(Progress in Low Temperature Physics 14) W.P. Halperin (Eds.)-Elsevier, Academic Press (1995)

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. Informa- tion 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 increas- ingly 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 pro- ceedings, preprints, periodicals, books, and popular journal articles have been joined by various electronic forms of dissemination of our research. In this envi- ronment, the book series Progress in Low Temperature Physics assumes a particular responsibility to continue the strong tradition of excellent reviews, guid- ing our reading of the literature and providing direction for future research pos- sibilities. 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 rota- tional 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 Ad- sorbed 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 magnet- ism 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 Multi- layer 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. Un- der their stewardship, we have come to expect that Progress in Low Tempera- ture 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 ........................................................................................................ vi i

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

Ch. I. The Landau critical velocity,

P.V.E. M c C l i n w c k and R.M. Bowley ................................................

1. Introduction ......................................................................................................................... 3

2. Quest for the Landau critical velocity ................................................................................. 5

2.1. The dispersion curve and excitation creation in He-II ................................................. 5

2.2. Critical velocity measurements in He-II ...................................................................... 8

2.3. Field emission in liquid helium .................................................................................... 11 2.4. Measurement of ionic drift velocities .......................................................................... 13 2.5. Observation of the Landau critical velocity ................................................................. 15

3. Theory of roton creation in He-ll ........................................................................................ 18

3.1. Early theories of supercritical dissipation .................................................................... 18 3.2. Roton creation by a light object ................................................................................... 19 3.3. Theory of single-roton creation .................................................................................... 23 3.4. Theory of roton pair creation ....................................................................................... 28

3.5. Comparison of the theory with experiment .................................................................. 33

3.6. A regime of negative resistance? ................................................................................. 35 3.7. Roton creation in extremely weak electric fields ......................................................... 38

4. Measurement of the Landau critical velocity ...................................................................... 40 4.1. Experimental details ..................................................................................................... 40

4.2. Velocity measurements in weak electric fields ............................................................ 46 4.3. The critical velocity ..................................................................................................... 50

4.4. The matrix element for roton pair creation .................................................................. 53

5. Roton creation at extreme supercritical velocities ............................................................... 54 5.1. Velocity measurements in high electric fields ............................................................. 54 5.2. Comparison with theory ............................................................................................... 55

6. Roton creation by "fast" ions ............................................................................................... 61 7. Conclusion ........................................................................................................................... 65 References ................................................................................................................................ 66

vii

viii C O N T E N T S

Ch. 2. Spin supercurrent and novel properties of NMR in 3He, Yu.M. Bunkov .................................................................................... 69

1. Introduction ......................................................................................................................... 71

2. Basic properties ................................................................................................................... 75 2.1. Spatially uniform NMR ............................................................................................... 79

2.2. Spin supercurrent ......................................................................................................... 85 3. Experimental methods ......................................................................................................... 93 4. NMR and spin supercurrent in 3He-B ................................................................................ 98

4.1. Pulsed NMR ................................................................................................................. 98 4.2. CW NMR ..................................................................................................................... 103 4.3. Processes of magnetic relaxation ................................................................................. 107

4.3.1. Spin diffusion and intrinsic relaxation .............................................................. 107 4.3.2. Surface relaxation .............................................................................................. 112 4.3.3. Catastrophic relaxation ...................................................................................... 114

4.4. HPD oscillations .......................................................................................................... 119 5. Steady spin supercurrent ...................................................................................................... 124

5.1. Spin supercurrent in a channel ..................................................................................... 124 5.2. Phase slippage .............................................................................................................. 128 5.3. Josephson phenomena .................................................................................................. 132 5.4. Spin supercurrent vortex .............................................................................................. 134

6. Spin supercurrent in 3He-A ................................................................................................ 138

6.1. Instability of homogeneous precession ........................................................................ 139 7. Spin supercurrent at propagating A-B boundary ................................................................ 146 8. Conclusion ........................................................................................................................... 152 Acknowledgments .................................................................................................................... 154 References ................................................................................................................................ 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. Introduction .......................................................................................................................... 161 2. Background of the B phase nucleation problem ................................................................... 163 3. Experimental history of the B phase nucleation problem ..................................................... 167

4. The recent experiments at Stanford ...................................................................................... 170 4.1. Experimental design ..................................................................................................... 170 4.2. Initial B phase nucleation observations ........................................................................ 174 4.3. B phase nucleation by irradiation ................................................................................. 177

4.3.1. Data acquisition ................................................................................................. 177

4.3.2. Dependence on radiation type ............................................................................ 179 4.3.3. Dependence on temperature and magnetic field ................................................ 181

4.4. Monte Carlo simulations .............................................................................................. 184 5. The baked Alaska model: theoretical considerations ........................................................... 190 6. Conclusions .......................................................................................................................... 200 Acknowledgments .................................................................................................................... 203 Appendix A: Probability of depositing energy E in a radius much less than the "typical"

radius R(E) ........................................................................................................................... 204

C O N T E N T S ix

Appendix B: Relaxation of the magnetization by flow ............................................................ 206 Appendix C: Analytical model of the thermodynamics of superfluid 3He .............................. 206 References ................................................................................................................................ 210

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

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

213

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 C O N T E N T S

5.2.2. Second layer solidification ................................................................................. 296

5.2.3. The second layer commensurate phase R2a ....................................................... 297

5.2.4. Remarks about the second layer density ............................................................ 300

5.2.5. The second layer intermediate region (0.178 ,~-2 to 0.26 ,~-2) ......................... 301

5.2.6. The second layer incommensurate phase above n = 0.26 ]~-2 ........................... 306

5.3. Mult i layer films ............................................................................................................ 308

6. Conclusions .......................................................................................................................... 312

References ................................................................................................................................ 314

Ch. 5. The propert ies o f multi layer 3He-4He mixture films, R.B. Hallock ...................................................................................... 321

1. Introduction ......................................................................................................................... 323

2. Bulk interfaces ...................................................................................................................... 324

2.1. The bulk free surface .................................................................................................... 324

2.2. The bulk-wall interface ................................................................................................. 329

2.3. Other surfaces ............................................................................................................... 333

3. Hel ium films ......................................................................................................................... 334

3.1. Theoretical overview .................................................................................................... 334

3.2. Thickness scales ........................................................................................................... 344

3.3. Energetics experiments ................................................................................................. 345

3.3.1. Heat capacity experiments ................................................................................. 345

3.3.2. Nuclear magnetic resonance experiments .......................................................... 355

3.4. Other experiments ......................................................................................................... 387

3.4.1. Third sound experiments .................................................................................... 387

3.4.2. Oscillator measurements .................................................................................... 416

3.4.3. Selected other experiments ................................................................................ 425

3.5. Future directions ........................................................................................................... 433

4. Summary .............................................................................................................................. 435

Acknowledgments .................................................................................................................... 435

References ................................................................................................................................ 436

Author Index ................................................................................................. 445

Subject Index ................................................................................................. 463

CONTENTS OF PREVIOUS VOLUMES

Volumes I -V I, edited by C.J Gorter

Volume I (1955)

I. The two fluid model for superconductors and helium II, C.J. Goner ............................................................................

II. Application of quantum mechanics to liquid helium, R.P. Feynman ...............................................................................

III. Rayleigh disks in liquid helium II, J.R. Pellam .................... IV. Oscillating disks and rotating cylinders in liquid helium II,

A.C. Hollis Hallett ................................................................ V. The low temperature properties of helium three, E.F.

Hammel ................................................................................ VI. Liquid mixtures of helium three and four, J.M. Beenakker

and K.W. Taconis ................................................................. VII. The magnetic threshold curve of superconductors, B.

Serin ..................................................................................... VIII. The effect of pressure and of stress on superconductivity,

C.F. Squire ............................................................................ IX. Kinetics of the phase transition in superconductors, T.E.

Faber and A.B. Pippard ........................................................ X. Heat conduction in superconductors, K. Mendelssohn ........ XI. The electronic specific heat in metals, J.G. Daunt ................ XII. Paramagnetic crystals in use for low temperature research,

A.H. Cooke ........................................................................... XIII. Antiferromagnetic crystals, N.J. Poulis and CJ. Gorter ........ XIV. Adiabatic demagnetization, D. de Klerk and M.J. Steen-

land ....................................................................................... XV. Theoretical remarks on ferromagnetism at low tempera-

tures, L. N6el ........................................................................ XVI. Experimental research on ferromagnetism at very low

temperatures, L. Weil ........................................................... XVII. Velocity and absorption of sound in condensed gases, A.

van Itterbeek ......................................................................... XVIII. Transport phenomena in gases at low temperatures, J. de

Boer ......................................................................................

1-16

17-53 54-63

64-77

78-107

108-137

138-150

151-158

159-183 184-201 202-223

224-244 245-272

272-335

336-344

345-354

355-380

3 81-406

xii CONTENTS OF PREVIOUS VOLUMES

Volume H (195 7)

II. III.

W~ V. VI. VII.

VIII. IX. X.

XI. XII.

XIII.

XIV

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

1-58 59-82

83-104 105-137 138-150 151-186

187-225 226-265 266-291

292-337 338-367

368-394

395-430

431-464

Volume III (1961)

I. II.

\III.

W~ V. VI.

VII.

VIII.

IX. X.

XI.

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

1-57 58-79

80-112 113-152 153-169

170-287

288-332

333-395 396--429

430--453 454-480

CONTENTS OF PREVIOUS VOLUMES xiii

Volume IV (1964)

II.

III.

IV.

V. VI. VII.

VIII.

IX. X.

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

1-37

38-96

97-193

194-264 265-295 296--343

344-383

384--449 450-479

480-514

Volume V (1967)

II.

III.

IV.

V. VI.

VII.

The Josephson effect and quantum coherence measurements in superconductors and superfluids, P.W. Anderson. Dissipative and non-dissipative flow phenomena in superfluid helium, R. de Bruyn Ouboter, K.W. Taconis and W.M. van Alphen ................................................................. Rotation of helium II, E.L. Andronikashvili and Yu.G. Mamaladze ........................................................................... Study of the superconductive mixed state by neutron- diffraction, 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. Been- akker and H.F.P. Knaap ........................................................

1--43

44-78

79-160

161-180 181-234

235-286

287-322

xiv CONTENTS OF PREVIOUS VOLUMES

Volume VI (1970)

II. III.

IV.

V~

VI.

VII.

VIII. IX.

X~

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

1-35 37-76

77-161

163-203

205-242

243-290

291-332 333-373

375-404

405-425

CONTENTS OF PREVIOUS VOLUMES XV

Volumes VII-XIII, edited by D.E. Brewer

Volume VII (1978)

.

~

, ,

Q

,

,

,

,

Further experimental properties of superfluid 3He, J.C. Wheatley ............................................................................... Spin and orbital dynamics of superfluid 3He, W.E. Brink- man 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. Thou- less ........................................................................................ 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 .......................

1-103

105-190

191-281

283-369

371-433

435-516

517-589

591-647

649-746

Volume VIII (1982)

~

2. 3. 4.

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

1-66 67-132

133-220

221-288

Volume IX (1985)

,

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

1-142

143-264

265-344

xvi CONTENTS OF PREVIOUS VOLUMES

~

2. 3.

.

~

2. 3. 4.

~

~

~

~

.

~

,

~

.

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

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. O t t . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , - "

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

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-T c superconductors, N.E. Phillips, R.A. Fisher and J.E. Gordon ................................................

1-72 73-137

139-370

371-424

1-73 75-125

127-188

189-214

215-289

1-44

45-114

115-194

195-296

271-339

1-90

91-165

167-218

219-265

267-357

CHAFFER 1

T I ~ LANDAU C R I T I C A L V E L O C I T Y

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 McClin- tock (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 discov- ery (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 wave- function. 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 pic- tures will usually be found more apposite to any given type of problem. In con- sidering 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, w 1

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, w 1 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 + hk /2m)~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 He- II, 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

P.V.E. McCLINTOCK and R.M. BOWLEY Ch. 1,w

I !

15

lO

5

/ s

J

s I

s s

s s

" I I

0 10 20 klnm-1

30

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)

Ch. 1, w THE LANDAU CRITICAL VELOCITY 7

a superfluid if/)L = 0. On the other hand, it should be noted that there is nothing in Landau's argument to preclude other possible forms of dissipation, for example through the production of non-elementary (metastable) excitations, perhaps at velocities lower than v L. Ignoring such complications, however, and assuming that the temperature is low enough for the drag caused by excitation scattering to be negligible, the simple Landau picture suggests that the drag on an object moving through He-II should vary with its velocity as indicated in fig. 2. The drag remains zero until the critical velocity u L is reached, above which dissipation sets in very abruptly; the theory makes no prediction about how the drag varies with velocity above rE, however, and the curves (a), (b) and (c) would all be equally consistent with Landau's picture. Similar arguments can, of course, be applied to the case of He-II flowing through a channel showing that, for velocities less than v L, there should be no viscous resistance to flow.

The temperature dependence of the roton parameters of He-II (Brooks and Donnelly 1977) implies that 1) L must also be temperature dependent. Below 1 K, however, the dependence is extremely weak and all the investigations to be described below refer, in effect, to u L in its T--> 0 limit. This is, of course, usually the most interesting regime because the drag on a moving object due to the normal fluid component (the excitation gas) is then very small and the onset of dissipation at/)L i s correspondingly dramatic. For the opposite situation, however, where the superfluid component moves and the walls (and normal fluid component) are stationary, the onset of dissipation at 1) L can be well defined even at temperatures near that of the lambda transition, Ta. In particular, Andrei and Glaberson (1980) were able to find evidence for a finite/)L through the investigation of fourth sound resonances in a highly packed powder that effectively clamped the normal fluid component. As T---> T;t, A --> 0, but/Co remains

drag

b C

v L velocity

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

8 P.V.E. McCLINTOCK and R.M. BOWLEY Ch. 1, w

finite and so one expects that VL ---> 0 from eq. (2.5). What Andrei and Glaber- son observed was that, for 1.2 < T < T~t, the velocity of fourth sound was mark- edly 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 well- defined 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 den- sities in superconductors can be related to pair-breaking above a Landau critical velocity (Tilley and Tilley, 1990) in a very similar way. Such phenomena are important and extremely interesting, but they lie beyond the scope of the present chapter, which is devoted to the problem of roton creation in He-II as originally formulated by Landau.

2.2. Critical velocity measurements in He-H

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

Ch. 1, w THE LANDAU CRITICAL VELOCITY 9

Awschalom and Schwarz, 1984) sets in at a relatively low velocity and effectively masks the onset of roton creation at v L. An excellent review of recent experiments on orifice flow, in which discrete dissipative events are observed, has been given by Varoquaux et al. (1991). The events in question occur for flow velocities that are relatively high (several ms-l), but are still much smaller than VL. They are associated either with vortex depinning/repinning or, more probably, with vortex nucleation ab initio (cf. III, IV, VI).

The other possible experimental approach is, of course, to move an object through stationary superfluid. Negative and positive ions constitute particularly convenient objects for this purpose. They can readily be injected into the liquid by a variety of different techniques, they can be moved through the liquid by application of electric fields, and their arrival at an electrode can be observed as a pulse of current. The so-called ions which can exist in liquid helium are, in fact semi-macroscopic objects with radii of ~1 nm and effective masses of ~100m4, where m 4 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 mod- ern discussion and critical analysis will be found in Donnelly (1991); see also I- VI. 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


- - 9 9 e 9 9 - . , .

6 0 - T'.505~ P=

5 0 , - " " -

=30 1

20

!, I 0 9

O _ I I I I ... ! f ! I 0 I0 20. 30 40 50 60 70 80

-F.., Volts/cm 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 Ray- field'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 THE LANDAU CRITICAL VELOCITY I I

~ 40 0 ,,,J t, aJ

"1 - , ~ I ! 9 " - . . . ,.,

" " "" " , . . . . , , ,.. , .

-- . . . . , .

V e l o c I t y Necessory F o r R o t o n C r e o t i o n

m

I . . _ , , i i I 2 0 I I

P R E S S U R E I N A T M O S P H E R E S

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 univer- sally 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 dras- tically reduced. The latter feature of the phenomenon leads to spacecharge- limited 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 ena- bling the ionic mobility to be deduced. Field emission and ionization in liquid 4He have been studied in detail by Phillips and McClintock (1975).

12 P.V.E. McCLINTOCK and R.M. BOWLEY Ch. 1,w

<

200 --

9 100

50

30 0

I I I ' " I I

(a) I t

I I I 1 I

10 20

IO-sP/Pa

I i

O.D..O,,O..D~ 0 ~ 0

(b)

~176 P = 25x105 Pa

_ 10SPa

-o-... --.o... _ "7-- O.

I I I , , . ! 0.3 0.4 0.5 0.6

7/K

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

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

The reason that elevated pressures are necessary to support currents of bare ions travelling at a velocity ~VL, limited by roton emission, can now be under-

Ch. 1, {}2 THE LANDAU CRITICAL VELOCITY 13

velocity

Vc

VL

I i i i i i I !

0 10 20 P (bar)

Fig. 6. Sketch to show the dependences on pressure P of the Landau critical velocity V L, and of the critical velocity v e for the creation of a charged vortex ring by a negative ion. Roton creation experiments are possible for pressures above ~10 bar where the two curves cross over.

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

2.4. Measurement of ionic drift velocities

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

14 P.V.E. McCLINTOCK and R.M. BOWLEY Ch. I,w

(a) I I I I I o I I I I I I I I I I I o

Gl G~_

I

I

I I I I I I I

G3C

(b)

V~ ~

i i o

! !

!

I

1 I

,,

I ! !

i i

X

!

!

,( i |

tl

o I i i ,, I o ! ! ,

I( o o

t2

' t I ! !

i o i i

, i

1 t

I

i

! o I

I !

: i

Ol I I

t3 t( t 5 t s

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

time-of-flight technique introduced by Schwarz (1972) is overwhelmingly the best and most straightforward, and the least likely to lead to ambiguities or yield artefacts. It was used, in modified form, for all of the experiments described below.

The method, slightly modified for use with a field emission ion source, is illustrated diagrammatically in fig. 7. The ionic velocity is determined by measurement of the transit time of a disk of ions across a region of uniform electric field, between G2 and G 3, whose length is known. A retarding electric field of a few kilovolts per metre is usually maintained between the gate grids Gl and G2, thus preventing the ions emitted from the field emission source S from entering the drift space G2-G 3. By applying a negative pulse of a few tens of volts to Gl the gate can, however, be opened momentarily, thus admitting a group of ions to the drift space. These then propagate across G2-G 3 at a characteristic speed which will depend on the electric field and on the temperature, pressure, and purity of the liquid helium. Their arrival at the collecting electrode C can be

Ch. 1, [}2 THE LANDAU CRITICAL VELOCITY 15

observed as a pulse of current whose transit time can be used to compute the velocity. The Frisch grid G 3 is required to screen the collector from the influence of the approaching charge.

In practice, to minimize heating effects, it was necessary to pulse the field emission source, rather than running it continuously; and, because the resultant switching transients saturated the signal processing system, it was necessary to delay the gate pulse relative to the start of the tip pulse as illustrated in fig. 7(b). The current Ic seen at the collector is shown, in somewhat idealized form, in the lowest diagram; the transients at X and Y arise from the emitter switching on and off; and those at t I and t2 are caused by the gate pulse. The rise in ic at t 3 indi- cates the first appearance of ions on the collector side of G 3, and t4 represents their arrival at C. The passage of the back end of the pulse through G 3 occurs at t5, and the last ions reach the collector at t6. The periods ( t4- t3) and ( t6- t5) represent the transit time of an ion across the space between G 3 and C. The transit time rD between G2 and G3, which we wish to measure, is (t5 - t2).

The signal at the collector is usually comparable with, or smaller than, the intrinsic electrical noise of the system, so that some method of signal enhance- ment is an essential feature of the single-pulse technique; which is why it could not be used effectively until digital signal averagers became available in the early 1970s. In the experiments to be described, Nicolet 1080 and 1280 data- processors were used for averaging the collector signal.

The roton creation experiments were carried out in three main stages. First, using natural helium (before the effect of 3He isotopic impurities had been ap- preciated) in a 10 mm cell mounted in a 3He cryostat, the Landau velocity was observed unambiguously for the first time. The results of these experiments are presented in the next sub-section, below; fuller details are given in I. Secondly, using a very large (100 mm drift space) precision made cell mounted in a dilution refrigerator, and with a sample of isotopically purified 4He, roton creation was investigated in extremely weak electric fields in order to make a precise determination of VL; this experiment is discussed in section 4. Thirdly, as discussed in section 5, roton creation was studied in the extreme supercritical regime of high electric fields, using a very short (1 mm drift space) cell. (Historically, the project on the extreme supercritical regime was completed before the determination of v L) Additional velocity data were obtained from a quite separate series of experiments (see III) on vortex nucleation, based on analyses of electric induction signals rather than time-of-flight measurements.

2.5. Observation of the Landau critical velocity

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


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

fig. 8. The ionic drift ve loci ty was de t e rmined by m e a s u r e m e n t of the interval

t 5 - t2 (see fig. 7). K n o w i n g the length L separa t ing grids G2 and G 3, the drift

ve loc i ty U = U ( t 5 - t2) cou ld be found immedia te ly . The main sys temat ic er ror

Ch. 1,w THE LANDAU CRITICAL VELOCITY

TABLE 1 Values of the electric field E, the number n of repetitions in each average, and the vertical calibration factor, for each of the signals illustrated in fig. 8 (Allum et al. 1977).

17

Signal E (kV m -l) N Ic/div (nA)

a 50 1 0.625 b 50 2048 0.625 c 65 1 2.50 d 65 128 2.50 E 100 64 12.5 f 200 32 25.0

lay in the de t e rmina t ion of L, and a m o u n t e d to --5%. Ful l detai ls o f the c r y o g e n -

ics, e lec t ron ics and e x p e r i m e n t a l p rocedures are g iven in I.

51 i I 1 i I

4 tl- .4 .0 K

2; 3

r~

X

0 2

0 0 0 0 0 O -- 0 0 0 0 0

o _ o

o

0.35 K o o

O

, O _

o

o

O

o

o

o

- o -

o

o

', o , ~ ', I I I ._. I I , ~ J ) Z l

0 20 40 60

~/(m s -~)

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


It was found that the signal became extremely weak at the 3He cryostat's base temperature of 0.29 K, especially for low electric fields; the lowest temperature at which good data could be recorded was 0.35 K. Some results are shown in fig. 9, plotted in the form of the average drag force (= eE) as a function of the average velocity U of the ion through the superfluid for more convenient comparison with the predictions of fig. 2, to which they bear a remarkably close correspondence. For comparison, the corresponding curve for a negative ion in normal (non-superfluid) liquid 4He, where it can be characterized by a constant mobility, is also plotted. The behaviour is seen to be qualitatively different in the two cases.

The 0.35 K data of fig. 9 clearly indicate that the drag on the moving ion ap- proaches zero near the value of VL calculated from accepted values of the roton parameters (as shown by the dashed line). They can thus be regarded as providing a satisfying and convincing vindication of Landau's excitation model and of his explanation of superfluidity in He-II. It is also evident that the ionic velocity readily exceeds v L in electric fields that are not particularly strong (3 x 105 V m -I at maximum in fig. 9). In order to measure VL from data of this kind it is clearly essential to construct a theory of roton emission that will enable the measurements to be extrapolated back to zero drag, i.e. to zero electric field, in order to identify the velocity at which the onset of dissipation first occurs.

3. Theory of roton creation in He-ll

3.1. Early theories of supercritical dissipation

The first detailed discussions of supercritical dissipation in a Bose superfluid appear to be those of Iordanskii (1968) and Volovik (1970). When Bowley and Sheard (1975) sought to explain the first experimental results (Allum et al. 1975) several years later, they were unaware of these earlier calculations. They found, however, that they were able to account for the data in considerable detail with a kinetic theory based on of the roton pair emission hypothesis to be described below in section 3.4. We return to consider Iordanskii's and Volovik's contributions in more detail in the context of the actual experiment, in section 3.7.

The only other theory of supercritical dissipation in He-II prior to the experiments was apparently that of Takken (1970), who treated roton creation on the basis of a classical wave radiation model, in analogy with Cerenkov radiation. He assumed therefore that the object would, as it travelled through the superfluid, be pushing at a point of fixed phase in a roton wave pattern. He concluded that an upper bound on the velocity of a negative ion would be

Ch. 1,w THE LANDAU CRITICAL VELOCITY 19

l)u.b. -" I ) L ( 1 + 10-16E2), (3.1)

where Vu.b. and VL are in ms -l and the electric field E is in V m -1, and therefore that, for realizable electric fields, it should be almost impossible to observe any increase in the ionic velocity beyond VL. This remarkable conclusion appeared at the time to be in accord with Rayfield's (1966, 1968) experimental data, but it is clearly at variance with the results of fig. 9. Furthermore, the shape of the measured U (E) curve, which is similar to (c) of fig. 2, is quite different from Tak- ken's prediction, which would be similar to (b). It seems clear, therefore, that Takken's theory is not applicable to the motion of negative ions through pressurized He-II below 0.5 K. The reason is probably connected with the relatively small mass of the ion.

The assumption that roton creation is a coherent process is in fact of doubtful validity unless the mass of the moving object is very large, because conservation of momentum dictates that, in emitting a roton, the forward momentum of the object be reduced by an amount Av 1 = hko/m. Unless m is very large, these events will have a tendency to destroy any incipient phase coherence in roton creation.

It is evident that a theory to describe the data of fig. 9 must make no assumption of phase coherence, must take explicit account of the recoil of the (relatively lightweight) ion each time it creates a roton, and must be valid in the weak-coupling limit in which the ion can exceed the threshold velocity by a substantial margin.

3.2. Roton creation by a light object

In the case of a light object, the second term in eq. (2.1) cannot safely be ignored, and the critical initial velocity v~ for the emission of a roton may become substantially greater than VL. The ionic velocity ~" which we measure experimentally is, however, a time-averaged value over many roton creation events: we shall see that ~ need not be significantly larger than VL, even though v' I may be. As discussed in more detail in section 3.3, we will assume that the influence on U of the fluctuating thermal velocity of the ion can, to a good approxima- tion, be ignored.

We consider first an object travelling through superfluid 4He at a very low temperature such that the drag arising from the scattering of thermal excitations can be neglected, and under the influence of a force which is sufficiently weak that roton emission will occur at a velocity negligibly larger than v' l . Under these conditions the angle 0 between the direction in which the object is travelling and that in which the momentum of the roton is directed, is zero, and the problem is essentially one-dimensional. We assume that the emitted roton has


energy e' l, m o m e n t u m h x ' l, and that the final veloci ty of the object, after emis-

sion is v'[. Then eq. (2.1) becomes

v' 1 - e ' l / h k ~ + h k ' l / 2 m . (3.2)

In order to conserve m o m e n t u m

m y ' l = mv'~ + h k ' l,

whence, using eq. (3.2),

pt t p

v 1 - e l / h k 1 - h k ' l / 2 m . (3.3)

F r o m eqs. (3.2) and (3.3), the a v e r a g e veloci ty for the onset of dissipat ion which

we measure exper imenta l ly , cor responding to the t ime taken to travel a given

dis tance ( typical ly several tens of mm) divided by that distance,

(3.4)

and the ins tantaneous velocity of the object as a function of t ime must be de-

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

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

O L . . . . . . . . . . . . . . .

U~ ' _ . . . . . . .

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

t

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

Ch. 1, {}3 THE LANDAU CRITICAL VELOCITY 21

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

1 de e~.---k--- hkl (3.5) -'h k'i = hkl 2 m - '

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

h(~'~ -~o) a h2(~ I -~o) 2 hkl = ~ - ~ ~ . (3.6) m r hk ' l 2 m r h k ' 1 2 m

Solving this equation numerically for k~, assuming the values of the roton parameters for 25 bar, for various masses m of the moving object, yields the results shown in fig. 11. It is clear that although the recoil of the object, as indicated by the difference between v~ and v'~, is significant even for effective masses as large as a few hundred times m 4, the critical time-averaged velocity ~cl hardly deviates from VL provided that m > 30m4.

The negative ion, which may be regarded as a non-localized electron trapped within a spherical void in the liquid, has an effective mass which is almost entirely hydrodynamic in nature and which therefore depends only on the ionic radius r i and the density p of the liquid. [Note that there is some evidence (Ellis et al. 1983) for a slight distortion of the ion at high velocities, perhaps caused by the Bernoulli pressure, leading to a very small p4 term in the ionic dispersion relation. In what follows, we shall ignore such effects.] Information concerning ionic radii at different pressures has been derived from a number of sources including, particularly, measurements of the mobility and of the trapping lifetime on superfluid vortices; and the data are found to be in satisfactory agreement with the simple bubble model first described by Kuper (1961) and subsequently developed by Springett et al. (1967). The available experimental information has been correlated, on the basis of this model, by Schwarz (1975) who has concluded that the ionic radius at 25 x 105 Pa is between 1.08 and 1.12 nm, depending on the precise value taken for the surface tension under pressure. Assuming a liquid density of 172 kg m -3, the corresponding hydrodynamics mass (= ZTr= 2ytr/3 p13) lies between 68 and 76m4. We shall therefore assume that, for 25 bar,

(Progress in Low Temperature Physics 14) W.P. Halperin (Eds.)-Elsevier, Academic Press (1995)

Documents

low temperature physics

department of physics

usa volume

low temperatures

elsevier science

elsevier amsterdam

volume of progress

superfluid helium