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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


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


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


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


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


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


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) 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


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


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


-

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
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


/)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


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


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


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


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 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


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 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

I

'

'

oo

Ol

~ff b d:)

70-

~/(ms-9 60-

5O

0

'

,J

,

,

I

50

,

i

,

,

I

100

i

,

i

i

I

150

'

,

,

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

/ 80

I

/

g

I,o,oop_ -

15 - 84 0

-

,,

o\

,, /o

I

," 70 S

/

0

,,

S~

,,"60

-

.

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

10

/ /

$/ -

s s

5

s

s s

s,

s

s

s

$s

j/

S s

t

~

\

s

S

i

s

~ s

s jS

s

-

s

S S~

.,

s~s S

f

..." -'v

~

/s

s SS s

,,t, t 7-'

s

js

i

,

LO

I.

20

!

i

i

i

I

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


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


1.10-,

0 u

i

'

' .

.

.

.

I

.

.

.

.

,

'

'

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


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


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


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


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


(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


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


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).

68

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


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


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


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


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


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


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


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


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


(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


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


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

Ch. 2, w


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

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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


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


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


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

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


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


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


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:


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).

Ch. 2, w


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

Ch. 2, w


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.

Ch. 2, w


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)

128

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


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

Ch. 2, w


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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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).

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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


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.

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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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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

> 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

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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-

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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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YU.M. BUNKOV

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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.

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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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Bunkov, Yu.M., V.V. Dmitriev and Yu.M. Mukharsky, 1984, Phys. Lett. 102A, 194. Bunkov, Yu.M., V.V. Dmitriev and Yu.M. Mukharsky, 1985, Soy. Phys. JETP 61,719. Bunkov, Yu.M., V.V. Dmitriev and Yu.M. Mukharsky, 1986, JETP Lett. 43, 168. Bunkov, Yu.M., V.V. Dmitriev, Yu.M. Mukharsky and G.K. Tvalashvily, 1988, Sov. Phys. JETP 67, 300. Bunkov, Yu.M., V.V. Dmitriev, Yu.M. Mukharsky, J. Nyeki and D.A. Sergatskov, 1989, Europhys. Lett. 8, 645. Bunkov, Yu.M., V.V. Dmitriev, D.A. Sergatskov, A. Feher, E. Gazo and J. Nyeki 1990a, Physica B 165, 53. Bunkov, Yu.M., V.V. Dmitriev, Yu.M. Mukharsky, J. Nyeki, D.A. Sergatskov and I.A. Fomin, 1990b, Physica B 165, 675. Bunkov, Yu.M., V.V. Dmitriev, A.M. Markelov, Yu.M. Mukharsky and D. Einzel, 1990c, Phys. Rev. Lett. 65, 867. Bunkov, Yu.M., S.N. Fisher, A.M. Guenault, C.J. Kennedy and G.R. Pickett, 1992a, Phys. Rev. Lett. 68, 600. Bunkov, Yu.M., V.V. Dmitriev and Yu.M. Mukharsky, 1992b, Physica B 178, 196. Bunkov, Yu.M., S.N. Fisher, A.M. Guenault, C.J. Kennedy and G.R. Pickett, 1992c, J. Low Temp. Phys. 89, 27. Bunkov, Yu.M., S.N. Fisher, A.M. Guenault and G.R. Pickett, 1992d, Phys. Rew. Lett. 69, 3092. Bunkov, Yu.M., V.V. Dmitriev, J.S. Korhonen, Y. Kondo, M. Krusius, Yu.M. Mukharskiy, U. Parts and E.V. Tuneberg, 1992f, Phys. Rev. B 46, 13983. Candela, D., N. Masuhara, D.S. Sherrill and D.O. Edwards, 1986, J. Low. Temp. Phys. 63, 369. Candela, D., D.O. Edwards, A. Heft, N. Masuhara, D.S. Sherrill and R. Combescot, 1987, Can. J. Phys. 65, 1330. Castelijns, C.A.M., K.F. Coates, A.M. Guenault, S.G. Mussett and G.R. Pickett, 1985, Phys. Rev. Lett. 55, 2021. Corruccini, L.R. and D.D. Osheroff, 1975a, Phys. Rev. Lett. 34, 564. Corruccini, L.R. and D.D. Osheroff, 1975b, Phys. Rev. Lett. 34, 695. Corruccini, L.R. and D.D. Osheroff, 1978, Phys. Rev. B 17, 126. Davis, J.C., J.D. Close, R. Zieve and R.E. Packard, 199 I, Phys. Rev. Lett. 66, 329. Einzel, D., 1981, Physica 108B, 1143. Eska, G., H.G. Willers, B. Amend and W. Wiedemaan, 1981, Physica B 108, 1155. Eska, G., K. Neumaier, W. Shoepe, K. Uhlig and W. Wiedemaan, 1982, Phys. Lett. 87A, 311. Fetter, A.L., 1986, in: Progress in Low Temperature Physics, Vol. 10, ed D.F. Brewer (NorthHolland, Amsterdam) p. 1. Feynmann, R.P., 1955, in: Progress in Low Temperature Physics, Vol. 1, ed C.J. Gorter (NorthHolland, Amsterdam) p. 17. Fishman, R.S. and J.A. Sauls, 1988, Phys. Rev. B 38, 2526. Fomin, I.A., 1976, Sov. Phys. JETP 44, 416. Fomin, I.A., 1978, J. Low Temp. Phys. 31,509. Fomin, I.A., 1979, JETP Lett. 30, 164. Fomin, I.A., 1980, Soy. Phys. JETP 51, 1203. Fomin, I.A., 1984a, JETP Lett. 39, 466. Fomin, I.A., 1984b, JETP Lett. 40, 1037. Fomin, I.A., 1985, Sov. Phys. JETP 61, 1207. Fomin, I.A., 1986, JETP Lett. 43, 171. Fomin, I.A., 1987a, JETP Lett. 45, 135. Fomin, I.A., 1987b, Sov. Phys. JETP 66, 1142. Fomin, I.A., 1988, Soy. Phys. JETP 67, 1148.

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Fomin, I.A., 1990, in: Helium Three, eds W.P. Halperin and L.P. Pitaevsky (Elsevier, Amsterdam) p. 609. Ginzburg, V.L. and L.D. Landau, 1950, Zh. Eksp. Teor. Fiz. 20, 1064. Giinnetta, R.W., E.N. Smith and D.M. Lee, 1981, J. Low Temp. Phys. 45, 295. Golo, V.L. and A.A. Leman, 1990, in Helium Three, eds W.P. Halperin and L.P. Pitaevsky (Elsevier, Amsterdam) p. 727. Golo, V.L., A.A. Leman and I.A. Fomin, 1983, JETP Lett. 38, 146. Gould, C.M., 1992, Physica C 178, 266. Gully, W.Y., C.M. Gould R.C. Richardson and D.M. Lee, 1976, J. Low Temp. Phys. 24, 563. Hakonen, P.J., O.T. lkkala, S.T. Islander, O.V. Lounasmaa, T.K. Markkula, P.M. Roubeau, K.M. Saloheimo, G.E. Volovik, E.L. Andronikashvily, D.I. Garibashvily and J.S. Tsakadze, 1982, Phys. Rev. Lett. 48, 1838. Hakonen, P.J., M. Krusius, M.M. Salomaa, J.T. Simola, Yu.M. Bunkov, V.P. Mineev and G.E. Volovik, 1983, Phys. Rev. Lett. 51, 1362. Hoyt, R.F., H.N. Scholz and D.O. Edwards, 1981, Physica 107B, 287. lkkala, O.T., G.E. Volovik, P.J. Hakonen, Yu.M. Bunkov, S.T. Islander and G.A. Haradze, 1982, JETP Lett. 35, 416. Ishikawa, O., Y. Sasaki, T. Mizusaki, A. Hirai and M. Tsubota, 1989, J. Low. Temp. Phys. 75, 35. Ivlev, B.I. and N.B. Kopnin, 1984, Sov. Phys. Uspekhy 27, 206. Josephson, B.D., 1962, Phys. Lett. 1, 251. Kapitza, P.L., 1938, Dokl. Akad. Nauk SSSR (in Russian) 18, 21. Kondo, Y., J.S. Korhonen, M. Krusius, V.V. Dmitriev, Yu.M. Mukharsky, E.B. Sonin and G.E. Volovik, 1991, Puis. Rev. Lett. 67, 81. Korhonen, J.S., Z. Janu, Y. Kondo, M. Krusius, Yu.M. Bunkov, V.V. Dmitriev and Yu.M. Mukharsky, 1989, QF&S, AIP Conf. Proc. 194, 147. Korhonen, J.S., V.V. Dmitriev, Z. Janu, Y. Kondo, M. Krusius and Yu.M. Mukharsky, 1990, Physica B 165, 671. Landau, L.D., 1957, Sov. Phys. JETP 5, 101. Leggett, A.J., 1975, Rev. Mod. Phys. 47, 331. Leggett, A.J., 1992, J. Low Temp. Phys. 87, 571. Leggett, A.J. and S. Takagi, 1977, Ann. Phys. 106, 79. Lee, D.M. and R.C. Richardson, 1978, in: Physics of Liquid and Solid Helium, part 2, eds K.H. Bennemann and J.B. Ketterson (Wiley, New York). Likharev, K.K., 1979, Rev. Mod. Phys. 51, 101. Maki, K., 1975, Phys. Rev. B 11, 4264. Markelov, A.V., 1988, Sov. Phys. JETP 67, 520. Movshovich, R., N. Kim and D.M. Lee, 1990, Phys. Rev. Lett. 64, 43 I. Nunes, Jr., G., C. Jin, D.L. Hawthorne, A.M. Putnam and D.M. Lee, 1992, Phys. Rev. B 46, 9082. Ohmi, T., M. Tsubota and T. Tsuneto, 1987, Jpn. J. Appl. Phys. 26, 169. Onsager, L., 1949, Nuovo Cimento 6, suppl. 2, 249. Osheroff, D.D., 1977, Physica B 90, 20. Osheroff, D.D. and W.F. Brinkman, 1974, Phys. Rev. Lett. 32, 584. Osheroff, D.D., R.C. Richardson and D.M. Lee, 1972, Phys. Rev. Lett. 29, 920. Pickett, G.R., 1988, Rep. Prog. Phys. 51, 1295. Pitaevsky, P.L., 1959, Soy. Phys. JETP 10, 1267. Poddjakova, E.V. and I.A. Fomin, 1988, JETP Lett. 47, 519. Sager, R.E., R.L. Kleinberg, P.A. Wirkeutin and J.C. Wheatley, 1978, J. Low Temp. Phys. 32, 263. Schopohl, N., 1982, J. LOw Temp. Phys. 49, 347. Scholz, H.N., 1981, Ph.D. Thesis, Ohio State University, Columbus, OH.

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Serene, J.W. and D. Rainer, 1977, in: Quantum Fluids and Solids, eds S.B. Trickey, E.A. Adams and J.W. Dufty, (Plenum, NY) p. 111. Silin, V.P., 1958, Soy. Phys. JETP 6, 945. Sonin, E.B., 1987, JETP Lett. 45, 747. Sonin, E.B., 1988, Soy. Phys. JETP 67, 1791. Timofeevskaya, O.D., 1991, Private communication. Varoquaux, E., O. Avenel, G. lhas and R. Salomelin, 1992, Physica B 178, 309. Vinen, W.F., 1961a, Proc. R. Soc. A 260, 218. Vinen, W.F., 1961b, in: Progress in Low Temperature Physics, Vol. 3, ed C.J. Gorter (NorthHolland, Amsterdam) p. 1. Vollhardt, D. and P. Wolfle, 1990, The superfluid phases of Helium-3 (Taylor and Francis, London). Vollhardt, D., K. Maki and N. Schopohl, 1979, J. Low Temp. Phys. 39, 79. Volovik, G.E., 1992, Exotic Properties of Superfluid 3He (World Scientific). Vuorio, M., 1976, J. Phys. C 9, L. 267. Webb, R.A., 1977, Phys. Rev. Lett. 39, 1008. Webb, R.A., R.E. Sager and J.C. Wheafley, 1975a, Phys. Rev. Lett. 35, 1010. Webb, R.A., R.E. Sager and J.C. Wheafley, 1975b, Phys. Rev. Lett. 35, 1164. Webb, R.A., R.E. Sager and J.C. Wheafley, 1977, J. Low Temp. Phys. 26, 439. Yip, S. and A.J. Leggett, 1986, Phys. Rev. Lett. 57, 345.

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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Ch. 3, w1

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

165

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

166

Ch. 3, w

P. SCHIFFER ET AL.

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


167

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-

168

P. SCHIFFER ET AL.

Ch. 3, w

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


169

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

170

ch. 3, w

P. SCHIFFER ET AL. 35

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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,


171

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-

172

P. SCHIFFER ET AL.

Ch. 3, w

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.


173

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

174

P. SCHIFFER ET AL.

Ch. 3, w

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


175

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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P. SCHIFFER ET AL.

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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.


177

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

P. SCHIFFER ET AL.

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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.


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

180

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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


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 (> 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 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-


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.

Ch. 3, w

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

I ....

~ ....

, ,,/,,

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

,

,

,

,

I

~

1.1

~

,

,

t

,

1.2

,

,

,

I

,

1.3

,

,

,

1.4

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.

Ch. 3, w

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-


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.

Ch. 3, w

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


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


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

P. SCHIFFER ET AL.

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;


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-


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-

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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.


211

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.

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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

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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

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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).

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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.

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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

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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.

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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


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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Fig. 2a,b.

1

2

3

T (K)

4

5

6

7

224

Ch. 4, w

H. G O D F R I N and H.-J. L A U T E R

10.80

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-

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-

Foam parallel (Uher and Sander, 1983)

10.75

it} =

o

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..-.,

10.70

>

10.65

"

10.60

=,._.

1/} o3 (9 o

o,,.,. t,..

o

(9 (9

10.55

...,.

10.50

0

1

2

3

4 T

E IE

tO

5

6

7

(K)

96.00

_ , , , , i , , , , i , , , , I , , ~ , i , , , , i , , , , 1 ~ , , , i , , , ,

95.5o

-

Foam perpendicular (Uher and Sander, 1983)

14'1 i

o

95.00

>., >

94.50

o...

(/) (/) (9

i._ 9 4 . 0 0

.

to n L_

~ 93.50 (9 (9

,,..--.

93.00

1

2

3

4 T

Fig. 2c,d

(K)

5

6

7

8

Ch. 4, w


1.460

225

,,,,i1,,,I,,,,i,,,,i,,,,i,,,,i,,i,i,,,, e

E E 0 = 1.453 b

-

ZYX parallel (Uher and Sander, 1983)

._> 1.445

OO O

O3 o-.

9

e,o

~

90

o

1.437

1.430

''l'

0

l,,Itll,,,I,,,,l,,,,I,,,,

1

2

3

I,,,,I,izl

T (K)

4

5

6

7

8

Fig. 2. Electrical resistivity of Grafoil, Foam and ZYX at low temperatures in the perpendicular and in the rolling directions, as indicated in the figures (from Uher and Sander 1983).

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


1.008

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/

-

9

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.,._.

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O0

000

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_

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(~ 9149149

0000

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o'} n"

0

#/,p 9

0.998 _

0.996

I

x

I

I

0.01

I i t J]

0.1

l

t

T (K)

I

i

I i r tl

l 1

,

,

,

,

,,,

10

Fig. 3. Very low temperatureresistivity of Grafoil in the rolling direction, normalized at 1 K. Filled diamonds (Rapp et al. 1985) (,o(1 K)= 14.23/~f~ m); open diamonds (Koike et al. 1984, 1985) (/9(1 K) = 17.49/tff~ m). The values at 1 K should be compared to that of Uher and Sander (1983) (fig. 2a) p(1 K) = 22.0/~Q m. The increase at low temperaturesis due to quantum localization and electron-electron interactions (sample dependent). Note the expanded scale. 2.2.8.

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


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


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 '

'

'

' '"'1

'

'

' ''"'i

'

'

'

229

''"'1

0 i__

>,

100

",-.~--

-

0

"

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

b

a

- -

0

0')

~~ "

c-

ilJ

Ill Ill

_

> ".,~

o 23 "13 r

$

10_--

s.~_----

c

I s o

I3

s J

E r-"

1 0.1

I

I

I

l iilil

i

1

I

1 I Illi[

I

I

!

10

i iliil

l

100

!

i

I

i l l i

1000

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


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


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


Ch. 4, w

50

o

|

.

v

,,l,,,,a

E

-50

i

o

~ o

:~ t~

t

-lOO

L

0

or}

"0

O (9

0.O60

R1b or DWS

~------ Rla a+F

0.050

0.040

, , , , ; , ; ] ~ , ; ~ ,

0

0.5

1

,,

....

1.5

, ....

2

~,,,,,

2.5

Temperature (K)

....

3

, ....

3.5

4

Fig. 32. Phase diagram of 3He adsorbed on graphite. The crosses indicate the location of melting peaks (Bretz et al. 1973; Hering et al. 1976). The low temperature phase boundaries correspond to the heat capacity results (Greywall 1990) and NMR data (Morhard et al. 1995). The system is fluid (F) at low coverages. An arrow indicates the commensurate phase R la coverage; coexistence between the F and Rla is seen between the onset of solidification (0.0434/~-2) and the pure Rla phase. The label R l b indicates a possible additional commensurate phase. This intermediate coverage region can be interpreted either as a coexistence regime between commensurate phases, or as a domain wall solid (DWS). At higher temperatures, a domain wall fluid (DWF) exists. At high coverages an incommensurate solid (IC 1) is found. rate phase and its melting transition; a quantitative evaluation of the kinetic energy (about 36 K) and the potential energy ( a b o u t - 1 7 0 K) per particle is given as a function of temperature. The nature of this phase is more convincingly demonstrated by neutron scattering measurements (see section 4.5). In this region of the phase diagram, the observed neutron diffraction signals have a peak at the wave vector q = 1.702 ~ - t , as expected for the ~/3R30 ~ registered phase (Nielsen et al. 1977" Lauter et al. 1980, 1987, 1990, 1991; Feile et al. 1982). The lattice parameter deduced from these measurements is that expected for the c o m m e n s u r a t e ~/3R30 ~ phase, a triangular structure of density 0.06366/~-2. Clearly, patches of c o m m e n s u r a t e phase of constant density, imposed by the substrate, grow from the fluid as the coverage is increased. This behavior is clearly seen in fig. 22. The intensity of the commensurate phase neutron scattering peaks grows as a function of coverage in the fluid-commensurate phase coexistence domain, reaches a m a x i m u m , then decreases slightly until a clear change in the position

282

Ch. 4, w


3O

9

9

9

9

,

9

9

9

9

x

25 q.. =

E

2O

",~..

15

i

)

10

57 )

O r

-5

,.m

0.0

0.5

1.0

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


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

'

'

'

'

I

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'

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. . . .

I

Ch. 4, w

. . . .

I

. . . .

0.35

I

"1"

I I !

0.3

I ! !

0.25

,y,

i

, !

-

0.2

t _ 1 %

E o.ls 0.1

0.3

I

t I

I

('3 0.25

', ~,-

- b\

O.2

0 ~-" ,.

-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


~

341

O

i'M

v

'

!

I

I

N

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 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

364

R.B. HALLOCK

Ch. 5, w

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.

Ch. 5, w


365

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.

366

R.B. HALLOCK

Ch. 5, {}3

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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367

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-

368

R.B. HALLOCK

Ch. 5, w

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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369

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-

370

R.B. HALLOCK

Ch. 5, w

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.

Ch. 5, w

PROPERTIES OF MULTILAYER 3He--4He MIXTURE FILMS

371

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,

372

R.B. HALLOCK

Ch. 5, w

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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373

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

374

R.B. HALLOCK

Ch. 5, w

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;

Ch. 5, w


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).

376

R.B. HALLOCK

Ch. 5, w

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),

Ch. 5, w


377

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 _ 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

384

ch. 5, w

R.B. HALLOCK

0.15 A

!

0.10

I

I

-

A

A

+

9

A

s

A

A

A

s

10 .2

A

10"4

O

10-6

cD o

Z 0.05

10 8 2

2.5

3

3.5

D

4

4.5

5

4

Fig. 51. The diffusion coefficient (triangles) and the Curie fraction (circles) for 0.1 monolayer of 3He on a 4He film as a function of coverage (from Sprague et al. 1994b, 1995a). The diffusion coefficient, Dmeas is related to the coefficient D through the tortuosity factor a, Dmeas = Dla (Sprague et al. 1992). The dashed line is the prediction of the theory of Miyake and Mullin (1983, 1984).

c~

I--

Q

0.1

9

,

,

,

i

i

L

1

1oo

T (mK')

Fig. 52. The measured diffusion coefficient, D(T), normalized by the predicted DFL, as a function of temperature for several values of the 4He coverage (from Sprague et al. 1995a).

Ch. 5, w


385

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

"'i

.

.

.

.

.

.

.

.

I

,t 1

0.01

I

E ,..I LI.

C) Fv

IvC~

I 0.001

l 0.1

I

i

D(T)ITDFLshowing

.

a

9

.

.

.

1

.

1 D

Fig. 53.

I

4

- D

c

(layers)

the scaling behavior observed by Sprague et al. (1995a).

386

R.B. H A L L O C K

Ch. 5, w

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

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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

R.B. HALLOCK

Ch. 5, w

~!APACITOR) NUCLEPORE

W

_J

COPPER

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


T(K) ,I

391

"...

!

--.

no (at'~':t) "-.. --

u

-

o

.0.34 90.37 o 0.40

-.., -.,...

1.2.

m

1.0_ \0~

"-. o";~~%

0.8_

....~ oO..~~

"~176176176

%%%% %%% "...

~

OL_ oO..'~ sS

0

1

l

O.2

I

_ ,1

0.4

1

1

0.6

.

l

.

X

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

c~ (mls) 40

-.0-----13--- - - 0

30 ~L~3L_

20

T (K) lO

! 0.2

,,! 0.4

1_ 0.6

Fig. 59. Temperature dependence of the third sound velocity at constant 4He coverage of 0.50/1.-2 for different average 3He concentrations: pure 4He (open squares), 0.156 (solid triangles), 0.33 (solid circles), 0.45 (open circles) (after Laheurte et al. 1981).

Ch. 5, {}3


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.

\

T(K)

\

\

\ \

1.5

\

\ \ \

\ k

lO

\

\\

..-k /

(15

~23

( ~"-.. 0

0

0.2

0.4

0.6

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


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.

396

Ch. 5, w

R.B. HALLOCK !

50 -

I

I

1

I

1

I

|

I

oO

A

40

I

U

30

~

t O00000000~

E

9149

---

20

A 9

9 Mixture 9 Pure

to

4He

0 to

8

v

oooooo~

i

E 6

6

0 0

~

4

9149149149

0

9 Mixture

'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


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


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

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,

0.05

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_1

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I

a

0.~5

.... 1

~

o.zo

!

,

1

0.2.5

l

-

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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


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

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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).

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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.

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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

Progress in Low Temperature Physics : Volume XIV (Progress in Low Temperature Physics)

Progress in Low Temperature Physics

Progress in Low Temperature Physics