ginzburg-Landau coherence length is inversely proportional to the distance to the HPD boundary. As a result the conditions for critical current are achieved first on the lower part of the orifice, thus provoking vortex formation. To cross the central part of the orifice, the length and consequently the vortex energy should increase. This energy possibly stabilizes the position of the vortex in the channel. The final jump of current is seen after the vortex crosses the central part of the orifice and is quickly displaced to the opposing wall. We have not succeeded in producing the same type of experiment in a channel with rectangular geometry. It is likely that the geometry of the channel is important for vortex stabilization. Let us describe now the specially designed experiment in which the state with one quantum of spin supercurrent circulation in the chamber was maintained and studied. This state corresponds to rotation of superfluid liquid with one quantum of circulation as in experiments by Vinen (1961 a) for 4He and Davis et al. (1991) for 3He-B, 6r a single Abrikosov vortex in a superconductor. To create the circular spin supercurrent in the HPD, an RF field with topology corresponding to the topology of the spin vortex was used. For this purpose the leads to the two parts of the RF coil were brought out of the cryostat separately, so we were able to change the connection from parallel to opposition (gradient coil). In the last case, the two parts of coil induce RF fields in opposite directions, so the RF field intensity was zero in the center of the cell. Its phase changes by 2.rt in the path around the center of the chamber. This RF field indeed excited HPD but with some different properties. The magnetic relaxation was higher than in conventional homogeneous RF field excitation, while the dispersion signal was lower. We believe that under these conditions the HPD with circular spin supercurrent was created. To prove this, an additional (nongradient) pick up coil was situated near the top of the cell. We received by this coil the induction decay signal after switching off the RF field. For the case of homogeneous RF field excitation, the pick-up coils detected the usual HPD induction decay signal which frequency and amplitude are shown in fig. 30a,c. In the case of the HPD maintained by gradient RF coils, the pick-up coils received low intensity nonregular induction decay signals, shown in fig. 30d, with frequency, shown in fig. 30b. These signals are the same as expected for the case of HPD with a spin vortex in the center. The small amplitude of the signals is the result of interference of RF radiation from different parts of the HPD, while the frequency corresponds to magnetic field on a moving boundary. The relaxation increase, measured by the frequency of the signal, is related to spin-diffusion relaxation in the region of the vortex core. It is clear also that during the time of HPD relaxation the spin vortex is stable, so an HPD with circular spin supercurrent does not convert to a homogeneous HPD. Furthermore the oscillations of signal intensity, seen in fig. 30d, can be the result of spatial oscillations (or rotation) of the spincurrent vortex-line.

138

Ch. 2,

YU.M. BUNKOV

w

N "1" o

I-

o8 U

o'o

1.0

cld II

=.o.a n

I I

f o

0.6

,-, 9 ..__

f

_~0.4

f

o.

EO.2

_.,-.

f

f

<

o.o~

'

16o ' 26o ' 36o Time, ms

'

6

'

16o " 2 6 0 " ' ~ 6 o Time, ms

Fig. 30. Frequency (top) and amplitude (bottom) of induction decay measured by a pick-up coil after switch of the HPD, maintained with an RF field from parallel connected coils (left) and oppositely (quadrupole) connected coils (right), 29.3 bar, 0.5Tc (after Borovik-Romanov et al. 1990).

As was mentioned before, the spin supercurrent described here is only one component of the spin current tensor so other types of vortices corresponding to other components of spin current, as well as combined spin and mass supercurrent vortices, are still waiting to be investigated. The combined spin and mass vortices were recently suggested by Volovik (1992).

6. Spin supercurrent in 3He-A The 3He-A spin dynamics differs from that of 3He-B owing to the different dipole-dipole interaction potential. This interaction is the cause of the instability of homogeneous precession as was shown in the second section of this review. Spin supercurrent transport the magnetization and in the process develop inhomogeneity. But at the beginning of this process a small inhomogeneity should

Ch. 2, {}6

SPIN SUPERCURRENT AND NMR IN 3He

139

exist. This starting inhomogeneity might be supplied by boundary conditions in the cell. As an example let us consider the case when all inhomogeneities are directed in the z direction. From eq. (2.24) one can write the expression for gradient energy averaged over fast rotation: F~ = " - ' L c~ [(1-cos fl)(3- cos fl)a '2 +fl,2 - ( 1 - c o s f l ) a ' r 1 6 2 4}, 2

'21.

(6.1)

After minimization of the dipole-dipole energy with respect to phase q~, we obtain the spin supercurrent J~ = - u-------~)[c2 Z(1(32),

u)a'].

(6.2)

The equations of motion for transverse magnetization (eq. (2.27)) for 3He-A have the form P = -VJp, (6.3)

&=Y OP-H =-yH-

8yH

(l+3cosfl).

It is easy to see that the inhomogeneity in the distribution of the angles a leads to a redistribution of P, that is, to inhomogeneity of ft. Due to the sign of the shift in the NMR frequencies, this process has a positive feedback. As a result exponential growth in the spatial inhomogeneities takes place. This is the mechanism of instability of homogeneous precession in 3He-A.

6.1. Instability of homogeneousprecession The mysterious aspect of pulsed NMR experiments in 3He-A is the very fast magnetic relaxation. It was observed to be one order of magnitude faster than predicted by the theory of intrinsic relaxation. It should be pointed out that all experimental studies of relaxation in 3He-A were carried out in a highly inhomogeneous magnetic field. In experiments on double pulsed NMR by Corruccini and Osheroff (1978) and by Eska et al. (1982) the inhomogeneity of the magnetic field was necessary in order to reduce the induction signal from the first pulse to the time of the second pulse. In experiments with a SQUID by Webb (1977) and by Sager et al. (1978) the inhomogeneity of the magnetic field was produced by the superconducting coil of a magnetic-flux transformer. As a

140

YU.M. BUNKOV

Ch. 2, w

result, the part played by the inhomogeneity of the magnetic field in the magnetic relaxation was unclear. To investigate relaxation processes under the most uniform conditions we designed an experimental cell in which the 3He-A was mainly in the region of the uniform radiofrequency field. This cell is shown in fig. 5a. We have measured and entered into a computer the distribution of the amplitudes of the radiofrequency fields in the chamber. This inhomogeneity corresponds to an inhomogeneity of the magnetization deflection after the pulse and, consequently, to inhomogeneity of the precession frequency. We were able to calculate the decay of the signal caused by inhomogeneity of the RF field for each specific experiment. In fig. 31 we show a typical digital record of the induction signal in 3He-A. The computer approximated every sixteen points of the recorded signal by a sinusoid of corresponding frequency, phase and amplitude and hence reconstructed the characteristics of the initial induction decay signal. It is possible to distinguish two parts of the signal. In the first the decay rate corresponds to a decay calculated from the RF field inhomogeneity. In the second the rate of decay accelerates. To study the intrinsic decay of the signal we have subtracted the dephasing caused by inhomogeneity of RF and external fields. We have investigated the signal of decay for different deflection angles. The intensity of the induction signal as a function of time for different angles, at a temperature T = 0.93Tc and at a frequency of 250 kHz, are collected in fig. 32a. The experimental curves are shown by the points. The continuous curve shows the decay of the induction signal for a spatially uniform relaxation process predicted by theory of intrinsic relaxation. To compare this curve with experimen-

Fig. 31. The 3He-A induction decay after an 81~ deflection pulse, stored at about four times per period. Upper curves shows the calculated decay due to inhomogeneity of the RF field (solid) and due to inhomogeneityof the external field (dashed) (after Bunkov et al. 1985).

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

141

Fig. 32. Decay (top) and change of frequency (bottom) of the induction signal in 3He-A, recorded after different tilting angles. The solid curve represents the Leggett-Takagi relaxation for the homogeneous solution. The beginning of each experimental sequence corresponds, as regards of tilting angle, with the theoretical curve (after Bunkov et al. 1985).

tal data, the initial points of the experimental curves obtained for different initial angles of deflection fl0 are placed in correspondence with the same angle on the theoretical curve so that their behavior can be followed over the same time scales. It can be clearly seen that for angles of deviation <38 ~ the transverse magnetization in 3He-A relaxes in good agreement with the Leggett-Takagi theory. For large angles of deviation, additional processes occur which lead to a much more rapid change in the transverse magnetization. We made a similar comparison for the time-dependence of the induction signal frequency (fig. 32b). In this case, we took as the theoretical curve the relationship oJ = to(fl(t)), obtained from the second equation in (6.3) and the same relaxation of angle fl as in the previous case. The experimental points are also placed in correspondence with the theoretical curve with respect to the initial value of spin deviation. It can also be seen that for angles of deviation of spins of <52 ~, the precession frequency follows the theoretical curve. For large angles of deviation of magnetization, the frequency of the precession is reestablished much more rapidly than expected from the Leggett-Takagi theory. It should be noted that when investigating the amplitude of the induction signal we obtain information on the change in the transverse magnetization of 3He-A, averaged over the dimensions of the chamber. When investigating the frequency of the induction signal we obtain information on the mean angle of deviation of the magnetization, assuming, of course, that the 3He-A is uniform. Knowing the

142

YU.M. BUNKOV

Ch. 2, {}6

angle of deviation of the magnetization and its transverse component we can establish the dependence of the change in the total magnetization of 3He-A averaged over the specimen. It turns out that for angles of deviation <38 ~ the total magnetization of 3He-A is conserved. In this case the magnetic relaxation process is in good quantitative agreement with the Leggett-Takagi theory. For large angles of deviation the total magnetization falls off in almost the same way as the transverse magnetization, since the change in the angle fl and the frequency lag in time behind the change in the transverse magnetization. We can conclude that the spatial homogeneity of the precession in 3He-A is considerably disturbed. The behavior of the induction decay signal of 3He-A corresponds to the theory of homogeneous precession instability in 3He-A and its decay into a precessing spatial texture. This theory was developed by Fomin (1979). In connection with our experiments, Fomin (1984) has investigated the effect of instability on the quantities measured in the experiment, namely, the amplitude and frequency of the induction decay signal. According to his theory a solution of the linearized equations of motion for small spatial perturbations of the homogeneous precession cosfl = cosfl0 + ~/~ (R),

P= P0 +~p(R),

a=ao(t)+6a(R),

(6.4)

~=~o +~r

has the form of texture waves 6 = 6 o x exp(ito k t

to k -

--

ikR),

(6.5)

I 4y2H Ckk 2 (Ckk--34f22A sin2 fl) ff2,~( 3g2-2Acos(1 +4COS C k +kfl) CfOlS+) ( fl) 1 + 4Ckk ] _ iDkk, (6.6)

where Ckk= ca#Zkck#, and Dkk = D~,tk~k-q. Let us consider now the influence of this solution on a spin system, precessing at an angle ft. In the case where k satisfies the condition Ckk< 3/4-QA2 sin2fl, the frequency of perturbation is clearly imaginary, and the solution of the equations of motion for the perturbations takes the form of an exponentially increasing function 6 = 6 o x e x p ( G k t - ikR),

(6.7)

Ch. 2, {}6

SPIN SUPERCURRENT AND NMR IN 3He

143

Fig. 33. Spatial distribution of the phase and amplitude of the precession and the angle $ as a spatial inhomogeneity develops.

where Gk = leakl. Hence, on a background of homogeneous precession a spatial inhomogeneity develops with a growth increment (Gk). This mode of texture waves is shown qualitatively in fig. 33. As we know, during the time that the magnetization S precesses around H, the order-parameter vector d moves along a trajectory similar to a figure-of-eight. The spatial orientation of the phase q~ coincides with the junction point of this trajectory. To obtain the effect of the developing inhomogeneities on the amplitude and frequency of the induction decay signal, it is necessary to carry out an appropriate calculation and integration over all k. Here we must bear in mind that, in general, the initial perturbation 60 also depends on k, and the problem becomes very complex. It was shown by Fomin (1984) that if we assume that the amplitudes of the initial perturbations are the same and are fairly small for all k (for example, thermal fluctuations), and take into account the fact that Gk has a maximum with respect to k, the values of the transverse magnetization and precession frequency observed should have the following time dependence:

(S•

S•

1-

(6.8)

As exp(2Gmt)l ' 42Gm t

(d-~ + 1 + 3ff22 ,, A (I+cosfl)

8yH

) Aa =~

~/2Gmt

exp(2Gmt),

(6.9)

where A s and Aa are constants corresponding to the initial perturbations of the system. If, for simplicity, we assume that Ckk is proportional to Dkk, then the maximum value of the disturbance increment G m can be written in the form a m = g m 3~'-22 sin 2/5,

16yH

(6.10)

144

YU.M. BUNKOV

Ch. 2, w

where Nm is the value of the function

N(x) = 2 x ( x - 1) 3 ( 1 + x ) - ( 1 +3x)cos fl - 4wDk~ ~ x , l+3x+(1-3x)cosfl Ckk

(6.11)

which is maximum with respect to x. Hence, if the initial distribution of the perturbations is uniform with respect to k, nonuniformities should develop in the experiment for which the growth increment is a maximum. It was shown in experiments by Bunkov et al. (1985) that the induction signal in 3He-A for angles of deflection of the magnetization >40 ~ falls in accordance with the relation I = I0[1- A exp(t / To )].

(6.12)

It was also shown that 1/T~ has the same dependence on the temperature, the magnetic field and the angle of deviation as 2G m, but it was substantially less than predicted from Fomin's theory. To investigate the reasons for the quantitative disagreement between the theoretical and experimental values, we carried out a more detailed investigation of the induction signal. The main contradiction between Fomin's theory and the experimental results is the value of the initial perturbation, which from the experimental results, lies in the range 10-3-10 -4, whereas the range of inhomogeneity produced by thermal fluctuations, according to Fomin's estimates, should be 10-7. Another source of initial inhomogeneity is the texture of the order parameter induced by walls of the chamber. Its value is of the order of ~A/a = 10-3 for k = 2ar/a = 10 cm -1, and falls as k increases (where ~A is the magnetic length and a is the dimension of the chamber along the direction of the magnetic field). In this case, under the experimental conditions, an instability having a wave vector less than the optimum value (kopt = 103 cm -1) and consequently a smaller growth increment, but a much greater value of the initial inhomogeneity, is obviously able to develop earlier. This is indicated indirectly by the results of an observation of the induction signal in chambers with plates placed along the magnetic field in practically the same way as in a chamber without plates, whereas in a chamber with plates oriented transverse to the magnetic field, the instability developed much more rapidly. Hence, it follows from the good qualitative agreement between the results of the experiments and the theory of the instability of homogeneous precession that homogeneous precession of magnetization in 3He-A at large angles of deflection, decays into a spatial by inhomogeneous texture. It is possible that under practical experimental conditions texture waves are produced on the walls of the cell and propagate within the chamber.

Ch. 2, {}6

SPIN SUPERCURRENT AND NMR IN 3He

145

In order to verify that the mechanism of instability of uniform precession considered here completely determines the processes of longitudinal relaxation in 3He-A, we compared the decay of uniform precession with the recovery of the longitudinal magnetization, measured using the two-pulse method. A 90 ~ pulse was applied to a spin system placed in an inhomogeneous magnetic field, and after a certain delay time a 7 ~ test pulse was applied. One can judge the value of the recovery of the longitudinal magnetization from the intensity of the induction signal after the 7 ~ pulse. It turned out in this case that this method can only be used at very high temperatures, since at lower temperatures the form of the induction signal after the 7 ~ pulse depends very much on the delay. As a result, even for very short recovery times, the receiving system is unable to extrapolate reliably the intensity of the induction signal after the test pulse to the time corresponding to the end of the RF pulse. In fig. 34 the decay of the intensity of the induction signal in a homogeneous field and the recovery of the magnetization in an inhomogeneous field are compared. It can be cle~:rly seen that the time scale of both processes is the same. Owing to the instability of homogeneous precession, a periodic texture of the order parameter of 3He-A develops in which the phase ~p also turns out to be modulated. Due to the spin-orbit interaction, a drain in the value of the magnetization of 3He-A appears, which determines the relaxation of the longitudinal magnetization. In conclusion it is possible to say that in the case of transverse NMR in 3HeA the process of inhomogeneity development is faster than long range transport

Fig. 34. Comparison of induction decay in a homogeneous magnetic field (O) and the recovery of the longitudinal magnetization, measured by the induction decay frequency (+) and by two pulse method in a highly inhomogeneous magnetic field (e) (after Bunkov et al. 1985).

146

YU.M. BUNKOV

Ch. 2, w

magnetization by spin supercurrent for any value of the gradient of magnetic field. Spatial structures with wave length of order 10-2-10-3 cm develop as a result of instability of the homogeneous precession determined by an enhanced magnetic relaxation process.

7. Spin supercurrent at propagating A-B boundary Recently, there has been much interest in the A-B transition boundary in superfluid 3He owing to the unique situation of co-existence of two quantum systems with different broken symmetries. Beyond specific low temperature physics applications, this problem has general relevance, for example, as a model for the hypothetical situation of the coexistence of two vacua with different sets of elementary particles (see the book by Volovik 1992). Also relevant is the interesting problem of the nucleation of the B phase from the hypercooled A phase (Leggett 1992). After nucleation, the B phase can expand very rapidly under the so-called hypercooled condition, i.e. when the latent heat of the A-B transition is lower than the heat capacity of the B phase. In this case the transition boundary velocity is not limited by the rate of removal of the latent heat. Furthermore, the so-called undercooling parameter, which is the ratio of the heat necessary to warm the stable phase back up to the temperature of the thermodynamic phase transition to the latent heat can be as large as 40. That is the largest value of the undercooling parameter for any known first-order transition. Pioneering experimental studies of the dynamics of the propagating A-B phase boundary have been conducted by Boyd and Swift (1990) in a cell, represented schematically in fig. 35. In their experiments the overcooling conditions for 3He-A were maintained inside the tube shape cell. Owing to the larger magnetic susceptibility of the A phase, the magnetic field shifts the equilibrium of the A-B transition to a lower temperature. Consequently one can stabilize the A phase at low temperatures by a relatively high magnetic field. Then the field in the cell can be decreased except for a region near the cell inlet. So overcooling conditions can be achieved in the cell, but the B phase cannot propagate from the heat exchanger due to the local high field region (magnetic valve). At a given moment of time the magnetic valve opens and the B phase boundary propagates into the region formerly occupied by the A phase. A series of SQUID sensors monitor the A-B boundary propagation by measuring the sharp decrease of magnetization of the B phase relative to the A phase. Two temperature regions were distinguished experimentally with different dynamic properties of magnetization. In the high temperature region (smaller overcooling conditions) the observed signals can be easily explained by the assumption that the magnetization corresponds to its equilibrium value for A and B phases. In the lower temperature region the signals become more complicated.

Ch. 2, {}7

SPIN S U P E R C U R R E N T AND NMR IN 3He

147

Fig. 35. Magnetization distribution after the A-B boundary begins to move at 4 ms (a), 8 ms (b), 16 ms (c) and 44 ms (d), and schematic representation of the experimental cell.

148

YU.M. BUNKOV

Ch. 2, w

For interpretation of these results an additional SQUID magnetometer with a single sensor (Boyd and Swift 1992) was used. With this magnetometer local behavior of the magnetization was clearly seen. The magnetization increased ahead of the moving boundary (precursor), indicated by a jump of magnetization at the A-B transition, and a decrease of magnetization remained far behind the boundary (slow moving signal). The observation of nonequilibrium magnetization in A and in B phases shows that magnetic relaxation is not localized behind the moving boundary and the full spin dynamics, including spin supercurrents, must be taken into account. The magnetization difference of A and B phases can be very large under the conditions of these experiments, where the ratio XB/XAcan be about 0.4. Consequently, the moving boundary deposits in an interval of time dt a nonequilibrium magnetization (ZA- XB)H at a distance VAB dt behind the boundary. While the processes of magnetic relaxation in the A and B phases are well known, the magnetic relaxation inside the boundary is unknown. However, we should emphasize that the boundary with thickness of order the coherence length, passes a fixed point in the liquid in a time of order 10-8 s, while the highest possible rate of magnetic relaxation is characterized by the energy of the dipole-dipole interaction of order 10--6 s. Therefore under the experimental conditions only 1% of the magnetization can be lost inside the boundary. Since we must deal with nonequilibrium longitudinal magnetization we should take its spin dynamics into consideration by using the first two equations of the system (eq. (2.27)). The first equation has a complicated form. Besides the spin supercurrent transport term JM there is the additional source of magnetization y(aV/Oq~). This dipoledipole term determines the magnetic behavior of the longitudinal magnetization at small excitation levels where q~ oscillates near its equilibrium direction. Consequently, the spatial transport term cannot play a significant role. At higher excitation, when the quantity (M~/x- H) is larger than the dipole-dipole term g2/y, d begins to rotate around M. This regime of longitudinal NMR has been studied experimentally by Webb et al. (1977). At large enough excitation, the influence of the source y(Ov/&p) on the equations of motion vanishes, owing to its average over the fast rotation of q~. Under these conditions we can consider the long distance transport of magnetization M~ by the spin supercurrent JM. We demonstrate that this case is appropriate to the conditions of fast A-B phasetransition propagation. We can rewrite expressions (eq. (2.34)) for the spin supercurrent JM appropriate to the A and B phases, assuming that all the variables depend only on one coordinate, z:

J~ = - ~ c 2 ~ ',

J~ = -~c~b',

(7.1)

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

149

where CB2= 2 c j . 2 - c 2 is the velocity, characteristic of the spin supercurrent in 3He-B, and CA2 the velocity, of the spin supercurrent in 3He-A, which depends on relative orientation of the orbital momentum 1. In order to consider spin dynamics in the region of the boundary, we should introduce the relation between the phases q~ of the order parameters of 3He-A and 3He-B. Let us assume that these phases are strongly correlated. We shall see that these conditions correspond to the experimental observations. What happens to the order parameter, when the boundary begins to propagate? The nonequilibrium magnetization excites the precession of the phase of the order parameter, ~. The spatial gradients of ~ create spin supercurrents which redistribute magnetization within the B phase and, because of our suggestion of strong correlation of the phases ~ on the two sides of the boundary, this also occurs in the A phase. (In the opposite case, if one allows phase slip at the boundary, there is no mechanism to transport the nonequilibrium magnetization ahead of the moving boundary.) The computer modeling of solution of eqs. (2.27) for the conditions of the experiments was done by Bunkov and Timofeevskaya (1992). They have shown that at lower temperature and consequently low magnetic relaxation rate, the nonequilibrium magnetization takes the form of two step solitons SA and SB, propagating out from the boundary into the A and B phases. They are reflected at both ends of the sample and propagate back crossing the A-B boundary. The result of computer simulations for the conditions of the experiments are shown in fig. 35 as a magnetization distribution at four specific times after the boundary begins to move. To compare the results of the calculations with experiment, the value of the longitudinal magnetization have been integrated in the region from 13 to 14 cm of the sample, i.e. the region of sensitivity the single sensor SQUID system in the experiment. Its time dependence is shown in fig. 36 by the solid line. One can see the passage of soliton SA, the soliton Sa, the second passage of soliton SA, reflected from the right end of sample and finally the passage of the A-B boundary. The same events can be seen in the experimental records measured by Boyd and Swift (1992), which are indicated in fig. 36 by the dashed lines. A similar comparison can be done for the 4 coil receiving system, but only for the beginning part of the signal. One can see good agreement between the theoretical and experimental curves, not only in terms of the time of the solitons passing but also in the absolute value of the signals. The reflection of the solitons from the ends of sample is a consequence of the longitudinal magnetization conservation law. In the experimental cell the heat exchanger is situated at the bottom of the tower. As a result, the nonequilibrium magnetization can be diluted in a higher volume of 3He-B on this side. However the numerical solution, assuming a closed end, fits the experimental record better. Possibly the distance between the experimental cell mouth and heat exchanger is very short in the real experimental set up, so that the nonequilibrium

150

YU.M. BUNKOV

.4

".-.

.

9

,

.

.

.

.

,

.

.

.

.

,

.

.

.

Ch. 2, {}7

.

,

.

.

.

.

= 1.2 0

Q

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Oo .8 .6

i/ 0

.

.

.

.

|

10

.

,

i

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20

.

.

.

.

!

30

.

.

.

.

|

40

.

_ .

.

.

50

Time, ms

Fig. 36. Magnetization calculated for a single sensor system as a function of time (solid line). The magnetization measured by a SQUID is also shown (dashed line). magnetization cannot be diluted considerably in the region of the heat exchanger. It is possible also that the switching of the magnetic valve created additional nonequilibrium magnetization in the bottom of the tower. Furthermore, one can see that the experimental record for the SB soliton is not as sharp as for the SA soliton which is possibly a result of the partially closed conditions at the left end of the sample. The calculations described above were done on the assumption of slow longitudinal magnetic relaxation. The magnetic relaxation for longitudinal NMR is well known and in our notation reads

/~lr~ -- . . . . .

Z T R ' M i _ Meq

,

(7.2)

w h e r e r R is a characteristic relaxation time. It is necessary to point out the very unusual relaxation term which is inversely proportional to the excitation level. As a result, the relaxation follows a square-root law and its rate decreases with increasing excitation. The full recovery time for A and B phases can be estimated as

TllA

=

T~ =

4 ~' 2 (M~ - Meq )2 ,t, R ~ 2 ~,-~4 A

- -

,

45

y2 (M~ - M ~ ) 2

16~, R

~ 2~'-~4 B

~7~,...,,

,

(7.4)

Ch. 2, w

SPIN SUPERCURRENT AND NMR IN 3He

151

Under the experimental conditions, if we assume the hydrodynamic value of r R of 0.15/zs, the recovery time is of the order of 10-4 s. On the other hand, to explain the observed phenomena one would have to propose a full recovery time for the solitons of order 10-2 s. This discrepancy can be considered as evidence of nonhydrodynamic depression of magnetic relaxation in the longitudinal NMR mode. As was shown experimentally for transverse NMR by Bunkov et al (1990c) the characteristic time r R for nonhydrodynamic conditions (f2rq >> 1) should be replaced by a frequency-dependent time, r(~o) =

rR(0)

1+ A(to.rq )2 '

(7.5)

where rq is a time of order the quasiparticle scattering time, ~2 the frequency of order parameter rotation and A a phenomenological constant of order unity. No one has yet considered the nonhydrodynamic correction for the relaxation of the longitudinal NMR mode, but to explain the experimental results one needs to assume such a correction. The velocity of the A-B phases boundary is determined by the difference of the Gibbs free energies in the two phases, AG, and the mutual friction, F, as k'AB = AG / F.

(7.6)

With decreasing temperature the value of AG and therefore the boundary velocity increases. Consequently the amplitude of the step solitons also increases. Nonhydrodynamic corrections should also increase very rapidly with temperature. As a result, the relaxation rate should decrease owing to the higher excitation and the nonhydrodynamic correction. At some temperature the spin dynamics should change abruptly from local relaxation near the boundary to soliton propagation. We believe that the two regimes of slow and fast boundary motion correspond to these conditions. After the A-B phase boundary crosses the cell, the B phase is left overmagnetized. The magnetization should than follow a relaxation process. An additional slow-moving magnetic step soliton was observed under these conditions. According to the equations considered above, any magnetic soliton in the B phase should propagate with velocity Ca, except in the region of very nonlinear dynamics, when M B -Meq----ZB~B]~. At this amplitude of excitation the frequency of the order parameter rotation falls virtually to zero, as was seen experimentally by Webb et al. (1977). The amplitude of the signal from the slowmoving soliton corresponds to this nonequilibrium magnetization. Possibly a domain structure with one domain with equilibrium magnetization and one

152

YU.M. BUNKOV

Ch. 2, w

overmagnetized domain appears under these conditions, similar to that observed for transverse NMR. However, to demonstrate this would require a specially designed experiment. In conclusion of this section we should emphasize that the studies of new modes of spin supercurrent, associated with A-B phase boundary propagation, are at the beginning stage. One should expect many new exciting phenomena at lower temperatures.

8. Conclusion

The frontiers of low temperature physics follow the latest achievements in cooling technique. The lowest temperatures now available can only be achieved artificially. Each new step along this road is usually accompanied by unpredicted discoveries. It is interesting to note that the discussion often arises that low temperature physics has achieved its final step and that physicists will not be able to find new phenomena by cooling further. Symptoms of this recession can be seen today. The result is a decrease in ultra low temperature physics at the most recent LT conferences. Partly, it can be explained by the existence of the rich uncle "High Tc". However, the main problem is the lack of new ideas for ultra low temperature adventures. For example, concerning NMR in 3He, the strange behavior of the relaxation particularly at 0.4To prevented its use at lower temperatures. Now, when we know about NMR in the Landau field and about spin dynamics with spin supercurrent, NMR again becomes a very useful tool for frontier studies. Recently, in Lancaster, we have observed an unbelievably long induction decay (see Bunkov et al. 1992e). The signal at a frequency of 1 MHz lives for 25 s in a highly inhomogeneous field. We have named this signal persistent, believing that we have seen the first magnetic coherent quantum state under conditions when magnetic interactions with quasiparticles have become negligible. We might suppose that this discovery will open the window of ultra low temperature modeling of elementary particle physics in the regime when quasiparticles no longer destroy the basic properties of the vacuum (condensate). This kind of process was studied theoretically by Volovik (1992). Beyond this main branch of development, there are many other very interesting experiments which can be done using the spin dynamical properties described in this review. First, the HPD appears to be a very useful tool for the study of quantum rotation in 3He-B. The basic motive for conducting HPD experiments in rotating 3He-B was to see the possible increase in magnetic relaxation due to additional magnetic inhomogeneity produced by vortices and counterflow. The first experiments of this kind were done in 1986 by Bunkov and Hakonen (published in 1991). The level of the CW NMR absorption and the dispersion signals from the HPD were studied as functions of rotation and accel-

Ch. 2, w

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153

eration. The large sensitivity of these signals to the acceleration in rotation and to the presence of vortices have been observed. The obvious advantages of the HPD method in comparison to the traditional NMR are: 1. Traditional CW NMR deals with the texture of 3He-B deformed by vortices or counterflow. The HPD produces a nearly homogeneous magnetic texture. Consequently the HPD signal contains information integrated over the whole region covered by the HPD. 2. The absorption and dispersion signals of the HPD contain information about magnetic relaxation and its change with rotation, which cannot be seen by CW NMR due to the inhomogeneous broadening of the line. 3. It is necessary to sweep the full CW NMR line to obtain information for the whole cell. This takes at least 1 min. Consequently, transient processes during a change of rotation speed can only be studied in comprehensive form by monitoring the HPD signal. As was shown experimentally, the HPD can be used for measurements of counterflow of superfluid and normal components of 3He-B. The first very interesting results of investigations of the quantum rotation by HPD were published recently by Korhonen et al. (1989, 1990), Kondo et al. (1991) and Bunkov et al. (1992f). The next very interesting problem is the experimental verification of the spin supercurrent phenomenon as a counterflow of two superfluids with opposite magnetic moments. Up to now this has only been a useful model for realizing mathematical solutions. New experiments, currently being prepared at the Kapitza Institute for Physical Problems, should demonstrate this counterflow as a changing of the Bernouilli pressure in a channel during spin supercurrent flow. Furthermore, many very interesting experiments can be carried out with different modes of spin supercurrent. Particularly, we have seen in the last section of this review, that the spin supercurrent pumps up the magnetization ahead of a moving A-B boundary. It may be possible to design an experiment in which longitudinal NMR and spin supercurrent pumps up the magnetization in the closed end of a cell to achieve new phases of superfluid 3He which may be stable at higher polarizations. There is currently a discussion as to whether a A-phase is indeed the axial phase. This question was posed recently by Gould (1992) as a result of a comprehensive analysis of all experimental results. We can add to this discussion the results of our very old, but unpublished, NMR experiments that possibly can be treated as experimental evidence of the incongruity of the real A-phase with its theoretical image. If the A-phase is indeed the axial phase, then during transverse NMR, the magnetization should precess in an elliptical orbit with orientation determined by the direction of the vector 1. Consequently, one should be able to drive this mode parametrically by a double frequency RF field directed parallel to the external field. The very first NMR experiments on 3He-A by our ,

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group were designed to search for this type of excitation. These experiments were made in a cell with a set of parallel plates, fixing the direction of 1. The usual RF pulses were used to excite the transverse precession. During the induction decay, we applied the parametric excitation field. We did not see any influence of the parametric excitation on the induction decay, while according to calculation, the effect should be order of 10% and our sensitivity was better than 1%. Possibly this negative result should be added to the discussion mentioned above.

Acknowledgments The major part of the experiments described above were done in Moscow at the Kapitza Institute for Physical Problems. Our group was established by academicians P.L. Kapitza and A.S. Borovik-Romanov in 1976 to study 3He by nonlinear pulsed NMR methods such as single pulsed echo (Bunkov et al. 1974), frequency modulation echo (Bunkov and Dumesh 1975), parametric echo (Bunkov 1976), developed for magnetically ordered materials. In addition to Borovik-Romanov and the author of this review, V.V. Dmitriev and Yu.M. Mukharsky joined our group as students at the beginning of the work. Later A. deWaard, D.A. Sergatskov, G.K. Tvalashvili, K. Flachbart and Y. Nieky took part in the experimental investigations. Many investigations were done hand-inhand with theoretical investigations by I.A. Fomin. The very good contact with G.E. Volovik, V.P. Mineev, V.L. Golo, D. Einzel, E. Poddyakova, O.D. Timofeevskaya and other theoreticians was very fruitful. The high skill and excellent hospitality of the Lancaster group led by A.M. Gu6nault and G.R. Pickett give the opportunity to perform NMR experiments as low as 0.12Te. Many experiments not described here, but which had influence on those described, took place in Helsinki in O. Lounasmaa's laboratory. I am very appreciative of all with whom I have had the pleasure to play in this fascinating game, NMR of superfluid 3He. This review have been written at the Kapitza Institute for Physical Problem, Moscow, Russia, at Lancaster University, England and at "Centre de Recherches sur les Tr~s Basses Temp6ratures"- CNRS, Grenoble, France. I am very grateful to D.I. Bradley, V.L. Golo and G.R. Pickett for helpful comments on the text and the Ministry of Education of France for a temporary Professor position in J. Fourier University, Grenoble.

References Anderson, P.W., 1966, Rew. Mod. Phys. 38, 298. Anderson, P.W. and P. Morel, 1961, Phys. Rev. 123, 1911.

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Andreev, A.F. and V.I. Marchenko, 1980, Sov. Phys. Uspekhi 23, 21. Avenel, O. and E. Varoquaux, 1985, Phys. Rev. Lett. 55, 2704. Balian, R. and N.R. Werthamer, 1963, Phys. Rew. 131, 1553. Bardeen, J., L.N. Cooper and J.R. Schriffer, 1957, Phys. Rev. 106, 162, 108. Barton, G. and M.A. Moore, 1974, J. Phys. C 7, 4220. Bogoliubov, N.N., 1958, Nuovo Cim. 7, 794. Borovik-Romanov, A.S. and Yu.M. Bunkov, 1990, Spin Supercurrent and Magnetic Relaxation in 3He, Sov. Sci. Rev. A Phys. 5, Harwood Acad. Borovik-Romanov, A.S., Yu.M. Bunkov, B.S. Dumesh and V.A. Tulin, 1974, Invited talk, 18 congress AMPERE, Proc., Nottingen p. 5. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev and Yu.M. Mukharsky, 1983, JETP Lett. 37, 716. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev and Yu.M. Mukharsky, 1984, JETP Lett. 40, 1033. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharsky and K. Flachbart, 1985, Sov. Phys. JETP 61, 1199. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev and Yu.M. Mukharsky, 1987, JETP Lett. 45, 124. Borovik-Romanov, A.S., Yu.M. Bunkov, A. deWaard, V.V. Dmitriev, V. Makrotsieva, Yu.M. Mukharsky and D.A. Sergatskov, 1988, JETP Lett., 47, 478. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharsky, E.V. Poddyakova and O.D. Timofeevskaya, 1989a, Sov. Phys. JETP 69, 542. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharsky and D.A. Sergatskov, 1989b, Phys. Rev. Lett. 62, 1631. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev and Yu.M .Mukharsky, 1989c, QF&S, AIP Conf. Proc. 194, 15. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharsky and D.A. Sergatskov, 1989d, QF&S, AIP Conf. Proc. 194, 27. Borovik-Romanov, A.S., Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharsky and D.A. Sergatskov, 1990, Physica B 165, 649. Boyd, S.T.P. and G.W. Swift, 1990, Phys. Rev. Lett. 64, 894. Boyd, S.T.P. and G.W. Swift, 1992, J. Low Temp. Phys. 86, 325, 87, 35. Bozler, H.M., M.E.R. Bernier, W.Y. Gully, R.C. Richardson and D.M. Lee, 1974, Phys. Rev. Lett., 32, 875. Bradley, D.I., A.M. Guenault, V. Keith, C.J. Kennedy, I.E. Miller, S.G. Mussett, G.R. Pickett and W.D. Pratt, 1984, J. Low Temp. Phys. 57, 359. Bunkov, Yu.M., 1976, JETP Lett. 23, 244. Bunkov, Yu.M., 1985, in: Low Temperature Physics, ed A.S. Borovik-Romanov. (MIR, Moscow). Bunkov, Yu.M., 1987, Invited talk, LT-18, Jpn. J. Appl. Phys. 26, 1809. Bunkov, Yu.M., 1989, Cryogenics 29, 938. Bunkov, Yu.M. and B.S. Dumesh, 1975, Sov. Phys. JETP 41,576. Bunkov, Yu.M. and S.O. Gladkov, 1977, Sov. Phys. JETP 46, 1141. Bunkov, Yu.M. and T.V. Maksimchuk, 1980, Sov. Phys. JETP 52, 711. Bunkov, Yu.M. and P.J. Hakonen, 1991a, J. LOw Temp. Phys. 83, 323. Bunkov, Yu.M. and O.D. Timofeevskaja, 1991b, JETP Lett. 54, 228. Bunkov, Yu.M. and O.D. Timofeevskaja, 1992e, Phys. Rew. Lett. 69, 3662. Bunkov, Yu.M. and V.L. Golo, 1993a, J. LOw Temp. Phys. 90, 167. Bunkov, Yu.M. and S.O. Zakazov, 1993b, unpublished. Bunkov, Yu.M., B.S. Dumesh and M.I. Kurkin, 1974, JETP Letters 19, 132. Bunkov, Yu.M., M. Krusius and P.J. Hakonen, 1983, JETP Lett. 37, 468; AlP conf. Proc. 103, 194.

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CHAgl~R 3

NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3HE: EXPERIMENTAL AND THEORETICAL CONSIDERATIONS BY

P. SCHIFFER* and D.D. OSHEROFF Physics Department, Stanford University, Stanford, CA 94305, USA

and

A.J. LEGGETT Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

* Present Address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA.

Progress in Low Temperature Physics, VolumeXIV Edited by W.P. Halperin 9 ElsevierScience B.V., 1995. All rights reserved

159

Contents 1. Introduction .......................................................................................................................... 2. Background of the B phase nucleation problem ................................................................... 3. Experimental history of the B phase nucleation problem ..................................................... 4. The recent experiments at Stanford ...................................................................................... 4. I. Experimental design ..................................................................................................... 4.2. Initial B phase nucleation observations ........................................................................ 4.3. B phase nucleation by irradiation ................................................................................. 4.3. I. Data acquisition ................................................................................................. 4.3.2. Dependence on radiation type ............................................................................ 4.3.3. Dependence on temperature and magnetic field ................................................ 4.4. Monte Carlo simulations .............................................................................................. 5. The baked Alaska model: theoretical considerations ........................................................... 6. Conclusions .......................................................................................................................... Acknowledgments .................................................................................................................... Appendix A: Probability of depositing energy E in a radius much less than the "typical" radius R(E) ........................................................................................................................... Appendix B: Relaxation of the magnetization by flow ............................................................ Appendix C: Analytical model of the thermodynamics of superfluid 3He .............................. References ................................................................................................................................

160

161 163 167 170 170 174 177 177 179 181 184 190 200 203 204 206 206 210

1. Introduction

The A and B phases of superfluid 3He are generally believed to correspond to two p-wave BCS states known as the ABM and the BW states, respectively. There have been many comprehensive reviews of the existing experimental and theoretical understanding of the two phases (e.g. Leggett 1975, Wheatley 1975, Anderson and Brinkman 1978, Lee and Richardson 1978, Vollhardt and W61fle 1990) so only a few of the outstanding characteristics of the phases are discussed here. The ABM state consists of equal spin Cooper pairs (up-up and down-down) aligned parallel and antiparallel to an external magnetic field. The free energy of the state is minimized when A phase pairs all have their orbital angular momenta (/) oriented in the same direction, making l a macroscopic quantity for a given sample of A phase. In equilibrium l is oriented normal to surfaces, and, in bulk samples, normal to the spin (field) direction. The order parameter and the energy gap of the A phase are anisotropic, varying across the Fermi sphere like sin 0 where 0 is the polar angle as measured from the direction of 1. The BW state consists of an equal mixture of the three possible symmetric spin states (up-up, down-down, and mixed) for l = 1. Unlike the A phase, the B

161

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Fig. 1. (a) Schematic of the equilibrium phase diagram of superfluid 3He (proportions are not scaled accurately). The top plane is the solid-liquid phase boundary. Notice that the B phase is preferred at lower temperatures than the A phase, but that the presence of high magnetic fields completely excludes the B phase (Greywall 1986, Osheroff et al. 1987, Hahn 1993). (b) Equilibrium phase diagram of superfluid 3He emphasizing the phase boundary between the A and B phases (reprinted from Hahn 1993). phase consists of Cooper pairs with an isotropic distribution of l, and the B phase energy gap is isotropic. Because of the mixed spin pairs, the B phase also has a lower magnetic susceptibility than the A phase. As shown in fig. l a,b, when liquid 3He is cooled in low magnetic fields, the fluid makes a transition from a normal Fermi liquid to the A phase at a pressure dependent temperature, Tc, where Tc = 2.49 mK at melting pressure (Greywall 1986). Both the A and B phases share a common Tc, as is true for all BCS states with the same orbital angular momentum, but just below Tc the A phase has the lowest free energy and hence is the stable phase. At a lower temperature, TAB (TAB = 1.93 mK at melting pressure) (Greywall 1986), the B phase becomes energetically preferred. Due to the susceptibility difference between the two phases, TAB is a quadratically decreasing function of magnetic field near melting pressure, and the B phase is excluded at all temperatures and pressures in fields above--0.6 T (Hahn 1993). Despite the B phase having a lower energy below

Ch. 3, w1 NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

163

the equilibrium transition temperature TAB, the first order AB transition does not ordinarily take place until the superfluid is much colder. This strong supercooling of the A phase has been observed since the first experiments (Osheroff 1972) that demonstrated the existence of the two phases. Experimentalists have found that the transition is supercooled by differing amounts each time an A phase sample is cooled, and that the level of supercooling varies widely, not only between different sample cells and as a function of pressure and magnetic field, but, in some cells, between different cooling runs from the normal phase in the same cell (Osheroff 1972, Alvesalo et al. 1973, Paulson et al. 1974, Hakonen et al. 1985, Swift and Buchanan 1987, Buchanan et al. 1986, Fukuyama et al. 1987). This article is a summary of progress to date in understanding the unique properties of the AB transition. In particular, a complete description is given of the recent experiments at Stanford University. Many of the results have appeared elsewhere: for theoretical background see Leggett (1992) and Leggett and Yip (1990) and for experimental results see Schiffer et al. (1992a, 1993) and Schiffer (1993).

2. Background of the B phase nucleation problem Supercooled first order phase transitions are most simply described in terms of homogeneous or Cahn-Hilliard (CH) nucleation theory in which a bubble of the energetically preferred phase (in this case the B phase) is nucleated by thermal fluctuations within the metastable supercooled phase (in this case the A phase). The energy, A, of the bubble is given by the sum of the positive surface energy per unit area of the interface between the two phases (OAB) and the negative bulk free energy difference per unit volume (AG = GB - GA) between them: A(R) = 4~R2trA8 + (4zt/3)R3AG, where R is the radius of the bubble. As is illustrated in fig. 2, A is such that the bubble is energetically constrained to shrink until it disappears unless R is greater than some critical radius R c = 2trAB/IAGI. For R > Rc the bubble will grow to fill the sample volume. The maximum energy of the bubble, A c = A(Re) = (2.rt/3)IAGI(Re) 3, is thus an energy barrier, which the system must overcome for the phase transition to occur. If the transition is assumed to be nucleated by a thermal excitation as is the usual case, the nucleation rate is proportional to exp(-Ae/kBT). When this theory is applied to the AB transition in superfluid 3He, the dynamics of the transition become quite intriguing. Osheroff and Cross (1977)

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Ch. 3, w

A

ACRc)

"

~ R

Fig. 2. Schematic of the total energy of a bubble of B phase in supercooled A phase as a function of the radius of the bubble. The energy increases with increasing bubble size for R < Re due to the surface tension and then decreases for R > Rc. measured the surface energy, OrAB, as a function of temperature at melting pressure. They also measured the depression of TAB with magnetic field which allows calculation of AG as a function of T and H from the susceptibility difference between the two phases. When their data are combined with measurements of AG, Rc is found to have a minimum value of about 0.45/tm at T = 0. At a temperature of 0.7Tc, where the B phase typically nucleates, Re " 1.5/tm. These values are enormous in comparison to a typical R e of about 50/~ for the waterice transition. More significantly, one finds that Ae/kBT--- 106 for T ~ 1.75 mK, so that the nucleation rate is given by to0exp(-106). Even taking a maximum physically reasonable value for to o ([the number of atoms in the sample] x [a typical atomic frequency] = 10 z3 x 1015 Hz) one would not expect the B phase to ever nucleate in the lifetime of the universe. This reasoning is directly contradicted by experimental observation of the AB transition within minutes whenever the fluid is cooled sufficiently below TAB. Although CH theory fails to predict the nucleation rate for the AB transition, it also fails for many other first order phase transitions. In most of these cases the failure is that the predicted nucleation rate is faster than that observed experimentally. This is due to factors which limit bubble growth for R > Re rather than factors which inhibit the initial nucleation of the bubble. When homogeneous nucleation theory does predict too slow a nucleation rate, for example with martensitic transitions in some metals, the failure can be attributed to crystal lattice defects which do not exist in superfluid 3He. Thus the failure of CH theory for homogeneous nucleation of the AB transition is a unique case and naturally leads to a search for other explanations of how the B phase nucleates. In the special case where the fluid has been warmed into the A phase from the B phase and cooled again, one can imagine that small pockets of the fluid (perhaps protected by some sort of surface topology) might remain in the B phase and

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

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therefore act as nucleation sites. Such "secondary nucleation" can explain the reduced supercooling observed under those conditions (Osheroff 1972, Kleinberg et al. 1974), but cannot explain the more general problem of how the B phase nucleates when the fluid has been cooled from the normal fluid (T > Tc), thus this mechanism is not discussed further. One possible explanation, first suggested by Leggett (1978), is that if the barrier to nucleation cannot be overcome classically, it might be possible to tunnel quantum mechanically from the A to the B phase. A rough estimate of the nucleation rate due to quantum tunneling (Leggett 1992), can be made by replacing the thermal energy kBT with an approximate zero point energy E 0 for a volume of size Re. Although Rc decreases with decreasing temperatures, Leggett found that E0 was so small that the resulting nucleation rate was even less than that expected from thermal fluctuations. A detailed calculation of a transition probability based on this mechanism (Bailin and Love 1980) also confirms the conclusion that such tunneling could not possibly account for the observed nucleation rates. If the AB transition cannot be nucleated in bulk, the next simplest possibility would be nucleation near the surfaces of the containing vessel. Discounting this possibility, however, is the fact that the A phase is preferred near smooth surfaces. The energetic preference is so strong that the AB transition is completely suppressed in liquid 3He which is confined between two narrowly separated fiat plates (Freeman et al. 1988). This effect is easily understood by considering the anisotropic nature of the A phase near surfaces. In the A phase, l can align normal to surfaces so that no Cooper pairs are broken by specular scattering from the walls (diffusive scattering as results when the surface is rough on a scale smaller than the superfluid coherence length will break the pairs). As illustrated in fig. 3a, some pairs are necessarily broken by a surface in contact with the isotropic B phase, requiring an increase in energy. In addition, the superfluid coherence length is shorter along l, so the superfluid wave function can drop to zero at a boundary at the cost of less energy to the A phase where 1 is normal to smooth surfaces. On the other hand, a surface which has local roughness on a scale larger than the coherence length may be less preferable for the A phase near such roughness, since the alignment of I normal to both sides of a sharp protrusion would lead to strong bending energies associated with the rapidly changing direction of 1 near the tip of the protrusion, as is illustrated in fig. 3b. The same problem would occur for a sharp crevice in the surface, and either sort of feature could perhaps reduce the barrier to B phase nucleation by raising the A phase free energy near the roughness. The random nature and the complexity of rough surfaces, however, inhibit calculation of the energies involved or the possible reduction of the nucleation barrier. As discussed below, however, textural singularities in the A phase order parameter can lower the barrier to nucleation so

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APhase

. ~~. . ~ ~ Bl Phase

l~~..

"//W//////////////////2a. The I vector near smooth surfaces in the A and B phases

APhase

b. The ! vector near a surface protrusion in the A

phase

Fig. 3. (a) The orbital angular momentum I near a smooth surface in both the A and B phases. In the A phase all of the pairs have l aligned to minimize pair breaking, while the B phase has isotropic I so that pairs are necessarily broken. (b) The effect of a sharp protrusion on the 1 vector in the A phase. The arrowheads indicate the normal direction to the surface. such singularities caused by arbitrary surface topologies seem to be reasonable candidates for nucleation mechanisms. Another heterogeneous mechanism which could increase the probability for B phase nucleation is based on the existence of textural singularities in the order parameter of the metastable A phase. Exactly how these might aid in nucleation has been detailed in the review by Leggett and Yip (1989), who concluded that, with one exception, they are rather poor candidates for explaining the nucleation puzzle. The exception is a surface singularity called a "boojum" (Mermin 1977, Leggett 1992), which is essentially a convergence of I in the A phase to a single point on the surface, known as the core. The isotropic B phase could possibly nucleate at the core, where the high bending energy required by the convergence of I would make the A phase energetically less favorable. Once nucleated, the same bending energy would make it favorable for the bubble of B phase to grow to a size comparable to R e, after which the bubble's bulk free energy would

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drive further growth. The energetics of nucleation by boojums have been discussed in detail by Leggett and Yip, who concluded that, while they could not strictly exclude this mechanism, every assumption they made in the calculation had to favor boojum based nucleation for it to be a possible mechanism. The final mechanism which has been suggested to account for the nucleation of the B phase is the baked Alaska model, proposed by Leggett (1984, 1985). This model depends on cosmic ray muons, with typical energies of a few GeV, or other forms of ionizing radiation or particles passing through the supercooled A phase. Such radiation produces many secondary electrons with energies of at least a few hundred electron volts which are stopped in the superfluid, warming small volumes (typically on the order of Re) well above Tc. Since the elementary thermal excitations in superfluid 3He are quasiparticles which have mean free paths long compared to R e below TAB and a velocity close to the Fermi velocity, the hot spot should evolve into a shell of quasiparticles traveling radially away from their origin. Inside such a shell, the superfluid would be left with only the ambient number of quasiparticles, which means it would be at the original temperature, well below TAB. Since this whole process happens in a very short time interval, the center would not necessarily form the A phase during the brief time when it was cooling through the temperature regime in which the A phase is stable. Instead, the fluid inside the shell could go immediately into the equilibrium B phase as it returns to the ambient temperature. If the center formed the B phase, it would be protected by the quasiparticle shell from surrounding A phase, and thus would not be forced to shrink by the relatively high surface energy. If the shell protected the B phase center until the radius was larger than R e, the B phase bubble would continue to grow even after the shell had dissipated and the phase transition would take place in the entire sample. The similarity of the hot quasiparticle shell surrounding the cold B phase interior to the gourmet dessert in which ice cream is baked inside a meringue shell led Leggett to name it the "baked Alaska" process. The exact nature of this mechanism is discussed in detail in a later section.

3. Experimental history of the B phase nucleation problem Until very recently, experimental data on the AB nucleation problem have been largely obtained during other experiments on 3He, and they are thus not surprisingly inconclusive. Starting soon after the discovery of the superfluid phases, many experimentalists noted the large and variable supercooling of the transition (Osheroff 1972, Alvesalo et al. 1973, Kleinberg et al. 1974). In 1985, AB transition data from the Helsinki rotating cryostat experiments were compiled for pressures between 18 and 29.3 bars (Hakonen et al. 1985). Their results at 29.3 bars showed a "catastrophe line" at about 0.67Tc, a narrow range of tem-

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peratures (-0.04Tc) in which the B phase always nucleated. The temperature of the catastrophe line, expressed as a fraction of T/Tc, had a strong pressure dependence, increasing with decreasing pressure. They interpreted their data to be in conflict with the baked Alaska model, since that model suggested a broader distribution of nucleation temperatures. Later calculations based on a revised theory (Leggett 1985, Leggett and Yip 1989) demonstrated, however, that the data were not inconsistent with the model. Within a somewhat limited amount of data, they observed no magnetic field dependence to the amount of supercooling between 28.4 and 56.9 mT. They also reported that, within their limited data set, the transition always took place while the sample was being cooled, not while it was in thermal equilibrium although this is probably attributable to the relatively long thermal relaxation time in their sample cell (Leggett 1985). They also detected no correlation between the rate at which the A phase was cooled (between 5 and 29/~K/min) and at what temperature the B phase nucleated. Further data on the transition were taken by Fukuyama et al. (1987) at melting pressure and in zero magnetic field in a sample cell that had a volume of -.-0.1 cm 3 (this was considerably smaller than that of the Helsinki group). They also found (within a rather limited set of data) that the cooling rate did not affect the temperature of B phase nucleation (between 2.3 and 14.9/tK/min) and that nucleation occurred within a somewhat less narrow temperature range (--0.1 To) than the Helsinki group. Furthermore, they compared the nucleation temperature in three different sample cells, each of which contained a different sintered heat exchanger (two made from 3400/~ Pt powder with surface areas of 0.5 and 1.2 m 2 and one made from 1000/~ Ag powder with a surface area of 6.3 m2). Within the 12 nucleations they observed, there was no difference in the nucleation temperatures for the three cells, which they interpreted as evidence against surfaces affecting the nucleation rate (since both the area and the surface topology varied significantly between their different cells). Since the pore size in the sinters must have been much less than Re, however, it is not clear that the full surface areas of the sinters could have aided in nucleating the B phase in the bulk of the samples (even if the B phase nucleated in the pores, the AB interface surface energy would prevent the phase boundary from traveling to the bulk region; see Osheroff and Cross 1977). Thus this result cannot be taken as completely ruling out the participation of surface effects in the AB nucleation process. The only experiments designed specifically to study the AB transition (before the very recent work at Stanford) were conducted at Los Alamos in the late 1980s (Buchanan et al. 1986, Swift and Buchanan 1987, Boyd and Swift 1992, 1993). They were primarily studying the propagation of the phase boundary between the A and B phases, but made several observations in regard to the nucleation problem as well. The Los Alamos sample cells were separated into two sections by a region of high magnetic field, called a "magnetic valve", in

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which the A phase was stable and hence the B phase was energetically excluded, as first suggested by Osheroff. This required the B phase to nucleate separately in each section of the cell. They found that the transition repeatedly occurred at a higher temperature in the section of their cell in contact with the sintered copper heat exchanger, than the other which was enclosed only by machined epoxy or metal. This could possibly be explained by the rough surface of the sinter aiding nucleation, but, since the two sections differed in volume and surface area, and there was no magnetic field applied to the sinter section of the cell while at least a 10 mT field was applied to other section (Buchanan et al. 1986), the correlation is not conclusive. In the sinter region, their data also showed a "catastrophe line" in that there was a rather narrow range of temperatures in which the B phase always nucleated for a given set of conditions (in the sinter-free section of the cell they found a wider distribution of nucleation temperatures; Boyd and Swift 1993), but they found (in contrast to the Helsinki group) that this line depended strongly on magnetic field. In the sinter-free section of the sample cell, based on time-of-flight measurements on the motion of the AB phase boundary, they also observed that the B phase seemed to nucleate preferentially at certain positions in their cell. These results suggested that cell surfaces or some unknown mechanism other than uniformly distributed ionizing radiation was responsible for B phase nucleation. In one experiment (Swift and Buchanan 1987) the Los Alamos group placed particle detectors above and below their sample cell to look for three-way coincidence of B phase nucleation with cosmic rays. They found no significant coincidence between the events, but they could not completely surround the cell with their detectors (-70% of the cosmic rays incident on their sample were undetected) or exclude effects from radiation naturally occurring in the surroundings (e.g. from lac in the epoxy walls of their cell). They interpreted their results as indicating that cosmic rays alone could not nucleate the B phase, however, they did not exclude the possibility that some form of the baked Alaska process was taking place, perhaps in conjunction with surface effects. The Los Alamos group (Boyd and Swift 1993) also conducted a careful study of the B phase nucleation temperature as a function of cooling rate. They warmed their samples (at 29.3 bars and 150 mT) well above Tc for a set amount of time and then cooled them at a constant rate. As shown in fig. 4, they did find that the nucleation temperature depended in a curious way on the cooling rate. When they cooled quickly (--30/tK/min) the B phase nucleated within a narrow temperature range. When they cooled more slowly (~6 /t K/min) the B phase nucleated over a broader range and at lower temperatures. One might expect the exact opposite dependence if the nucleation rate depended on temperature only, since the samples would then spend more time (and thus have a higher probability for nucleation) at the temperature where the nucleation probability was reasonably high. One possible explanation is that the rate at which the sample is

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ch. 3, w

P. SCHIFFER ET AL. 35

' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

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cooled through Te determines the density of textural singularities which aid in nucleation. The difference between this finding and the Helsinki and Tokyo groups' results (that the nucleation temperatures were independent of cooling rate) could be a simple matter of statistics since the Los Alamos group collected many (5-12) data points at each of 5 cooling rates, or of differing cell geometries. The Los Alamos data were also taken in a much higher magnetic field, which might somehow account for the difference.

4. The recent experiments at Stanford

4.1. Experimental design In 1991-1992 a new set of experiments on the AB transition were conducted at Stanford University. Based on previous workers' results, the sample cell (fig. 5) in which these experiments were conducted was designed with the expectation that either surface effects or textural singularities were responsible for nucleating the B phase. As explained above, although the A phase is stable near smooth surfaces, a rough surface topology could be involved in the nucleation process. Consistent with this theory, the different surfaces of various sample cells could possibly account for the poor reproducibility of the AB transition temperature from cell to cell, since microscopic surface irregularities are unpredictable by nature. The presence of singularities in the superfluid order parameter, such as boojums or vortices, also varies with sample cell design and with the rate at which samples are cooled (Mermin 1977, Awschalom and Schwarz 1984), suggesting them as candidate nucleation mechanisms. In order to test whether these mechanisms were responsible for nucleating the B phase,

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experimental regions were created where the AB transition would be suppressed by eliminating the specific factors that could aid nucleation. In order to create an environment where surface roughness could not nucleate the B phase, the 3He was contained in extruded fused silica cylindrical tubes (1 mm i.d.). The surfaces of such tubes are smooth on the scale of 100/~, which is on the order of the coherence length of the superfluid and much smaller than Re. Scanning electron micrographs of the inner surfaces confirmed them to be smooth to at least 200/~ including the sealed ends which had been melted shut (for a detailed description of the sealing technique, see Schiffer (1993)). To exclude the possibility that dust would enter the tubes and create surface roughness, the tubes were flushed in a class 1000 clean room with filtered ethanol using a long-needled syringe. While in the clean room, the open ends of the tubes were capped with 0.1/~m pore filters. This prevented dust from entering the tubes after they were removed from the clean room but retained a relatively large open crossection to provide for thermal contact to the sample through the 3He. To study the AB transition in these tubes, the B phase which nucleated elsewhere in the sample cell needed to be prevented from passing into the 3He in the experimental tubes. This was especially important since the superfluid made thermal contact with the nuclear demagnetization refrigerator through a sintered silver heat exchanger, which necessarily has very rough surfaces. Thus, as is shown in fig. 5, the open ends of the tubes were inserted through holes in a NdFeB permanent magnet. The field inside the holes was measured to be about 0.6 T by measuring ~V dt across a small diameter coil which was pulled rapidly from inside the holes. Since the B phase is excluded from regions of high magnetic field (H > 0.59 T) (Gould 1991), this magnet acted as a "magnetic valve" similar to that of Swift and Buchanan (1987) requiring that the B phase nucleate in each tube separately. The continuous cylinder of superfluid which passed through the magnet assured adequate thermal contact between the sintered silver heat exchanger and the 3He in the tubes. Three tubes were used in the experiment as described below. Each tube had an NMR coil wound on a spool placed around it. The static NMR field was applied in the horizontal plane, perpendicular to the tube axes. Tube 1. This tube was ~ 10 cm long. It was wrapped with about 5 turns/cm of 25/~m platinum-tungsten wire, which allowed heating in order to nucleate solid 3He in it at low temperatures. Tube 2. This tube was ~20 cm long, twice the length of the other two tubes. The bottom end was inserted through a second equivalent permanent magnet. This tube was intended to prevent B phase nucleation through boojums since it is theorized that boojums would preferentially reside in the end of a cylindrical container (which is the point of maximum surface curvature). Since the end was in a region of high magnetic field where the B phase could not nucleate, a boo-

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Fig. 5. Sample cell used in the AB nucleation and NMR experiments. The drawing on the left has been distorted for labeling purposes, while the drawing on the right shows the correct aspect ratio (reprinted from Schiffer et al. 1992a).

jum sitting in its minimum energy end position could not nucleate the B phase in the bulk of the tube where the NMR coil was located. To insure that boojums would migrate to this point, the tube should have been conical, a shape which was unfortunately not experimentally feasible. The lower magnet's dipole moment was oriented parallel to that of the upper magnet so as to reduce the field gradients in the NMR region necessarily caused by the upper valve magnet. All of the NMR coils were precisely centered between the two magnets so that the NMR linewidths would be minimized. Tube 4. This tube was used for thermometry, not for studying the AB transition directly. The bottom 3 cm of the tube were filled with 3 ~ m Pt powder (filling factor --28%) for use as a Curie law thermometer.

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The tubes were epoxied into a nylon feedthrough into which the valve magnet fit tightly. The tight fit was necessary to prevent the magnet from rotating in the external NMR field and breaking the tubes. The nylon body was attached with a grease seal onto a silver feedthrough to a variable volume cell and a capacitance strain gauge, described in detail elsewhere (Feng 1991, Feng et al. 1991), which allowed us to control and measure the 3He pressure to within a millibar. The 3He was thermally linked to a 60 mole copper wire nuclear demagnetization refrigerator through a sintered silver heat exchanger whose characteristics have also been described previously (Feng 1991, Osheroff and Richardson 1985). The fused silica tubes were surrounded by a brass heat shield which also rigidly held the lower permanent magnet against motion in the NMR field. All of the data in these experiments were taken with the NMR coils around the sample tubes and incorporated in parallel within a conventional cw NMR spectrometer to observe the 3He absorption signals. Except where otherwise noted, the data were taken at a magnetic field of 28.2 mT applied normal to the tube axes. This field led to Larmor frequencies of about 914 kHz for the 3He and linewidths in the normal Fermi liquid phase of about 250 Hz FWHM. The external NMR magnet was rotated about the vertical axis (Feng 1991) so that field gradients from the permanent magnets separated the Larmor frequencies of the two sample tubes by about 800 Hz, allowing the two to be observed independently. The temperature was determined from the Pt susceptibility or from the demagnetization field, both calibrated from the A transition in the 3He samples and confirmed by the temperature dependence of the A phase NMR frequency shifts closely matching previous measurements at melting pressure between Tc and 1.5 mK (Osheroff and Brinkman 1974). A complete discussion of the thermometry is given by Schiffer et al. (1992b) and Schiffer (1993). The supercooled A phase and the AB transition were observed through the NMR absorption signal. The susceptibility of the A phase is nearly that of the normal phase and is independent of temperature. The resonant frequency is shifted above the Larmor frequency by a temperature and pressure dependent amount. The shift is almost linear in temperature near Tc and approaches a maximum value at the lowest temperatures of almost 7 kHz (at melting pressure) at 28.2 mT. The B phase has a smaller susceptibility than the A phase and its NMR absorption is spread out over a relatively large range of frequencies by textural effects in the tubes' narrow geometry, with the largest portion of the signal at the Larmor frequency. These two effects combine to make the AB transition correspond to an effective disappearance of the resonance signal from its shifted A phase frequency. Data were taken by sweeping repeatedly over the frequency range that included the shifted A phase resonances from both tubes. The resonant frequencies changed with the temperature of the samples, becoming constant when the

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Fig. 6. Typical data sheet from the AB nucleation experiment. The NMR spectrometer was swept back and forth across a frequency range of 4 kHz that included the resonances of both sample tubes. The pen was slowly moved in the vertical direction so that each successive sweep (which lasted for 5 min) would be offset. Thus the vertical axis corresponds to time. The two NMR signals are from tubes 1 and 2 in the A phase. At the bottom of the figure the changing frequency shift of the signals shows the samples coming into thermal equilibrium (indicated by A), and the disappearance of the signals towards the top indicates that the B phase (with its smeared and relatively unshifted signal) has nucleated in the tubes (indicated by B and C for tubes 1 and 2). The separation in time between the two nucleations demonstrates the efficacy of the top magnetic valve. tubes were in thermal equilibrium. F r o m the temperature d e p e n d e n c e of the shift, one could m e a s u r e the temperature of the A phase (within the N M R coils) i m m e d i a t e l y before the B phase nucleated, even when the samples had not reached thermal equilibrium. S h o w n in fig. 6 is a typical data sheet taken at 34.2 bars and at an equilibrium temperature of ~ 1.1 mK. 4.2. Initial B phase nucleation observations A l t h o u g h the B phase did nucleate in both sample tubes w h e n e v e r the superfluid

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was cooled sufficiently in this cell, the A phase could be supercooled over a broad range of pressures to significantly lower temperatures than had previously been possible. Near the melting pressure, the A phase could be routinely cooled below 0.5Tc (the lowest previously recorded temperature for the A phase (Fukuyama et al. 1987)), and could typically be maintained at T-0.4Tc for several hours before the B phase nucleated. At such pressures the A phase could occasionally be supercooled to temperatures as low as 0.36 mK, or 0.15Tc, where it was stable for up to 30 min (this was the lowest temperature to which the cryostat could cool liquid 3He at melting pressure). The transition was studied at 5, 12, 21, and 29.3 bars as well as near the melting pressure and the supercooling of the A phase (measured against reduced temperature, i.e. T/Tc) decreased with decreasing pressure. This is demonstrated in fig. 7 where the lowest temperature in which the A phase reached thermal equilibrium during these experiments is plotted versus pressure. The filled lozenges in the figure indicate the temperature to which the A phase could be held sufficiently stable to measure the temperature from the Pt thermometer (either in complete thermal equilibrium or in dynamic equilibrium during slow cooling (-0.5-5 ~K/min) at the lower pressures). The filled squares indicate the minimum temperatures to which the A phase was cooled by rapidly decreasing the temperature (-3050/tK/min). The temperatures in these cases are based on the NMR frequency shift in the A phase and actually indicate the maximum temperature of the superfluid in the tubes since the greatest supercooling was in tube 1 where the NMR coil was situated at the end of the tube farthest from the heat exchanger. The temperature dependence of the shift is uncalibrated at these low temperatures but should be roughly proportional to [1 - T/Tc] in the temperature range studied (T> 0.65Tc). The pressure dependence of the supercooling is consistent with the observations of earlier workers who studied the AB transition as a function of pressure, although they saw nucleation at higher temperatures at all pressures as indicated in fig. 7. Without irradiating the samples, several observations could be made as to the nature of the nucleation process. The A phase was less stable in the longer tube (tube 2) than in the shorter tube (tube 1). This was indicated by the B phase nucleating first in tube 2 about twice as often as in tube 1. As had been observed in earlier studies, the B phase would nucleate at a different temperature each time the samples were cooled. If the samples were cooled to the same temperature repeatedly, the time interval before nucleation occurred would vary widely. The B phase nucleated not only while the samples were cooling, but also after they had been in thermal equilibrium for several hours. Nucleation occurred between 0.36 and 1.3 mK near melting pressure, demonstrating a broad temperature range over which the nucleation rate was not vanishingly small, and suggesting that the temperature dependence of the nucleation rate was not as strong as in previous experiments where nucleation occurred in a comparatively

176

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narrow temperature range (Hakonen et al. 1985, Swift and Buchanan 1987). This suggests that perhaps the mechanism responsible for B phase nucleation in the smooth walled tubes was different from that through which nucleation occurred in previous workers' sample cells. The A phase could be supercooled farthest by letting the superfluid come into thermal equilibrium at a temperature where it was quasi-stable against the AB transition, and then demagnetizing rapidly to low temperatures. This effect is demonstrated in fig. 7 by the difference in the temperatures of the filled squares and lozenges as described above. This finding again contradicts the earlier results that the nucleation temperature was independent of the cooling speed, (Hakonen et al. 1985, Fukuyama et al. 1987). Furthermore this is exactly the opposite dependence on cooling rate from what the Los Alamos group observed in their cell, again suggesting the existence of a new nucleation mechanism in the smooth walled cell. The effect of the cooling rate is also a further indication that the probability of B phase nucleation is not an extremely steep function of temperature (this result is demonstrated quantitatively below). Several unsuccessful attempts were made to stimulate the AB transition while the samples were deeply supercooled to about 0.4Tc. High resonant RF levels in the detection coils were used to saturate the A phase NMR line. A sufficiently high voltage was used to heat the samples by hundreds of microkelvin (which may have actually been due to Joule heating in the NMR coil), but there was never any evidence that B phase nucleation was associated with such excitation.

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Other unsuccessful attempts to nucleate the B phase were made by creating acoustical noise and by hitting the cryostat (gently) while the A phase was strongly supercooled. On one occasion, a crystal of solid 3He was grown in tube 1, nucleated by the heater wire wrapped around the tube and observed by the large solid NMR signal at the Larmor frequency. The A phase remained deeply supercooled (--1.15 mK) while the crystal was being grown and while the fluid/solid mixture was in thermal equilibrium for about 30 min. 4.3. B phase nucleation by irradiation 4.3.1. Data acquisition Having failed to observe any relationship between ordinary external stimuli and B phase nucleation, the baked Alaska model was tested. Rather than correlating nucleation with the passage of cosmic ray muons through the supercooled A phase, as had been tried with negative results by Swift and Buchanan (1987), the baked Alaska mechanism was simulated by producing many more high energy electrons in the sample than would naturally occur. A 1.9 mCi 6~ source was obtained which emits gamma rays at 1.17 and 1.33 MeV. Such gamma rays easily penetrate the dewar and the various heat shields of the cryostat, and about 5000 gammas per second were incident on tube 1 with the source in place (Schiffer 1993). While most of the gamma rays would pass straight through the apparatus, a few interacted with the electrons in the fused silica and the 3He samples through Compton scattering and photo-ionization, creating secondary electrons in the 3He. Placing the unshielded 6~ source near the cryostat caused a dramatic reduction in the lifetime, r, of the metastable supercooled A phase. To measure r, the A phase was allowed to come into thermal equilibrium in both tubes. The source was then removed from its shielded container and placed at a fixed position near the cryostat. The time between placement of the source and the AB transition was taken to be the lifetime of that sample in the presence of the source. After nucleation, the samples were warmed well above TAB and usually into the normal Fermi liquid phase. This was done to avoid memory effects in the A phase sample that would alter the nucleation process when they were cooled again for the next measurement of r. Due to the time consuming nature of these experiments, r was measured only near melting pressure, at 34.2 _+ 0.1 bars. The A phase in the presence of the source again displayed a wide range in lifetimes, even when measured at a constant temperature. It was found that for a given temperature, the number of trials for which the A phase remained after a time, t, followed an exponential decay behavior which corresponded to nucleation of the B phase being a single stochastic process. This distribution is such that if one were to start with NO samples of supercooled A phase at time t = 0,

178

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Fig. 8. Exponential distribution of the A phase lifetime at 1.18 mK in the presence of the 60Co source. The dashed line is an exponential fit to the data, the solid line assumes the average of the lifetimes for the slope of the exponential.

then at time t = t0, the number of samples that had not made the AB transition would be given by N = No exp(-to/r) where r is the average of the individual measurements of the A phase lifetime in the limit of No ---) oo. This distribution was observed for the collected lifetimes at each temperature at which samples were exposed to the 6~ source. Thus r(T) is obtained by averaging the individual lifetimes at several temperatures T. A Z 2 test of the exponential distribution using these averaged values for r showed that the fits to the distribution were in the range of 20-60% confidence levels, which is reasonable given the relatively small (10-25) number of data points at each temperature. A comparison of the predicted distribution using this average value of r and an actual data set is shown for T = 1.18 mK in fig. 8. Given the temperature dependence of r (see below), it was necessary to reduce the flux of gamma rays at low temperatures to keep r long enough to be precisely measured. This was accomplished by attenuating the gamma flux with sheets of lead between the source and the cryostat. The known attenuation factor of Pb for 6~ gamma rays was used to correct the measured values of r for the attenuation, assuming that r should be inversely proportional to the flux of gamma rays. This assumption was tested at 1.18 mK, and all of the data are presented as for the gamma flux of the unshielded source. The error in the individual lifetimes was small, typically less than +_5%, except in the occasional trial when the B phase nucleated within a few seconds of putting the source in place.

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

179

Thus the total error in determining r was given by _+r/(No)it2 where No is the total number of measurements of r that were made at a given temperature (note that this expression was misprinted in Schiffer et al. (1992a)). Since No was usually between 10 and 25, the uncertainty is about __.20-30% which is small in comparison with the several orders of magnitude that r varied over the temperature range. With the gamma source in place, r as measured in tube 1 was consistently --2.0 times that measured in tube 2 at the same temperature. This is consistent with the relative size (both volume and surface area) of the two tubes, so the two data sets were combined and are presented here for the size of tube 1. The raw data for r as a function of temperature were adjusted for various imperfections in the data acquisition (these corrections are described in detail in Schiffer (1993)). The temperatures themselves, as taken from the Pt susceptibility, were corrected for a small heat leak into the Pt powder. The values of r were also corrected for heating due to absorption of the 6~ gamma rays, which was evidenced by a decreased shift in the A phase NMR frequency after the source was exposed. A final adjustment was made to correct for small variations in the temperatures (a few/~K) at which the lifetime was measured. The final compiled values for r as a function of temperature are presented in the next section. 4.3.2. Dependence on radiation type The lifetime of the supercooled A phase was measured in the presence of the 6~ source between 0.91 and 1.33 mK as shown in fig. 9. The data fit well to a strong exponential function of temperature as suggested by the baked Alaska model (discussed in the next subsection and shown as a solid curve in fig. 9). In order to quantify the increase in the nucleation rate due the presence of the 6~ source, r was also measured in the presence of only background radiation at a field of 28.2 mT. Even for the lowest temperature at which r was measured, the lifetime in the absence of an additional radiation source was significantly longer than the thermal equilibration time of the tubes. Thus complete thermal equilibrium, defined by a constant NMR frequency, was taken to be the starting point of the lifetime (this assumption should not affect the results due to the stochastic nature of 3). These results are also shown in fig. 9, for a temperature range between 1.2 and 0.87 mK. The curve drawn through the data is the same functional form as was used to fit the 6~ data, multiplied by a factor of 1650. That both data sets can be well fit by the same functional form is important in that it suggests that the same mechanism is responsible for nucleation of the B phase in both cases. This implies that radiation, from either cosmic rays or radioactive decay in materials in or around the sample cell, was responsible for all of the AB transitions observed in the Stanford experiment at 28.2 mT, even those without an additional radioactive source nearby. The relatively weak temperature dependence of r that is observed at low temperatures explains the

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Fig. 9. The lifetime of the metastable A phase as a function of temperature in a magnetic field of 28.4 mT with different ionizing radiation incident on the samples. All three data sets are fit to the same functional form as discussed in the text and shown by the solid lines. These were obtained by fitting the functional form to the gamma ray data and then multiplying by constant factors of 7 and 1650 to fit the neutron and background data, respectively. ability to supercool the A phase the most by cooling the sample quickly, as was discussed above. If the nucleation rate had increased strongly with decreasing T at all temperatures below TAB, supercooling would have been limited to temperatures above which the lifetime became much shorter than the equilibration time and all nucleation would take place within a narrow temperature range. This was suggested to be the case for the Helsinki group' s data (Leggett and Yip 1990) and the explanation for the "catastrophe line" which they observed. The lifetime was also measured as a function of temperature in the presence of a PuBe thermal neutron source (moderated by paraffin). Because of their charge neutrality, upon leaving the paraffin the thermal neutrons traveled easily through the shielding around the sample cell and thus were incident on the samples, with a flux of 2.5 __. 1 thermal neutrons per second incident on tube 1 (Schiffer 1993). Thermal neutrons have a large cross section for capture by 3He through the reaction, 3He + n ~ 3H + H + 0.764 MeV, with an absorption length of about 100/tm. While high energy electrons (such

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

181

as those created by gamma rays or muons passing through the samples) deposit their energy sparsely until the last few keV which are deposited within a few micrometers, the 0.764 MeV kinetic energy is dissipated over a pathlength of about 4 0 / t m by the resultant proton and triton which travel in opposite directions due to momentum conservation. The nature of the microscopic heating of the supercooled A phase due to neutron capture should therefore be very different than that which naturally occurs due to cosmic ray muons, and one might expect that the B phase would somehow be nucleated differently, if at all, by the neutrons. The measured values of r in the presence of the PuBe source, as displayed in fig. 9, were again much shorter than in the absence of a radioactive source. While the PuBe source also produces a small flux of 2.2 MeV gamma rays in addition to neutrons, this was measured to be about a factor of 60 lower than the flux from the 6~ source with a Geiger counter. Furthermore, the Pt thermometer showed essentially no heating in the presence of the PuBe source (<5/tK) while the 6~ source caused hundreds of microkelvin of heating. This confirms that the gamma flux due to the PuBe source was insignificant, and that the shortened r in its presence can be attributed to the thermal neutrons. The data taken with the neutron source are intriguing in several ways. The functional form which is fit to the neutron data in fig. 9 is again the fit to the 6~ data, multiplied by a constant factor of 7. That the nucleation rate should display the same temperature dependence for the two sorts of energy deposition is quite surprising and cannot easily be explained. The fact that the neutrons reduce the lifetime at all is rather conclusive evidence that radiation alone can cause nucleation of the B phase. The additional verification after the observed effect of the 6~ gamma rays was necessary to exclude the possibility that the gamma rays were causing nucleation only through heating of the fused silica tube walls, in which many of the gammas scatter and in which the resultant electrons deposit energy more densely than in the 3He. A further important feature of the neutron data is the rather short lifetime given the low flux of neutrons incident on the samples. That only an average of about 150 incident neutrons were necessary to nucleate the B phase at 1.0 mK rather strongly discounts the possibility that radiation is required to act in concert with some sort of textural singularity for nucleation to take place (since the neutron absorption creates a heated region a few tens o f / t m long and a f e w / t m across, the ~ 150 neutrons necessary to nucleate the B phase only involved about 10-8 of the sample volume). The possibility of this sort of mechanism will be further discussed below. 4.3.3. Dependence on temperature and magnetic field Although the lifetime of the A phase was clearly reduced by the presence of a radioactive source, the question remained as to whether the observed B phase nucleation was, in fact, due to the baked Alaska effect. The strong observed

182

P. SCHIFFER ET AL.

Ch. 3, w

temperature dependence of the A phase lifetime, r, was the obvious choice to compare with the model's predictions. As discussed in section 5, in the baked Alaska model the temperature dependence of r is almost entirely through that of the surface energy and free energy difference between the A and B phases in the form of the critical radius Rc. One can evaluate these energies at a given temperature and magnetic field through previous measurements of the surface tension between the A and B phases and other thermodynamic quantities, and then calculate the expected nucleation rate. The exact temperature and field dependence is, however, quite complicated and does not show exact agreement with the data, so in this section we will revert to the earlier, simpler, functional form suggested by Leggett and Yip (1990) and leave detailed comparison of the data with the theory to be discussed later in the context of the exact theory. The form we will use for the moment is r-

Co exp[a(RclRo)n],

where CO is in seconds, a is a dimensionless parameter, Ro is the value of Rc in the limit of low temperature and low field, and n was estimated to be between 3 and 5. Given that R c = 2O'AB/AG, where O'AB is the surface tension between the two phases and AG is the bulk free energy difference as before, we can use the known temperature dependences of those parameters to fit the above form to our data. We take the temperature dependence of O'ga to be trAa(T)--O'AB(T= 0) ( 1 - T/Tc) u2 which is consistent with the lowest temperature data of Osheroff and Cross (1977). Since the Osheroff and Cross data have only a very small intersection with the temperature range of our data and no other data are available, we cannot verify the accuracy of this assumption. Variation of the functional form of the temperature dependence of OrgB did not, however, significantly alter the results discussed below. We assume A G ( H - 0) is proportional to (1 - T/TAB), to be consistent with the quadratic depression of TAB with magnetic field (Scholz 1981, Osheroff 1972). The field dependence of AG comes mainly from the difference in the susceptibilities between the A and the B phases and is accounted for by the term (Heff/Hc)2 in the final expression: R e = Ro(1 - T / T c ) I I 2 / ( 1

- T/TAB - (Heff/Hc)2).

Here Heff = (ZA/ZB)I/2H, as we assume the nucleation occurs at constant magnetization (see the discussion in section 5 on this point), and H c - 0 . 6 3 T to be consistent with the measured dependence of TAB on magnetic field (Scholz 1981). This is not equivalent to the critical field which is necessary to completely suppress the B phase at T = 0, because AG approaches its T - 0 value as T ~. We use the value of (ZA/ZB) = 3, as determined by NMR measurements which agree with existing theory, even though this differs somewhat from the

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

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measured value of the static magnetization (Hahn et al. 1995). We take R0 = 0.45 p m from extrapolation of the measured values of the surface energy (Osheroff and Cross 1977) and from the critical field (Gould 1991). This expression for R e is probably good to 30% in our temperature range. The A phase lifetimes in the presence of the 6~ source at 28.2 mT were then fit well by the expression

exp[5.25(Re/Ro) 3/2]

r = 0.000211

as shown by the solid line in fig. 9. Although the exponent of 3/2 is far from Leggett and Yip' s estimate of 3-5, the exponential form of the data is consistent with the baked Alaska model, and the data would certainly allow n to be as high as 2 without seriously impairing the fit. Unfortunately, the curvature in the data is sufficiently small that any number of functions of the form exp[f(T)] fit the data adequately. For example, the data can be fit by something as simple as ro exp[aTn], just as well as the above function. However, if r is in fact a function of only Re, the field dependence should be predicted by that of Re. To test this hypothesis, we measured r as a function of temperature at fields of 14 mT and 100 mT as well as 28.2 mT. The upper curve in fig. 10 was determined using the fit to the 28.2 mT data given above

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184

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and our expression for Rc with Heff = 0.173T, consistent with a field of 100 mT and the relative susceptibilities, thus using no free parameters. As is clear from the figure, this agrees well with the 100 mT data, supporting the model and the conjecture that nucleation occurs at constant magnetization. Similar analysis for a magnetic field of 14 mT predicts very little deviation from the 28.2 mT fit in our temperature range as is shown by the dashed curve in fig. 10. This does not agree with our 14 mT results at the higher temperatures where the measured values for r are significantly below the 28.2 mT results. We suspect that in low fields a parallel nucleation mechanism may become important, or that this simple functional form somehow does not properly account for the physics which is taking place (the more exact form given in section 5 does not, however, improve this situation greatly). Regardless of this shortfall, the rather good fit of the data to theoretical expectations is in rather strong support of the baked Alaska model.

4.4. Monte Carlo simulations To better understand the interaction of radiation with the sample cell and samples, the Stanford experiment was simulated with the EGS4 Monte Carlo program, developed at the Stanford Linear Accelerator Center (Nelson et al. 1985). These simulations were intended to determine the rate at which a given flux of radiation creates "baked Alaska events", i.e. deposits a minimum energy in a small volume (~Rc) of the supercooled A phase to raise its temperature well above Tc and thus be a candidate for the baked Alaska process. More importantly, these simulations could be combined with the experimental data to calculate the efficiency t; of the baked Alaska process, defined as the fraction of baked Alaska events which result in nucleation of the B phase. Furthermore, the interaction of gamma rays with the sample cell could be compared with that of cosmic ray muons, and the relative efficiency of the two sorts of radiation in creating the right conditions for the baked Alaska process could be compared with the experimental measurements of the nucleation rates. Due to the complex geometry of close packed tubes surrounded by various heat shields and other parts of the sample cell, only electrons which were created by radiation interacting with the fused silica of an individual tube and with the 3He in that tube (rather than those created in the surrounding materials) were considered. This is not strictly accurate since the radiation will create secondary electrons in the surrounding materials of sufficiently high energy (a few hundred keV) to penetrate the fused silica tubes and deposit energy in the 3He. The error induced by not considering all of the electrons in the final results of the Monte Carlo calculations is at most a factor of 2, which does not significantly alter the results discussed below due to their approximate nature. The EGS4 program simulates photons and electrons traveling through matter,

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

185

given any geometry of matter, the nature of the matter, and an incident flux of either photons or electrons, or both. It cannot simulate heavier particles such as muons, or the protons and tritons which result from neutron capture. Muons, however, interact with the 3He through high energy secondary electrons, so they were simulated by writing separate computer code to calculate the flux and energy distribution of these electrons and then submitting them as incident particles to EGS4. The details of how EGS4 treats the electrons are discussed in Schiffer (1993) and more completely in Nelson et al. (1985). Since the baked Alaska process requires at least a few hundred eV of energy, a cutoff energy of 500 eV was employed so that only electrons which had energy greater than 500 eV were considered. The number of electrons created by ionizing radiation is roughly proportional to the density of the material. Thus many more electrons are created in the fused silica than in the 3He which has about a factor of 20 lower density. All numbers cited are based on simulations of 64372 muons and 106 gamma rays being incident on a 10 cm vertical quartz tube (1 mm inner diameter and 2 mm outer diameter). The gamma rays are assumed to be traveling horizontally, as was the case in the Stanford experiment's geometry, with a flux of 5000 gammas/s on the tube (Schiffer 1993). The muons were taken to have the empirical cos20 angular distribution and a flux of--1.8 muons/s incident on a 10 x 10 cm horizontal square above the tube as is appropriate at sea level (Particle Data Book; Rossi 1948). Correcting the energy distribution for the shielding of the several floors of building above the experiment (taken to be equivalent to 18 inches of concrete), the resultant secondary electrons were calculated using the Bethe-Bloch energy loss formula. We did not consider the incidence of cosmic ray showers, which would have significantly increased the nucleation rate expected from cosmic radiation. The baked Alaska model depends not on the energies of the electrons themselves, but on how their energy is deposited in the superfluid, since, in order for the B phase to nucleate, the model requires a baked Alaska event as described above. As will be shown in section 5, for a volume of superfluid of order (Re)3, the necessary minimum energy (Emin) is a few hundred eV. This assumes that only about 1/3 of the energy deposited by an electron in the liquid helium actually is converted to heat in the liquid, the rest being radiated away in the form of ultraviolet light (Leggett and Yip 1990). To estimate the rate (R) of baked Alaska events from the Monte Carlo, the 3He space in the simulation was divided into 0.5 btm cubes (comparable to Re ~ 0.45/~m at T = 0). As the electrons in the simulation were stepped through the helium, the energy they lost was taken to be deposited in the cube through which they were traveling. A histogram was created of how often different amounts of energy were deposited within cube boundaries. The number of times (N) an amount of energy greater than Emin was deposited into a cube was then equivalent to the number of baked Alaska events that would occur for the radiation flux. Taking Emin = 500 eV and

186

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Cutoff Energy (keV) Fig. 11. The number of baked Alaska events per second (R) predicted by the Monte Carlo simulations as a function of the cutoff energy for 0.5/~m cubes. The data for cosmic ray muons is multiplied by 10 000 in order to fit on the same scale. The horizontal lines indicate the values of R suggested by the number stopped method of analysis (their horizontal position is chosen for comparison with the other data). the fluxes of gamma rays and muons as given above, R was about 170 events per second from the g a m m a rays and about 0.025 events per second from the muons. Using a cube size of 2/~m rather than 0.5/~m increases these numbers by about 30% for the muons and 15% for the gamma rays. As can be seen in fig. 11, however, N is a steeply decreasing function of Emin below 1 keV which implies that R is also a steep function of Emi,. Thus R is only roughly estimated by this method, and an exact efficiency of the baked Alaska process in nucleating the B phase is not determined. These results do, however, indicate that on the order of 100 baked Alaska events are necessary to nucleate the B phase (e ~ 10 -2) since the A phase lifetime was at least 1-2 s in the presence of the 6~ source, even at the lowest temperatures. Another method for estimating R is to count the number of electrons which are stopped in the helium for a given amount of incident radiation (referred to hereafter as the "number stopped method"). The number of electrons stopped in the helium is essentially the same as the number of cubes into which more than a few hundred eV were deposited, since the rate of energy loss by the electrons (dE/dx) was found to be much less than 1 keV/ym for electron energies greater than several keV, and electrons with only a few keV will be stopped within a few micrometers. The characteristic energy loss is demonstrated in fig. 12 which shows the deviation of electrons of various energies as they are stopped by the surrounding liquid 3He. This method has an advantage over the previous method

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

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Fig. 12. How far low energy (keV) electrons deviate from their initial positions as they are stopped in liquid 3He. A distribution of starting energies is shown, but they universally lose the last few keV within a region of--1/~m before stopping. Since the electrons lose energy more slowly at higher energies, this suggests that each electron that is stopped in the helium corresponds to a single baked Alaska event. Such events require -500 eV to be deposited within a fraction of a micrometer. in that there is no possibility of double counting a single event which happens to deposit energy in two different cubes. The number stopped method predicts values for R of 45 and 0.012 baked Alaska events per second for the gamma rays and for the cosmic ray muons respectively indicated by the horizontal lines in fig. 11. These values are somewhat smaller than the other estimates based on Emin = 500 eV, but they do not significantly change the estimate of e. There is no simple way to predict the uncertainty in the "number stopped" values of R, but they are reasonably close to the values obtained through the other method suggesting that the results are at least self-consistent. Although the predictions of R and the efficiency of the baked Alaska process cannot be experimentally verified, one number which can be compared between the simulations and the experimental data is a, the ratio of the rates of baked Alaska events that are expected to occur from the gamma rays to that expected from the cosmic ray muons. When plotted as a function of Emin, as shown in fig. 13, a is independent of Emin below 1.5 keV. Since Emin is likely to be <1 keV, this value of a ~ 6500 seems reasonable. A similar value of a = 3750, obtained from the number stopped method, is also plotted in fig. 13. Experimentally, as discussed above, the ratio between the A phase lifetimes in the presence and absence of the 6~ source (which should be equivalent to a ) was temperature independent at about 1650 as is also shown on the figure. This is a factor of 2-4

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that the ratio is roughly constant for cutoff energies below ~1 keV. The horizontal lines indicate the experimental ratio and the ratio obtained from the number stopped method. below the Monte Carlo value, which is fairly good agreement given the assumptions that entered the Monte Carlo calculations. Since the temperature dependences of the nucleation rates seemed to be identical in the absence and presence of the 6~ source, the discrepancy with experiment is probably not due to some alternate nucleation mechanism which is not based on ionizing radiation. It is most easily explained by the presence of other radiation sources near the samples. For example, all experimental 3He samples are obtained from decaying tritium and thus contain tritium impurities, which beta decay with a half life of about 12 years. Even as low a concentration as one part in 1015, which is well within reasonable expectations for even the cleanest 3He (Buchal et al., unpublished), would be sufficient to account for the discrepancy between the ratios if the tritium was carried into the fused silica tubes. This problem could have been prevented by filling the cell slowly only at very low temperatures so that the tritium would plate out on the walls of the filling capillary, but, during the experimental run in question, 3He was introduced to the cell at 77 K and some 3He remained in the cell while it was cooled to low temperatures. Other radioactive isotopes in the materials surrounding the cell also must have contributed to the relatively high background nucleation rate, although fused silica should be relatively free of such

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

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contamination. As mentioned above, another source of radiation which the Monte Carlo did not take into account was the flux of high energy electrons created when the muons and gamma rays passed through the materials around the tubes. The electron fluxes from the two types of radiation covered quite different energy ranges. The flux from the muons could not penetrate the fused silica tubes, while the flux from the gamma rays could. In addition, the muons and gamma rays were incident with different angular distributions. Thus the resultant errors in the Monte Carlo results are unlikely to cancel. Unfortunately, because of the complex geometry surrounding the tubes, there is no simple way to calculate this contribution accurately. A similar comparison of the Monte Carlo results for the gamma rays to the experimental data can be made from the neutron irradiated data. Since the B phase nucleation rate with the neutrons was much higher (a factor of-230) than with background radiation only, the effects of the poorly understood sources of additional radiation should be minimal in this comparison. The value of R for the neutron irradiation is estimated by Rneutro n =

(neutrons/second) x (baked Alaska events/neutron).

Since the protons and tritons which result from neutrons being absorbed in the 3He are more massive and have higher kinetic energy than electrons, they will deposit energy much more densely and over a longer path than the electrons created by the 6~ gamma rays and cosmic rays. In addition, the resultant secondary electrons have low energies (~ 1 keV) and should be stopped within a few micrometers. Since the resultant heating of the 3He will be more like a "hot sausage" than the point of the idealized baked Alaska model, it is difficult to predict how many baked Alaska events will result from each neutron. One approach is to assume that each charged byproduct will be equivalent to a single baked Alaska event (as in the above analysis of the electrons). This is reasonable since the proton and triton travel in opposite directions, and (for kinematic reasons) the secondary electrons produced by ionization along the path will have maximum energies of ~ 1 keV and will closely follow the path of the proton or triton which created them. In this case each neutron absorbed will produce two events, which is a minimum estimate since the high density of energy deposition could produce multiple baked Alaska events. Using the neutron estimated flux of 2.5 __. 1 neutrons per second incident on the 3He and assuming that all incident neutrons are absorbed due to the large cross section, 5 +_2 baked Alaska events per second would be expected under neutron irradiation. This is a factor of - 7 - 1 5 less than the low estimate of R for the gamma rays and a factor of -25-55 less than the high estimate. The ratio is in reasonable agreement with the factor of 7 difference between the experimental nucleation rates in the presence of the 6~ and PuBe sources, especially considering that the value for

190

P. SCHIFFER ET AL.

Ch. 3, w

Rneutron is almost certainly a minimum estimate. Taking into account the gamma rays which are associated with the neutron source would only improve the agreement with experiment, although the heating caused by the neutron source suggests that this contribution was small.

5. The baked Alaska model: theoretical considerations

The experimental results reported in the previous section prompt two obvious questions: (1) What is the mechanism ("MI") by which gamma rays, neutrons and perhaps background radiation induce the nucleation of the B phase? (2) Is the mechanism ("M2") of the nucleation at higher temperatures which has been seen in all previous experiments a variant (perhaps strongly "assisted") of the "background" form of M1, or something completely different? In either case, why is it not effective in the Stanford experiments? We will devote most of this section to question (1); some brief and speculative remarks about (2) are made in section 6. Whatever the details of the mechanism we have called M1, it appears virtually certain that it is associated not with the "primary" incident particles (photons, neutrons, muons) but with the fast secondary electrons they produce. As evidence for this claim we note (a) the very small cross-sections for those processes of interaction of the primary probe with the liquid which do n o t produce secondary electrons (e.g., for gamma rays, low-momentum-transfer scattering off the 3He nuclei), (b) the similarity in the temperature-dependence of the nucleation rates due to gamma rays, neutrons and background, and (c) the relatively close coincidence, which appears unlikely to be an accident, between the ratio of the nucleation rates observed in the presence and absence of a 6~ source and the calculated ratio of secondary electrons produced by the source and by the cosmic-ray background (see section 4.4). Let us note for future reference some relevant features of the production process (see also section 4): Muons in the 1-2 GeV range (the main component of the cosmic radiation at sea level) produce secondary electrons by ionization (Perkins 1972, ch. 2), with an energy distribution which varies roughly as E -2 dE over the energy range (say ~0.1-10 keV) which will be of relevance in what follows; the mean distance between production events in 3He is about 0.05 cm. (Because of the relatively low energies of the majority of secondary electrons, production in the SiO2 is likely to play a minor role - the vast majority of electrons produced there do not reach the liquid). Gamma-rays in the energy range (~1 MeV) of those emitted in the decay of 6~ produce secondary electrons primarily by the photoelectric and Compton effects (Perkins 1972); in either case the typical energy of the secondary electron is itself of order 1 MeV. Because such electrons have long mean free paths, this means that a significant contribution

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

191

probably comes from those which are produced in the SiO2 walls and thereafter enter the liquid; their energy distribution is a complicated function of the experimental geometry, etc., but we can see intuitively that it is likely to be more heavily weighted towards high energies than in the muon case. Finally, a thermal neutron incident on liquid 3He will rapidly undergo the charge exchange reaction n + 3He ~ 3H + p. The ionization thereafter is produced by both the proton, which carries away about 570 keV in kinetic energy, and the triton (~ 190 keV); the energy distribution in the relevant regime is to a good approximation the same as for the muon, namely dEIE 2, but with the important difference that for kinematic reasons there is an upper cutoff at about 1.1 keV (or less as the particles slow down). On the other hand, the total production is much greater, and in fact the average distance between production events is initially of order 1/~m (and decreases as the particles slow down). In the light of the above it seems reasonable to consider, for a first approach, the problem of the nucleation of 3He-B by a single secondary electron. A conjecture as to how this might happen (the so-called "baked Alaska" scenario) was presented in section 5 of Leggett and Yip (1990), and it appears that with some modifications it may be consistent with the main features of the experimental results given in section 4. Of course, this consistency does not in itself preclude alternative scenarios; however, in view of the failure of attempts (see section 4.2) to induce nucleation of the B phase by various kinds of "diffuse" energy input (heat pulses, mechanical vibration, NMR pulses...), it seems very likely that one feature these must have in common with the "baked Alaska" picture is that the strongly localized nature of the energy input (see below) is an essential ingredient. In the following we first briefly review the original "baked Alaska" scenario in the form presented in Leggett and Yip (1990), and then consider what modifications to it may be necessary. Consider an electron of energy E in the kilovolt range incident on liquid 3He at an initial temperature of 1-2 mK. (As regards electrons of higher energy, such as those produced by gamma rays, we shall assume that their important role begins only when they have been slowed into the kilovolt energy range.) Such an electron will lose energy to the atoms of the liquid by ionization and excitation and will also be scattered by elastic Coulomb collisions. All these processes are of course highly stochastic in nature, and at best we can talk about "typical" behavior. In view of the fact that both the mean free path against large-angle scattering and (approximately) the range scale as the square of the energy, it seems reasonable to assume that the typical radius R of the region in which most of the energy is deposited does so too; thus we assume R(E) = kE2

(1)

and use the results of the detailed calculation of Tenner (1963) to fix the con-

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stant k to be approximately 4000 A keV -2. (Here we note that the molar density of liquid 3He at the melting curve is almost identical to the density of 4He at atmospheric pressure, the system considered by Tenner (1963).) We shall be interested in the following in the probability PE(r) of a secondary electron depositing all or most of its energy in a region r which is small compared to R(E). In Appendix A we show that this is indeed given, up to logarithmic corrections, by the expression which was stated as a plausible ansatz in Leggett and Yip (1990), namely, PE(r) ~ exp[-(aR(E)lr)],

(2)

where a is an unknown numerical constant of order unity. It should however be emphasized that this formula is strictly valid only in the limit R(E)>> r; for R(E) _< r it gives at best a qualitative description. What is the final result of this deposition of energy? For the specific case of a 400 eV electron with a "typical" trajectory ( r ~ R(E)), a detailed calculation has been carried out by Tenner (1963), with the conclusion that after a time which is at most of order 0.5/ts (and could actually be considerably shorter), about 2/3 of the initial energy has been radiated off in the ultraviolet (and hence is of no further interest to us) while the rest is converted into kinetic energy of neutral atoms. We shall assume that the result is qualitatively similar for other energies and trajectories in the regime of interest, even when r << R(E) (since there seems no reason why trajectories having this feature should be in themselves in any way special); note however that for E ~, 10 keV x (r/R) 3 the deposition volume is heated above boiling point, so that new considerations (e.g. involving shock waves) may come in. We will ignore these complications in what follows, assuming that they do not much affect the later stages of the process. So far, everything has been based relatively firmly on known atomic physics. The next stage involves a conjecture - the so-called "baked Alaska" scenario. We assume that the mechanism by which the extreme thermal inhomogeneity created by the secondary electron is dissipated is essentially ballistic in nature, with a "shell" of hot quasiparticles traveling outwards from the hot spot at a velocity nearly equal to the Fermi velocity, and the inside nearly as cold as the original ambient liquid. The feature of this scenario which is crucial for our purposes is that so long as the hot shell remains above the superfluid transition temperature Tc until the interior region has dropped below it, the latter can form an independent matrix for nucleation of the superfluid phase, with no "information" from the outside bulk A phase to tell it which pairing configuration is required. Under these conditions, and given the extremely small volume and ultra-rapid cooling rates involved, a B phase configuration which is initially formed, as it were, by mistake in the interior will find no viable local mechanism in the temperature regime between Tc and the first-order transition TAB to

Ch. 3, {}5 NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

193

convert it to the more stable A phase. Once the temperature of the interior has fallen below TAB, of course, the B phase is the stable one globally. However, if at this stage the bubble is too small, the surface tension associated with the AB interface will outweigh the advantage gained from the bulk energy difference, the bubble will collapse back to zero radius and nothing will have been gained. (For details of the argument, including the appropriate definition of "temperature" in this context, see Leggett and Yip 1990). It is thus essential that the original secondary electron should provide sufficient energy to maintain the hot "shell" of the baked Alaska configuration until the bubble radius is larger than the critical value Rc at which the volume energy difference (proportional to R 3) starts to outweigh the surface tension term (proportional to R2), and beyond which the bubble can expand freely and fill the whole cell. Moreover, in order to get the baked Alaska temperature profile at all, it is essential that this energy be deposited within a radius which is some fraction, say ),, of R e (or more accurately of the maximum baked Alaska radius, see below). It is easy to see that, rather generically, the combination of these two requirements leads to an extremely sharp dependence of the effectiveness of the mechanism on the critical bubble radius R c and hence on the thermodynamic parameters. For suppose that the minimum energy Emin(P) which the secondaryelectron must supply in order to maintain a baked Alaska configuration of radius p with a shell temperature above Tc is given by

En~n(P ) = bp".

(3)

According to eq. (2) above, the probability of depositing the minimum energy necessary to form the critical bubble within the necessary fraction y of Rc is P = exp{--ctR[Emin(Rc)]/YRc}.

(4)

Using eqs. (1) and (3), and introducing the notation,

Ro = (y/ab2k)l/(2n-1),

(5)

we can write eq. (4) in the form

p = exp{_[(RclRo)Zn-1] }.

(6)

As we shall see, the dependence on the thermodynamic parameters of P and hence of the overall nucleation rate comes overwhelmingly from Re, with the variation of Ro playing only a minor role. The argument of the last paragraph is, of course, naive. What we should really do is to consider all possible bubbles of (mean) radius larger than R e, and also all possible shapes, distributions of energy, etc. However, it is easy to con-

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vince oneself that unless the dependence on these variables is very pathological, the effect of such complications will be at most to multiply the expression (6) by an algebraic prefactor whose variation is negligible in the limit Re >> Ro. That this last condition is well fulfilled over at least the bulk of the experimental region is suggested by the fact that even at the lowest temperatures the overall efficiency of the baked Alaska mechanism, as estimated (in section 4.4) from a comparison of the observed nucleation rate with the calculated rate of production of secondary electrons, is only of order 10-l - 10-2 (and for higher temperatures is of course very much less than this); cf. below. Thus, the expression (6) should be a very good approximation for our purposes, except perhaps at the lowest temperatures. Up to now our discussion has been consistent with that of Leggett and Yip (1990). We shall however diverge from the latter in our choice of the parameters n, Ro and Re. Consider first the question of what should be the value of n in eq. (3). In the cited reference it was noted that we should expect this parameter ( - 3 - n in the notation of that reference) to lie in the range of 2-3, and "for concreteness" it was chosen to be 3. However, with the virtue of hindsight it actually seems more consistent with the spirit of the baked Alaska scenario to choose n = 2, on the grounds that it is only the shell of the baked Alaska, not the whole volume, which has to be heated above Tc. In fact, since the shell, to provide effective "insulation" of the interior in the context of its superfluid transition, must have a minimum width of the order of the zero-temperature coherence length S~o, we should expect that the energy Emin required is given by an expression of the form Emin(P) = 12~.6"E(T).~.p 2,

(7)

where E(T) is the energy needed to heat unit volume of the A phase liquid from the starting temperature T to Tc, and 6 is a numerical factor of order of, but probably somewhat larger than, unity. (The right-hand side of eq. (7) incorporates a factor 3 to allow for the fact that only about 1/3 of the original secondary electron energy ends up as kinetic energy of neutral 3He atoms, see above.) Note that at low temperatures, where the temperature variation of ~ is small, eq. (7) also represents, within a numerical factor, the surface tension energy of a quasistatic bubble of B phase of radius p. With the choice (7), the parameter Ro is defined by the expression. Ro = [(7 / ak)(12ztr~oE(T)) -2 ]1/3.

(8)

At the low temperatures of experimental interest the temperature variation of E(T) and hence of R o is small and the dependence on magnetic field is negligible. Thus, we may reasonably estimate Ro from its zero-temperature, zero-field

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

195

value: note that the magnitude of the latter is (at melting pressure) approximately 10r/~m, where the lumped fudge factor r - (71a62) 1r3 is difficult to estimate reliably but is almost certainly considerably smaller than unity. Thus, a reasonable guess is that Ro is of order 1/tm. On the other hand the variation of Ro with pressure is substantial and needs to be taken into account in fitting data taken over a range of pressures. We also note that the critical energy Ec = bRc 2 is of order 256Re 2 eV, where Re is in micrometers. The expression (6) gives (within possible algebraic corrections, see above) the probability of a secondary electron of energy greater than Ee actually forming an adequate baked Alaska configuration. To obtain the nucleation rate for unit incident flux of a particular kind of radiation, we need to multiply by (a) the probability of producing a secondary electron of energy greater than Ec, and (b) the probability that the B phase rather than A phase nucleates as the interior of the baked Alaska cools through Te (and is afterwards locked in). From the considerations given in section 5 of Leggett and Yip (1990) it is plausible that the second factor is approximately independent of all the thermodynamic parameters (indeed, one would guess that it is universally approximately 0.5). The first factor may have a weak dependence on Ec, i.e. on Re, which may be different for the different types of incident radiation, but we shall neglect this since it at most contributes an algebraic prefactor of the type we have already discarded. (In the case of neutrons it is possible that Ee exceeds the kinematic cutoff. On the other hand, under these conditions the production density of secondary electrons is great enough that several may cooperate.) We thus conclude that for a fixed flux of a given type of radiation and at fixed pressure, the dependence of the lifetime 1: on temperature and magnetic field is well approximated by the expression In r = const.

+

(Re(T,H)/Ro) 3.

(9)

The crucial question, therefore, is: What is the dependence of the critical bubble radius on temperature and field? Leggett and Yip (1990) assumed without discussion that Re is just the "Cahn-Hilliard" critical radius, that is, the value of R at which a bubble of B phase in the bulk A phase would be in unstable thermodynamic equilibrium (section 2). As is discussed in section 2, the formula for the Cahn-Hilliard critical radius is Re(cH) = 2t7 AB / AGAB,

(1o)

where AGAB(T,H) is the bulk Gibbs free energy difference between the A and B phases and trAB is the usual equilibrium surface tension. It is easy to verify that substitution of Re(ca) for Re in eq. (9) would give, inter alia, a considerably weaker dependence of In(r) on H than observed experimentally.

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This discrepancy should not disturb us, since with hindsight it is easy to see that the identification of Re with Rc(CH)is in fact quite unrealistic. Let us consider the timescales involved in the baked Alaska nucleation process. As we saw above, the time taken for (part of) the energy of the original secondary electron to be transformed into kinetic energy of neutral atoms is at most of order 0.5 kts. The time taken for the hot shell to expand to a radius of order Re is about Rc/vF, and hence for Re of the order of a few micrometers (cf. below) is of the same order or rather smaller. Finally, the time taken for the superfluid AB interface to form as the shell falls below Tc and back to the original temperature is difficult to estimate, but cannot be many orders of magnitude longer than the normalquasiparticle relaxation time (--4 x 10-8 s at Tc). Thus, the crucial stage at which energetic considerations decide whether the B phase bubble shall contract to zero or expand and fill the whole volume is reached already after a time of the order of a microsecond at most after the incidence of the original secondary electron. This time is long compared to the timescale (presumably --R/cs, where cs is the speed of sound) for establishment of pressure equilibrium between the inside and the outside of the bubble; however, it is certainly short compared to the timescale for relaxation of the magnetization (> 1 ms even in the superfluid phase (Corruccini and Osheroff 1975); for the effects of spatial inhomogeneity, see Appendix B) and may also be comparable to the time needed to establish thermal equilibrium across the interface, cf. below. Thus the competition between the newly formed interior B phase and the external A phase takes place not at constant field but at constant magnetization. Provided that the B phase as well as the A phase susceptibility may be taken as independent of temperature in the region of interest (a good approximation for all the points taken in a 100 mT field, which are the only ones for which the field correction is substantial), the effect of this constraint is to add to the (so far unidentified) zero-field energy difference AEAB(T,0) which replaces AGAB in eq. (10) a term of the form V2(ZA/XB)(ZA-XB)H2, which is a factor XA/,~B- 3 greater than the corresponding term under constant-field conditions. It is convenient to use the fact (Scholz 1981) that the nonlinearity of the susceptibility is very small in the relevant field regime to write this expression in terms of the zero-field, zero-temperature free energy difference AGAB(0) between the two phases and the zero-temperature critical field He (0) (-5.5 kG) for the AB transition; thus, we obtain the result AEAB(T,H) = AEAB(T,0)- AGAB(O)'(H/Hc(O))2(ZAIXB).

(~1)

There is a prima facie difficulty concerning the meaning of the "surface tension" trAB in the equation which replaces (10). As we shall see, the final temperature Tf of the interior B phase, in so far as it exists at all, may be appreciably higher than the initial temperature T, which is presumably still the approximate temperature of the ambient A phase liquid. A complete calculation of the appropri-

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

197

ate "surface tension" under these highly nonequilibrium conditions would appear to be a substantial project in its own right. Fortunately, however, this effect is substantial, if at all, only in the lower-temperature part of the experimental regime, where Tf is small enough that the B phase normal density is small, and calculations (Modgil and Leggett 1993)of the equilibrium surface tension in this regime have shown that its temperature-dependence arises, as one might intuitively expect, overwhelmingly from the A phase side of the interface. Thus, even if such temperature disequilibration occurs, it should be adequate to identify the O'ABwhich appears in the revised eq. (10) with O ' A B ( T , H ) , the equilibrium surface tension at the initial field and temperature. There is at present no direct experimental information on this quantity in the regime of interest; however, theoretical considerations (Modgil and Leggett 1993) indicate that its field-dependence should be very much weaker than that of AEAB(T,H) and therefore negligible for our purposes. Thus we can replace eq. (10) by the equation

Rc(T,H) =

2tTAB(T)/AEAB(T,H ),

(12)

with the field-dependence of AEAB given by eq. (11). Since, moreover, all the factors occurring in the quantity R0 of eq. (8) (including the thermal energy E(T) necessary to warm the A phase from T to Tc) should depend only negligibly on H, we can rewrite eq. (9) in the form In r(T,H) = A' + B'((OAB(T))3.(E(T))2/(AEAa(T,H)) 3 },

(13)

so that the only dependence on field is through AEAB(T,H) as given in eq. (11). We can go a little further if we observe that both OAa(T) and E(T) are expected theoretically to behave at low temperatures as (1 -const.(T/Tc)4), where the constant is in each case close to 1. Thus we may rewrite eq. (13) in the approximate form In r(T,/-/) = A' + B"'(1

-

~(T]Tc)4)5/(AEAB(T,H)) 3,

(14)

where ~ = 1. Although the variation of the numerator is substantially less than that of the denominator, it is sufficiently important to be kept. We must now address the critical question of the identification of the energy difference AEAB(T,0) which drives the expansion of the B phase bubble. To see why this cannot automatically be identified with the ordinary Gibbs free energy difference AEAB(T,0), we use the calculations of the macroscopic thermal resistance of the AB interface given by Yip (1985), together with the standard results (Vollhardt and W61fle 1990) for the B phase specific heat in the lowtemperature limit, to make a rough estimate of the thermal relaxation time r

198

P. SCHIFFER ET AL.

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rth ~ (Aa/kT) 5t2 "R x 10-8 s,

(15)

where AB is the B phase gap and R is in micrometers. This is precisely of the order of the timescales which we have estimated for the "critical stage" of the expansion process! Thus it is not clear, even modulo the point about magnetization discussed above, that thermal equilibrium is attained by the critical stage. On the other hand, in the limit where there is no heat flow at all across the interface, either before or during the critical stage, (and hence Tf > T), it is clear that the relevant AEAB is the enthalpy difference, which under these circumstances is automatically zero, so that no nucleation is possible. Although it is possible to envisage more complicated scenarios (e.g. thermal equilibration before, but not during, the critical stage), the simplest ansatz would therefore seem to be that the "effective" AEAB is, for any given temperature, some fraction 2(7) of AGAa(T,0); according to eq. (15), we would expect 2(T) to be a (fairly weakly) increasing function of T, but obviously its precise form is difficult to calculate. In view of this we shall compare the data with the theoretical predictions under the assumption that 2(T) - 1; that is, we fit the data to the formula In z'(T,H) = A + B[ 1 -

~(T]Tc)4]5/[AGAB(T)

- (cn2)]

3.

(16)

Note that the assumption 2(T)= const. ~:1 would lead (via adjustment of the constant B in eq. (16)) to an identical temperature dependence: however, the "effective" coefficient of H 2 would be modified. Because it is the Gibbs free energy difference which occurs in eq. (16), and this quantity is only --3% of the free energy of either phase separately, it is necessary to treat the thermodynamics of the superfluid phases quite carefully. Unfortunately, direct experimental information on AGAa(T) is not available for the whole of the relevant regime. We therefore use the analytic two-parameter model described in Appendix C, with the two parameters fixed directly from the pressurization-curve measurements which should be very accurate; this model appears to be consistent with all existing data. Since it predicts the value = 1.15 as regards the E(T) contribution, we use this in the fit; however, the latter is only fairly weakly sensitive to the value of ~ when the latter is close to 1. The constant C, which is not a fitting parameter, is determined as explained in Appendix C. Thus, in fig. 14 we show the fit of the gamma-ray data to eq. (16) with ~ = 1.15, C = 0.155 ergs/cm 3 kG 2. The values of A and B, are chosen so as to optimize the fit to the seven 28 mT points plus the two lower-temperature 14 mT ones: (A = 1.02, B = 1.758 cm 9 erg-3); no other free parameters are then used in fitting the rest of the data. While there are some discrepancies, they do not seem to be systematic except at low temperatures, and the general trend of

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He .

61 "U"

....

i ....

tO cO

~1

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~ ....

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o

14.0

mT

d"

~

28.2

mT

/I

i ....

04

199

!

'

E

".~ ~

J~1 0 2 c-

,

0.8

,

,

,

I

,

0.9

,

,

,

I

1

,

,

,

,

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~

1.1

~

,

,

t

,

1.2

,

,

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,

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Temperature (mK) Fig. 14. Temperature dependence of the lifetime of the metastable A phase in the presence of the 60Co source at three different magnetic fields. The curves are fits to the data of the form given in eq. (16).

the results is encouraging; in particular the strong field-dependence lends support to the idea that, whatever the details of the nucleation process, it certainly takes place over a timescale short compared to the magnetization relaxation time. The two higher-temperature points at 14 mT are clearly anomalous from any point of view, and one is tempted to suspect the onset here of a different mechanism, perhaps connected with "M2" and possibly involving textural effects (which are the only features which one would think prima facie would be sensitive to the difference between 14 and 28 mT). At the lowest temperatures the nucleation rate is considerably faster than predicted by eq. (16), and it is tempting to attribute this to the fact that many of our approximations may be expected to break down when exp[-(Rc/Ro) 3] becomes comparable to unity (cf. below); however, given our current complete ignorance of the detailed dynamics of the "baked Alaska" process this must remain a matter of speculation. We do not show a fit to the neutron and background data, since in each case there are only three data points and at least one of them falls in the range where eq. (16) appears to be unreliable. Let us conclude by briefly examining some orders of magnitude and the internal consistency of the scenario proposed. If we take the zero-temperature value of the surface tension CrABtO be the value extrapolated from the highertemperature data, then the quantity Re(0,0) comes out to be of order 1/~m, and from the fit to the data Ro(0) then comes out to be also of order 1/~ - precisely in the range estimated above on a priori grounds. The quantity exp[-(RJRo) 3] is

200

P. SCHIFFER ET AL.

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then small compared to one over most (although not all) of the whole experimental regime, as assumed above. The minimum energy Ec of a secondary electron which can create an adequate baked Alaska is, as above (a few times) 12.7tRc2~o.E(T), and thus is of the order of a few hundred eV as assumed; finally, the "typical" radius R(E) of deposition for this energy is about 4000E 2 ,~ keV -2, so that the condition R(E)>> Re which is assumed in Appendix A should be fairly well fulfilled for at least the higher-temperature nucleations. Thus all the assumptions made in the above argument should be reasonably well fulfilled except at the lowest temperatures, and the scenario is consistent.

6. Conclusions The primary conclusion which can be drawn from the Stanford experiments is that ionizing radiation can lead to nucleation of the B phase in superfluid 3He. That the nucleation rate increased by more than three orders of magnitude when the 6~ source was placed nearby is incontrovertible evidence that radiation has this effect. The further observation of a similar effect in the presence of a thermal neutron source severely limits the likelihood that the observed effect was due to heating of the sample tube walls which accompanies irradiation with the 6~ source. The question does arise as to whether the increased nucleation rate is, in fact, due to the baked Alaska model. While one might argue that radiation could be causing nucleation through some other mechanism, it is difficult to imagine another and none has been proposed in the 18 years that this problem has been outstanding. In any case, such an alternative mechanism would undoubtedly involve much of the physics on which the baked Alaska model is based. Furthermore, the model's prediction of the magnetic field dependence of the nucleation rate was verified between 28.2 and 100 mT without any adjustable parameters. The 14 mT data did deviate from the prediction, but only at higher temperatures which is consistent with the possibility of a parallel nucleation mechanism and not in conflict with the essential correctness of the baked Alaska model. The possibility has been suggested (Buchanan et al. 1986, Leggett and Yip 1990) that the baked Alaska mechanism would require the coincidence of ionizing radiation with some sort of textural singularity in the A phase. While this cannot be ruled out completely as the cause of nucleation in the Stanford experiments, it seems quite unlikely based on the relatively low average number of ionizing events that are necessary for nucleation to occur. At the lowest temperatures, the thermal neutrons would cause nucleation in about 70 s, which corresponds to no more than a few hundred neutrons being incident on the tube. Similarly, while the Monte Carlo simulations cannot calculate the exact number of expected baked Alaska events, again only a few hundred at most are pre-

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

201

dicted to take place within the -2 s lifetime of the A phase at the lowest temperatures. Assuming that nucleation could occur only if the energy deposition were within 1/~m of a singularity, the probability of such a coincidence for a given ionizing particle would be -10-1~ where N was the number of point singularities in the fused silica sample tubes, expected to be at most several thousand. Additionally, based on the equilibrium vorticity in He II (Awschalom and Schwarz 1984) which should be considerably higher than that in the A phase, coincidence of a baked Alaska event with a vortex line is also of negligible importance, even with the much larger probability due to the onedimensional nature of the vortex. The Stanford results certainly do not, however, rule out the possibility of texture assisted radiation based nucleation in other experimental geometries as discussed below. The ability to deeply supercool the A phase in a smooth-walled sample cell was the second major result obtained from these experiments. In every other reported result, experimenters saw the B phase nucleate at temperatures well above those observed in the Stanford experiments. Even taking into account that other sample volumes were up to 100 times larger, radiation alone could not have been responsible for previous nucleation at higher temperatures, due to the steeply increasing values of r observed with increasing temperature. The most significant difference between the Stanford cell and others was the isolation of the samples from rough surfaces, therefore surfaces probably can aid B phase nucleation. Nucleation in those cases could have been associated with spinoidal decomposition occurring in the presence of textural singularities (Leggett and Yip 1990) when the sample temperatures fell below some "catastrophe line." Such singularities could presumably have been caused by rough surfaces or complex cell geometries which were absent from the open fused silica tubes in the Stanford cell. Since radiation has now been shown to definitely nucleate the B phase, a scenario in which radiation interacting with a textural singularity leads to nucleation seems to be as much a possibility as some mechanism through which the singularity alone leads to nucleation (although the former sort of mechanism would require radiation to be always incident rather close to the singularities). As discussed in section 2, calculations suggest that, even in the absence of radiation, the singularity known as a "boojum" could potentially result in a B phase nucleation rate close to that observed experimentally in conventional cells (although every assumption in the calculation needs to favor such nucleation in order to obtain such a result). The open geometry of the Stanford experiments would also minimize the number of singularities: the most stable texture has a single boojum sitting at the closed end of the tube. The supposition that some surface or geometry induced textural singularity aids nucleation is supported by the Los Alamos time-of-flight data (Swift and Buchanan 1987), since they found that the B phase nucleated preferentially at certain positions in their sample cell. This result could also be attributed to a

202

P. SCHIFFER ET AL.

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localized high concentration of some radioactive isotope, such as 14C in the epoxy of their cell, which produced nucleation predominantly at those positions, but the texture-based explanation seems much more probable since one would expect such isotopes to be rather uniformly distributed. Another possible explanation for the B phase not nucleating at higher temperatures in the Stanford sample cell is based on the elongated geometry of the sample tubes. In more open geometries, hydrodynamic heat flow could have created vortex tangles while cooling, which could perhaps have led to nucleation alone or in combination with radiation. The Los Alamos group (Boyd and Swift 1992), however, saw "high temperature" nucleation in their 3 mm diameter magnetically valved sample tubes (with rough walls) at cooling rates of only 25 nK/s, which are too slow to have created vortex tangles, discounting this mechanism as a likely candidate to explain the data. On the other hand, their observed dependence of the nucleation temperature on cooling rate does suggest that A phase textures created while the superfluid is cooled are somehow involved in the nucleation process. Yet another possible high-temperature nucleation mechanism that would not have been active in the Stanford experiments is somewhat related to the baked Alaska scenario. It is likely that in other cells there have always been pockets in the cell surfaces ("lobster pots" in the language of Leggett and Yip (1990)) which, given the various rates of cooling, could independently condense into the B phase on passing through Tc as in the baked Alaska scenario, and thereafter, through isolation from the rest of the sample cell caused by a narrow connecting channel, would preserve it down to a temperature (presumably below TAB) where it becomes energetically favorable for the interface to pop out of the hole. This mechanism, mentioned briefly in Leggett and Yip (1990), would be consistent with Boyd and Swift's (1993) observed cooling rate dependence of the degree of supercooling since faster cooling through Tc would presumably lead to such a situation more frequently. It is, however, somewhat difficult to imagine that all of the sample cells used in previous experiments have had a sufficient number of such "lobster pots" to insure that at least one of them would always be guaranteed to nucleate the B phase - especially taking into consideration that the B phase is depressed in the presence of walls. Furthermore, since Boyd and Swift also saw high temperature nucleation when they cooled the samples extremely slowly, it is difficult to attribute all high temperature nucleations of the B phase to a lobster pot model. A "fringe benefit" of the Stanford results is the ability to deeply supercool the A phase in smooth-walled sample cells. This opens up the low temperature and low field portion of the phase diagram to experimental study of the A phase. Low temperature A phase NMR measurements, made in parallel with the Stanford experiments described in this paper, led to the first confirmation of the theoretically predicted low temperature limiting behavior of an ABM state and

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

203

to the first experimental evaluation of the zero temperature A phase energy gap (Schiffer et al. 1992b). Other experiments which take advantage of the suppression of the AB transition in superfluid contained by smooth walls are currently being planned (Krusius 1993). Although great progress has been made toward understanding how the B phase nucleates, several unanswered questions remain. Primary among these, as indicated above, are the questions of what role rough surfaces play in the nucleation process and what mechanism led to the high temperature nucleation that other groups observed. Future experiments might study how surfaces affect the nucleation rate by containing either an arbitrary rough surface or very well defined surface irregularities within a smooth-walled chamber. A heater could also be included to nucleate solid 3He and study the effect of solidification and melting within the smooth tubes. Such experiments are being designed at Stanford as of the time of this writing. Further investigation in low magnetic fields might lend clues as to an alternate mechanism, since the baked Alaska model does not account for the high temperature results at 14 mT in the Stanford data. Ideally, future experiments might take place deep underground, to reduce the background radiation as much as possible. Care could be taken to freeze tritium impurities out of the 3He, by passing the sample through a 4 K cold trap before allowing it into the fridge, and SiO2 tubes would be used again to minimize radiation from the surrounding cell walls. The nucleation process has only been studied carefully at pressures near melting pressure where the AB transition happens in an intermediate temperature range, well below TAB, but not in the low temperature limit. Future experiments could also be conducted at lower pressures where supercooling of the transition is quite limited and nucleation occurs close to TAB- Such studies could also test the pressure dependence of the nucleation rate suggested by eq. (8) in section 5. Lower pressure experiments would have the advantage of shorter thermal relaxation times due to the higher thermal conductivity and lower heat capacity at lower pressures and to the smaller temperature ranges through which the samples would need to be cycled to "reset" the phase between runs. Clearly much study of this unique phase transition remains to be done.

Acknowledgments The authors are grateful to M.T. O'Keefe and Hiroshi Fukuyama who participated in the Stanford experiments and to M.D. Hildreth for his help in running Monte Carlo programs. We are grateful to S.T.P. Boyd and G.W. Swift for sharing their unpublished data with us, and for detailed descriptions of the Los Alamos experiments. Additional useful discussions were held with B. Cabrera,

204

P. SCHIFFER ET AL.

Ch. 3, w

C.M. Gould, D. Modgil, and J.P. Schiffer. P.S. and D.D.O. were supported by NSF grant DMR-9110423 and P.S. was also supported by AT&T Bell Labs.. A.J.L. was supported by NSF grant DMR-9214236.

Appendix A: Probability of depositing energy E in a radius much less than the "typical" radius R(E) According to standard formulae (Perkins 1972, ch. 2) an electron of energy E large compared to the helium ionization energy but small compared to mc 2 has a mean free path for nuclear Coulomb scattering through an angle greater than O0 given by lc(E) = a ' E 2,

a'=

tan 2 1~0 [ 2 4 ~rnaZ R 2 '

(A.I)

where R o is the Rydberg and ao the Bohr radius (both for hydrogen). The energy loss by ionization is formally described by an "inelastic mean free path"/in(E)E/(-dE/dx) given by /in (E) = f l ' ( E ) E 2 ,

fl'(E) = 1 . ln(E / Eat )-1 _= /30 , ln(E / Eat ) 4 4zna2o R o2

(A.2)

where Eat is approximately 1/4 of the "average" ionization energy of the helium electrons, that is, about 10 eV. Thus, for | = at/2 and E -- 1 keV, we have a ' -16fl' and a ' is, reassuringly, exactly of the order of the value of the coefficient k = R(E)IE 2 which we deduced in section 5 from the calculation of Tenner (1963), namely 4000 ]k (keV) -2. In order to deposit all its energy in a radius r which is much less than R(Eo) (where Eo is its original energy) the electron must repeatedly make short "runs" of length < r terminated by large-angle (O o > ~/2, say) elastic scattering events. In the following we shall assume that r/R(Eo) is small enough that the fractional loss of energy on each run (except perhaps for the last few) is small. Then the probability P(0) of doing this on the first step is given approximately by

P(O) = r/lc(Eo)= r/a'Eo 2.

(A.3)

To calculate the probability P(n) of doing the same on the nth step, we note that we have approximately

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

dE/dn = (-r/~' (E)]E =l.

205

(A.4)

The solution is a logarithmic integral which for large values of mately given by

2nr

2

E2(n) = E 2 - - T - l n ( E o / Eat ) ~ E o ( 1 - n / no), Po

E/Eat is approxi-

(A.5)

where no --floEo21(2r ln(Eo/Eat)) is approximately the total number of runs. Now since/c(E) o~ E-2, we have ln(P(n)/P(0)) = -In(1 -

n/no).

(A.6)

Using eqs. (3) and (6), we see that the total probability P of making no runs of length < r in this way is given by t, no

In P = n o In P ( 0 ) - J01n(l - n / n o )dn = n o (In P(0)+ 1)

=_fl, E2o { l n ( E o / E l ) _ l r

ln(E o / Eat)

1

}

(A.7)

2 ln(E o / Eat)

where El - ( a ' / r ) 1/2. Since r is by hypothesis much less than R(E), which in turn is of order a ' E 2, the quantity Eo/E l as well as Eo/Eat is large compared to 1, and hence to logarithmic accuracy we have Pr(E) = exp{-fl'

E2/r} = exp{-const.(R(E)/r) }

(A.8)

as stated in section 5. The precise definition of R(E) is, of course, irrelevant to the above argument, provided only that it scales as E 2. It should be emphasized that the argument only really makes sense to the extent that the condition no >> 1 is satisfied, so that we should expect eq. (8) to be strictly valid only asymptotically, for the "most improbable" nucleation processes. Clearly a more accurate treatment, either analytical or computational, of the regime relevant to the lower-temperature data is desirable, but in the absence of such a treatment we shall take eq. (8) as at least a qualitative guide there also.

206

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Ch. 3, w

Appendix B: Relaxation of the magnetization by flow The timescale for relaxation of the magnetization quoted in the text (~ 1 ms) is that appropriate to a spatially homogeneous situation. In the scenario envisaged here, the magnetization itself of course starts spatially homogeneous and is presumably left so by the (spin-insensitive) baked Alaska process. Nevertheless, because the "effective field" OEIOMis different for the inside B and outside A phases, it is necessary to consider the effects of possible flow of magnetization in restoring equilibrium. Such flow may be carried either by the normal component or by the superfluid, that is by "spin supercurrents " (Vuorio 1974). Considering first the normal process, we note that the relevant relaxation time ~rM is likely to be considerably larger than the thermal relaxation time rth of eq. (15), both because the contribution of the higher-energy quasiparticles which can easily cross the interface is weighted in rth-~ by a factor of E 2 relative to their contribution to rM-~, and because of the large enhancement of the susceptibility relative to the specific heat. Thus, if thermal (i.e. temperature) equilibrium between inside and outside is attained at all, this happens while the excess B phase magnetization is largely unrelaxed, and the latter, because of the low temperature relative to Tc, is then overwhelmingly carried by the superfluid component, so that further "normal" diffusion is negligible. What of the spin supercurrent process? At first sight, the relevant relaxation time should be of order R/csp, where Csp is the spin-wave velocity; although this time is longer by a factor -3 than the time R/VF for the bubble expansion, it is not necessarily long compared to the overall timescale of the scenario, and thus the assumption of "constant magnetization" would be dubious. However, if we assume that the velocity of magnetization flow through the interface cannot exceed the experimentally observed critical velocity vc --0.02 cm/s of the A phase (Wheatley 1975), then we find that for a field of 100 mT the time tM is substantially longer than the above estimate, in fact >2/~s. Note, moreover, that substantial supercurrent flow can start only when the AB interface is stabilized with a superfluid density of the order of that in the bulk. Thus the assumption made in the text that the bubble expansion takes place "at constant magnetization" seems reasonable.

Appendix C" Analytical model of the thermodynamics of superfluid 3He In this appendix we shall construct a simple analytical model of the thermodynamic properties of the A and B phases at melting pressure. Any such model should ideally be consistent with (at least) the following data: (1) the pressurization-curve measurements of the latent heat and specific heat discontinuity at the AB transition;

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

207

(2) Greywall's direct measurements of the B phase specific heat down to -0.85 mK; (3) the A phase specific heat measurements of Halperin et al. (1976) down to ~ 1 . 2 mK; (4) the AB Gibbs free energy differences as inferred from measurements of the susceptibilities XA, XB(T) and transition field Hc(T); (5) the theoretically predicted value of the A phase specific heat in the lowtemperature limit. We shall regard it as most important to preserve consistency with (1) and (2); as regards (3), the scatter of the A phase data is appreciable, use of (4) requires accurate measurement of the absolute normal-state susceptibility (cf. below), and (5) involves an estimate of the zero-temperature rms A phase gap Ao, which is not directly measured. Nevertheless we shall see that a reasonable degree of consistency with all five constraints is attainable. In the following we assume that the A phase specific heat is not a function of magnetic field, as is strongly suggested by theory and consistent with experiment. It is convenient to measure temperatures t =- T/TAB in units of the first-order (zero-field) AB transition temperature TAB= 1.932 mK, specific heats in units of the specific heat C ~ of the B phase at TAB, and energies in units of C ~ TAB ---100 ergs/cm 3. Then, from fig. (18i) of Greywall (1986), we represent CB(T) in the approximate form Ca(t) = t 3,

t m < t,

CB(t) = t3"(2(t/tm) - 1),

CB(t) = 0,

t < tm/2,

(C. l a)

tin/2 < t < tm

(C.lb)

(C. 1c)

where tm is the value of t at which the specific heat begins to deviate appreciably from a T3 form: by inspection of the cited figure, this is approximately 0.55. Equation. (C.la) slightly underestimates CB in the higher-temperature regime, and, of course, eq. (C.lc) is not strictly correct, but since the true specific heat falls off exponentially at low temperature (except for the very small spin-wave contribution) the corrections due to this should be very small. We now need an ans~itze for the A phase specific heat. For temperatures close to TAB we know from the data of Greywall (1986) and Halperin et al. (1976) that CA(t) is well approximated by the form (1 - a ) t 3, where a is the discontinuity CO - C ~ in the specific heat at TAB in units of C ~ . Also, theory indicates that a t3 form should also be valid in the limit T ~ 0. However, use of the form

208

P. SCHIFFER ET AL.

Ch. 3, w

( 1 - a)t 3 for all T < TAB would lead to thermodynamic inconsistency with the measured value of the latent heat L at the AB transition. Thus we make the ansiitze CA(T) = (1 - a)t 3,

t'm < t,

CA(T) = (1 - a ' ) t 3,

(C.2a)

t < t' m,

(C.2b)

when t' m is a suitably chosen crossover temperature. The choice of t' m most consistent with fig. 5 of Halperin et al. (1976) probably lies in the range 0.650.7, but since the quantity we are eventually interested in, AGAB(T), actually turns out to be very insensitive to small variations in t' m we shall simplify the model by choosing t' m = t m. (Taking t' m= 0.7 actually improves our fit to the data somewhat, but the improvement is only about 5% and does not justify going into this complication here.) Thus consistency of the entropy difference at TAB with the measured latent heat L leads to the relation 3L+a

a'=a-

3 tm

15} 32

(C.3)

and, using the fact that the Gibbs free energies are equal at TAB, we find for the zero-temperature free energy difference AGAB(0) the expression (in units of

C~ TAB)

AGAn(O) = (L + l a l - ( l (a-

31 ")t4. a')'+" 3 ~ ) m

(C.4)

(Taking t m - 0.7 would make only a difference of about 5% in this quantity.) Directly from the melting-curve pressurization data we have a = 0.083 and, if we take CB(O)TAB = 100ergs/cm 3 as above from the data of Greywall, L = 0.0059. Thus we obtain the numerical values a ' = -0.0535,

(C.5)

AGAB(0) = 0.0147

(C.6)

(i.e., in real units, AGAB(0)= 1.47 ergs/cm3). The value of a ' is consistent with theory (1 - a ' = (14zt2/15)(Toa/Ao2Tc)) if Ao/kBTc has the reasonable value 2.02, (a somewhat smaller value (1.85 _+0. l) was estimated by Schiffer et al. (1992b) from their low-temperature NMR data), while the value (6) of AGAB(0) is con-

Ch. 3, w NUCLEATION OF THE AB TRANSITION IN SUPERFLUID 3He

209

sistent with the (reasonable) zero-temperature transition field of 5.5 kG, cf. below. One might ask, by the way, whether it would not be more appropriate, rather than letting the A phase specific heat have the above rather artificial discontinuity at t' m, to choose (say) the continuous form

(c.7)

CA(t) = (1 - a ' ) t 3 - (a - a ' ) t 4.

However, if one chooses this particular form, it is easy to verify that the calculated value of AGAB(0) is about 3.6 ergs/cm 3, which is well outside the bounds set by the susceptibility measurements. With the ans~itze (C.1) and (C.2) we find that for T > Tm we have the simple result for AGAB (in units of CB(O)TAB): AGAB (t)= (L + ffa)(1- t ) - ( l a ( 1 - t 4 )).

(c.8)

This result is of course independent of the choice of Tin. For Tin/2 < T < Tm we have (setting T' m = T m as above)

AGAB (t) = A G A B ( 0 ) +

1 t5

1

10 t m

12

1 tat -

( 2 - a')/4 + -9-6

1

4

3 - ~ tm

(C.9)

with AGAB(0) and a ' given by eqs. (C.4) and (C.3), respectively. For T < Tm/2 we have, of course, simply AGAB(t)= AGAB(0)- (1/12)a't 4, but none of the experimental data points fall in this regime. Finally we must discuss the choice of the coefficient C of H 2 in eq. (16). According to the arguments of the text, we have C = ~(ZA / ZB)(ZA -- Za) ------(ZA / Z B ) ( A G A B ( T ) I H 2 ( T ) )

(c.10)

(the second equality being valid to the extent that nonlinearity of the B phase susceptibility can be neglected (cf. Scholz 1981)), so that in principle we could simply rewrite the denominator of the expression in (16) as AGAB(T)[1(XA/"ZB(T))(H/Hc(T)) 2] and use directly measured values of the Xs and of the AB transition field He(T). However, it is not clear that the published measurements of these quantities are thermodynamically consistent with the specific-heat data. We therefore use the fact that not only XA but also ;tB is nearly constant in this (low-temperature) regime to approximate C by a constant. The exact value of this constant appears to be somewhat controversial at present; in the text we have taken it to be 0.155 ergs/cm 3 kG 2. Given our value of AGAB(0) this corre-

210

P. SCHIFFER ET AL.

Ch. 3, w

sponds to a zero-temperature value of Hc of approximately 5.5 kG, which seems a reasonable extrapolation of the Hc(T) values reported by Scholz (1981). A lower limit on Hc(0) is presumably given by Scholz's largest measured value, 4.77 kG, and the upper limit can hardly be greater than 6.3 kG; while the right hand side of eq. (16) at fixed B is very sensitive to variation of Hc(0) (i.e. of C), for any given Hc(0) in this range adjustment of B (and A) allows a fit to the data which is not much worse than that shown in fig. 14a. It will be of interest to revisit this question, and indeed the whole subject-matter of this appendix, in the light of very recent thermodynamic data (Hahn 1993, Gould 1993) on the A B transition. However, the point we wish to emphasize is that if the factor )(,A/)(,B in eq. (C.10) above is set equal to 1 (corresponding to the hypothesis that the magnetization has completely relaxed, see section 5), then there is no choice of the parameters A and B which will give a reasonable fit to the 1 kG data for any value of He(0) which is even remotely plausible.

References Ahonen, A.I., M. Krusius and M.A. Paalanen, 1976, J. Low Temp. Phys. 25, 421. Alvesalo, T.A., Yu.D. Anufriyev, H.K. Collan, O.V. Lounasmaa and P. Wennerstrfm, 1973, Phys. Rev. Lett. 30, 962. Anderson, P.W. and W.F. Brinkman, 1978, in: The Physics of Liquid and Solid Helium, Part II, eds K.H. Bennemann and J.B. Ketterson (Wiley, New York). Awschalom, D.D. and K.W. Schwarz, 1984, Phys. Rev. Lett. 52 49. Bailin, D. and A. Love, 1980, J. Phys. A 13, L271. Boyd, S.T.P. and G.W. Swift, 1992, J. LOw Temp. Phys. 87, 35. Boyd, S.T.P. and G.W. Swift, 1993, private communication and unpublished. Buchal, C., M. Kubota, R.M. Mueller and F. Pobell, unpublished. Buchanan, D.S., G.W. Swift and J.C. Wheatley, 1986, Phys. Rev. Lett. 57, 341. Corruccini, L.R. and D.D. Osheroff, 1975, Phys. Rev. Lett. 34, 564. Feng, Y.P., 1991, Ph.D. Thesis, Stanford University. Feng, Y.P., P. Schiffer and D.D. Osheroff, 1991, Phys. Rev. Lett. 67, 691. Freeman, M.R., R.S. Germain, E.V. Thuneberg and R.C. Richardson, 1988, Phys. Rev. Lett. 60, 596. Fukuyama, H., H. Ishimoto, T. Tazaki and S. Ogawa, 1987, Phys. Rev. B, 36, 8921. Gould, C.M., 1991, 1993, private communication. Greywall, D.S., 1986, Phys. Rev. B 33, 7520. Hahn, I., 1993, Ph.D. Thesis, University of Southern California. Hahn, I., S.T.P. Boyd, H. Bozler and C.M. Gould, 1995, Proc. Symp. Quantum Fluids and Solids, Physica B, in press. Halperin, P.J., C.N. Archie, F.B. Rasmussen, T.A. Alvesalo and R.C. Richardson, 1976, Phys. Rev. B 13, 2124. Hakonen, P.J., M. Krusius, M.M. Salomaa and J.T. Simola, 1985, Phys. Rev. Lett. 54, 245. Hensley, H.H., Y. Lee, P. Hamot, T. Mizusaki and W.P. Halperin, 1992, J. Low Temp. Phys. 89, 501. Kleinberg, R.L., D.N. Paulson, R.A. Webb and J.C. Wheatley, 1974, J. Low Temp. Phys. 17, 521. Krusius, M., 1993, private communication.

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Lee, D.M. and R.C. Richardson, 1978, in: The Physics of Liquid and Solid Helium, Part II, eds K.H. Bennemann and J.B. Ketterson (Wiley, New York). Leggett, A.J., 1975, Rev. Modem Phys. 47, 331. Leggett, A.J., 1978, in: Proc. 15th Int. Conf. Low Temp. Phys. in: J. Phys. (Paris) Coll. C 6, 1264. Leggett, A.J., 1984, Phys. Rev. Lett. 53, 1096.. Leggett, A.J., 1985, Phys. Rev. Lett. 54, 246. Leggett, A.J., 1992, J. Low Temp. Phys. $7, 57 I. Leggett, A.J. and S.K. Yip, 1989, in: Superfluid 3He, eds L.P. Pitaevskii and W.P. Haiperin (North Holland, Amsterdam). Mermin, N.D., 1977, in: Quantum Fluids and Solids, eds S.B. Trickey, E.D. Adams and J.W. Duffy (Plenum Press, New York). Modgil, D. and A.J. Leggett, 1993, unpublished. Nelson, W.R., H. Hirayama and D.W.O. Rodgers, 1985, The EGS4 Code System, Stanford Linear Accelerator Center Report no. 265, Stanford, CA. Osheroff, D.D., 1972, Ph.D. Thesis, Comell University. Osheroff, D.D., 1974, Phys. Rev. Lett. 33, 1009. Osheroff, D.D. and W.F. Brinkman, 1974, Phys. Rev. Lett. 32, 548 and unpublished. Osheroff, D.D. and M.C. Cross, 1977, Phys. Rev. Lett. 38, 905. Osheroff, D.D. and R.C. Richardson, 1985, Phys. Rev. Lett. 54, 1178. Osheroff, D.D., H. Godfrin and R. Ruel, 1987, Phys. Rev. Lett. 58, 2458. Particle Data Book, part 2, 1992, Phys. Rev. D 45. Paulson, D.H., H. Kojima and J.C. Wheatley, 1974, Phys. Rev. Lett. 32, 1098. Perkins, D.K., 1972, Introduction to High Energy Physics, ch. 2 (Addison Wesley, Reading, MA). Richardson, R.C., 1993, private communication. Rossi, B., 1948, Rev. Modem Phys. 20, 537. Schiffer, P., 1993, Ph.D. Thesis, Stanford University. Schiffer, P., M.T. O'Keefe, M.D. Hildreth, H. Fukuyama and D.D. Osheroff, 1992a, Phys. Rev. Lett. 69, 120. Schiffer, P., M.T. O'Keefe, H. Fukuyama and D.D. Osheroff, 1992b, Phys. Rev. Lett. 69, 3096. Schiffer, P., M.T. O'Keefe, H. Fukuyama and D.D. Osheroff, 1994, Proc. 20th Int. Conf. Low Temp. Phys., Physica B 194-196, 807. Scholz, H.R., 1981, Ph.D. Thesis, The Ohio State University. Swift, G.W. and D.S. Buchanan 1987, in: Proc. 18th Int. Conf. Low Temp. Phys., Jpn. J. Appl. Phys. 26-3, 1828. Tenner, A.G., 1963, Nucl. Instrum. Methods 22, 1. Vollhardt, D. and P. Wtilfle, 1990, The Supeffluid Phases of Helium 3, (Taylor and Francis, London). Vuorio, M., 1974, J. Phys. C 9, L267. Wheatley, J.C., 1975, Rev. Modem Phys. 47, 415. Wheatley, J.C., 1978, Further experimental properties of supeffluid 3He, in: Progress in Low Temperature Physics, Vol. VII, ed D.F. Brewer (North-Holland, Amsterdam). Yip, S., 1985, Phys. Rev. B 32, 2915.

This Page Intentionally Left Blank

CHAgl~R 4

EXPERIMENTAL PROPERTIES OF 3HE ADSORBED ON GRAPHITE BY

H. GODFRIN l Centre de Recherches sur le Trds Basses Temperatures, Centre National de la Recherche Scientifique, Laboratoire associ~ ~ l' Universitg J. Fourier, BP 166, 38042 Grenoble, Cedex 9, France

and

H.-J. LAUTER 21nstitut Max von Laue - Paul Langevin, BP 156, 38042 Grenoble, Cedex 9, France

Progress in Low Temperature Physics, Volume XIV Edited by W.P. Halperin 9 Elsevier Science B.V., 1995. All rights reserved 213

Contents 1. Introduction .......................................................................................................................... 2. Graphite substrates ............................................................................................................... 2.1. Exfoliated graphite ....................................................................................................... 2.2. Physical properties of exfoliated graphite .................................................................... 2.2.1. General properties of different exfoliated graphites ........................................... 2.2.2. Chemical impurities ........................................................................................... 2.2.3. Structural properties ........................................................................................... 2.2.4. Specific area ....................................................................................................... 2.2.5. Electronic properties .......................................................................................... 2.2.6. Specific heat ....................................................................................................... 2.2.7. Electrical conductivity ....................................................................................... 2.2.8. Thermal conductivity ......................................................................................... 2.2.9. Magnetic susceptibility ...................................................................................... 3. Physical adsorption of 3He on graphite ................................................................................ 3.1. Adsorption potentials .................................................................................................... 3.2. Interaction potential and zero point energy in adsorbed layers .................................... 3.3. Layering ........................................................................................................................ 3.4. Coverage scales ............................................................................................................ 4. Experimental techniques of surface Physics at low temperatures ........................................ 4.1. Experimental details ..................................................................................................... 4.1.1. Experimental cells .............................................................................................. 4.1.2. Preparation of the adsorbed 3He sample ............................................................ 4.2 Adsorption isotherms ..................................................................................................... 4.3. Heat capacity ................................................................................................................ 4.3.1. Guide to the literature ........................................................................................ 4.3.2. Techniques ......................................................................................................... 4.4. Nuclear magnetic resonance ......................................................................................... 4.4.1. Guide to the literature ........................................................................................ 4.4.2. Techniques ......................................................................................................... 4.5. Neutron scattering ......................................................................................................... 4.5.1. Guide to the literature ........................................................................................ 4.5.2. Techniques ......................................................................................................... 4.6. Other techniques ........................................................................................................... 5. Structure and phase diagram of the adsorbed films .............................................................. 5.1. Submonolayer coverages .............................................................................................. 5.1.1. Very low coverages ............................................................................................ 5.1.2. The fh'st layer fluid phase .................................................................................. 5.1.3. The commensurate phase ................................................................................... 5.1.4. The intermediate coverage region ...................................................................... 5.1.5. The incommensurate phase ................................................................................ 5.2. Second layer ................................................................................................................. 5.2.1. The second layer fluid phase .............................................................................. 5.2.2. Second layer solidification ................................................................................. 5.2.3. The second layer commensurate phase R2a ....................................................... 5.2.4. Remarks about the second layer density ............................................................ 5.2.5. The second layer intermediate region (0.178 ]k -2 to 0.26/~-2) ......................... 5.2.6. The second layer incommensurate phase above n = 0.26/~-2 ........................... 5.3. Multilayer films ............................................................................................................ 6. Conclusions .......................................................................................................................... References ................................................................................................................................

214

215 215 216 217 217 217 218 219 219 220 221 226 228 229 230 233 235 237 240 241 241 245 247 248 248 252 253 253 256 261 261 262 269 270 270 270 272 279 285 288 292 292 296 297 300 301 306 308 312 314

1. Introduction 3He films of atomic thickness adsorbed on graphite substrates exhibit remarkable properties at milliKelvin temperatures, due to the Fermionic character of the 3He atom and to the two-dimensional (2D) nature of these systems. Their study has led to the development of a new research field, low temperature surface Physics, in rapid and continuous evolution. A fascinating variety of structural phases has been discovered and studied, providing results connected to several fields of research: phase transitions, statistical Physics, quantum mechanics of low dimensionality and inhomogeneous systems. In recent years twodimensional nuclear magnets have been discovered, which constitute excellent model systems of two-dimensional ferromagnets and antiferromagnets. Present understanding of the properties of 3He films adsorbed on graphite substrates is based on adsorption isotherms, heat capacity, NMR and neutron scattering measurements. We discuss in this chapter the experimental techniques of surface Physics at milliKelvin temperatures and the structural properties observed in these systems. Nuclear magnetic properties will be discussed in a forthcoming article.

2. Graphite substrates Two-dimensional helium films are obtained by physical adsorption of helium gas onto a solid substrate. High quality substrates are essential to obtain even moderately good results in surface studies. The reason can be easily understood: line defects have a strong influence in two dimensions, since they destroy long range order. If such defects are located, for instance, at an average distance of 100/~, 10% of the atoms will be directly affected in two-dimensions. It is therefore not surprising that early studies of gases adsorbed onto powders, known to have very small crystalline facets, gave results which are still not well understood. A good substrate for low temperature surface Physics studies must satisfy at least four requirements: large surface area, homogeneity, good cleaning characteristics and good thermal properties. Large specific areas are needed to obtain a reasonable signal from adsorbed atoms; typical figures are in the range 11000 m 2 in experimental volumes of 1-100 cm 3. The number of adsorbed atoms is then on the order of 1020, several orders of magnitude below that usually found in condensed matter studies. The second condition excludes samples pre215

216

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

senting different crystallographic facets exposed for adsorption, and hence a complicated distribution of adsorption potentials. Powders of solids with cubic symmetry (like MgO) give excellent results, since all the facets are identical; unfortunately, their thermal properties are not well adapted to very low temperature studies. The other possibility is to use systems where one of the crystalline orientations is dominant: in graphite powders, for instance, the (0002) planes are preferentially exposed for adsorption. Graphitized carbon black has a large specific surface area, and its adsorption potential is rather homogeneous, as evidenced by adsorption isotherm techniques. Even better results were obtained by Duval and Thomy (1964) and Thomy and Duval (1969, 1970) on exfoliated graphite; this material is presently widely used for surface studies. Its properties are discussed in the next section. Cleaning the substrate means removing the adsorbed contaminants (air, water, oil, etc.) from the surface. This is usually done by pumping the experimental cell under a secondary vacuum and at elevated temperatures during several hours. This procedure is very efficient with graphite, which also presents the convenient feature that once cleaned at about 800~ its contamination when exposed to air is small, and can be removed by pumping at room temperature. This is essential for low temperature experiments, where plastic cells and delicate thermometers need to be attached to the sample. Finally, the substrate must have reasonably good thermal properties to be cooled down to milliKelvin or sub-milliKelvin temperatures. This is hardly the case for insulating powders, even compacted: Their thermal conductivity is low and their heat capacity may be high, resulting in large time constants and thermal decoupling under residual heat leaks. Sintered silver powder has been used in very low temperature studies, but it does not present a good homogeneity and it is difficult to clean. Exfoliated graphite can be used at very low temperatures, despite its rather poor thermal conductivity, using adequate techniques.

2.1. Exfoliated graphite Exfoliated graphite is presently the most extensively used material in classical physisorption studies as well as in very low temperature surface physics experiments. It is manufactured by Union Carbide in the United States of America and commercialized under the names of "Grafoir' and "ZYX", and in France by Le Carbone Lorraine under the trade mark "Papyex". Several types of each variety exist, which present substantially different properties. The manufacturing procedure (see for instance Dash 1975) is the following. The base material consists of natural graphite flakes, placed in a strongly oxidizing medium to form an intercalation compound then rapidly heated to exfoliate the material. The expanded material has a very low density (2-6 g/dm 3) and a

Ch. 4, w

3He ADSORBED ON GRAPHITE

217

large specific area (about 80 m2/g). It is subsequently compressed, forming a self sustaining material of very low density. This material is rolled into binderless sheets with a density on the order of 1 g/cm 3, about one-half that of crystalline graphite. The sheets are black, soft, flexible, and little flakes peel off relatively easily using, for instance, adhesive tape. Optical inspection reveals the relatively high degree of orientation of the crystallites. Typical specific areas are in the range 1-20 m2/g. We discuss below the physical properties of exfoliated graphite substrates.

2.2. Physical properties of exfoliated graphite Even though exfoliated graphite cannot be expected to have sample-independent properties, it turns out that the manufacturing procedure seems to give rather reproducible results within some limits. The data given below can therefore be considered as representative. 2.2.1. General properties of different exfoliated graphites GTA grade Grafoil is available in sheets of thickness > 0.1 mm. The apparent density of Grafoil is on the order of 0.9 g/cm 3. The material is relatively inexpensive. Papyex "N" exists in several forms. As seen later, its physical properties are better than those of Grafoil for surface studies, but it is substantially more expensive. It is available in foils and rolls of thicknesses from 0.2 to 3 mm. Its apparent density is about 1.1 g/cm 3. This value increases under pressure: 1.5 g/cm 3 at 100 bar, and 2 g/cm 3 at 400 bar. UCAR-ZYX is a high quality exfoliated graphite, particularly expensive. This material is produced from stress-annealed highly oriented pyrolitic graphite, exfoliated but not severely rolled. It is available in the shape of plates of thickness 0.6 mm. Its apparent density is on the order of 0.5 g/cm 3. Foam is a low density exfoliated graphite, obtained at an intermediate stage of the preparation of Grafoil, before the final rolling procedure. Its apparent density is on the order of 0.2 g/cm 3. 2.2.2. Chemical impurities Grafoil: according to the manufacturer, typical levels of impurity are (in ppm) A1 = 40, Mn = 30, Fe - 20, Si = 80, with a total ash residue of 0.1%. Bretz et al. (1974) found smaller values: Si = 10, Mn - l, Fe = 10 and Cu = 1. The impurities are mainly in the form of inclusions in the initial material, and not at the surface (Dash 1975). Surface analysis (R.E. Rapp, private communication) detects an amount (not quantitatively determined) of contaminants probably introduced during the exfoliation process (Cl, Fe, Zn).

218

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Papyex: the maximum ashes residue is given by the manufacturer as 0.1%, and the maximum content of chlorides is 30 ppm. 2.2.3. Structural properties The microscopic structure of the crystallites is that of graphite: the atoms are arranged in hexagonal layers with an interlayer distance of 3.37/~. The nearest neighbor distance is 1.42/~. The stacking of the layers is of the type abab, but stacking faults are frequently found. Exfoliated graphite is a compressed powder of these graphite crystallites. Due to the lamellar nature of graphite, the aspect ratio of the crystallites is large, with large dimensions for the (0002) planes. Exfoliated graphite is indeed a stack of flakes of different sizes and orientations. Typical lateral dimensions of crystallites according to electron microscope pictures are on the order of a few thousand Angstroms, and their thickness, determined by neutron scattering is on the order of a few hundred ]kngstroms (Kjems et al. 1976; Schildberg and Lauter 1989). The voids in Grafoil are thin and parallel to the sheets, as evidenced by NMR experiments (frequency shift) on protons in ethyl alcohol soaked Grafoil (Hickernell et al. 1974). Electron microscope pictures show a preferential alignment of the basal plane parallel to the exfoliated graphite sheet surface and this can be quantitatively studied by neutron diffraction. The degree of orientation of the crystallites depends on the type of exfoliated graphite, and the same is true for the size of the atomically flat regions within each crystallite. There is however no preferential orientation for the directions perpendicular to the c-axis. We provide hereafter typical values for the structural properties of exfoliated graphites; details about their experimental determination are given in section 4.5. The distribution of orientation of the c-axis angle with respect to the normal to the exfoliated graphite sheets (mosaic spread) is found to be on the order of 30 ~ for Grafoil and for Papyex, and 10~ for ZYX. The fraction of randomly oriented crystallites is on the order of 50% for Grafoil (Kjems et al. 1976; Schildeberg and Lauter 1989) as seen by neutron scattering and also by NMR lineshifts in 3He measurements at ultralow temperatures (Godfrin et al. 1988a, and unpublished). This point is discussed further in section 4.5. An important parameter for surface physics studies is the quality of the sample surface at a microscopic level. The relatively large crystallites present surface defects, and the dimensions of the "microscopically fiat" regions are much smaller, although considerably larger than those found in other powders. The typical length is called "coherence length", and it can be measured by neutron diffraction experiments on adsorbed gases in the solid phases, and in particular in the commensurate phases where the substrate dominates the coherence of the diffraction spectra. Typical coherence lengths are 200/~ for Grafoil, 1900/~ for ZYX and 300 ,& for Papyex.

Ch. 4, w

3He ADSORBED ON GRAPHITE

219

2.2.4. Specific area The specific area of Grafoil (GTA grade Grafoil is typically 18-24 m2/g. Note that in early experiments (Bretz et al. 1973) values around 24 m2/g were found, whereas recent measurements on the material before bonding (Greywall 1990 and H. Godfrin, unpublished) yield values on the order of 18 m2/g, probably showing an evolution in the manufacturing process. The procedure described in section 4.1 allowing bonding of exfoliated graphite to metallic foils causes a reduction of the specific area; the value 17.1 m2/g has been observed for Grafoil bonded to copper (Franco 1986; Rapp and Godfrin 1993) and 14.0 m2/g by Greywall (1990) for Grafoil on silver. The surface area reduction is not surprising, considering that pressure is used in the process to manufacture Grafoil to bond the graphite flakes. It is difficult to estimate the surface contamination introduced during the bonding process due to evaporated metal. The quality of the results obtained by Greywall on his low specific surface sample tends to prove that the dominant effect is surface reduction, and not contamination due to the evaporation of the metal, which was a priori more likely with silver than with copper. ZYX has a specific area on the order of 2 m2/g, one order of magnitude smaller than that of Grafoil. The specific area of Papyex is similar to that of Grafoil. The surface area is determined experimentally for a given sample using adsorption isotherms, neutron scattering, heat capacity and NMR techniques, discussed later. When the required accuracy is better than about 5% a more elaborated discussion is needed in order to define the "surface area" of the sample (see section 3.4 on coverage scales). 2.2.5. Electronic properties Graphite is a semi-metal, with rather peculiar properties (Spain 1971, 1981; Ziman 1972; Ashcroft and Mermin 1976; Kelly 1981). To a first approximation each layer is a covalently bonded two-dimensional crystal, only held to the next layer by van der Waals forces; the system has two bands: the lower one is filled, the upper one is empty, and they touch each other at the Fermi level with a zero value of the density of states. This would be a zero gap semi-conductor. In fact, a small interaction between second-nearest planes in the abab structure gives rise to a small overlap of the valence and the conduction band at the Fermi level. The density of states at the Fermi level has thus a small value (McClure 1957) and the carrier density is only of about 3 x 1018 cm -3 for electrons and for holes. The Brillouin zone has the shape of a hexagonal pillbox. Elongated pockets of electrons and holes are located along the six edges, overlapping each other as expected for a semi-metal. It has been pointed out (van der Hoeven and Keesom 1963) that any acceptor or donor states due to physical defects or chemical impurities will shift the Fermi level into regions of higher density of states. The density of states curve is

220

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

steep where the bands overlap, and therefore small changes in the Fermi energy give substantial changes in the density of states at the Fermi level. Stacking faults are expected to reduce the band overlap, due to the reduced coupling between second-neighbor planes. Therroopower measurements at low temperatures by Uher (1982) on Grafoil and ZYX samples indicate an excess of 20% of holes with respect to electrons. The excess carrier density of holes is attributed to impurities introduced during the exfoliation process. Electronic states at the surface are affected by the presence of defects like steps, as evidenced in recent scanning tunneling microscopy (STM). The existence of localized states at the Fermi level has been theoretically calculated (Kobayashi 1993)explaining the observation of anomalous STM pictures. The electronic properties of exfoliated graphite can therefore vary substantially depending on the amount of defaults and impurities of the samples, as observed experimentally in graphite and graphitized carbon black. 2.2.6. Specific heat The specific heat of graphite has been extensively studied more than two decades ago, due to the interesting lamellar structure of this material. Lattice and electronic contributions are expected to be present in the specific heat at low temperatures. For pure natural graphite single crystals the specific heat is given by the expression C = yT + cT3, with an electronic contribution y = 13.8/~J/ mol K and a lattice contribution c = 27.7/~J/mol K4 below 1 K (De Sorbo and Nichols 1958; van der Hoeven and Keesom 1963). The last value corresponds to a Debye temperature of 413 K. In samples of worse quality, the experimental situation is complicated by a large influence of defects and size effects on both terms. The lattice specific heat of graphite samples without stacking faults is expected to follow a T3 behavior below 2 K due to out-of-plane vibrations, a T2 law above 20 K where the frequency spectrum is dominated by longitudinal and transverse modes in the graphite planes, and a transition region in between (van der Hoeven and Keesom 1963). The effect of the stacking faults is to increase the number of low frequency modes, thus increasing the low temperature specific heat, as indeed observed in graphite samples. A similar behavior is observed in Grafoil (Rapp et al. 1979). In the case of graphite samples with small crystallites (on the order of 100/~), an excess heat capacity proportional to the temperature is found; this anomalous contribution has been attributed (Fujita and Bugl 1969) to bending modes of platelike crystallites. Numerical estimates give an order of magnitude similar to that of the electronic contribution. The existence of this type of modes in exfoliated graphite is very likely, given the structure of these materials. It is however difficult to test this hypothesis, due to the uncertainties on the electronic heat capacity.

Ch. 4, w

3He ADSORBED ON GRAPHITE

221

The low temperature (T < 1 K) lattice specific heat is hence usually described by the expression Cph = a T + c T 3, where a and c depend on the presence of structural defects. The conduction electrons specific heat is given by the expression Cel = ~'T. The value of ), for graphite is (van der Hoeven and Keesom 1963) 7/= 13.8/zJ/mol K 2, close to the theoretical band structure calculation @' = 12.6/~J/mol K2). In exfoliated graphite, however, it is not clear that this value can be used due to the modifications in the electronic properties discussed in section 2.2.4. Van der Hoeven et al. (1966) performed experiments on boronated samples (boron acts as an acceptor) in order to test the dependence of the specific heat on the density of states. No clear conclusion could be drawn on the origin of the linear term in the specific heat. An unusual contribution to the heat capacity, with a 7~ dependence, has been observed in a recent experiment at temperatures below I K by Viana et al. (1994). It has been attributed to localized electronic states associated with defects in the crystal structure of Grafoil, similar to those observed in phosphorous-doped silicon. Due to its weak temperature dependence below 0.5 K, this contribution dominates at very low temperatures. A dependence on the magnetic field is expected. The general expression for the specific heat of exfoliated graphite at low temperatures is therefore C -

Cel +

Cph

-

(a + ),)T + b7 ~ + cT 3

(1)

Experimentally, (a + ~,) is found to be equal to 42/zJ/mol K 2, b = 1.2/tJ/ mol Ka§ a = - 0 . 6 and c = 49/zJ/mol K4 for Grafoil in the temperature range 0.1-0.75 K (Viana et al. 1994). Typical specific heat data of graphite samples are shown in fig. 1. 2.2.7. Electrical conductivity Transport properties are strongly affected by the anisotropy of the graphite microscopic structure and by that of the Grafoil macroscopic structure. The electrical conductivity of pure graphite has been the object of several studies (see Spain 1971). The high anisotropy ratio observed between c-axis conduction and that in the basal planes is an intrinsic phenomenon. The electrons have a free motion in the basal planes, but are localized in the layers of carbon atoms. At high temperatures conduction along the c-axis is due to thermal activation, while low temperature conduction is ascribed to extrinsic effects. Structural misalignments in natural graphite crystals have a profound effect on the electrical resistivity (Bhattacharrya and Dutta 1981). Exfoliated graphite is a complicated system from the point of view of its electrical conduction. It can be viewed as a random network of graphite pieces connected through relatively high contact resistances. The majority carriers are

222

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Fig. 1. Specific heat of graphite samples as a function of temperature. (a) Canadian natural graphite; (b) boronated Canadian natural graphite; (c) graphitized lampblack; (d) pile graphite (De Sorbo and Nichols 1958); (e) Madagascar natural graphite (van der Hoeven and Keesom 1963); (f) boronated Madagascar natural graphite (van der Hoeven et al. 1966). Open circles, Grafoil (Rapp et al. 1979); diamonds, Grafoil (Viana et al. 1994). probably holes (Uher 1982), as discussed in section 2.2.5, due to impurities introduced during the exfoliation process. Room temperature values of the resistivity are given by Uher (1982) for different exfoliated graphites in the parallel and perpendicular orientations: Grafoil parallel 1.11 x 10-3 ff~cm; Grafoil perpendicular 1.14 x 10-1 f2cm; Foam parallel 5.87 x 10 .-3 ff]cm; Foam perpendicular 5.48 x 10-2~cm; ZYX parallel 8.16 x 10-4 ~cm. The temperature dependence has been measured in the range 1.5-300 K, both in the parallel and in the perpendicular orientation by Uher and Sander (1983) and Hegde et al. (1973), and at low temperatures (0.1-6 K) in the rolling direction by Rapp et al. (1985). Data are shown in figs. 2 and 3. Below 3 K the data can be represented by the law p = P o § A T . For Grafoil, in the rolling direction, Po is on the order of 0.2 p ~ m (sample dependent, probably associated with a density difference) and A = 2.15 x l0 -8 ~ m / K (same value in both experiments). In the perpendicular direction, for Grafoil, po = 1.938 x 10 -3 ff2m, A =4.267 x 10-6f~m/K; in the parallel direction, for foam P o - 1 . 0 5 x

Ch. 4, w

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3He ADSORBED ON GRAPHITE

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10 -4 f~m, A = 3.396 • 10-7 f~rn/K; in the perpendicular direction, for foam P0 = 9.35 • 1 0 - a r m , A = 2.17 • 10-6Qm/K; in the parallel direction, for ZYX P0 = 1.43 • 10-5ff~m, A = 3 . 0 4 • 10 -Sf~m/K (Uher and Sander 1983). At high temperatures the data (Uher and Sander 1983) can be fitted well by considering two conduction mechanisms acting in parallel: an ordinary metallic mechanism and a hopping-like one with a temperature dependence exp T -~/4. The coexistence of several conduction processes is probably due to the spatial heterogeneity of the material and the large influence of defects. At milliKelvin temperatures an increase of the electrical resistivity due to quantum localization in a disordered conductor has been observed by Koike et al. (1984, 1985) and Rapp et al. (1985). The conductivity at milliKelvin temperatures is proportional to T It2, and the magneto-conductivity varies as H ~/2, features that can be understood in terms of quantum localization and electronelectron interactions (figs. 3 and 4). Note that the electrical resistance is relatively high and almost constant at low temperatures, an important condition for NMR experiments in the milliKelvin temperature range.

226

Ch. 4, w

H. GODFRIN and H.-J. LAUTER

1.008

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Thermal conductivity

The thermal conductivity x of exfoliated graphite (fig. 5) is dominated by lattice excitations above 1 K, and by the electronic contribution below this temperature. It is highly anisotropic: the conductivity along the planes is two orders of magnitude larger than that perpendicular to the planes. The magnitude of the thermal conductivity below 1 K is smaller than 1 mW/Km; due to this very poor conductivity special care must be taken in order to thermalize this material at very low temperatures. The parallel conductivity of Grafoil below 0.7 K is found to be proportional to temperature, with a magnitude in good agreement with that deduced from the electrical conductivity using the Wiedemann-Franz law x = LT/p with the standard Lorentz number L = 2.45 x 10-8(V/K) 2. Above 1 K the in-plane thermal conductivity is due to lattice excitations and is found to be proportional to T 2"76 (Uher 1980; Dillon et al. 1985) (fig. 5). This temperature dependence is similar to that found in pyrolitic graphites (Slack 1962; Klein and Holland 1964; Holland et al. 1966; Nihira and Iwata 1975). The parallel conductivity data of Hegde et al. (1973) do not follow this general trend; they are probably affected

Ch. 4, w

3He ADSORBED ON GRAPHITE

227

Fig. 4. Electrical conductivity of Grafoil in the rolling direction as a function of temperature, for different values of the magnetic field (Koike et al. 1984, 1985).

by a heat leak along the measuring leads (see Uher 1980). Due to the low density of foam its thermal conductivity in the rolling direction is about one order of magnitude smaller than that of Grafoil. The parallel thermal conductivity of ZYX is very close to that of Grafoil. In the perpendicular direction, the smaller density of foam is compensated by a larger degree of misalignment of the crystallites, favoring the conductivity by the contribution of the in-plane conduction mechanism. Similar conductivities are therefore observed for foam and Grafoil in the perpendicular direction (Uher 1980). The perpendicular conductivity data for Grafoil of Hegde et al. (1973) are somewhat lower (fig. 5), but they follow the same general trend, and overlap well with lower temperature data of Dillon et al. (1985). ZYX is a special case; the perpendicular conductivity is dominated by macroscopic defects in the sample. Experimental data are shown in fig. 5. The anisotropy of the thermal conductivity is an important factor for the design of surface Physics experiments at milliKelvin temperatures. Data for Grafoil and Foam are shown in fig. 6 (Uher 1980; Dillon et al. 1985). Note that the conductivity is affected by the density of the sample: Uher's data correspond to a Grafoil sample of density 0.82 g/cm 3, whereas Dillon's sample density is on the order of 1 g/cm 3. The bonding procedure between Grafoil and metals de-

228

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Fig. 5. Thermal conductivity of graphite samples as a function of temperature, in the parallel and perpendicular directions. (a) Grafoil, parallel; (b) ZYX, parallel; (c) Foam, parallel; (d) Foam, perpendicular; (e) Grafoil, perpendicular (Uher 1980); (f) Grafoil, perpendicular (Hegde et al. 1973); (g) Grafoil, parallel; (h) Grafoil, perpendicular (Dillon et al. 1985).

scribed in section 4.1 requires pressure, and is expected to cause an increase in the sample density and thermal conductivity. 2.2.9. Magnetic susceptibility NMR measurements (frequency shift) on protons in ethyl alcohol soaked Grafoil (Hickernell et al. 1974) show that the parallel susceptibility is much smaller than the perpendicular susceptibility; the latter is equal to -1.6 x 10 -4 MKS per unit volume, or -1.73 x 10-5 emu/g; the density used for the unit conversion is 0.73 g/cm 3, determined for their Grafoil sample including the Mylar foil spacers. The density of pure graphite is 2.25 g/cm 3. The perpendicular susceptibility of bulk graphite is dominated by the free electron contribution, and very sensitive to the band structure; it is diamagnetic and large, i.e. on the order of - 3 • 10 -5/g at low temperatures (Ganguli and Krishnan 1941). The parallel susceptibility, on the other hand, is small, close to the free atom v a l u e - 5 x 10-7 emu/g.

Ch. 4, w

3He ADSORBED ON GRAPHITE '

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T (K) Fig. 6. The ratio of the parallel and the perpendicularthermal conductivity as a function of temperature. (a) Grafoil (Uher 1980); (b) Grafoil (Dillon et al. 1985); (c) Foam (Uher 1980). References to the literature on this subject as well as results about other forms of graphite (C6o and nanotubes) are given by Heremans et al. (1994). The NMR line broadening due to the demagnetizing field of the graphite substrate is discussed in section 4.4.

3. Physical adsorption of 3He on graphite Adsorption of atoms onto a solid surface is a general phenomenon due to attractive forces experienced by an atom in the vicinity of the surface of a solid material. Depending of the magnitude and nature of the attractive forces the effect is described as chemisorption or physisorption. In the first case the process involves a transfer of electric charge, and the energies involved are on the order of thousands of degrees Kelvin. Physisorption corresponds to much weaker interactions, and typical energies are on the order of a few hundreds Kelvin. The attractive forces in this case are van der Waals forces due to fluctuating electrical dipole moments. At short distances the interaction is repulsive due to the overlap of electrons from the substrate with those of the adsorbed atom. A detailed discussion of physical adsorption can be found in Steele (1974), Zangwill

230

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

(1988) and Kohn (1990). A review of data on physical adsorption potentials has been recently published (Vidali et al. 1991). A simple empirical potential for the He-graphite interaction has been proposed recently (Joly et al. 1992). Note that the graphite potential for 3He and 4He is the same, since both isotopes are identical from the point of view of their electronic structure; the energy of the bound states, however, will differ due to the mass difference.

3.1. Adsorption potentials A 3He atom located at a distance z from a graphite surface experiences a potential that varies rapidly as a function of distance, especially in the vicinity of the substrate (within atomic dimensions); it has a minimum at a distance z0, and then increases smoothly at large distances. The substrate structure provides an additional dependence of the adsorption potential along the surface; this "corrugation potential" is often described by its Fourier components due to the periodicity of the substrate. The potential minimum for 3He on graphite has a magnitude of about -200 K, and the corrugation potential has its largest Fourier component at a wavevector of about 0.4/~-1 corresponding to the periodicity of the graphite lattice, with an amplitude of about 40 K. Simple approximations have been used to describe the adsorption potentials. Assuming 6-12 Lennard-Jones interactions between the 3He atom and the graphite atoms, and additivity of the pairwise interactions, one obtains after integration over the graphite semi-infinite half space a 3-9 potential

U12 V(z) = 4a'eno'/~"~9 L45Z

or6 1 6Z 3

(2)

as a function of the coordinate z in the direction normal to the substrate. The values of e and tr are usually fit to experimental data or obtained from combination rules using the He-He and C-C interaction parameters (Steele 1974; Cole and Klein 1983). A frequently used expression for the Lennard-Jones adsorption potential (see for instance Vidali et al. 1991, Treiner 1993, Cheng 1993) is

V(z) =

4 C a / 27 D2

Z9

C3 , Z3

(3)

where D is the depth of the potential well and C3 the van der Waals adsorption strength parameter.

Ch. 4, w

3He ADSORBED ON GRAPHITE

231

Bound state resonances for both helium isotopes on a graphite substrate can be determined experimentally using atomic collision and spectroscopic data (for 3He, see Derry et al. 1979, 1980). Several resonances are observed; disagreements between the experimental data and calculated values motivated substantial theoretical research on the adsorption potential of helium on graphite. A detailed review has been given by Cole et al. (1981) (see also Vidali et al. 1983; Chung et al. 1986; Ruiz et al. 1986; Ihm et al. 1987). Recently a simple model potential for helium on graphite has been proposed by Joly et al. (1992) and used in theoretical studies of low density phases of 3He adsorbed on graphite (Brami et al. 1994): V ( z ) = A exp(-az)

C3

C4 ,

Z3

z4

(4)

with A = 195.315 eV, a = 3.715/~-x, C3 = 157.7 meV ,~3 and Ca = 888.07 meV/~4. This potential is shown in fig. 7. The constants have been adjusted to reproduce the experimental spectra for 4He atoms and the spectroscopic properties calculated for 3He are in good agreement with experimental data. Due to the small mass of the 3He atom a substantial kinetic energy remains even in the ground state. This zero point energy is due to the strong localization of the atom in the z direction, according to the Heisenberg uncertainty principle. The adsorption energy is therefore considerably smaller than what could be inferred from the depth of the potential well; the binding energy is calculated to be -11.74 meV (-136.2 K) (Joly et al. 1992; Brami et al. 1994), and the experimental value is -11.62 meV (-134.8 K) from spectroscopy (Derry et al. 1979, 1980), and -11.73 meV (136.2 _ 2 K) (Elgin et al. 1978; Cole et al. 1981) from thermodynamic data. The substantially larger value (-179 K) determined by Ezell et al. (1981) by dynamic gas diffusion techniques is probably due to heterogeneity (an average binding energy at low coverages is in fact determined by this experiment). Note that for 4He on graphite the adsorption potential is the same, but the kinetic energy is substantially smaller due to the larger mass of this isotope. Hence, the adsorption energy is larger for 4He than for 3He. Preferential adsorption for 4He is well known from experiments on liquid 3He/aHe mixtures confined in porous materials at low temperatures (see for instance Brewer 1970; Thompson 1978; Ezell et al. 1981); this effect has been used to preplate substrates with a 4He layer. The excited states of a 3He atom in the graphite adsorption potential are well separated from the ground state level. The energy of the first excited state is -5.38 meV (-62.4 K), and that of the third state is-1.78 meV (-20.7 K) (Derry et al. 1979, 1980). Calculated values are -5.57 meV (-64.7 K); -2.23 meV

232

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

50

o

|

.

v

,,l,,,,a

E

-50

i

o

~ o

:~ t~

t

-lOO

L

0

or}

"0

<

-15o

_

-200

, , i I,,, 0

2

I,,,

I,,,

I,,,

4

6

8

z

(/~)

I, 10

,, 12

Fig. 7. The adsorption potential for a helium atom on a graphite surface averaged along the lateral coordinates (formula given in the text; Joly et al. 1992). The bound dates for 3He are shown in the figure by thin lines.

(-25.9 K); -0.716 meV (-8.3 K); and -0.166 meV (-1.9 K) for the first four excited states ( Joly et al. 1992). The mean position of the 3He atom above the graphite surface calculated by Joly et al. 1992, is (z) = 2.88/~; due to the large zero point energy, this value is substantially larger than the distance corresponding to the potential minimum. Note that the experimental value for 4He ((z) = 2.85/~, Carneirol et al. 1981) was used in the calculation of the potential. The dependence of the potential on the lateral coordinates has been studied theoretically by Carlos and Cole (1978, 1979, 1980) and references therein (fig. 8). The amplitude of the corrugation potential is on the order of 40 K according to experiments (Boato et al. 1979). The corrugation potential is not strong enough to localize a single adsorbed helium atom, the atomic motion of which is close to that of a free particle. Band

Ch. 4, w

3He ADSORBED ON GRAPHITE

233

Fig. 8. The adsorption potential for a helium atom on a graphite surface for different lateral coordinates (Cole et al. 1985); the different curves correspond to the positions above the graphite substrate indicated in the picture.

effects due to the periodic substrate potential are expected (fig. 9), and should be incorporated in the calculation of the binding energy (Carlos and Cole 1980). An effective mass increase of 3% has been predicted for 3He adsorbed on graphite, unfortunately somewhat small to be verified experimentally. Assuming that the potential consists of a region of constant attractive potential adjacent to a corrugated hard wall Boato et al. (1978, 1979) give a corrugation distance of 0.21/~, and a different analysis (Carlos and Cole 1980) gives 0.29 ]k, thus a relatively large fraction of (z), the mean distance of the particle relative to the substrate.

3.2. Interaction potential and zero point energy in adsorbed layers For systems of finite density it is necessary to take into account other effects, in addition to the adsorption potential described above: the van der Waals interaction between the 3He atoms, the in-plane zero point kinetic energy, and quantum statistics. The simplest description of the interactions is given by the Lennard-Jones potential

234

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Fig. 9. Band structure of 3He adsorbed on graphite (Carlos and Cole 1980). The free particle behavior expected in the absence of corrugation is indicated by a dashed line.

(5)

where e = 10.22 K and tr = 2.55 ,~ (de Boer and Michels 1938; Hirshfelder et al. 1954); the potential minimum corresponds to the distance 2.869/~. More sophisticated potentials are found in the literature. Recent work is based on the Aziz potential (Aziz et al. 1979, 1987). There is some evidence that the effective interaction between helium atoms adsorbed on graphite is modified by screening effects due to the substrate, and, to a smaller extent, by the averaging over the third dimension (Vidali and Cole 1980). The adsorption and interaction potentials are the starting point of theoretical calculations of the structure and excitations of adsorbed helium films. Even though the potentials are rather accurately known, calculations are particularly delicate. Monte Carlo techniques presently allow investigations of the structure of 3He films as a function of coverage and temperature. Brami et al. (1994) studied the low-submonolayer coverage region; the simulations by Abraham and Broughton (1987) apply to the submonolayer coverage range, but at temperatures on the order of 2 K; Abraham et al. (1990), using path integral and varia-

Ch. 4, w

3He ADSORBED ON GRAPHITE

235

tional wave-function techniques, investigated the stability of a bilayer film structure. Variational many-body theories developed in recent years have been applied to study the growth of liquid 4He and 3He films on different substrates (see for instance Clements et al. 1993; Pricaupenko and Treiner 1994). At finite densities, i.e. when the area available per particle is of atomic dimensions, the zero point kinetic energy becomes relatively large, reaching for example values on the order of 40 K in the solid phase. In this phase each particle is surrounded by six neighbors, forming a small cage; the central particle can be viewed as vibrating in the effective potential well due to the neighbors. Twodimensional solid 3He is a strongly anharmonic system, where the delocalization of a 3He atom may be as large as 30% of the lattice parameter. The magnitude of the zero point energy can be inferred from heat capacity data, since it is approximately equal to the Debye temperature. Surprisingly, the numerical calculations mentioned above do not explicitly give the results for the zero point energy. The physical state of 3He at a given density is determined by the adsorption potential (including corrugation), the interaction potential, the zero point kinetic energy and the quantum statistics. Several structural phases exist in adsorbed 3He systems (gas, liquid, commensurate or incommensurate solids, domain wall solids or liquids...) and their energies may differ by very small amounts: phase transitions are observed at temperatures well below 1 K. It is therefore necessary to evaluate the wave-functions and energies very accurately in order to obtain a realistic result.

3.3. Layering It has been implicitly assumed in the preceding discussion that at finite densities the 3He atoms do not form three-dimensional clusters. This assumption is physically sound, due to the strong substrate potential compared to the weak heliumhelium interactions. The growth of multilayer films, however, is not a simple problem and considerable attention has been given in recent years to the problem of wetting and layering in adsorbed films. Non-wetting behavior has been observed, for instance, on substrates with anomalously weak adsorption potentials, like cesium (see Cheng et al. 1993 and references therein; Ketola and Hallock 1993). The detailed structure of 3He films adsorbed on graphite is described in the next section. We focus here on the problems associated with layer formation. The simple picture is the following: when the areal density of a 3He film reaches values as high as about 0.11 atoms/~-2, it becomes energetically favorable to place an extra atom in the "second layer", on top of the previously formed "first layer".

236

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Fig. 10. Density profile of 4He films calculated for a strong substrate (like graphite) (Epstein and Krotsckek 1988) for several 4He coverages. The higher potential energy is compensated by the lower kinetic energy both in- and out-of-plane. Note however that the "second layer" particle wave function may overlap substantially that of the "first layer" atoms. Experimentally one observes that the first layer density increases progressively as the second layer is populated. More sophisticated (and more realistic!) descriptions are now available; theoretical calculations treat the quantum mechanical problem of n particles (Epstein and Krotscheck 1988; Clements et al. 1993; Pricaupenko and Treiner 1994 and references therein) qualitatively reproducing the layering effects. In the case of liquid 3He films, the filling of "layers" is associated with the completion of "Fermi disks" (Guyer 1990; Pricaupenko and Treiner 1994). In the calculation of the energy, the quantum statistics plays a significant role; computation is particularly difficult for the case of Fermions due to the required antisymmetrization. The evolution of the spatial distribution of the particles in a multilayer film is shown in fig. 10. Clearly, well defined "layers" do not exist strictly speaking, since the probability does not go to zero periodically. The peaked structure, however, shows that one can use qualitatively a "layer" model. In particular, the first layer is remarkably well defined, and the second layer would probably be found to be better defined in more quantitative calculations; further layers are probably well described by the theoretical models. The simple "statistical layer model" is still particularly useful to interpret experimental data

Ch. 4, w

3He ADSORBED ON GRAPHITE

237

on adsorbed helium, even on heterogeneous substrates (see for instance Brewer 1970).

3.4. Coverage scales In order to compare results of experiments performed on the same type of substrate, but on different samples, it is necessary to use some procedure to determine the surface areas: one is mainly interested in comparing data taken at the same areal density. The amount of adsorbed gas can be accurately measured: the precision is usually much better than 1%, usually limited by the care taken to calibrate the volumes of the adsorption gas handling system. The surface area of a graphite substrate is not so easily determined, and several procedures have been used experimentally. Adsorption isotherms with conveniently chosen gases display features at particular coverages which can be associated with a known areal density. Nitrogen vapor pressure isotherms at temperatures on the order of 74 K display a sub-step indicating the coexistence of fluid and commensurate phase (Chan et al. 1984). The areal density of the commensurate phase is given by the underlying graphite substrate lattice parameter as 0.06366/~-2, a value which has been verified by neutron scattering. The difficulty is to identify in the adsorption isotherm the exact coverage corresponding to the pure commensurate phase. As seen in fig. 11, where the nitrogen adsorption isotherm used by Greywall is shown, this can be done with an accuracy of about 3%. Heat capacity techniques provide interesting features at the commensurate coverage for adsorbed 3He and 4He. A maximum in the melting temperature and a minimum in the low temperature heat capacity are observed at the coverages that correspond to the perfect registered phase. Some inconsistencies are found however, between the amounts of different gases needed to fill the commensurate phase (Bretzl et al. 1973; Greywall 1993). Typical discrepancies are on the order of 2-3%. Neutron scattering experiments in the coverage range around the commensurate phase show that the lattice parameter remains constant at the expected commensurate phase value within this coverage range, but the amplitude of the diffraction peaks has a maximum at a coverage which is identified as corresponding to perfect registry. NMR experiments have also been shown to display a feature at the commensurate phase coverage (see Richards 1980). Identification of the commensurate phase coverage allows in principle, in all these experiments, to determine the area of the sample. There are two difficulties, however: one is associated with the heterogeneity of the samples, and the second to the sensitivity of different techniques. Heterogeneity probably explains the difference in absolute values found for perfect registry using different

238

H. GODFRIN and H.-J. LAUTER 60

|

-

(3_ i-O3

o 40 0

'

/ sL.

I

~

Ch. 4, {}3

i

r

50

i

1

E o > L_

47

20

o O"J "a

44

o

'

'

;.

2

'

Pressure (Torr)

3

8

Fig. 11. Nitrogen adsorption isotherm on Grafoil at 74 K (Greywall 1990). The surface area deduced from this measurement is 203 m 2.

gases. The asymmetry between the properties of the incomplete and the overfilled commensurate phase is seen differently by different techniques, resulting in shifts of the determined characteristic coverage. These limitations cause an uncertainty which seems to be on the order of 2-4%. Since the surface area determined following the techniques described above is based on the identification of the commensurate phase, we refer to the coverages determined using this area as measured in a "commensurate scale". Another coverage scale can be deduced from neutron diffraction measurements in the submonolayer incommensurate phase of 3He adsorbed on graphite. The measured areal density (see section 4.5 and 5.1.5) has been found to be larger than the one determined in the commensurate scale by as much as 8.5% (Lauter et al. 1990; see section 5.5). This may be due to heterogeneity: the effective area available for the commensurate phase is smaller than that of the incommensurate phase due to defects of the exposed area, or to adsorption on other planes than the basal ones. Similar discrepancies have been found in NMR experiments (Godfrin and Rapp 1993). The larger effective area in the incommensurate phase defines another scale of coverages: the incommensurate scale. Second layer promotion has also been used as a marker for determining areal densities. This is based on the accident observed in adsorption isotherms, NMR susceptibility, heat capacity, neutron scattering, etc., as the monolayer is completed and some atoms begin to populate the second layer. This determination, however, is very sensitive to the particular technique used. Heat capacity and NMR linewidth measurements have an excellent sensitivity to detect a small

Ch. 4, w

3He ADSORBED ON GRAPHITE

239

number of second layer atoms, but this promotion occurs in a coverage range where considerable compression of the first layer is taking place; it is therefore difficult to associate a particular absolute value of coverage to the observed feature. The result is also dependent on the sensitivity of the experimental setup, an effect particularly important in NMR studies (see the discussion by Godfrin and Rapp 1993). Finally, the density attributed to the coverage where promotion is observed is indirectly determined from another measurement, the latter being necessarily derived from either the commensurate or the incommensurate scale. Determining coverages from second layer promotion only is therefore an uncertain procedure. Therefore, despite the relatively good homogeneity of the exfoliated graphites, particular care must be taken when assigning a particular density to a given amount of adsorbed of gas. The existence of regions of higher and lower adsorption energy compared to that of the perfect basal planes implies uncertainties in the areal density of the adsorbed phases, and, as described later, modifies the structural evolution as a function of coverage. The determination of the surface area of a substrate used in a particular experiment may therefore be done in several ways, not necessarily yielding the same result. It is recommended to perform at least a 3He adsorption isotherm measurement at 4.2 K: using the data of the Seattle group, the "commensurate scale" area is determined by conveniently scaling the isotherms (fig. 12). Nitrogen or krypton adsorption isotherms are also suitable, although not so frequently used (see fig. I 1). Low temperature heat capacity (see Greywall 1990) is particularly sensitive to the full solidification into the commensurate phase (yielding to a direct determination of the area), or to layer promotion (indirect determination. Neutron scattering also provides a direct determination, as discussed above, but the technique is obviously very specialized. Other techniques to determine the area of the experimental substrate have been used in recent years: it is often possible to relate some feature of the data to a "coverage" determined by another technique. An example of this procedure has been given by Rapp and Godfrin (1993): the NMR signal due to the first layer solid in the partial second layer completion regime corresponds to an areal density that can be inferred from neutron scattering. Due to the increasing number of results in the field, it is now possible to refer to previous results: Schiffer et al. (1993), for instance, evaluated the coverages in their very low temperature experiments by conveniently comparing the coverage dependence of their NMR data to that found by Franco et al. (1986) around the "ferromagnetic peak" feature. It should be pointed out that problems related to coverage scales have been explored quantitatively only recently. Differences in coverages of several percent are typically found for the same system when determined by different techniques.

240

Ch. 4, {}4

H. GODFRIN and H.-J. LAUTER 0.50

i

i

1

j

i

1

i

I

!

I

i

i

l

i

i

i

i

i

i

_ t'M

E

in Ico v

o o

0.45

_

/

E o >

0.40

"o 11)

3He/Graf0il

at T=4.2

K

t.___

o "o 0.35

, 0

,

,

I 2

t

,

I

I

,

4

pressure

,

,

I

z

6

i

I

1

t

I

8

J

10

(mbar)

Fig. 12. 3He adsorption isotherm on Grafoil. Data from Bretzl et al. (1973), scaled to the total area of the sample (deduced from the commensurate phase coverage determined by the heat capacity data). This plot is used to determine the area of a graphite sample by simply measuring an adsorption isotherm.

4. Experimental techniques of surface Physics at low temperatures In three-dimensional systems, structural properties are often easily characterized by X-rays or neutron scattering measurements; thermodynamical or dynamical techniques provide detailed complementary information, for instance on the nature of phase transitions, defects, kinetics, etc. This is not the case for adsorbed 3He samples at low temperatures. Direct structural information is scarce due to technical limitations. X-rays scattering, for instance, involves high power dissipation in the graphite samples, a situation hardly compatible with the poor thermal conductivity of the samples at low temperature. Neutron diffraction experiments, as discussed later, also suffer from severe limitations compared to experiments performed on bulk samples. Other methods, such as heat capacity or NMR, are therefore particularly valuable in this research field. They have been adapted to the particular situation of adsorbed films, usually by increasing their sensitivity to take into account the small number of atoms. New techniques were needed to investigate helium adsorbed on graphite at milliKelvin, and even lower temperatures: the tasks of designing sample cells and an adequate thermometry, measuring in equilibrium (both in density and temperature), character-

Ch. 4, w

3He ADSORBED ON GRAPHITE

241

izing the substrate, and many others involve particular care. We discuss in the following paragraphs some techniques adapted to surface Physics at low temperatures.

4.1. Experimental details 4.1.1. Experimental cells The experimental set-up usually includes a graphite substrate, a metallic support that ensures convenient thermal contact of the graphite to a cold source, a cell surrounding the substrate to confine the gas at high temperatures, thermometers, heaters, eventually NMR coils, etc. The graphite substrate has been described in detail in section 2. It consists typically of a few cm 3 of exfoliated graphite foils, of thickness on the order of 0.3 mm, and lateral dimensions on the order of a few cm. A monolayer of 3He corresponds to a total number of atoms on the order of 1 mmol. In order to cool down the adsorbed 3He layers, which can have very large heat capacities in some coverage ranges (see section 4.3), it is important to use thin foils of graphite, since the thermal path necessarily involves the substrate. The time constant can reach values as high as 30 min (fig. 13) at coverages where the 3He molar specific heat has values on the order of R. It is necessary to achieve a good thermal contact between the graphite foil and the cold source (dilution refrigerator or nuclear demagnetization stage). The technique used by the Seattle group (Bretzl et al. 1973) consists of an initial deposition of a 0.5-/zm thick copper film, then adhesion to a massive copper part is achieved by intercalating thin copper powder, and finally heating the assembly in vacuum to about 10~~ Sputtering is a very convenient technique to deposit relatively thick copper films onto graphite; the quality of the Seattle data seems to indicate that there is no substantial contamination of the substrate by the copper vapor during the process. A different technique was developed by the Grenoble group (Franco et al. 1984; Rapp and Godfrin 1993), leading to graphite to metal bonding by a direct procedure. Copper and Grafoil foils were pressed together in a simple stainless steel device, consisting of two steel plates held by two steel screws, using springs and flexible washers to keep a moderate pressure. The assembly is baked at temperatures on the order of 500~ for a few hours in vacuum, and cool-down is performed in a 4He atmosphere. The experimental parameters (temperature, pressure, etc.) are not critical. Very good bonding is achieved over all the contact area. Bonding of the Grafoil onto the stainless steel is also very efficient, and Grafoil foils are used as spacers to protect the "sample". Bonding Grafoil to silver foils (Greywall 1990) also provides excellent results. Some reduction of the surface area of the substrate is observed after bonding, an effect most likely due to the applied pressure and not to con-

242

H. GODFRIN and H.-J. LAUTER

Ch. 4, {}4

40 35

T

3o

~

25

.

10 5 0

0

1

2

3

4

5

6

7

8

T (mK)

Fig. 13. Thermal time constant measured (Godfrin et al. 1988b) for 3He adsorbed on a Grafoil sample at a coverage n = 0.233 A-2, at a field B = 6.44 mT (circles) and B = 14.21 mT (squares). The maximum at about 2 mK is due to a similar feature in the heat capacity of the 2D 3He ferromagnetic system. tamination, since similar effects are observed with copper and silver. The metal foils are usually welded to a copper or silver post ensuring thermal contact to the refrigerator. The direct bonding technique it is now widely used to manufacture experimental cells for very low temperature studies. The elevated temperatures used in this process, under vacuum, are sufficient to clean the sample, eliminating most of the surface contamination. Cleaning graphite is usually performed by heating the material under a secondary vacuum, at temperatures in the range 800-1200~ Adsorption isotherm measurements performed on samples as delivered by the manufacturer, pumped only at room temperature, give surface areas about 30% lower than that found after the cleaning procedure. In many experiments, in particular in those requiring plastic cells, it is very difficult to avoid exposing the graphite sample to air; obviously, the cell cannot be heated up to several hundred degrees Celsius, but fortunately experience shows that it is sufficient to pump the experimental cell for a few days with a nitrogen trapped diffusion pump to remove the additional impurities adsorbed on a sample that has been in contact with air, if the graphite was originally cleaned at elevated temperatures. An interesting technique has been reported by Greywall (1990): a short copper tube was used to allow a proper evacuation of the cell, and then crimped

Ch. 4, {}4

3He ADSORBED ON GRAPHITE

243

closed, whereas a very small diameter capillary (0.1 mm) was used for the 3He sample gas admission during the experiment. At temperatures well below 1 K the 3He is entirely adsorbed onto the substrate; at temperatures on the order of Kelvins, however, only a fraction of the 3He is adsorbed, the rest being in the 3D gas phase. A small cell is therefore needed to minimize the non-adsorbed fraction; this is particularly important for a correct annealing of the films. For the same reason, a small cell reduces the corrections needed to the adsorption isotherms (section 4.2). The material used to build experimental cells depends on the particular experiment: plastic cells are used in NMR measurements, whereas thin metallic cells are normally used for heat capacity experiments, as discussed later. The walls of the cell do not contribute significantly to the total area of the adsorption cell, dominated by that of the graphite sample; problems related to diffusion of 3He in plastic cells, however, can be found if the sample is allowed to warm up to room temperature. The cell is connected to the room temperature gas handling system by capillaries. Small diameter capillaries are used to reduce the dead volumes at low temperatures and to reduce the thermomolecular corrections needed for low pressure adsorption isotherms. Tables for the thermomolecular corrections on 3He pressure measurements are given by Watkins et al. (1967), Freddi and Modena (1968) and McConville (1969). ~ In practice, capillaries on the order of 1 mm are normally used at low temperatures. The diameter is increased to 2 or 3 mm at nitrogen and room temperature, since dead volumes at the warm parts of the apparatus do not contribute significantly to the non-adsorbed gas corrections, and small diameters would unnecessarily limit the conductance of the line. Heat leaks at very low temperatures constitute a serious problem, which should be addressed during the design of the cell. In addition to the standard problems usually found at low temperatures, like vibrations leading to internal friction of plastic supports, here it is necessary to take into account that even picowatt heat leaks flowing in a thermal path involving the graphite will cause a temperature gradient in the sample, usually rising the 3He temperature by a few milliKelvin compared to that measured on the metallic parts of the cell, well connected to the cold source. This effect can be estimated using the thermal conductivities given in section 2.2. This problem is particularly important in NMR and neutron scattering experiments. A special construction was used in the first experiments performed at Grenoble on adsorbed 3He at milliKelvin temperatures: the cell was immersed inside the mixing chamber of the dilution refrigerator, in order to use the mixtures as a viscous damper for vibrations, and ensure sub-picowatt heat leaks. This solution, although very efficient, has not Note that the sign of the term "0.15823" in eq. (2) in Freddi and Modena should be negative.

244

H. GODFRIN and H.-J. LAUTER

Ch. 4, {}4

been used in later experiments; cells supported by a rigid frame, in vacuum, were successfully used by Godfrin et al. (1988a) down to 0.5 mK, and more recently sample temperatures as low as 0.075 mK, could be reached by Schiffer et al. (1993). A typical experimental cell for low temperature surface studies is shown in fig. 14. A special problem arises when using hydrogen isotopes to preplate the graphite substrate: the heat released by the ortho-para conversion is large, leading to a substantial heating of the graphite. HD has been used by Siqueira et al. (1993) at temperatures below 1 mK; in this case, the heat of conversion is due to the H2 and D2 present as impurities at the percent level. The fact that they did

Fig. 14a.

Ch. 4, w

3He ADSORBED ON GRAPHITE

245

Fig. 14. (a) Experimental set-up used for NMR measurements on 3He adsorbed on Grafoil (Morhard et al. 1995). (b) Experimental cell used for heat capacity studies of 3He adsorbed on Grafoil (Greywall 1990).

not observe heating effects may be due to the relatively fast conversion kinetics on the graphite substrate. Let us mention finally some problems related to the diffusion of the 3He gas inside the graphite samples at high temperatures (several degrees Kelvin), where the mass transport is ensured by the 3D gas at pressures on the order or larger than 1 Pa. Bretz et al. (1973) drilled 1/16 inch diameter holes through the Grafoil sheets and scribed cuts along their surface to accelerate the equilibrium times for adsorption. This procedure was also adopted by Greywall (1990), Schiffer et al. (1993) etc. It is likely that this is needed in large cells, like those used in the heat capacity experiments; there is no evidence for gas diffusion limited time constants in the small cells used at Grenoble, where the graphite foils had not been perforated. Long equilibrium times, however, have been found in large cells with high packing fractions for the Grafoil sheets.

4.1.2. Preparation of the adsorbed SHe sample Equilibrium times are very long, in the low temperature regime: when the pres-

246

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

sure of the 3D gas in equilibrium with the adsorbed film is below 1 Pa, mass transport is ensured by the diffusion along the substrate surface. High precision 3He adsorption isotherms at 4.2 K performed at Grenoble for coverages slightly below monolayer completion show that the equilibrium pressure tends logarithmically to the equilibrium value, indicating a slow diffusion in the surface film. This process is very long, in particular when the adsorbed phase is solid or commensurate with the substrate. In order to prepare uniform samples, it is therefore necessary to anneal them at convenient temperatures. The importance of the annealing for helium films is clearly shown by heat capacity data of the Seattle group. Bretz et al. (1973) observed truncated heat capacity peaks in rapidly cooled samples; also, sharp peaks associated to the order-disorder phase transition around the commensurate phase coverage are strongly reduced in poorly annealed samples (Dash 1975). Greywall and Busch (1990) observed non-equilibrium densities in submonolayer liquid 3He films below the onset of the commensurate phase formation, and attributed this effect to insufficient annealing. Note that the heat of adsorption of a substantial monolayer fraction is large at submonolayer coverages, leading to a large heating of the system when the gas is introduced in the experimental cell. This effect is often used as an initial annealing procedure. Bretz et al. (1973) showed that temperatures on the order of 7-8 K were needed to anneal monolayer films. Higher temperatures (on the order of 17 K) are needed at low coverages; a thorough discussion on the annealing criterion is given by Elgin and Goodstein (1974). Usually the sample temperature is reduced slowly (several hours), particularly in the regime where the gas pressure is still above 1 Pa, or around solidification transitions. It is common practice in this field to leave the sample at 4.2 K overnight, and then continue cooling the sample down to below 1 K very slowly (again, several hours), with particular care around phase transitions. The situation is more delicate at coverages above one complete monolayer: in principle temperatures above 10 K would be needed to melt the first layer, but then a substantial amount of the sample is desorbed. It is therefore convenient to minimize the dead volume of the experimental cell to reduce the amount of desorbed gas in order to obtain reasonably annealed films. Evidence for some degree of reconstruction in multilayer films has been obtained at Grenoble: a 5 layers film always kept at very low temperatures displayed somewhat different magnetic properties after several months (a 30% increase of the Curie-Weiss temperature was observed), despite rather careful preparation of the film. Annealing problems, clearly, are not completely understood presently.

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4.2 Adsorptionisotherms 3He atoms adsorbed onto a graphite substrate may form several phases, in thermodynamical equilibrium among themselves, but also with the bulk gas; the chemical potential/~ is the same for all phases. The bulk gas is usually almost an ideal gas, and its pressure P is simply related to the chemical potential by the expression

~=kBTlnl P

h2 ] + B P T5/2 2azm3/2

(6)

where kB is the Boltzmann constant, h the Planck constant, m the atomic mass, and B the second virial coefficient (see for instance Landau and Lifshitz 1967; Elgin and Goodstein 1974). The measurement of the pressure of the 3D gas at a given temperature T immediately provides the chemical potential of the adsorbed film as a function of coverage. Measuring adsorption isotherms for a large range of temperatures and coverages is a standard technique in surface Physics. Detailed phase diagrams have been obtained for rare gases adsorbed on graphite (see for instance Thomy et al. 1981; Zangwill 1988). Also, a step-like increase of the adsorbed amount as a function of pressure is seen, revealing the formation of the successive layers that can be modeled by a simple lattice-gas, Ising-like, calculation (de Oliveira and Griffiths 1978). Measurements of helium adsorption isotherms on graphite are usually performed at temperatures on the order of 4.2 K; the completion of the first layer is seen at a pressure on the order of 0.5 mbar (see fig. 12). Typical 3He isotherms in the temperature range 4.2-7 K can be found in Bretz et al. (1974), Goellner et al. (1975), Hegde and Daunt (1978), Godfrin et al. (1988b) and Rapp and Godfrin (1993). Daunt et al. 1981, reported measurements at lower temperatures, down to 0.88 K; the data demonstrate the expected scaling of the measured pressure with the saturated vapor pressure at the relevant temperature. They also provide values for the isosteric heat of adsorption as a function of coverage, which allow estimating the binding energies at high coverages. Multilayer adsorption should give rise to steps in the adsorption isotherms, as seen previously for other rare gases. Unfortunately, this is observed for helium at rather low temperatures, and hence at extremely low pressures. Data for 4He have been recently published (Zimmerli et al. 1992), where layer by layer growth through at least seven layers is reported. They correspond to a temperature of 0.639 K and the measured pressures are below 1 Pa. Data (for 4He) at very low pressures are also given by Taborek and Goodstein (1979). Although no direct structural information has been yet obtained from 3He adsorption isotherms on graphite, they are particularly interesting from the point of

248

H. GODFRIN and H.-J. LAUTER

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view of transferring coverage scales (section 3.4), estimating binding energies (see above and section 3.1), characterizing the substrate heterogeneity (section 5.1.1) and estimating desorption corrections in "high temperature" heat capacity studies (section 4.3). It would therefore be desirable to have a complete set of 3He isotherms on graphite.

4.3. Heat capacity 4.3.1. Guide to the literature Heat capacity is a powerful tool to investigate adsorbed gases; this technique has a unique sensitivity to phase transitions (marked by peaks of characteristic shape), to the nature of the adsorbed phase (deduced from the temperature dependence); also, phase coexistence regimes, heterogeneity effects, annealing properties are most easily studied by this method. Detailed investigations on 3He adsorbed on graphite substrates were performed at Seattle (Hering et al. 1970; Bretzl et al. 1973, 1974; Hering and Vilches 1973; Dash 1975, 1978; van Sciver and Vilches 1975, 1978; van Sciver 1978). In addition to published results, a large amount of valuable information can be found in unpublished Ph.D. thesis manuscripts (Bretz 1971; McLean 1972; Hering 1974; van Sciver 1976, etc.). This pioneering work provided the first structural information about these systems; the measurements were careful and extensive, and the results are still today the main source of information about the "high temperature "phase diagram. The low temperature heat capacity has been investigated more recently, at Bell Laboratories; high quality data have been obtained by Greywall and Busch (1989, 1990) and Greywall (1990). This work completes the exploration of the structural aspects initiated by the Seattle group, and provides unique information on the aspects related to quantum statistics and nuclear magnetism of 3He films. A summary of presently available data is given in figs. 15 and 16. Their interpretation is not straightforward; some simple ideas, however, allow us to understand the main features. The molar heat capacity of solid phases is approximately given by the expression

CsoI = 28.85R

+~ R

,

(7)

where the first term represents the phonon contribution, and the second that of the spin exchange heat capacity. The phonon contribution dominates at high temperatures, whereas the spin one becomes visible only below a few milli-

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Fig. 15. (a) Heat capacity isotherms as a function of areal density in the low-coverage submonolayer regime for temperatures (from bottom to top) 0.05; 0.4; 0.6; 0.8; 1; 1.2; 1.4; 1.6 and 1.8 K (McLean et al. unpublished; Vilches, private communication). (b) Heat capacity isotherms as a function of areal density in the intermediate coverage submonolayer regime for temperatures (from bottom to top) 0.2; 0.4; 0.6; 0.8 and 1 K (Hering, unpublished; Vilches, private communication).

250

H. GODFRIN and H.-J. LAUTER

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Kelvins; due to their different temperature dependence, these contributions are easily separated. When liquid is present, it is easily identified by its characteristic Fermi liquid temperature dependence below 100 mK: Cli q -)pT; at higher

Fig. 16a.

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3He ADSORBED ON GRAPHITE

251

Fig. 16. (a) Heat capacity isotherms as a function of areal density in the submonolayerregime and at low temperatures (Greywalland Busch 1990b). (b) Heat capacity isotherms as a function of areal density in the multilayerregime and at low temperatures (Greywall 1990).

temperatures the molar heat capacity tends to the classical value Cli q = R, but a typical intermediate plateau is seen at about 300 mK, corresponding to the nondegenerate quantum regime. The heat capacity of the liquid is much larger than that of the same amount of solid around or above 50 mK, up to a few Kelvin: heat capacity isotherms, therefore, are dominated by the liquid signal in this temperature range. At milliKelvin temperatures, on the other hand, the liquid heat capacity is small compared to that due to spin exchange in the solid; therefore, large contributions in the heat capacity isotherms at milliKelvin temperatures are associated with the presence of "highly magnetic" solid phases. Finally, phase coexistences may be inferred from the heat capacity isotherms: the heat capacity varies then linearly with coverage in some range. A detailed discussion appropriate to the different structural phases of 3He adsorbed on graphite is given in section 5.

252

H. GODFRIN and H.-J. LAUTER

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Heat capacity measurements of 3He adsorbed on other substrates have been performed in several laboratories: let us mention the results on neon (Wennerstr6m et al. 1978), and on silver powder (Greywall and Busch 1988). A special, and particularly interesting case is that of 3He-aHe mixture films, not discussed here; heat capacity data can be found in Hickernell et al. (1976) and Bhattacharrya and Gasparini (1985). See also the review by Hallock, to be published. Data for 4He adsorbed on graphite are also of interest, due to the close similarity for both isotopes in the phase diagrams and heat capacity signatures at temperatures above 1 K, where statistics are not important, but where zero point quantum effects still play an important role. Experimental results can be found in Bretzl et al. (1973), Bretz (1977), Polanco et al. (1978), Polanco and Bretz (1978), Ecke and Dash (1983), Ecke et al. (1985), Chae and Bretz (1989), Greywall and Busch (1991), Zimmerli et al. (1992) and Greywall (1993). 4.3.2. Techniques The conventional quasi-adiabatic calorimetry technique has been used in all the heat capacity experiments on adsorbed 3He. Large samples (on the order of 10 cm 3) are used, providing surface areas on the order of 100 m 2. The empty cell heat capacity is due to a large extent to the metallic walls and foils (copper or silver) used to improve thermal diffusion in the sample (section 4.1.1 and to the graphite itself (section 2.2.6); at milliKelvin temperatures the addenda (thermometers, heaters) may provide substantial contributions (see Greywall 1990), but graphite has a large heat capacity due to localized electronic states, much larger than the contribution expected from conduction electrons (section 2.2.6). The background heat capacity, nevertheless, is substantially smaller than that due to the adsorbed 3He in most of the coverage and temperature ranges presently investigated. The large 3He heat capacities found at some coverages lead to experimental problems associated, as seen is sections 2.2.8 and 4.1.1, with long time constants due to the poor thermal conductivity of exfoliated graphite. Using thin exfoliated graphite foils conveniently thermalized on metallic foils allows reducing the internal equilibrium time of the composite "sample" to less than a few minutes (see fig. 13, and Greywall 1990). This figure gives an idea of the requirements imposed on the thermometry, and on the adiabaticity of the system. The main difficulty with this technique comes from the fact that one is interested in measuring accurately, and systematically, in a large temperature and coverage range. The heat capacity may change by several orders of magnitude, but it may also display rather weak anomalies that can only be found by high precision measurements. Although standard thermometry (carbon or germanium resistances, for example) is sufficient for most of the experiments, severe difficulties appear at milliKelvin temperatures. A CMN thermometer has been used

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by Greywall (1990) down to 2 mK, but the heat capacity addenda and the large internal time constant limit its applicability to low temperatures. Except for the obvious subtraction of the empty cell background, heat capacity data at low temperatures do not require corrections. At higher temperatures, where the evaporation of the film is not negligible, desorption corrections (Dash et al. 1970; Elgin and Goodstein 1974; Daunt et al. 1979) are needed, an effect particularly important for multilayer films. Finally, heat capacity measurements, as all surface studies, are faced with the problem of absolute coverage scales; although this technique provides several "markers" of the commensurate phase, determining precisely the amount of gas corresponding to perfect registry is not a trivial problem, as seen in sections 3.4 and 5.1.3.

4.4. Nuclear magnetic resonance 4.4.1. Guide to the literature Nuclear magnetic resonance (NMR) is a standard tool in Surface Physics (see for instance the review articles by Tabony (1980) and Duncan and Dybowski (1981)). The large nuclear magnetic moment of the 3He nucleus makes it an excellent candidate for NMR studies, and the first experimental investigations of adsorbed 3He by this technique were carried three decades ago. A summary of the early work can be found in Richards (1980). The principles of the measurements are discussed in well known NMR books (Abragam 1961; Slichter 1978). The parameters determined from NMR experiments are the nuclear magnetic susceptibility, the spin-lattice relaxation time T~ and the spin-spin relaxation time 7'2. Extensive measurements on 3He adsorbed on Grafoil have been performed as a function of coverage and temperature in the range 0.3 < T < 4.2 K by Grimmer and Luszczynski (1977, 1978); Cowan et al. (1977, 1987); Owers-Bradley et al. (1978); Hegde and Daunt (1978, 1979); Widom et al. (1979); Richards (1980) and Satoh and Sugawara (1980) and references therein. It was expected at that time that substantial information could be obtained from the analysis of the relaxation times, due to the different scale of the correlation times in the different adsorbed phases. In fact, a complicated behavior of Tl and /'2 as a function of coverage, temperature, nature of the substrate and magnetic field was found. Nevertheless, large progress was achieved in the understanding of these phenomena; in particular, the existence of quantum exchange in the 2D solid phases was demonstrated by Cowan et al. (1977); in a subsequent publication Cowan et al. (1987) presented a theory allowing to use Tl data to determine quantitatively the exchange frequencies as a function of coverage. Exchange in

254

H. GODFRIN and H.-J. LAUTER . . . .

I

99

3.5

|

. . . .

|

. . . .

q--

.

.

.

Ch. 4, w .

f r e q u e n c y = 10 M H z temperature = 1K

,,

,?,

"~" 2.5 (D 03

E ~. I--

e', , I 9 I!

2

! !

! I

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, #

,

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areal density (atoms//~ 2) Fig. 17. Spin-lattice relaxation time T2 as a function of the areal density at submonolayer coverages determined by Satoh and Sugawara (1980). The minima correspond to the commensurate phase and to monolayer completion.

solid 3He films gives rise to effective magnetic interactions, which are discussed later. 7'2 data display a coverage dependence (fig. 17) where features associated with the commensurate phase and monolayer completion are seen; the accuracy of this identification is not well established (see section 3.4). As another potential use of NMR, let us mention the results of Widom et al. (1979) who suggested that melting of incommensurate solid monolayers could be explained by a Kosterlitz-Thouless mechanism. This hypothesis, based on relaxation time data, seems however to conflict with neutron data (Feile et al. 1982). Nuclear susceptibility measurements provide useful information, in this temperature range, only for liquid phases, since the solid susceptibility simply follows a Curie law. Negative deviations with respect to the Curie law are observed for liquids at temperatures below 1 K (Owers-Bradley et al. 1978; Hegde and Daunt 1978) corresponding to the onset of Fermi degeneracy in a Fermi gas with hard core interactions. For the purpose of determining the phase diagram of adsorbed 3He, these high temperature studies basically confirmed heat capacity results. It must be pointed out, however, that they also set the basis for the later developments of

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3He ADSORBED ON GRAPHITE

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the field, both from the theoretical and the experimental point of view, and motivated further work at lower temperatures. NMR studies at much lower temperatures, in the milliKelvin range, were initiated at Grenoble (Franco et al. 1984) to investigate the possibility of nuclear ordering of the 3He spins in solid phases. Large quantum exchange effects were observed in multilayer 3He films (Franco et al. 1985, 1986); they were attributed initially to the liquid, and finally to the second (solid) layer. The solid phases, now considered as particularly interesting two-dimensional nuclear magnets, have been the object of detailed studies; this is also the case for the twodimensional fluid phases. The research has been performed using bare graphite substrates, as well as preplated substrates, in several laboratories: at Grenoble (Godfrin 1987; Godfrin et al. 1991, 1993, 1994a,b; Rapp and Godfrin 1993), Bell Laboratories (Godfrin et al. 1988a,b, 1990), London (Saunders et al. 1990, 1991; Lusher et al. 1991a,b; Siqueira et al. 1992, 1993, 1994) and Stanford (Schiffer et al. 1993, 1994). These experiments have determined the coverage dependence of the nuclear magnetization, also providing valuable information about the structural changes in the films. Recent reviews have been given by Godfrin et al. (1987, 1991); Saunders et al. (1991); Greywall (1994) and Godfrin and Rapp, to be published in Advances in Physics. An overview of the experimental data is presented in fig. 18. The signal from liquid phases at temperatures T < 1 K displays the characteristic Fermi fluid susceptibility, independent of temperature below a temperature TF** on the order of 200 mK, whereas solid phases give rise to very large signals, following a Curie law down to mK temperatures, where ferro or antiferromagnetic deviations can be observed. The nuclear susceptibility isotherms at temperatures on the order of 20 mK, therefore, directly provide the amount of 3He atoms in the solid (commensurate or incommensurate) phases; the difference with respect to the non-interacting spins Curie law corresponds to the amount of liquid. The susceptibility isotherms at low milliKelvin temperatures are dominated by the solid contribution, and in particular by the nuclear exchange contribution. Low temperature deviations from the Curie law directly provide information about the ferromagnetic or antiferromagnetic sign of the interactions. The magnetic signature of the different phases is discussed in section 5. Pulsed and continuous-wave (CW) NMR experiments have been performed on SHe adsorbed on heterogeneous substrates (jeweler's rouge, Vycor glass, Zeolite, etc.). One of the motivations for such studies was the problem of spin relaxation of bulk 3He at the walls of experimental cells, associated with the problem of boundary resistance at very low temperatures and also with that of polarized liquid 3He. Short spin-lattice relaxation times (T1 on the order of milliseconds) are found in adsorbed layers, and the total relaxation time (for bulk and surface 3He) is proportional to the surface Tl. It is unfortunately impossible to discuss here this problem, particularly important for low temperature

256

H. GODFRIN and H.-J. LAUTER

Ch. 4, {}4

Fig. 18. Magnetization times temperature as a function of the areal density for 3He films adsorbed on graphite, at temperatures 3, 5, 10 and 30 mK. The behavior expected for free (non-interacting) spins is indicated by the straight line. The step-like increase observed in the 30 mK curve is due to the solidification of the second layer. The deviations from the free spin behavior that appear only at the lowest temperatures in the second solid layer are due to antiferromagnetic interactions and above 0.2 A-2, to ferromagneticinteractions. Note the large "ferromagneticpeak" at about 0.23 A2 (Franco et al. 1986; Godfrin 1987; Rapp and Godfrin 1993).

Physics; references can be found in Cowan (1983), Franco et al. (1984), Hammel and Richardson (1984), Schuhl et al. (1987) and Bunkov et al. (1992); for the related topic of magnetic coupling between 3He and nuclei belonging to the substrate, see also Chapellier (1982), Richardson (1984) and Egorov et al. (1990). Let us mention, finally, that NMR measurements have been performed on a related system: 3He adsorbed onto liquid 4He films on different substrates. In these "mixture films" the 3He atoms occupy surface (Andreev) states, providing excellent examples of 2D 3He liquids. References to this topic can be found in Sprague et al. (1994). 4.4.2. Techniques NMR measurements on 3He adsorbed on graphite substrates have been performed using both continuous-wave (CW) and pulsed NMR methods. Most of

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the data obtained in the temperature range above 0.3 K correspond to pulsed methods (except for the work of Hegde and Daunt 1978, 1979), while low temperature data (except for the work of Franco et al. 1984, and unpublished) were taken using CW NMR. The choice of pulsed NMR for high temperature work was justified by the fact that the relaxation times provide the most useful information in this case; also, pulsed methods had a better sensitivity than CW methods at that time. More recent work at milliKelvin temperatures, on the other hand, could benefit from the development of high sensitivity CW techniques; the important experimental parameter, the magnetic susceptibility, could be measured with better precision with this technique, also avoiding heating problems. The experiments are made at frequencies on the order of 1 MHz, and hence at magnetic fields of about 30 mT, since the gyromagnetic ratio of the 3He nucleus is ~,/2~ = 32.435 MHz/T. In this low frequency regime the nuclei are coupled to the radio-frequency (rf) field by means of a small coil surrounding the graphite sample, of typical dimensions on the order of a few cm 3 (see section 4.1.1), and inductance on the order of 1 mH. The latter is determined by the fact that the coil resonates with the capacitance of the coaxial line, on the order of 100 pF, forming the NMR tank circuit. The equivalent parallel impedance of the circuit is on the order of 100 kf~; a high impedance pre-amplifier with a low capacitance input and an adequate noise figure must be used. Q values are on the order of 100; a large Q is needed for measuring very small signals (at high temperatures), whereas much smaller Q values must be used to avoid nonlinearity when the 3He signal becomes large compared to the carrier amplitude (ferromagnetic solid at low temperatures). Some problems are found when using higher Q values: delicate tuning, phase noise and oscillations. Non-resonant rf losses in the graphite determine the quality factor of the circuit in this limit. The number of 3He nuclei in the NMR coil is smaller by more than two orders of magnitude compared to that used in 3D samples. It is therefore particularly important to maximize the signal-to-noise ratio. It is well known that this figure is proportional to the 3/2 power of the external magnetic field. It could therefore look surprising that experiments are performed at fields as low as 1 MHz; in fact the linewidth, as discussed later, increases with field. Also, the penetration depth of the rf field in the graphite sample becomes smaller than the cell's dimensions above that frequency (Hegde et al. 1973; Hegde and Daunt 1978). Cylindrical cells and coils have been used by Richards and co-workers to improve the filling factor in high temperature (1 K) experiments, where signals are extremely small. The liT temperature dependence of the 3He susceptibility (which holds even for the liquid down to about 300 mK) renders the requirements less stringent at lower temperatures. Below 10 mK, however, the applied rf field must be reduced substantially, due to non-resonant radio-frequency heating of the samples. Typical voltages on the NMR coils in CW experiments

258

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

are on the order of 1 mV, and the NMR absorption signal has an amplitude two to three orders of magnitude lower. Another type of heating effects, due to preamplifier oscillation at very high frequencies (GHz), has been often observed at Grenoble for different spectrometers, using either home-built or commercial amplifiers. Note that pW heat leaks directly fed into the graphite are sufficient to warm up the 3He sample at mK temperatures (see section 4.1.1). Heating effects have been the source of problems in early pulsed NMR measurements, even at Kelvin temperatures, due to the high excitations used (see Husa et al. 1979). It is possible, however, to perform Tj measurements at lower temperatures using small pulses of long duration; this approach is limited by the relatively large linewidth of adsorbed 3He, and the requirement to "cover the line" imposed on the rf field. Radio-frequency heating can be reduced by choosing a parallel orientation of the graphite foils relative to the rf field (see section 2.2.7); usually, the external magnetic field is also parallel to the graphite foils, in order to reduce eddy current heating. Many experiments have been performed in this configuration; however, since dipole interactions between 3He nuclei adsorbed onto a substrate depend on the orientation of the sample with respect to the external field, this parameter affects the dynamical properties (T l, T2), as well as the type of ordering expected in 2D solid phases at very low temperatures and fields. The first effect has been discussed in detail in the literature (see Mullin et al. 1976; Satoh and Sugawara 1980); the second has been explored by Bozler and co-workers (see Bozler 1991). Also, substrate induced line broadening discussed below, depends on the sample orientation with respect to the external field (see previous references and Schiffer et al. 1993, 1994). Let us now discuss briefly the problem of the NMR lineshape. The linewidth is controlled by the dipole-dipole coupling of neighboring 3He atoms, which creates a local field on the order of 0.1 mT on each spin. Motional narrowing, however, is expected to be substantial both in the liquid and solid phases, as is indeed observed. In the 2D solid phases, the linewidth increases with density, following qualitatively the behavior expected for dipole coupled 3He atoms undergoing quantum exchange. The lineshape, however, has a rather peculiar aspect, that can be described by a stretched Lorentzian (Rapp and Godfrin 1993), particularly near monolayer completion (fig. 19), instead of the quasi-Gaussian shape predicted by Mullin et al. (1976). The 2D-solid NMR lines, with narrow center and broad, low intensity tails, are difficult to measure. Although motional narrowing should lead to extremely narrow lines for 2D liquid 3He, in practice a substantial width is found in CW measurements. This spurious line broadening is due to the anisotropic diamagnetic susceptibility of the substrate together with the distribution of orientations of the graphite crystallites in exfoliated graphite (Hickernell et al. 1974; see section 2.2.9). Not surprisingly, T2 obtained from

Ch. 4, w

3He ADSORBED ON GRAPHITE

T-7.64

mK

259

T=10.37 m K

X=0.910 1

---

X=0.990

i---,---.--.,---,--

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(a)

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~

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(c)

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X=,1.128

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Fig. 19. NMR lines measured by continuous wave techniques. The radio-frequency absorption signal of 3He nuclei is represented as a function of the external magnetic field sweep. The signal corresponds to the dense first layer of adsorbed 3He for 0.91 < x < 1.128 layers, where x = 1 indicates monolayer completion. The second layer fluid signal is negligible at the mK temperatures of the measurement. Note the broadening of the line near x = 1, the development of a stretchedLorentzian shape and the narrowing effect of the liquid layer at high coverages (Rapp and Godfrin 1993).

spin-echo measurements is then found to be much longer than 7"2*, the i n h o m o g e n e o u s dephasing time deduced from the inverse of the C W linewidth. This substrate induced broadening is proportional to the squared magnetic field, and inversely proportional to the diffusion coefficient (Cowan et al. 1977; H e g d e and Daunt 1978). Magnetic impurities are expected to lead to a field independent broadening ( C o w a n et al. 1977); this effect is not well d o c u m e n t e d .

260

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

Narrowing of the NMR lines as solid phases melt has been reported by Rollefson (Phys. Rev. Lett. 29, 410, 1972); this result is incorrect, as clearly demonstrated by several other works (see for instance Hegde and Daunt 1979): the linewidth due to the substrate diamagnetism is already substantially larger than that determined by Rollefson. This experiment is unfortunately often quoted as an example of the potential use of NMR to detect phase transitions. CW NMR measurements are usually performed recording the voltage on the NMR tank circuit, conveniently amplified, as the magnetic field is swept. Lockin detection allows a simultaneous measurement of the absorption and dispersion signal. A rf bridge configuration has been used in the Grenoble experiments (see Rapp and Godfrin 1993); the coil voltage and a reference signal adjusted in phase and amplitude are fed into a high impedance differential preamplifier, followed by a lock-in amplifier. The system is particularly stable, the output of the balanced bridge being insensitive to generator amplitude or phase fluctuations. A reference attenuation is used to make the system independent on variations in the amplifier gain and lock-in range. Computer data acquisition involves typically 500 points/line; averaging of 10 or more lines is usually necessary above 30 mK. Other CW works have been performed using Robinson oscillators, modulating the frequency (Siqueira et al. 1994) or the magnetic field (Hegde and Daunt 1978). The broad wings of the NMR lines imply that field sweeps on the order of 1 mT are needed to ensure an adequate measurement of the whole NMR line: determining the correct base line for the signal is a difficult task when an accuracy on the NMR line area better than 5% is needed. This area is proportional to the total magnetic moment (TMM) of the sample. Note that this is an extensive magnitude, which should not be confused with the magnetization or susceptibility; there is no reason to normalize the values in a 2D system to unit volume. The terms magnetization and susceptibility are, however, traditionally used in this field to designate the TMM, because data are taken at constant field and are often expressed in arbitrary units. A "Curie constant per atom", or "the monolayer Curie constant" are often used to normalize NMR data in a simple way; the Curie law is written as TMM = nC/T, where n is the number of atoms (layers), C the Curie constant per atom (per layer). This is not adequate for very low temperatures, high fields or in presence of ferromagnetic interactions, where the adsorbed layer is highly polarized: the saturation TMM is here the convenient reference value. Since the number of atoms is known in an adsorbed 3He experiment, the measured TMM can be expressed in absolute (non-arbitrary) units using for calibration data at a coverage where the average magnetic moment per atom can be calculated (Curie law at monolayer coverage, for instance).

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4.5. Neutron scattering 4.5.1. Guide to the literature Neutron scattering is a common technique for studying the structure and excitations in condensed matter. Due to the very small nuclear cross-section of most elements, neutrons are only weakly scattered, and are therefore able to penetrate inside matter. The scattering process can be either elastic, or inelastic. In the former case the angular dependence of the scattered neutrons displays diffraction peaks associated with the structure of crystalline samples. Inelastic scattering, on the other hand, corresponds to creation or destruction of elementary excitations (phonons, for instance), which can be quantitatively studied by analyzing, in addition, the energy of the scattered neutrons. Neutron scattering is therefore a microscopic probe of the structure and dynamics of bulk matter. Surface studies, however, can also be performed using this technique; in spite of difficult experimental conditions, this method has provided valuable microscopic information about the nature of the adsorbed phases, complementary to that obtained from thermodynamical techniques. Several books on neutron scattering describe in detail the theoretical framework and the general experimental techniques (see Bacon 1975; Lovesey 1984). The interaction of neutrons with adsorbed molecules has been reviewed by several authors (White et al. 1978, Thomas 1982; Sinha 1987). The fundamental scattering formulae for twodimensional systems can be found in Kjems et al. (1976) and Carneiro (1977) and Sinha (1987); references on more technical aspects (interference with substrate, lineshape, etc.) are given later. Neutron diffraction studies of 3He adsorbed on graphite were performed by Nielsen et al. (1977) at Riso, and by Feile et al. (1982), Lauter et. al. (1980, 1987, 1990, 1991) at the Institut Laue-Langevin (ILL), Grenoble. Only one inelastic study on adsorbed 3He has been reported (Frank et al. 1990, 1991), about the phonon gap in the commensurate phase. The structural phase diagram of 4He is similar to that of 3He; useful neutron diffraction data on this isotope have been obtained by Carneiro et al. (1976, 1981), Carneiro (1977) and Lauter et al. (1980, 1983, 1987, 1991 ). Several inelastic neutron scattering measurements on 4He adsorbed on different graphite substrates have been performed in recent years; the relevant Physics, however, is mainly related to the elementary excitations of a superfluid 4He film (Carneiro et al. 1976; Thomlinson et al. 1980; Lauter et al. 1981, 1987, 1991, 1992), and they are not directly relevant to the present subject. A closer connection exists, however, with the inelastic studies of 4He at submonolayer coverages conducted by Frank et al. (1990, 1991). A few investigations of nucleation of bulk helium induced by graphite substrates have been carried on using neutron scattering (Tiby et al. 1981; Wiechert et al. 1982).

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Data on adsorbed hydrogen isotopes are particularly precise and detailed, and are often compared to the results obtained for the helium isotopes (see Lauter et al. 1987, 1991; Schildberg et al. 1987, Wiechert et al. 1987). It should be pointed out that results on other gases adsorbed on Grafoil are now rather numerous, after the pioneering experiments by Kjems et al. (1976), using N2; see the review articles quoted above. The phase diagrams of these "classical gases" bear some similarity with those of the "quantum" systems described here, and the experimental techniques involved in these studies are almost identical. 4.5.2. Techniques Knowing that samples of volume on the order of 1 cm 3 are needed for bulk studies, it is not obvious a priori that neutron scattering techniques could be applied to surface investigations: the number of adsorbed atoms is two to three orders of magnitude smaller than that of the substrate and the experimental cell! It is, however, possible to measure diffraction signals from adsorbed monolayers in some favorable conditions, using suitable techniques. Surface studies of helium films require a large flux of sub-thermal energy neutrons; hence, most of the experiments have been done at the high-flux reactors at Riso, Brookhaven, and ILL. The principle of the experimental set-up is simple. A cold source moderates the neutrons initially available at the reactor core: their distribution is shifted to longer wavelength 2 (for thermal neutrons ;t = 1.9/~, whereas values larger than 4/~ will be needed for surface studies). The neutron beam is then directed towards a monochromator, a pyrolytic graphite crystal that provides, by Bragg reflection, a beam of neutrons of well defined energy within a small range, determined by the choice of the crystal "quality". The monochromatic beam passes through a monitor (small efficiency detector providing a count rate proportional to the neutron flux), needed to normalize count rates, and then hits the sample. The neutrons diffracted different angles are counted by a detector that rotates around the sample center position. In practice, arrays of detectors or multi-wire counters are frequently used in diffraction work. The crystalline structure of the sample leads to Bragg reflections, observed as peaks in the count rates at particular angles (see discussion below). Inelastic scattering requires, in addition, an analysis of the scattered neutrons energy. This can be done by two methods: three-axis, or time-of-flight spectrometers. In the first, the outgoing beam for a given direction is Bragg reflected by a second pyrolytic graphite crystal ("analyzer"), and then detected by a detector rotating around the analyzer. The various angles are conveniently set in order to count only the neutrons that underwent a particular change of energy and momentum. In the time of flight technique, the beam incident on the sample is chopped and the time of arrival of neutrons to different detectors is monitored.

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Knowing the arrival time and the angle of the detector one can infer the energy and momentum change of the neutron. These changes correspond to the creation or annihilation of elementary excitations in the sample and their properties are determined from the observation of peaks in the count rate, proportional to the inelastic structure factor, that appear at defined energies for some points in the reciprocal space. In addition to the fundamental monochromator-sample(analyzer)-detector set-up, a neutron scattering spectrometer incorporates several other features (see fig. 20). A filter (a liquid nitrogen cooled polycrystalline beryllium filter is used for wavelengths larger than 4/~), with a convenient Bragg cut-off, allows suppressing high-order contamination of the beam (halfwavelength neutrons are efficiently removed from the beam, but nominal wavelength neutrons are not scattered). The monochromator (and eventually the analyzer) can often be focused on the sample (or detector) horizontally (in the scattering plane) or vertically. In both cases, the increased count rate is obtained at the expense of the spectrometer resolution. The horizontal divergence of the beam at the different places of the spectrometer can be adjusted by collimators (Sollers slits; 30-60 ~ of arc are typical values). A spectrometer is a very flexible instrument; since the count rate is usually very small, optimizing the adjustment parameters is an essential part of neutron studies on adsorbed helium films. Let us now examine the specific case of diffraction by two-dimensional solid phases adsorbed on exfoliated graphite. 4.5.2.1. Signal. One certainly needs a large number of diffracting nuclei to work in favorable conditions. Since the size of the sample is usually limited by that of the beam, high specific area substrates are needed. As discussed later, a compromise between the specific area and the mosaicity of the substrate must be found. Exfoliated graphites are particularly adequate substrates from this point of view. The diffraction signal is proportional to the coherent scattering cross section of the adsorbed species; the most favorable case corresponds to 36Ar, with 77.9 barns, followed by nitrogen (32.8 barns at Q = 1.703/~l, Q-dependent for a diatomic molecule), whereas the corresponding value for 3He is 4.42, and only 1.34 for 4He. Also, the signal is reduced by the quantum fluctuations of the helium atoms around their equilibrium position: the attenuation due to the DebyeWaller factor is given by the expression exp(-2W(q) = 89

2,

(8)

where q is the momentum transfer vector and u the atomic displacement vector from the equilibrium position; parentheses indicate the averaged value. The mean squared displacement can be evaluated using the formula

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Fig. 20. (a) Neutron diffractometer D16 of the Institut Laue-Langevin (Grenoble) used for surface studies of adsorbed helium films (Lauter et al. 1990, 1991 and references therein). (b) Neutron time-of-flight spectrometer IN6 of the Institut Laue-Langevin (Grenoble) used for surface studies of adsorbed helium films (Lauter et al. 1992a,b and references therein).

Ch. 4, w

u2 _ 9

3He ADSORBED ON GRAPHITE

h/ mk a ~ ,

265

(9)

40DW where ODW is on the order of the Debye temperature (see the discussion given by Glyde 1994), and borrowing the Debye temperatures from the heat capacity results of Bretz et al. (1973). In condensed helium this factor may lead to a substantial reduction of the diffraction signals, particularly for the low density solids found in the second atomic layer (see Abraham and Broughton 1987). Clearly, helium is a weak scatterer. Not surprisingly, it, is necessary to use the spectrometer in a low resolution configuration with substantial focusing and weak collimation. Focusing the monochromator vertically does not degrade the resolution substantially for usual preferentially-oriented graphite samples in the configuration parallel to the scattering plane. A particular problem arises in neutron studies of 3He: this nucleus has an enormous absorption cross section, proportional to the neutron wavelength (5333 barns at 2 = 1.7981/~). In bulk 3He condensed phases, the resulting neutron penetration inside the sample is limited to a few tenths of mm. This means that for adsorbed 3He samples of much lower density, but of a few cm diameter, the absorption corrections are substantial, leading to background subtraction anomalies (see below). 4.5.2.2. Line position a n d lineshape. The reciprocal lattice of twodimensional crystals consists of Bragg rods normal to the crystal plane. If the scattering plane coincides with the crystal plane, one obtains the usual symmetric diffraction peaks. However, with a random orientation of the normal to the crystalline planes the diffraction signals have a very characteristic asymmetric lineshape, with a sharp edge at low angles, and a slow decrease for angles larger than that of the peak (see fig. 21). This is simply due to the fact that in the "tilted" reciprocal lattice a larger Q is needed to connect different Bragg rods. The rise of the line is controlled by the coherence length of the graphite crystallites, whereas the slow decrease is obviously related to the mosaic spread. In addition, the shape of measured lines is also affected by the finite resolution of the neutron spectrometer. Note that due to these effects the angular position of the peak is not exactly located at the Bragg value, and hence some data analysis is required even to extract the lattice parameter of the 2D crystal from a diffraction spectrum. Several models have been developed to analyze the measured neutron lineshapes. The first calculations are due to Warren (1941), and the name "Warren lineshape" is therefore commonly used. They were followed by extensive work (Ruland and Tompa 1968; Dutta et al. 1980; Dutta and Sinha 1981; Weling and Griffin 1981; Stephens et al. 1984; Sinha 1987; Schildberg and Lauter 1989). Recent calculations include the effect of the distribution of the size of the 2D crystallites, that leads to an additional foot at the leading edge

266

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Fig. 21. Neutron diffraction line of the first layer of 3He adsorbed on graphite at a total coverage of 0.178/~-2. The asymmetric shape is characteristic of 2D solids. A constant background has been subtracted from the raw data provided by a multidetector counter (Lauter et al. 1990, 1991). of the diffraction line (Schildberg and Lauter 1989). Coherence lengths and mosaic spreads determined by fitting diffraction spectra for different gases adsorbed on Grafoil, Papyex and ZYX are given in section 2.2.3. The orientation of the crystallites is discussed further in the context of the background (see below). The detailed analysis provides a precise determination of the lattice parameter of a solid two-dimensional film. It should be pointed out that in principle the complete diffraction spectrum is needed to determine the structure of the adsorbed film; unfortunately, one only observes in helium films the most intense reflection, and it is therefore necessary to assume a given two-dimensional lattice to infer the lattice parameter. In practice, however, triangular structures are the only ones providing areal density values consistent with those obtained from other methods. Neutron diffraction measurements provide a detailed picture of the structural evolution of the film as a function of coverage. A locking of the lattice parameter clearly identifies a commensurate phase, while a density increase proportional to the coverage is characteristic of an incommensurate phase. Melting is associated with a disappearance of the diffraction peak. Domain walls, that may be present in some phases, display typical satellites, as observed in D2 films; in helium films, unfortunately, their low intensity has precluded their direct obser-

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vation. Note also that neutron scattering allows a direct determination of the areal density of a homogeneous 2D solid film. The following formula are useful to analyze data corresponding to a triangular lattice (/9 is the areal density, a the nearest-neighbor distance, Q the neutron wavevector, q the wavevector transfer, and 0 the diffraction angle): 2

p = (a2 f ~ ) ,

(10)

q = 2Q sin 0,

(11 )

4~

q= a.~"

(12)

4r3q 2 P"-

8.7lr

(13) "

The last formula explains why neutron data are often represented as plots of the q measured at the diffraction peak as a function of the square root of the coverage. An overview of the results for 3He adsorbed on ZYX is presented in fig. 22; these data are discussed in detail in section 5. 4.5.2.3. Background. Neutrons interact with the substrate, as well as with the walls of the experimental cell, of the cryostat, etc. Fortunately, the lattice parameter of 2D helium solids is much larger than those usually found in bulk matter; diffraction peaks will therefore be located in a region (in angle, or, equivalently, in Q-space) almost free of bulk matter reflections. The use of long wavelength neutrons (2 on the order of 4.5/~) allows avoiding Bragg reflections from most solid bodies. In particular, the aluminum used for the cryostat and cell is then almost transparent to neutrons. An unfortunate exception is the 002 line of graphite, which will contaminate the measured spectra for q = 1.88/~-1. This line would not be present if the graphite sample were perfectly oriented with its c-axis perpendicular to the scattering plane of the spectrometer. It is therefore the mosaic spread of exfoliated graphites that gives rise to the 002 peak observed in the experimental spectra. The probability distribution for the orientation of crystallites in a particular orientation is easily determined performing "rocking curves"; note, however, that this distribution does not necessarily coincide with that of the surfaces exposed for adsorption. The probability distribution of the orientation of the c-axis angle with respect to the normal to the exfoliated graphite sheets may be described by the expression

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Fig. 22. Neutron diffraction data on 3He adsorbed on graphite. The density of the first and second layers is derived from the measured lattice parameter, as a function of the total coverage. At low coverages, a constant value is found, characteristic of the commensurate phase. Note the step-like shape of the first layer density around first and second layer completion. Diffraction peaks for the second layer solid have only been observed at high coverages. The insert shows the first layer diffraction-peak splitting (filled symbols) at coverages around 0.19 ,~-2 (Lauter et al. 1990, 1991).

p(O) sin 0

= A exp[-(0 / a)2 ]

(14)

(A is a normalization constant). The mosaic spread is defined by the expression 0 mos = 20" 1~"2".

(15)

The mosaic spread is found to be on the order of 30 ~ for Grafoil and for Papyex, and 10 ~ for ZYX (Stephens et al. 1984). Besides this relatively oriented fraction of the crystallites, it is necessary to include in the probability distribution a substantial amount of randomly oriented crystallites, which therefore contribute to p(O) as B sin(0), where B is a constant. This fraction is on the order of 50% for Grafoil (Kjems et al. 1976; Schildberg and Lauter 1989) as seen by neutron scattering and also by NMR lineshifts in 3He measurements at ultralow temperatures (Godfrin et al. 1988b and unpublished).

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Samples with a small mosaic spread can be used to investigate the region of the diffraction spectrum around the (002) peak, the loss of surface area associated with the high quality sample being usually over-compensated by the reduced and less structured background. A convenient feature of the graphite substrates is their moderate scattering cross-section and very small capture cross-section; also, the unavoidable (002) line finds some utility as a reference during the spectrometer calibration procedure. Among the miscellaneous problems associated with the background, note that small angle diffraction due to the size of the sample crystallites, as well as diffusescattering in the air, may be a problem in some experiments. Diffuse scattering from the cryostat aluminum enclosures, obviously excentric with respect to the sample, leads to unexpected counts at angles determined geometrically, hence predictable: multiple scattering can bring Bragg reflected neutrons into a detector that they should never have reached; this effect can be traced and corrected by computer simulations. Inelastic spectrometers can be helpful for diffraction studies, using the energy analysis to separate the elastic signal from the diffuse background originating from the cryostat, cell, etc. Even in optimized experimental conditions, the background signal accounts for a significant part of the measured signal. It is common practice to perform before the "measurement scan", a "background" or "empty cell scan", i.e. without adsorbed gas onto the graphite. The measurement and empty cell data, normalized to the same number of monitor counts, should in principle be directly subtracted; however, due to the 3He adsorption, a part of the background signal is masked in the measurement. One can either subtract an adjustable fraction of the background, determined in a convenient angular range where the signal is expected to be low, or, better, use Monte Carlo simulations to determine quantitatively the necessary corrections (in particular, when they are qdependent). Special care is required for the case of data around the (002) graphite peak. Its amplitude and position are affected by the adsorption of the adatoms, leading to typical oscillations in the difference spectra (Marti et al. 1977; Taub et al. 1977, Carneirol et al. 1981).

4.6. Other techniques Even though most of the results on adsorbed 3He have been obtained using heat capacity, NMR and neutron scattering techniques, there exist other interesting experimental possibilities. Capacitive techniques have been used at Kelvin temperatures to study the spreading pressure in 4He-films (Rabedeau 1989), and similar measurements could be performed on 3He films. Torsional oscillator

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Ch. 4, w

techniques have been applied to investigate helium films (Crowell and Reppy 1993 and references therein; Mohandas 1992). The latter reference includes a study of the melting transition in 3He films. The possibility to perform third sound measurements in adsorbed 4He leads to important results on the structure of films (see Zimmerli et al. 1992). Also, the density profile of the free surface of 4He has been determined by X-rays (Lurio et al. 1992).

5. Structure and phase diagram of the adsorbed films Our present knowledge of the structural properties of adsorbed 3He films originates, as seen in the previous section, from partial information originating from many different experimental techniques. Discovering the subtle properties of the systems described below has required imagination and physical intuition due to the relatively scarce experimental information and the richness of the physical systems. Some parts of the phase diagram are rather well known now, but other parts are only approximately understood. Many ideas about possible structures have substantially evolved with time, as new experimental results appeared, rendering the literature on the subject particularly difficult to follow. We summarize in the next paragraphs the present situation, trying to separate well established facts from what is still speculative.

5.1. Submonolayer coverages 5.1.1. Very low coverages A single 3He atom adsorbed onto the basal plane of a graphite substrate experiences an attractive potential on the order o f - 2 0 0 K with a corrugation of about 40 K at a wavevector of 0.4/~-i due to the periodicity of the graphite lattice; the binding energy of an atom in the graphite potential is about-136 K (see section 3.1). Classically, the atom would be trapped in the potential minima located at the centers of the graphite hexagons, but quantum mechanical effects allow a delocalization of the atoms and the formation of a band of relatively large width (about 10 K). The tunneling motion between substrate sites is so rapid that the experimental properties of low density adsorbed 3He gas correspond in principle to those of an ideally flat substrate. Band effects are thought to be important in the determination of the binding energy and should also lead to an effective mass increase of 3% with respect to the 3He atomic mass (Carlos and Cole 1980). In real graphite substrates, only a fraction of the surface area available for adsorption corresponds to the perfect graphite potential. A fraction of the sites

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271

(about 2-4% of a monolayer) has stronger binding energies than the average. Therefore, at very small coverages (<0.002/~-2) most of the adsorbed atoms are localized, a behavior first seen by heat capacity techniques (Bretz et al. 1973). The heat capacity temperature dependence, fitted by a Debye expression, leads to very low Debye temperatures (about 5 K), but it is not clear that this fit represents adequately the underlying physics. The localized atoms probably form patches at substrate defects, located preferentially along the steps of the graphite platelets, whereas the homogeneous phases occupy the central part of the platelets. Quantitative results were obtained by Elgin et al. (1974, 1978) and Cole et al. (1981) using adsorption isotherms at low coverages. Their model of the adsorption potential included heterogeneity through a distribution of binding energies, providing a satisfactory fit to the experimental data. The binding energy of the nth site is given by the expression

E(n) = Eg [1 + 1 +

n

1,

(16)

where Eg is the uniform graphite binding energy (136 K) and no, a characteristic coverage of 0.0023 atoms/~-2 found to describe the strongly bound fraction in low density films. Additional evidence for the existence of a localized fraction in the adsorbed film has been obtained by pulsed NMR techniques at Kelvin temperatures (Cowan and Kent 1984): about 0.5% of a monolayer were found to be strongly localized, and an additional amount on the order of 1.5% less tightly bound. Saunders et al. (1990) observed in low temperature NMR susceptibility measurements a solid-like signal corresponding to 0.022/~-2. Recent low temperature NMR measurements (Morhard et al. 1995) show that the amount of atoms described by a Curie law susceptibility, and hence in a solid-like structure, increases with coverage, reflecting the distribution of adsorption energies. The data (fig. 23) agree with previous results at low coverages, where the typical 2% of a monolayer are found, but this value reaches 4% of a monolayer at a coverage of about 0.04/~-2, leading to substantial corrections when determining the true density of submonolayer films. The presence of the strongly localized atoms may influence the properties of liquid and solid phases adsorbed on graphite. It is possible that the large, rather constant heat capacity measured by Greywall in fluid layers adsorbed on silver and on graphite substrates (Greywall and Busch 1988, 1990a) is due to spins localized at heterogeneous substrate sites, coupled by a distribution of exchange interactions which may involve the liquid, an effect already discussed for the electronic defects in the graphite substrate (see section 2.2.6). Evidence for such

272

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H. GODFRIN and H.-J. LAUTER

1

' ' ' ' I ' ' ' ' I ' T ' ' I ' ' ' '

I''''

I ''''

,,,.

.

I''-''

0.8

D

0.6

0.4

0

0

0.01

0.02

0.03

0.04

0.05

i, 0.06

0.07

total coverage (atoms/A2) Fig. 23. The amount of 2D solid, normalized to that corresponding to the commensurate phase, as a function of coverage, for temperatures in the mK range. The solid observed at low coverages is due to localization at heterogeneous sites. The linear increase above an onset coverage is interpreted as a coexistence of 2D liquid and commensurate solid. The open diamond indicates the value expected assuming the existence of a vacancy solid. (Morhard et al. 1995).

a behavior has been found for 3He adsorbed on Vycor (A. Golov, private communication). An anomaly with a characteristic temperature Tk of about 3.2 mK was found in heat capacity data that could be due to spin-glass freezing (Elser, unpublished, Greywall 1994). This is in good agreement with magnetization measurements (Siqueira et al. 1994) for 3He adsorbed on preplated graphite at submonolayer coverages which also display an anomaly at 3 mK. Other interpretations for this effect ate given in the next section. 5.1.2. The first layer fluid phase A fluid phase is observed up to a coverage (at T = 0) of about 0.043/~-2, where a fluid-commensurate phase coexistence region begins. The existence of this fluid phase is proven at high temperatures (above 1 K) by heat capacity measurements (Bretz et al. 1973): this property is temperature independent and its magnitude is very close to the expected classical value of kB per atom. At temperature on the order of 1 K the data can be described by virial expansions. At lower temperatures, quantum effects lead to a characteristic plateau in the heat

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Fig. 24. Heat capacity of submonolayer 3He films, dominated by the contribution of the twodimensional liquid; note the characteristic plateau around 200 mK and the beginning of the linear Fermi liquid regime below 100 mK. The heat capacity is small in the commensurate phase, around n = 0.06366/~-2. Data from Bretz et al. (1973). capacity (fig. 24), similar to that observed in bulk liquid 3He. This quantum nondegenerate regime has been investigated theoretically by Dyugaiev (1990). Quantum degeneracy effects were observed (Bretz et al. 1973) at temperatures of about 0.1 K and studied in detail by Greywall and Busch (1988, 1990) and Greywall (1990). A linear temperature dependence of the heat capacity is measured in the low temperature regime (fig. 25), as expected for a twodimensional Fermi liquid. The analysis of the data yields the effective mass of the 3He quasi particles as a function of the total density m* m

3h 2 C ~zk2 m A T

(17)

where A is the surface area and C the heat capacity (Greywall 1990). Note that the true liquid density is smaller than the average density after correcting for

274

H. GODFRIN and H.-J. LAUTER

I ......

M'~

I

First layer -..

v

v

I

P1 = 0 . 0 4 t

I k

/

ato

Ch. 4, w

0

~ 0

--

't--

t~

0

0

l,.

10

!

20

!

30

!

4O

.,

50

t e m p e r a t u r e (inK) Fig. 25. Heat capacity of the first layer fluid as a function of coverage, in the low temperature regime. The linear behavior is characteristic of a Fermi fluid. Note that an anomaly is found at about 3.2 inK, as well as a constant contribution (Greywall and Busch 1990a).

heterogeneity (Morhard et al. 1995; fig. 26). The coverage dependence of the effective mass has been attributed by Chubukov and Sokol (unpublished; see Greywall 1994) to p-wave pairing, dominant with respect to s-wave interactions. Note, however, that the subtle differences between the scattering theory in twodimensions with respect to that in three dimensions (Averbuch 1986) are not taken into account. The overall temperature dependence of the heat capacity is very similar to that measured in 3D liquid 3He. However, the 2D nature of the adsorbed 3He fluid is revealed in heat capacity data by two features: in the "classical" high temperature regime a limiting value of kB per atom is observed, and in the "quantum regime", at low temperatures, the heat capacity is proportional to the total surface area A, but not to the number of adsorbed atoms at low coverages. NMR measurements of the susceptibility of the submonolayer fluid have been performed at London and Grenoble (Saunders et al. 1990; Morhard et al. 1995). The data can be described by a two-dimensional Fermi liquid expression, and also by the phenomenological formula (Dyugaev 1990) (fig. 27). The Fermi temperature is expected to be proportional to the density in the "Fermi gas" regime (in two dimensions) TF = zth2p/mkB, where p is the areal density of the fluid. In practice, the signal is too small to be accurately measured at such small coverages, and there are no susceptibility data in this range. For coverages larger than 0.01/~-2 the effective Fermi temperatures are on the order of 0.2 K; they

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3He ADSORBED ON GRAPHITE

i

l

l

i

l

l

l

l

l

l

l

l

l

l

l

l

l

i

i

l

275

i

l

l

l

3.5

E

3

m

2.5

m

o

E

-

2 -

m

1.5-

m

l

1

l

0

l

l

i

l

l

l

l

0.01

P liquid

l

l

l

0.02

l

l

l

l

l

l

0.03

l

l

l

l

l

0.04

(atoms/A 2)

l

0.05

Fig. 26. Coverage dependence of the effective mass of 2D liquid 3He adsorbed on graphite in the first layer (Greywall and Busch 1990a; Greywall 1990); the data corrected to take into account the atoms localized at heterogeneous adsorption sites are indicated by filled squares (Morhard et al. 1995). decrease as a function of density (fig. 28), as seen in bulk liquid 3He. At the density where the commensurate phase begins to coexist with the liquid, the maximum susceptibility enhancement compared to the non-interacting Fermi gas is on the order of ten, and the effective mass is about three times the bare 3He mass. This coverage range corresponds therefore to a strongly interacting regime, that has attracted considerable theoretical interest. Within the Landau picture of a Fermi liquid, the susceptibility enhancement is

Z(0._.....~)= ~.(m* Im) Xo(0) l+F0 a

(18)

This expression suggests a simple way to eliminate the explicit density dependence of the parameters (known experimentally but not theoretically) in order to

276

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Fig. 27. Susceptibility as a function of temperature for an adsorbed 3He sample at a coverage 0.0523 atomsL/k2, in the liquid-commensurate phase coexistence regime. Filled diamonds denote the total susceptibility. Open diamonds correspond to the 2D Fermi fluid susceptibility, fitted by the theoretical expressions given in the text (Morhard et al. 1995). compare the theoretical predictions to the data by combining the susceptibility and the mass enhancements, as shown in fig. 29. The validity of a "Fermi liquid" description for such strongly interacting systems is not obvious; Bouchaud and Lhuillier (1989) suggested a possible dimerization of the 3He particles and the disappearance of the Fermi surface. It should be pointed out that the prediction of the "quasi-ferromagnetic" paramagnon model (BEal-Monod et al. 1968, 1980)) as well as that of the "quasi-localized" model (Vollhardt 1984) agree rather well with the data. The limiting value at large effective mass also corresponds well with the prediction of the "spin-glass" model of Castaing (1980). It is unfortunate that models based on very different physical pictures lead to very similar results. The low density regime should be investigated experimentally, despite the technical difficulties, to understand the evolution between the liquid and gas regimes. Note that the susceptibility data do not display any particular feature corresponding to the 2D nature of the films. The linear density depend-

Ch. 4, w

4ooi

3He ADSORBED ON GRAPHITE

277

/I

,oof / I 250

,,

200

I.-

15o 100 50 0 0

0.01

0.02

P total

0.03

0.04

0.05

0.06

(at~

Fig. 28. Coverage dependence of the magnetic Fermi temperature of 2D submonolayer fluid 3He films adsorbed on graphite. Note that the constant value at high coverages correspond to the coexistence of the liquid and the commensurate phase (Morhard et al. 1995).

ence of the Fermi temperature would be a good indicator but this behavior is expected at densities below the presently investigated range (fig. 28). Using TF as a scaling parameter leads to almost identical results for the experimental susceptibilities of 2D and 3D 3He. The fluid phase at submonolayer coverages is presently interpreted as a homogeneous fluid even down to zero temperature, in agreement with theoretical predictions (Miller and Nosanow 1978). It should be pointed out that recent variational Monte Carlo calculations (Brami et al. 1994), however, suggest that this phase might be unstable at low temperatures against the formation of a self bound fluid phase with a density on the order of 0.02/~-2 (see sections 5.2.1 and 5.3 for a discussion of the second and third layers fluid). The experimental data described above favor a description in terms of a homogeneous phase. The reasons for the discrepancy are probably associated with approximations in the potentials; the calculation strongly suggests, however, that the 2D fluid is very close to a phase separation instability. At milliKelvin temperatures an anomalous (temperature independent) heat capacity has been observed, as well as a sharp feature at 3.2 mK (Greywall and

278

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Ch. 4, {}5

Fig. 29. The Landau parameter 1+F0a as a function of the inverse effective mass of 2D fluid 3He films. Filled squares: first layer fluid (Morhard et al. 1995); open squares: second layer fluid in 4He preplated films (Lusher et al. 1991); effective masses are taken from heat capacity data (Greywall 1990). Crosses. bulk liquid 3He. The theoretical predictions for the quasi-localized model (dashed line), the paramagnon model (dot-dashed line) and the limit for the spin-glass model (dot) are also shown.

Busch 1990a). A small peak at 3 mK has also been observed recently in the susceptibility of fluid phases of 3He adsorbed onto preplated graphite samples (Siqueira et al. 1994). The origin of the "constant" term may be related to the localized spins (see section 5.1.1.). The low temperature anomaly may be attributed to a superfluid transition (Greywall and Busch 1990a; Greywall 1994; Chubukov and Sokol unpublished), to a gas-liquid or fluid-registered phase transition (Brami et al. 1994), or to a freezing of the localized spins (see section 5.1.1.). The fact that the N M R measurements do not detect any shift or linewidth change at the low temperature anomaly, weaken the arguments favoring the possibility of a p-wave superfluid transition.

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Finally, let us mention that the 3.2 mK anomaly discussed above could be related to a transition to the commensurate phase below some temperature in the milliKelvin range (Brami et al. 1994). It is difficult to draw a conclusion about the nature of the low coverage films at very low temperature; present evidence suggests that the ground state corresponds to a homogeneous fluid. The observed anomalies are probably related to atoms localized at heterogeneous sites, magnetically coupled by random exchange interactions; this initiates a new research line on "dirty Fermions" which should expand in the next few years. The question of p-wave superfluidity in the 2D liquid remains open. 5.1.3. The commensurate phase The lowest density solid-like phase observed at submonolayer coverages is the commensurate phase. This structure is very stable, as can be deduced from the considerable region of the phase diagram occupied by this phase. Even though the graphite potential is not strong enough to localize the 3He atoms at low densities, the increase of the two-dimensional spreading pressure with the fluid density is sufficient to induce a transition to a state where each 3He atom is bound to the substrate lattice site. A commensurate phase results, with one 3He atom occupying every third hexagonal site of the graphite lattice, with an areal density of 0.06366/~-2 determined by the graphite lattice parameter (fig. 30). This ~/3R30 ~ phase (also called "1/3", "registered", or, following Greywall's notation, Rla ) is quite generally found for several gases adsorbed on graphite. The existence of this phase for 3He has been first demonstrated by heat capacity experiments (Bretz et al. 1973). Large heat capacity peaks were observed,

Fig. 30. The commensurate phase of 3He on a graphite substrate. The 3He atoms occupy one out of three sites provided by the graphite hexagons.

280

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Fig. 31. Heat capacity of the commensurate phase (areal density on the order of 0.064 ~-2) as a function of temperature (Bretz et al 1973).

and correctly attributed to an order-disorder transition between the commensurate phase and the fluid phases (fig. 31). These careful measurements allowed to establish the boundaries of the phase diagram shown in fig. 32 for temperatures on the order of 1 K. The observed second order transitions were shown to be consistent with the two-dimensional 3-states Potts model prediction (Alexander 1975; den Nijs 1979, 1988), as expected from the observation that three equivalent sublattices can be occupied on the graphite substrate. In particular, the critical exponent 0.36 derived experimentally (Bretz 1975, 1977; Dash 1978; Vilches 1980) is rather close to the 1/3 value expected from theory, and identical to that found in para-hydrogen adsorbed on graphite (Motteler and Dash 1985). Note, however, that Chae and Bretz (1979) find for 4He films adsorbed on different types of graphite considerable deviations from the Potts exponent: the transition becomes almost Ising-like under the influence of the graphite surface defects acting as nucleation centers. Path-integral Monte Carlo calculations (Abraham and Broughton 1987) lead to a nice visualization of the commensu-

Ch. 4, w

3He ADSORBED ON GRAPHITE

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282

Ch. 4, w

H. GODFRIN and H.-J. LAUTER

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Energy {THzl Fig. 33. Neutron inelastic spectrum of 3He adsorbed on graphite for Q = 1.70 A-1" the coverage is 90% of that corresponding to the full commensurate phase and the temperature 0.85 K. The full line corresponds to a lattice dynamical model. Note the existence of a gap in the energy spectrum due to the commensuration with respect to the substrate (Frank et al. 1991). of the peak is seen at a coverage associated with the transition into the intermediate phase. The coverage where the maximum intensity is found is associated with "the best commensurate phase", i.e. that where the number of atoms is equal to the number of graphite sites. This measurement determines the "neutron scattering commensurate coverage scale" (see section 3.4). In principle the intensity at lower coverages provides information about the amount of solid formed at each coverage. However, the intensity also depends on the size and the distribution of the solid "islands" on the graphite platelets. Commensurate phase signals have been observed by neutron scattering (Lauter et al. 1990, 1991) at coverages as low as 0.050/~-2, in agreement with the phase diagram of fig. 32. The commensurate nature of this phase is also revealed by inelastic neutron scattering experiments (see section 4.5). The lack of translational invariance produces a gap A at the zone center in the acoustic branch of the dispersion re-

Ch. 4, w

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283

lation; the magnitude of the gap is related to the corrugation of the adsorption potential. The value directly determined by neutron scattering (Frank et al. 1991; Lauter et al. 1991) is 10.9 K (fig. 33). The data also show that the width of the density of states is relatively large, reflecting the fact that interactions among helium atoms play an important role in the effective potential that localizes the 3He particles in the registered phase. It is interesting to compare the behavior of 3He with that of 4He at the same coverages. The gap value determined by neutron scattering for this isotope is identical to that found for 3He (Frank et al. 1991; Lauter et al. 1991). Recent heat capacity data for 4He adsorbed on Grafoil (Greywall 1993) can be analyzed considering that a significant contribution is due to defects and the regular part to a gap A = 10.5 K in the phonon spectrum. The agreement with the neutron scattering value A = 11 K is remarkable. Theoretical calculations predict values substantially larger than the experimental ones; successive refinements of the calculations, however, lead to a better agreement, with values on the order of 16 K (Bruch and Gottlieb 1990). The fluid-commensurate phase boundary of the 3He phase diagram, at low temperatures, was not investigated in detail in the early heat capacity measurements by the Seattle group. However, unpublished measurements located the onset of formation of commensurate phase roughly at a coverage on the order of 0.04-0.05 A -2. Saunders et al. (1990) observed by NMR techniques the formation of commensurate solid at much lower coverages (0.032 ~-2); this has not been confirmed by later measurements, and the effect was understood as originating from annealing problems (see section 4.1.2). Greywall and Busch (1990b) observed by heat capacity the coexistence of liquid and commensurate phases; heat capacity isotherms as a function of coverage display a characteristic linear dependence in this range (see fig. 16). The onset was located at a coverage 0.043 ]k-2, and it was suggested that the coexistence involved a commensurate solid containing 6% of vacancies. A similar suggestion was made by Guyer (1977), who predicted vacancy induced ferromagnetism, and by Carneiro (1981), in order to explain the high temperature heat capacity of the Seattle group. Recent NMR measurements, however, provide a different picture (fig. 23, Morhard et al. 1995). They locate the onset coverage at 0.0449/~-2 on Grafoil, and estimate the onset density on a perfect graphite substrate to occur at 0.0434 ~-2. The vacancy solid hypothesis is not verified by the NMR measurements: instead, the data show that the solid coexisting with the liquid is that having the highest melting temperature, which is most likely the pure commensurate phase. The discrepancy with Greywall's scale is thought to originate from the fact that the commensurate phase coverage was associated to the minimum observed in the heat capacity isotherm in the vicinity of the commensurate phase coverage, and not to the extrapolation of the linear coexistence regime to zero

284

H. GODFRIN and H.-J. LAUTER 0.35

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heat capacity. The presence of a small amount of liquid at small binding energy heterogeneous sites even when the full commensurate phase has been achieved in the regular graphite regions explains the shift in the coverage location of the minimum heat capacity. This subtle effect shifts Greywall's coverage scale by about 4%. This possibility was already mentioned by Greywall and Busch in their original article (1990b). The commensurate phase is also observed in a little coverage range above the theoretical value 0.06366 ~-2. This can be seen from the heat capacity data of Seattle (fig. 32), in the low temperature data of Greywall and Busch (1990b), in the neutron scattering data of Nielsen et al. (1977) and Lauter et al. (1990, 1991) (fig. 22); see also the spreading pressure measurements done for 4He by Rabedeau (1989). As explained above, the liquid at low binding energy sites solidifies in this range; it is therefore not surprising that the commensurate phase exists in the regular sites for coverages about 4% higher than the pure commensurate phase value. Other effects, however, can be expected. The presence of interstitials has often been invoked to describe an "overfilled" registered phase. The nature of the defects present in the compressed commensurate phase is not

Ch. 4, w

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clear, and their study is complicated by the fact that the heterogeneity effects are of the same magnitude for presently available substrates. The heat capacity isotherms at high temperatures (Hering, Ph.D. thesis, unpublished) indicate the heat capacity increases linearly with coverage in this regime, with a relatively strong temperature dependence, probably corresponding to a thermal activated process. In the low temperature data of Greywall and Busch (fig. 16) this contribution has become smaller than that of the remnant liquid, leading to the shift in the coverage scale mentioned above. A small dip in the transverse relaxation time/'2 has been observed in pulsed NMR measurements at Kelvin temperatures (Richards 1980; Satoh and Sugawara 1980). This feature can be interpreted as a reduction of the motional narrowing for the pure commensurate phase: the "defects" (vacancies, interstitials) present at neighboring coverages increase the relaxation rates. The correlation time is governed by quantum exchange of neighboring particles; its magnitude has been estimated to be on the order of 30 ns (Cowan et al. 1987). The magnetic interactions due to quantum exchange are rather small; ferromagnetic deviations from the Curie law of non-interacting spins have been observed by NMR at milliKelvin temperatures around the commensurate phase coverage (Godfrin 1987; Godfrin et al. 1990, 1991; Siqueira et al. 1992; Rapp and Godfrin 1993). The magnetic (exchange) heat capacity contribution observed (fig. 16) in the data of Greywall and Busch (1990b) at milliKelvin temperatures displays several features at coverages that correlate well with those corresponding to structural transitions. At the pure commensurate phase coverage, the spin contribution to the heat capacity varies as the inverse of the temperature (Greywall and Busch 1990b), instead of following the T -2 behavior found at higher coverages. This suggests that more information about the structure of the films in this coverage range could be obtained from experiments at submilliKelvin temperatures. 5.1.4. The intermediate coverage region Some features characteristic of the commensurate phase, as discussed above, persist at coverages slightly above "the best commensurate phase". It is clear from the experimental point of view, however, that the commensurate phase does not directly transform into the incommensurate phase as the density is increased. A complicated region of the phase diagram called the "intermediate region", is found at densities between 0.064 and 0.078/~-2. This region of the phase diagram (fig. 32) was first explored by Hering et al. (1976), who showed that it is characterized by narrow melting peaks at about 1 K. Above this temperature, the "re-entrant" fluid exhibits some peculiar properties. According to path-integral Monte Carlo calculations (Abraham and Broughton 1987), the probability contours in this high density fluid display characteristic domain walls, as suggested by the experimental data of Motteler (1986) and in agree-

286

H. GODFRIN and H.-J. LAUTER

Ch. 4, {}5

ment with the renormalization group calculations of Halpin-Healy and Kardar (1986). What is the structure of the low temperature phases? The answer to this question is not obvious. The system could choose to form commensurate phases for adequate coverages, with phase coexistence for intermediate coverages. Alternatively, the formation of domain walls of different kinds would allow to accommodate the adsorbed atoms through a continuous variation of the wall separation as the density is increased. The domain wall picture has been often favored by experimental and theoretical investigators in the field; recently, however, the existence of a commensurate phase Rib in this intermediate coverage region has been suggested by GreywaU and Busch (1990b). We describe in the following paragraphs the experimental evidence (certainly not conclusive) supporting these interpretations. The identification of the intermediate phase as a domain-wall solid is supported by the calculations of Halpin-Healy and Kardar (1986). The phase diagram of Halpin-Healy and Kardar, however, is only qualitative; since it corresponds to 4He, it can only be considered as a reasonable approximation for 3He at temperatures above 1 K, where quantum statistics is not relevant. Several phases are predicted: above the commensurate phase density the domain-wall fluid, a striped phase, and a hexagonal domain wall phase, separated by coexistence domains. The identification of the intermediate regime as a domain wall phase results from neutron scattering measurements (Lauter et al. 1990, 1991). The neutron diffraction peaks are very similar to those measured on D 2 adsorbed on graphite, where the larger magnitude of the D 2 spectra allow the identification as a striped super-heavy domain wall phase (Freimuth et al. 1990). The much less favorable scattering conditions in the case of 3He did not allow the observation of satellites, which constitute the signature of the domain walls. The variation of the diffraction angle (see fig. 22) and the general shape of the diffraction signals is, however, very suggestive of a similar behavior for both quantum systems. The phase diagrams of the hydrogen isotopes are substantially different; in particular, in the intermediate regime several phases have been observed in D 2 by neutron scattering and low energy electron diffraction, which correspond to a striped super-heavy domain wall phase, followed as a function of density by a hexagonal density-modulated phase that undergoes locking transitions into higher order commensurate phases. The phase diagram of hydrogen, on the other hand, seems to be simpler: only the striped domain wall phase is presently known at low temperatures in the intermediate coverage regime. Present neutron data for 3He or 4He are unfortunately not as detailed as those for D2 and H2. A comparison with theoretical models is certainly difficult at the present stage; the case of the hydrogen isotopes has been discussed in detail, including a summary of the theoretical situation, by Freimuth et al. (1990). A

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general discussion of the domain wall theory of the commensurateincommensurate transitions has been given by den Nijs (1988). A different interpretation was given to recent heat capacity data on 4He and 3He at low temperatures. Let us discuss first the most recent results for 4He (Greywall and Busch 1991; Greywall 1993). The coverage range immediately above the Rla phase, up to 0.070/~-2, is found to be consistent with the striped super heavy domain wall phase proposed by Halpin-Healy and Kardar, but coexistence between two commensurate phases is certainly not excluded. A dramatic change in the heat capacity isotherms is observed at this last coverage, suggesting either a change of the nature of the walls from striped to hexagonal, or the beginning of a coexistence region with a second commensurate phase corresponding to an areal density of 0.0764/~k -2. This phase is assumed to be a 2/5 commensurate phase named Rib, where this fraction means that two out of every five graphite hexagons are occupied by helium atoms. Indeed, in this coverage range a maximum melting temperature and the largest heat capacity peak amplitude are found at this coverage for 4He. The order-disorder transition is found to be of first order, contrary to the Rla commensurate phase, which is of second order. Motteler and Dash (1985, unpublished) in fact proposed a similar commensuration for 4He and hydrogen isotopes, consisting basically of an array of domain walls corresponding to an average density 0.0764/~-2. Note that this is not a domain wall in the sense of the theory of Halpin-Healy and Kardar, where the domain wall density varies continuously; in this case, the domain walls lock to the substrate in a particular structure. Evidence for a third commensurate phase (Rio) corresponding to a structure where 3/7 of the graphite sites are occupied is found for adsorbed 4He in a narrow coverage range dose to 0.082/~-2, for temperatures lower than 0.6 K (Greywall and Busch 1991; Greywall 1993). The interpretation of heat capacity results for 3He is similar, but less clear (Greywall and Busch 1990b). A maximum in the spin contribution is observed at approximately this location; it is placed by these authors in the coverage region above the hypothetical registered phase Rib, but may well be attributed to this phase within errors in coverage scales. A coexistence is assumed between the Rib phase and the incommensurate phase at higher coverages, i.e. there is no evidence for the existence of the 3/7 phase, possibly seen in the 4He data. Several remarks can be made to conclude this section. It is clear, both theoretically and experimentally, that a strong modulation of the adsorbed layer is imposed on the layer by the substrate corrugation potential. The formation of domain walls is also a physically sound concept; pinning of these walls, however, accompanied by structural relaxation, can stabilize in principle a variety of commensurate phases. This is particularly the case, for simple topological reasons, for hexagonal domain walls. This type of wall is also favored at high tem-

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Ch. 4, w

perature due to the large entropy associated with low energy breathing models of a hexagonal network (Villain 1980). Also, finite size and impurities can have profound effects on the nature of the domain wall phases (see den Nijs 1988 and references therein). Frustration effects play a significant role in the structural organization of the adsorbed layers. The problem can be mapped onto that of the response of arrays of Josephson junctions as a function of the external magnetic field. In this case domain-wall superlattice states have been observed (Th6ron et al. 1994), corresponding to either striped or hexagonal domain walls; a transition to a vacancy two-dimensional superlattice has been predicted. Numerical calculations seem to be a promising technique to gain further insight into the family of ground states of these frustrated systems.

5.1.5. The incommensurate phase At coverages larger than 0.078 ,~-2 the first layer structure is much simpler, and understood. It corresponds to an incommensurate (or "floating") solids of triangular structure, as shown by heat capacity and neutron scattering studies mentioned above. The decoupling from the substrate results from the increase of the zero-point energy as the coverage is increased: the depth of the corrugation potential is about 40 K whereas the Debye temperature of the films, a measure of the zero-point energy, evolves from about 17 K at low coverages to values above 50 K near monolayer completion, as seen in fig. 35 (Hering et al. 1976, Hering, Ph.D. Thesis, unpublished; Ecke and Dash 1983; Greywall 1990). It is thus reasonable to expect that the properties of the incommensurate phase are not strongly affected by the substrate corrugation, except perhaps at the lowest densities. Interestingly, at the density 0.080/~-2 corresponding to a possible commensurate phase in 4He films (Greywall 1993), some anomalies in the heat capacity isotherms can be seen in the Seattle data for 3He (graphs communicated by O.E. Vilches, data from Hering, Ph.D. Thesis, unpublished). This may indicate that the incommensurate phase at the lowest coverage could still be influenced by the substrate. In all this coverage range, the lattice parameter measured by neutron scattering (fig. 22) corresponds to areal densities proportional to the total coverage, showing that this phase is uniform. This fact was used to establish the "incommensurate coverage scale" (see section 3.4). The incommensurate solid phase melts at high temperatures (fig. 32) by a mechanism which is not well understood. According to heat capacity data (Hering et al. 1976; Vilches 1980; Ecke and Dash 1983), 3He melting peaks are narrow, while those corresponding to 4He are rather rounded. Note that at the melting temperatures of several Kelvin, the effects of quantum statistics on the transition should be small. Near layer promotion, however, the 4He melting peaks sharpen considerably. This effect has been explained by Elgin and Good-

Ch. 4, {}5

3He ADSORBED ON GRAPHITE 55

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Fig. 35. Debye temperature of submonolayer 2D incommensurate solid 3He films adsorbed on Grafoil as a function of the areal density, determined by heat capacity (Hering et al. 1976; Hering and Vilches 1973; Hering 1974, Ph.D. Thesis, unpublished).

stein (1974): the increase in spreading pressure associated with the melting transition, characteristic of a process at constant surface area, leads to promotion of some atoms to the second layer; the enhanced heat capacity reflects the difference in binding energy of the different layers. The heat capacity of the solid near melting displays considerable deviations with respect to the low temperature Debye values. This has been interpreted as being due to the creation of thermally activated defects, like vacancies (Ecke and Dash 1983), with activation energies on the order of 20 K. Neutron measurements of the coherence length, the peak intensity, and the structure factor (Feile et al. 1982) are consistent with a defect-mediated melting mechanism. A Kosterlitz-Thouless transition was inferred from NMR relaxation time measurements (Widom et al. 1979), but a different interpretation of these data, in terms of vacancies, has been given by Ecke and Dash (1983). A first order transition, broadened by substrate heterogeneity, has been claimed in 4He films (Ecke and Dash 1983; Hurlbut and Dash 1984, 1985); however, a continu-

290

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Ch. 4, w

ous transition can also explain these results (Strandburg et al. 1985). The fact that the melting peaks become narrower as the coverage is increased (Bretz et al. 1972; Hering et al. 1976) has led Elgin and Goodstein (1974) to suggest that second layer promotion may enhance the heat capacity peaks and perhaps participate to the melting mechanism. Detailed discussions can be found in the review of melting theories in two dimensions and experimental results published by Strandburg (1988). The nuclear magnetic properties of the 2D incommensurate solid have been investigated by several techniques; the results agree well with the picture of a floating incommensurate solid described above for this coverage range. Pulsed NMR measurements (Cowan et al. 1987, and references therein) showed that quantum exchange of 3He particles leads to a strong density dependence of the NMR linewidth, due to the modulation of the dipolar interaction, similar to that observed in bulk solid 3He. The spin heat capacity at low temperatures (Greywall 1990) displays characteristic T-2 tails, but the order of magnitude of the inferred exchange interactions (J = 0.1 mK) is substantially smaller than those obtained from pulsed NMR. Direct effects on the nuclear magnetization (Godfrin 1987) provide even smaller values (J < 0.05 mK). The origin of the discrepancy is probably that the analysis of the data within the framework of the Heisenberg model is inadequate. A discussion of the nuclear magnetism has been given by Godfrin and Rapp (Advances in Physics, in press). At a coverage on the order of 0.108/~-2 (Hering et al. 1976), perhaps 2% smaller (Bretz et al. 1973; van Sciver and Vilches 1978), or 0.109/~-2 (Greywall 1990) promotion of atoms to the second layer is observed by heat capacity techniques, very sensitive to the presence of a second layer fluid contribution. According to neutron scattering, the measured first layer density is substantially smaller (certainly smaller than 0.105/~-2) when promotion to the second layer begins, as seen by a deviation of the measured density versus coverage with respect to a lineal dependence (fig. 22). This illustrates the discussion on coverage scales given in section 3.4: the real density of the first layer is smaller than what could be inferred from coverages calibrated at the commensurate phase coverage. The first layer is compressed further as the total coverage increases, an effect clearly seen in the neutron data of fig. 22. The density of the first layer at coverages between 0.11/~-2 and 0.15 ,~-2 is 0.107/~-2. In the coverage range 0.160.21/~-2 a rapid compression of the first layer is observed by neutron scattering, precisely in the range where heat capacity experiments place solidification of the second layer and promotion to the third layer (this point is discussed in the next section). The shape of the diffraction peaks is better described here by a double peak, possibly indicating a slight distortion of the unit cell due to a one directional registry with the substrate, or some effect associated with the solidification and commensuration of the second layer. Unfortunately, the uncertainty

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in coverage scales does not allow a clear correlation with the most recent heat capacity data on the second layer solidification (Greywall 1990). The maximum density for the first layer is practically reached at a coverage of 0.2/~-2 (neutron commensurate scale), and its value (0.1106/~-2 measured by diffraction) may correspond to an 8 x 8, over-structure with respect to the graphite; it remains constant at least until coverages as high as 0.33/~-2. Note that the value 0.114/~-2 for the compressed monolayer density has been used to analyze recent heat capacity results; this value is determined by using a 5% compression (taken from neutron scattering) of the "monolayer coverage" n = 0.109 ~-2 found in heat capacity experiments based on the commensurate coverage scale. As seen in fig. 36, the first layer densities used to analyze heat capacity data are substantially different from the microscopic neutron values: the nonuniformity of the substrate leads to an "effective area" of the substrate that increases slightly with coverage. This error in the absolute value of the coverages causes serious problems in the interpretation of second layer commensurate phases, as seen below.

Fig. 36. Density of the 3He layers adsorbed on graphite as a function of coverage. Neutron scattering data (Lauter et al. 1990, 1991) are shown using open symbols; a splitting of the neutron peaks (symbol X) is seen at second layer completion. Filled symbols correspond to the values used by Greywall (1990) to analyze heat capacity experiments. The symbol + for the "second layer" corresponds to NMR measurements on a related system (3He adsorbed onto 4He preplated graphite; Lusher et al. 1991).

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Ch. 4, w

5.2. Second layer 5.2.1. The second layer fluid phase At coverages above that of monolayer completion, a fluid phase is observed in the second layer (fig. 37): the compressed monolayer acts as a rather smooth, weakly attractive substrate, with a characteristic binding energy of about 25 K (van Sciver and Vilches 1978). This second layer fluid is of particular interest, since it exists in a large coverage range. The properties of the second layer fluid have been investigated by heat capacity measurements in the temperature range 45 m K - 4 K by van Sciver and Vilches (1978), and at lower temperatures, between 2 and 50 mK by Greywall (1990) and Greywall and Busch (1990a). The high temperature data are consistent with the quantum non-degenerate Fermi fluid behavior known for bulk 3He and already discussed for the first layer fluid (section 5.1.2). In particular, a

Fig. 37. Phase diagram of the second layer of 3He adsorbed on graphite as a function of the total coverage and temperature. The diamonds identify heat capacity peaks (van Sciver and Vilches 1978; van Sciver 1978); their nominal coverages have been divided by 1.045 to agree with Greywall' s scale (see text) and allow a comparison with the low temperaturephase transitions (Greywall 1990). The shaded area corresponds to the fluid coexistence with the R2a registered phase. A possible second commensurate phase may exist at higher coverages (R2b). IC2 is an incommensurate solid phase. The striped lines indicate third and fourth layer promotions. The phase boundaries are only drawn as a guide to the eye.

Ch. 4, w

3He A D S O R B E D O N G R A P H I T E

293

Fig. 38. Heat capacity of the second layer fluid determined by van Sciver and Vilches (1978) as a function of temperature, for different coverages indicated on the figure. See also the results for the first layer liquid (fig. 24).

Fig. 39. Heat capacity of the second layer fluid as a function of coverage, in the low temperature regime. The linear behavior is characteristic of a Fermi fluid. Note that an anomaly is found at about 3.2 mK, as well as a constant contribution (Greywall and Busch 1990a). See also the data for the first layer, fig. 25.

294

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

rounded plateau is seen at temperatures on the order of 0.1 K (fig. 38), and at lower temperatures the heat capacity tends toward a linear behavior. This regime is clearly seen in the data of Greywall and Busch (fig. 39). The density dependence of the effective mass (Greywall 1990; see section 5.1.2) is similar to that of the first layer fluid at the same density (fig. 40). It should be pointed out, however, that the density of the fluid is not well known in this coverage range; Greywall and Busch assumed that the first layer density remains constant, whereas neutron scattering data show a clear compression of the first layer above second layer promotion. Since a substantial fraction of the atoms added in this regime go into the first layer, the real second layer densities are smaller than the reported ones by about 10%; unfortunately, it is not obvious to correlate the heat capacity and neutron scattering coverage scales to determine the correct liquid densities. The heat capacity of the liquid is large compared to that of the solid at low temperatures; it is therefore easy to subtract the small contribution of the dense solid first layer. In NMR measurements, on the other hand, the magnetic susceptibility of the second layer is difficult to measure, since its value is small compared to that of the first layer solid. Early experiments at Grenoble at very low temperatures (Godfrin 1987) showed that in this coverage range the susceptibility of the second layer atoms is very small, in agreement with a fluid second layer in the degenerate regime. In order to suppress the first layer contribution, Lusher et al. (1991a), preplated the substrate by a convenient amount of 4He (a solid monolayer). Due to the preferential adsorption of this isotope (see section

.. o

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1

. this work

Van Sciver &Vilches

" McLean 1

0.02

004

I

0.06

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0.08

Fig. 40. Effective mass of two-dimensional liquid 3He in the second layer (Greywall 1990); note that the coverages values are only approximate (see first layer results, fig. 26).

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Fig. 41. Magnetic Fermi temperature of the liquid 3He fraction found in adsorbed 3He layers on top of a graphite substrate preplated by one monolayerof 4He, as a function of the total helium density n = n4 + n3, where n4 is 0.114/~-2 (Lusheret al. 1991).

3.1), 3He atoms remain in the second layer, with a binding energy of about 30 K (Cheng et al. 1993), slightly larger than that discussed above for pure 3He films. A difficulty with this preplating technique is that some fraction of the 3He atoms enters the first layer at high coverages; a small increase of the susceptibility at low temperatures is indeed observed (Saunders et al. 1991; Siqueira et al. 1994). The magnetic susceptibility of the fluid films could be described by a modified Fermi gas expression, and effective Fermi temperatures were found to be very close to those of the first layer liquid (fig. 41). Due to the high densities achievable for the second layer fluid, it is possible to obtain highly correlated systems with a susceptibility enhancement on the order of 20 with respect to the noninteracting gas value (Lusher et al. 1991a; Saunders et al. 1991). These authors presented the Landau parameter for the fluid 3He on preplated graphite as a function of the effective mass taken from pure 3He films heat capacity data, suggesting that the data supported the quasi-localized model (Vollhardt 1984); in fact, as discussed in section 5.1.2 for the first layer fluid, similar agreement is found with several theoretical descriptions of liquid 3He (see fig. 29). Recent data for pure 3He films (Rapp and Godfrin 1993, Morhard et al., to be published) confirm, within relatively large error bars due to the subtraction of the first layer signal, that the second layer fluid Fermi temperatures are similar to

296

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

those determined in the preplated system. Similar conditions have been reached for 3He on graphite substrates preplated with HD (Siqueira et al. 1994). The second layer liquid phase is thought to be uniform: see, however, the discussion in section 5.1.2 about the prediction of condensation in a weak adsorption potential, and section 5.3 on the possible observation of this effect for the third layer fluid. 5.2.2. Second layer solidification The solidification of the second layer fluid has been the object of a long debate; as a result, the literature on this subject is difficult to follow. Early heat capacity data (van Sciver and Vilches 1978) suggested that at coverages around 0.186/~-2 the second layer solidifies and at the same time promotion to the third layer begins. Nuclear magnetization measurements at low temperatures (Franco et al. 1986) showed that for coverages above 0.179/~-2 the magnetization in the 20-30 mK range was very close to free spin value, as is expected from a solidlike phase, and magnetization data were initially analyzed under this assumption (Franco 1985). However, this behavior was also observed below the coverage 0.186/~-2 mentioned above, in a range where heat capacity data indicated a liquid-like structure; also, early neutron scattering experiments did not detect any second layer diffraction peaks. This discrepancy led to the suggestion (Franco et al. 1986) that the layer could be a fluid with a very low degeneracy temperature (below 10 mK); the experiments described below, however, showed that errors in coverage scales were responsible for the discrepancy, and clarified the experimental interpretation of the initial data. We summarize in the following paragraphs the present interpretation of the experimental results. Further neutron scattering experiments detected the very small diffraction signal of the second layer at high coverages (Lauter et al. 1987; note that their fig. 4 is original, and does not correspond to reference 3, as misprinted). Convincing evidence for the solidification of the second layer at low coverages was provided by the low temperature heat capacity measurements of Greywall and Busch (1989) and Greywall (1990). A fluid solid coexistence was observed for the coverage range 0.169--0.178/~-2 (Greywall's coverage scale); here the high temperature heat capacity, dominated by the liquid contribution, vanishes, and the low temperature one, dominated by the solid spin contribution, increases (fig. 42). Although solidification is almost complete at this last density, about 4% of the second layer is found to remain fluid at this coverage, an effect probably due to the sample heterogeneity. Note that van Sciver and Vilches (1978) found that complete solidification occurs at a coverage 0.186/~-2 (Seattle coverage scale); comparing this figure to that given by Greywall (0.178/~-2) shows that the coverage scales differ by 4.3%. This has to be kept in mind during the following discussion of the second layer phase diagram.

Ch. 4, w

297

3He ADSORBED ON GRAPHITE

5 i" "1 4

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i

I

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Fig. 42. Isotherms of heat capacity versus coverage (Greywall 1990). The solidification of the second layer is seen by the reduction of the fluid contribution in the 200 mK isotherm and the increase of the spin heat capacity of the solid in the 2.5 mK isotherm. Third layer promotion is seen through the increase of the fluid contribution at the coverage indicated by a dashed line.

5.2.3. The second layer commensurate phase R2a The solid phase coexisting with the liquid phase is thought to be commensurate with the first layer. There is no direct proof for this assumption, but convincing evidence. First, its density is very low (on the order of 0.06 ,~-2) and no incommensurate phase exists in the first layer (even for 4He) at such low coverages. Second, the heat capacity peaks, observed by van Sciver (1978) and attributed by Roger and Delrieu (1987) and Greywall (1990) to melting of a commensurate solid phase, are observable in a narrow range of coverages, a situation similar to that found in submonolayer commensurate phases. The second layer phase diagram (fig. 37) is indeed similar to that of the first layer. It is therefore likely that a commensurate phase (R2a following Greywall's notation) exists in the second layer at coverages around 0.178/~-2. According to the heat capacity coverage scale, the second layer coverage is then in the range 0.55---0.64/?k-2, whereas the first layer density is about 0.114 A -2. The second layer density is close to that of the first layer commensurate phase Rla; however, since the graphite corrugation potential vanishes exponentially away from the substrate, it does not seem reasonable to assume that commensuration with respect to the substrate corrugation could take place for the second layer. The R2a commensurate phase corresponds to a ratio between the second and first layer coverages of about 0.56, a figure which is not very sensitive to cover-

298

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

age scale adjustments, and hence reasonably accurate. It is very close to the 4/7 value corresponding to the phase proposed by Elser (1989), shown in fig. 43. Such a structure has been shown theoretically to be stable (Abraham et al. 1990). Elser's structure possesses several appealing features. The lattice is triangular, the most compact lattice in two-dimensions, thus ensuring a minimum zero-point energy. Among the four atoms in the unit cell, three are located in the most favorable locations (minima of the first layer potential). The remaining atom is placed at a potential maximum (above a first layer atom), but its wave function is certainly spread on the surrounding regions of lower potential energy. Even though the structure of this commensurate second layer is not firmly established experimentally, it may be considered as a reasonable assumption. In particular, this 4 n commensurate phase is consistent with heat capacity data on 4He adsorbed on graphite (Greywall 1993). Recent calculations by Bernu (unpublished) demonstrate that some commensurate structures, for a triangular lattice both in the first and second layer, are more likely to be found: they correspond to the ratios 4/7 and 3/4. For arbitrary coverages, frustration is certainly a key element in the description of the ground state of complex structures involving domain walls, as already discussed in the context of the first layer commensurate phase (section 5.1.3). One should not forget, however, that the structure of the first layer also evolves in this coverage range (see fig. 36), and therefore a reduction of the second layer lattice parameter associated with the first layer compression is expected as a function of coverage. The nuclear magnetic susceptibility measured for the second layer atoms at this coverage (Franco 1985; Franco et al. 1986; Godfrin et al. 1991) is antiferro-

Fig. 43. The commensuratephase proposed by Elser (1989) for the second layer of 3He on top of a high density first 3He layer. The ratio of areal densities is 4/7. Shaded circles correspond to second layer atoms, which can be of two types. A, in a potential minimum and B, centered on a first layer atom.

Ch. 4, w

3He A D S O R B E D ON G R A P H I T E 0.35

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30

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(mK)

Fig. 44. Inverse magnetization of the second layer of 3He adsorbed on Grafoil at a coverage n = 0 . 1 8 7 / 1 - 2 (Franco et al. 1986; Godfrin et al. 1991). The antiferromagnetic Weiss temperature corresponds to exchange interactions on the order of 1 mK. 2.5

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0.45

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Fig. 45. The nuclear magnetic properties of multilayer 3He films, characterized by the effective magnetic exchange interaction parameter J, evolve from antiferromagnetic to ferromagnetic as a function of coverage, reflecting density and structural changes in the second layer solid (diamonds, NMR, Franco et al. 1986; Godfrin et al. 1994a; triangle, NMR, Godfrin et al. 1988a; squares, heat capacity, Greywall 1990; crosses, NMR, Lusher et al. 1991, 4He preplated film).

300

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

magnetic, with a Curie-Weiss temperature on the order o f - 5 mK (figs. 44 and 45). The low density solid 3He layer adsorbed on 4He or HD preplated graphite substrates is also antiferromagnetic, with similar exchange interactions (Lusher et al. 1991; Siqueira et al. 1993). The susceptibility data do not display any sharp feature that could be associated with the commensurate nature assumed for this phase. Schiffer et al. (1993, 1994), however, pointed out that the low temperature susceptibility of their films showed an additional Curie contribution from the second layer spins that could be interpreted as the signal from the noncoupled spins (1/4 of the second layer atoms) of Elser's model. Such a strong decoupling, however, is not seen in numerical simulations (Bernu, private communication). The spin heat capacity, on the other hand, displays an anomalous T -l dependence, similar to that observed in the submonolayer commensurate phase (fig. 46) Greywall has suggested that this behavior could be a signature of commensurate phases; it is more likely that this temperature dependence is the result of competing ferromagnetic and antiferromagnetic exchanges of large magnitude, owing to the low density of these solid phases. The third layer is not yet populated and hence in-plane multiple spin exchange processes similar to those found in bulk solid 3He are expected to be dominant. The low density solid displays magnetic properties of particular interest, since it constitutes a novel example of twodimensional nuclear antiferromagnet (B. Bernu private communication; Rapp and Godfrin, Advances in Physics, in press). Melting of this phase at a temperature on the order of 1 K is demonstrated by well defined heat capacity peaks (van Sciver 1978); unfortunately, only a few coverages were measured in this range. The melting mechanism is not understood; the exponential nature of the heat capacity, however, suggests a defect mediated mechanism. Similar measurements for 4He adsorbed on graphite (Greywall 1993), at coverages corresponding to the R2a, phase, suggest that the melting transition is second order, or very weakly first order. The shape of the melting peaks, rather broad, is very similar for both systems. This is in conflict with the first order transition expected for the 4/7 phase predicted by Elser. An improved agreement with the data can be obtained assuming that the chirality of the structure favors an Ising-like symmetry and hence a softer transition. 5.2.4. Remarks about the second layer density Promotion of atoms to the third layer begins, according to heat capacity data (Greywall 1990), at a coverage of 0.182 ]k-2. The second layer density increases progressively as the total coverage is increased, a situation already found in the first layer near completion. Unfortunately, neutron scattering data for the second layer exist only at two coverages around 0.3/~-2 (figs. 22, 36). Assuming that the density dependence of the melting temperature should be the same in the first and second layers, and using the melting temperatures measured by the

Ch. 4, w

3He ADSORBED ON GRAPHITE

301

Seattle group, Greywall (1990) obtained an empirical expression for the second layer density which agrees rather well with neutron data at high coverages. This empirical curve shown in fig. 36, however, is only an approximation, as discussed below. The other source of information about the density of the second layer is the susceptibility measurements at low temperatures. In the range around 30 mK the susceptibility of the liquid is small, whereas that of the solid is large, and both components can be separated by a fit, due to their different temperature dependence. In the case of pure 3He films, it is necessary to subtract the contribution of the first layer, which is difficult to evaluate. In order to suppress this contribution, experiments were performed at London University on a system where the first layer is 4He instead of 3He; the measurements provide the density of the second layer as a function of coverage (Lusher et al. 1991). The results display considerable structure, and also a larger variation range than that given by Greywall's empirical expression, as seen in fig. 36. Similar experiments on pure 3He films performed at Grenoble (unpublished), however, support Greywall's data indicating a smoother variation than that found in 4He preplated films, but on the other hand a dip is found at 0.24/~-2, qualitatively similar to that found in the preplated system. The origin of the discrepancy is not clear. It is possible that 4He preplating affects the second layer more than is presently believed. In this respect it is instructive to observe that the 3He densities found in London are close to those measured by neutron scattering in pure 4He films (Lauter et al. 1991). An alternative explanation is that second layer densities from susceptibility data may be inaccurate due to the simplified analysis. The first layer density measured by neutron scattering is constant for coverages above 0.20/~-2, excluding large variations of the second layer density above this coverage; this fact provides some support to Greywall's values, and probably disagrees with the results of Lusher et al. On the other hand, a clear compression of the first layer is seen on the neutron scattering data as the second layer is completed, suggesting that the step-like increase of the second layer density reported by Lusher et al. is a real effect, simply related to the completion of the fluid layers. Clearly, the structure of 4He preplated films and pure 3He films should be the object of further investigations. 5.2.5. The second layer intermediate region (0.I 78/~-2 to 0.26 ~-2) Let us discuss now in more detail the experimental information about the structure of the second layer for coverages in the intermediate region, located between the R2a and IC2 coverages (0.178/~-2 to 0.26/~-2), where IC2 denotes the incommensurate phase found at high coverages. In this range the second layer density varies substantially, from about 0.064 to about 0.083 ]k-2, according to heat capacity and neutron scattering results (fig. 36).

302

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

The second layer density variation begins to saturate in the range 0.1780.193/~-2. Some atoms go to the third layer, as seen by heat capacity at 0.182/~-2, and also to the first layer, according to the 2% increase in the first layer density measured by neutron scattering in this range (fig. 22). The total Curie constant measured at Grenoble (Franco et al. 1986 and unpublished data) shows that the liquid fraction is certainly small in this coverage range. It has been pointed out by Greywall that about 5% of the second layer remains liquid at a coverage where complete transformation into the REa could be expected. Complete solidification only occurred at a coverage of 0.184 ,k-E, an effect very likely due to heterogeneity. The evolution of the second layer structure at coverages above 0.184 ,~-2 is not well understood. Different interpretations have been proposed, none of which is presently strongly favored from the experimental point of view. Van Sciver (1978) observed small peaks at temperatures on the order of 1 K in this coverage range (fig. 37), and suggested that the peaks observed above second layer completion were due to evaporation of three-dimensional clusters. Indeed, the heat capacity of the films as a function of coverage tends asymptotically to that of bulk 3He, but this can be easily explained by the contribution of two-dimensional fluid 3He in the third layer. The heat capacity peaks, on the other hand, have an exponential shape, suggesting an activation mechanism with a characteristic energy comparable to that of bulk liquid 3He; note, however, that a similar result can be obtained for the layer promotion mechanism. The melting peaks of the second layer are expected to be strongly affected by the mechanism proposed by Elgin and Goodstein (1978) (see section 5.1.5), due to the small binding energy difference between the second and third layers. It is likely that the peaks observed at coverages above 0.23/~-2 correspond to melting of the incommensurate solid second layer (Greywall 1990); the strongest argument supporting this interpretation is that the second layer melting temperatures agree well with those expected from the study of the melting transition of incommensurate solids at submonolayer coverages. We are thus left with the task to explain the origin of the heat capacity peaks found around 0.2/~-2 at temperatures on the order of 0.7 K. This corresponds to the region of the phase diagram located between the low density (commensurate?) solid phase and the high density incommensurate phase. Detailed arguments have been presented by Greywall (1990) to support a model based on the existence of a second commensurate phase at a total coverage 0.193/~-2. The density of this R2b phase is supposed to be about 0.074/~-2, i.e. close to 2/3 of that of the first layer. In the coverage range 0.178--0.193/~-2 coexistence of the Rza and R2b phases is proposed; this identification is based on the coverage dependence of the spin heat capacity (fig. 46) as a function of the second layer coverage. It has been speculated (Franco et al. 1986) that a coexistence in the coverage

Ch. 4, w

3He ADSORBED ON GRAPHITE 10

--

-

i

i

I

I

I

I II

I

I

I

303 I

I

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II_

c ~ l / T

,,,F

E

-" 0

1

_--

-_

.io

0 -

-

0 0

k

-

0 Q) c-

O0

-

\

0.184

o,

8

-

~

~176 o

00 o

-

\

9 1

i

l I I ~ i ,ll t0

i

l

oo

k

\

0.1

_

~o

\\

1

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\

-

~oc \

\

I 1 1 1

100

T(mK)

Fig. 46. Nuclear spin contribution to the heat capacity for coverages corresponding to the second layer commensurate phase R2a. Note the anomalous liT temperature dependence. The high temperature additional bump may be due to liquid at heterogeneous adsorption sites (Greywall 1990). range 0.2-0.24/~lt - 2 between a low magnetization phase (a commensurate solid, according to Greywall 1990) and the incommensurate phase IC2 may explain the continuous evolution of the exchange contribution to the magnetic susceptibility. This point of view is supported by the very low temperature NMR measurements of Schiffer et al. (1993, 1994), where the ferromagnetic component is found to grow linearly with the second layer density in the coverage range 0.20.24/~-2. The data could therefore be explained by assuming that the second commensurate phase is antiferromagnetic, the incommensurate phase ferromagnetic, and that the intermediate densities correspond to a coexistence region. We consider, however, that the evidence for the existence of this second commensurate phase is extremely weak. Neutron scattering data in this coverage range show that the solidification of the second layer is accompanied by a substantial degree of compression and a simultaneous distortion of the first layer solid (fig. 22). According to heat capacity data, promotion to the third layer begins in the same coverage range, at 0.182/~-2 (see section 5.2.4). In this complicated region of the phase diagram, the only evidence for commensuration is the spin heat capacity very slow dependence on a second layer density which is not precisely known, and ferromagnetic effects which appear at coverages higher than that corresponding to second layer full solidification. It is difficult, if one

304

H. GODFRIN and H.-J. LAUTER

Ch. 4, w

assumes the existence of two commensurate phases, to understand their very similar magnetic behavior, seen in magnetic susceptibility and low temperature heat capacity data. The spin heat capacity (fig. 16) and the Curie-Weiss temperature of the second layer (fig. 45) remain rather constant in this coverage range, although it is known that nuclear magnetism of two-dimensional 3He is very sensitive to the interatomic distance and to the structure of the solid. In addition, heat capacity measurements in 4He films (Greywall and Busch 1991; Greywall 1993) are consistent with the existence of only one commensurate phase, the 4/7 phase mentioned before, transforming at higher coverages into an incommensurate phase, without intermediate phases. According to theoretical calculations (B. Bernu, private communication) the suggested 2/3 ratio corresponds to a particularly unlikely value for achieving commensurability of two triangular lattices. Note that the next favorable ratio (see section 5.1.3) is 3/4, which corresponds, using densities measured by neutron scattering, practically to the dense incommensurate phase. Let us now examine the alternative models for the second layer structure in this intermediate region. In the range 0.18-0.24/~-2 the spin heat capacity evolves in a rather complicated manner, from the T -l behavior observed in the commensurate phase to a more conventional 7'-2 dependence (fig. 18). Within the framework of the interpretation discussed above, this indicates the transformation of an antiferromagnetic solid in a ferromagnetic incommensurate solid. Greywall (1990) observed that the excess heat capacity with respect to the results at a coverage 0.184/~-2 displayed a simple behavior; a rounded maximum was observed at a few milliKelvin, that shifted gradually towards lower temp (a toms/,~, 2 ) 0.t75 81

0.180 I

3 rcl layer

0.190

0.200

I

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0.240

I

I ~

I

maximum--"'''" 2 nd layer Plt l

6; E 4 ._=

IP'~" Pha s e b ~ nd a r 'es""~i

//\11

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I

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305

3He ADSORBED ON GRAPHITE

Ch. 4, w

,,e' v

7

E

v

E

tt3 O4 II ~"

6

5

>, ~

4

t~ "1-

3

0

0.02

0.04

0.06

third layer density

0.08

0.1

0.12

(atoms/,/k 2)

Fig. 47. (a) Nuclear spin heat capacity isotherms as a function of the second layer density. Dashed lines indicate the coverages corresponding to the second layer commensurate phases R2a and R2b. Fluid-R2a coexistence is assumed at low coverages, R2a-R2b, coexistence between the dashed lines and R2b-incommensurate solid coexistence at high coverages. The incommensurate solid exists as a pure phase only at the highest coverages (Greywall 1990). Another interpretation is given in (b). (b) The same data as in (a), represented as a function of the third layer coverage (Godfrin et al. 1994b). The linear behavior suggests that the fluid in the third layer condenses and governs the magnetic heat capacity; this argument provides an alternative interpretation of the second layer phase diagram with respect to that described in the caption of the previous figure (see text). peratures as the coverage was increased. The data are interpreted supposing that the R2a c o m m e n s u r a t e phase partially melts into a high density fluid phase, allowed in principle by the low density of the c o m m e n s u r a t e phase. The dense liquid hypothesis is based on the observation that the excess heat capacity observed with respect to that of the c o m m e n s u r a t e phase has a liquid-like temperature dependence, but its m a x i m u m is located at a few milliKelvins, instead of a few hundreds. Note, however, that the same general temperature variation is also found in simple 2D magnetic systems, and the similarity with the liquid phase is therefore a weak argument. In addition, the N M R susceptibility of the

306

H. GODFRIN and H.-J. LAUTER

Ch. 4, {}5

second layer varies substantially in the same coverage and temperature range; it is therefore unlikely that a simple subtraction of the second layer spin heat capacity could be an adequate analysis of the heat capacity data. It should be pointed out that in the previous models, the third layer liquid does not participate in the magnetic phenomena. An alternative analysis has been proposed by Godfrin et al. (1994), assuming that the third layer liquid is self-bound. This is more likely for the third layer than for the first or second layer liquid: the theoretical calculation of Brami et al. (1994), clearly shows that condensation is favored by a weak confinement in the third dimension, as physically expected from the crossover from two to three dimensions. Indeed, the heat capacity of the third layer fluid determined by Greywall (1990) is well described by the hypothesis of a self-bound liquid: the coefficient of the linear term in temperature varies proportionally to the coverage of this layer, contrary to what is observed for the first and second layer liquid (see sections 5.1.2, 5.2.1, and 5.3). A similar coverage dependence is found for the magnetic properties (spin heat capacity, low temperature and ultra-low temperature magnetization). Magnetic susceptibility and spin heat capacity would reflect, within this picture, ferromagnetic effects induced in the second layer by a self-bound liquid forming islands of increasing size. In other words, the ferromagnetic effects are not necessarily associated with a structural phase transition (see section 5.2). This interpretation of the data requires the existence of only one solid phase, and supposes that its properties are modified by the presence of liquid. The existence of one commensurate phase in addition to that of the incommensurate phase must probably be invoked, however, to take into account the high temperature part of the phase diagram. 5.2.6. The second layer incommensurate phase above n = 0.26 ,~-e Neutron scattering data (Lauter et al. 1990, 1991) show that at very high coverages the second layer structure is triangular, and its density is about 0.083 A -2. The structure is not necessarily incommensurate with the first layer; the density variation of the melting peak measured by the Seattle group, very similar to that observed for the first layer incommensurate phase, makes this assumption reasonable. Neutron data for 4He show, however, that the structure of the second layer may present some locking with respect to the first layer at coverages which correspond to the same region of relative coverages. We assume in the following that an incommensurate phase (IC2) exists in the second layer. This phase is believed to exist as a pure phase only for total coverages above 0.26 A -2 (Greywall 1990), since only above this value the spin heat capacity follows a simple T -2 behavior at low temperatures (fig. 48), and an associated feature (change in slope) is observed in the low temperature susceptibility isotherms (fig. 18) at 0.25 A -2 (the difference between these values is exactly that found systematically between Greywall's coverage scale and the Seattle scale

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Fig. 48. Nuclear spin contribution to the heat capacity for coverages corresponding to the second layer intermediate coverage regime (lower graph); note the anomalous temperature dependence. The upper graph corresponds to the second layer incommensurate solid; the T-2 dependence is that expected for magnetic exchange interactions (Greywall 1990). used by the Grenoble group). The density of the second layer at this coverage is about 0.082/~-2 in this coverage range, according to Greywall's analysis (fig. 36). This value agrees very well with neutron scattering data (figs. 22, 36); such a good agreement is somewhat unexpected, considering the different values found for first layer coverages. Evidence for the compression of the second layer incommensurate solid, as the coverage is increased, is obtained from the reduction of the exchange inter-

308

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actions as a function of density, an effect well known in bulk solid 3He. The largest magnetic effects are seen at a coverage of about 0.24/~-2. These are the ferromagnetic peak observed by Franco et al. (1986) (fig. 18) and the spin heat capacity peak seen by Greywall (1990) (fig. 16). As expected, at higher coverages the magnetic contributions to the heat capacity and to the NMR susceptibility are strongly suppressed. The high temperature phase diagram of the incommensurate solid (fig. 32) has been briefly discussed in section 5.2.5. The nature of the melting transition of the second layer in multilayer films has not yet been investigated in detail. The influence of layer promotion and the fact that the spreading pressure increases as a function of temperature should play an important role here. Heat capacity data for 4He films at high coverages (Greywall 1993) display rather sharp peaks at about 2 K attributed to the melting transition of the second layer incommensurate solid phase; no analysis of the peak shape was reported, probably due to the difficulties encountered in subtracting the fluid 4He background.

5.3. Multilayerfilms The discussion of the preceding section already involved multilayer films, since the second layer density increases with coverage, even after fourth layer promotion. It is clear that solidification of the third and following layers does not occur even at high coverages, as demonstrated by susceptibility measurements in films up to 9 layers (Franco et al. 1986; Godfrin and Rapp, unpublished), by heat capacity data in films up to 5 layers (Greywall 1990) and by susceptibility results in bulk liquid 3He confined inside Grafoil (Bozler et al. 1978a,b, 1991; Godfrin et al. 1978). For 3He adsorbed on a substrate preplated by a 4He layer (Lusher et al. 1991), the observed amount of solid 3He grows with coverage in a step-like way, that can be accounted for by reasonable densities of the second helium layer (first 3He layer), as interpreted by the authors, and not to a partial solidification of the third layer. A similar situation is encountered in 3He adsorbed on Grafoil preplated by two atomic layers of HD (Siqueira et al. 1993): only the first 3He layer solidifies. The magnetic properties of the solid 3He layer on preplated substrates follow the general behavior observed for the second 3He layer in pure 3He films: low density solid layers formed on low binding energy substrates (i.e. all systems studied presently except the first layer on bare graphite) are remarkably similar. These "second layer" solids are less affected by the substrate corrugation potential, and can be considered as more ideal 2D solids. On the other hand, due to the low binding energy of second layer atoms, the decoupling of the

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characteristic energies for in-plane and out-of-plane interactions is reduced with respect to that observed in the first layer, degrading the 2D nature of these solid films. Liquid 3He films are observed in the third layer of 3He adsorbed on graphite (van Sciver 1978, Greywall 1990). The heat capacity of the fluid is linear in temperature at milliKelvin temperatures. However, contrary to what has been observed for the first and second layers liquid (figs. 26, 40), the coefficient of the linear term does not display the characteristic rapid increase as a function of coverage due to the increase of the effective mass of the interacting Fermi fluid. Instead, it grows rather linearly with coverage. In fact, it is found to be proportional to the amount of liquid in the third layer (fig. 49); this has been interpreted by Godfrin et al. (1994) as the signature of a condensation of the third layer fluid. This "puddling" model provides a consistent interpretation of experimental data on the magnetism of the second layer in 3He films: within this picture, the regions of the second layer covered by the fluid puddles are the object of ferromagnetic interactions, while the naked second layer areas remain antiferromagnetic. Other interpretations of these magnetic phenomena, however, are also possible, which may include or not the participation of the liquid to the magnetic exchange processes. The condensation of the third layer fluid, how0

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ever, is in line with the theoretical prediction of Brami et al. (1994) that a soft adsorption potential favors condensation whereas a strictly 2D 3He system (see section 5.1.2) should remain in a uniform state, according to the theoretical calculations of Miller and Nosanow (1978). Note that 4He 2D fluid, which does not display such an extreme quantum behavior as 3He, does indeed condense, as evidenced by recent heat capacity measurements (Greywall and Busch 1991; Greywall 1993) and theoretical calculations (Clements et al. 1993; Treiner, private communication). The theoretical prediction is that layer completion followed by puddling should be reproduced in the successive layers. According to the experimental data, analyzed within the puddling model, the 3He liquid islands in the third layer merge to form a uniform fluid at third layer coverages on the order of 0.04 ,/k-2, which corresponds to a total coverage of about 0.24 ,/k-2. Fourth layer promotion occurs at about 0.26 ]k-2, and fifth layer promotion at 0.39 ]k-2. The concept of layer promotion is certainly not well defined for liquid layers; we discuss this point further in the next paragraphs. These values, however, are significantly different from those deduced by Greywall (1990)" 0.24/~-2 and 0.29/~-2, respectively, assuming that the third layer fluid is uniform and attributing all heat capacity anomalies to transitions in the solid. Related investigations involving low-density 3He liquid films have been conducted in two-dimensional 3He-aHe mixture films adsorbed onto a variety of heterogeneous substrates, by heat capacity, third sound and NMR techniques. The physics of these systems is governed by the existence of surface states for the 3He atoms, by an additional hydrodynamic contribution to the effective mass of the 3He quasi particles, and by the Fermi statistics. A description of these phenomena is given in the review article by Hallock in the same edition of Progress of Low Temperature Physics. Experimental details on NMR measurements can be found in the articles by Brewer et al. (1970) (Vycor glass); Hallock (1991, 1995); Sprague et al. (1994) (Nuclepore); heat capacity data on Nuclepore are given by DiPirro and Gasparini (1980); Bhattacharrya et al. (1984); third sound experiments on Nuclepore are described by Hallock (1991). References to the theoretical work on the 3He Andreev surface states in 4He can be found in Andreev (1966), Edwards and Saam (1978), Krotscheck et al. (1988), Treiner (1993), Belic et al. (1994) and references therein. Another type of liquid 3He systems has attracted considerable attention in recent years: these are multilayer 3He films which display a characteristic step-like structure in all their thermodynamic properties as a function of coverage. For instance, the heat capacity isotherms at temperatures on the order of 100 mK measured by Greywall (1990) for 3He films adsorbed on graphite, shown in fig. 16, display three oscillations as a function of coverage, which correspond to "layer promotion"; features associated with three liquid layers are seen in the investigated coverage range. The magnitude of the increase of the heat capacity

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is consistent with the value expected for 2D liquid 3He, and tends to the bulk 3He value. NMR data for 3He adsorbed on graphite (Lusher et al. 1991; Saunders et al. 1991) extending to higher coverages allowed the observation of five characteristic oscillations in the nuclear magnetization (fig. 50). The magnetization tends roughly towards the bulk liquid 3He value, and the oscillations as a function of coverage are well correlated with those observed in the heat capacity. In fact, this oscillating behavior was first observed in 3He films adsorbed on other substrates than Grafoil. The smoothing effect of the first solid layers, or eventually of a suitably chosen preplating by other atoms, allows the investigation of thick liquid films adsorbed onto heterogeneous substrates. Oscillations in the magnetization corresponding to two liquid layers were observed in NMR measurements in multilayer films adsorbed on Nuclepore (Higley et al. 1989). Results similar to those discussed above for the heat capacity of 3He multilayers adsorbed on Grafoil were obtained for 3He adsorbed on a silver powder substrate (Greywall and Busch 1988). The properties of multilayer liquid 3He films have been studied from the theoretical point of view by Guyer and co-workers (Guyer et al. 1989, 1990; Guyer 1990, 1991). This Fermi liquid approach takes into account the finite thickness of the film; quantum size effects result from the Fermi statistics of the

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312

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particles in a finite "box". The occupation of the energy levels proceeds by completing Fermi discs associated with the finite size of the box in the direction perpendicular to the substrate, leading to oscillations in the thermodynamic properties and also in the film density profile. The helium interactions, as well as helium-substrate interactions, are not taken into account into this "free Fermion gas" model. The effect of the interactions (Pricaupenko and Treiner 1994) can be described by introducing a density of states and effective masses for each Fermi disk; the steps in the heat capacity and magnetization are associated with the increase in the density of states expected as a new continuum becomes occupied. The model predicts that the step-like structure will be smeared out when the coherence length of the substrate is not much larger than the Fermi wavelength corresponding to the Fermi disks. The coherence length of exfoliated graphite substrates is therefore convenient for such studies, since the Fermi wavelength is on the order of one typical interatomic distance of liquid 3He. The accidents interpreted as the formation of "liquid layers", therefore result from the completion of Fermi disks which occur at areal densities kF2/3:rt, similar to the density expected for one liquid layer (see ref. 20 in Pricaupenko and Treiner 1994). Let us mention, finally, that theoretical calculations predict that 3He wets all substrates down to the absolute zero temperature, and that prewetting is expected to occur on very weakly attracting substrates, like Cs, Rb and K (Cheng et al. 1993; Pricaupenko and Treiner 1994). The properties of multilayer 4He films have been studied theoretically in more detail (Epstein et al. 1988, 1990; Clements et al. 1993; Treiner 1993; Wagner and Ceperley 1994 and references therein), and many experimental results have been obtained using different techniques: heat capacity (Greywall and Busch 1991; Greywall 1993); third sound (Zimmerli et al. 1992); torsional oscillator (Crowell and Reppy 1993); inelastic neutron scattering (Lauter et al. 1991, 1992a,b) etc. We shall not attempt to describe here the subtle properties of these films, which include superfluidity, layering, wetting, and the existence of unusual elementary excitations. Note that the chemical potential of 4He films also displays several plateaux as a function of coverage, associated with the formation of "liquid layers", as demonstrated by recent quantitative calculations (Cheng et al. 1993). Similar results for 3He films would be helpful.

6. Conclusions

The extraordinary physical properties of exfoliated graphite samples make them particularly useful as substrates for surface physics studies. At very low temperatures, physisorbed helium atoms are confined to two-dimensions, opening

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3He ADSORBED ON GRAPHITE

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the possibility of investigating experimentally the nature of quantum fluids and solids in reduced dimensionality systems. The structural phase diagram of the helium isotopes adsorbed on graphite is particularly rich. Both isotopes display large zero-point energy quantum effects and are submitted to the same interaction potentials; as a result, their phase diagrams are similar, but nevertheless strongly modified with respect to those of classical adsorbed gases. Several structural phases have been experimentally observed as a function of the thermodynamical conditions of the 2D films, defined by the areal density of the adatoms and the temperature. At zero temperature, as a function of density, the system evolves from a 2D gas phase for 3He (a gas-liquid coexistence for 4He) to a highly compressed incommensurate solid; commensurate phases are observed at intermediate coverages. Locking to the substrate is therefore observed in some coverage ranges where the average inter-particle spacing is on the order of the dominant Fourier component of the substrate corrugation potential, while "floating" phases are preferred at extreme coverages. Detailed experimental phase diagrams for the first and second layers of 3He- and 4He adsorbed on reveal the subtle spatial distribution chosen by the adsorbed atoms; these effects, in fact, are governed by very small free energy differences between structural phases of different topological nature. There is presently no theory able to predict the complete structural phase diagram of these systems. Approximate solutions have been proposed, often limited to some aspects of the structural properties. The theory of commensurateincommensurate transitions has been used extensively to describe physisorbed films, introducing the properties of domain walls and the concept of domain wall fluids; melting of the commensurate phase of helium and hydrogen isotopes adsorbed on graphite, for instance, is presently understood as belonging to the universality class of the three-states Potts model. The melting mechanism of incommensurate phases is not so well understood theoretically; the experimental interpretation of heat capacity data is uncertain: both a first order transition broadened by heterogeneity, and a Kosterlitz-Thouless transition have been reported to fit experimental data well. In fact, the heterogeneity of the substrates as well as the thermal promotion of atoms to excited states in the third dimension lead to severe experimental problems. The theoretical description of the solid phases, except for a few numerical simulations, is still rather limited, due to the difficult task of incorporating the full effects of a strong potential and corrugation. On the other hand, considerable progress has been achieved in the description of fluid layers, a situation corresponding to weak adsorption potentials; these inhomogeneous quantum systems are amenable to quantitative calculations of their density distribution functions and elementary excitations, using techniques originally devoted to nuclear matter studies. Spectacular layering effects observed experimentally in liquid helium films are well described by these theories. They give rise to oscillations in the ther-

314

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Ch. 4, w

modynamical properties as a function of coverage with a periodicity of about one atomic layer. This feature, which might be surprising for a liquid, is the result of a combination of hard core effects and quantum statistics. At low temperatures, the properties of helium films are dominated by the quantum statistics: radical differences are observed between the behavior of the 3He fermions and the 4He bosons in 2D, a fact that is well known for the bulk systems. 3He liquid films are indeed very similar to bulk liquid 3He, and display all the characteristic properties of Fermi fluids: linear specific heat, constant magnetization, etc. The 2D Fermi fluids, however, can be studied in a large range of densities, not allowed for the bulk system; a new insight on the properties of highly correlated Fermi systems is emerging from such studies. Quantum statistics also lead to large effects in the two-dimensional solid phases of 3He: exchange interactions among the spin 1/2 3He particles do not give rise to magnetic order at milliKelvin temperatures, as in bulk solid 3He; instead, an exponential divergence of the correlation length is found, as expected for two-dimensional magnets with a T = 0 ordering temperature. This is probably the most spectacular consequence of the two-dimensional nature of adsorbed 3He systems, as theoretically expected from the Mermin and Wagner theorem. The existence of solid phases in two-dimensions should be forbidden by the same arguments; in this case, however, the loss of positional correlation is logarithmic, unobservable with typical experimental substrates where the coherence length is microscopic. The logarithmic divergence predicted theoretically for many observables (collision cross-sections, Fermi fluid properties, etc.) in two-dimensional systems results often in minute effects in experimental systems, due to finite size effects, weak three-dimensional couplings, and surface defects. The improvement of the quality of the substrates and of the experimental techniques, together with a more accurate consideration of these effects by theorists, is beginning to yield a consistent picture of quantum fluids and solids in two dimensions; the present results have opened the way, most of the work remains to done.

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CHAPTER 5

T H E P R O P E R T I E S OF M U L T I L A Y E R 3HE - 4HE MIXTURE FILMS BY ROBERT B. HALLOCK Laboratory for Low Temperature Physics, Department of Physics and Astronomy, University of Massachusetts at Amherst, Amherst MA 01003, USA

Progress in Low Temperature Physics, VolumeXIV Edited by W.P. Halperin 9 Elsevier Science B.V., 1995. All rights reserved 321

Contents 1. Introduction ......................................................................................................................... 2. Bulk interfaces ......................................................................................................................

323 324

2.1. The bulk free surface .................................................................................................... 2.2. The bulk-wall interface ................................................................................................. 2.3. Other surfaces ...............................................................................................................

324 329 333

3. Helium films .........................................................................................................................

334

3.1. Theoretical overview .................................................................................................... 3.2. Thickness scales ........................................................................................................... 3.3. Energetics experiments ................................................................................................. 3.3.1. Heat capacity experiments ................................................................................. 3.3.2. Nuclear magnetic resonance experiments .......................................................... 3.4. Other experiments .........................................................................................................

334 344 345 345 355 387

3.4.1. Third sound experiments .................................................................................... 3.4.2. Oscillator measurements .................................................................................... 3.4.3. Selected other experiments ................................................................................ 3.5. Future directions ........................................................................................................... 4. S u m m a r y ..............................................................................................................................

387 416 425 433 435

A c k n o w l e d g m e n t s ....................................................................................................................

435

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

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322

1. Introduction

In this paper, we review some of the history and our current understanding of multilayer 3He-aHe mixture films. The story begins with a discussion of the properties of 3He-aHe bulk solutions and the free surface of bulk 4He in the presence of a 3He impurity. This is reasonable, since interest in the case of 3He impurities in 4He films was a natural extension following the emergence of an understanding of the bulk case. The paper is organized as follows: We begin with a brief review of the free surface of 4He in the presence of a 3He impurity. Next we briefly discuss the interesting situation of bulk mixtures in the vicinity of a containing wall. We then, for the major part of the review, discuss the case of mixture films. Here the majority of the work which has contributed to our knowledge of the subject has come from heat capacity, NMR and to some extent from third sound investigations. The most important development in mixture films has been the emergence of our understanding of the energetics of the 3He impurity in the 4He environment; it is this subject which forms a major focus of this review. Our approach is at times historical and is often from an experimental point of view, but we draw freely from the relevant theory to reference the evolution of theoretical understanding and progress. We conclude with a discussion of a number of current questions and point to anticipated future work in the field. Those who are primarily interested in the energetics of the film may focus on the sections on heat capacity and NMR measurements which are reasonably self-contained. There are a number of interesting areas of study which are close to the areas we describe, but which are not discussed in any significant detail here. These are (1) the case of very thin pure 3He films (for one or two monolayers on strong binding substrates), (2) the case of pure 3He films which are thick enough to support superfluidity and (3) the case of monolayer and submonolayer films. For some of these, there is an extensive literature, including the current review by Godfrin (1995) in this volume. Any review will, of necessity, fail to discuss work which some may feel is important, or emphasize certain work over other. The author has made an effort to be accurate, fair and comprehensive. He has made the choices made with no intention to slight any of the work in the field and takes responsibility for any omissions which may occur.

323

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R.B. HALLOCK

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2. Bulk interfaces

2.1. The bulk free surface Consider a container which is sealed and, for example, half filled with pure 4He. In the presence of gravity the bulk 4He occupies the lower half of the container and is separated from the vapor pressure above by the surface of the fluid. Measurements of the temperature dependence of the surface tension of the 4He (Allen and Meisener 1938) made by a number of workers at temperatures below T= 1 K show that the surface tension increases with falling temperature (Edwards et al. 1975, Eckardt et al. 1977). This behavior is now understood on the basis of the thermal excitation of quantized capillary waves known as ripplons first described by Atkins (1953). Under the assumption that ripplons provided the only temperature dependence, Atkins (1953) derived an expression for the change in surface tension with temperature as tr4(T) = tr4(0 ) - (l/4x)F(7/3)~(7/3)[nom4/tr4(0)h]2r3T 7t3 where tra(T) is the surface tension at temperature T, a4(0) is the surface tension at T = 0, no is the number density of pure 4He and m4 is the bare 4He mass. Measurements by Atkins and Narahara (1965) were in general agreement with this result, although at the time, a competing prediction by Singh (1962) of a T2 dependence could not be ruled out. For the case of a 0.05% solution of 3He in 4He, Atkins and Narahara (1965) observed an intriguing maximum in the surface tension at about 0.6 K (fig. 1). Andreev (1966), motivated by the experimental observations, suggested that the addition of 3He to a 4He liquid should have the effect of increasing the surface tension with falling temperature unless there were a bound surface state for the 3He at the free surface of the 4He in which case the effect would be to decrease the surface tension with falling temperature. Andreev showed that if the 3He atom behaved as an excitation with energy E = - A + p212m in the body of the fluid, and was bound to the free surface by a binding energy eo, such that for an atom at the free surface E = - A - e0 + q212~, then the surface tension would be reduced from the surface tension for the case of pure 4He so that tr = o'4(T) - c3noh(fl]m3)(2xT]m4) It2 exp(eo/T) where c3 is the concentration of 3He and/~ is the 3He effective mass. When combined with the surface tension expression for tra(T), this yields a maximum in the surface tension in the case of mixtures. The physical reason for the existence of the surface state, as discussed by Chang and Cohen (1973), is this:

Ch. 5, w

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325

Fig. 1. Surface tension measurements for a bulk 3He-4He mixture with a 3He concentration of 0.05% as measured by Atkins and Narahara (1965). Since the 3He atom has a smaller mass than the 4He atom, it is able to lower its kinetic energy (and its total energy) by moving to a region of lower density than that provided by an environment of pure 4He. It is unable to escape to vacuum, however, since there it would no longer feel the attractive potential provided by the 4He and its total energy would rise. Thus, the 3He impurity moves to the location of lowest available 4He density and is bound to the free surface of the 4He. 3He atoms in such a surface state, as we shall see, act as an interacting twodimensional Fermi system. The spreading pressure of this two-dimensional 3He reduces the surface tension. Zinov'eva and Boldarev (1969), motivated by Andreev (1966), conducted careful capillary rise measurements on bulk 3He-aHe mixtures for concentrations from 0 to 0.22 and showed that the presence of the 3He depresses the surface tension at low temperatures. Comparison to the Andreev (1966) theory revealed e 0 = 1.7 ___0.2 K and /~ = (0.9 +__0.1)m3. It was concluded that for T greatly above the degeneracy temperature little 3He was bound to the surface while at low tempeatures a monolayer was bound.

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Motivated by the experiments of Zinov'eva and Boldarev (1969) and the theoretical ideas of Andreev (1966), Edwards and his co-workers set about studying the low temperature behavior of the surface tension in considerable detail. Since a thorough review of Edward's studies of the effect of 3He on the free surface of 4He is available (Edwards and Saam 1978), our description of this work is brief, with a focus on the key experimental results. In the first study, Guo et al. (1971, 1975) studied the surface and interfacial tension with a capillary rise technique and measured the energy and effective mass of the surface state. They also studied the approach to and presence of phase separation by noting the appearance of a thick surface layer and measured the 3He-3He interaction in the surface state and found it to be weakly negative. In an effort to search for a new hydrodynamic mode, surface sound, predicted by Andreev and Konpaneets (1972), Edwards and collaborators (Eckardt et al. 1974, Edwards et al. 1975) utilized a thermal source and bolometers positioned so as to be partially immersed in the 4He free surface. Two coverages of 3He were used, each too small to correspond to a complete monolayer at the lowest temperatures. Pulses of thermal energy from the source caused the propagation of a surface disturbance on the bulk free surface and this was the predicted new surface sound mode. In the same experiment, the capillary rise of the mixture was also carefully measured by a parallel plate capacitive technique. The results of these experiments conclusively demonstrated the existence of the predicted surface sound in the surface layer of 3He and allowed a more precise determination of the 3He binding energy (2.28 __.0.03) K and the effective mass for the 3He in the surface state (1.3 _+0.1)m 3. An interesting additional result of the experiments was the measurement of the effective interaction between the 3He quasiparticles; this was found to be weakly repulsive with the likely value in the r a n g e - 1 • 10-31 to 3 x 10-31 erg cm 2. Andreev and Konpaneets (1972) also predicted that there should be a first order phase transition on the free surface due to the condensation of the surface 3He into a liquid. An expected change in the slope of the surface tension with temperature was searched for but not found. The conclusion was that no such phase transition was present, the data all being explained by an almost ideal Fermi gas model. While these experiments were underway, Lekner (1970) put the suggestion of Andreev (1966) on a more firm theoretical basis by means of the Feynman variational method. He assumed the geometry of a free surface in the x - y plane and adopted a ground state wave function ~0 for N 4He atoms with ~0 the eigenstate of the Hamiltonian N

H0 =

N N ~ V(Ir~ - rjl), i=1 i<j

(-hi 2m4)~V j2 + ~ j=l

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

327

where m4 is the bare mass of a 4He atom. Replacement of one 4He atom by a 3He atom results in a new Hamiltonian H = H o - h2/8m3Vl 2, and a new ground state of the form ~ =f(z)~0 with f ( z ) = n(z)-ln~ where n(z) is the number density. Use of the variational principle then results in an equation of the form ~p + ( 2 - U(z))tp = 0. Lekner thus reduced the problem to that of the motion of a 3He particle in a one-dimensional potential h2U(z)/2m3 where m 3 is the bare mass of a 3He atom. Saam (1971), through the use of a phenomenological potential for U(z), completed the solution and obtained a value for the binding energy. Chang and Cohen (1973) calculated from first principles the effective potential which the 3He sees (Shih and Woo 1973). These calculations did not take into account the effect of 3He interactions. This was done by Edwards et al. (1975) and the results compared with detailed measurements of the surface tension as a function of temperature for several values of the concentration with the excellent agreement shown in fig. 2. The picture one has then is that of a 3He impurity which at low temperatures occupies a surface state at the 4He free surface. The binding energy and phase space are such that as the temperature is raised the 3He dissolves into the bulk 4He. Following the successful general understanding of the mixture problem, refinements were made to the theory. Yim and Massey (1979) adopted a simple microscopic approach and determined the binding energy and also showed how the effective mass of the 3He could be obtained (Woo et al. 1969, Massey et al.

12

!

r

T. . . . . . .

l . . . .

T. . . . .

--T"~-

* 0 3i8 E

I

!

~ 0221

8

-0155

~)

O 0/071 O 01045

"~

9

-

0.039

~4

-

_~O l

T(K)

0.2

O5

Fig. 2. The decrease in the surface tension of 4He as a function of temperature due to adsorbed 3He. Several different concentrations are shown and denoted in terms of fractional monolayers at T = 0 K. The smooth curves are theoretical calculations based on a quasiparticle gas model as described by Edwards and Saam (1978).

328

R.B. HALLOCK

Ch. 5, w

1970). Guyer and Miller (1979) studied the Euler-Lagrange equation for the case of the hydrogen isotopes and compared them to the case of 3He as did Mantz and Edwards (1979). An entirely different approach to the free surface was taken by Krotscheck and his colleagues. In the first such calculation, Krotscheck et al. (1985) carried out a variational calculation of the 4He system in the HNC approximation for an inhomogeneous case. The case studied in numerical detail was a system homogeneous in two dimensions but of finite extent in the third, z. For large enough z, the "thick slab" geometry was of constant density at the center and thus each face represented the free surface of bulk 4He, thus allowing a determination of the 4He surface profile. This calculation provided the seeds for later inhomogeneous calculations in the presence of a substrate. An alternate approach to the bulk mixture problem is based on density functional theory. Dalfovo and Stringari (1988) pointed out that density functional formalism has the benefit of accounting for non-local effects which arise due to the interactions. The effective mass is thus modified and brought into closer agreement with experiment. This approach showed the extreme sensitivity of the 3He surface state to the profile of the free surface of the 4He fluid and noted that for a sufficiently broad surface profile the 4He surface would contain two surface states for the 3He. More recent density functional work by Pavloff and Treiner (1991a--c) included hard core and finite range interaction effects missing in the Dalfovo and Stringari (1988) work. The technique is contrasted with the inhomogeneous approach of Krotscheck et al. (1985) where a layering effect of adjacent atoms plays an essential role. The Pavloff and Treiner (199 l a) calculation missed the characteristic distance of the repulsive hard core and this prevented oscillations of the density (layering). This was overcome in later work (Pavloff and Treiner 1991b) in applications to films where such structure imposed by the substrate is of major importance. This deficiency has a small effect in the discussion of the bulk free surface. A primary result of the Pavloff and Treiner (1991a) work is the confirmation that, given a relatively wide surface profile for the 4He, there will be at least two (rather than just one) states at the free surface and that these persist for finite 3He coverage. At T = 0 only the lowest state is occupied at small coverages of 3He, but when the coverage exceeds a critical value, the second state becomes occupied. This occupation is expected to give rise to a change in slope of the surface tension as a function of coverage. Unfortunately, the surface tension data of Edwards and his collaborators did not extend to coverages as high as would be necessary to check this prediction. The needed high coverage surface tension results are not yet available. The occupation of two surface states is expected to lead to steps in the heat capacity. The existing heat capacity data (Crum 1973) was taken on too large a grid of 3He coverages to allow this step structure to have been seen. As we shall describe shortly, additional heat capacity data for thick 4He films decorated with 3He by Gasparini and his colleagues and NMR work by Hallock and his colleagues, is

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

329

not inconsistent with this two-state bulk surface prediction, but does not confirm it.

2.2. The bulk-wall interface There is another "surface" of relevance in the case of bulk mixtures and this is at the interface between the bulk mixture and the walls of the container which holds the liquid. As this introduces a substrate into the problem, we review developments in this area. In the history of bulk mixture studies, extraneous heat leaks in experiments which were designed to study pure 3He were sometimes attributed to the existence of a 4He impurity superfluid phase in the mixture. This superfluid phase was thought to give rise to a superfluid film which would coat the walls of the container and flow up the walls through the 3He rich phase and reach higher and warmer regions of the apparatus and thereby, by refluxing, introduce a heat leak. The early experiments were those of Wheatley's group and belatedly reported in the review by Abel et al. (1965a) and Wheatley (1970). Other early work was that of Brewer and Keyston (1962). Keyston and Laheurte (1967) and Laheurte and Keyston (1971) were able to document the existence of this effective thermal conductivity and observed its onset to be above the bulk phase separation temperature, lending support to the possibility that a film of superfluid 4He could form on the walls of the container. Evenson et al. (1968) reported adsorption isotherm measurements of 3He-aHe mixtures adsorbed in Vycor and concluded that the adsorption isotherms were different for the mixtures than they were for the pure fluids, due to the preferential adsorption of 4He to the walls. Further evidence for this conclusion was the behavior of the vapor pressure when the pores in the apparatus were filled which was interpreted to mean that 4He could take the place of 3He at the walls, causing an expulsion of the 3He. In the work of Black et al. (1971) measurements of the thermal equilibrium times for CMN in the presence of bulk 3He with various 4He impurities were made as a function of temperature (Abel et al. 1965b, 1966). In the case of 4He impurity concentrations greater than adequate to supply enough 4He atoms to cover the CMN with at least a monolayer of 4He, the thermal equilibration times in the apparatus were very long. For cases for which the amount of 4He was inadequate to form a monolayer of 4He, the equilibration times were relatively short. This was strong indirect evidence for the existence of the 4He film adjacent to the boundaries in 3He-4He bulk solutions. To further document this, Laheurte and his colleagues carried out a lengthy series of detailed studies of the nature of the helium adjacent to the walls in a container which held bulk mixtures of various concentrations. In an early one of these experiments (Laheurte 1972), the presence of a superfluid film was noted

330

R.B. HALLOCK

Ch. 5, w

by measuring the heater power required to maintain a constant temperature at a given point on a filling capillary. Also in the cell was a capacitor used to measure the bulk dielectric constant. By this capacitive technique, bulk phase separation could be monitored and one could clearly see that the onset of the presence of the film adjacent to the substrate was a different event from bulk phase separation. The experiments gave the connection between the 4He molar concentration and the film formation temperature and information on the effective binding energy and mass of the 4He in the surface layer. In later theoretical work Laheurte et al. (1974) showed preferential location of the 4He at the walls due to the smaller volume taken by the 4He and suggested that the superfluid phase could nucleate at the walls (under some conditions) before it nucleated in the bulk. Romagnan et al. (1978) reported detailed measurements of the presence of the 4He layer near the substrate in experiments which relied on interdigitated capacitor plates (Laheurte and Romagnan 1974) located on the container walls. In these experiments one could detect the shifts in dielectric constant at the surface associated with the growth of the 4He film. An important conclusion claimed in the work was that phase separation with a 4He rich layer can occur in the boundary region adjacent to the container walls before the appearance of superfluidity in the 4He rich region. It was also concluded that the 4He films produced in such an apparatus behaved in a manner similar to unsaturated 4He films and thus there was a universal nature to the films. Various theoretical arguments and predictions were also made concerning the thickness of the 4He layer as a function of concentration and temperature. Gearhart and Zimmermann (1974) added further evidence for the existence of superflow at an anomalous location on the 3He-aHe phase diagram by their studies of the presence of superflow in a superleak connecting two reservoirs of different concentrations of 3He-aHe mixture. Their work showed that superfluidity existed in the system (by the relaxation of pressure differences across the superleak) at anomalous points on the phase diagram. This was interpreted as evidence for a 4He film moving along the walls. The data points extended the earlier work of Laheurte to lower concentrations. At about this same time, Eggington and Moore (1974) carried out a theoretical calculation which included the effect of the van der Waals field for mixtures confined to cylindrical pores (Vycor) and concluded that the 3He experiences a repulsion from the walls which has ramifications for the specific heat in general accord with experiments due to Brewerl et al. (1971) (Brewer 1970). Further evidence for the existence of a layer of 4He at the surface of a container with mostly 3He in it came from the work of Ahonen et al. (1976). In this work an enhancement of the susceptibility of 3He between sheets of Mylar with 4/tm spacing was observed at temperatures near 2 mK. This enhancement of the susceptibility could be completely removed by the addition of enough 4He to the sample to allow for about two atomic layers of 4He on the walls of the apparatus and the Mylar surfaces.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

331

5He Rich (normol)

-'WvV'-

4He Rich (superfluid) III

Fig. 3. Schematic representation of the beaker-style apparatus used by Peshkov (1975) to demonstrate the presence of a superfluid 4He film adjacent to the substrate in the 3He rich phase of a bulk mixture.

The conclusion was that the 4He indeed coated the Mylar, removed the proximity of the 3He to the surface and erased the susceptibility enhancement. Perhaps the most direct and elegant experiment to document the existence of a superfluid film on the surface in contact with the 3He rich phase of bulk 3He4He mixture was one performed by Peshkov (1975). In this experiment a beaker was immersed in a bulk 3He-aHe mixture such that at low temperature when phase separation had taken place, the beaker would contain a phase separated mixture as shown in fig. 3 with the top of the beaker immersed in 3He rich normal phase. The application of heat to the beaker caused the position of the phase boundary in the beaker to shift. It was concluded, by adding up to 2000 copper wires of 35/~m diameter to the "flow path" so as to increase the area, that there was indeed a film of 4He flowing along the beaker (and wire) surface. From the known geometry, it was concluded that the critical flow velocity was about 15 cm/s, a value consistent with the critical flow velocity of saturated (thick) films of pure 4He (Jackson and Grimes 1958, Atkins 1959). An additional experiment by Borovikov and Peshkov (1976) studied the interaction between an oscillating solid wall and a 3He-4He solution. It was concluded that (1) very close to the solid boundary 10-15/~ of superfluid is present and (2) that there is

332

R.B. HALLOCK !

Ch. 5, w

!

.,,~______.____._...-----4

>x k_

El t/3

tO (1) CL U3

(:5

O9 ~

I

~

!

6

9

4He Ioyers Fig. 4. The specularity at 3 mK for bulk 3He in the presence of a silicon surface as a function of 4He coverage as measured by Tholen and Parpia (1993) showing the abrupt onset of specularity in the vicinity of two adsorbed 4He layers.

a film about 100 atoms thick which forms on all solid bodies in the 3He rich phase in which the 3He concentration is near to that in the 4He rich phase. Recent experiments carried out by Tholen and Parpia (1993) utilized torsional oscillator techniques to study the nature of the scattering of the 3He from the surfaces for the case of pure 3He and for the case in which enough 4He was present in the apparatus to allow up to several monolayers of the 4He to plate onto the walls. It was concluded that the nature of the scattering changed from diffuse to specular when the amount of 4He exceeded about 2 atomic layers (fig. 4). An increase of the pressure in the cell caused the scattering to switch back from specular to diffuse and this was interpreted as being due to the solidification of the 4He; the scattering returned to what it was with no 4He in the apparatus. It was concluded that enough 4He had to be present to be superfluid for the scattering to become specular. The apparatus was not sensitive enough to observe the superfluid transition in the 4He directly. The superfluid 3He film flow work of Steel et al. (1990) also addressed the question of specular reflection with the conclusion that the addition of a single monolayer of 4He was adequate to cause a crossover from diffuse to specular scattering. Earlier work by Freeman et al. (1988, 1990) and others (Kim et al. 1990) concluded that at least two monolayers were necessary.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

333

2.3. Other surfaces There are other "surfaces" which should be mentioned in a discussion of 3He at the surface of 4He. For example, the notion of an Andreev surface state was invoked by Dahm (1969) in an effort to explain the fact that the critical velocity for vortex ring production due to the motion of an electron bubble was influenced by the presence of the 3He (Rayfield 1968). Dahm argued that this might be due to the condensation of the 3He at the surface of the electron bubble where it would affect the surface tension and thereby change the size of the bubble. It would also change the boundary condition at the bubble surface. Another surface to consider is the surface akin to the bulk free surface, but one which exists on the surface of a cluster of helium atoms. For large droplets, one has the limit of the bulk free surface, but for small droplets, this is not the case. Dalfovo (1989) has carried out a theoretical study of 3He at the surface of small clusters of 4He as a function of the size of the helium cluster using the Feynman-Lekner formulation and also using the density functional formulation. The results of the two theoretical approaches are similar and show that, at least for the drops in excess of 100 atoms which were studied, a bound state for the 3He exists on the surface of the droplet (fig. 5), with an energy which depends on the size of the

Fig. 5. Energies of the three lowest l = 0 states for 3He on the surface of a 4He cluster of size N as a function of N -1/3 showing the energy shift as a function of cluster size. The squares are based on a density functional calculation, the triangles use the Lekner method. The intercept points are for a planar free surface (from Dalfovo 1989).

334

R.B. HALLOCK

Ch. 5, w

droplet. Experiments in this direction have not yet been done, although the methods of helium cluster production now seem within reach (Bucheneau et al. 1990, Jiang and Northby 1992). Yet another surface of sorts is that provided by a vortex core in 4He. Dalfovo (1992) has studied this case recently as have a number of others earlier (Ohmi et al. 1969, Kuchnir et al. 1972, Ostermeier and Glaberson 1976, Williams and Packard 1978, Muirhead et al. 1985). Finally, the surface of solid 4He in the presence of 3He impurities is also and interesting interface which under some conditions includes the presence of a bound state for the 3He. The presence of such 3He impurities at the solid 4He surface greatly influences crystal interfacial growth and roughening phenomena (Landau et al. 1980, Carmi et al. 1985, 1989, Keshishev and Andreeva 1991, Wang and Agnolet 1992, 1994, Burmistrov and Dubovskii 1993, 1994).

3. Helium films 3.1. Theoretical overview

Next, we turn our attention to the case of helium film on a substrate. Consider a chamber at finite but low temperatures, sealed except for a fill capillary. If helium is slowly admitted to the chamber, atoms will, except for the case of certain alkali metals (Cheng et al. 1991, 1992, 1993) adhere to the container walls. They are attracted there by the van der Waals force between the helium and the substrate. If we maintain the vapor pressure in the cell, P, below the saturated vapor pressure, Po, at the temperature of interest, the chemical potential will be /z = A/z +/z o where /z0 is the chemical potential at bulk saturation and Alz = k T ln(P/Po) = f~(d) where Q(d) is the potential a distance d away from the container walls (substrate). It has been traditional (Putterman 1974) to write ~(d) =--aid3(1 + fld) -l where a and fl characterize the Van der Waals potential imposed by the substrate. For relatively thin 4He films retardation effects become less important (Anderson and Sabisky 1970, Sabisky and Anderson 1973) and one has ff2(d) = --a/ae. It has recently become common to express this more explicitly as ff2(d) = - A C 3 / d 3 with AC 3 = C 3 - CHe where C 3 is the coefficient of the van der Waals potential provided by the substrate and CHe is the coefficient which would be provided by a helium substrate. Modern work (Cheng et al. 1993) assumes various forms for the potential in detailed calculations. For very small coverages, one presumes a submonolayer film to grow on a smooth substrate. For such a case it is of interest to ask about the nature of the s u b m o n o l a y e r - is it a gas, a liquid or a solid? The answer depends on the strength of the substrate. For increasing liquid coverage, or, if the film wets, it thickens, and eventually or exceeds the critical value, or*, at the KosterlitzThouless (Kosterlitz and Thouless 1973, Nelson and Kosterlitz 1977) transition,

Ch. 5, {}3

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

335

where tr*lTc = 2kBm2/~h 2, with Tc the transition temperature; for tr > tr* the film is a superfluid for T < Tc. It is this environment into which 3He atoms are introduced to form a mixture film. There have been many theoretical efforts directed at an understanding of the 3He-aHe mixture film. These efforts can be broken into a number of categories, including calculations directed at the true two-dimensional case, the behavior of helium in a single monolayer, the study of several-layer helium films by the use of microscopic techniques and by the use of density functional and other techniques. Most of these theoretical efforts are directed at the notion of phase transitions although the more recent are concerned with the specifics of the 3He impurity - where it is and what the energetics and the effective mass are. We begin with a brief discussion of the true two-dimensional case and then focus on multilayer films. In the true two-dimensional limit, the work of Cardy and Scalapino (1979) and Berker and Nelson (1979) was directed at a study of two-dimensional phase separation in the presence of a smooth substrate with no underlying corrugation potential. Each theory used a Migdal-Kadanoff (Migdal 1975, Kadanoff 1976, Jose et al. 1977) renormalization approach. Phase separation was found with distorted phase diagrams, with the lambda line meeting the phase separation curve at much lower concentrations than in the bulk case. No tricritical point was found for any range of the parameters studied. The universal character of the superfluid density jump at the transition was preserved in the calculations. These two calculations were very similar except for the fact that Cardy and Nelson and Scalapino (1979) used a square lattice and Berker and Nelson (1979) used a triangular lattice. Hickernell et al. (1976) carried out specific heat studies of submonolayer coverages of 3He-4He films on Grafoil. Four 4He coverages were studied with a number of 3He concentrations at each coverage. The results showed that there was apparently uniform mixing of the isotopes over the range of parameters studied (0.04--4.4 K). A shoulder on some of the heat capacity data taken as a function of temperature was consistent with a phase transition, but the presence of such a transition was considered unlikely. Thus it was concluded that there was no isotopic phase separation in the two-dimensional environment provided by the Grafoil. More recent theoretical work by Carneiro and de Mello (1987) studied mixtures of 3He and 4He adsorbed on graphite in terms of a Potts latticegas model (Berker et al. 1978) and resulted in qualitative phase diagrams. The comparison to the work of Hickernell et al. (1976) was "encouraging but inconclusive". Earlier thin film mixture work on Vycor is that of Brewer et al. (1970) (Brewer 1970). More recent heat capacity studies for thin mixture films are those of Steele et al. (1992). A two-dimensional calculation based on zero temperature equations of state obtained from a variational calculation was carried out by Miller (1978). Finite

336

R.B. HALLOCK

Ch. 5, [}3

temperature behavior was obtained by use of simple models. A graphite strength potential was assumed which was expected to hold the adatoms in a twodimensional plane close to the substrate. The calculation showed that at zero pressure the 3He density is zero; thus, the 3He is a gas. This means that the situation is a two-dimensional 4He liquid in equilibrium with a two-dimensional 3He gas. So, the "phase-separation and the liquid-gas transition become intermingled" (Miller 1978). This theory predicts that the system phase separates at low enough temperature, but at finite temperature the 3He dissolves into the 4He. Under some conditions at low pressure one expects to have binary phase separation with 4He-rich liquid in equilibrium with 3He-rich gas. At high pressure, there will be a homogeneous mixed phase. There is a strong disagreement between the predictions and the data of Hickernell et al. (1976). One of several suggestions put forward to explain this is the presence of strong modulation in the substrate potential which exists in the experiment, but is not included in the theory. Mon and Saam (1981) considered a two layer generalization of the earlier work of Cardy and Scalapino (1979) and Berker and Nelson (1979). The motivation was to examine how deviations from the pure two-dimensional case would affect the predictions. The results, including the effects of the substrate potential were far more rich than in the original calculations. The resulting phase diagrams had considerably more complexity, including re-entrant behavior (for example, fig. 6). The finite coverage film case of central importance to the work we describe here is not a two-dimensional situation and developments were independent of those for the two-dimensional case. We will now begin to present some of the ideas of relevance to films which are not strictly two-dimensional and which show a gradient in structure which is imposed by the presence of a substrate. Following the success of understanding of the energetics of 3He in the bulk 4He free surface, it was natural to ask what might be the quantum behavior of a 3He atom in the context of a thin film of pure 4He which was not in the true twodimensional limit, but instead was of multilayer coverage. Gasparini et al. (1984) (DiPirro and Gasparini 1980) were the first to do this by use of a variational procedure based on that followed by Mantz and Edwards (1979) for the case of Hydrogen, Deuterium and Tritium on the bulk 4He free surface. To treat the case of the thin film, Gasparini et al. (1984) introduced into the potential a term which accounts for the potential due to the substrate. This has a major effect: it introduces a boundary for the 3He at the substrate. This, coupled with the boundary introduced by the presence of the free surface, confines the 3He to a box perpendicular to the substrate and introduces discreteness into the allowed states for the 3He. Although a number of approximations were made in this first calculation, the results were in reasonable agreement with heat capacity experiments which we describe shortly. A refinement of the Gasparini et al. (1984)

Ch. 5, {}3

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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=

337

2.0

1.2 .

~ 0.8

O.4

0

0

9

O. I

02

0.3

.4

0.5

0.6

0.7

0.8

9

t.O

Xs

Fig. 6. A typical temperature versus concentration phase diagram as computed by Mon and Saam (1981) which shows strong deviations from the three-dimensionalcase. For this example, there are two 2 lines, one indicated by a dashed line and the other (between the two lobes) by a short solid line. calculation was carried out by Sherrill and Edwards (1985). This refinement removed two difficulties with the original theory; it correctly incorporated the fact that the binding energy should increase with decreasing film thickness and it removed the need to introduce an adjustable parameter to smoothly join the potential in the near field to that in the far field. Since this calculation is particularly relevant, we review its essential features here. We assume that the coordinate z extends from the substrate up through the film, with the substrate present for z < 0. Beyond a solid 4He layer adjacent to the substrate an effective Van der Waals potential is assumed, --.as/Z 3. Sherrill and Edwards (1985) assume a ground state of N 4He atoms on an area A of the form ~o(rl ..... r,) of energy E0. For an impurity atom of mass m replacing one 4He atom we have

~ ( r l ..... rn) = [q~(rl)/a(rl)l~Po(rl ..... r.), where a2(rl)=p(rl)/po(O ) is the normalized density profile of a pure 4He film

338

R.B. H A L L O C K

Ch. 5, w

with wave function V)0. The Euler-Lagrange equation which minimizes the energy E is a single particle equation for ~P(rl), (-h212m)Vl 2 + V(Zl)]qb(rl) = eq~(rl), where V(Zl) is an effective potential of eigenvalues e = E - E0 -

0.3

T

L4

given by

- - 1 I

0.2 v

2

N

"--~ 0. I

C

]

. . . .

I

-2 A

3c12

>

-6

v,

-8 !

P _ ply(o).~l

I00 o a.

I

5O

A

o q.

(o) 0

.

0

I0 z

(~)

20

30

Fig. 7. Results of calculations by Sherrill and Edwards (1985) for a d = 20 A 4He film as a function of z, the distance from the substrate. (a) The pressure P and the density p normalized to the density of bulk 4He at zero pressure,o4(0). (b) The effective potentials V3(z) for a 3He atom, and V4(z) for a 4He atom. The dashed line shows V40(z), the 4He effective potential neglecting the variation of density in the film. (c) The normalized probability densities ~(z)l 2 for the two lowest 3He states in the film.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

339

V(z) = h21(2m)aTa + [maim- 1]t(zl) - L4 with t(Zl) the 4He kinetic energy. The exact ground state for pure 4He has e = - L 4, the 4He chemical potential in the film. In the theory L4 depends on the 4He thickness and is taken to be L4---L4~ = ( a s - a)/d 3 where L ~ is the binding energy to bulk liquid and -a3/z 3 is the van der Waals potential above the bulk surface. Since there is no lateral modulation of the potential, ~(rl) = ~(zl) exp(ikrl) where k is parallel to the surface. A density profile for the free surface of the film is chosen patterned after that for the bulk, p(z) = po(P)/[ 1 + exp(p(z))] 2 where P is the local pressure and p(z) is a parametrized function. The kinetic energy is approximated as t(z)= toaE~(z) - (hE/(2m4))a"/a and the potential chosen as Va(Z)= V4(z)+ (ma/m4- 1)toaE~(z) with V4(z) = (h2/(2m4))"/a- L4. Some of the results of this calculation are shown in figs. 7 and 8 (Sherrill and Edwards 1985). A bound state exists at the free surface of the film. Excited states exist within the film and the energetics depend on the film thickness. Krotscheck has had a major impact on our understanding of the nature of helium films and 3He impurities in such films through his series of variational calculations. The variational model was successfully applied to inhomogeneous

9

|

'

,.

>-. (.9 nILl Z ILl

-4

-6

-8

m

0

1 0.02

0.04

I/D

0.06

0.08

0.1

(/~")

Fig. 8. Energies of the lowest 3He states el,e 2..... as a function of lID, where D is the "mass thickness" of the film and incorporates the fact that extra density is present in the first layer or so adjacent to the substrate. In terms of D, the areal density of the film is 0.0218D/~2.

340

R.B. HALLOCK

Ch. 5, w

Fig. 9. Density profiles for helium films from the early work of Krotscheck (1985), for the strong substrate potential model of helium on graphite, for particle densities n = 0.14, 0.16, 0.18, 0.20, 0.22, 0.24, and 0.26 ]k-2. The substrate is located at z < 0. bulk helium in a theory developed by Krotscheck et al. (1985). It was then applied (Krotscheck 1985) to the structure of helium films adsorbed onto surfaces of various strengths. In this application, the Euler-Lagrange equations were solved for one and two body correlations for a model potential for the helium-substrate interaction in three manifestations: a strong version reminiscent of the helium-graphite interaction strength, a somewhat weaker version chosen to mimic about 10/~ of hydrogen adsorbed to a glass substrate and a very weak potential chosen to model the surface of a thick helium film. The one-body densities show modulation perpendicular to the substrate and this modulation has a strength which depends on the strength of the potential chosen. A typical example of behavior from this work is shown in fig. 9. The modulations result from the "geometrical restrictions acting on the helium atoms and the compression of the liquid due to the attractive substrate potential." In this work Krotscheck examines the effect of a single 3He impurity in the 4He film on the basis of the improved wave function he has available. This work differs from the earlier work by others in that here the explicit layering of the system is imposed by the substrate potential. In this theory the collective

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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Fig. 10. Occupation probabilities of three bound states of a 3He impurity (solid, long-dashed, and short-dashed lines with increasing energy) are shown for the film with specific 4He coverage n = 0.26/~-2 for the case of the strong substrate potential of helium on graphite (Krotscheck 1985). Also shown is, for comparison, the 3He impurity potential (short-dashed line in the lower half of the figure). The normalization of the occupation probability, 6pl(3)(z) = IW(z)l2, is arbitrary here. excitations can also be calculated. The position of the lowest lying mode is momentum dependent. For the case of a 3He impurity the physical location of the 3He depends on the strength of the substrate potential. For a strong potential the 3He sits at the free surface and the next excited state sits closer to the substrate (fig. 10). A third state is located in the bulk of the film and is expected to evolve into the continuum of states available as the system becomes bulk-like as the film thickness grows. As in the earlier work, it was not possible to handle the solid near the substrate and so a modified potential is used which results in no solid layer. The only apparent effect of this is that the theory is applicable only to the liquid portions of the film. Krotscheck et al. (1988) followed this early work with a more systematic and detailed variational theory of the properties of impurity atoms (3He, hydrogen, deuterium and tritium) adsorbed to 4He films of various coverages on a substrate. This work relaxed the assumption made earlier that the two body correlations between the impurity and the background atoms were the same as between the individual background atoms. This had been attempted by Mantz and Ed-

342

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R.B. HALLOCK

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wards (1979) and Guyer and Miller (1981) but these efforts did not take account of the layered structure of the 4He film itself. In Krotscheck's work three or more bound states were found for the 3He and the interpretation was that two of these states are at the free surface. As with the work of Dalfovo and Stringari (1988), the presence of two states at the free surface was interpreted as being due to the width (Stringari and Treiner 1987) of the surface profile. Following this calculation, Epstein et al. (1990) included 3He-background correlations and 3He-3He interactions and considered the effective and hydrodynamic mass of the 3He as a function of 4He coverage. A typical result of this work is shown in fig. 11 which includes both the effective mass and the hydrodynamic mass. The inaccuracy of the HNC calculation in getting the density correct was noted. It was predicted that it would be hard to observe variations in the 3He effective mass as a function of 4He coverage for the higher coverages. These theoretical results will later be compared in more detail with Gasparini's

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PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

343

experiments and the NMR work of Valles et al. (1988) and Sprague et al. (1995a) (Sprague 1993). More recently Clements et al. (1993a) have more carefully examined the structure of 4He helium films as the coverage is increased for two of the model potentials previously studied: (1) two solid 4He layers on graphite and (2) about 10/~ of H2 adsorbed to glass with parametrized "tuning" of the potential in each case, and for alkali metals (Saarela et al. 1993). The calculations are done in the HNC approximation with contributions from the pair distribution function and four and five body interactions. Triplett correlations are also included. The resuits reveal that film growth is far from uniform (fig. 12). As the coverage is increased on these "weak" substrates, for which solid is already present or will never grow, for the lowest coverages cluster growth occurs followed by monolayer completion (Clements et al. 1993a,b, Saarela et al. 1993). Even before monolayer completion, it is energetically favorable for cluster growth to begin in what will become the second layer. The growth of clusters in the second layer is accompanied by further growth of the first layer with an increase in density there. For example, it is found (with some sensitivity to tuning of the potential) that the theory has "no solutions" for certain coverages. As seen in fig. 13 there are regions of coverage for which the incompressibility, mc 2, where c is the velocity of sound, cannot be computed. A stronger substrate potential tends to produce more highly compressed layers, but has only modest effect on the width in coverage of the regions of instability. The recent observation of re-entrant superfluidity for 4He as a function of coverage on graphite by Crowell and Reppy (1993, 1994) would appear to be consistent with the growth mechanism contained in this theory, although an alternate explanation for the re-entrant su0.06 -

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344

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perfluidity based on a Bose-Hubbard model has been proposed by Zimanyi et al. (1994). These more detailed calculations have been applied to the case of thicker films of 4He, as studied by Pavloff and Treiner (1991b), with the conclusion that the Pavloff and Treiner functional is too repulsive in the thin film limit. A detailed comparison of the HNC/EL and non-local density functional predictions for identical potentials has recently been carried out by Clements et al. (1994). There is presently some controversy between the density functional and the microscopic communities over the proper description of thin helium films. This level of HNC/EL theoretical work is currently being applied to mixture films (Krotscheck 1994) where the implications could be significant. Much of the recent theoretical work, in particular the recent detailed calculations of the energetics to be mentioned later, has been motivated by experiments, particularly those which revealed the heat capacity and NMR properties of the mixture films. Following a brief discussion of coverages scales, we turn to a description of some of these experiments and a comparison of the results to the predictions of theory.

3.2. Thickness scales 3He and 4He coverages have been expressed by different authors in different ways. Theorists tend to use atoms//~ 2 while experimentalists often use other conventions. For 3He coverages one atomic layer (denoted d3 when expressed as an equivalent thickness at bulk density) is defined to be n 3 = 0.0647 atoms/~2;

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PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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Thus, by this convention we have one atomic layer of 3He when there are 6.47 x 1014 atoms/cm 2 on the surface. In some cases it is natural to express the distance from the substrate to the surface of the 4He film as d4 in atomic layers with one such layer defined to be a distance of 3.6/~. Third sound measurements are often reported this way since third sound measurements provide a measure of the thickness of the helium 4He film, d4. For 4He at bulk density our convention here will be 1 layer = 12.82/~mol/m 2 = 0.0772 atoms//~ 2 = 7.72 x 1014 atoms/cm2; thus, the thickness in bulk density equivalent layers will be denoted D4, with n 4 = 12.82/~mol/m 2 per layer x D 4. Differences in descriptions concerning the thickness of the 4He film among different authors often result from different assumptions about the density of 4He in the film in the first two layers adjacent to the substrate. Here we will generally discuss experimental results by use of the conventions adopted by the original authors. Thus, to compare the results from different types of measurements requires a bit of care with the coverage scales. In particular, heat capacity results for mixture films are often reported in terms of the coverage of liquid above a solid 4He base. To obtain the total coverage one needs to account for the 4He in this immobile base.

3.3. Energetics experiments 3.3.1. Heat capacity experiments The beginning of experimental understanding of the energetics of 3He in 3He4He mixture films came with the work of Gasparini and his colleagues in the context of heat capacity measurements. We describe these experiments and they will be compared with more detailed recent results obtained primarily by NMR later in this review. To obtain adequate signal in a reasonable volume it was necessary to select a substrate material which had considerable surface area. The material chosen, Nuclepore, used as a biological filter, is a polycarbonate material of thickness 10/tm threaded by roughly cylindrical holes of nominal diameter 2000/~. More detail on this material and its characterization can be found elsewhere (Chen et al. 1977, Godshalk and Hallock 1987, Chen et al. 1990, Crawford et al. 1992). These heat capacity experiments (DiPirro 1979, DiPirro and Gasparini 1980, Bhattacharyya 1983, Bhattacharyya et al. 1984) were carried out in two different experimental cells generally illustrated in fig. 14. To enhance the surface area so as to provide an adequate two-dimensional signal, the substrate consisted of 2000 filters with a surface area of 1.1 x 106 cm 2 as determined by N2 adsorption. In an effort to ensure good thermal contact to the filters, each filter was heat sunk to the copper calorimeter by gold wires which were epoxied to the edges of the filters. As with any heat capacity measurement, the heat capacity of the system of interest must be obtained from the total heat capacity measured in

346

R.B. HALLOCK

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Fig. 14. A schematic of the calorimeter arrangement used by Bhattacharyya et al. (1984) for heat capacity studies of thin mixture films. C, CMN thermometer; F, copper wire connecting calorimeter to heat switch connected to mixing chamber of dilution refrigerator; G, germanium thermometer; H, heater; W, cotton thread suspension; N, Nuclepore filters; B, gold wires; S, squeeze connector; P, quartz capillary section of filling line; K, light shield anchored at mixing chamber; L, light shield anchored at 1 K; V, vacuum jacket. the experiment. For mixture film experiments, this in general consists of three contributions: (1) the system of interest, (2) the 4He which is adjacent to the substrate and (3) the body of the calorimeter and the associated thermometers, heaters etc. In the work of Bhattacharyya et al. (1984), it was shown that for 4He film thickness values above a certain value the dominant contribution to the 4He heat capacity was from surface excitations on the liquid film and these were too small to measure. Thus, the observable heat capacity signal was from the impurity 3He atoms in the 4He film environment. The 3He-4He phase space covered by these measurements (Bhattacharyya et al. 1984) is shown in fig. 15. Results similar to those shown in fig. 16 were obtained as a function of temperature for various coverages of 3He for a number of different thicknesses of 4He film. The increase above the Boltzmann value, shown in dotted lines near the right axis in the example of fig. 16, is due to excitations of the 3He to states

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PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS I

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above the ground state in the film. At low temperatures, ignoring interaction effects, one expects the heat capacity to be that of an ideal two-dimensional Fermi gas (Bloom 1975). In two dimensions this is special with the heat capacity being independent of the number of atoms in the apparatus. To see this, we consider a simple picture. For an ordinary Boltzmann system, one expects the energy to depend on the number particles in the system, E ~ N k b T with C = 6 E / 6 T . For a Fermi system, of course, only a fraction, kbTleF, of the particles in the system are close enough to the Fermi surface to accept excitation, thus, we have E ~ N k b T x (kbT/ef) and since in two dimensions we have ef ~ N, the heat capacity is independent of N. In the fully degenerate regime, we have C = (Ztkb2/3h2)mAT,

where A is the area decorated by the particles of mass m. To analyze the data, Bhattacharyya et al. (1984) adopted the simple model (Gasparini et al. 1984) of the film described earlier in which the system is assumed to be comprised of a finite set of single particle states. For simplicity in comparison to the data, it was assumed that a ground state, a first excited state and a free particle state were present. Thus, the model allowed the parameters of the energy ei and the mass m i of the 3He in the ground state and the first excited state. An interative procedure was used in which e i and mi were adjusted so that the best least squares fit to the heat capacity data was obtained. Except for the case of the thickest 4He film studied, this set of states was adequate to completely characterize the data and values for the effective mass and energy of the two lowest states were obtained. As expected, in most cases these quantities tended toward their known values for the bulk surface as the thickness of the

348

R.B. HALLOCK

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Fig. 16. Heat capacity for 3He in a 10 A (liquid) 4He film (Bhattacharyya et al. 1984). The data is plotted here on a linear scale and with emphasis on the low-temperaturebehavior. C is in erg/mK. The dashed lines are the Boltzmann values. The solid lines are fitted to the data. 4He film was increased. An example of this is shown in fig. 17 where we illustrate the effective mass of the ground state and the difference in energy between the ground state and the first excited state for the case of 0.28--0.29 layers of aHe on the 4He film. Data for the energy of the individual states is also available and these are shown as a function of the 3He coverage in fig. 18 with the effective masses shown as a function of 3He coverage in fig. 19. The energy differences and the effective mass ratio as a function of coverage is shown in fig. 20. Note here that the effective mass ratio is substantially greater than unity. The 3He in the excited state has a considerably enhanced effective mass above that found for the ground state at the free surface of the film. The effective mass in the ground state was observed to show a dramatic change with 3He coverage near 0.2 monolayer as shown in fig. 19. This interesting effect is not yet understood and is a feature which does not appear in the most complete theories.

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P R O P E R T I E S O F M U L T I L A Y E R 3He-4He M I X T U R E F I L M S

349

Fig. 17. The ratio of the ground-state effective mass to the 3He mass, and the excitation energy between the ground state and the first excited state, e I - r 2, for 0.28-0.29 layer of 3He on various films of 4He as determined from heat capacity measurements by Bhattacharyya et al. (1984). The dashed lines represent the value of these parameters for the bulk surface.

Fig. 18. The energies e I and e 2 as a function of 3He coverage for two films of 4He as determined by Bhattacharyya et al. (1984). The coverage numbers 10, 12.3/~ refer to the thickness of 4He liquid present. Lines are drawn to guide the eye.

350

R.B. HALLOCK

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Fig. 19. The ratio of mI and m2 to the 3He bare mass as a function of 3He coverage (in layers) for various liquid 4He film thickness values (from Bhattacharyya et al. 1984). Lines are drawn to guide the eye. Reference to fig. 17 shows one of the conclusions of the work. One expects that the energy difference between the ground state and the first excited state should appear to approach the value it has on a bulk 4He film as the 4He film thickness gets large; a thickening film should eventually look like the free surface on bulk. The effective mass is seen to be similar to that on the bulk free surface, but the difference in energy appears to be rather low. The second notable conclusion reached is that the thinning of the film appeared to have a substantial effect on this difference in energy. We will return to these heat capacity results later in the context of the recent more detailed NMR results. Bhattacharyya and Gasparini (1982, 1985) have reported an interesting and unexpected observation of the heat capacity of 3He in 4He films for a narrow range of 3He coverages and 4He film thicknesses. The basic observation is shown in figs. 21 and 22 where the heat capacity is seen to display a sudden transition to a linear behavior at low temperatures. As already noted, for a degenerate two-dimensional Fermi system, one expects that the low temperature heat capacity will be given by C = ztk2BmeATI3h 2 where A is the area covered by the 3He and m e is the effective mass of the 3He. For data taken at a variety of

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PROPERTIESOF MULTILAYER 3He-4He MIXTURE FILMS

351

Fig. 20. The ratio m21mI and the excitationenergy as a function of 3He coverage for various liquid 4He films (from Bhattacharyyaet al. 1984). The lines drawn are to guide the eye. coverages, one would expect that the low temperature slope of the heat capacity would be independent of 3He coverage. This is because for a Fermi system in two dimensions the heat capacity is independent of the density. Data from the experiment which shows this expected behavior of the slope for higher 3He coverages is shown in fig. 23. The dashed-dot line assumes that the 3He is homogeneously distributed over the surface of the 4He. Clearly, the experimental observation is that the slopes for many of the coverages studied are not the same and the value of the slope depends linearly on the coverage. These data were interpreted by Bhattacharyya and Gasparini (1982, 1985) as indicative of a transition in the film from homogeneous two-dimensional behavior to a condensed higher density phase which occupies increasing area on the substrate as the amount of 3He is increased. Accompanying the transition was a small anomaly (jump downward) in the heat capacity. Below the transition, the heat capacity is considerably reduced below that expected for a homogeneous system, which implies an unexpected missing entropy. These authors suggested several reasons why the missing entropy was not observed and proposed a number of possible alternative explanations for the behavior of the heat capacity. One of these, the appearance of a three-dimensional region of 3He in the apparatus, termed a "bubble", was dismissed due to lack of quantitative agree-

352

R.B. HALLOCK

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Fig. 21. Heat capacity of 3He at various coverages(expressed as fractions of a monolayer)on a 4He film of liquid thickness 10-~ as determinedby Bhattacharyyaand Gasparini (1985). The solid lines represent a fit of the data under the assumption that the 3He is homogeneously spread out over the surface of the film. The dashed lines are drawn to guide the eye. ment with the data. As noted recently by Tiwari and Glaberson (1990), the experiment contains some other discrepancies with the phase transition interpretation which Bhattacharyya and Gasparini (1984) noted: the behavior of the heat capacity at low temperatures does not support the existence of a low 3He density phase, the presumed phase separation is not accompanied by a jump in the heat capacity and, again, there is entropy missing at low temperatures. In passing, we should note here the more recent work by Greywall and Busch (1990) (Greywall 1994) at much lower temperatures in which the heat capacity of 3He adsorbed onto graphite displays perhaps similar behavior near 3 mK in the first and second fluid layer heat capacity. It resembles the Bhattacharyya and Gasparini (1984) data in the characteristics of its temperature dependence. Whether the physics of the two dramatically different temperature regimes is similar has not been established. In an effort to explain these unexpected heat capacity results due to Bhattacharyya and Gasparini (1985), Guyer (1984) developed a model of the 3He-aHe mixture film which considered the energetics of the 3He in the 4He film system and concluded that at T = 0 K the 3He should sit atop thickened regions of the 4He termed "mesas" with the surrounding "plains" relatively devoid of 3He. The

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PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

353

Fig. 22. Heat capacity of 3He at various coverages on a 12.3-/~ (liquid-thickness) 4He film. The solid lines through the two lowest coverages are drawn at the expected Boltzmann value. The solid line through the squares is the expected behavior of the data if the 3He were to remain homogeneously spread out over the 4He film. The dashed lines are a guide to the eye (from Bhattacharyya and Gasparini 1985). model was extended to finite temperatures with the result that as the temperature was increased, the "plains" became more decorated with 3He accompanied by a reduction of the amount of 3He on the "mesas". A critical point was present in the theory above which the "mesas" disappeared. This calculation has been criticized by Sherrill and Edwards (1985) for its neglect of exchange effects and for the definition used to define the film thickness in the case of a mixture. As we shall describe in more detail later, this disagreement was investigated by Valles et al. (1986) (Valles and Hallock 1987) who carried out measurements of the velocity of third sound as a function of the amount of 3He and 4He in a third sound resonator. Shifts in the velocity allowed a measurement of the quantity t~2e/t~D4 2, where e is the energy of 3He ground state. Comparison to the theory of Sherrill and Edwards (1985) showed agreement. Anderson and Miller (1989) later carried out a semiphenomonological calculation which fixed the parameters of the calculation by appeal to experiment; they found no evidence for coexisting phases and thus did not support the phase transition interpretation of the heat capacity experiments.

354

R.B. HALLOCK

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Fig. 23. Low-temperature slope of the heat capacity at various coverages and for two liquid 4He film thicknesses. The dashed-dotted line is the slope that all the data would be expected to have, if the 3He were homogeneously spread out over the 4He film surface. An effective mass of 1.7m3 was used to obtain this line (from Bhattacharyya and Gasparini 1985).

The experimental observation (figs. 21,22) remains unexplained and has not been supported by other measurements using other techniques. For example, measurements of the magnetic susceptibility might be expected to show anomalous behavior if a condensation such as that proposed were present. The susceptibility is given by x(T) = X30[1 - exp(-Tr./T)], where X30 = (yEI4zt)Amh and TF = zrhEN31mhkBA, with mh the hydrodynamic mass to be described in more detail later. This has the same dependence on area and independence of the density of the 3He in the apparatus as does the heat capacity. Given the smooth dependence of the susceptibility on temperature seen in the NMR experiments, it is clear that no abrupt transition is seen in the susceptibility (Valles et al. 1988). If C ~ N3, such a condensation model predicts that x(T= 0) should be proportional to N3 as well, but the NMR data shows that this is not the case. This difference in behavior between the heat capacity and NMR experiments remains unexplained. We will return to this point later in the discussion of the NMR results.

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3.3.2. Nuclear magnetic resonance experiments Relatively recent NMR studies have considerably expanded the details of our knowledge of the mixture film system. To begin the discussion, we first mention early and largely forgotten NMR measurements on pure 3He in Vycor by Brewer and Rolt (1972). These experiments and their interpretation are relevant to later work in the context of mixtures and are discussed here to recollect the work and help to put the later work on 3He-aHe mixtures into better historical perspective. In these early experiments, the interest was to examine the behavior of 3He in a geometry with considerable surface area in an effort to examine what effect the proximity of a substrate might have on the 3He NMR behavior. The results of this work for the case of filled Vycor are shown in fig. 24. An effort to explain these results in terms of a statistical model was made (Brewer et al. 1971). In this model, the first two 3He layers were assumed to have higher density than bulk and were assumed to act independently of each other and of the remaining liquid in the system. Each region, the first layer, the second and the remaining liquid was thus assumed to behave as bulk liquid with the relevant

Fig. 24. Main diagram, linear plot of xTIC versus T; the solid line through the points represents the experimental results, the dashed curve is for bulk liquid (Thompson et al. 1970, Ramm et al. 1970). Inset, the same data plotted semilogarithmicallyto bring out the low-temperaturepart. The lowest curve is for bulk liquid; the upper dashed curve represents the prediction of the statistical layer model described by Brewer et al. (1971) (from Brewer and Rolt 1972). The points are the experimental results.

356

R.B. HALLOCK

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local density. While the model accounted for the susceptibility down to 0.4 K, it was unable to account for the data at lower temperatures; the observed susceptibility was considerably higher than the predictions of the model. In an effort to explain the observed behavior, Brewer and his colleagues proposed two alternate possibilities, each of which retained the notion that the separate layers of 3He might have different behavior. The first speculation was that the second layer susceptibility might behave as a two-dimensional Fermi system with degeneracy temperature to be determined. The susceptibility of the second layer was assumed to behave as

ZIg(T) = xIC = [1 - exp(-To**lT)]/TD **, where To** replaces the usual ideal gas degeneracy temperature. A reasonable fit to the data was obtained with TD**--50 mK. The second speculation was that the second layer might behave as if part of it was comprised of localized 3He atoms which obeyed Curie's law while the rest acted like ordinary bulk fluid. As we shall see, each of these suggestions is relevant to some of the physics of current NMR measurements with mixture films. For completeness we should note here the recent NMR work on pure 3He films for temperatures in the range 1.3 < T< 4.2 by Swanson et al. (1988).

Fig. 25. Schematic diagram of the apparatus used for sequential NMR and third sound experiments (after Hallock 1991).

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Recent experiments on the NMR properties of mixture films are the continuing series of experiments by Hallock and his co-workers. The sample cell (Hallock 1987, 1991) which has been used for this work was designed for simultaneous third sound and NMR studies and is shown schematically in fig. 25. The NMR substrate consisted of 400 Nuclepore filters of thickness 10 pm, each perforated by -4 x 108/cm 2 nearly cylindrical pores of nominal pore diameter 2000 ~. The filters provided a large surface area (~ 1.77 m 2) in the ~ 1 cm 3 volume inside the NMR coil. The Nuclepore filters were fit onto a copper post and after cooling each was in good thermal contact with the mixing chamber of a dilution refrigerator. The choice of Nuclepore was motivated by a desire to use a substrate with adequate surface area for the NMR and also to have a substrate in common with the heat capacity work of Gasparini and his co-workers. The magnetization (susceptibility) of the 3He was measured in thermal equilibrium by pulsed NMR techniques in a 2 T magnetic field for which the Larmor frequency was 62.9 MHz; the applied magnetic field was oriented perpendicular to the average direction of the pores of the Nuclepore. The height of a spin echo, E(v) was typically obtained following a z~/2-r-zt pulse sequence for a number of different values of v. The extrapolation to v = 0 provides a measure of the magnetization. The magnetization was calibrated to an uncertainty of about 6% by Curie law data taken at the lowest coverages studied. In the first of these experiments, Valles et al. (1988) found that for low coverages of 3He the magnetic susceptibility of the real physical system closely matched that predicted for a nearly ideal two-dimensional Fermi gas. For such a system, the susceptibility is given by x(T) = (y214~)Amh[1 - exp(-TrJT)],

where Tr = zth2N3/mhkb A, )1 is the 3He gyromagnetic ratio, and where mh is the hydrodynamic mass (the mass of a 3He atom interacting with the surface of the 4He, but not interacting with any other 3He atoms). Temperature dependent data was fit to this expression for several small 3He coverages with a 4He coverage d4 = 2.8 layers with the conclusion that rnh/m 3 = 1.8. For higher coverages of 3He on a d4 = 9.5 layer 4He film (which should be reminiscent of the bulk 4He surface) deviations from ideal two-dimensional behavior were seen and since these deviations became stronger with increasing 3He coverage they were attributed to the 3He-3He interactions. For an interacting Fermi system, one expects that the susceptibility will be enhanced by the interactions. In the context of a Landau Fermi liquid theory (Pines and Nozieres 1966, Baym and Pethick 1971, Freedman 1978, Havens-Sacco and Widom, 1980) model, the susceptibility is expected to behave as z(T)]x(0) = (mh]m3)(1 + FlS/2)/(1 + F0a),

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Fig. 26. Susceptibility as a function of inverse temperature for several 3He coverages and a relatively large 4He coverage of 9.5 layers (3.6 ]k each) (from Valles et al. 1988). 3He coverages are in equivalent layers (One aHe layer corresponds to an areal density of 6.45 x 10-2 atom/~2): squares 0.088; triangles, 0.176; diamonds, 0.352; circles, 0.440. The dashed lines are extrapolations used to obtain values of X(0) which are plotted as a function of 3He coverage in the inset, where the dashed line is a linear fit to the data. where the two Fermi liquid parameters, Fl s and F0 a are expected to depend on the amount of 3He present. The data (Valles et al. 1988) (fig. 26) showed that the combination (1 + FIV2)/(1 + F0a) increases linearly with 3He coverage for relatively low 3He coverages. This linearity was consistent with interpolations from earlier experiments by Brewer et al. (1971) in which the 3He concentration was kept fixed while the total coverage was changed. The dependence of the susceptibility on the thickness of the 4He film was shown to be a decreasing function of increasing film thickness. The physical basis of this resides in the hydrodynamic mass. For thinner films the 3He-4He interactions in the (more structured) vicinity of the free surface are stronger than they are for the case of thicker films and thus mh increases as the 4He film thickness decreases. We will return to this point later. This system was studied in some detail theoretically by Krotscheck et al. (1988b) who, as mentioned earlier, have developed a microscopic theory of the quasiparticle interactions between two 3He atoms adsorbed to the surface of the

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4He film. The theory assumes that the 3He particles occupy the ground state (the surface state at the surface of the 4He film). The quasiparticle interaction is computed and from this the Fermi liquid parameters can be computed for the case of dilute 3He coverage. Two effects are found to contribute to the effective mass of the 3He: the interaction with the 4He (the hydrodynamic mass effect) and the Fermi effective mass which arises from the Fermi motion of a 3He atom relative to the other 3He atoms in the system which is related to the Fermi liquid parameter Fl s. The total effective mass which can be computed directly is then m* = mh(1 + FlS/2). The magnetic susceptibility has an additional Fermi liquid parameter Fo a, X/X30 = mh/m3(1 + FlS/2)/(1 + Fo a) = (m*/m3)/( 1 + F0 a) and Krotscheck et al. (1988b) predict Z using values for mh]m3 taken from the

Fig. 27. The magnetic susceptibilityX/g30of a 3He film as a function of areal density n in atom//~2 and coverage in layers. The uppermost curve corresponds to a 4He coverage of 0.30 atom/]k2. The squares are the experimental results of Valles et al. (1988) for a 4He film of about 34/~ thickness. The circles are susceptibility data for a 3He density of 0.088 layer on 4He films of 10.0, 12.2, and 17.1/~ thickness. A hydrodynamic mass of 1.26m3 was assumed in the calculations for all cases (Krotscheck et al. 1988b).

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experiments of Valles et al. (1988). The results as a function of the 3He coverage for several different values of the 4He coverage are shown in fig. 27. The experiment of Valles et al. (1988) was designed to cover some of the same area in the 3He-aHe coverage plane as did the heat capacity experiments of Gasparini and his co-workers. Indeed, the choice of substrate material was motivated in part by the desire to use NMR techniques to study in more detail the liquid-gas (or puddling) transition which was inferred from the heat capacity results. Given the smooth dependence of the susceptibility on temperature which was observed, no sign of an abrupt transition was evident in the susceptibility measurements. The slope of the heat capacity dC/dT(O) was linear in the amount of 3He in the heat capacity cell. The puddle model unambiguously predicts that the low temperature susceptibility X(0) should also be proportional to N3. The lowest temperature data of Valles et al. (1988) were not proportional to N3 and thus not consistent with the notion of a condensation of puddling transition in the 3He on the surface of the 4He film. No clear explanation for this discrepancy in behavior has emerged. Recent work by Anderson and Miller (1992, 1993) relaxed their earlier theoretical constraint that there was only a single state in the problem located at the surface of the 4He film. The presence of higher-lying transverse states does not, however, result in a self-bound 3He film. In more recent work, Anderson and Miller (1995) have shown that the level spacing for mixture films has a dependence on 3He coverage. They speculate that the heat capacity results may signal a phase separation rather than a condensation event. If true, this might resolve the apparent disagreement between the two techniques. It is possible that new approaches to the growth of helium films may eventually enhance our understanding of the differences between the heat capacity and the susceptibility results. For example, as mentioned earlier, the recent work of Clements et al. (1993a,b) showed that the growth of 4He films can, under some circumstances, proceed by the formation of two-dimensional liquid clusters. It is possible that in the case of mixtures such a mechanism might also exist. The effect of clusters in the mixture case awaits theoretical study from this perspective. In an effort to more thoroughly explore the 3He coverage dependence of the susceptibility, Higley et al. (1989) carried out NMR measurements over a wide range of 3He coverage, 0.005 < d 3 < 4 layers. The results of these experiments revealed striking steps in the magnetization which are shown in fig. 28 (Higley et al. 1989, 1990). Shown are the low temperature magnetization data available from detailed measurements of the magnetization of the type shown in fig. 29. For 0.1 < d3 < 0.65 layer, the magnetization is degenerate at 40 mK and its increase with density is ascribed to the increase in Fermi liquid interactions among the 3He as the 3He areal density increases. The data can be linearly extrapolated to the zero coverage limit where there are no quasiparticle-quasiparticle interactions and it is found that mh = 1.38 __0.22 for a 4He coverage of 44/~mol/m 2 (a

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Fig. 28. Magnetization at T = 40 mK versus 3He coverage, d 3, showing steps in the magnetization at d 3 = 0.8 and 1.5 layers, and an increase in magnetization with the same slope as for bulk liquid above d 3 = 2 layers (Higley et al. 1989). The dashed line extrapolates low-coverage data to obtain the hydrodynamic mass m h = 1.38m 3.

thickness of d4 = 2.14 layers). Such low temperature extrapolations provide a direct measure of the hydrodynamic mass of the 3He in its ground (surface) state. Subsequent experiments (Higley 1991) for thickness values of 1.77 and 2.94 layers resulted in mh values of 1.64 and 1.34, respectively, consistent with the expected increase of mh as the coverage is decreased. The slope of the magnetization with an increase of 3He coverage is in good agreement with the theoretical work of Krotscheck et al. (1988b). For 0.7 < d3 < 0.9 a step-like doubling of the magnetization occurs (fig. 30); a second less prominent step is evident (fig. 28) near d 3 = 1.5. These steps imply occupation of higher discrete energy levels in the film. Indeed, the positive value of dM/dT (fig. 29) for 3He coverages preceding the first step suggests thermal promotion into a second energy level. For large coverages, the magnetization appears to increase with the same rate per particle as should bulk 3He. The picture here is that at the higher coverages, additional 3He simply joins other 3He which have 3He neighbors on all sides and the system appears bulk-

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Fig. 29. Magnetization versus 1/T of 3He for the coverages (in layers) shown on the figure at fixed 4He coverage 44/amol/m2 (Higley et al. 1989). The positive dMIdT at d3 = 0.65 layers precedes the step in magnetization. The curves for d3 < 0.2 layer are 2D Fermi-gas fits, and the curves for d3 > 0.65 layer are a guide to the eye. like. In the work of Higley et al. (1989) a simple model of the system was adopted in which the system was assumed to have only two levels and an approximation to extract a value of the energy separation, e~2, due to HavensSacco and Widom (1980) was used. This, in spite of being applied beyond the weak- interaction limit region of validity, resulted in a good fit to the temperature and coverage dependent magnetization with el2 = 1.8 K. To account for the temperature independent slope of the magnetization at the step for T < 60 mK, a parameter to account for an assumed variation, tr, in the value of e 12 across the substrate (due to inhomogeneity) was introduced. The best fits to the data resuited in tr --- 0.1 K, thus tr << el2, and substrate inhomogeneity was therefore presumed to play only a small role in the energetics of the 3He revealed by the experiments. The physical picture one should have in mind here as the basis for this step structure begins with an ideal two-dimensional Fermi gas and draws on some of our earlier discussion concerning effective mass. Consider first a collection of

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Fig. 30. Magnetization versus d3 at 40, 100, and 250 mK. Also shown is a fit by a simple two-level model having energy gap el2 --- 1.8 K near the step at d3 = 0.74 layer and a distribution in el2 of cr -- 0.10 K (Higley et al. 1989). non-interacting two-dimensional levels at T = 0 which can be populated by ideal non-interacting Fermi particles. For such a system, the magnetization (normalized by M o, the ideal two-dimensional Fermi gas magnetization) is a constant (unity) independent of the number of Fermi particles in the system. As the 3He coverage is increased, TF increases; ultimately the chemical potential reaches the energy of the first excited state. Further addition of particles begins the occupation of a second two-dimensional level which also has a magnetization independent of coverage. Thus the total magnetization doubles at the coverage where the occupation of the second level begins and the magnetization displays a step. In the actual non-ideal system several additional processes are at work: individual 3He atoms interact with the 4He substrate and become "dressed" (m 3 goes to mh > m3); interactions among the 3He quasiparticles increase the effective mass due to "Fermi" effects and the magnetization and energy levels change with 3He coverage. In this physical manner we can envision the reasons behind the "tilted plateau" appearance and the step of the magnetization with an increase in 3He coverage (Hallock 1991). These detailed magnetization measurements confirmed the existence of discrete energy states in the system and provided considerable detail about the properties of the 3He in those states. It is presently uncertain whether the ground state and the first excited state are both in the free surface of the film. Some have argued that this is the case. For this to be so, the surface of the film has to

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Fig. 31. Magnetization, T1, and T2 versus 3He coverage d3 at the indicated temperatures for a thin 4He film of thickness d4 = 1.77 layers. The structure evident in the magnetization is also evident in the relaxation times (from Alikacem et al. 1992). have a relatively wide profile. Evidence that this is the case has been presented by Lurio et al. (1992, 1993) based on X-ray studies of the helium surface. Measurements of the NMR Tl and T2 (Hallock 1991) showed that the structure in the magnetization as a function of 3He coverage had a counterpart in the relaxation time measurements. An example of this behavior (Alikacem et al. 1992) is shown in fig. 31 where there is a clear correlation between the relaxation time measurements, and the magnetization. The origin of this behavior in the relaxation time must lie with the occupation of the various quantum states in the system, but the specific reasons for the detailed structure are as yet unknown. One might imagine that as the 3He coverage is increased from low values, the 3He-3He interactions will increase as the coverage increases. As the density increases, the strength of the dipole-dipole interaction between the 3He atoms will increase, perhaps driving down the 7"2 as is seen. But, why the value of 7"2 should increase at the first step in the magnetization is not clear. One might imagine that a new parallel relaxation process would further enhance the relaxation rate, but the opposite seems to be true. Apparently promotion to the second state effects interactions in an unexpected manner. Another interesting correlation exists between these magnetization measurements and earlier measurements on the Q of third sound (Ellis and Hallock 1984). We discuss these third sound measurements and this correlation later.

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An interesting contrast exists between these data and the surface tension data on the surface of bulk 4He. In the experiments of Edwards and his colleagues (Guo et al. 1971), the entropy of 3He on the surface of 4He as determined from the surface tension measurements approaches the bulk slope for S/T versus d3 beyond d3 = 0.5 layer without any steps. Although it would be desirable to have these data for larger SHe coverages, it would appear that the presence of the confining van der Waals potential which serves to enforce the two-dimensional behavior of the film system is the origin of the difference in behavior. Alternatively, the absence of such steps may argue against the existence of two bound states in the bulk surface which have been predicted theoretically (Pavloff and Treiner 1991 a).

Fig. 32. Isotherms of the relaxation time T 1 versus 4He coverage for 30 mK (circles) and 200 mK (triangles) for Nuclepore substrates without (open symbols) and with (solid symbols) 0.8 layer of 0 2 preplating for a 3He coverage of 0.098 layers (Sprague 1993, Sprague et al. 1995b). The temperature dependence is observed to saturate for high coverages.

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The next series of measurements which involved NMR was a set of experiments on the temperature dependence of the relaxation rate 1/T~ for these films. Simultaneous new measurements of the magnetization were also carried out. As we shall see, these measurements have resulted in a rather detailed knowledge of the energetics of the 3He in the mixture films as a function of the 4He and 3He coverage. A discussion of the temperature dependence of the relaxation rate llTl breaks naturally into the temperature range above 250 mK where the behavior is thermally activated and the range 30 mK < T < 250 mK where there is no thermal activation. We begin with a discussion of the lower temperature range. Figure 32 shows T 1 for a 3He coverage of 0.098 layers as a function of 4He coverage D 4 at various temperatures, 30 mK < T < 250 mK (Sprague et al. 1991, Sprague 1993). Data from two different experimental situations are shown here: 3He on Nuclepore (open symbols) and 3He on Nuclepore in the presence of -0.8 monolayer of preplated oxygen (solid symbols). When oxygen is present, a distinct peak of width less than one-half layer of 4He is seen in the 30 mK isotherm of Tl centered at n 4 0.23/~-2 n4. ~ ?las" n4s is the coverage at which superfluidity appears at this temperature (as confirmed by a third sound resonator in the apparatus). No corresponding structure is seen (Sprague 1993, Sprague et al. 1995) in the coverage dependence of the magnetization (fig. 33). With the =

_

Fig. 33. Zero temperature extrapolations of the magnetization as a function of 4He coverage for 3He coverages of 0.0985 (solid circles) and 0.0982 (solid triangles) layers for the case of Nuclepore substrates without (triangles) and with (circles) 0.8 layer of 02 preplating (Sprague 1993, Sprague et al. 1995a).

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Fig. 34. (a) T 1 for a 3He coverage of 0.098 layers versus 4He coverage in the presence of 0.8 layer of 0 2 preplating for several temperatures: 30 mK (solid triangles), 50 mK (squares), 100 mK (open triangles), 150 mK (circles), and 250 mK (diamonds). (b) T2 versus n4 for two temperatures: 30 mK (solid triangles) and 250 mK (diamonds) (from Alikacem et al. 1991). pre-adsorbed oxygen absent, the relaxation times are enhanced considerably at low 4He coverages. The temperature dependence seen in 1/T 1 for n 4 > n4* (fig. 34) is apparently robust to the presence of oxygen. For n 4 < n4*, T 1 has a weak linear dependence on temperature, T l --- A-l(1 + yT), y -- 1 K -l. For n 4 = n4*, TI is seen to deviate from linearity in T, and for n 4 > n4* the temperature dependence of T1 is dramatically different; 1/T1 --- B(na)/~T (fig. 35). Similar ~/T dependence of relaxation times seen (Owers-Bradley et al. 1978, Himbert and Dupont-Roc 1989) in films near 1 K is attributed to the temperature dependence of the thermal velocity of classical 3He. Here, the ~/T behavior of Tl is preserved well into the degenerate regime of the two-dimensional 3He (inset, fig. 35). The relaxation rate 1/Tl is apparently composed of two independent relaxation processes; 1/Tl = WA + WB over the full range of n 4. W A is found to be a weak function of temperature, W A ~ A/(1 + ),T) and Wa "-B(na)/~T. Since, for larger coverages, the mild temperature dependence of WA is obscured by the stronger temperature dependence of WB, it is assumed that the form of the tem-

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Fig. 35. I/TI-I versus I/T -1/2 for various n4 > 0.247 ~-2. The dashed lines are straight line fits to 1/T1-1~ lIT-1/2. Several coverages are shown: n4 = 0.248 A-2 (solid triangles), 0.267 (squares), 0.290 (diamonds), 0.325 (open triangles) and 0.369 (circles). The inset shows representative twodimensional Fermi fits to the magnetization which show the spins to be degenerate for T < 100 mK (from Sprague et al. 1991). perature dependence of WA does not change significantly for n 4 > n 4.. T h u s , to characterize the data (Sprague 1993, Sprague et al. 1995a), the temperature was scaled by TF** and the data were fit by Tl -l --- A l/(1 + yT) + Bl(TF**/T) 1/2 for all n 4, with A l, B1, and ), functions of n 4. The effect of setting ), = 0 for n 4 > n4* does not significantly change the fit values of A l and B l, except near n4* where y =_ B-21. Figures 36 and 37 show the coverage dependence of the slopes and intercepts of the fitted rates A I and Bl for the case of no oxygen (fig. 36) and for the case of oxygen (fig. 37) on the surface. The behavior is similar except for the low coverage region. WA is seen to decrease relatively smoothly with increasing n4, and is at most weakly perturbed by the onset of WB; this supports the conclusion that these are two independent relaxation mechanisms. The mechanism which gives rise to the observed ~/T temperature dependence of WB is presently not understood. The normal cores of vortices may be populated by 3He (Ostermeier and Glaberson 1975), providing a reservoir of non-degenerate spins with which the degenerate 3He could interact. However, it is not clear why the existence of bound versus unbound pairs would influence the coverage dependence of the relaxation. Next, we discuss the 4He coverage dependence of Bl. For n 4 < h a * , B l _= 0. For ha* < n 4 < 0.32/~-2, BI increases approximately linearly with n4 and satu-

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Fig. 36. Comparison of rates A and B from the data taken for clean Nuclepore. Shown are fits to the longitudinal relaxation data T ! (top) and fits to the transverse relaxation data T2 (bottom). Here n4 = 12.82/~mol/m2 per layer x D4. rates for n 4 > 0.33/~-2. The range of n 4 over which this linear increase occurs corresponds to about 1 layer of 4He. The onset of the relaxation rate B1 at n4* may be correlated with the onset of superfluidity at n4s and its saturation may be related to the maximum extent of overlap between the 3He ground state wavefunction of size A and the superfluid, about one layer. As the superfluid grows from zero with coverage at fixed temperature, the amount of superfluid sampled by the 3He wavefunction increases and this overlap will naturally saturate as the thickness of the superfluid blanket exceeds A. The physical mechanism which motivates this relaxation is not available. Similar behavior is observed for the coverage and temperature dependence of T2 for these two data sets (lower figs. 36 and 37, parameters A 2 and B2, and also fig. 38). The relaxation rate 1 / T 2 displays some features in common with 1 / T 1. In particular, for D 4 _>4 both become rather independent of D 4 (fig. 3 8 ) and show little temperature dependence. In an effort to further explore the thickness dependence of the relaxation rates in the presence of oxygen and perhaps explain why A l and A 2 seem to grow with decreasing coverage only when oxygen is present (Sprague 1993), Sprague et al. (1995a) looked for a power law depend-

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Fig. 37. Comparison of rates A and B from the data taken with 0.8 monolayer of 0 2 on the Nuclepore. Shown are fits to the longitudinal relaxation data T1 (top) and fits to the transverse relaxation data T2 (bottom). Here n4 = 12.82/~mol/m2 per layer x D 4. ence. Al(d4) =aid~ was found with al = 7.78 _+0.04 Hz and 1 / = - 1 . 2 8 + 0.04. This exponent is considerably smaller than might be expected for paramagnetic impurity relaxation. Next, consider the temperature range T >_250 mK where the temperature dependence of the relaxation rate is quite different. Figure 39 shows measurements of the spin-lattice relaxation rate, 7'1-~, as a function of 1/~IT(0.03 < T < 0.60 K) for several 4He coverages (Alikacem et al. 1991). For T > 0 . 2 5 K , T1-1(T) increases dramatically with temperature. For simplicity, it is assumed that the observed deviation of the relaxation rate from the low temperature behavior, WLT = A + BLOT, is due to the addition of another relaxation rate, Wrrr, which is associated with different mechanisms of relaxation for T > 0.25 K. Thus, in general, T1-1 ~ WET + WHT. The rate Wm is well described by an exponential, WriT-" exp(-AJT), with A dependent on n 4. The values of Wrrr shown in fig. 40 are obtained from data like that shown in fig. 39 by subtraction" WriT = T l - l ( T ) - WET, where it is assumed that WET retains its ~/T character as deduced at low temperatures even for T > TF. 7"2-l shows a similar behavior for T > 0.25 K.

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Fig. 38. Logarithm of T1 and T2 as a function of the 4He coverage. For the case of no oxygen: T1 at 30 mK (open circles), T1 at 200 mK (open triangles), T2 at 30 mK (solid circles), T2 at 200 mK (solid triangles). For the case of a preplate of 0.8 layer of 02: T1 at 30 mK (open squares), T1 at 200 mK (x), T2 at 30 mK (solid squares), and T2 at 200 mK (+). (Sprague 1993, Sprague et al. 1995b). To clarify a physical understanding of this relaxation rate, T1-1(T), connection is made to the quantum states available to the 3He and it is assumed that WLX ~ noWo, where no and W0 are respectively the temperature-dependent density and relaxation rate of the 3He spins in the ground state; Wo(T) retains the form a + b,A[T (Sprague et al. 1990, Alikacem et al. 1991). Similarly, Wrrr "" nlWl; where nl and W1 are respectively the density and the relaxation rate of 3He spins in the first excited state. As the temperature is increased, a fraction of the 3He spins are thermally promoted into an excited state in the film, providing an additional channel for relaxation (fig. 41). Assuming that the exchange rate,

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Fig. 39. T1-1 versus T -1/2 for several 4He coverages: 0.217/~-2 (solid triangles), 0.290/~-2 (circles), 0.339 ,~-2 (diamonds), 0.362 ]k-2 (squares) and 0.400 A-2 (open triangles). The dashed lines are fits to T1-1 ~ T -1/2 (from Alikacem et al. 1991).

Fig. 40. LOgl0 WI.[I. versus 1/T for several 4He coverages: 0.217/~-2 (solid triangles), 0.290/~-2 (circles), 0.339 A-2 (diamonds), 0.362/~-2 (squares) and 0.400/~-2 (open triangles). WHT--exp(A/T) (from Alikacem et al. 1991).

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Fig. 41. Schematic illustration of the lowest two energy states available to the 3He in the 4He film. n i and W i and respectively the density and the relaxation rate of the 3He spins in the ground (i = 0)

and first excited (i = 1) state. The exchange rate between the two states, WE is assumed to be much greater than wither W i (from Alikacemet al. 1991). WE, between the two states is faster than W0 and W 1, T I - 1 ( T ) -, n o W o + nl W l . By modeling the film to have two discrete energy levels eo and el, one can solve (Guyer et al. 1989) for the chemical potential of the 3He, using a Fermi distribution, at fixed number, N, and energy separation 6e = el - Co. For temperatures which are not too high, two available energy states are adequate to describe the data where n0(T)-- 1 - exp(-A/T) with nl(T)--exp(-A/T). A is approximately the energy separation between the Fermi level and the excited state. Consequently, since for the temperatures of interest no "- 1, the observed relaxation rate can be written as: TI-I(T)--- Wo(T)+ W1(T)exp(-AJT). The rate Wl, which characterizes the mechanisms of relaxation from the excited state, may be a function of temperature. It is assumed that any anticipated temperature dependence in Wl is weak compared with the exponential behavior of nl(T) (Alikacem et al. 1991). Thus, for the purpose of extracting the energies, A, it is assumed that WI(T) is independent of temperature. It is perhaps reasonable to choose a power of T, such as T2, as expected for a bulk Fermi liquid, or ~/T, as observed for the case of Wo(T), for W1(T). It is found that such a choice changes A by only ~ 10% and does not change significantly the quality of the exponential fit to data such as that shown for example in Fig. 40 (Sprague 1993). The relaxation rate in the excited state is found to be typically much larger than that of the ground state, Wl -- 50Wo. There has been no clear explanation for this, although if the excited state is located closer to the substrate than the ground state, it might be expected that W~ > 1410. In fig. 42 the energies, A, are shown as a function of 4He coverage, n 4, determined from several experiments of this type (Alikacem et al. 1991, Sprague 1993, Sprague et al. 1995b). A has structure as a function of n4. By adding the Fermi energy (determined from the magnetization) to A, the energy separation

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Fig. 42. Energy gap, A, as determined from T1 data as a function of 4He coverage. Shown are data for the cases of oxygen present (circles) and absent (squares) and no systematic difference is seen in the energy values (Sprague 1993, Sprague et al. 1995b). Asterisks are from an early analysis of some of the data (Alikacem et al. 1991). between the ground and the excited state 6e - e l - e0 was obtained from the data. It is observed that in this temperature range, the longitudinal relaxation rate is also thermally activated, 1/T2 ~ 14102+ W12 exp(-6/T). 6 is not as well determined as is A, but it is found that 6 -- A. The ratio of the rates, WlE/Wo2 ~ 200, is similar to the ratio TE/T l for T < 250 mK. Sprague et al. (1994) have shown that for a properly designed cell, the magnetization can be used to sensitively monitor the evaporation of 3He from the film. Measurements of this, coupled with the measurements of the energy differences A, and the Fermi energy, allow an absolute measurement of the 3He energetics. At the coverage used for these measurements, [n 3 = 1.08/tmol/m 2 (----0.1 monolayers)] 3He-aHe interactions are small (Higley et al. 1989, Krotscheck et al. 1988b) and the energetics apparently remain largely unaffected by the - 1 0 0 m K variations in the substrate potential (Higley et al. 1989), thus ensuring relatively uniform coverage of the substrate. The coverage of the adsorbed 4He superfluid film was varied over the range 2.6 < D 4 < 7 layers, where D 4 is the thickness in terms of bulk-density layers, such that n 4 - 12.82/tmol/m 2 per layer x D4, so that the absolute energies could be determined as a function of coverage by a process we now describe. As has been indicated earlier, the temperature dependence of the magnetization is well approximated (Valles et al. 1988) by an ideal 2-D Fermi gas scaled by the degeneracy temperature, TF**, and Curie constant, C = NIt2H/kB;

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MTF** = C(1 - exp(-TF**/T)).

375 (1)

T h e quality of fits of this expression to these recent (Sprague et al. 1994a) data where only TF** is allowed to vary with coverage can be seen in fig. 43, where the magnetization curves for many 4He coverages with 2.6 < D 4 < 7 layers are seen to collapse on a universal curve with 90 < TF**(D4) < 300 mK. This is rather different from the case of very thin helium films adsorbed to Grafoil, where the Fermi liquid interactions are stronger and universal behavior does not follow this expression for MTr:** (Lusher et al. 1990, 1991). For the very thin

Fig. 43.3He magnetization at 79 separate 4He coverages, 2.5 < D4 < 7.5 layers, scaled by the Curie constant, C = N3u2H/kB,and the degeneracy temperature, TF**, plotted against the reduced temperature, TF**/T,follows a universal curve. The solid curve is (CITF**)exp(-TF**/T).Inset: fractional deviation of the magnetization from the Fermi gas fit. Large deviations at high temperatures are due to 3He evaporation from the film into the vapor phase (from Sprague et al. 1994a).

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case, the substrate apparently imposes a more stringent two-dimensional behavior which enhances the interactions. For temperatures T > 300 mK evaporation of the 3He from the 4He film surface occurs, and as 3He leaves the NMR coil as vapor, a dramatic reduction in the magnetization signal is seen (see fig. 43 inset). Use of the magnetization to measure the binding energy in these experiments was possible because at least 99% of the dead volume in the experimental cell was outside of the NMR coil, while nearly 98% of the surface area was inside the coil. Thus, as the temperature was raised, only 3He which remained in the film contributed significantly to the measured 3He magnetization. The Fermi energy was determined at each coverage from the low temperature temperature dependence of the magnetization, thus, the binding energy was the sole adjustable parameter to fit the magnetization in the temperature range where evaporation occurs. The binding energy of the 3He to the film surface was obtained by equating the chemical potentials of the film and the vapor phase, in analogy with Andreev' s (1966) treatment of the evaporation of 3He from the surface state into the 4He bulk. For the 3He in the vapor phase one can assume that Boltzman statistics can be applied and use a concise form for the chemical potential of an ideal gas,/~v = kB T In(p32T3), where P3 is the 3He number density of the vapor, and 2T = ~2,rth21mkBT is the thermal de Broglie wavelength. For the 3He in the film, a simple dispersion relation e ( p ) - eB + p2/2m* where p is the two-dimensional momentum in the film is used. At temperatures T > TF* occupation of the two-dimensional planar momentum states in the film assume approximately a Boltzman distribution. In this limit the chemical potential is to good approximation/~ =eB + kBTln(n32T*2). Here 2T* is the thermal de Broglie wavelength where the 3He effective mass in the film has been substituted for the bare mass. Equating these chemical potentials/~v =/~, with the constraint that the total number of atoms N 3 = An 3 + Vp3 in the sealed sample cell be fixed, the fraction of spins which remain in the film during evaporation is

(

/'

n3(T) = l + ~V ~ m e cBIknT ~'Ta m * n 3(0)

where A is the total surface area and V is the total volume of the experimental cell. If the magnetization were purely Curie-like in its temperature dependence, then it would be possible to apply the expression for n3(T)]n3(O) directly. However, as can be seen in fig. 44 deviations from Curie behavior of--10% exist at the evaporation temperatures. It is necessary then to make the approximation that the effective mass will not be significantly changed as n 3 decreases. The measurements of m*/m by Bhattacharryya et al. (1984) (Bhattacharyya 1993),

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Fig. 44. Magnetization at high temperatures for two 4He coverages, 3.513 layers (triangles) and 4.358 layers (circles), compared with the fit Fermi gas magnetizations (dashed curve). The thin solid curves are the theoretical fits to the evaporation (from Sprague et al. 1994a). indicate that m*/m --- 1.5 for coverages 0.1 < D 3 _<0.3 and 4 < D 4 < 16. Thus, the approximation is made that m*/m ~ 1.5 at all coverages, and that the effective mass changes little during the evaporation process. This allows the additional temperature dependence seen in the magnetization to be ascribed to the Curie constant prefactor through the loss of 3He from the film; C(T)= n3(T)/n3(O)C(O). Fits of this type prove to be very effective in describing the high temperature regime of the magnetization curves (fig. 44). Figure 45 shows the ground state binding energies, e 0, of the 3He as a function of the 4He coverage combined with measurements of the energy gap (Alikacem et al. 1991, Sprague et al. 1994) A and the Fermi energy to give e I. The effect on the binding energy of increasing the 3He coverage has been measured and it was found (Sheldon et al. 1994, Sheldon and Hallock 1994, Sprague et al. 1994a) that for d 3 < 0.26 layers, e o is constant to within experimental un-

378

R.B. HALLOCK

Ch. 5, w

Fig. 45. Binding energies of the ground state e0 (solid circles) and the first excited state e I (open circles and open triangles of 3He bound to the 4He film surface as determined by Sprague et al. (1994a). e0 and e I values were not obtained for every coverage. Theoretical curves (labeled Ei) from Pavloff and Treiner (1991b) for the ground state (solid curve), first excited state (dashed curve), and second excited state (dotted curve) at zero temperature, pressure, and 3He coverage are shown. The energy gaps between the ground and first excited states determined form heat capacity measurement from Bhattacharyya et al. (1984) have been combined with the theoretical ground state energies at D3 = 0.1 (+) and D3 = 0.285 layer (x) and are shown for comparison. certainty. These data indicate that the binding energies reported for 0.1 layers of 3He can be taken to be in the zero coverage limit. The curves in fig. 45 are recent theoretical predictions of Pavloff and Treiner (1991b) (with no adjustments m a d e to the coverage scale) for the energies of the three lowest lying bound states. At D 4 = 7 layers, e l, is well below the binding energy, E, of 3He dissolved in bulk 4He, e 0 is within 2% of the value of the known bulk surface state. This is in a g r e e m e n t with theory; e I is predicted (Pavloff and Treiner 1991b) to remain below E as n 4 increases, consistent with a prediction by D u p o n t - R o c et al. (1990) that there exist two bound surface states of 3He on bulk 4He. Shown in fig. 46 (dashed lines) is very recent work due to Pricau-

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

379

Fig. 46. Experimental data for the ground state e 0 (solid circles) and the first excited state e I (open circles and open triangles) of 3He bound to the 4He film surface as a function of 4He coverage as determined by Sprague et al. (1994a). Also shown are preliminary unpublished values for the energies as determined by Princaupenko and Treiner (1994) (dashed curves) and upper limits by Krotscheck (1994) (solid curves, thickness scale shifted 0.25 A-2).

penko and Treiner (1994) for the energetics of 3He on 4He in a substrate potential appropriate to Nuclepore. The lowest dashed curve is a predicted "substrate" state (for which the 3He is predicted to reside close to the substrate), with the next higher state the nearly degenerate surface state analogous to the Andreev state for a bulk liquid. It is perhaps relevant to note here that the presence of oxygen on the substrate for some of these data (figs. 45 and 46) has no measurable effect on the energetics. Although the variational theories of Kotscheck and his collaborators result in upper limits for the energetics, comparison of those predictions to the experimental values is useful to see the general structure. At this writing, new predictions for the HNC/EL work are emerging. Also shown in fig. 46 is the preliminary predicted energy for a 3He impurity for the ground and first excited states from this new theoretical work (Krotscheck 1994). Measurements of the temperature dependence of the NMR relaxation time Tl and of the magnetization as a function of 3He coverage (Sheldon and Hallock 1994, Sprague et al. 1994a, 1995a) have recently been made and from these values for e0 and el determined in a manner similar to that done for 0.1 3He monolayer as a function of 4He coverage (fig. 47). Few theoretical predictions for the absolute energetics of the 3He in the 4He film as a function of 3He coverage for finite 3He coverage are presently available. In recent work, Anderson

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(,~-2) 3 Fig. 47. The ground and first excited state energy for 3He in the environment of a 4He film as a function of 3He coverage as determined from NMR measurements by Sprague et al. (1994) and Sheldon and Hallock (1994). Solid lines am theory of Anderson and Miller (1993) for a similar film. Dashed lines am theory of Pavloff and Treiner (1991a) for 3He on bulk 4He. and Miller (1992, 1993, 1995) have generalized their earlier semiphenomological approach (Anderson and Miller 1989) to the problem of 3He on the surface of a 4He film so as to allow multiple discrete states for the 3He. They find that the explicit areal density dependence of the 3He self-energy reduces the gap between the ground state and the first excited state. This is a different mechanism from gap changes induced by the broadening of the film surface profile considered by Stringari and Treiner (1987). Anderson and Miller (1993, 1995) have concluded that the level spacing for the 3He states in mixture films depends on the 3He coverage in a non-trivial manner and find that the energy gap levels off with increasing 3He coverage, with the Fermi energy meeting e~ from below at n 3 = 0.045/~-2, in general agreement with the location of the step in the magnetization versus 3He coverage seen by Higley et al. (1989) (Figs. 28, 30). The numerous NMR measurements as a function of 3He coverage (Higley et al. 1989, Alikacem et al. 1991, Sprague et al. 1994a, 1994b, 1995a) which have been done for several values of the 4He coverage (e.g. Fig. 48) allow a determination of the slopes of the magnetization as a function of 3He coverage (Fig. 48, inset) (Higley 1991, Sprague 1993, Sprague et al. 1995a). This allows a large body of susceptibility data taken as a function of D 4 with D 3 = 0.1 to be extrapo-

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

381

Fig. 48. Data for the magnetization at T = 40 mK as a function of 3He coverage for several 4He films (Higley 1991, Sprague 1993). Fits to M / M 0 = m h = s D 3, where M0 is the ideal 2D Fermi gas magnetization, over the range 0.1 < D3 < 0.4 layers yield the slopes shown in the inset.

lated to zero 3He coverage, which results in a determination of the hydrodynamic mass as a function of 4He coverage. These mh values are shown in fig. 49 as a function of 4He coverage. These hydrodynamic mass values are compared with the predictions of Epstein and Krotscheck (1990) in fig. 50 where they are shown as a function of inverse film thickness. Also shown on the figure are heat capacity and torsional oscillator measurements of the effective mass for a 3He coverage of 0.3 layers (Wang and Gasparini 1988). The theory predicts modulation of the mh values due to the layering structure of the 4He density imposed by the ideal two-dimensional substrate. These modulations are not seen in the experiments presumably due to modest substrate inhomogeneity. The peak in mh was identified by Wang and Gasparini (1988) as possibly due to the formation of the first superfluid layer. NMR results to be mentioned shortly suggest, due to the behavior of 7'2, that the peak in mh may instead be coincident with solid layer completion. Recent measurements of the 3He spin diffusion (Sprague et al. 1992, Sprague 1993, Sprague et al. 1994b, 1995a, 1995b) coefficient, D, for a submonolayer

382

R.B. H A L L O C K

Ch. 5, w

Fig. 49. Hydrodynamic mass, mh, values as a function of 4He coverage for clean Nuclepore (triangles) and 0 2 preplated Nuclepore (circles).

Fig. 50. A comparison of the hydrodynamic mass values determined from NMR experiments, circles and crosses (Higley et al. 1989, Sprague 1993, Sprague et al. 1995a), heat capacity, asterisks (Bhattacharyya et al. 1984), and torsional oscillator measurements, diamonds (Wang and Gasparini 1988). Shown as the solid line are the predictions of Epstein and Krotscheck (1990).

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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coverage of 3He in the environment of the 4He film at low 4He coverages have been made. In this work measurements of the susceptibility were made by the use of Hahn spin echos and the inversion recovery technique. Generally field gradients <10 G/cm in the 2T field were used. An NMR image of the 3He in the sample cell was obtained and this agreed with the known cell geometry, including the copper post which threaded the Nuclepore filters (Higley 1991). For the diffusion coefficient, stimulated echos and spin echos were both used. This allowed measurements on timescales out to t ~ TI as stimulated schoes are not limited by T2. This was useful since T2 ~ 0.01T I. The work was primarily done for a 3He coverage of 1.08/tmol/m 2 (-0.098 monolayer) for a range of 4He coverages, 2.10 < D 4 < 5.15 layers. For a second data set, the 3He coverage was varied from 0.098 < d3 < 0.235 layers. Since the topology of the surface is relevant in diffusion measurements, a tortuosity factor, a, determined from third sound measurements (Sprague et al. 1992), is included, so that Dmeas = Diet, with a ~ 16. Dramatic 4He coverage dependence to both the diffusion and the magnetization was observed. This recent work falls naturally in two regimes of 4He coverage, above and below a coverage of Dc = 2.66 (bulk density) layers. As usual, for the Nuclepore substrate, the coverage can be expressed as n 4 = 12.82/tmol/m 2 per layer x D4. For D 4 > Dc interesting and unexpected diffusion was seen. For D4 < Dc a mobility edge for the 3He, perhaps somewhat analogous to what is seen in two-dimensional electron systems, was observed and interpreted as being due to the confinement effect (i.e. localization) of the 3He when the film gets thin enough to have little or no 4He fluid present. In addition, unexpected scaling behavior was found in the data. We now discuss these two coverage regimes in more detail. For 4He coverages D4 > De, degenerate susceptibility was observed with degeneracy temperatures, TF**, in the range 100 < TF**< 300 mK and the temperature dependence was well fit by an ideal Fermi gas scaled by TF**= TF*(1 + F0a). In this coverage range the 3He diffusion coefficient (mobility) was seen (fig. 51) to be an increasing function of the 4He coverage. In this region, it is expected that at sufficiently low temperature scattering between quasiparticles at the Fermi surface will dominate the spin diffusion. Miyaki and Mullin (1983, 1984) have shown that quasiparticle-quasiparticle scattering confined to a twodimensional Fermi surface exhibits a logarithmic temperature dependence in addition to T2. The low polarization, low temperature limit of their theory for the diffusion constant, DEE, can be expressed in terms of experimental parameters Z, TF* and Foa as

DFL = (Zo/Z )3( TF./T)2(Trh/m )/ IFoa l2 In(TF*/T). The use of Z/X0 measured at each4He coverage and Krotscheck's values of in the Miyake and Mullin (1983, 1984) expression yields a coverage depend-

F0 a

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Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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ence which is similar (primarily due to the role of the hydrodynamic mass in both the susceptibility and the diffusion) to that of the measured diffusion, but the relative magnitudes differ by at least an order of magnitude (fig. 51) (Sprague et al. (1995a) and the temperature dependence in the measured diffusion is seen to be much weaker than 1/T2. The measured spin diffusion, D, when scaled by the predicted microscopic diffusion DFL, was seen to follow a simple power law (fig. 52 for temperatures T < 100 mK, D(T, Da)/DFL = f i t u where for all coverages/z = 1. Furthermore, the coefficients, fl, can be expressed as fl ,,~ ( D 4 - Dc) v where v = 1. Thus all of the low temperature diffusion data follows D(T, D a ) = y ( D 4 - De)TDFL where y = 0.29 layers -1 mK -1 (fig. 53). This behavior is unpredicted and not presently understood. For thinner 4He films even more interesting behavior was found (Sprague et al. 1994b, 1995a). For D 4 < 2.66 the magnetization contains a Curie component to the lowest temperatures investigated, ( T > 2 4 mK) (see fig. 51). M = " ~ + ColT. T h e Curie fraction, nc/N = Co/(C + Co), is likely CITF**(1 - e - TF* *i1) associated with spins localized at the surface of the 4He solid layer in a manner somewhat reminiscent of the proposal by Brewer and Rolt (1972) for the case of the second layer of pure 3He. For coverages D 4 -- 2 layers these localized 3He

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Fig. 54. The NMR relaxation time T2 as a function of 4He coverage for the two cases of clean (closed triangles) and 0 2 preplated Nuclepore (open triangles) from Sprague et al. (1995a).

Fig. 55. Phase space of measurements where NMR measurements have been taken for coverages of 4He at and above about two atomic layers.

Ch. 5, {}3 PROPERTIESOF MULTILAYER 3He-4He MIXTURE FILMS

387

account for ~ 10% of the total 3He coverage in the film. For D4 > 2.6, nc/N --40. The vanishing of the Curie component for D 4 > 2.66 layers is likely due to the liberation of localized 3He from the solidified second layer of the 4He film. Below this coverage, spin diffusion D -- 10-6 cm2/s was measured (fig. 51), and long (-10-100 s) Tl relaxation times were seen, both consistent with 3He participating in a solidified second layer. As the 4He coverage is increased, 4He apparently replace the 3He which reside in the relatively immobile solid-like layer. It is also apparent from the temperature dependence of the T~ data that there is a dramatic reduction in the spin correlation time as D4 is increased beyond 2.66 layers. A sharp peak in 7'2 is evident, apparently near solid layer completion (fig. 54). This is consistent with a sudden broadening of the excitation spectrum responsible for relaxation. For D4 < 2.43, T2 is an increasing function of temperature and for D 4 > 2.43 T2 becomes temperature independent. The NMR experiments, coupled with the earlier heat capacity work have provided a wealth of information on the detailed microscopic properties of 3He in the environment of a 4He film. The coverage dependence of the energetics is now known and allows a critical test of recent and emerging theoretical work. The phase space explored by these measurements is shown in fig. 55. Further exploration of the dynamics of the 3He, particularly the apparent mobility edge, should provide an interesting contrast to localization studies of electron systems in two dimensions.

3.4. Other experiments 3.4.1. Third sound experiments We next turn to an entirely different technique, third sound, which is able to measure the macroscopic hydrodynamic behavior of the helium film and the effect on the hydrodynamic modes brought about by changes in the 4He thickness, the 3He impurity concentration and the temperature. One of the interesting ramifications of the absence of viscosity in the superfluid phase of liquid 4He is the ability of helium films to move and to support long wavelength surface excitations, waves, known as third sound (Atkins 1959). These waves are analogous to tidal waves on the ocean for which the wavelength is much greater than the depth of the medium. For tidal waves on the ocean, the wavelength, 2 is much greater than the depth of the ocean, H, and to good approximation the velocity, v, of such waves is given by v2= gH where g is the acceleration of gravity, the force per unit mass. In the case of superfluid films, the analogous equation for the velocity of third sound, C, is C2 =fhps/p where f is the van der Waals restoring force per unit mass provided by the substrate, h is the distance between the substrate and the free surface of the film and

388

R.B. HALLOCK

Ch. 5, w

,osl,o is the superfluid fraction in the film. This expression is the long wavelength approximation to the expression valid for all values of 21h, c2= [(,os2f12~,o ) + (,os2g/2,rr,p) + (2~tr/p2)] tanh(2~h/2), which includes restoring forces due to the van der Waals force, gravity and surface tension, or. The velocity of third sound can be written more generally as the derivative of the chemical potential, ~, with respect to film coverage,

C = h((,Os)/p)(6,u/rh ), where we have incorporated (,Os)/p = ps/,O(1 -Ds/d), the empirical effective superfluid fraction in the film, with Ds a parameter related to the immobile film thickness adjacent to the substrate (Kagiwada et al. 1969, Putterman 1974). Third sound was first predicted by Atkins (1959) and first observed in his laboratory by Everitt et al. (1962, 1964). Measurements of the third sound velocity provide information on the superfluid fraction of the film, the structure of the film and impurity effects due to the presence of 3He. Third sound consists of a thickness fluctuation of the film and (for T greater than zero) a fluctuation of the temperature. This connection was apparent in the early theory of Atkins, but was described in more accurate detail by Bergmann (1969, 1971, 1975). In general, for relatively thin films, the film thickness fluctuation, 6h, is related to the accompanying temperature fluctuation, c~T, by ~ShhST =-if~L, where L is the latent heat. The advent of the superconducting strip bolometer (by which the thermal fluctuations of third sound can be measured) first introduced to the study of third sound by Rudnick et al. (1968), led to rapid progress in the field and many of the properties of thin 4He films have been investigated by means of this technique. A limitation of the technique is the need to operate the bolometer close to its superconducting transition temperature for maximum sensitivity. The application of a magnetic field or bias current allows the bolometer to be used over a reasonable range of temperature, but for operation over a wide temperature range, a different technique is employed. Since the third sound wave consists of a thickness fluctuation, this alternate technique involves a measurement of the thickness directly by capacitive techniques. These techniques can be applied to the measurement of third sound pulses, or in a resonance mode for which the frequency and, importantly, the dissipation of the resonator can be measured. An example of a resonator is shown in fig. 56 (Hallock 1987). The operation of this device is typical of resonators in general and will be briefly described here. The helium coats the inner walls of the cylindrical pancakeshaped resonator. In this case, there is a small hole in the resonator which allows, for example, the 3He/4He ratio to be changed by the addition of helium at

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

389

Fig. 56. Schematic representation of a third sound resonator used to study mixture films. The capacitor gap is -20/~m with a 2 cm diameter pancaked-shaped open volume (after Hallock 1987). low temperatures. The capacitor plates in the resonator are in parallel with a superconducting coil and the LC combination is operated by a back diode circuit that operates near 20 MHz. The third sound fluctuations in the film thickness modulate the capacitance of the capacitor and the resulting modulations of the frequency of the back diode circuit are detected by FM demodulation techniques and fed to a lock-in amplifier where the in phase and quadrature amplitudes can be measured. The best sensitivity presently available for a fiat plate resonator with no porous material present is about 100/~/~. For a substrate with substantial surface area between the plates of the capacitor (provided by porous material), the sensitivity is greatly enhanced. For example, for the resonator shown in fig. 57 for which the capacitor plates are evaporated directly onto the Nuclepore, the sensitivity to film thickness fluctuations is 2.5/t/~ (Hallock 1991). The use of resonant devices allows both the frequency and the damping (l/Q) to be measured. As we shall see, the damping appears to provide a connection to the film energetics in the case of mixture films. The detailed physical understanding of why this is the case is not yet clear. Some of the earliest experiments to study 3He-aHe mixture films by the techniques of third sound were those of Ratnam and Mochel (1974). The apparatus used was a third sound resonator (Ratnam and Mochel 1970a) consisting of

390

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Fig. 57. Third sound resonator made from a rectangular piece of Nuclepore. In the case shown here, the drive is thermal and the detection via capacitance techniques (after Hallock 1991 ).

thin glass plates flame sealed at the edges with a small amount of argon inside (which would provide the substrate for the mixture film at low temperatures) which employed bolometric detection of temperature fluctuations, and which had been previously used for detailed studies (Ratnam and Mochel 1970a,b) of 4He. The amount of helium could be changed by diffusion through the glass plates at room temperature. These experiments showed that the addition of 3He to a 4 layer 4He film caused the third sound frequency and Q to be reduced. Beyond this observation, there was little quantitative information available from the experiment. Additional early work was that of Downs and Kagiwada (1972) which, as described by Bergmann (1975), showed that the addition of 3He to a 4He film reduced the third sound velocity. A thermodynamic analysis of the properties of mixture films which grow on surfaces in the presence of the van der Waals force field of the substrate in contact with a mixture vapor was made by Chester et al. (1976). One interesting result from this calculation was that for relatively high temperatures it was predicted that the mixture film is normal adjacent to the substrate and the superfluid floats on top; this in spite of the fact that the 4He is apparently more pure closer to the substrate. This was explained as being due to the fact that there is also a pressure gradient in the film and this is the predominant effect. At lower temperatures (below about 0.8 K) the theory calls for the superfluid part of the film to be adjacent to the substrate with the normal part on the top furthest from the substrate. This raises the interesting possibility of a dynamical change in the film properties as a function of temperature as the physical location of the su-

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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Fig. 58. Superfluid transition temperatures as a function of total sample concentration for thin film mixtures for three different 4He coverages, n4 = 0.345, 0.375 and 0.401/~-2. The bulk phase diagram is represented in dotted line (after Laheurte et al. 1980). perfluid component changes with temperature 9As we shall see presently, interesting behavior has been seen in the behavior of the third sound as a function of temperature for mixture films by Hallock and Laheurte and their co-workers, but it is not yet clear whether this behavior is related to these predictions. A sequence of third sound measurements on 3He-4He mixture films was carried out by Laheurte and his collaborators beginning with the work of Laheurte et al. (1980). The experiments were done on a glass substrate with Millipore filters also in the apparatus so as to provide a large reservoir of film. The limits of the global system phase diagram were deduced for three different values of the pure 4He thickness and for each of these for several values of the 3He con-

392

R.B. HALLOCK

Ch. 5, w

centration. The onset values of third sound determined coordinates on the concentration-temperature plane and these points mapped out the phase boundary of the superfluid-normal transition for the several films studied. A roughly linear reduction in the transition temperature was observed (fig. 58) with an increase in 3He concentration and the transition temperature was seen to be lower for a given concentration for a thinner starting 4He film (Laheurte et al. 1980). The results for the value of the jump in the superfluid density at the Kosterlitz-Thouless transition agree with predictions and with the quartz oscillator work of Webster et al. (1979). The Kosterlitz-Thouless number for OslT thus was found to be robust to the addition of 3He. No evidence was seen for the two-dimensional phase separation predicted by Berker and Nelson (1979), although it was suggested that the films may have been too thick for this. The claim was made that there was no observation of a phase separation event in the films in spite of the fact that the superfluid onset points actually enter the forbidden bulk phase diagram. Thus, it was likely that the films must have had a concentration gradient in them. The temperature dependence of the

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Ch. 5, {}3

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

393

third sound velocity was measured from the superfluid onset temperature down to 0.5 K. In subsequent work, Laheurte et al. (1981) replaced the Millipore by crushed glass and the capillary condensation properties of the crushed glass were studied for pure 4He. Then, mixtures of total coverage smaller than that which caused capillary condensation in the pure 4He film were studied. Measurements were taken as a function of decreasing temperature. For fixed coverage but different concentrations, a dramatic rise in the third sound velocity as the temperature was lowered was observed (fig. 59). Fifth sound (Jelatis et al. 1979, Williams et al. 1979) was ruled out as a possible mechanism. An alternate explanation was offered, that there was phase separation perpendicular to the walls and this resulted in a thinner superfluid film as the temperature was lowered. Further work was carried out at fixed concentration as a function of coverage. A similar result was obtained. The experiments were hysteretic with hysteresis observed on warming through the temperature range 0.18-0.28 K.

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Fig. 60. Experimental determination of the superfluid onset temperature in thin mixture films for four different constant coverages equivalent to thickness values (at T = 0 K) of 6.0 (diamonds), 5.1 (solid circles), 4.2 (open triangles) and 3.4 (open circles) atomic layers. The dashed lines are curves which correspond to constant amounts of 4He (labeled with the equivalent pure 4He film thickness).

394

R.B. HALLOCK

Ch. 5, w

Further onset measurements for third sound in 3He-aHe mixture films were carried out by Romagnan and Noiray (1984) with the conclusion that addition of 3He to a 4He film caused the superfluid onset temperature to be depressed linearly with an increase in the 3He concentration. This linear dependence is interpreted as supportive of complete phase separation in the film at zero temperature; at finite temperature a complete layered situation is not supported by the measurements. Laheurte et al. (1986) extended these onset measurements to T 0.1 K and found the nearly linear depletion with total 3He concentration to be preserved (fig. 60). A different series of experiments involving third sound and mixture films was that of Ellis et al. (1981, 1984) (Ellis and Hallock 1984) where the emphasis was on the structure of the film as a function of added amounts of 3He at various temperatures as deduced from measurements of the third sound velocity. For these experiments two different apparatus were used. In one (to be discussed in another context later), a waveguide arrangement (fig. 61) was used in which both the temperature signature, AT and the capacitive film thickness signature Ah associated with the passage of a third sound pulse were recorded. The presence of capillary condensation in a film reservoir in this apparatus limited its utility for the study of 3He-aHe mixtures, but led to the development of second and third generation apparatus, shown schematically in fig. 62 (Ellis et al. 1981, 1984, Valles et al. 1986). For the experiments conducted at T > 0.3 K one could

Fig. 61. Representation of a third-sound time-of-flight apparatus (Ellis et al. 1984). The symbols S1, $2, $3, and $4 refer to AI film bolometers; C1 and C2 are capacitive detectors, and $5 is a silver strip heater. Silver solder tabs (visible here) were evaporated around the edges of the glass slides so electrical contact would not interfere with the capacitor gaps. The capacitor gap was --7/~m.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

395

Fig. 62. Schematic diagram of two experimental cells used to study third sound in mixtures by Ellis etal. (1984) for(a) T>0.3 K and for (b) T> 0.050 K.

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'1' 2

9 Pure

0

4He

0 -

,,.-...

C

0.3 0 I--

9

Mi xlure

9 Pure a~ L..

4He

0.2

0"}

n

9&

0.1

&

A

A

000 9

0 ~

AAA AA &A

O

0.2__5 0 . 5 0

O

0.?5

Temperolure

O

.

I.OO

1.25

1.50

(,K)

Fig. 63. Typical operation of a sealed cell apparatus as a function of temperature showing the vapor pressure, the film thickness as deduced from capacitance measurements, and the observed third sound velocity. These data are for pure 4He with a thickness of -5.7 layers, and for a mixture with 3He added so as to result in a total concentration of 41.6%. As the temperature is increased, preferential evaporation results in a varying film concentration (after Ellis et al. 1984).

simultaneously measure the third sound velocity, the vapor pressure in the cell and the film thickness directly with a capacitive thickness monitor (fig. 63). In

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

397

addition, the geometry of the cell was designed so as to allow the measurement of the velocity of ordinary sound in the vapor (Ekholm and Hallock 1980c), thus allowing an in situ determination of the 3He-aHe concentration in the vapor. With this apparatus, measurements by Ellis et al. (1981, 1984) were made over a large range of 3He concentrations and temperatures. For low temperatures, where there was little vapor, the most significant results were those for the velocity of third sound as a function of the amount of 3He in the cell for fixed amounts of 4He. These results are shown in fig. 64 in comparison to theoretical predictions for two possible extreme cases for the configuration of 3He in the 4He film (Ellis et al. 1984). These predictions, for the velocity in a mixture film, C3 compared to the velocity in a pure 4He film, C30, based on hydrodynamic arguments, were that for a completely mixed film, C321C302 = (1 - h3/h4)-4(1 + n3h3/n4h4) (1 + n3h3m3/n4h4m4) -1 ,

while for the case of complete isotopic layering one expects, C32/C302= 1 + (n3/n4)[(1 + h3lh4) - 4 - 1],

where n 3 and n 4 are the number densities and h i a r e the equivalent pure phase thicknesses in the films (Ellis et al. 1984). As seen in fig. 64 the data are in general agreement with the predictions for the case of complete, or nearly complete, 1.0

0.8

•i _

A

1

I

I

1

9

9

tx

\~o,,

~

(96

\. ~, \\ "i

1

0.40 K

l

0.45K

1

0.50K

0

0.60 K

,,

\lNk~

O.6

~o

I

-

0.55K

o ~

oO~o

\

0.4

0.2

,

0.0 0.0

I 0.2

l

I

J

0.4

I O. 6

I

I 0.8

I I .0

h31h4

Fig. 64. (C3/C30) 2 where C 3 and C30 are the third-sound velocities on the mixture and pure 4He film, respectively, as a function of the ratio of the number of 3He to 4He in the apparatus (after Ellis et al. 1984).

398

R.B. HALLOCK

Ch. 5, w

isotopic layering. Thus, the picture available from these third sound experiments is that for low temperatures (T < 0.5 K), if 3He is added in relatively large amounts to a relatively thin 4He film which resides on a glass surface, the 3He goes predominantly on the 4He film surface and as the amount of 3He is increased, a macroscopically thick layer grows on the surface. A more detailed linearized hydrodynamic calculation of layered superfluid systems was carried out by Guyer and Miller (1981) (Monarkha and Sokolov 1981, 1982). In this work, the layer above the 4He superfluid was itself assumed to be either normal or superfluid. The calculation was done in the context of the possibility of creating superfluid hydrogen at the surface of 4He. The equations have generally been confirmed for the case of a normal overlayer of 3He by the experiments of Ellis and Hallock (1984), (Ellis et al. 1981), although some of the work of Laheurte et al. (1980) does not agree. These more detailed predictions are that the third sound velocity is given by (Guyer and Miller 1981):

C3•

1/2{ [c342(1 + A) + C312] __.[C342(1 + A ) - C312]F},

where the various terms in this equation are defined as F44 = 3ct4(h4/a)[y(a/(D + h 4 + hi) 4 + (1 - y)(a/(D + h4))4],

FI1 = 3 ( h l l a ) a l ( a l ( D

+ h4 +

hi)) 4,

A = --y[ 1 - (D + h4)4(O + h 4 + hl) 4,

C342=

3(a4/m4)(h4/a)(a/(D +

h4)) 4,

C312--" 3 ( a l / m l ) ( h l / a ) ( a / ( D + h I + h4)) 4, with n4h4F41 = n l h l F l 4 = nlh4Fl 1. Here F = { 1 + (4F41F14]m4ml)[(C342(1 + A ) -

C312)} 1/2.

For the case where the surface layer is normal, C31 = 0 and we have Ca2 = C342(1 + A). In the case that both films are superfluid, there are two modes at each value of k; the coupling produces a shift of the ordinary third sound velocity and there appears a new mode in the overlayer of the second superfluid. In later work, Guyer and Miller (1982) extended the calculation and relaxed the constraint of incompressibility; Puff and Dash (1980) had relaxed this condition earlier in the context of a pure 4He film. Monarkha and Sokolov (1982) have

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

399

also studied this s y s t e m and included interaction effects b e t w e e n the stratified layers while n e g l e c t i n g F e r m i liquid effects. O n e peculiar o b s e r v a t i o n from this g e n e r a t i o n of pulsed third sound experim e n t s d e s e r v e s m e n t i o n (Ellis et al. 1984). G e n e r a l l y the a m p l i t u d e of the thermal third s o u n d was o b s e r v e d to g r o w with the a m p l i t u d e of the third s o u n d drive v o l t a g e used to create the third sound pulse. H o w e v e r , in cases w h e r e capillary c o n d e n s a t i o n was present e l s e w h e r e in the apparatus, (which presum a b l y had an effect on the 3He c o n c e n t r a t i o n in the film) an unusual d e v e l o p m e n t of the pulse a m p l i t u d e with an increase in drive voltage was o b s e r v e d (fig.

B

0.35

0.30

0.25 0.20

0.10 0.00

-

0.00

0.05

Fig. 65. (A) Third-sound (and ordinary sound) pulse shapes as a function of time for various drive voltage for XT = 40% and T = 0.705 K. For these data if the 3He were removed the 4He would be 5.8 layers thick. Here C3 = 17.83 m/s, u x = 55.70 m/s (vapor sound visible in the early part of the traces), and the energy per 20-kHz drive pulse is AE = 1.6 x 10-TV2 J, where V is the indicated drive voltage in volts. Each bolometer had a resistance of -350 Q. Four bolometers were deposited on the plate with separation 0.475 cm. (B) Data for the same mixture and under the same conditions except for T = 0.850 K. Below T = 0.8 K capillary condensation was present in the reservoir in the apparatus. Note the different behavior of the third sound pulses as a function of drive (after Ellis et al. 1984).

400

R.B. HALLOCK

Ch. 5, {}3

Fig. 66. Square of the relative third-sound velocity for a mixture film as a function of 3He concentration at T=0.1 K as determined by Ellis and Hallock 1984. Shown here is (flf4)2=(C31C30) 2 where C3 and C30 are the third-sound velocities on the mixture and pure 4He film, respectively. The coverage ~ presumes one completed 3He monolayer is of density o = 6.45 • 1014 atoms/cm2. The curves are predictions based on a layered (solid), mixed (dashed), or mixed to 6.4% 3He (dashdot) model for the morphology of the film for 3He added to a pure 4He film of 5.3 layers. 65). The reason for this remains a mystery. More recent work by Sheldon and Hallock (1994) and Sheldon et al. (1994) to be described shortly has shown an unexplained behavior of the phase difference between the drive and the detected amplitude for mixture films in a resonator. However, in this latter work, no capillary condensation was present and the observations may be unrelated. In subsequent work carried out with a third sound resonator (fig. 62b) it was possible to carry the measurements of the velocity as a function of 3He coverage to much higher precision and simultaneously measure the damping (Ellis and Hallock 1983, Hallock 1987). Results for the resonance frequency and Q from these measurements are shown in figs. 66 and 67 (Ellis and Hallock 1984). The data taken at T = 0.1 K are in extremely good agreement with the isotopic layering prediction mentioned above. Although the mixing of components is likely to occur at higher temperatures, the conclusion of this work was that at this low temperature no more than 1% of the added SHe could be anywhere but at the surface of the 4He. This conclusion has been substantiated by the detailed N M R energetics measurements mentioned earlier in this review. An interesting observation in this work was that the damping in the system showed considerable structure as a function of 3He coverage even though there was no hint of such structure in the frequency. This structure, in the excess damping (i.e. damping above that present for the pure 4He film) is shown in fig. 68 (Ellis and Hallock 1984). Although an effort was made to understand the general damping in the system (Guyer 1985), no complete explanation as to why this structure was pre-

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS 30 000

1

I

f

,

1

,

I

i

401

,

T = O.IOK

20 0 0 0 "1 9 9

9

o

9

9

o~

9

I 0 000 --

9

9 9

9

9 O0

oo

~o

9

l'" 82

0

9

0.0

l

,,

l

L

~

1.0

0.5

,

l

1.5

2.0

,

25

Fig. 67. Values for the Q of third sound for 5.6 layers of 4He as a function of the 3He concentration expressed as monolayers of added 3He .as determined by Ellis and Hallock (1984).

sent was available. It is still not clear what the detailed physical mechanism is for the influence of the 3He on the damping, but, as reference to figs. 28 and 67 makes clear, the structure correlates with the structure seen in measurements of the magnetization and the relaxation times TI and T2. Thus, it is likely that its origin has to do with the effect that occupation of the available energy states in the film has on the third sound damping. Further theoretical work is necessary to

-

0

0

2

-3

-

!

1

u

m

u

9

I

'

O

I I

0

9

--4 v 0

o

-5 T:

-6

O. I O K I

-I.0

l

l

-0.5 tog

I

0.0

lO

l

0.5

(~)

Fig. 68. The excess third sound dissipation as a function of the 3He coverage (after Ellis and Hallock 1984).

402

R.B. HALLOCK

Ch. 5, w

more fully understand this. For large 3He coverages an increase in temperature to T = 0.25 K has a strong effect on the 3He coverage dependence of the Q, but only a subtle effect on the third sound velocity (Hallock 1987). Heinrichs (1985) carried out a further series of detailed measurements in the silver plated and glass resonator (fig. 62b) in which measurements of the temperature and 3He coverage dependence of the frequency and damping of three modes of the resonator were made for two 4He coverages, 3.6 and 5.3 atomic layers. Since these measurements are not well known, we mention some of the more important results here. One of the results of this experiment was an unusual behavior for the temperature dependence of the excess damping of the third sound in the presence of the 3He in the cell. The damping was observed (Heinrichs and Hallock 1983a, Hallock 1987) to be thermally activated, 1 / Q 1/Qo = A + B exp(-C/T) where Q0 is the Q of the resonator for pure 4He, for T > 0.2 K, for 3He coverages above about 0.7 layers (fig. 69). For these data the activation energy C was found to be in the range 1-1.5 K. The thermal activation was thought to be related to the promotion of 3He atoms into an excited state in the film. It was presumed that the 3He atoms in the excited state degraded the Q by an amount proportional to their population. Thus, the temperature dependence of the Q was thought to be a measure of the difference, A, in

-

4

_

-

~

1

Y

~ - T

-

~

r

. . . .

l ......

d~:

d4 "5 30 -6

I

r--------

I28

9

0 (2

7

-r . . . . . . . . . .

9 -8

IZ ...J

|~_

-IO

|

9 9

0

5

IO

lIT

I

15

20

25

(K -~ )

Fig. 69. The excess damping in a 3He - 4He mixture above that present for a pure 4He for the case of 5.3 layers of 4He and 1.28 added layers of 3He, as a function of temperature. Thermal activation is present (after Heinrichs and Hallock 1983a, Hallock 1987).

Ch. 5, w

PROPERTIESOF MULTILAYER 3He-4He MIXTURE FILMS

T25

"1

I

v

'

1

'

1

'

1

_

Z20

'

I

403

T

(:14:6.50

--

d3

AA

:

I

76

A

_

,-...

715

--

710

-

705

-

I',4

I

---9

700

' 0.00

1

,

0.05

!

_1

o.to

I

a

0.~5

.... 1

~

o.zo

!

,

1

0.2.5

l

-

0.30

0.3.5

T(K)

7OO

I

u

]

,

1

~

I

i

I

1

v

d,

680

t

:6..50

-

d3:3.76

--

_

&

660

-

640

-

620

-

N

I "-

A

6OO

0.00

j

0.0,,5

,

l

o.to

J

1

a

o.t5

I

0.z0

A

,

1

0.2.5

9

,

1

030

-

s

0 3.5

T(K)

Fig. 70. Unusual changes in the frequency of the third-sound resonator observed with changes in the temperature for two 3He coverages (after Hallock 1987).

energy between the ground state Fermi level and the energy of the excited state (Heinrichs 1985, Hallock 1987). No measure of e~ was possible. These results were in general agreement with the heat capacity measurements and served to help to motivate the NMR experiments we have described earlier. These data also showed an unusual temperature dependence to the resonance frequency for larger coverages of 3He. For low coverages (say d 3 less than 1 layer) the temperature dependence of the resonant frequency was consistent with the excitation of 3He into states within the film. For large coverages, however, the temperature dependence of the resonant frequency became dramatic (fig. 70) (Heinrichs 1985, Hallock 1987). A peak in the resonance frequency appeared near T= 0.15 K for coverages which exceeded about 1 layer and the peak grew

404

R.B. HALLOCK

Ch. 5, w

more pronounced and asymmetric as the 3He coverage was increased. When the peak is present the third sound velocity appeared to be nearly the same on the high and low temperature sides of the peak. It has been speculated that this may imply that the 3He configuration in he film is the same at the temperatures on either side of this peak, but that it is different in the temperature window of the peak (Hallock 1987). For large coverages of 3He (say three layers or so) the peak evolves into a step with the frequency considerably lower on the high temperature side of the peak than on the lower side. Again, it was speculated that large 3He configuration changes might be taking place in the film as a function of temperature with the possibility that the 3He might be purged out of the surface layer (Hallock 1987). This work and the parallel work by the Laheurte group (e.g. Laheurte et al. 1981) in which a much steeper temperature dependence was seen in the third sound velocity for mixtures, is quite suggestive, but not definitive on the issue of configurational changes in the film. Inconsistencies exist, perhaps with their origin in the particular substrates used. Recent work for low 3He coverages (Sheldon et al. 1994, Sheldon and Hallock 1994), to which we will return later, documents the existence of a gentle peak near T= 200 mK, but the origin of this remains unclear. Experiments by Draisma et al. (1994) also find evidence for this gentle peak although in their case it is drive-power dependent. The measurements of Heinrichs (1985) as a function of 3He and 4He coverage were used by Valles et al. (1986) to make conclusions about theoretical predictions. The prediction which follows from the Lekner formulation of the problem, that the square of the third sound velocity should decrease linearly with the 3He coverage, is confirmed by experiment. Shown in fig. 71 is the relative shift in the third sound velocity as a function of 3He coverage for T = 0.10 K and we see that for coverages below about 0.1 layer of 3He, the shift is indeed linear (Valles et al. 1986, Hallock 1987). A second conclusion was that the thickness dependence of the binding energy of the 3He to the 4He film predicted by Sherrill and Edwards was correct. The theory of Sherrill and Edwards (1985) resulted in values for the binding energy as a function of film thickness. The third sound directly measured the relative shift in the velocity of third sound as a function of 4He coverage. In the Lekner (1970) formulation, the binding energy, El, is the sum of the chemical potential in a pure 4He film,/t a0, and the energy required to change the mass of a single 3He atom to that of a single 4He atom in a 4He film, El, el =/~4o + E~. The introduction of a small amount of 3He into an otherwise pure 4He film shifts the chemical potential by the amount //4(N3,N4)

- ~40(0,N4)

--

(N3[A)[6 ~,1/b(N4/A)],

where N4 is the number of 4He atoms present in the system of area A. The accompanying change in the third sound velocity at T = 0 K is

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

O

0

O.I

0.2

d~(loyets)

0.4

405

0.5

-0.1

to o tO

Od i,"1

-0.2

I IMIr

tO

-0.3

-.

1

1

1

1

1

Fig. 71. Third-sound velocity values as a function of the amount of 3He added to the experimental cell as determined by Valles et al. (1986). For amounts of 3He below about 0.1 atomic layer the behavior is linear in 3He coverage. Two 4He coverages are shown at T - 0 . 1 K, 3.58 layers (triangles) and 5.27 layers (circles) (after Valles et al. 1986).

C32(N3,N4)- C302(0,N4) _- (NsNg/M4AE)[d2e,/d(N4/A)2],

where C30 is the velocity of third sound for the pure 4He film. To facilitate comparison with theory, the expression for the third sound velocity may be rearranged as

(An4M4/RN3)(C32-

C302) = D4t~2e I/c~D42,

where R is the reduced superfluid fraction in the film, R = ((,Os}/p)(1- D/d4), and where n 4 is the 4He number density and D 4 is the ratio of the areal 4He density to the bulk 4He density, n 4 DaA = N4. Because of the density gradient in the film, D 4 > d 4 where d4 is the distance between the substrate and the surface of the film. A comparison between the data of Valles and Hallock (1987) and Valles et al. (1986) and the predictions of Sherrill and Edwards (1985) is presented in fig. 72 where the solid line is the theoretical prediction for a simple layered-model hydrodynamic calculation for an incompressible film with m*/m = 1.0 (Ellis et al. 1981, Guyer and Miller 1981). The dash-dot curve is the

406

Ch. 5, w

R.B. HALLOCK v.v

u

-0.4

~

I

~

1

i

iii

-

!

~m t.)

-0.8

/. /

-t.2 33

4.0

50

6.0

d (Ioyers)

Fig. 72. Fractional change in the third sound velocity divided by the 3He coverage, d3, as a function of the 4He film thickness at T= 0.1 K. For these data the 3He coverage was ---0.05 atomic layer (Valles et al. 1986, Valles and Hallock 1987). The dashed-dotted curve represents the Sherrill and Edwards model with the choice m3,1m3 = 2.1, independent of 4He coverage. prediction of Sherrill and Edwards (1985) for a compressible film with m * / m = 2.1; the dashed curve is the same for m * / m = 1.0. The third sound data supported of the Sherrill and Edwards (1985) approach, but pointed to the need for accurate m * / m values. Further study of the velocity of third sound in 3He-4He mixtures was undertaken by Noiray et al. (1984). In these experiments, a Nuclepore resonator was used and the third sound frequency was determined as a function of the amount of 3He and 4He in the apparatus for T = 0.4 K where it was assumed that the amount of 3He in the film remained fixed as the 4He converge was changed. As shown in fig. 73, for small values of the 4He thickness (d4 = 2.9 layers) the 3He apparently mixed into the film and the behavior of the third sound velocity as a function of the amount of 3He present followed closely the mixed-model expression. For a large amount of 4He (d 4 = 6.2 layers) in the apparatus, the opposite was true with the third sound velocity clearly showing a stratification of the film with the 3He on top of the 4He. To further study the possible crossover from one regime to the other, measurements were made as a function of the 4He coverage for 1 and 2 layers of 3He in the apparatus. The results of this, shown in fig. 74, demonstrated that there was a smooth evolution in behavior from the mixed mode for thin films to the stratified mode for thicker films (Noiray et al. 1984). The effect was concluded to be dependent on of the amount of 4He and not of the amount of 3He in the apparatus. Unfortunately these experiments did not

Ch. 5, {}3

P R O P E R T I E S O F M U L T I L A Y E R 3He-4He M I X T U R E F I L M S

407

Fig. 73. Square of the relative third-sound velocity at 0.4 K as a function of 3He coverage. The full lines and dashed lines represent respectively complete phase separation and perfect mixing for each 4He coverage. Experimental values were obtained from a time-of-flight technique on a glass substrate (circles) and with use of a resonator filled with Nuclepore filters (triangles and stars) (from Noiray et al. 1984).

Fig. 74. Square of the relative third-sound velocity at 0.4 K as a function of 4He coverage. The full lines and dashed lines represent respectively layered and mixed films for each 3He coverage. Experimental values are obtained by use of a resonator filled with Nuclepore filters: d 3 = 1 (triangles) and d 3 = 2 (stars) (from Noiray et al. 1984).

408

R.B. HALLOCK

Ch. 5, w

reach below 0.4 K and thus some thermal population effects must have been present. In the work of Heinrichs (1985) (fig. 66) measurements taken at T = 0.1 K for 4He coverages of 5.3 and 3.6 atomic layers show good agreement with the bi-layer model for the 3He coverages studied, 0 < d3 < 2 atomic layers (Hallock 1987). No evidence for crossover behavior was seen at T = 0.1 K. Further work on this system would be useful. A qualitative speculation proposed that these apparent crossover effects are due to interface fluctuations which get constrained as the film thickness is reduced (Noiray et al. 1984). For larger film thickness values, the interface between the lower 4He and the 3He on top can fluctuate in thickness freely, but for thinner 4He films, the presence of the underlying substrate constrains the fluctuations and introduces velocity gradients which lead to perturbations in the concentration profiles in the 3He. In a later publication Laheurte et al. (1986) explain the theory in more detail and report additional experimental results. The theory involves the displacement amplitude of the fluctuations and the introduction of a parameter to define the crossover regime. The result is that for very thin films, we have stratification, for thicker films there is a homogenous regime and for quite thick films, once again the stratification regime reappears; so, the stratification should be re-entrant. The experiments probe the homogeneous regime at relatively low coverages, show a crossover and then for thicker films show stratification. (Apparently the lowest coverages studied do not directly show the stratification since the films studied were not thin enough). In an effort

Fig. 75. Temperature dependence of the third sound velocity for films with a constant amount of 4He (2.9 atomic layers) with 3He thickness values ranging from 0 to 2.0 atomic layers. The large symbols define a "temperature", T*, at which Laheurte et al. (1986) interpret stratification in the films to take place.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

409

to search for the onset of stratification at the lower critical thickness of the film, the temperature was varied (since the lower critical thickness is expected to be temperature dependent). One would expect to see an increase in the third sound velocity as the film moved from homogeneous to stratified at the lowest temperatures. This is observed by Laheurte et al. (1986) (see fig. 75) for the case of 2.9 layers of 4He with 3He coverages ranging form 0 to 2 atomic layers. No quantitative comparison to theory was made. A new parameter, do is introduced such that for thickness values above do the fluctuation magnitude is negligible. This is necessary since without this suppression, the theory and the data do not agree. Unfortunately, the manner in which the claimed stratification at low temperatures matches the prediction of the stratified theory developed earlier (Ellis et al. 1981, Guyer and Miller 1981) is not discussed. This fluctuation speculation has not been widely adopted and the authors themselves describe it as "limited and crude", yet it is consistent with their measurements. An alternate explanation for this behavior might be due to the effect of finite temperature and the population of states available to the 3He which are imposed by the particular substrate. A more systematic series of measurements over a wider range of temperature and concentration and perhaps for different substrates would be useful. Draisma et al. (1994) and Draisma (1994) have reported the results of several experiments with pure 4He and mixture 3He-4He films for 4He coverages of 3.61, 4.21 and 5.85 atomic layers with 3He coverages <1.25 atomic layers. Measurements were made of the third sound velocity via time of flight techniques with capacitive detection as a function of temperature for a sealed cell (Draisma et al. 1994). For enhanced amounts of 3He in the film an anomaly appears in the third sound velocity at the lowest temperatures; there is a small peak and for larger concentrations it is stated that this turns into a step. This is similar to what was seen earlier (Hallock 1987). This peak as a function of temperature centered near T = 200 mK is observed to be dependent on the energy in the third sound drive pulse, and nearly disappears for the lowest pulse energies used. As will be described next, recent measurements by Sheldon and Hallock (1994) used resonance techniques with lower power levels and also observed this weak peak, but in this case the relative size of the peak was independent of the power level. The results for the velocity ratio as a function of the 3He/aHe ratio are interpreted by Draisma et al. (1994) to show that the film is either completely mixed or that there is possibly a substrate state (Pavloff and Treiner 1991b). As noted by Hallock (1987) the evolution of the frequency may be indicative of a configurational changes in the film. However, the present evidence for the existence of a substrate state is not strong. For example, experiments with bulk mixtures in sintered powders of silver and copper by many workers show no evidence of the binding of 3He to the surfaces. Preliminary analysis of recent work with bulk mixtures in Nuclepore also finds no evidence for such a sub-

410

R.B. HALLOCK

Ch. 5, w

strate state (Sheldon and Hallock 1995). More definitive experiments with a variety of substrates, including weak binding ones, are necessary. More extensive third sound data on 3He-aHe mixture films have been taken recently by Sheldon and Hallock (1994) (Sheldon et al. 1994). These data were taken in the same apparatus as some of the NMR data reported in previous sections of this review. This work covered some of the same ground as the earlier work of Heinrichs (1985), but in addition concentrated on smaller coverages of 3He. In this work, the third sound frequency, Q and phase, q~, between the thermal third sound drive and the detected amplitude were measured by use of the Nuclepore capacitive-detection resonator shown in fig. 57 for a range of 3He coverages and temperatures with n 4 = 0.389/~-2 (d4 = 3.65 layers). For each of eight 3He coverages in the range 0.007 < n 3 < 0.049/~-2 (0.099 < d3 < 0.721 layers) , measurements were made in the temperature range 40 < T < 500 mK and for drive powers 0.044 < P~ < 440 nW. As we saw earlier, a simple quantitative description for the third sound velocity in a mixture film system of 3He and 4He is derived from a set of linearized hydrodynamic equations by assuming that there are two layers; the lower layer (1) contains all the superfluid and some concentration of 3He, and the upper layer (u) is a normal fluid blanket of a 3He-aHe mixture (Ellis et al. 1981, Guyer and Miller 1981, Monarkha and Sokolov 1981). The theory assumes that the helium is incompressible and that there are no interaction effects between the two layers. Interaction effects were included in a later calculation by Monarkha and Sokolov (1982). The result for the third sound velocity in a mixture film at T = 0 of Guyer and Miller (1981) can be written as C32 = C302(m41ml)(d41dl)4(1 - p u l p l + (,Ou/pl)/(1 + du[dl)4), where C30 is the third sound velocity for a film of pure 4He of thickness d 4 and d4 is the thickness of the film if there were no 3He present, m4 is the mass of an 4He atom, m ! is the average mass per particle in the lower layer, Pu and,o I are the average number densities in the upper and lower layers, and du and dl are the thicknesses of the layers. Temperature dependence is added to the film thicknesses du, dl by assuming that as the temperature increases, the 3He is both excited into its higher energy state inside the 4He film and evaporated into the vapor. The extraction of the energetics from the frequency data requires an ability to compute the 3He population in each energy state in the film for all temperatures. For lower 3He densities it is adequate to use a Boltzmann distribution for a two-state system so that nlA/N = exp(-AJk~T),where A + eF= e I - e o. For higher densities, the film has the properties of a two-dimensional Fermi gas. It is necessary to use the chemical potential for each energy state in the film, ~ i = Ei + ll2niVo s +/Un:G i, where e i, n i and/~IFci are the energy, areal density and ideal Fermi gas chemical potential of state i, respectively.

Ch. 5, w

411

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

460

42~ 440

~"

1.01

400

It

380

360

o

~

30 2.0

I

0.0

o.o I"

-o2

L ~.

",

It

T

8 i

(b)

o

0 ,-

9

,i~~

9

t

a v

1.00

9

D3 0.39 0.55 0.61 0.~ 0.70 0.72

* 9 * * * =

~,

-04 -0.6

A o o

,

0.99

9 0.74

-0.8

v

0.76

9

9 087

.10 -1.2

v

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-

i~"

" 9

9

V

+

II

1 9

o

,

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0

100

200

300

T (inK)

400

500

o

I

~ <

100

I 200

9- -

0.98 30O

T (rnK)

Fig. 76. Temperature dependence of (a)fl and (b) Rex (Rex = 1/Q- 1/Q4 where I/Q 4 is the damping for the lowest 3He coverage) for each 3He coverage: n 3 =0.007 ( x ) , 0.016 (+), 0.025 ( v ) , 0.036 (*), 0.040 ( ~ ) , 0.044 (zx), 0.046 (O) and 0.047 ]k-2. Typical fits are shown to the frequency and damping data from which the energetics are determined. The inset shows the single exponential behavior of the excess damping. (c) Phase versus temperature for a single 3He coverage of 0.046 ]k-2 at four drive powers. Lines are drawn as a guide to the eye (from Sheldon and Hallock 1994). (d) Enlarged view of the frequency peak region shown as fl/fTmin versus temperature, where Tmin is 40 mK for all but the data at D 3 = 0.39 layers for which Tmin = 70 mK.

The temperature dependence of the third sound frequency is shown in fig. 76 for eight different 3He densities. As seen previously, at low temperatures the third sound frequency has the expected dependence on 3He thickness for a layered film (the lower layer being nearly all 4He and the upper, 3He). The film is expected to be completely phase-separated at T = 0 as determined by the energetics because the ground state of the 3He is at the surface of the 4He film (Pavloff and Treiner 1991c). At low temperatures (T < 150 mK) the 3He is in its ground state and the frequency has little structure. There is a small maximum at 150 mK which gets relatively larger as the 3He density is increased (fig. 76d). This peak, was earlier seen to be much more dramatic at considerably higher 3He densities (Heinrichs and Hallock 1983b, Heinrichs 1985, Hallock 1987, Sheldon et al. 1994). In the case of Draisma et al. (1994) the amplitude of the peak was drive power dependent. This was not the case in the work of Sheldon

412

R.B. HALLOCK

Ch. 5, w

and Hallock (1994) (Sheldon et al. 1994), where very low drive powers were used. The origin of this local maximum in the velocity remains unexplained. The frequency decrease with increasing temperature in the range 150 mK < T < 300 mK is attributed to the 3He being excited into a higher energy state (Heinrichs and Hallock 1983b). The application of the equation for C3 with the temperature dependence included in the layer thicknesses leads to the expectation of a more gradual decrease in frequency with increased temperature than is observed for any reasonable energy difference A between the first two states. Therefore it is concluded that the expression is not adequate to describe these measurements. This inadequacy of the theory in this case is consistent with the difficulty in explaining the velocity shift seen earlier by the Laheurte group and seen by Hallock and co-workers. It is clear that intriguing behavior is present in the temperature dependence of the third sound velocity near T = 0.2 K, behavior which has been documented but not yet fully understood. For T> 300 mK the frequency increases due to 3He evaporating from the film. The expression for C3 can be fit to the increasing part of the frequency versus temperature curve, with the binding energy in the probability density of the vapor as a parameter. In the region where evaporation is prevalent, the 3He evaporating from the top of the 4He dominates the effect of the 3He mixing into the 4He film, so the phase-separated form is used, ( C 3 ( T ] C 3 o ( T ) ) 2 --

1-

p3/P4[

1 - (1 + d3(T)/dl)-4],

where dl is the lower layer thickness. From the fits of this equation to data, one can determine the ground state binding energy eo of 3He to the 4He film for each 3He density (see fig. 77). In addition to frequency data, data for the attenuation is also available. The excess attenuation is the additional attenuation as 3He is added to the 4He film, Rex =- 1 / Q - llQo, where Qo is the attenuation for a pure 4He film. At this writing this Qo attenuation is not yet available. Thus, Qo is approximated to be the attenuation for the lowest 3He density, n 3 = 0.007/~-2 for which data exists. The excess attenuation, Rex, is observed to be thermally activated with an empirical form Rex = R o + R1Pl, where R0 and R 1 a r e parameters and Pl is the first excited state probability distribution for a two state system with activation energy A'. For low 3He densities, Pl takes the form of a Boltzmann distribution and for higher 3He densities it is necessary to use the Fermi distribution discussed earlier. By fitting the appropriate form of Rex to the excess attenuation data, Sheldon and Hallock (1994) extract an activation energy, A' for each 3He density in much the same manner as for the NMR experiments. From the fits to frequency data of the sort shown in fig. 76 Sheldon and Hallock (1994) obtain the ground state binding energy eo and, from the fits to the excess damping data, obtain an activation energy, A'. Along with the Fermi

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

'

I

'

I

'

I

'

I

"

413

I

E1

-4

-

X

xx Ix -5X

-6

0.00

.

I

0.01

.

I

0.02

,

I

0.03

i

I

0.04

.

I

0.05

0.06

n 3 (A -2) Fig. 77. Ground, e 0, and first excited state, e l, dependence on 3He density of the binding energies of 3He to a thin 4He film. Energies from third sound (squares) are compared to the energies from NMR (triangles). Crosses and plusses are data from Bhattaracharyya et al. (1984) for a similar film (from Sheldon and Hallock 1994).

energies from NMR magnetization measurements (which were carried out at the same coverages), one can derive an excited state energy el if the activation energy A' is assumed to be related to A as A' = A. The two energies, e0 and el, are compared in fig. 77 to the ground and first excited state energies derived from NMR (Sprague et al. 1994). The agreement is good. Also included in fig. 77 are the energies from the earlier NMR experiments and re-analyzed heat capacity data of Bhattacharyya et al. (1984) for similar film configuration (d4 = 3.4 layers) on Nuclepore for some low 3He densities; agreement is reasonable for both energy states. There is to date no clear theoretical explanation as to why the third sound damping is sensitive to A'. Apparently the reason has its origin the location of the impurity in the excited state, but the detailed mechanism as to how this influences the damping has not been described. An additional property of interest available from the third sound measurements is the phase between the drive pulse and the 4He film thickness fluctuations. It is predicted by Bergmann (1969, 1970) that there should be a phase difference of ~ between the changes in temperature and the changes in 4He film thickness for third sound in pure 4He films. When the temperature of the film is

414

R.B. HALLOCK

Ch. 5, w

highest, the film thickness is predicted to be at its thinnest. The quantitative relationship between the magnitude of the temperature and film thickness oscillation predicted by Bergmann (1969, 1970) has been measured (Ellis et al. 1984, Brooks et al. 1978, Laheurte et al. 1991) for third sound in a pure 4He film as will be discussed further below. For the recent measurement on mixture films by Sheldon and Hallock (1994), within experimental error, for all the 4He densities, at the lowest drive powers and below 200 mK, there is indeed a phase difference of at. (One 3He coverage is shown in fig. 76c.) Then, abruptly near 200 mK, the phase changes to zero, indicating that the drive and response are in phase. Below 200 mK, the higher the drive power, the closer the drive and response are to being in phase. Above 200 mK there is no drive power dependence. This general behavior may be related to the unusual behavior of the third sound velocity in the vicinity of 200 mK and perhaps also to the curious pulse inversion seen in earlier third sound work at much higher coverages (Ellis et al. 1984). From a number of third sound experiments it is apparent that for mixture films something quite interesting, but presently not understood, is happening near T= 200 mK. We conclude the discussion of third sound in mixtures by noting that recently Laheurte et al. (1991) have simultaneously measured the magnitude of the thickness and temperature oscillations of third sound in experiments which were patterned after earlier experiments by Brooks et al. (1978). In these earlier experiments, carried out in the apparatus shown schematically in fig. 61, the readout of the capacitive film thickness detector was by means of tunnel diode techniques with a resolution of transients of about 0.01 A, (Ellis et al. 1984). The theory of Bergmann (1969, 1971, 1975) predicts that AT/Ah =-fT/L where f is the van der Waals force at the surface of the film and L is the latent heat. The results (Ellis et al. 1984) for a range of 4He film thickness values were in general agreement with the predictions as shown in fig. 78. An interesting feature of the experiment was the observation that the presence of a third sound wave on one plate induced the creation of a twin wave on the opposite facing plate within a propagation distance of one wavelength of third sound. This was the result of coupling between the two waves by the vapor when the mean free path in the vapor was longer than the plate spacing. In the recent work of Laheurte et al. (1991), the apparatus was very similar, but the emphasis was on a measurement of the temperature (rather than the thickness) dependence of the ratio AT/Ad and also the effect of small quantities of added 3He. For pure 4He the ratio AT/Ad was found to be governed by the van der Waals force in agreement with earlier work (Brooks et al. 1978, Ellis et al. 1984) and as predicted by Bergmann (1969, 1971, 1975). Brooks et al. (1978) and Ellis et al. (1984) found that the ratio increased above the prediction as the temperature was raised and this was confirmed by the recent measurements of Laheurte et al. (1991). In these recent experiments a wider temperature range was used and the departure from the

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS 10'5

415

-.

T = 1.3.5 K

o

10 4

0o~ oO/

E r

o

o/

,,r =L m

-

BERGMAN

//

"u <1 I.-<1

o

i0 z

ATKINS m

I0

.

,

10 4

I

l

IO s

C 3z

.

.

.

I

10 6

-

-

,

or

(cmZ/sec z ) Fig. 78. The relative amplitude IATIAdl as a function of the square of the third-sound velocity as determined by Ellis et al. (1984). The solid line is the prediction of Bergmann (1969, 1971); the dashed line represents the prediction of the early theory due to Atldns (1959).

predicted magnitude of AT/Ad was observed to begin at T ~ 0.7 K (fig. 79). The thickness dependence is as predicted by Bergmann, but the temperature dependence deviates at higher temperatures. When small amounts of 3He are added to the film, at T = 0.6 K for two thickness values of the 4He (5.4 and 7.3 atomic layers), the temperature oscillation increases with added 3He, but the thickness oscillation remains about constant as a function of the added 3He. The enhancement of the temperature oscillation amplitude is greater for the thicker 4He film. The enhancement of AT caused by the addition of 3He is assumed to be due to the entropy associated with the 3He and an analysis based on this yields the observed 3He coverage dependence for AT/Ad (Laheurte et al. 1991). For completeness we should note that recent theoretical work by Brouwer et al. (1992) has extended the Bergmann treatment of third sound for pure 4He to a parallel plate geometry and is apparently capable of explaining two previously unexplained observations. The theory suggests that two different types of third sound modes can propagate in such a geometry, a symmetric mode and an antisymmetric mode. When the effects of the two modes are added together, the

416

R.B. HALLOCK

Ch. 5, w 1

~ 16 O 1

8'
6 ~

8 4

0.00.Z

A

..

0.4

-.O

0.6

0.8

1.0 1.2 T(K)

1.4

Fig. 79. IAT/Adl as a function of temperature for a 4He film thickness of 8.3 atomic layers (from Laheurte et al. 1991). The solid line represents the predictions of the Bergmarm theory.

observations of Laheurte et al. (1991) (that the ratio AT/Ad rises substantially above the Bergmann prediction above about 0.6 K) seem to be explained on the basis of the theory. In addition, Brooks et al. (1978) and Ellis et al. (1984) observed that AT/Ad rises modestly with frequency in the range 1 kHz to 5 kHz. This is also in qualitative accord with the predictions of the new theory. 3.4.2. Oscillator measurements Bishop and Reppy (1978, 1980) carried out a detailed study of the behavior of the superfluid mass and the dissipation in thin 4He films. The experimental technique was patterned after the modification of the Andronikashvili (1946, 1948) torsion oscillator introduced by Berthold et al. (1977). In these experiments, an extremely high Q torsional oscillator was used in conjunction with a "jelly roll" of Mylar on which the 4He could be adsorbed. The moment of inertia of the system could be resolved to better than a few parts in 109 and the superfluid mass resolved to one part in 104. The technique was an improvement over the quartz crystal resonance used by Chester and Yang (1973) in that the Q was higher (and measured) and the surface area larger; these improvements resulted in enhanced reselution of the transition region. In a typical experiment a helium charge is adsorbed into the cell and the temperature scanned over a narrow region in the vicinity of the Kosterlitz-Thouless (1973) transition. The period of the oscillator and the Q are measured, the background period is extrapolated and the period shift is computed. The period shift in these torsional oscillator and quartz crystal resonance experiments is a direct measure of the superfluid fraction. In the Bishop and Reppy (1980) work, a limited amount of data was taken with a 3He-4He mixture. The period shift and the damping for a pure 4He film was first studied (fig. 80) and then a small amount of 3He was added (10% of the 4He amount). Two major effects were seen: the major structure of the period shift moved to lower temperature and a second structure appeared in the data

Ch. 5, {}3

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

417

Fig. 80. The superfluid period shift and the dissipation for a torsional oscillator experiment for the case of pure 4He (lower plot) and the same film with the addition of 10 at.% of 3He (upper plot). The presence of the 3He moves the Kosterlitz-Thouless transition to lower temperatures and causes the appearance of a different "transition" at a higher temperature (from Bishop and Reppy 1980). which began at a slightly higher temperature than for the pure 4He case. T h e conclusion was that the addition of 3He apparently produced not one but two superfluid transitions in the apparatus. The lower temperature transition case

418

R.B. HALLOCK

Ch. 5, w

was a Kosterlitz-Thouless transition with an associated jump in the superfluid density. These data confirmed the earlier conclusion that the addition of 3He to the 4He maintained the Kosterlitz-Thouless jump at its pure 4He size (Webster et al. 1979). The second transition was apparently of a different type with no jump in the superfluid density; rather, the superfluid density evolved in a smooth manner with no associated dissipation peak. Additional work (Smith et al. 1978) of a limited nature was done on thin films adsorbed to Vycor with the observations that the transition temperature increased with added 4He and upon the first addition of pure 3He at fixed 4He the transition temperature rises. The corresponding superfluid mass extrapolated to T = 0 K showed a decrease upon addition of the first shot of 3He. Subsequent additions of 3He lowered the transition temperature and further reduced the extrapolated superfluid mass at T = 0 K. More recent work on mixture films in Aerogel show that small concentrations of 3He reduce Tc and the superfluid density at zero temperature, but that the overall shape of the superfluid mass versus Te is independent of 3He concentration (fig. 81) (Crowell 1994, Crowell and Reppy 1995). Webster et al. (1979) measured the mass loading on a quartz oscillator as a function of the amount of 3He in the apparatus up to 30% as a follow up to ear-

Fig. 81. The period shift in a torsional oscillator experiment for pure 4He (n4) and for added 3He. Coverages are given in gmol on an area of 9.2 m2 for Aerogel of density 0.2 g/cm3 (from Crowell 1994).

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

419

Fig. 82. Measured values of the superfluid jump at the Kosterlitz-Thouless transion for pure 4He (right) from a number of investigators and for mixture films at T = 1.3 K as a function of concentration as reported by Webster et al. (1979). fl is the 3He concentration. lier measurements by the same technique for the case of pure 4He. The conclusion of the work was that (1) the Kosterlitz-Thouless relationship for the jump at the superfluid transition is universal (fig. 82), and (2) there was no evidence for any phase separation in the apparatus, no lateral separation (as had been predicted by several theorists) is present and no vertical phase separation is present either. Webster et al. (1980a) carried out additional measurements and interpreted the data to show that the films were quite uniform and thus the data provided strong evidence that no droplet or puddle coalescence mechanism was operating at the transition and thus that it was indeed a Kosterlitz-Thouless transition. They also showed that very close to the superfluid transition the differential superfluid fraction (the rate of change of the areal superfluid density with coverage) assumes its bulk value (Webster et al. 1980b). This was true for all the concentration values studied. Agnolet et al. (1984) studied the response of a torsional oscillator with a superfluid 4He coverage of 0.24 layers as a function of added amounts of 3He. This was done in an effort to test the effect of the addition of the 3He on the universal jump prediction and to test the mixture system for the presence of distorted phase diagrams which had been predicted by Cardy and Scalapino (1979) and Berker and Nelson (1979). The results confirmed the universal nature of the jump for these very low coverage films. Also noted in the experiment was an

420

R.B. HALLOCK

Ch. 5, w

apparent increase in the zero temperature superfluid mass for 3He coverages above about 0.2 monolayer. No clear evidence for phase separation (no kink in the transition temperature versus concentration curve) was seen. The increase in the superfluid mass at zero temperature did allow the speculation that phase separation might be taking place with the 3He expelled from the 4He. McQueeney et al. (1984) extended the torsional oscillator measurements of Bishop and Reppy (1980) and Agnolet et al. (1984) to measurements of thin 4He coverages with large coverages of 3He. In these experiments a fixed amount of 3He (about 12 atomic layers) was introduced into the torsional oscillator cell and 4He was added until the first evidence of a superfluid transition was seen at low temperature. This requires more 4He than necessary in the case of pure 4He (so the presence of the 3He suppresses the superfluidity to some extent). Increasing amounts of 4He showed an increase in the transition temperature, but with the temperature and the jump obeying the Kosterlitz-Thouless prediction. Thus, the superfluid with a thick overlayer of 3He appears to be a good example of a twodimensional superfluid. Below the superfluid transition temperature the dissipation remains finite representing the presence of dissipation which is not understood. Temperature effects are seen in the temperature dependence of the period shift and attributed to 3He moving into the 4He and, at higher temperatures, 4He moving into the 3He. A key result of the work is that the effect of the 12 over"'

' i

-- '"

I

"' I

....

I

(

4 gl

0 EL.

2

25

30

55

40

45

4Ne coverage (p.moles/m 2)

50

Fig. 83. The period shift of torsional oscillator data for pure 4He (open circles) and for 12 additional layers of 3He (closed circles). The added 3He requires that additional 4He must be present before superfluidity takes place (from McQueeneyet al. 1984).

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

421

layers of 3He is to require an additional roughly 1/2 layer of 4He before superfluidity is evident (fig. 83). The results also suggest that solution effects may be important. That is, the layers do not remain pure as the temperature increases. This is generally consistent with what is known about the energetics of thin films from heat capacity and NMR measurements. Recent work by Crowell (1994) with large concentrations of 3He on Aerogel is consistent with the work of McQueeney et al.; an excitation attributable to the 3He depletes the superfluid density at low temperatures. Perhaps motivated by the work of McQueeney et al. (1984) third sound experiments on Nuclepore were later extended by Romagnan et al. (1988) to a wide range of 3He coverages and temperatures and the analysis expanded to allow for partial mixing of the films. In this work, for 2.6 and separately 3.0 layers of 4He, the superfluid transition temperature was measured as a function of 3He coverage. The transition temperature was observed to decrease with 3He coverage and to saturate for high 3He coverages. One of the coverages was the same as used by McQueeney et al. (1984) and thus, the saturation (which reproduced the temperature value of McQueeney et al.) showed that beyond a critical coverage, further addition of 3He did not further lower the superfluid transition temperature. Detailed study of the temperature dependence of the third sound velocity for d4 = 3 layers as a function of added 3He showed that the addition of 3He caused a lowering of the third sound velocity, but that saturation set in for 3He thicknesses beyond 4 layers. The film was assumed to be stratified for 3He coverages beyond this. The onset values agreed with those found by McQueeney et al. (1984). Data at fixed 4He of 3 atomic layers and fixed 3He as a function of temperature show (fig. 84, see also fig. 75) that as the temperature is increased above 0.2 K there is a dramatic fall in the third sound velocity which can only be partially explained by assuming that the superfluid phase in the film is not pure 4He. The temperature dependence (fig. 84) is in accord with the superfluid density measurements of McQueeney et al. (1984), but to get the magnitude of the velocity correct for temperatures above 0.2 K requires a substantial shift in the restoring force. This shift is toward a thicker film and might be explainable if additional 3He mixes into the 4He film but the absence of detailed energetics data at the time precluded quantitative discussion. None of the explanations offered to date seem capable of explaining the temperature dependence in the region of the sharp drop in the velocity, or of the magnitude of the velocity shift. The shifts are large and this experiment too points to behavior in the vicinity of 200 mK which is, as is the case for other experiments discussed, incompletely understood. In the above experiments, the description of how the 3He enters the superfluid is discussed in terms of the notions introduced by McQueeney et al. (1984). The strongest contribution comes from the AT2 dependence of the dissolution of 3He into the 4He. Since the coefficient here is found (McQueeney et al. 1984) to be independent of the thickness of the 3He

422

R.B. HALLOCK

Ch. 5, w

Fig. 84. Measured third sound velocity in mixture films with a 4He coverage of 3 atomic layers for 3He coverages of 4 (open triangles) and 4.5 (open squares) atomic layers. Assuming the superfluid phase is pure 4He yields the horizontal prediction for the velocity. Adding superfluid mass measurements from McQueeney et al. (1984) produces the solid curve. The dashed curve fits the data above 0.2 K but requires a very weak restoring force (from Romagnanet al. 1988). layer it is argued that there is an interface effect as mentioned earlier (Romagnan et al. 1988). To return to the discussion of oscillator experiments, Wang and Gasparini (1986, 1987a,b, 1988) carried out a series of measurements to further explore the mixture observation due to Bishop and Reppy (1980) using the technique of the torsional oscillator. Wang and Gasparini report observations of the period shift and dissipation seen for fixed but selectable coverages of active 4He (i.e. under this is immobile 4He) with the addition of up to about a monolayer of 3He. Several experimental runs are reported with, in the case of one of the early measurements, roughly one fluid 4He monolayer in the cell (the Tc being 0.91 K), clear evidence for two transitions is seen with the lower temperature "A" transition of the traditional Kosterlitz-Thouless type and the higher temperature "B" transition moving eventually to temperatures higher than that of the pure 4He transition (figs. 85, 86). These two transitions were interpreted as resuiting from regions of different thickness and concentration in the apparatus and the suggestion was made that it was the addition of the 3He which caused the two transitions since the pure 4He data in the underlying film showed only a single transition. Repeated runs with this and a second cell resulted in more ambiguous results (with the B transition not readily visible) and a general lack of reproducibility apparently produced by non- uniformity or inhomogeneity of the underlying 4He film. The Bishop and Reppy (1980) observation was in reasonable accord with the first of these experimental results. The two-step transi-

Ch. 5, w

PROPERTIESOF MULTILAYER 3He--4He MIXTURE FILMS

3 2r

M~........~..__.._.

: ....

,

,

423

,

---._<

.

24

20

L.._-

3He(Ioyer]

0.OO OO4 0.09 O.14 O. 17 0.24 O.31 0.39 0.47 0.52 0.58 O.65 0.74

16

%1~ o •

12

t'M

"

0.82

,~ ..~

0.91 1.06 1.25 1.51 1.79 2.43 0 -Q92

~

-2 IO.3 '

~"

------' ' .5 0.7 TEMPERATURE o'

'

(K)

I.I'

Fig. 85. Period variations for a pure 4He film, 1.2 active layers, and various coverages of 3He. Data have been offset by 0.92 x 10 -6 for clarity. The region between the two steps is one of coexistence of a superfluid phase, B, and normal phase, A (from Wang and Gasparini 1988).

tion suggests that the 3He-4He mixture system may have phase separated. In spite of this, it was not clear that the two transition picture was that described theoretically by Mon and Saam (1981) or that of the deformed film as described by Guyer (1984) (and discussed by Sherrill and Edwards, 1985). The thickness values for these experiments which did show two transitions were such that very near a single active layer of 4He was in the apparatus, with a resulting Kosterlitz-Thouless transition temperature of about 1 K. For coverages somewhat different from one active layer of 4He or in situations in which the position of the Kosterlitz-Thouless transition for pure films suggested a lack of homogeneity, the evidence for a second transition was greatly suppressed. This behavior has not been adequately explained by theory nor has it been repro-

424

R.B. HALLOCK

Ch. 5, w

Fig. 86. The temperature of maximum dissipation Tc, normalized by the value for the pure film, as a function of 3He concentration (from Wang and Gasparini 1988).

duced by other workers. Extension of the re-entrant superfluidity studies of Crowell and Reppy (1993) to mixtures may be relevant here. Laheurte and co-workers carried out a third sound experiment on a Nuclepore substrate explicitly to search for the presence of the two transitions seen by torsional oscillator techniques (Laheurte et al. 1987). In this work a third sound generator and detector were evaporated directly onto the Nuclepore substrate as had been done earlier by Smith et al. (1987). The third sound velocity was then measured as a function of temperature (presumably decreasing temperature) for four films of constant 4He thickness (2.7 atomic layers) and different amounts of 3He. The regular Kosterlitz-Thouless transition was seen, but no third sound evidence was seen for the "B" transition at higher temperature. It was noted in this work that the dramatic rise in the third sound velocity as the temperature fell in the vicinity of 0.2-0.3 K (depending on the amount of 3He in the cell), consistent with earlier work of the Laheurte group, has no signature in the torsional oscillator work. It was thus argued that the "transition" (i.e. steep temperature dependence) in the third sound velocity observed by the Laheurte group must take place at constant superfluid density per unit area. As has been mentioned,

Ch. 5, w

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425

other third sound work has seen shifts in the third sound velocity, but of smaller magnitude. Additional torsional oscillator measurements were carried out by Tiwari and Glaberson (1990) for films with transition temperatures between 1.3 and 1.9 K. In all cases studied the effect of the addition of 3He was to decrease the transition temperature and no evidence was seen for the existence of phase separation or other second transition in the film. The jump in the superfluid density at the transition was in accord with the universal value predicted by Nelson and Kosterlitz (1977). For some films the transition temperature, while falling below the pure 4He value, was larger than the transition temperature for a similar concentration in the three-dimensional bulk case. This was explained as likely being due to the expulsion of the 3He from the vicinity of the substrate. The presence of the 3He results in a broadening of the dissipation peak associated with the transition in addition to resulting in a reduction of the transition temperature. There is some disagreement among the various torsional oscillator results. A more comprehensive experimental examination of this "two transition" behavior would be of value, especially if it could be done with a well characterized substrate. 3.4.3. Selected other experiments We mention here a selection of additional topics which involve mixture films. (1) Effect of 3He on film flow. The effect of the addition of 3He on the behavior of persistent currents of 4He flowing in a closed loop on an annular ring has been investigated by Ekholm and Hallock (1980). In these experiments, third sound was used as a tool to measure the speed of flowing currents of 4He with addition of 3He. The decay rate of the metastable flows was measured as a function of the 3He concentration in the apparatus. The addition of 3He also affected the speed of the third sound on a film which was at rest. The change of the third sound speed on a film at rest with the addition of 3He could be explained generally by the assumption that the additional 3He (at these relatively high temperatures, T> 1 K) acted as an enhanced normal component distributed throughout the film. This simple effect was, however, not adequate to explain fully the persistent current decay; the 3He had a more profound effect, perhaps affecting the vortex dynamics. Much earlier work by van den Meijdenberg et al. (1961) concluded that the presence of 3He in a driven film flow situation decreased the film transfer rate with an increase in concentration by a greater amount than could be explained by the corresponding decrease in the superfluid density. This was in contrast with even earlier work by Esel'son et al. (1958) which concluded that the film transfer rate was proportional to the superfluid density in the presence of various concentrations of 3He. This work of van den Meijdenberg et al. (1961) confirmed and expanded on earlier work by Fairbank and Lane (1950) that the addition of 3He caused the film flow rate to be reduced.

426

R.B. HALLOCK

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Even earlier work by Daunt (1947) concluded that the presence of 3He in a reservoir did not result in the transfer of the 3He to a second reservoir by film flow; i.e. the helium three did not move with the flowing 4He film. Work by Crum et al. (1974) which went to much lower temperatures observed that the flow rate of the 4He film was depressed by the addition of 3He for the quantities of 3He used; the flow rate was observed to be proportional to the square root of the surface tension and this was preserved by the addition of the 3He except for the case of high coverages and temperatures T < 0.1 K for which the flow rate was found to be an increasing function of falling temperature, perhaps due to degeneracy effects in the 3He surface layer. Crum et al. (1975) also established the rate of 3He transfer along the 4He film. In this work, a polished stainless steel beaker was used and level differences were created and observed to flow towards equilibrium. Following the 4He film flow, a concentration difference remained and this relaxed due to the motion of 3He along the (static) 4He film. The presence of Atkins oscillations (Atkins 1950, Hallock and Flint 1973, Flint and Hallock 1974) between the two flow reservoirs did not impede the flow of the 3He. (2) Thermal conductance. Finotello et al. (1986) carried out measurements of the effective thermal conductance of thin films of pure 4He and 3He-aHe mixtures as a function of concentration and temperature using techniques developed earlier for pure 4He films (Maps and Hallock 1981, 1983, Agnolet et al. 1981, Hess and Muirhead 1982, Joseph and Gasparini 1982). The universal sharp divergence of the conductivity, K, is observed; K = h-lf(T)a2/D)exp(4~tb-lt-1/2), where t = T/To-1, h is the film thickness, f(T) is a regular function of temperature which involves the latent heat, D is the diffusion constant, a is the vortex core parameter and b is a non-universal constant. From the measurements of the conductance, the parameters Dla 2 and b were determined and found to have a dramatic dependence on concentration at low concentrations, ones for which the number of 3He crosses over the number of free vortices estimated to be present in the film (fig. 87). A non-linear shift in Tc with concentration is observed. It was suggested that the presence of 3He causes free vortices to be present in the film even for T less than To, a result consistent with the earlier persistent current decays of Ekholm and Hallock (1980). Mantz et al. (1980) have made measurements of the spreading pressure and effective thermal conductance of 3He on the surface of a 4He film. In this work, a Mylar substrate (coated with neon of unreported thickness) is in contact with enough 4He so as to form a saturated film on the Mylar. The thermal conductance between two heaters located along the Mylar was measured. The effective thermal conductance, Keff, as a function of 3He surface coverage, Ns falls inversely with an increase in Ns (fig. 88). From the results, the 3He-ripplon scattering rate is determined. (3) Wetting phenomena. From the beginning of studies of helium adsorption it was generally accepted that helium was the "universal wetting agent" which

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would wet any material with which it was brought into contact (Goodstein 1975). It was thought that the attraction of helium to itself was weaker that the attraction of helium to all other substances. Based on information on interaction potentials (Zaremba and Kohn 1976, 1977), this belief was challenged in 1991 by Cheng et al. (1991,1992) who argued that for certain of the alkali metals the electron orbital extended far enough away from the nucleus so as to prohibit a helium atom from experiencing the depth of the alkali-helium interaction. Thus, it was argued that the helium-helium interaction was stronger than, for example, the helium-cesium interaction with the resulting prediction that helium should not wet cesium. An approximate quantitative relationship is to note that wetting will take place when 2trey < -Po ~ V(z) dz. That is, wetting takes place when the cost to create the liquid-vapor and the solid-liquid interfaces is less than the energy gain by the film due to the potential, V(z), imposed by the substrate. Here, following Cheng et al. (1991), we have approximated the liquid-solid surface tension, tr]s, to be equal to the liquid-vapor surface tension trnv. In response to this prediction, Nacher and Dupont-Roc (1991) carried out an experiment which utilized a hollow glass tube, sealed with a charge of helium inside and a thin ring of cesium coating the inside perimeter of the tube at one location along its length. Measurements of the thermal conductivity of the tube as a function of temperature were made and from these it was concluded

428

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!

J

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Fig. 88. The two-dimensional thermal conductivity Keff as a function of the 3He coverage. The straight line represents KeffNs = 4.5 x 10l0 erg K-! cm-2 s-1 (from Mantz et al. 1980). that no superfluid helium film transport took place across the cesium. The conclusion was that that under certain conditions, 4He would indeed fail to wet cesium. Ketola et al. (1992) utilized a different approach and conducted a third sound experiment on a glass substrate with one-third sound driver and several detectors. The surface of the substrate between the driver and one set of detectors was composed of clean glass. Between the driver and some of the other detectors, the surface of the glass substrate was plated in situ with cesium evaporated from a dispenser. For low values of the chemical potential such that a thin 4He film resided on the clean glass, third sound propagated across the glass substrate, but no third sound signal propagated across the cesium region. A third sound pulse was seen to reflect from the edge of the cesium region. As a function of increasing chemical potential, an abrupt appearance of a third sound signal (fig. 89) was visible on the third sound detector located beyond the cesium plated region, thus demonstrating (1) the ability of 4He to fail to wet cesium "normally" and (2) a prewetting transition for 4He on cesium. More recent work by Mukherjee et al. (1992) has verified the non-wetting of 4He to a cesiated graphite substrate. More detailed measurements on the 4He-cesium system have been made by Rutledge and Taborek (1992) (Toborek and

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Fig. 89. Behavior of third sound pulse amplitudes as a function of the thickness of helium film on a glass substrate, with PIPo values shown for reference. Data are normalized to the amplitude, AD, of third sound received on a detector for the trajectory Z-D. A a is the amplitude received for the trajectory Z-A, A R is the amplitude received for the trajectory Z-edge-D for which the third sound reflects from the edge of the cesium. The insert shows the schematic of the substrate used for the measurements with the third sound driver in the center, and a cesium region to the left (from Ketola et al. 1992). Rutledge 1993a, 1993b, 1994) who, by use of quartz crystal oscillator techniques, have shown the details of the prewetting behavior (fig. 90), and have established a surface-dependent wetting temperature - 2 K for 4He on cesium. While the precise value of the wetting temperature may be somewhat uncertain at this point due to differences in the quality of the various cesium surfaces (roughness and impurities can shift and enhance the wetting behavior by changing the effective binding), there is no question about the observation of prewetting and non-wetting in these various experiments with cesium substrates. Pettersen and Saam (1993) extended the theoretical study to the case of 3He impurities in the 4He film and predicted a phase diagram with re-entrant wetting

430

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Ch. 5, w

Fig. 90. Quartz crystal frequency shifts for different temperatures as a function on 4He added to the cell. At low temperatures there is non-wetting; at high temperatures the kink in the curve illustrates the pre-wetting of the 4He to the cesium (from Rutledge and Taborek 1992). transitions. The essence of the prediction is seen by reference to the temperature dependence of the surface tension. For pure 4He Crlv is a monotonic increasing function of decreasing temperature; 4He fails to wet cesium for T < Tc. At low temperature the presence of 3He in the surface state depresses the surface tension such that the inequality is again satisfied. Thus, the wetting is predicted to be re-entrant, with the specific details sensitive to the surface tension and the energetics of the bound states of 3He on a 4He film and to the energetics of a possible substrate state which has been predicted (Pavloff and Treiner 1991b, Treiner 1993) in the vicinity of the 4He-Cs interface. Ketola and Hallock (1993a,b) carried out measurements of the adsorption of 3He-aHe mixture films as a function of temperature and concentration, X = N3/(N3 + N4), where Ni is the number of iHe atoms in the apparatus. The use of quartz crystal oscillator techniques (Lee et al. 1984) showed clearly that wetting is induced at low temperatures by the presence of the SHe. In a typical experiment a fixed (but selectable) concentration of 3He was sealed into an experimen-

Ch. 5, {}3

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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tal cell and the temperature dependence of the resonance frequency was measured for two cases: gold plated quartz and cesium coated gold plated quartz. The dependence of the crystal resonance frequency for a 3He-aHe mixture film on a gold substrate is similar to that for the pure 4He film except at the lower temperatures where a condensation of the 3He is observed. An example of this for a 3He concentration X = 0.045 is shown in fig. 91 for the case of the fl harmonic of a gold crystal. Beginning at about T = 0.8 K the enhanced mass due to the 3He in the film causes the frequency to fall as the temperature is lowered. At the lowest temperatures studied, a rise in the frequency is observed. It is believed that this is due to a shift in the location of the 3He from a general occupation of the film to finite occupation of the lowest energy states available

Fig. 91. (a) The behavior of the frequency (of the first harmonic) of a quartz crystal with gold electrodes as a function of temperature for a film which saturates at about 1.45 K in a sealed cell. (b) The behavior of the first harmonic for a crystal which has a coating of cesium (from Ketola and Hallock 1993a).

432

R.B. HALLOCK

Ch. 5, w

to the 3He, presumably those at the free surface of the film. Further study of the behavior in this low temperature regime is required. In fig. 91 the behavior of the fl harmonic of the Cs plated crystal for the case of the same 4.5% mixture film as was measured with the gold substrate is also shown. Note that in this case cooling through T= 1.6 K shows no KosterlitzThouless transition on the Cs coated substrate. Thus, the mixture film (which is nearly pure 4He at this temperature) is not thick enough at T = 1.6 K to be a superfluid. This is consistent with the absence of wetting. Cooling to T= 1.5 K shows a shift in the resonance frequency followed by a weakly temperaturedependent value of the frequency in the range 0.7 < T < 1.5 K. The shift near T = 1.5 K is present for the fl harmonic of the cesiated crystal for the case of pure 4He in the apparatus; presumably this is due to helium decoration of the non-cesiated, less sensitive, parts of the crystal. Below 0.7 K a more dramatic shift in the frequency is seen which is interpreted as the wetting of 4He induced by the presence of 3He and an increasing film concentration of 3He as the temperature continues to fall. The frequency shift is in this case larger than in the case of the gold substrate for T < 0.8 K. Further cooling results in an upturn at the lowest temperatures, a behavior which is qualitatively similar to that seen on the gold crystal. On warming hysteresis is observed. Near the point labeled Tc the warming data deviate markedly from the trajectory followed on cooling. This is taken as evidence for a metastable wetting film comprised mostly of 4He for T > Tc, perhaps augmented by an occupied substrate state which would enhance the stability relative to a pure 4He film. As warming continues, the characteristic decrease in the frequency as the amount of superfluid is reduced with an increase in temperature is observed. This signature confirms the presence of 4He in the wetting film. This behavior ends at about T = 1.5 K as evaporation becomes a dominant phenomena and the film becomes quite thin. Recent work by Stefanyi et al. (1994) has confirmed that the addition of 3He causes the wetting of 4He to take place at temperatures where it would not for pure 4He. (4) Mixtures in aerogel. Recently Kim et al. (1993) and Ma et al. (1993) have reported torsional oscillator measurements for bulk 3He-aHe mixtures in Aerogel. From these measurements they conclude that the phase diagram is distorted (fig. 92) such that there exists a miscible superfluid at high 3He concentrations. Since 4He is favored at the surface of the SiO2 strands which comprise Aerogel, one might argue that the "miscible superfluid" signal is instead due to a 4He superfluid film which decorates the strands of SiO2. In an effort to explore this possibility Kim et al. (1993) carried out experiments on pure 4He films with coverages equivalent to those which would be present if the 4He in the mixture case had phase separated against the SiO 2. The superfluid signatures as a function of temperature were entirely different than the signatures for the mixture cases. Further work of various kinds (heat capacity, NMR, ultrasonics) is un-

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derway in several laboratories to further explore this. It seems clear that very interesting behavior is present, behavior which is not yet fully understood. It is possible that the 3He-4He mixture in the Aerogel system is one in which the effect of the SiO2 is to provide a simple surface effect for which, due to the extremely small dimensions, the proximity of surfaces influences the relative distribution of the helium isotopes. This point of view has been advanced recently by Pricaupenko and Treiner (1995). An alternate possibility is that the randomness imposed by the SiO2 is the more important effect. This point of view has been taken by Maritan et al. (1992) and Falicov and Berber (1995) in the context of a random-anisotropy Blume-Emery-Griffiths model (Blume et al. 1971).

3.5. Future directions Areas of interesting possible future progress in the field of mixture films include (1) further systematic studies of the behavior of the third sound velocity as a function of 3He and 4He coverage as a function of temperature, (2) the determination of the presence or the absence of a "substrate state" for 3He near specific substrates, (3) the dynamics of 3He in the "slush" of 4He for the very thin 4He coverage case, (4) the propagation of spin waves in two-dimensional films, (5) a more detailed understanding of the behavior of bulk mixtures and mixture films in the presence of surfaces with complicating geometry such as aerogel, (6) the

434

R.B. HALLOCK

Ch. 5, w

as yet unexplored situation of mixtures in the presence of weak binding substrates such as hydrogen where unusual and incompletely understood (Gabay and Kapitulnik 1993, Zhang 1993) "transitions" have been seen by Chen et al. (1992) (Mochel and Chen 1994) as a function of temperature for pure 4He, and (7) combined studies of the susceptibility and heat capacity so as to determine the two-dimensional Fermi liquid parameters (each measurement provides a different combination of Fl s and F0a). Also interesting would be a careful study of the surface tension of bulk mixtures at low temperatures for coverages high enough to shed light on the possible existence of two states at the free surface. An exciting potential area for progress with mixture films is (8), the possibility of observing new superfluid transitions among the 3He on the 4He film surface. Bashkin (1980), studied theoretically the possibility of 3He dimers and the possible transition to the superfluid state for the 3He on the surface of a 4He film or along the core of a vortex filament. The calculation showed that under conditions of weak attraction, there are bound states of the 3He and the formation of a Bose system of 3He dimers. At lower temperature there is a phase transition in which the 3He dimers condense from a gas into a liquid and the possibility of a Kosterlitz-Thouless (1973) transition of the dimers into a superfluid state. Andreev and Konpaneets (1973) had considered a possible phase transition for 3He on the bulk surface (see also Ostgaard and Bashkin 1992). For a surface coverage of 2 x 1013cm -2 the superfluid transition was predicted to be as high as 35 mK under some conditions. In work described by Edwards (1982) this possibility for the bulk 4He surface was studied with some care. It was suggested for that case by Bashkin (1980) and separately by Antsygina and Slyusarev (1980) that the 3He might form dimers in the two-dimensional surface of the 4He film. For the case of two dimensions (and also the case for one dimension) there is always a bound state for two 3He no matter how weak the potential well. As noted by Edwards, the onset of dimerization would be observable by a decrease in the spreading pressure of the 3He and thus should be observable by measurements of the surface tension. Measurements down to 0.020 K showed no such onset of a decrease in the surface tension. For Cooper pairing in a non-zero angular momentum state, the difficulty of the potential is removed and Edwards et al. (1977) calculated that a transition temperature of 1 mK might be relevant. The numerical value depends importantly on some of the parameters which describe the 3He, such as the effective mass. Recently Kagan and Chubukov (1988,1989) (Chubukov and Kagan 1991) have predicted that 3He-aHe mixture films may undergo a phase transition at a temperature as high as 13 mK in the presence of a magnetic field. It is also possible that the "sandwich" configuration in which there is 3He decorating a surface state as well as a substrate state (perhaps adjacent to a weak binding substrate) may undergo a superfluid transition (Ostgaard and Bashkin 1992). In this case the coupling between the two two-dimensional layers might be controlled by ad-

Ch. 5, w

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

435

justments of the 4He film thickness. At this writing, few experiments with mixture films have been carried out to temperatures below about 25 mK, although several experiments have been proposed. One exception is the recent work of Shirahama and Pobell (1994) in which a torsional oscillator was used to search for a superfluid transition in the 3He on a 4He film on a silver substrate. The study was of 3He coverages of 0.00608, 0.0254 and 0.0305/~-2 on a 4He coverage of 0.158 A-2 and was carried out to a minimum temperature in the vicinity of 900 mK with no sign of a superfluid transition in the 3He.

4. Summary In this review we have attempted to bring forward a number of the important experiments and theoretical ideas which have led to our present understanding of the physics of 3He-aHe mixture films. There has been a lot of progress in this understanding, particularly in recent years. This recent progress has focused on the energetics and the properties of the 3He atoms as an impurity. The advent of systematic measurements of the heat capacity and the magnetic susceptibility and the NMR relaxation times have provided a wealth of information, most of which is now relatively well understood. The data awaits a detailed theory for the case of relatively large amounts of 3He. In the case of third sound and other measurements, the mixture picture presented by the experiments is more complicated with some apparent disparity among the experiments. There is as yet no complete theoretical understanding of the dynamics of these films as revealed by experiment. It may be the case that substrate effects are present and responsible for some of the disparity which exists. Future work is likely to expand our understanding of the dynamics of the 3He in the case of extremely thin 4He films where the 3He will become constrained by the "solid" 4He underlayer. NMR experiments are likely to lead in this, but simultaneous NMR and heat capacity work would be of value. Wetting phenomena will continue to be a rich field for the mixture film problem, particularly when difficulties in the preparation of the alkali metal substrates are overcome. A most exciting direction will be the search for the several new superfluid phases which are predicted to exist in the context of mixture films.

Acknowledgments It is a pleasure to acknowledge the numerous helpful remarks which I have received from many colleagues who have made contact with the mixture problem over the years and in the context of the preparation of this review, knowing that in the listing, inadvertent omissions may occur: G. Agnolet, R. Anderson, E.P.

436

R.B. HALLOCK

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Bashkin, M.H.W. Chan, E. Cheng, B. Clements, M. Cole, P. Crowell, J. DupontRoc, D.O. Edwards, G. Frossati, F.M. Gasparini, H. Godfrin, R.A. Guyer, J. Harrison, R. Jochemsen, E. Krotscheck, J.P. Laheurte, H.J. Lauter, C. Lusher, M.D. Miller, W.J. Mullin, P.J. Nacher, D. Osheroff, J. Parpia, N. Pavloff, M. Petersen, J.D. Reppy, R.C. Richardson, J.P. Romagnon, W. Saam, J. Saunders, M. Schick, J. Treiner, H. van Beelan, and O. Vilches, and the several collaborators with whom I have worked on various aspects of the mixture film problem" N. Alikacem, J.S. Brooks, D.T. Ekholm, F.M. Ellis, R.M. Heinrichs, R.H. Higley, B.R. Johnson, K.S. Ketola, T.A. Moreau, P.A. Sheldon, D.T. Sprague, J.M. Valles, Jr. and J. Vithayathil. In some cases this review has quoted freely from work with my collaborators and occasionally presented data prior to publication. The work on mixture films at the University of Massachusetts has been supported by the National Science Foundation, by Research Trust Funds administered by the University of Massachusetts, and facilitated by a NATO travel grant. It is also a pleasure to acknowledge the support of the J.S. Guggenheim Memorial Foundation during a portion of the time in which this review was prepared and flexible time provided by receipt of a University Faculty Fellowship. I also wish to acknowledge with gratitude the patience of the editor of this volume and other contributors during the period of gestation of this review.

References Abel, W.R., A.C. Anderson, W.C. Black and J.C. Wheatley, 1965a, Physics 1,337. Abel, W.R., A.C. Anderson, W.C. Black and J.C. Wheatley, 1965b, Phys. Rev. Lett. 15, 875. Abel, W.R., A.C. Anderson, W.C. Black and J.C. Wheatley, 1966, Phys. Rev. Lett. 16, 273. Agnolet, G., S.L. Teitel and J.D. Reppy, 1981, Phys. Rev. Lett., 47, 1537. Agnolet, G., D. McQueeney and J.D. Reppy, 1984, in: Proc. 17th Int. Conf. on Low Temperature Physics, eds U. Eckern et al. (North-Holland, Amsterdam)p. 965. Agnolet, G., D.F. McQueeney and J.D. Reppy, 1989, Phys. Rev. B 39, 8934. Ahonen, A.I., T. Kodama, M. Krusius, M.A. Paalanan, R.C. Richardson, W. Schoepe and Y. Takano, 1975, J. Phys. C 9, 1665. Allen, J.F. and A.D. Misener, 1938, Proc. Cambridge Philos. Soc. 34, 299. Alikacem, N., D.T. Sprague and R.B. Hallock, 1991, Phys. Rev. Lett. 67, 2501. Alikacem, N., R.B. Hallock, R.H. Higley and D.T. Sprague, 1992, J. Low Temp. Phys. 87, 279. Anderson, C.H. and E.S. Sabisky, 1970, Phys. Rev. Lett. 24, 1049. Anderson, R.H. and M.D. Miller, 1989, Phys. Rev. B 40, 2109. Anderson, R.H. and M.D. Miller, 1992, J. Low Temp. Phys. 89, 665. Anderson, R.H. and M.D. Miller, 1993, Phys. Rev. B 48, 10426. Anderson, R.H. and M.D. Miller, 1995, in: Recent Progress in Many Body Theories, Vol. 4, eds E. Schachinger et al. (Plenum Press) p. 79. Andreev, A.F., 1966, Zh. Eksp. Teor. Fiz. 50, 1415; [Sov. Phys. JETP 23, 939]. Andreev, A.F. and D.A. Konpaneets, 1972, Zh. Eksp. Teor. Fiz. 61, 2459; [Sov. Phys. JETP 34, 1316]. Andreev, A.F. and D.A. Konpaneets, 1973, Zh. Eksp. Teor. Fiz. Pis'ma Red. 17, 379; [JETP Lett. 17, 268].

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Ch. 5

PROPERTIES OF MULTILAYER 3He-4He MIXTURE FILMS

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AUTHOR INDEX Abel, W.R. 329, 436 Abragam, A. 253, 314 Abraham, F.F. 234, 265, 280, 285, 298,

Anufriyev, Yu.D., see Alvesalo, T.A. 163, 167,210

Archie, C.N., see Halperin, W.P. 207, 208,

314

210

Agnolet, G. 419, 420, 426, 436 Agnolet, G., see McQueeney, D. 420, 422,

Ashcrofl, N.A. 219, 314 Atkins, K.R. 324, 325, 331,387, 388, 415, 426, 437 Atkins, K.R., see Everett, C.W.F. 388, 439 Avenel, O. 133, 155 Avenel, O., see Varoquaux, E. 9, 68, 128,

441

Agnolet, G., see Wang, C.L. 334, 443 Ahonen, A.I. 8, 66, 330, 436 Alexander, S. 280, 314 Alikacem, N. 364, 367, 370-374, 377, 380,

158

Averbuch, P.G. 274, 314 Awschalom, D.D. 9, 66, 170, 201,210 Aziz, R.A. 234, 315

436

Alikacem, N., see Sheldon, P.A. 377, 400, 404, 4 1 1 , 4 4 2

Alikacem, N., see Sprague, D.T. 256, 310, 319, 343, 365, 366, 368, 369, 371,373375, 377-386, 413,442 Allen, J.F. 3, 66, 324, 436 Allum, D.R. 4, 7, 12, 14, 16-18, 20, 22, 24, 30, 33-35, 66 Allum, D.R., see Ellis, T. 4, 56, 57, 59, 67 Alvesalo, T.A. 163, 167, 210 Alvesalo, T.A., see Halperin, W.P. 207, 208, 210 Amend, B., see Eska, G. 97, 109, 156 Anderson, A.C., see Abel, W.R. 329, 436 Anderson, C.H. 334, 353,436 Anderson, C.H., see Sabisky, E.S. 334,

Bablidze, R.A., see Esel'son, B.N. 425, 439

Bacon, G.E. 261,315 Bailin, D. 165, 210 Bakharev, O.N., see Egorov, A.V. 256, 316 Balfour, L., see Landau, J. 334, 440 Balian, R. 72, 155 Banavar, J.R., see Maritan, A. 433, 441 Bardeen, J. 7 1 , 1 5 5 Bartolac, T., see Bozler, H.M. 308, 315 Barton, G. 72, 155 Bashkin, E.P. 434, 437 Bashkin, E.P., see Ostgaard, E. 434, 441 Batrouni, G.G., see Zimanyi, G.T. 344, 443 B~iuerle, Ch., see Godfrin, H. 255, 305, 306, 309, 317 Baym, G. 357, 437 B6al-Monod, M.T. 276, 315 Belic, A. 310, 315 Bergmann, D. 388, 390, 413,414, 415, 437 Bergmann, D., see Kagiwada, R.S. 388,

442

Anderson, P.W. 72, 136, 154, 161, 210 Anderson, R.H. 360, 379, 380, 436 Andreev, A.F. 80, 155, 310, 314, 324, 325, 326, 333, 376, 434, 436 Andreeva, O., see Keshishev, K.O. 334, 440

Andrei, E.Y. 7, 66 Andronikashvili, E.L. 416, 437 Andronikashvily, E.L., see Hakonen, P.J. 134, 157 Antsygina, 434 437

440

Berker, A.N. 335, 336, 392, 419, 437 Berker, A.N., see Falicov, A. 433,439 Bemier, M.E.R., see Bozler, H.M. 109, 155 445

446

AUTHOR INDEX

Berthold, J. 416, 4 3 7 Berthold, J., see Smith, E.N. 418, 424, 4 4 2 Betts, D.S., see Wilks, J. 8, 68 Bhattacharya, R. 221,315 Bhattacharyya, B.K. 252, 310, 315, 342, 345-354, 376, 378, 382, 413, 4 3 7 Bhattacharyya, B.K., see Chen, T.P. 345, 438

Bhattacharyya, B.K., see Gasparini, F.M. 336, 347,439 Bird, R.B., see Hirshfelder, J.O. 234, 317 Birgenau, R.J., see Strandburg, K.J. 290, 319

Bishop, D.J. 416, 417, 420, 422, 4 3 7 Bishop, D.J., see Berthold, J. 416, 4 3 7 Bishop, D.J., see Smith, E.N. 418, 424, 4 4 2 Bishop, J.H., see Black, W.C. 329, 4 3 7 Black, W.C. 329, 4 3 7 Black, W.C., see Abel, W.R. 329, 4 3 6 Bloom, P. 347,437 Blume, M. 433,437 Boato, G. 232, 233, 3 1 5 Bogoliubov, N. 3, 66 Bogoliubov, N.N. 76, 155 Boldarev, S.T., see Zinov'eva, K.N. 325, 326, 4 4 3 Bomchil, G., see White, J.W. 261,320 Borovikov, A.P. 331,437 Borovik-Romanov, A.S. 74, 85, 98, 104, 111, 113, 125, 126, 129, 133, 136, 138, 155

Bossy, J., see Godfrin, H. 255, 305, 306, 309, 3 1 7 Bossy, J., see Morhard, K.-D. 271,274, 318

Bouchaud, J.P. 276, 315 Bowley, R.M. 4, 13, 18, 23, 27, 30, 33, 5356, 59, 6 7 Bowley, R.M., see Allum, D.R. 4, 7, 12, 14, 16, 17, 20, 22, 24, 30, 34, 35, 66 Bowley, R.M., see Ellis, T. 4, 21, 53, 56, 57, 59, 6 7 Bowley, R.M., see Hendry, P.C. 4, 13, 6 7 Bowley, R.M., see Nancolas, G.G. 4, 60, 68

Bowley, R.M., see Sheard, F.W. 35-37, 68 Boyd, S.T.P. 88, 146, 148, 149, 155, 168170, 202, 2 1 0 Boyd, S.T.P., see Hahn, I. 183, 210 Bozler, H., see Hahn, I. 183,210

Bozler, H.M. 109, 155, 258, 308, 315 Bradley, D.I. 93, 155 Brami, B. 231,234, 277-279, 306, 310, 315

Brami, B., see Joly, F. 230-232, 318 Bretz, M. 217, 219, 237, 240, 241,245, 246, 248, 252, 265, 271-273, 279-281, 290, 3 1 5 Bretz, M., see Chae, H.B. 252, 280, 3 1 5 Bretz, M., see Duta, P. 265, 316 Bretz, M., see Polanco, S.E. 252, 3 1 9 Brewer, D.F. 231,237,310, 315, 329, 330, 335, 355, 358, 385,437 Brewer, D.F., see Evenson, A. 329, 4 3 9 Brewster, P.M., see Black, W.C. 329, 4 3 7 Brinkman, W.F., see Anderson, P.W. 161, 210

Brinkman, W.F., see Osheroff, D.D. 173, 211

Brooks, J.S. 6, 7, 51, 53, 67, 414, 416, 4 3 7 Brooks, J.S., see Ellis, F.M. 394--399, 414416, 4 3 9 Broughton, J.Q., see Abraham, F.F. 234, 265, 280, 285, 298, 3 1 4 Brouwer, P.W. 415, 4 3 7 Brown, D.S., see Stephens, P.W. 268, 319 Bruch, L.W., see Gottlieb, J.M. 283, 317 Brundobler, S. 28, 31, 35, 38, 6 7 Buchal, C. 188, 2 1 0 Buchanan, D.S. 168, 200, 210 Buchanan, D.S., see Swift, G.W. 163, 168, 169, 171,176, 177, 201,211 Buchenau, H. 334, 4 3 7 Bugl, P., see Fujita, S. 220, 317 Bunkov, Yu.M. 74, 76, 77, 85, 88, 89, 91, 93, 97, 103, 107, 111-116, 118, 119, 121-123, 140, 141,144, 145, 149, 151153, 155, 156, 256, 3 1 5 Bunkov, Yu.M., see Borovik-Romanov, A.S. 74, 85, 98, 104, 111, 113, 125, 126, 129, 133, 136, 138, 155 Bunkov, Yu.M., see Godfrin, H. 255, 299, 306, 309, 3 1 7 Bunkov, Yu.M., see Hakonen, P.J. 91, 157

Bunkov, Yu.M., see Ikkala, O.T. 91,134, 157

Bunkov, Yu.M., see Korhonen, J.S. 113, 157

Burmistrov, S.N. 334,

437

AUTHOR INDEX Busch, P.A., see Greywall, D.S. 246, 248, 251,252, 271,273-275, 278, 283-287, 292, 293,296, 304, 310-312, 317, 352, 439

Candela, D. 90, 110, 118, 1 5 6 Candela, D., see Swanson, D.R. 356, 4 4 2 Cantini, P., see Boato, G. 232, 233, 3 1 5 Cardy, J.L. 335, 336, 419, 4 3 7 Carley, J.S., see Aziz, R.A. 234, 315 Carlos, W.E. 232, 233, 234, 270, 3 1 5 Carlos, W.E., see Derry, G. 231,316 Carmi, Y. 334, 4 3 7 Carneiro, G.M. 232, 261,283, 315, 335, 437

Cameiro, G.M., see Taub, H. 269, 319 Carneiro, K. 232, 261,269, 315 Cameiro, K., see Taub, H. 269, 319 Castaing, B. 276, 315 Castelijns, C.A.M. 97, 1 5 6 Ceperley, D.M., see Wagner, M. 312, 320

Chae, H.B. 252, 280, 315 Chan, M.H.W. 237, 315 Chan, M.H.W., see Kim, S.B. 432, 433, 440

Chan, M.H.W., see Ma, J. 432, 441 Chan, M.H.W., see Mukherjee, S.K. 428, 441

Chan, M.H.W., see Zimmerli, G. 247, 252, 270, 312, 3 2 0 Chang, C.C. 324, 327, 4 3 7 Chapellier, M. 256, 315 Chapellier, M., see Godfrin, H. 308, 317 Chapellier, M., see Schuhl, A. 256, 319 Chen, M.T. 434, 4 3 7 Chen, M.T., see Mochel, J.M.,434, 441 Chen, T.P. 345, 437, 4 3 8 Cheng, E. 230, 235, 295, 312, 315, 334, 427, 4 3 8 Chester, M. 390, 416, 4 3 8 Chester, M., see Webster, E. 392, 418, 419, 443

Chester, M., see Webster, G.D.L. 419, 4 4 3 Chubukov, A.V. 434, 4 3 8 Chubukov, A.V., see Kagan, M.Y. 434, 440

Chung, S. 231,315 Cieplak, M., see Maritan, A. 433,441 Clark, W.G., see Godfrin, H. 308, 317

447

Clements, B.E. 235, 236, 310, 312, 315, 343, 344, 360, 4 3 8 Clements, B.E., see Saarela, M. 343,442 Close, J.D., see Davis, J.C. 137, 1 5 6 Coates, K.F., see Castelijns, C.A.M. 97, 156

Cohen, M., see Chang, C.C. 324, 327, 4 3 7 Cohen, M.H., see Springett, B.E. 21, 68 Cole, M.W. 230, 231, 271,315 Cole, M.W., see Carlos, W.E. 232-234, 270,315

Cole, M.W., see Cheng, E. 235, 295, 312, 315, 334, 427, 4 3 8 Cole, M.W., see lhm, G. 231,318 Cole, M.W., see Vidali, G. 230, 234, 3 2 0 Colella, N.J., see Strandburg, K.J. 290, 319 Collan, H.K., see Alvesalo, T.A. 163, 167, 210

Combescot, R., see Candela, D. 90, 110, 156

Cooper, L.N., see Bardeen, J. 71,155 Corruccini, L.R. 73, 98, 103, 109, 139, 156, 196, 2 1 0 Coulomb, J.P., see Marti, C. 269, 318 Cowan, B.P. 253, 256, 259, 271,285, 290, 316

Cowan, B.P., see Lusher, C.P. 255, 278, 291,294, 295, 300, 301,308, 311,318, 375, 441 Cowan, B.P., see Mohandas, P. 270, 318 Cowan, B.P., see Mullin, W.J. 258, 318 Cowan, B.P., see Owers-Bradley, J.R. 253, 254, 3 1 9 Cowan, B.P., see Saunders, J. 255, 271, 274, 283, 295, 311, 319 Cowan, B.P., see Siqueira, M. 244, 255, 260, 272, 278, 284, 285,295, 296, 300, 308,319

Cowley, R.A. 58, 6 7 Cowley, R.A., see Smith, A.J. 56-58, 68 Crawford, G.P. 345,438 Creswell, D.J., see Brewer, D.F. 310, 315, 330, 355,437 Creswell, D.J., see Mullin, W.J. 258, 318 Croset, B., see Marti, C. 269, 318 Cross, M.C., see Osheroff, D.D. 163, 168, 182, 1 8 3 , 2 1 1 Crowell, P.A. 270, 312, 316, 343,418, 421,424, 4 3 8 Crowell, P.A., see Zimanyi, G.T. 344, 4 4 3

448

AUTHOR INDEX

Crum, D. 328, 426, 4 3 8 Cunsolo, S. 61, 6 7 Curtiss, C.F., see Hirshfelder, J.O. 234, 317

DiPirro, M.J. 310, 316, 336, 345, 4 3 8 DiPirro, M.J., see Bhattacharyya, B.K. 310, 315, 342, 345-351,376, 378, 382, 413,

D'Amico, K.L., see Strandburg, K.J. 290,

DiPirro, M.J., see Chen, T.P. 345, 437, 4 3 8 DiPirro, M.J., see Gasparini, F.M. 336, 347,439 Dmitriev, V.V., see Borovik-Romanov, A.S. 85, 98, 104, 111, 113, 125, 126, 129, 133, 136, 138, 155 Dmitriev, V.V., see Bunkov, Yu.M. 85, 93, 103, 111,114, 121,122, 140, 141,144, 145, 151,153, 156, 256, 3 1 5 Dmitriev, V.V., see Kondo, Y. 153, 157 Dmitfiev, V.V., see Korhonen, J.S. 113,

437 319

Dahm, A.J. 333, 4 3 8 Dalfovo, F., see Belic, A. 310, 315 Dalfovo, F.Z. 328, 333, 334, 342, 4 3 8 Dandache, H., see Laheurte, J.P. 398, 4 4 0 Dash, J.G. 216, 217, 246, 248, 253,280, 316

Dash, J.G., see Bretz, M. 219, 237, 240, 241,245, 246, 248, 252, 265, 271-273, 279-281,290, 315 Dash, J.G., see Ecke, R.E. 252, 288, 289, 316

Dash, J.G., see Hurlbut, S.B. 289, 318 Dash, J.G., see Kjems, J.K. 218, 261,262, 268, 3 1 8 Dash, J.G., see Motteler, F.C. 280, 287, 318

Dash, J.G., see Puff, R.D. 398, 442 Daunt, J.G. 3, 67, 253, 316, 426, 4 3 8 Daunt, J . G . , s e e Ezell, E.F. 231,316 Daunt, J.G., see Goellner, G.J. 247, 317 Daunt, J.G., see Hegde, S.G. 222, 227,228, 247, 253, 254, 257, 259, 260, 317 Daunt, J.G., see Hickemell, D.C. 228, 258, 317

Daunt, J.G., see Husa, D.L. 258, 318 Davis, J.C. 137, 156 de Boer, J. 234, 315 de Mello, E.V.L., see Carneiro, G.M. 335, 437

de Oliveira, M.J. 247, 319 de Souza, E.P., see Rapp, R.E. 220, 222, 319

den Nijs, M. 280, 287, 288, 316 Denenstein, A., see Everett, C.W.F. 388, 439

Derry, G. 231,316 DeSorbo, W. 220, 222, 316 Deutsch, M., see Lurio, L.B. 270, 318, 364, 441

deWaard, A., see Borovik-Romanov, A.S. 133, 155 Dillon, L.D. 226-229, 316 Dillon, L.D., see Rapp, R.E. 222, 225, 226, 319

157

Doake, C.S.M. 3, 61, 62, 64, 6 7 Doane, J.W., see Crawford, G.P. 345, 4 3 8 Donnelly, R.J. 8, 9, 51, 52, 56, 57, 67 Donnelly, R.J., see Brooks, J.S. 6, 7, 51, 53,67

Donnelly, R.J., see Muirhead, C.M. 334, 441

Donnelly, R.J., see Springett, B.E. 54, 68 Downs, J.L. 390, 4 3 8 Draisma, W.A. 404, 409, 411,438 Draisma, W.A., see Brouwer, P.W. 415, 437

Druist, D.P., see Mukherjee, S.K. 428, 441 Dubovskii, L.B., see Burmistrov, S.N. 334, 437

Dumesh. B.S., see Borovik-Romanov A.S. 74, 104, 155 Dumesh. B.S., see Bunkov Yu.M. 97, 155 Duncan, T.M. 253, 316 Dupont-Roc, J. 378, 4 3 8 Dupont-Roc, J., see Cheng, E. 235, 295, 315, 334, 4 3 8 Dupont-Roc, J., see Himbert, M. 367,440 Dupont-Roc, J., see Nacher, P.J. 427, 441 Duta, P. 265, 316 Dutta, A.K., see Bhattacharrya, R. 221, 315

Duval, X. 216, 316 Duval, X., see Thomy, A. 216, 319 Dybowski, C., see Duncan, T.M. 253,316 Dyugaev, A.M. 273, 274, 316 Eckardt, J.R. 324, 326, 4 3 8 Eckardt, J.R., see Edwards, D.O. 324, 326, 327,438

AUTHOR INDEX Ecke, R.E. 252, 288, 289, 316 Eden, V.L. 62, 64, 65, 67 Edwards, D.O. 310, 316, 324, 326, 327, 342, 434, 438 Edwards, D.O., see Candela, D. 90, 110, 118, 156 Edwards, D.O., see Crum, D. 426, 438 Edwards, D.O., see Eckardt, J.R. 324, 326, 438

Edwards, D.O., see Guo, H.N. 326, 365, 439

Edwards, D.O., see Hoyt, R.F. 118, 157 Edwards, D.O., see Landau, J. 334, 440 Edwards, D.O., see Mantz, I.B. 328, 336, 342, 426, 428,441 Edwards, D.O., see Sherrill, D.S. 337-339, 353, 404, 405, 406, 423,442 Edwards, D.O., see Swanson, D.R. 356, 442

Eggenkamp, M.E.W., see Draisma, W.A. 404, 409, 411,438 Eggington, M.A. 330, 438 Egorov, A.V. 256, 316 Einzel, D. 128, 129, 156 Einzel, D., see Bunkov, Yu.M. 111, 114, 151,156 Ekholm, D.T. 397,425, 426, 438, 439 Elgin, R.L. 231,246, 247, 253, 271,288, 290, 302, 316 Ellenson, W.D., see Carneiro, K. 261, 315 Ellenson, W.D., see Nielsen, M. 261, 281, 318

Ellis, F.M. 364, 394-401,405,409, 410, 414--416, 439 Ellis, F.M., see Brooks, J.S. 414, 416, 437 Ellis, T. 3, 4, 21, 36, 41, 42, 44, 45, 47-51, 53, 56, 57, 59, 64, 67 Ellis, T., see Nancolas, G.G. 4, 60, 68 EI-Nasr, L.A., see Cowan, B.P. 285, 290, 316

Elser, V., see Abraham, F.F. 234, 298, 314 Emery, V.J., see Blume, M. 433, 437 Epstein, J.L. 236, 312, 316, 342, 381,382, 439

Epstein, J.L., see Clements, B.E. 343,344, 438

Epstein, J.L., see Krotscheck, E. 310, 318, 341,342, 358, 359, 361,374, 440 Esel'son, B.N. 425,439 Eska, G. 97, 109, 139, 156

449

Evenson, A. 329, 439 Evenson, A., see Brewer, D.F. 335, 358, 437

Everett, C.W.F. 388, 439 Ezell, E.F. 231,316 Fairbank, H.A. 425, 439 Falicov, A. 433,439 Fantoni, S., see Belic, A. 310, 315 Fardis, M., see Cowan, B.P. 285, 290, 316

Fatouros, P.P., see Eckardt, J.R. 326, 438 Fatouros, P.P., see Edwards, D.O. 324, 326, 327, 438 Feder, J.D., see Edwards, D.O. 434, 438 Feher, A., see Bunkov, Yu.M. 114, 156 Feile, R. 254, 261, 281,289, 316 Feile, R., see Lauter, H.J. 261, 281,318 Felcher, G.P., see Boato, G. 232, 233,315 Feng, Y.P. 173, 210 Fetter, A.L. 9, 13, 67, 134, 156 Feynman, R.P. 3, 67 Feynmann, R.P. 134, 156 Finotello, D. 426, 427,439 Finotello, D., see Crawford, G.P. 345, 438 Finotello, D., see Steele, L.M. 335, 442 Fisher, S.N. 8, 67 Fisher, S.N., see Bunkov, Yu.M. 76, 77, 89, 107, 113-116, 118, 122, 156 Fishman R.S. 119 156 Hint, E.B. 426, 439 Hint, E.B., see Hallock, R.B. 426, 439 Fomin, I.A. 74, 81, 83, 85, 91, 92, 95, 98, 100, 119, 120, 123, 128, 134, 142, 143, 156, 157

Fomin, I.A., see Bunkov, Yu.M. 114, 156 Fomin, I.A., see Golo, V.L. 85, 157 Forbert, H., see Clements, B.E. 360, 438 Fozooni, P., see Lee, M.J. 430, 440 Franco, H. 219, 239, 241,255-257, 296, 298, 299, 302, 308, 316 Frank, V.L.P. 261,282, 283, 316 Frank, V.L.P., see Lauter, H.J. 238, 261, 262, 264, 266, 268, 281-284, 286, 291, 301,306, 312, 318 Frankl, D.L., see Chung, S. 231,315 Frankl, D.R., see Cole, M.W. 231, 271, 315

Frankl, D.R., see Derry, G. 231,316 Fraser, J.C., see Kagiwada, R.S. 388, 440

450

AUTHOR INDEX

Fraser, J.C., see Rudnick, I. 388, 4 4 2 Freddi, A. 243, 3 1 6 Fredkin, D.R., see B6al-Monod, M.T. 276, 315

Freedman, R. 357, 4 3 9 Freeman, M.R. 165, 210, 332, 4 3 9 Freimuth, H. 286, 317 Freimuth, H., see Schildeberg, H.P. 218, 262, 265, 268, 319 Freimuth, H., see Wiechert, H. 262, 3 2 0 Frossati, G., see Brouwer, P.W. 415, 4 3 7 Frossati, G., see Draisma, W.A. 404, 409, 411,438

Frossati, G., see Godfrin, H. 308, 317 Fujita, S. 220, 317 Fukase, T., see Koike, Y. 225-227, 318 Fukuyama, H. 163, 168, 175, 176, 2 1 0 Fukuyama, H., see Schiffer, P. 163, 172, 173, 179, 203, 208, 2 1 1 , 2 3 9 , 244, 245, 255, 258, 300, 303, 3 1 9 Gabay, M. 434, 4 3 9 Gaeta, A.A., see Chen, T.P. 345,437 Ganguli, N. 228, 317 Garibashvily, D.I., see Hakonen, P.J. 134, 157

Gasparini, F.M. 336, 347, 352, 4 3 9 Gasparini, F.M., see Bhattacharyya, B.K. 252, 310, 315, 342, 345-354, 376, 378, 382, 413,437 Gasparini, F.M., see Chen, T.P. 345, 437, 438

Gasparini, F.M., see DiPirro, M.J. 310, 316, 336, 345,438 Gasparini, F.M., see Eckardt, J.R. 324, 326, 4 3 8 Gasparini, F.M., see Edwards, D.O. 324, 326, 327, 4 3 8 Gasparini, F.M., see FinoteUo, D. 426, 427, 439

Gasparini, F.M., see Joseph, R.A. 426, 4 4 0 Gasparini, F.M., see Wang, X. 342, 381, 382, 422-424, 4 4 3 Gazo, E., see Bunkov, Yu.M. 114, 156 Gearhart, Jr., C.A. 330, 4 3 9 Gerhold, J. 55, 6 7 Germain, R.S., see Freeman, M.R. 165, 210, 332, 4 3 9 Giinnetta R.W. 98 1 5 7 Ginzburg V.L. 91 157

Glaberson, W.l., see Andrei, E.Y. 7, 6 6 Glaberson, W.J., see Ostermeier, R.M. 334, 368, 441 Glaberson, W.J., see Tiwari, R. 352, 425, 443

Gladkov, S.O., see Bunkov, Yu.M. 1 5 5 Glyde, H.R. 265, 317 Godfrin, H. 218, 238, 239, 242, 244, 247, 255, 256, 268, 284, 285, 290, 294, 298, 299, 305,306, 309, 317, 323,439 Godfrin, H., see Franco, H. 239, 241,255257, 296, 298, 299, 302, 308, 316 Godfrin, H., see Frank, V.L.P. 261,282, 283,316

Godfrin, H., see Lauter, H.J. 238, 261,262, 264, 266, 268, 281-284, 286, 291,296, 301,306, 312, 3 1 8 Godfrin, H., see Morhard, K.-D. 271,274, 318

Godfrin, H., see Osheroff, D.D. 162, 211 Godfrin, H., see Rapp, R.E. 219, 222, 225, 226, 239, 241,247, 255, 256, 258-260, 284, 285, 295, 319 Godfrin, H., see Tiby, C. 261,320 Godfrin, H., see Viana, R. 221,222, 3 2 0 Godshalk, K.M. 345,439 Godshalk, K.M., see Smith, D.T. 424, 4 4 2 Goellner, G.J. 247, 317 Golo, V.L. 85, 1 5 7 Golo, V.L., see Bunkov, Yu.M. 119, 155 Goodstein, D. 427, 4 3 9 Goodstein, D.L., see Cole, M.W. 231, 271, 315

Goodstein, D.L., see Elgin, R.L. 231,246, 247, 253, 271,289, 290, 302, 316 Goodstein, D.L., see Taborek, P. 247, 319 Gottlieb, J.M. 283, 317 Gould, C.M. 153, 157, 171,183,210 Gould, C.M., see Gully, W.Y. 109, 1 5 7 Gould, C.M., see Hahn, I. 183,210 Greif, J.M., see Elgin, R.L. 231, 271,316 Greywall, D.S. 162, 207, 208, 210, 219, 237-239, 241,242, 245, 246, 248, 251253, 255, 271-275, 277, 278, 281,283288, 290-294, 296--300, 302-312, 317, 352, 4 3 9 Gribbon, P.W.F., see Doake, C.S.M. 3, 61, 62, 64, 6 7 Griffin, A., see Weling, F. 265, 3 2 0 Griffiths, R.B., see Blume, M. 433, 4 3 7

AUTHOR INDEX Griffiths, R.B., see de Oliveira, M.J. 247, 319

Grimal, R., see Laheurte, J.P. 330, 4 4 0 Grimes, L.G., see Jackson, L.C. 33 l, 4 4 0 Grimmer, 253 317 Guenault, A.M., see Bradley, D.I. 93, 155

Guenault, A.M., see Bunkov, Yu.M. 76, 77, 89, 107, 113-116, 118, 122, 1 5 6 Guenault, A.M., see Castelijns, C.A.M. 97, 156

Gu6nault, A.M., see Fisher, S.N. 8, 6 7 Guidi, C., see Boato, G. 232, 233, 315 Gully, W.Y. 109, 1 5 7 Gully, W.Y., see Bozler, H.M. 109, 155 Guo, H.N. 326, 365,439 Guyer, R.A. 236, 283, 31 l, 317, 328, 342, 352, 373,398, 400, 405,409, 410, 423, 439

Guyer, R.A., see Ellis, F.M. 397-399, 405, 409, 410, 4 3 9 Guyon, E., see Rudnick, I. 388, 4 4 2 Haensel, R., see Lauter, H.J. 261,262, 281, 296, 3 1 8 Haensel, R., see Schildeberg, H.P. 218, 262, 265,268, 319 Hahn, I. 162, 183,210 Hakonen, P.J. 91,134, 157, 163, 167, 176, 210

Hakonen, P.J., see Bunkov, Yu.M. 91,152, 155

Hakonen, P.J., see lkkala, O.T. 91, 134, 157

Hallock, R.B. 310, 317, 356, 357, 363, 364, 388-390, 400, 402-404, 408,409, 411,412, 426, 4 3 9 Hallock, R.B., see Alikacem, N. 364, 367, 370-374, 377,380, 4 3 6 Hallock, R.B., see Brooks, J.S. 414, 416, 437

Hallock, R.B., see Ekholm, D.T. 397, 425, 426, 4 3 8 HaUock, R.B., see Ellis, F.M. 364, 394401,405, 409, 410, 414--416, 4 3 9 Hallock, R.B., see Flint, E.B. 426, 4 3 9 Hallock, R.B., see Godshalk, K.M. 345, 439

Hallock, R.B., see Heinrichs, R.M. 402, 411,412, 4 4 0

451

Hallock, R.B., see Higley, R.H. 311, 317, 360--363,374, 380, 382, 4 4 0 Hallock, R.B., see Ketola, K.S. 235,318, 428--431,440 Hallock, R.B., see Maps, J. 426, 441 Hallock, R.B., see Sheldon, P.A. 377, 379, 380, 400, 404, 409, 411-414, 4 4 2 Hallock, R.B., see Smith, D.T. 424, 4 4 2 Hallock, R.B., see Sprague, D.T. 256, 310, 319, 343, 365, 366, 368, 369, 371,373375, 377-386, 413, 4 4 2 Hallock, R.B., see Valles, Jr., J.M. 343, 353, 354, 357-360, 374, 394, 404-406, 443

Halperin, W.P. 207, 208, 210 Halperin, W.P., see Hensley, H.H. 2 I0 Halpin-Healy, T. 286, 317 Hammel, P.C. 256, 317 Hamot, P., see Hensley, H.H. 210 Haradze, G.A., see Ikkala, O.T. 91, 134, 157

Harrison, J.P., see Steel, S.C. 332, 4 4 2 Hata, T., see Kim, D. 332, 4 4 0 Haubach, W.J., see Watkins, R.A. 243, 320

Havens-Sacco, S.M. 357, 4 3 9 Hawthorne, D.L., see Nunes, Jr., G. 91, 157

Heft, A. 90, 110, 156 Hegde, S.G. 222, 226-228, 247, 253, 254, 257, 259, 260, 317 Hegde, S.G., see Daunt, J.G. 253, 3 1 6 Heinrichs, R.M. 402, 403, 404, 408, 410412, 439, 4 4 0 Heinrichs, R.M., see Valles, Jr., J.M. 343, 353, 354, 394, 404, 405,406, 4 4 3 Hendry, P.C. 4, 13, 40, 6 7 Hendry, P.C., see Williams, C.D.H. 63-65, 68

Heremans, J. 229, 317 Hering, S.V. 248, 28 l, 285,288-290, 317 Hess, G.B. 426, 4 4 0 Hickemell, D.C. 218, 228, 252, 258, 317, 335, 336, 4 4 0 Hickemell, D.C., see Bretz, M. 219, 237, 240, 241,245, 246, 248, 252, 265, 271273,279-28 l, 290, 315 Hickemell, D.C., see Husa, D.L. 258, 3 1 8 Higley, R.H. 31 l, 317, 360-363,374, 380, 382, 383,440

452

AUTHOR INDEX

Higley, R.H., see Alikacem, N. 364, 367, 436

Higley, R.H., see Valles, Jr., J.M. 357-360, 374, 4 4 3 Hildreth, M.D., see Schiffer, P. 163, 172, 179,211

Himbert, M. 367, 4 4 0 Himbert, M., see Dupont-Roc, J. 378, 4 3 8 Hirai, A., see lshikawa, O. 85, 113, 1 5 7 Hirayama, H., see Nelson, W.R. 184, 185, 211

Hirshfelder, J.O. 234, 317 Holland, M.C. 226, 3 1 8 Holland, M.C., see Klein, C.A. 226, 318 Horn, P.M., see Stephens, P.W. 268, 319 Horn, P.M., see Strandburg, K.J. 290, 319 Hoyt, R.F. 118, 1 5 7 Hrubesh, L.W., see Ma, J. 432, 441 Huff, G.B., see Bretz, M. 248, 290, 315 Hurlbut, S.B. 289, 318 Husa, D.L. 258, 3 1 8 Husa, D.L., see Hickemell, D.C. 228, 258, 317

Hussein, A., see Cowan, B.P. 253, 290, 316

lannacchione, G.S., see Crawford, G.P. 345,438 lhas, G., see Varoquaux, E. 128, 1 5 8 lhas, G.G. 61--63, 6 7 lhas, G.G., see Sanders, T.M. 65, 68 lhm, G. 231,318 lhm, G., see Vidali, G. 230, 3 2 0 lkkala, O.T. 91,134, 1 5 7 lkkala, O.T., see Hakonen, P.J. 134, 1 5 7 lordanskii, S.V. 18, 26, 38, 6 7 lshikawa, O. 85, 113, 1 5 7 lshikawa, O., see Kim, D. 332, 4 4 0 Ishimoto, H., see Fukuyama, H. 163, 168, 175, 176, 2 1 0 Islander, S.T., see Hakonen, P.J. 134, 1 5 7 Islander, S.T., see lkkala, O.T. 91,134, 1 5 7 Ivantsov, V.G., see Esel'son, B.N. 425, 4 3 9 lvlev, B.I. 128, 1 5 7 lwata, T., see Nihira, T. 226, 319 Jackson, L.C. 331,440 Janu, Z., see Korhonen, J.S. 113, 1 5 7 Jarvis, J.F., see Thompson, J.R. 355,443 Jelatis, G.J. 393, 4 4 0

Jewell, C.I., see Ellis, T. 3, 6 7 Jiang, T. 334, 4 4 0 Jin, C., see Nunes, Jr., G. 91,157 Jochemsen, R., see Brouwer, P.W. 415, 437

Jochemsen, R., see Draisma, W.A. 404, 409, 411,438 Johnson, B.R., see Valles, Jr., J.M. 357360, 374, 4 4 3 Joly, F. 230, 231,232, 3 1 8 Joly, F., see Brami, B. 231,234, 277, 278, 306, 310, 3 1 5 Jones, H., see Allen, J.F. 3, 6 6 Jonsson, H., see Ruiz, J.C. 231, 319 Jortner, J., see Springett, B.E. 21, 6 8 Jose, J.V. 335, 4 4 0 Joseph, R.A. 426, 4 4 0 Josephson, B.D. 132, 1 5 7 Kadanoff, L.P. 335,440 Kadanoff, L.P., see Jose, J.V. 335, 4 4 0 Kagan, M.Y. 434, 4 4 0 Kagan, M.Y., see Chubukov, A.V. 434, 438

Kagiwada, R.S. 388, 4 4 0 Kajiwada, R.S., see Downs, J.L. 390 Kajiwada, R.S., see Rudnick, I. 388, 4 4 2 Kapitulnik, A., see Gabay, M. 434, 4 3 9 Kapitza, P. 3, 6 7 Kapitza, P.L. 7 1 , 1 5 7 Kara, A., see Chung, S. 231,315 Kardar, M., see Halpin-Healy, T. 286, 317 Keesom, P.H., see van der Hoeven, J.C. 219-222, 3 2 0 Keesom, W.H. 3, 6 7 Keith, V., see Bradley, D.I. 93, 155 Keller, W.E. 8, 6 7 Kelly, B.T. 219, 3 1 8 Kennedy, C.J., see Bradley, D.I. 93, 1 5 5 Kennedy, C.J., see Bunkov, Yu.M. 76, 77, 114-116, 118, 122, 1 5 6 Kennedy, C.J., see Fisher, S.N. 8, 6 7 Kent, A.J., see Cowan, B.P. 271,316 Keshishev, K.O. 334, 4 4 0 Ketola, K.S. 235,318, 428-431,440 Ketterson, J.B., see Kuchnir, M. 334, 4 4 0 Keyston, J.R.G. 329, 4 4 0 Keyston, J.R.G., see Brewer, D.F. 329, 4 3 7 Keyston, J.R.G., see Laheurte, J.P. 329, 440

AUTHOR INDEX Kim, D. 332, 4 4 0 Kim, H.-Y., see Vidali, G. 230, 3 2 0 Kim, N., see Movshovich, R. 118, 1 5 7 Kim, S.B. 432, 433,440 Kirkpatrick, S., see Jose, J.V. 335, 4 4 0 Kjems, J.K. 218, 261,262, 268, 318 Kjems, J.K., see Taub, H. 269, 319 Klein, C.A. 226, 318, 3 2 0 Klein, C.A., see Holland, M.C. 226, 318 Klein, J.R., see Cole, M.W. 230, 315 Klein, J.R., see Vidali, G. 230, 3 2 0 Kleinberg, R.L. 165, 167, 210 Kleinberg, R.L., see Sager, R.E. 139, 1 5 7 Klier, J., see Stefanyi, P. 432, 4 4 2 Knuth, E.L., see Buchenau, H. 334, 4 3 7 Kobayashi, K. 220, 318 Kodama, T., see Ahonen, A.I. 330, 4 3 6 Kodama, T., see Kim, D. 332, 4 4 0 Kohn, W. 230, 318 Kohn, W., see Krotscheck, E. 328, 340, 440

Kohn, W., see Zaremba, E. 427, 4 4 3 Koike, Y. 225-227, 318 Kojima, H., see Kim, D. 332, 4 4 0 Kojima, H., see Paulson, D.H. 163, 211 Kokko, J., see Ahonen, A.I. 8, 6 6 Kolableva, S.L., see Egorov, A.V. 256, 316 Kondo, Y. 153, 1 5 7 Kondo, Y., see Bunkov, Yu.M. 153, 1 5 6 Kondo, Y., see Korhonen, J.S. 113, 1 5 7 Konpaneets, D.A., see Andreev, A.F. 326, 434, 4 3 6 Kopnin, N.B., see Ivlev, B.I. 128, 1 5 7 Korhonen, J.S. 113, 153, 1 5 7 Korhonen, J.S., see Bunkov, Yu.M. 153, 156

Korhonen, J.S., see Kondo, Y. 153, 1 5 7 Korshunov, see Th6ron R. 288, 319 Kosowsky, S.D., see Lurio, L.B. 270, 318, 364, 441 Kosterlitz, J.M. 334, 416, 434, 4 4 0 Kosterlitz, J.M., see Nelson, D.R. 334, 425, 441

Krishnan, K.S., see Ganguli, N. 228, 317 Krotscheck, E. 310, 318, 328, 340-342, 344, 358, 359, 361,374, 379, 4 4 0 Krotscheck, E., see Clements, B.E. 235, 236, 310, 312, 315, 343, 344, 360, 4 3 8 Krotscheck, E., see Epstein, J.L. 236, 312, 316, 342, 381,382, 4 3 9

453

Krotscheck, E., see Saarela, M. 343, 4 4 2 Krusius, M. 203, 210 Krusius, M., see Ahonen, A.I. 330, 4 3 6 Krusius, M., see Bunkov, Yu.M. 91, 153, 155, 156

Krusius, M., see Hakonen, P.J. 91,157, 163, 167, 176, 2 1 0 Krusius, M., see Kondo, Y. 153, 1 5 7 Krusius, M., see Korhonen, J.S. 113, 1 5 7 Kubota, M., see Buchal, C. 188, 210 Kuchnir, M. 334, 4 4 0 Kuper, C.G. 21, 6 7 Kurkin, M.I., see Bunkov, Yu.M. 97, 155 Kusmartsev, F.V., see Saarela, M. 343, 442

Laheurte, J.P. 329, 391-394, 398, 404, 408, 414-416, 424, 4 4 0 Laheurte, J.P., see Chester, M. 390, 4 3 8 Laheurte, J.P., see Keyston, J.R.G. 329, 440

Laheurte, J.P., see Noiray, J.C. 406-408, 441

Laheurte, J.P., see Romagnan, J.P. 330, 421,422, 4 4 2 Landau, J. 334, 4 4 0 Landau, L. 247, 318 Landau, L.D. 3, 4, 5, 12, 65, 67, 68, 75, 157

Landau, L.D., see Ginzburg V.L. 9 1 , 1 5 7 Lane, C.T., see Fairbank, H.A. 425, 4 3 9 Lauter, H.J. 238, 261,262, 264, 266, 268, 281-284, 286, 291,296, 301,306, 312, 318

Lauter, H.J., see Clements, B.E. 235, 236, 310, 312, 315, 343,360, 4 3 8 Lauter, H.J., see Feile, R. 254, 261, 281, 289,316

Lauter, H.J., see Franco, H. 241,255-257, 316

Lauter, H.J., see Frank, V.L.P. 261,282, 283,316

Lauter, H.J., see Freimuth, H. 286, 317 Lauter, H.J., see Godfrin, H. 255, 285,298, 299, 3 1 7 Lauter, H.J., see Schildeberg, H.P. 218, 262, 265, 268, 319 Lauter, H.J., see Tiby, C. 261,320 Lauter, H.J., see Wiechert, H. 261,262, 320

Lawson, N.S., see Hendry, P.C. 4, 13, 6 7

454

AUTHOR INDEX

le Pair, C., see van den Meijdenberg, C.J.N. 425,441 Lee, D.M. 74, 1 2 4 , 1 5 7 , 1 6 1 , 2 1 1 Lee, D.M., see Bozler, H.M. 109, 155 Lee, D.M., see Giinnetta R.W. 98, 157 Lee, D.M., see Gully, W.Y. 109, 157 Lee, D.M., see Movshovich, R. 118, 157 Lee, D.M., see Nunes, Jr., G. 91,157 Lee, D.M., see Osheroff, D.D. 71,157 Lee, M.J. 430, 4 4 0 Lee, Y . , see Hensley, H.H. 2 I0 Leemann, Ch., see Th6ron, R. 288, 319 Leggett, A.J. 3, 68, 72, 73, 79, 108, 146, 157, 161,163, 165-168, 180, 182, 185, 191-195, 200-202, 211 Leggett, A.J., see Modgil, D. 197, 211 Leiderer, P., see Frank, V.L.P. 261,282, 283,316

Leiderer, P., see Lauter, H.J. 238, 262, 264, 266, 268, 281-284, 286, 291, 301,306, 312,318

Lekner, J. 326, 404, 4 4 0 Leman, A.A., see Golo, V.L. 85, 157 Lemer, E., see Daunt, J.G. 253,316 Lemer, E., see Goellner, G.J. 247,317 Lemer, E., see Hegde, S.G. 222, 227, 228, 257, 3 1 7 Lemer, E., see Rapp, R.E. 220, 222, 319 Lemer, E., see Viana, R. 221,222, 320 Leung, P.W., see Abraham, F.F. 234, 298, 314

Lhuillier, C., see Bouchaud, J.P. 276, 315 Lhuillier, C., see Brami, B. 231,234, 277, 278, 306, 310, 315 Lhuillier, C., see Joly, F. 230, 231,232, 318

Li, Z.R., see Chan, M.H.W. 237, 315 Lifshitz, E., see Landau, L. 247, 318 Lifshitz, E.M. 38, 68 Likharev, K.K. 133, 157 Lindqvist, T., see Wennerstr/Sm, P. 252, 320

Lipson, S., see Carmi, Y. 334, 4 3 7 Lipson, S., see Landau, J. 334, 4 4 0 London, F. 3, 68 Lounasmaa, O.V., see Alvesalo, T.A. 163, 167,210

Lounasmaa, O.V., see Hakonen, P.J. 134, 157

Love, A., see Bailin, D. 165, 210

Lovesey, S.W. 261,318 Luey, K., see Bozler, H.M. 308, 315 Lurio, L.B. 270, 318, 364, 441 Lusher, C.P. 255, 278, 291,294, 295,299301,308, 311,318, 375, 441 Lusher, C.P., see Mohandas, P. 270, 318 Lusher, C.P., see Saunders, J. 255, 271, 274, 283,295, 3 1 1 , 3 1 9 Lusher, C.P., see Siqueira, M. 244, 255, 260, 272, 278, 284, 285, 295,296, 300, 308, 319 Luszczynski, see Grimmer 253, 317 Ma, J. 432, 441 Ma, J., see Kim, S.B. 432, 433,440 Ma, S.K., see B6al-Monod, M.T. 276, 315 Maattan, L., see Landau, J. 334, 4 4 0 Macwood, G.E., see Keesom, W.H. 3, 6 7 Maegawa, S., see Schuhl, A. 256, 319 Maki, K. 85, 157 Maid, K., see Vollhardt, D. 129, 158 Makrotsieva, V., see Borovik-Romanov, A.S. 133, 155 Maksimchuk, T.V., see Bunkov, Yu.M. 97, 155

Mantz, I.B. 328, 336, 341,426, 428, 441 Maps, J. 426, 441 Marchenko, V.I., see Andreev, A.F. 80, 155

Maritan, A. 433, 441 Markelov, A.M., see Bunkov, Yu.M. 111, 114, 151,156 Markelov, A.V. 133, 157 Markkula, T.K., see Hakonen, P.J. 134, 157

Marlow, I. 261,320 Martel, P., see Smith, A.J. 56-58, 68 Marti, C. 269, 318 Martinoli, P., see Th6ron, R. 288, 319 Massey, W.E. 327,441 Massey, W.E., see Woo, C.W. 327,443 Massey, W.E., see Yim, M.B. 327, 443 Masuhara, N., see Candela, D. 90, 110, 118, 156 Maynard, J. 51, 68 Maynard, J.D., see Jelatis, G.J. 393, 4 4 0 Mc Tague, J.P., see Carneiro, K. 261, 315

McCall, K.R., see Guyer, R.A. 311, 317, 373,439

AUTHOR INDEX McClean, E.O., see Hickemell, D.C. 335, 336, 4 4 0 McClintock, P.V.E. 4, 40, 6 8 McClintock, P.V.E., see Allum, D.R. 4, 7, 12, 14, 16-18, 20, 22, 24, 30, 33-35, 6 6 McClintock, P.V.E., see Bowley, R.M. 4, 13,67

McClintock, P.V.E.,

see

Eden, V.L. 62, 64,

65,67

McClintock, P.V.E., see Ellis, T. 3, 4, 21, 36, 41, 42, 44, 45, 47-51, 53, 56, 57, 59, 64, 6 7 McClintock, P.V.E., see Hendry, P.C. 4, 13, 40, 6 7 McClintock, P.V.E., see Nancolas, G.G. 4, 60, 6 8 McClintock, P.V.E., see Phillips, A. 3, 10, 11,68

McClintock, P.V.E., see Williams, C.D.H. 63-65, 6 8 McClure, J.W. 219, 3 1 8 McClure, J.W., see van der Hoeven, J.C. 221,222, 3 2 0 McConville, G.T. 243,318 McConville, G.T., see Aziz, R.A. 234, 315 McCourt, F.R.W., see Aziz, R.A. 234, 315 McLean, E.O., see Bretz, M. 219, 237, 240, 241,245, 246, 248, 252, 265, 271273, 279-281,290, 315 McLean, E.O., see Hickemell, D.C. 218, 252, 3 1 7 McQueeney, D. 420, 422, 441 McQueeney, D., see Agnolet, G. 419, 420, 436

McTague, J.P.,

see

Nielsen, M. 261, 281,

318

McTague, J.P., see Taub, H. 269, 319 Meisel, M.W., see Schuhl, A. 256, 310 Mendelssohn, K., see Daunt, J.G. 3, 6 7 Mermin, N.D. 166, 170, 211 Mermin, N.D., see Ashcroft, N.A. 219, 314 Meyer, H., see Ramm, H. 355,442 Meyer, H., see Thompson, J.R. 355, 4 4 3 Meyer, L. 3, 9, 10, 6 8 Meyer, L., see Neeper, D.A. 10, 12, 13, 68 Meyer, L., see Reif, F. 25, 6 8 Michels, A., see de Boer, J. 234, 315 Migdal, A.A. 335,441 Migone, A.D., see Chan, M.H.W. 237, 315 Miki, T., see Kim, D. 332, 4 4 0

455

Miller, I.E., see Bradley, D.I. 93, 155 Miller, M.D. 277, 310, 318, 335, 336, 441 Miller, M.D., see Anderson, R.H. 535, 360, 380, 4 3 6 Miller, M.D., see Ellis, F.M. 397-399, 405, 409, 410, 4 3 9 Miller, M.D., see Guyer, R.A. 328, 342, 398, 405, 409, 410, 4 3 9 Mineev, V.P., see Hakonen, P.J. 91,157 Miner, K.D., see Chan, M.H.W. 237,315 Misener, A.D., see Allen, J.F. 324, 4 3 6 Mistura, G., see Zimmerli, G. 247,252, 270, 312, 3 2 0 Miyake, K. 383, 384, 441 Mizusaki, T., see Ishikawa, O. 85, 113, 1 5 7 Mochel, J. 434, 441 Mochel, J., see Chen, M.T. 434, 4 3 7 Mochel, J., see Ratnam, B. 389, 4 4 2 Mochrie, S.G., see Strandburg, K.J. 290, 319

Modena, I., see Freddi, A. 243,316 Modgil, D. 197, 211 Mohandas, P. 270, 318 Mon, K.K. 336, 337, 423,441 Monarkha, Y.P. 398, 410, 441 Moncton, D.E., see Stephens, P.W. 268, 319

Moncton, D.E., see Strandburg, K.J. 290, 319

Moore, M.A., see Barton, G. 72, 155 Moore, M.A., see Eggington, M.A. 330, 438

Morel, P., see Anderson, P.W. 72, 154 Morelli, D.T., see Heremans, J. 229, 317 Morhard, K.-D. 271,274, 3 1 8 Morhard, K.-D., see Godfrin, H. 255, 299, 305,306, 309, 317 Morita, S., see Koike, Y. 225-227, 318 Moss, F.E., see Bowley, R.M. 4, 13, 6 7 Mota, A.C., see Black, W.C. 329, 4 3 7 Motteler, F.C. 280, 285, 287, 3 1 8 Movshovich, R. 118, 1 5 7 Mueller, R.M., see Buchal, C. 188, 210 Muirhead, C.M. 334, 426, 441 Muirhead, R.J., see Hess, G.B. 426, 4 4 0 Mukharsky, Y.M., see Bunkov, Yu.M. 256, 315

Mukharsky, Yu.M., see Borovik-Romanov, A.S. 85, 98, 104, 1I1, 113, 125, 126, 129, 133, 136, 138, 155

456

AUTHOR INDEX

Mukharsky, Yu.M., see Bunkov, Yu.M. 85, 93, 103, 111, 114, 121,122, 140, 141, 144, 145, 151,153, 1 5 6 Mukharsky, Yu.M., see Kondo, Y. 153, 157

Mukharsky, Yu.M., see Korhonen, J.S. 113,157

Mukherjee, S.K. 428, 441 MuUin, W.J. 258, 318 MuUin, W.J., see Cowan, B.P. 253, 259, 316

Mullin, W.J., see Miyake, K. 383, 384, 441 Mussett, S.G., see Bradley, D.I. 93, 155 Mussett, S.G., see Castelijns, C.A.M. 97, 156

Nacher, P.J. 427,441 Nain, V.P.S., see Aziz, R.A. 234, 315 Nakanomyo, T., see Koike, Y. 225, 226, 227,318

Nancolas, G.G. 4, 60, 68 Nancolas, G.G., see Bowley, R.M. 4, 13, 67 Narahara, Y., see Atkins, K.R. 324, 325, 437

Nayak, V.U., see Mantz, I.B. 426, 428, 441 Neeper, D.A. 10, 12, 13, 68 Nelson, D.R. 334, 425,441 Nelson, D.R., see Berker, A.N. 335,336, 392, 419, 4 3 7 Nelson, D.R., see Jose, J.V. 335, 4 4 0 Nelson, W.R. 184, 185, 211 Neumaier, K., see Eska, G. 109, 139, 156 Nichols, G.E., see DeSorbo, W. 220, 222, 316

Nielsen, M. 261, 281,284, 318 Nielsen, M., see Duta, P. 265, 316 Nihira, T. 226, 319 Noiray, J.C. 406, 407, 408, 441 Noiray, J.C., see Laheurte, J.P. 391-394, 398, 404, 408, 414--416, 424, 4 4 0 Noiray, J.C., see Romagnan, J.P. 330, 394, 421,422, 4 4 2 Northby, J.A., see Buchenau, H. 334, 4 3 7 Northby, J.A., see Jiang, T. 334, 4 4 0 Nosanow, L.H., see Miller, M.D. 277, 310, 318

Novaco, A.D., see Kjems, J.K. 218, 261, 262, 268, 3 1 8 Nozieres, P., see Pines, D. 357,441

Nunes, Jr., G. 91,157 Nyak, V.S., see Edwards, D.O. 434, 4 3 8 Nyeki, J., see Bunkov, Yu.M. 114, 1 5 6 O'Keefe, M.T., see Schiffer, P. 163, 172, 173, 179, 203, 208, 2 1 1 , 2 3 9 , 244, 245, 255, 258, 300, 303, 319 Ocko, B.M., see Lurio, L.B. 270, 318, 364, 441

Oestereich, T., see Webster, E. 419, 4 4 3 Oestereich, T., see Webster, G.D.L. 419, 443

Ogawa, S., see Fukuyama, H. 163, 168, 175, 176, 2 1 0 Ohmi, T. 110, 112, 157, 334, 441 Olk, C.H., see Heremans, J. 229, 317 Ondris-Crawford, R., see Crawford, G.P. 345,438 Onsager, L. 134, 157 Osheroff, D.D. 71, 123, 157, 162, 163, 165, 167, 168, 173, 182, 183, 211,239, 244, 245, 255, 258, 300, 303, 3 1 9 Osheroff, D.D., see Corruccini, L.R. 73, 98, 103, 109, 139, 156, 196, 2 1 0 Osheroff, D.D., see Feng, Y.P. 173, 2 I0 Osheroff, D.D., see Godfrin, H. 218, 242, 244, 247, 255, 268, 284, 285, 299, 3 1 7 Osheroff, D.D., see Schiffer, P. 163, 172, 173, 179, 203, 208, 21 I, 239, 244, 245, 255, 258, 300, 303, 3 1 9 Ostermeier, R.M. 334, 368, 441 Ostgaard, E. 434, 441 Ostlund, S., see Berker, A.N. 335,437 Owers-Bradley, J.R. 253, 319, 367, 441 Owers-Bradley, J.R., see Widom, A. 253, 254, 289, 3 2 0 Paalanen, M.A., see Ahonen, A.I. 8, 66, 330, 4 3 6 Packard, R.E., see Davis, J.C. 137, 1 5 6 Packard, R.E., see Williams, G.A. 334, 443

Papoular, M., see Romagnan, J.P. 421, 422, 4 4 2 Parpia, J.M., see Tholen, S.M. 332, 4 4 3 Particle Data Book, 185 211 Parts, U., see Bunkov, Yu.M. 153, 1 5 6 Passell, L., see Carneiro, K. 232, 261,269, 315

Passell, L., see Duta, P. 265, 316

AUTHOR INDEX Passell, L., see Kjems, J.K. 218, 261,262, 268, 3 1 8 Passell, L., see Taub, H. 269, 319 Passell, L., see Thomlinson, W. 261,319 Paulson, D.H. 163, 211 Paulson, D.N., see Kleinberg, R.L. 165, 167,210

Pavloff, N. 328, 344, 365, 378, 380, 409, 411,430, 441 Pavloff, N., see Dupont-Roc, J. 378, 4 3 8 Pedroni, P., see Ramm, H. 355, 4 4 2 Peierls, R.E., see Dash, J.G. 253, 3 1 6 Perkins, D.K. 190, 204, 211 Pershan, P.S., see Lurio, L.B. 270, 318, 364, 4 4 1 Peshkov, V.P. 331,441 Peshkov, V.P., see Borovikov, A.P. 331, 437

Pethick, C., see Baym, G. 357, 4 3 7 Pettersen, M.S. 429, 4 4 1 Phillips, A. 3, 10, 11, 6 8 Phillips, A., see Allum, D.R. 4, 7, 12, 14, 16-18, 20, 22, 24, 30, 33, 34, 66 Pickett, G.R. 93, 97, 1 5 7 Pickett, G.R., see Bradley, D.I. 93, 155 Pickett, G.R., see Bunkov, Yu.M. 76, 77, 89, 107, 113-116, 118, 122, 1 5 6 Pickett, G.R., see Castelijns, C.A.M. 97, 156

Pickett, G.R., see Fisher, S.N. 8, 6 7 Pines, D. 357,441 Pinkse, P.W.H., see Draisma, W.A. 404, 409, 411,438 Pinske, P.W.H., see Brouwer, P.W. 415, 437

Piott, J.E., see Hickemell, D.C. 228, 258, 317

Pitaevskii, L.P. 58, 6 8 Pitaevskii, L.P., see Lifshitz, E.M. 38, 6 8 Pitaevsky, P.L. 7 1 , 1 5 7 Pobell, F., see Buchal, C. 188, 210 Pobell, F., see Shirahama, K. 435,442 Poddyakova, E.V., see Borovik-Romanov, A.S. 104, 126, 129, 133, 155 Polanco, S.E. 252, 319 Pollock, F., see Ezell, E.F. 231,316 Polturak, E., see Carmi, Y. 334, 4 3 7 Pratt, W.D., see Bradley, D.I. 93, 1 5 5 Pricaupenko, L. 235, 236, 312, 319, 379, 433, 441

457

Puff, R.D. 398, 4 4 2 Pumam, A.M., see Nunes, Jr., G. 91,157 Putnam, F.A., see Berker, A.N. 335, 4 3 7 Putterman, S.J. 334, 388, 4 4 2 Qian, G.X., see Krotscheck, E. 328, 340, 440

Quaterman, J.H., see Polanco, S.E. 252, 319

Rabedeau, T.A. 269, 284, 319 Rabedeau, T.A., see Lurio, L.B. 270, 318, 364, 441 Rainer, D., see Serene, J.W. 77, 1 5 8 Ramm, H. 355, 4 4 2 Ramm, H., see Thompson, J.R. 355,443 Rapp, R.E. 219, 220, 222, 225, 226, 239, 241,247, 255, 256, 258-260, 284, 285, 295,319

Rapp, R.E., see Dillon, L.D. 226-229, 316 Rapp, R.E., see Franco, H. 239, 255, 256, 296, 298, 299, 302, 308, 316 Rapp, R.E., see Godfrin, H. 238, 239, 255, 284, 285, 298, 299, 305,306, 309, 317

Rapp, R.E., see Viana, R. 221,222, 3 2 0 Rasmussen, F.B., see Halperin, W.P. 207, 208,210

Ratnam, B. 389, 4 4 2 Rayfield, G.W. 3, 9, 11, 19, 61, 68, 333, 442

Reif, F. 25, 68 Reif, F., see M e y e r , L. 3, 9, 1O, 68 Reif, F., see Rayfield, G.W. 9, 61, 68 Reppy, J.D., see Agnolet, G. 419, 420, 426, 436

Reppy, J.D., see Berthold, J. 416, 4 3 7 Reppy, J.D., see Bishop, D.J. 416, 417, 420, 422, 4 3 7 Reppy, J.D., see Crowell, P.A. 270, 312, 316, 343, 418, 424, 4 3 8 Reppy, J.D., see McQueeney, D. 420, 422, 441

Reppy, J.D., see Smith, E.N. 418, 424, 4 4 2 Retz, P.W., see Lee, M.J. 430, 4 4 0 Richards, M.D., see Owers-Bradley, J.R. 253,254, 319 Richards, M.G. 237, 253,285, 3 1 9 Richards, M.G., see Cowan, B.P. 253,259, 316

458

AUTHOR INDEX

Richards, M.G., see Owers-Bradley, J.R. 367, 4 4 1 Richards, M.G., see Widom, A. 253, 254, 289, 3 2 0 Richardson, R.C. 256, 3 1 9 Richardson, R.C., see Ahonen, A.I. 8, 66, 330, 4 3 6 Richardson, R.C., see Bozler, H.M. 109, 155

Richardson, R.C., see Freeman, M.R. 165, 210, 332, 4 3 9 Richardson, R.C., see Gully, W.Y. 109, 157

Richardson, R.C., see Halperin, W.P. 207, 208,210

Richardson, R.C., see Hammel, P.C. 256, 317

Richardson, R.C., see

Lee,

D.M. 74, 124,

157, 1 6 1 , 2 1 1

Richardson, R.C., see Osheroff, D.D. 71, 157, 173,211 Roach, P.R., see Kuchnir, M. 334, 4 4 0 Roberts, P.H., see Donnelly, R.J. 52, 6 7 Rodgers, D.W.O., see Nelson, W.R. 184, 185,211

Roesler, J.M., see Chen, M.T. 434, 4 3 7 Rolt, J.S., see Brewer, D.F. 355, 385, 4 3 7 Romagnan, J.P. 330, 394, 421,422, 4 4 2 Romagnan, J.P., see Chester, M. 390, 4 3 8 Romagnan, J.P., see Laheurte, J.P. 330, 391-394, 398, 404, 408, 414--416, 424, 440

Romagnan, J.P., see Noiray, J.C. 406, 407, 408, 441 Rosenbaum, R., see Williams, G.A. 393, 443

Rossi, B. 1 8 5 , 2 1 1 Roth, J.A., see Jelatis, G.J. 393,440 Roubeau, P.M., see Hakonen, P.J. 134, 1 5 7 Rouille, G., see Laheurte, J.P. 414--416, 440

Rudnick, I. 388,442 Rudnick, I., see Kagiwada, R.S. 388, 4 4 0 Rudnick, I., see Williams, G.A. 393, 4 4 3 Ruel, R., see Osheroff, D.D. 162, 211 Ruel, R.R., see Godfrin, H. 218, 242, 244, 247, 255, 268, 299, 3 1 7 Ruiz, J.C. 231,319 Ruland, W. 265, 319 Rutledge, J.E. 428, 430, 4 4 2

Rutledge, J.E., see Taborek, P. 429, 4 4 2 Saam, W. 327, 4 4 2 Saam, W., see Cheng, E. 334, 427, 4 3 8 Saam, W., see Edwards, D.O. 326, 327, 342, 4 3 8 Saam, W., see Mon, K.K. 336, 337, 423, 441

Saam, W., see Pettersen, M.S. 429, 441 Saam, W., see Romagnan, J.P. 330, 4 4 2 Saam, W.F., see Cheng, E. 235, 295, 312, 315

Saam, W.F., see Edwards, D.O. 310, 3 1 6 Saarela, M. 343,442 Saarela, M., see Clements, B.E. 343,344, 360, 4 3 8 Saarela, M., see Epstein, J.L. 312, 316, 342, 4 3 9 Saarela, M., see Krotscheck, E. 310, 318, 341,342, 358, 359, 361,374, 4 4 0 Sabisky, E.S. 334, 4 4 2 Sabisky, E.S., see Anderson, C.H. 334, 4 3 6 Sachrajda, A., see Steel, S.C. 332, 4 4 2 Sager, R.E. 139, 1 5 7 Sager, R.E., see Webb, R.A. 84, 85, 109, 148, 151,158 Saloheimo, K.M., see Hakonen, P.J. 134, 157

Salomaa, M.M., see Hakonen, P.J. 91,157, 163, 167, 176, 2 1 0 Salomelin, R., see Varoquaux, E. 128, 158 Sander, L.M., see Uher, C. 222, 225, 226, 320

Sanders, T.M. 65, 68 Sanders, T.M., see lhas, G.G. 61-63, 6 7 Sarwinski, R.E., see Crum, D. 426, 4 3 8 Sarwinski, R.E., see Guo, H.N. 326, 365, 439

Sasaki, Y., see lshikawa, O. 85, 113, 1 5 7 Satoh, K. 253, 254, 258, 285, 319 Sauls, J.A., see Fishman R.S. 119, 1 5 6 Saunders, J. 255, 271,274, 283, 295, 311, 319

Saunders, J., see Lusher, C.P. 255, 278, 291,294, 295, 300, 301,308, 311,318, 375,441 Saunders, J., see Mohandas, P. 270, 318 Saunders, J., see Siqueira, M. 244, 255, 260, 272, 278, 284, 285,295,296, 300, 308, 3 1 9

AUTHOR INDEX Scalapino, D.J., see Cardy, J.L. 335, 336, 419, 4 3 7 Scalettar, R.T., see Zimanyi, G.T. 344, 443 Schaab, J., see Tiby, C. 261,320 Schiffer, P. 163, 171-173, 177, 179, 180, 185, 203,208, 2 1 1 , 2 3 9 , 244, 245, 255, 258, 300, 303, 319 Schiffer, P., see Feng, Y.P. 173,210 Schildberg, H.P. 218, 262, 265,268, 319 Schildberg, H.P., see Freimuth, H. 286, 317

Schildberg, H.P., 281,296, 318 Schildberg, H.P.,

see

Lauter, H.J. 261,262,

see

Wiechert, H. 262,

320

Schoepe, W., see Ahonen, A.I. 8, 66, 330, 436

Scholz, H.N. 118, 157 Scholz, H.N., see Hoyt, R.F. 118, 157 Scholz, H.R. 182, 196, 209, 211 Schopohl, N. 77, 157 Schopohl, N., see Vollhardt, D. 129, 158 Schriffer, J.R., see Bardeen, J. 71,155 Schuhl, A. 256, 319 Schwarz, K.W. 9, 14, 21, 68 Schwarz, K.W., see Awschalom, D.D. 9, 66, 170, 201,210 Scoles, D., see lhm, G. 231,318 Scoles, G., see Ruiz, J.C. 231, 319 Serene, J.W. 77, 158 Sergatskov, D.A., see Borovik-Romanov, A.S. 104, 126, 129, 133, 155 Sergatskov, D.A., see Bunkov, Yu.M. 114, 156, 256, 315 Sheard, F.W. 35, 36, 37, 68 Sheard, F.W., see Bowley, R.M. 4, 13, 18, 23, 27, 30, 33, 53-56, 59, 6 7 Sheldon, P.A. 377, 379, 380, 400, 404, 409, 410-414, 442 Sheldon, P.A., see Sprague, D.T. 256, 310, 319, 343, 365, 366, 368, 369, 371,373375, 377-386, 413,442 Shen, S.Y., see Eckardt, J.R. 324, 326, 438

Shen, S.Y., see Edwards, D.O. 324, 326, 327, 4 3 8 Sherrill, D.S. 337-339, 353,404--406, 423, 442

Sherrill, D.S., see Candela, D. 90, 110, 118, 156

459

Shih, Y.M. 327, 442 Shirahama, K. 435,442 Shoepe, W., see Eska, G. 109, 139, 156 Shu, Q.-S., see Ecke, R.E. 252, 3 1 6 Shvets, A.D., see Esel'son, B.N. 425, 4 3 9 Silin, V.P. 89, 158 Silvera, I.F., see Lurio, L.B. 270, 318, 364, 441

Simola, J.T., see Hakonen, P.J. 91,157, 163, 167, 176, 210 Simond, J.B., see Th6ron, R. 288, 319 Singh, A.D. 324, 442 Sinha, S.K. 261,265,319 Sinha, S.K., see Duta, P. 265,316 Siqueira, M. 244, 255, 260, 272, 278, 284, 285, 295,296, 300, 308, 319 Slack, G.A. 226, 319 Slichter, C.P. 253,319 Slyusarev, see Antsygina 434, 4 3 7 Smith, A.J. 56, 57, 58, 68 Smith, D.T. 424, 442 Smith, E.N. 418, 424, 442 Smith, E.N., see Giinnetta R.W. 98, 157 Solla, S.A., see Strandburg, K.J. 290, 319 Solokov, S.S., see Monarkha, Y.P. 398, 410, 441 Sonin, E.B. 91, 92, 93, 158 Sonin, E.B., see Kondo, Y. 153, 157 Sornette, D., see Laheurte, J.P. 394, 408, 440

Sornette, D., see Noiray, J.C. 406, 407, 408, 441 Spain, I.L. 219, 221,319 Specht, E.D., see Strandburg, K.J. 290, 319 Sprague, D.T. 256, 310, 319, 343, 366, 365, 368, 369, 371,373,374, 375, 377386, 413,442 Sprague, D.T., see Alikacem, N. 364, 367, 370-374, 377, 380, 4 3 6 Sprague, D.T., see Guyer, R.A. 311, 317, 373, 4 3 9 Sprague, D.T., see Higley, R.H. 311, 317, 360-363, 374, 380, 382, 4 4 0 Sprague, D.T., see Sheldon, P.A. 377,400, 404, 410, 411,442 Springett, B.E. 21, 54, 68 Stamp, P.C.E., see Bowley, R.M. 4, 13, 6 7 Steel, S.C. 332, 442 Steele, L.M. 335,442 Steele, L.M., see Crawford, G.P. 345, 4 3 8

460

AUTHOR INDEX

Steele, W.A. 229, 230, 319 Stefanyi, P. 432, 4 4 2 Stephens, P.W. 265, 268, 319 Stewart, G.A., see Dash, J.G. 253,316 Strandburg, K.J. 290, 319 Straub, W.D., see Holland, M.C. 226, 318 Stringari, S. 342, 380, 4 4 2 Stringari, S., see Belie, A. 310, 315 Stringari, S., see Dalfovo, F.Z. 328, 342, 438

Stiihn, B., see Wiechert, H. 26 I, 3 2 0 Sugawara, T., see Satoh, K. 253,254, 258, 285,319

Sullivan, T.S., s e e Ecke, R.E. 252, 3 1 6 Suter, R.M., see Strandburg, K.J. 290, 319 Swanson, D.R. 356, 4 4 2 Swift, G.W. 163, 168, 169, 171,176, 177, 201,211 Swift, G.W., see Boyd, S.T.P. 88, 146, 148, 149, 155, 168-170, 202, 2 1 0 Swift, G.W., see Buchanan, D.S. 163, 168, 200,210

Swift, M.R., see Maritan, A. 433,441

Teitel, S.L., see Agnolet, G. 426, 4 3 6 Tenner, A.G. 191, 192, 204, 211 Th6ron, R. 288, 319 Theumann, A., see B6al-Monod, M.T. 276, 315

Tholen, S.M. 332, 4 4 3 Thomas, R.K. 261, 319 Thomas, R.K., see White, J.W. 261,320 Thomlinson, W. 261, 319 Thomlinson, W., see Carneiro, K. 232, 261,269, 315 Thompson, A.L., see Brewer, D.F. 330, 335, 355, 358, 4 3 7 Thompson, A.L., see Evenson, A. 329, 439

Thompson, A.L., see Owers-Bradley, J.R. 367, 441 Thompson, J.R. 355,443 Thompson, J.R., see Ramm, H. 355, 4 4 2 Thompson, K. 231,320 Thomson, A.L., see Bozler, H.M. 308, 315 Thomson, A.L., see Brewer, D.F. 310, 315 Thomson, A.L., see Cowan, B.P. 253,259, 316

Tabony, J. 253, 3 1 9 Taborek, P. 247, 3 1 9 , 428, 4 4 2 , 4 4 3 Taborek, P., s e e Rutledge, J.E. 428, 430, 442

Taeonis, K.W., see van den Meijdenberg, C.J.N. 425,441 Tagirov, M.S., see Egorov, A.V. 256, 316 Takagi, S., see Leggett, A.J. 108, 1 5 7 Takano, Y., see Ahonen, A.I, 8, 6 6 Takano, Y., see Ahonen, A.I. 330, 4 3 6 Takken, E.H. 7, 18, 28, 40, 6 8 Tan, H.T., see Massey, W.E. 327,441 Tan, H.T., see Woo, C.W. 327,443 Tarvin, J.A., see Thomlinson, W. 261, 319 Tatarek, R., see Boato, G. 232, 233, 315 Taub, H. 269, 319 Taub, H., see Carneiro, K. 232, 261,269, 315

Taub, H., see Kjems, J.K. 218, 261,262, 268, 3 1 8 Taub, H., see Lauter, H.J. 261,264, 266, 281,318 Taylor, W.L., see Aziz, R.A. 234, 315 Taylor, W.L., see Watkins, R.A. 243, 3 2 0 Tazaki, T., see Fukuyama, H. 163, 168, 175, 176, 2 1 0

Thomson, A.L., see Owers-Bradley, J.R. 253,254, 319 Thomy, A. 216, 3 1 9 Thomy, A., see Duval, X. 216, 316 Thorel, P., see Marti, C. 269, 318 Thouless, D.J., see Kosterlitz, J.M. 334, 416, 434, 4 4 0 Thoulouze, D., see Franco, H. 241,255257,316

Thoulouze, D., see Godfrin, H. 308, 317 Thuneberg, E.V., see Freeman, M.R. 165, 210, 332, 4 3 9 Thwaites, T., see Derry, G. 231,316 Tiby, C. 261,320 Tiby, C., see Lauter, H.J. 261, 318 Tilley, D.R. 8, 6 8 Tilley, J., see Tilley, D.R. 8, 6 8 Timofeevskaja, O.D., see Bunkov, Yu.M. 88, 149, 152, 155 Timofeevskaya, O.D. 93, 1 5 8 Timofeevskaya, O.D., s e e BorovikRomanov, A.S. 104, 126, 129, 133, 155 Tiwari, R. 352, 425, 4 4 3 Toennies, J.P., see Buchenau, H. 334, 4 3 7 Toigo, F., see lhm, G. 231,318 Toigo, F., see Maritan, A. 433,441

461

AUTHOR INDEX Tompa, H., see Ruland, W. 265, 319 Ttime, A., see Wennerstr/Sm, P. 252, 320 Tough, J.T., see Guo, H.N. 326, 365, 439 Treiner, J. 230, 310, 312, 320, 430, 443 Treiner, J., see Cheng, E. 235,295, 312, 315, 334, 427, 438 Treiner, J., see Dupont-Roc, J. 378, 438 Treiner, J., see Pavloff, N. 328, 344, 365, 378, 380, 409, 411,430, 441 Treiner, J., see Pricaupenko, L. 235, 236, 312, 319, 379, 433,441 Treiner, J., see Stringari, S. 342, 380, 442 Trewem, T., see White, J.W. 261,320 Tsakadze, J.S., see Hakonen, P.J. 134, 157 Tsubota, M., see Ishikawa, O. 85, 113, 157

Tsubota, M., see Ohmi, T. 110, 112, 157 Tsuneto, T., see Ohmi, T. 110, 112, 157, 334, 441 Tulin, V.A., see Borovik-Romanov, A.S. 74, 104, 155 Tuneberg, E.V., see Bunkov, Yu.M. 153, 156

Uher, C. 220, 222, 225-229, 320 Uhlig, K., see Eska, G. 109, 139, 156 Usui, T., see Ohmi, T. 334, 441 Valles, Jr., J.M. 343, 353,354, 357-360, 374, 394, 404--406, 443 van Beelan, H., see Brouwer, P.W. 415, 437

van den Meijdenberg, C.J.N. 425, 441 van der Hoeven, J.C. 219-222, 320 van Sciver, S.W. 248, 290, 292, 293, 296, 297, 300, 302, 309, 320 van Sciver, S.W., see Hering, S.V. 248, 281,285, 288, 289, 290, 317 Varoquaux, E. 9, 68, 128, 158 Varoquaux, E., see Avenel, O. 133, 155 Viana, R. 221,222, 320 Vidali, G. 230, 231,234, 320 Vidali, G., see Derry, G. 231,316 Vilches, O.E. 280, 288, 320 Vilches, O.E., see Bretz, M. 219, 237, 240, 241,245, 246, 248, 252, 265, 271-273, 279-281,290, 315 Vilches, O.E., see Dillon, L.D. 226, 227, 228, 229, 316 Vilches, O.E., see Ecke, R.E. 252, 316

Vilches, O.E., see Hering, S.V. 248, 281, 285,288-290, 317 Vilches, O.E., see Hickemell, D.C. 218, 252, 317, 335, 336, 440 Vilches, O.E., see van Sciver, S.W. 248, 290, 292, 293,296, 320 Villain, J. 288, 320 Vinen, W.F. 3, 68, 134, 137, 158 Vinen, W.F., see Muirhead, C.M. 334, 441 Vithayathil, J., see Sheldon, P.A. 377, 400, 404, 411,442 Vollhardt, D. 74, 108, 114, 129, 158, 161, 197, 211,276, 295,320 Volodin, A.G., see Egorov, A.V. 256, 316 Volovik, G.E. 18, 38, 39, 68, 134, 138, 146, 152, 158 Volovik, G.E., see Hakonen, P.J. 91, 134, 157

Volovik, G.E., see lkkala, O.T. 91, 134, 157

Volovik, G.E., see Kondo, Y. 153, 157 Vora, P., see Duta, P. 265,316 Vuorio, M. 73, 158, 206, 211 Wagner~ M. 312, 320 Wagoner, G., see van der Hoeven, J.C. 221,222, 320 Wang, C.L. 334, 443 Wang, S., see Ketola, K.S. 428, 429, 440 Wang, X. 342, 381,382, 422-424, 443 Warren, B.E. 265, 320 Watkins, J.L. 65, 68 Watkins, R.A. 243, 320 Webb, R.A. 84, 85, 103, 109, 139, 148, 151,158 Webb, R.A., see Kleinberg, R.L. 165, 167, 210

Webster, E. 392, 418, 419, 443 Webster, E., see Webster, G.D.L. 419, 443 Webster, G.D.L. 419, 443 Webster, G.D.L., see Webster, E. 392, 418, 419, 443 Weling, F. 265, 320 Wennerstr/Sm, P. 252, 320 Wennerstrtim, P., see Alvesalo, T.A. 163, 167,210

Werthamer, N.R., see Balian, R. 72, 155 Wesner, D., see Derry, G. 231,316 Wheatley, J.C. 161,206,211,329, 443 Wheatley, J.C., see Abel, W.R. 329, 436

462

AUTHOR INDEX

Wheatley, J.C., see Black, W.C. 329, 4 3 7 Wheatley, J.C., see Buchanan, D.S. 163, 168, 200, 2 1 0 Wheatley, J.C., see Kleinberg, R.L. 165, 167,210

Wheatley, J.C., see Paulson, D.H. 163, 211 Wheatley, J.C., see Sager, R.E. 139, 1 5 7 Wheatley, J.C., see Webb, R.A. 84, 109, 148, 151,158 White, J.W. 261,320 Widom, A. 253,254, 289, 320, 362 Widom, A., see Havens-Sacco, S.M. 357, 439

Wiechert, H. 261,262, 3 2 0 Wiechert, H., see Feile, R. 254, 261, 281, 289, 3 1 6 Wiechert, H., see Freimuth, H. 286, 317 Wiechert, H., see Lauter, H.J. 261,262, 281,296, 3 1 8 Wiechert, H., see Schildeberg, H.P. 218, 262, 265, 268, 319 Wiechert, H., see Tiby, C. 261,320 Wiedemaan, W., see Eska, G. 97, 109, 139, 156

Wilks, J. 8, 6 8 Willers, H.G., see Eska, G. 97, 109, 156 Williams, C.D.H. 63-65, 68 Williams, C.D.H., see Hendry, P.C. 4, 13, 67

Williams, G.A. 334, 393,443 Williams, G.A., see Watkins, J.L. 65, 6 8 Winkler, C., see Buchenau, H. 334, 4 3 7 Wirkeutin, P.A., see Sager, R.E. 139, 1 5 7 Wolfle, P., see Vollhardt, D. 74, 108, 114, 158

Wtilfle, P., see Vollhardt, D. 161, 197, 211 Wong, C.C.K., see Aziz, R.A. 234, 315 Woo, C.W. 327,443 Woo, C.W., see Massey, W.E. 327, 441 Woo, C.W., see Shih, Y.M. 327, 4 4 2 Woods, A.D.B., see Cowley, R.A. 58, 6 7 Woods, A.D.B., see Smith, A.J. 56-58, 6 8 Wyatt, A.F.G., see Stefanyi, P. 432, 4 4 2 Yang, L.C., see Chester, M. 416, 4 3 8 Yeager, C.J., see Crawford, G.P. 345, 4 3 8 Yeager, C.J., see Steele, L.M. 335,442 Yim, M.B. 327, 4 4 3 Yip, S. 197, 2 1 0 Yip, S., see Leggett, A.J. 163, 166, 168, 180, 182, 185, 191-195, 200, 201,202, 211

Yu, Y.Y., see Finotello, D. 426, 427,

439

Zakazov, S.O., see Bunkov, Yu.M. 89, 155 Zangwill, A. 229, 247, 3 2 0 Zaremba, E. 427, 4 4 3 Zawadowski, A. 58, 68 Zawadzki, P., see Steel, S.C. 332, 4 4 2 Zhang, S.C. 434, 4 4 3 Zieve, R., see Davis, J.C. 137, 156 Ziman, J.M. 219, 3 2 0 Zimanyi, G.T. 344, 4 4 3 Zimmerli, G. 247, 252, 270, 312, 3 2 0 Zimmerman, Jr., W., see Gearhart, Jr., C.A. 330, 4 3 9 Zimmerman, Jr., W., see Varoquaux, E. 9, 68

Zinov'eva, K.N. 325, 326, 4 4 3 Zmuidzinas, J.S., see Watkins, J.L. 65,

68

SUBJECT INDEX

-

Curie behavior 376, 384, 385 Curie constant 374, 377 Curie fraction 384

A-B boundary propagation 146 magnetic soliton 147 magnetic relaxation 150 adsorption isotherm 238, 240, 247 adsorption potential 230 Aerogel 418, 421,432, 433 annealing of 3He films 246 antiferromagnetism of 3He films 299

-

damping, third sound 400, 401,402 Debye temperature 265,288 Debye-Waller factor 265 degeneracy temperature 356, 368, 374, 375, 383 degenerate system 350 density functional 328 density profiles 340, 343 diffuse reflection 332 Dipole-dipole interaction - in 3He-A 78, 81 in 3He-B 78, 82 dispersion curve, for He II (see also "roton parameters") 5, 25, 38, 65 drag on a moving object in He II 7, 17

baked Alaska model 167, 182-184, 190--200 beaker flow 331,332 Bernoulli pressure, on ion moving in He II 21 bolometer 388 Boltzmann transport equation 27, 33, 36 boojum 166, 172 Bose-condensed system 3 bound state 333,338, 341

-

Cahn-Hilliard theory 163 calorimeter 345, 346 capillary rise 326 Cerenkov radiation, analogy to roton creation 18 cesium 427--432 charge multiplication, in He II 11 chemical potential 334, 376, 404, 410 cluster 333,334, 343 commensurate phase 279, 297 configuration changes 390, 404, 406 cosmic rays 169, 179, 185-188, 190 coverage scales 237 critical current density in superconductors 8 critical dissipation in He II 20, 22 critical radius 163-164, 185, 193-200 critical velocities in He II - in flow 8 for vortex creation by ions 9, 13 cross-correlation technique, for ionic velocity measurement 45-50

effective mass 324, 326, 344, 348,349, 350, 351,359, 377, 406 effective mass of 3He films 275,294 effective potential 338 electric induction technique, for ionic velocity measurement 15 electrical conductivity of graphite 221 electrical discharge in He-4 vapour, for creation of "fast" ions 61 electron bubble 333 energetics 345,349, 351,378, 413 energy gap 362, 373,374, 381,402 energy gap, in superfluid He-3 8 energy, binding 324, 326, 339, 348, 349, 351,376, 377,378, 379, 380, 413 Euler-Lagrange 338, 340 exchange interaction 284, 299 excitations in He II 3, 5, 6, 10 creation of (see also "roton creation") 10 scattering of 10, 19, 23, 35

-

-

-

463

464

SUBJECT INDEX

- self-energy at large momenta 58 exfoliated graphite 216

interface fluctuations 408 interface, bulk-wall 329 ionic velocities, in He II - distribution of 13 - measurement of average (see also "ions in liquid helium") 13ff, 40ff ions in liquid helium 9, 21 - effective mass of 21, 53 - "fast" 5, 9, 61--65 - in superfluid He-3 8 in He II - "intermediate mobility" 62-65 - velocity measurements 15-18, 34, 40-50 mobility of 21 - in very weak electric fields 46-50 - spacecharge spreading of 40 - transition to charged vortex rings 4, 8, 9, 60, 61 - trapping lifetime of 21 isotopic purification of He-4 15, 40

"fast" ions in He II, see "ions" Fermi's golden rule 25, 28, 30, 35 applicability of 28, 31 Fermi energy 380, 410, 413 Fermi fluid (2D) 273 Fermi gas 326, 347, 357, 362, 374, 410 Fermi liquid parameters 357, 358 F0 a 118 - F2 a 118 - Landau field 76, 117 Fermi momentum, in superfluid He-3 8 Fermi temperature 277, 357, 374 ferromagnetism of 3He films 256 field emission in He II 11, 12, 14, 41 field ionization in He II 11 film flow 425,426 film thickness 396 foam 217 fourth sound in He II 8 -

-

gamma rays 177, 185-188 Gradient energy 86 Grafoil 217, 335,375 He-3 isotopic impurities in He II, effect of 13, 23, 35 heat capacity 345, 350 - of graphite 222 - of 3He films 248, 273, 274, 280, 293, 303, 305, 307, 309 Heisenberg uncertainty principle 39 HNC 328, 343, 344 homogeneous film 351 homogeneous nucleation 163-164 homogeneously precession domain formation 99, 104 oscillations 119 in rotated 3He 153 hydrodynamic flow, of He II 5, 9 hydrodynamic mass 342, 357, 358, 360, 361, 381,382

-

9

-

Kosterlitz-Thouless 334, 392, 416, 418, 419, 420, 422, 423 Landau critical velocity 3ff - for light object 19ff - for massive object 6 - generalised 8 - measurement of 40-52 observation of 15, 17 - pressure dependence of 6, 13, 52 temperature dependence of 7 layered film 340, 397, 398, 400, 406, 421 layering 235 layering 340, 343 Lekner formulation 326, 327, 333,404 lobster pots 202 localized spins 385, 387 -

-

-

-

-

incommensurate phase 288, 306 inelastic neutron scattering data, need for in He II 52 interaction potential 233 interactions 342

macroscopic wave-function, in He II 3 magnetic relaxation in 3He by spin diffusion 103, 107, 109 - non-hydrodynamic correction 112 intrinsic relaxation 108 - non-hydrodynamic correction 112, 151 surface relaxation 112 catastrophic relaxation 114 magnetic susceptibility 355, 356, 357, 359 of graphite 228

-

-

-

-

-

SUBJECT INDEX - of 3He films 255, 272, 276, 284, 292, 299, 311 magnetization 362, 366, 375, 377 matrix element for roton creation - for "fast" ion 63-65 single-roton process 25, 36, 37 - roton-pairs 30, 35-37, 50, 53, 54, 60 momentum dependence of, in pair-creation 55, 58 mesa 352 mixed film 397, 406, 407 molecular ions in He II 65 monolayer, 4He 329, 330, 331,332 Monte Carlo 184-190 mosaic spread 268 multi-electron bubbles in He II 65 Mylar 330, 416

465

creation of 4, 8, 9, 60, 61 discrete dissipative events 9 quantum mechanical tunnelling 165 quasiparticle 358, 359 -

-

re-entrant superfluidity 343 re-entrant wetting 429, 430, 431 relaxation rate (see relaxation time) relaxation time T 1 364, 365, 367, 368, 369, 370, 371,372, 379, 387, 401 relaxation time T2 364, 367, 371,374, 386, 387, 401 resonator 356, 388, 389, 390, 400, 402, 406 retardation effects 334 roton creation in He II 9, 10 - at extreme supercritical velocities 54-61 - effect of thermal fluctuations 23, 24, 25 in classical wave radiation model 18, 28, 40 - by ions in very weak electric fields 38-40 by a light object 19, 23 - by moving object 9, 10 - kinetic theory of, for single rotons 23-28 - kinetic theory of, for roton pairs 28-33, 38 single-roton and roton pair processes compared 35-38 - strong-coupling model 38-40 roton parameters 5, 51, 52, 56, 57 (see also "dispersion curve") rotons (see also "excitations in He II") 5, 6

-

-

-

negative resistance regime in He II 35-38 neutron scattering in 3He films 261 NMR 173-174, 354, 355-387 NMR in 3He longitudinal 84 transverse 83 NMR of 3He films 253 non-linear phenomena 38 Nulepore 345, 357, 389, 390, 406

-

-

-

-

onset 392, 393, 420 orifice flow of He II 9 pair-breaking, in He-3 8 Papyex 217 parametric excitation 153 period shift 417, 418 phase 410, 411,413,414 phase diagram 336, 337 - of 3He films 281,292 phase difference 400 phase separation 326, 335, 336, 352, 353, 360, 392, 393, 394, 407, 412, 419, 420, 425 phase space 347, 386 pole strength, for high momentum excitations in He II 55, 58 puddling 351,360, 419 pulse, third sound 399 Q, third sound 364, 389, 401,402, 411, 412 quantized vortices in He II

single-pulse time-of-flight technique 13, 14, 40ff spacecharge spreading, of ions in He II 40 specific area 219 specific heat (see heat capacity) 347 specular reflection 332 spin diffusion 89-91, 112, 381,383, 384, 385, 387 - in normal 3He 90 - in3He-A 142,145 in 3He-B 101,109, 112 - anisotropy 129 spin-echo 357 steps, magnetization 360, 361,362, 363 stratification 408, 421 submonolayer films 334, 335 substrate state 379, 409, 410 superconductors 8 -

466

SUBJECT INDEX

supercooling 168, 175-176 supercritical dissipation in He II - early theories of 18 through roton pair emission 30 through single-roton emission 24 superfluid density 388 superfluid film 329-331,335, 420 superfluid helium-3 free energy 163-164, 195-198, 208-210 heat capacity 197, 207-210 - magnetic susceptibility 206-209 phase diagram 161-163 surface energy 163, 197-199 - thermodynamics 206-210 superfluidity 3ff, 420, 421,423, 424, 434 - criteria for occurrence of 3 breakdown of (see also "roton creation") 3 surface nucleation 165 surface profile 328, 342, 364 surface sound 326 surface state 324, 328 surface tension 324, 327 symmetric mode 415 -

-

-

textural nucleation 165-167, 200-202 thermal conductance 426 thermal conductivity of graphite 226 thermal neutrons 180-181, 189-191 thickness scales 444 third sound 353,364, 387-416, 421,424, 428, 429 torsional oscillator 416--425 traditions, differing, in research on He-4 3

-

-

-

-

van der Waals 330, 334, 339, 365, 387, 390 vapor pressure 396 vapor sound 397 variational calculation 328, 339 vibrating wires, in superfluid He-3 8 vortex ring 333 vortices, see "quantized vortices" Vycor 330, 334, 355, 418 waveguide 394 wetting 426--432 zero-point energy 233 ZYX 217

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