No title

PROGRESS IN LOW TEMPERATURE PHYSICS IX

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PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

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

VOLUME IX

1986

NORTH-HOLLAND AMSTERDAM. OXFORD. NEW YORK .TOKYO

@ Elsevier Science Publishers B.V., 1986

All rights reserwd. 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 prior permission of the publisher, Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division), P.O. Box 103, loo0 AC Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered wirh the Copyright Clearance Cxnfer Inc. (CCC), Salem, Massachusens. lnformation can be obtained from the CCC about conditions under which photocopies of parts of rhis publication may be made in the U.S.A. AN other copyright questions. including photocopying outside of the USA, should be referred to the publisher. ISBN: 0 444 86971 9

PUBLISHED B Y :

NORTH-HOLLAND PHYSICS PUBLISHING A DIVISION OF

ELSEVIER SCIENCE PUBLISHERS B . V . P.O. BOX 103 1000 AC AMSTERDAM THE NETHERLANDS s o u DISTRIBUTORSFOR THE u s A AND CANADA

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue New York. N . Y . 10017 U.S.A.

PRINTED IN THE NETHFRLANIXi

PREFACE

In t h e Preface to Volume VII of this Series I noted the tendency for review articles to become longer but more limited in scope. Volume VIII saw a slight reversal of this tendency but it has resumed in the present volume: the average number of pages is 114, compared with 81 in Volume VII and 75 in Volume VIII (- and 23 in Volume I). This presumably reflects t h e more detailed understanding we have acquired, and as such it is to be welcomed, although it inevitably means that the breadth of an individual’s interest must decrease. Volume IX contains three articles, two of which are on superfluid hydrodynamics and the third on glasses. As an illustration of the comments made above, Glaberson and Donnelly’s article on vortices i n superfluid 4He has had to be restricted even within this particular subject, and several topics of great current interest in vortices have had to be excluded. The problem of the generation of vorticity in helium I1 is still not understood, and will doubtless form t h e subject of a future article. A further illustration of the increasing length to breadth ratio is provided by Hall and Hook’s article on t h e hydrodynamics of superfluid 3He which is made additionally intricate by the complex structure of its order parameter. It builds o n the Brinkman and Cross review of spin and orbital dynamics in Volume VIIa but now it has been necessary to restrict it to orbital dynamics; again, we expect to deal with spin dynamics in superfluid 3He in a future article. Finally, Hunklinger and Raychaudhuri’s article on glasses provides some balance to what would otherwise have been a volume on superfluid helium, and reminds us that some disordered condensed systems, which used to be shunned by many experimental and theoretical physicists, are now regarded as tractable problems. The problem of balance within a particular volume is of course greatly exacerbated by the length-to-breadth effect. Without restricting the authors unreasonably, the only solution is to provide balance over successive volumes, which I hope to do. I shall also repeat my promise to produce t h e next volume in a shorter interval of time, with a confidence I believe to be better founded than last time. As usual, I am grateful to many colleagues for discussion of articles for this Series, to the authors for writing them, and to the publishers, in particular Professor P. de Chitel, for their help.

D.F. Brewer

Sussex, 1985 V

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CONTENTS VOLUME IX Preface

V

Ch. 1. Structure, distributions and dynamics of vortices in helium Il, William I. Glaberson and Russell J. Donnelly . . , . . . . . . . . . .

1

1. Introduction . . . . . .. . . ........... ......._ 5 2. The structure of quantized vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. Equilibrium vortex distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4. Vortex dynamics-steady state . . . . . . . . . . . . . . . . 5. Vortex dynamicswaves . . . . . . . . , . . . . . . . . . . . . References . Note added in proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Ch. 2. The hydrodynamics of superfluid 'He, H.E. Hall and J.R. Hook ............................................

143

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ . . . . _ . . . . . . . . . . . . . 2. The thermodynamic basis of hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . 3. The interaction between flow and textures in 'He-A . . . . ..... . 4. SuperRow in 'He-A and 'He-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Uniformly rotating 'He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. Measure rrnodynami rameters .. .......... ........................ References

145 146 171 200 236 248 259

Ch. 3. Thermal and elastic anomalies in glasses at low temperatures, S. Hunklinger and A.K. Raychaudhuri . . . . , . . . . . . . . . . . . . . 265 ...........

. . . . . . . . . . . _ . _ . 267 . . . . . 269 Metallic glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 "Glassy properties" of disordered crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Origin of the tunneling systems-theoretical attempts . . . . . . . . . . . . . . . . . . . . . . . . . 329 Connection between low temperature anomalies and the glass transition temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

1. Introduction 3. 4. 5.

6. 7.

Author I n d e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 ,

Subject Index

,

,

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

vii

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

359

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CONTENTS OF PREVIOUS VOLUMES

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

Volume I(1955)

I

111 IV V

VI VII VIII IX X XI XI1 XI11 XIV

xv XVI XVII XVIII

The two fluid model for superconductors and helium 11, C.J.Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of quantum mechanics to liquid helium, R.P. Feynman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rayleigh disks in liquid helium 11, J.R. Pellam . . . . . . . Oscillating disks and rotating cylinders in liquid helium 11, A.C. Hollis Hallett . . . . . . . . . . . . . . . . . . . . . . . . . . . The low temperature properties of helium three, E.F.Hamme1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 specificheat in metals, J.G. Daunt . . . . Paramagnetic crystals in use for low temperature research, A.H. Cooke . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiferromagnetic crystals, N.J. Poulis and C.J. Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic demagnetization, D. de Klerk and M.J. Steenland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical remarks on ferromagnetism at low temperatures, L. NCel . . . . . . . . . . . . . . . . . . . . . . , . . . . Experimental research on ferromagnetism at very low temperatures, L. Weil . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity and absorption of sound in condensed gases, A.vanItterbeek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transport phenomena in gases at low temperatures, J. de Boer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1-16 17-53 54-63 64-77 78-107 108-137 138-150 151-158 159-183 184-201 202-223 224-244 245-272 272-335 336-344 345-354 355-380 381-406

x

CONTENTS OF PREVIOUS VOLUMES

Volume I1 (I 957) I I1 111

IV V

VI VII VIII IX X XI XI1

XI11 XIV

Quantum effects and exchange effects on the thermodynamic properties of liquid helium, J . de Boer Liquid helium below 1°K. H.C. Kramers . . . Transport phenomena of liquid helium I1 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, V.A. 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 83-104 105-137 138-150 151-186 187-225 226-265 266-29 1 292-337 338-367

368-394 395-430 431-464

Volume III (1 961)

I 11 111

1v V

v1 VII

Vortex lines in liquid helium 11, W.F. Vinen . . . . . . . . . Helium ions in liquid helium 11, G. Careri . . . . . . . . . . . The nature of the h-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 11, W.J. Huiskamp and H.A. Tolhoek . . . . . . . . . . . . . . . , Solid state masers, N. Bloembergen . . . . . . . . . . . . . . . . ,

VIII

IX

1-57 58-79

80-1 12 113-152 153-169 170-287 288-332 333-395 396429

CONTENTS OF PREVIOUS VOLUMES

X XI

The equation of state and the transport properties of the hydrogenic molecules, J.J.M. Beenakker . . . . . . . . . . . Some solid-gas equilibria at low temperatures, Z.Dokoupi1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

430453 454-480

Volume IV (1964) I

I1 I11 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. Ntel, 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 296343 344-383 384-449 450479 480-5 14

Volume V (1967)

I I1 I11

IV

V

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

143

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

xii

VI VII

CONTENTS OF PREVIOUS VOLUMES

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

235-286 287-322

Volume VI (1970)

I I1 I11

1v V

VI

VII

Vlll

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 He3 and dilute solutions of He3 in superfluid He' 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. H u h , 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.GloverII1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . 40-5-425 ..

Volumes V t t , VIM, edited by D.F. Brewer

Volume VII (1 978) 1

2 3

Further experimental properties of superfluid 'He, J.C.Wheatley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin and orbital dynamics of superfluid 'He, W .F. Brinkman and M.C. Cross . . . . . . . . . . . . . . . , . . . Sound propagation and kinetic coefficients in superfluid 3 H e , P . Wolfle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The free surface of liquid helium, D.O. Edwards and W.F.Saam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

4

,

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

CONTENTS OF PREVIOUS VOLUMES

5 6

7

8 9

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. Gruner and A. Zawadowski .................... Application of low temperature nuclear orientation to metals with magnetic impurities, J. Flouquet . . . . . . . .

...

Xlll

371-433

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

Volume VZII (1982) 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

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CHAPTER 1

STRUCTURE, DISTRIBUTIONS AND DYNAMICS OF VORTICES IN HELIUM 11* BY

William I. GLABERSON Department of Physics and Astronomy, Rutgers University, Piscata way, NJ 08903, U S A and

Russell J. DONNELLY Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403, USA

* Research supported by the Low Temperature Physics Program of the National Science Foundation, under grants DMR 83-19941 and DMR 83-13487.

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

Contents

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

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

3

. . . . . . . . . . . . . . . . . . 19 2.5. The bound excitation model.. ......................... . . . . . . . . . . . . . . 29 2.6. The Hills-Roberts theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7. Stochastic behavior of ions in the presence of vortices.. . . . . . . . . . . . . . . . . . . . . 41 ................................... 47 ............................ 49 ................................... 56 4. Vortex dynamics-steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1. Mutual friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2. Thermorotation effects . . . . . . . . . . . . . . . . . . . 74 4.3. Vortex pinning . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4. Spin-up and the v o r t e x s . . . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . 95 4.5. Vortex dynamics in thin 5. Vortex dynamics-waves . . . . ................................... 101 5.1. Isolated vortex lines . . . ..................... 101 5.2. Collective effects-infinite vortex arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 126 5.3. Collective effects-finite vortex arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. A vortex instability . . . 5.5. Thermally induced vortex w 5.6. The effect of mutual friction on References . . . . . . . . . . . . . . . . . . . . . Note added in p r o o f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

List of symbols A A

a B B' d d0

D D D' D, E E F

F"S

f fD fa

f, fM f P

G h, h k kB

Gorter-Mellink mutual friction coefficient A Helmholtz free energy used in healing calculations vortex core parameter dimensionless mutual friction parameter dimensionless transverse mutual friction parameter film thickness width of vortex free strip in rotating helium I1 vortex diffusivity in films microscopic mutual friction coefficient transverse microscopic mutual friction coefficient D' - p s ~ internal energy frequency factor in mutual friction theory Helmholtz free energy mutual friction force temporary symbol for bulk superfluid density drag force per unit length of vortex line drag force due to excitations on a vortex line Iordanskii force per unit length of vortex line Magnus force per unit length of vortex line pinning force per unit length of vortex line Gibbs free energy Planck's constant, Planck's constant divided by 2 7 ~ wave vector Boltzmann's constant reduced stiffness constant in thin films kinetic energy of vortex ring angular momentum mass of 3He atom mass of 4He atom pressure momentum of elementary excitations impulse of vortex ring roton momentum

K K L m3 m P P P Po 4 us- V" Q heat flux

W.I. GLABERSON AND R.J. DONNELLY

1

R vortex ring radius S entropy per gram T absolute temperature lambda transition temperature Kosterlitz-Thouless transition temperature TA- T potential energy velocity of a vortex line velocity of normal fluid velocity of superfluid group velocity of rotons vortex excitation probability - 0, w evaluated at the Landau critical velocity

V"

attenuation of second sound owing to vortices (TA

-

energy of elementary excitations in helium I1 as ; i function of momentum p Euler's constant drag coefficient per unit length of vortex line drag coefficient per unit length of line drag coefficient per unit length of line displacement length in healing theory semiminor axis of elliptical motion in Tkachenko waves semimajor axis of elliptical motion in Tkachenko waves roton energy gap coefficient of bulk viscosity lattice sum quantum of circulation (= h/rn,) roton effective mass chemical potential ( ~ / 4 nIn(b/a) ) coherence length viscous vortex-boundary force coefficient healing length stress tensor total density of liquid helium total density of liquid helium at the lambda point superfluid density normal fluid density superfluid areal density in a film parallel scattering length for vortex lines transverse scattering length from vortex lines

[Ch. 1

Ch. 1, 511 7

VORTICES IN HELIUM I1

5

relaxation time

4 phase of the condensate wave function +b wave function for condensate

0 angular velocity of rotation angular frequency V x us+ 20, vorticity in the laboratory frame 6 radius of small vortex rings w w

1. Introduction

The idea of quantized vortices in superfluid helium was first put forth to students and colleagues at Yale University by Lars Onsager about 1946. The announcement to the scientific world was in a celebrated remark following a paper by Gorter on the two-fluid model at the Conference on Statistical Mechanics in Florence in 1949. He said, in part, “Thus the well-known invariant called hydrodynamic circulation is quantized; the quantum of circulation is h l m . . . In case of cylindrical symmetry, the angular momentum per particle is R . . .” (Onsager, 1949). It is difficult to recall any single remark in the history of science which has had more far-reaching consequences. Progress in Low TemperaturePhysics has recorded much of the progress in this field, beginning with Feynman’s seminal paper (1955) in Volume I on the application of quantum mechanics to liquid helium. It is also difficult to recall many review articles as influential on a field of physics as that one. Further articles have been a review of vortex lines in liquid helium I1 by Vinen (1%1) in Volume 111, an article on critical velocities and vortices by Peshkov (1964) in Volume IV, an article on flow phenomena in superfluid helium by de Bruyn Ouboter et al. (1967) in Volume V, and a further article on the rotation of helium I1 by Andronikashvilli and Mamaladze (1%7) in the same volume, an article on intrinsic critical velocities by Langer and Reppy (1970) in Volume VI, an account of two-dimensional systems including vortices in thin films by Kosterlitz and Thouless (1978) in Volume VII@), and an article on superfluid turbulence by Tough (1982) in Volume VIII. Other articles, of course, have used the concepts of quantized vortices freely. The authors believe that it is not generally appreciated how much knowledge about the structure and behavior of quantized vortices has been accumulated. We note in passing, however, that Fetter’s valuable review of ions and vortices published in 1976 listed 512 references! It is manifestly impossible in 1985 to review every article published on vortices, and we have not even attempted to do so. Instead we have concentrated on a few areas which we hope will be of interest in

6

W.I. GLABERSON AND R.J. DONNELLY

[Ch. I, 31

illustrating the directions our field has taken in recent years. The plan we have adopted is to organize this knowledge in five sections. The introduction contains a few remarks about quantized vortices and the various configurations that are thought t o occur. The structure of vortices is discussed in section 2 including the interaction with impurities. Section 3 deals with equilibrium vortex distributions such as are induced by rotation. Sections 4 and 5 have to do with vortex dynamics. The first of these discusses steady state effects: friction on vortices, the interaction of heat flow and rotation in one particular geometry, vortex pinning, spin-up and finally vortices in thin films. The final section treats vortex waves in the individual and collective limits as well as several specialized topics: vortex stability in the presence of axial flow, thermal fluctuations, and the effect of friction on vortex waves. These restrictions mean that we shall not be able to comment on some topics of considerable current interest, such as superfluid turbulence and thermal nucleation of vortices. Both these topics, however, are represented in other articles in this series referred to above. If helium I1 is contained in a rotating bucket it is known experimentally that at low speeds of rotation the normal fluid will rotate with the container and the superfluid will remain at rest. At a certain critical angular velocity of rotation a single quantized vortex line will appear in the center of the container. If the core is assumed to be hollow and of radius a then the energy per unit length of the vortex line will simply be the kinetic energy of rotation of the fluid about the core: E

=

[ &,u:

d2r = @ , ~ ~ / 4 In(bla) 7~) ,

J

where pr is the superfluid density, b is the radius of the bucket, and K = hlm is the quantum of circulation. This is an enormousoenergy: assuming bla = lo’, it amounts to 1.85x lO-’erg/cm or 13.4 KIA at low temperatures. Any energy, or more properly free energy, associated with t h e core structure can be absorbed into the core parameter. The centrifugal force on each ring of fluid surrounding the core is balanced by a pressure gradient: dpldr = p p f l r = p , ~ ~ / 4 . r r ~ r ’ . A sketch of the velocity and pressure distributions around a vortex line is shown in fig. 1.1. Next suppose we have isolated pairs of vortex filaments a distance d apart. If the circulations of the pair are in opposite directions the energy per unit length of the pair is given by &

= (PSK2/27r)In(d/a)

(1.3)

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7

VORTICES IN HELIUM I1 v s (m/s)

-5

-4

-3

-2

-;I

1

I

2

,

3

4

5 4

-eo P (bars) Fig. 1.1. Supeduid velocity andopressure about a rectilinear vortex line located at r = 0, having a hollow core of radius 1 A and circulation K = h/m. The radius at which the Landau critical velocity W, is reached is marked by the outer dashed line.

whereas if the circulations are parallel the energy per unit length is given by &

= (p,K2/2T) ln(b2/ad).

(1.4)

The situation is shown in fig. 1.2. Neglecting the mass associated with the cores, each vortex filament will move with the velocity from the opposite one so that the parallel filaments will rotate about a point halfway between them with an angular velocity w = K / ? r d z whereas the antiparallel filaments will move through the fluid with velocity t, = ~ / 2 ~ The d . “impulse” of the lines, or the effective momentum per unit length associated with the vortex motion is p,Kd for the oppositely directed pair. Suppose a bucket of helium I1 is rotated at relatively high rates of speed 0.Under these circumstances it is known experimentally that the fluid rotates as a solid body, and therefore the average vorticity of the

[Ch. 1, $1

W.I.GLABERSON AND R.J. DONNELLY

V

V

V

V

Fig. 1.2. Behavior of parallel vortex filaments in each other’s flow fields: (a) filaments with circulation in the same direction, (b) filaments with opposite circulation.

superfluid must be 2 0 . Equating this vorticity with the effect of a uniform array of n quantized vortices per unit area parallel to the axis of rotation we obtain n~ = 20. n corresponds to about 2000 vortices per cm2 at a rotation rate of 1 radian per second. Another arrangement of vortices occurs in very thin films of helium 11 where it is conjectured that there is a distribution of antiparallel pairs of vortices so that seen from above the distribution looks like a two dimensional Coulomb gas. This distribution of vortices is thermally activated. We shall return to this discussion in section 4.5. Experiments on helium I1 in wide channels by Awschalom and Schwarz (1984) suggest that there are residual vortices present upon cooling through the lambda transition. Fig. 1.3 shows a conjectural sketch of how such vortices might look, pinned to protuberances of various sizes on the boundaries of t h e channel. Heat induced counterflow (beyond some critical heat flux) down a channel open to a bath at one end and closed by a heater at the other

Fig. 1.3. Conjectural drawing of residual vortices in a channel.

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VORTICES IN HELIUM I1

9

Fig. 1.4. Conjectural drawing of a three dimensional vortex tangle projected on a plane.

produces turbulence in the superfluid component which is thought to consist of a tangle of quantized vortex lines. A conjectural sketch of the appearance of such a tangle in a counterflow is shown in fig. 1.4. Vortex rings are produced at low temperatures by the motion of negative and positive ions (Rayfield and Reif, 1 W ) . When control grids are properly arranged, the ion, which is trapped in the core of the vortex ring, can generate vortex rings so large that the size of the generating ion is negligible. We show in fig. 1.5 a vortex ring of radius R and hollow core radius a. The energy, velocity and impulse for a ring where R s=u are given by

E

= ~ p , ~ ~ R I l n ( 8 R-/ 2 a1),

(1 -5)

u = (~/4?rR)[ln(8R/a)- 11,

(1.6)

P =pS~?rR2.

(1.7)

Fig. 1.5. A quantized vortex ring in helium I1 of radius R and core size a.

I0

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, $1

An important generalization of the vortex ring calculation was obtained by Arms and Hama (1%5). Its importance lies in making possible approximate calculations of the motion of arbitrary configurations of the vortex line. The fluid velocity at some point in space, induced by a vortex line, is given by the Biot-Savart law:

where r is the distance from the point considered to a point on the vortex line and ds is a segment of the vortex line. The integral is over the entire length of the line. Arms and Hama pointed out that, as long as the vortex line radius of curvature is much larger than the core radius, a vortex line bent into any shape will have a velocity induced at a point on the line, due to neighboring line segments, which can be approximated by

where R is the radius of curvature of the line at the point considered, and L is of the order of some characteristic macroscopic length-R, the wavelength of a line perturbation, or the interline spacing, whichever is smaller. This locally induced velocity is perpendicular to t h e plane of the curve. An example of a modified ring is the vortex loop which is believed to be formed on an ion moving through the fluid at a critical velocity at very low temperatures. This situation, a matter of considerable current interest, is illustrated in fig. 1.6, where the loop is shown just at the moment of nucleation.

Fig. 1.6. Conjectural drawing of a vortex loop on an ion of radius R , . The circulation of the loop IS shown, the direction of the ion is into the page. (After Muirhead et al., 1984.)

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11

In spite of the variety of vortex configurations mentioned above, we shall restrict our discussion primarily to vortex lines, arrays of lines, and rings. Most of the fundamental properties of quantized vortices can be understood from these simple configurations, and we believe that insight into other configurations will be enhanced by a thorough understanding of the simplest cases.

2. The structure of quantized vortices

There is a somewhat complacent view in the low temperature community that quantized vortices in helium I1 are “understood”. We shall show in this section, and the corresponding sections on the experimental situation, that in fact very little is known about the structure of quantized vortices, either experimentally or theoretically. It is a conservative statement to say that no fully satisfactory theory exists for a vortex structure, applicable at all temperatures. There are certainly useful hints from several directions and we shall discuss some of them. We shall describe a highly tentative compromise view which we believe is consistent with the experimental evidence and which permits one to make definite calculations at all temperatures and pressures in helium 11. 2.1. DYNAMICS OF CLASSICAL VORTEX

RINGS

It is possible to gain some insight into vortex dynamics and t h e structure of the cores of vortices of constant circulation by considering the classical expressions for the energy and velocity of vortex rings. We shall need to use such classical expressions in order to interpret the results of experiments in helium I1 to be described in the following section. We shall show that such rings can be described by a total energy formally equivalent to a Hamiltonian E and that the velocity and impulse of the vortex rings are connected by Hamilton’s equation (Roberts and Donnelly, 1970) v=

aE/aP.

(2.1.1)

The classical expressions for the kinetic energy K, t h e velocity v and the impulse P, of a circular vortex ring with a hollow core are (Hicks, 1884): = iprc2R[1n(8R/a)- 21 ,

(2.1.2)

u = (~/4d?)[ln(8R/a)- i ]

(2.1.3)

K

12

W.I. GLABERSON AND R.J. DONNELLY

P = prtrR’,

[Ch. 1. 82

(2.1.4)

where R is the radius of the ring, a the radius of the core, presumed infinitesimal compared to R, and p is the density of the fluid. The approximation a 4R allows us to neglect the ellipticity of the core and other details not germane to the present discussion. For a modern discussion of classical vortex rings see Fraenkel (1970). A vortex ring having fluid in the core of the same density as that outside, but rotating as a solid body, is described by the equations (Lamb, 1945)

K

= ;p~’R[ln@R/a)-7/41,

(2.1.5)

The problem with eqs. (2.1.2)-(2.1.6) is that if K is considered t h e total energy, then Hamilton’s equation is not obeyed. This problem can be resolved by paying attention to the details of t h e core structure, which we shall do in the following paragraphs. In order to clarify the pressure relationships near a vortex ring, consider the flow near enough to a vortex that it may be considered to be straight as shown in fig. 1.1. Suppose the pressure inside the core is pc, the surface tension of the fluid is u and the pressure at distances far from the ring is pm.The forces acting at r = a are due to the external pressure ps and the surface tension which exerts a pressure u[(l/a)+ (l/R)] = u/a since a R. This is opposed by the pressure gradient of t h e circulating fluid given by eq. (1.2). Thus

pl + cr/a - pc =

laz

(pK2/4n2)r-3dr= prc*/8m*a*.

(2.1.7)

Suppose the vortex ring is considered hollow, the surface tension and core pressure both zero and the fluid is assumed incompressible. Then, if t h e volume V of fluid plus core is kept constant, V = Vi+V, = constant. To the order we are working V, = 2 r 2 a 2 Rand d V = d(2sr’a’R)

=

0

(2.1.8)

since V , is constant. Upon applying an impulse d P to the ring, the radius will increase by dR according to eq. (2.1.4), but a will decrease according to eq. (2.1.8). The operation of keeping t h e volume fixed has increased P=, but for an incompressible fluid this does not change t h e energy of

Ch. 1, 921

VORTICES IN HELKJM I1

13

motion. The total energy of the system E = K to within an additive constant and the velocity comes from differentiating E at constant volume which can be done by writing

E = Ip~~R[ln(87r(2R~)’~vr”) - 21

(2.1.9)

and

which reproduces eq. (2.1.3). Thus we see that eqs. (2.1.2) and (2.1.3) apply to hollow vortex rings with pc= u = 0, whose core volume is constant. Suppose the vortex core is considered to contain uniformly rotating fluid as in eq. (2.1.5). We note that the same particles must always move with the vortex filament, so that again the volume of the core is preserved. The moment of inertia of t h e core, considered as a line of length 27rR is I = p.nZRa4,and the angular velocity of the fluid in the core is (to leading order) o = ~ / 2 7 r u ~ Thus . the angular momentum of the core is ~ ~ K T Rwhich u ~ , must be conserved, and this conservation is equally expressed by eq. (2.1.8). The kinetic energy of the core is floZ = 1/8p~’R which is the difference between the kinetic energies of eqs. (2.1.2) and (2.1.5). The velocity u = (aE/aP), is given by eq. (2.1.6). Thus we have learned that the classical expressions (2.1.2), (2.1.3), (2.1.5) and (2.1.6) apply to constant volume cores. Now consider the case of constant pressure and a hollow core of fixed radius a. We take pc = u = 0 and hence keep p- fixed upon applying an impulse dP. As the core is lengthened the external surface of the liquid is displaced against the pressure p-, doing work which may be retrieved when the core is shortened. This means the system has, in a formal sense, a potential energy U given by

u = p , ~ ,= f p ~ ’ ~ .

(2.1.11)

Using eq. (2.1.2) for K

E

=K

+ U = ~prcZR[In(8R/a)- i]

and the velocity of the ring is given by

again recovering (2.1.3).

(2.1.12)

14

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 22

Finally, consider the case of a core with surface tension, t h e model originally quoted by Feynman (1955) to estimate the core size of a quantized vortex line. Here p, = p=, and from (2.1.7) we find (+

(2.1.14)

= prc2/8.rr2a

and the core radius is determined entirely by surface tension. Again we have a constant core radius model. Now the core has a surface energy cAc where A,=4.rr2aR is t h e surface area of the core. The surface entropy T(dc/d T ) A , is neglected. Considering the surface energy formally as a potential energy (2.1.15) the total energy and velocity of a vortex governed by surface tension is E

=

K+U

c

=

(aE/aP), = ( ~ / 4 v RIn(8Rla). )

= fprc2R[In(8R/a)-

11, (2.1.16)

Consider now rectilinear vortices. For a hollow vortex in a container of radius b, the kinetic energy p e r unit length is given by eq. (l.l), and this is the total energy unless one wishes to include the work done against external pressure in creating t h e core, in which case U = p m m 2= prc2/87r, and

E = (p~~/4.rr)[In(b/a)+ 51 .

(2.1.17)

Similarly a fluid core rotating as a solid body gives (2.1.18) Roberts and Donnelly’s demonstration that large circular rings (a 4 R ) obey Hamilton’s equation (2.1.1) leads one to enquire whether the equation of motion of rings of arbitrary a/R can also be written in canonical form. Rings of moderate a/R do not have circular crosssections, and it is necessary to redefine a and R : for example the cross-sectional area can be taken to be m2,and R can be chosen to be the mean of the closest and farthest points of the vortex core from the axis of symmetry (Fraenkel, 1970). Roberts (1972) demonstrates that eq. ( 2 1 . 1 ) i s obeyed exactly for circular vortex rings, for all a/R, no matter what their core structure may be. Moreover, Roberts shows that the

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15

governing equations of motion for a set of vortex rings separated by distances large compared to their dimensions can also be written in canonical form. [A few results on interacting vortex rings have been reported for liquid helium (Gamota et al., 1971; Hasegawa and Varma, 1972).] Dyson (1893) and Fraenkel (1972) showed that for one particular core structure, the so-called “standard model” where vorticity in the core increases with radius out to the edge of the core, that a sequence of vortex rings exist, which range continuously from a large circular vortex at one extreme to Hill’s spherical vortex at the other. Further, Fraenkel showed that for all sufficiently small a/R, similar families of vortex rings exist for a wide variety of core structures. When a / R starts to become large, the expressions quoted in this section for energy and velocity are not applicable and one should consult the papers by Fraenkel (1970,1972), and Norbury (1973) for details. Plots of energy and impulse for the classical rings of Norbury (1973) are shown in fig. 2.1.1. We shall return to the question of small rings again in section 2.4. This section of our review should not be closed without a few remarks on the stability of classical vortex rings. Vortex rings in nature have generally been considered to be both stable and persistent flows. Indeed the work of Kelvin (Thomson, 1867), J.J. Thomson (1883) and others was directed towards establishing the stability of vortex rings and calculating their frequencies of oscillation in order to investigate atomic structure. In practice, however, vortex lines (such as aircraft trailing vortices) and vortex rings are often unstable. A recent and accessible review of the subject has been given by WidnaIl (1975). The instability of a thin vortex ring of “uniform” vorticity (r-’ curl u =constant, where r is measured from t h e center of the core) in an ideal fluid has been considered in detail by Widnall and Tsai (1977). They find that such a vortex is unstable to short azimuthal bending waves (ka = 2.5 appears to be dominant, where k is t h e wavenumber) but the lowest mode which is unstable is the so-called second radial mode where t h e core moves in a direction opposite to the outer flow. A photograph of such an instability is shown in Widnall (1975), Widnall and Tsai (1977). Quantum vortices are, however, known experimentally to be as stable as Lord Kelvin could have wished. There is no experimental evidence for their breakup owing to internal instabilities. While the theoretical basis for the following statement is lacking, it seems reasonable to assume that the instability described by Widnall’s group is associated with vorticity in the core and therefore will not be expected to occur either for hollow classical vortices, or for quantum vortices. We direct the reader interested in learning more about classical vortices

[Ch. 1, $2

W.1. GLABERSON AND R.J. DONNELLY

16

250

2 00 1s o

w 100

50 0 1

0

2

3

4

5

3

4

S

R/a

5 00

400 300

a 200

I00

0 0

I

2 R/a

Fig. 2.1.1. (a) Energy E and (b) impulse P for the classical rings of Norbury are shown for vortex rings as a function of R/n. The dashed lines represent the equations E = 27r'R (In(SR/o)- 7/4} and P = 27r2R2/a2.The units of energy and momentum are pu2a/4rr2 and prta'/2n to make these plots directly comparable to those for quantum rings in fig. 2.4.2. The circled points are rhe calculated results. (After Muirhead et al., 1984.)

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VORTICES IN HELIUM I1

17

to a remarkable survey of classical vortex motion which has recently been published (Lugt, 1983). 2.2. VORTEX RINGS IN HELIUM 11 Rayfield and Reif (1964), in their paper on quantized vortex rings, used the rotating core model given by eqs. (2.1.5) and (2.1.6) to analyze their data for evidence of quantization of circulation. Their data on the energy and velocity of vortex rings made possible the first absolute determination of the core parameter and quantum of circulation. The analysis of a vortex ring experiment requires the adoption of one of the models of section 2.1 above to describe the energy and velocity of vortex rings. Let us write the general case as

u = K/(47rR)[ln(8R/a)- /3]

.

(2.2.2)

Writing the product of the expressions for energy and velocity yields an expression for their product

( u E ) ’=~B”{ln E - In[(uE/B)”+ i ( p - a)]} -I-B’”[ln(l6/p,~~a):(a + p ) ] ,

(2.2.3)

a

where B = p,~~/87r. Using Q = f, p = Rayfield and Reif plotted their original data as a function of the term in curly brackets in eq. (2.2.3). The data formed a straight line with slope B”’,which determined K. The intercept then determined a. Since t h e term in curly brackets requires knowledge of B, they used successive approximations to fix B. The results were K

=

(1.00*0.03~10-3)cm2s-’,

a = (1.28kO.13)A.

(2.2.4)

Since the theoretical value of K = h/m = 9.97 x cm2s-’, the result for circulation was gratifyingly close to Onsager’s (1949) prediction. The problem with Rayfield’s and Reif’s choices of a and p is that they actually belong to the rotating core model, which is a constant core volume model. Since vortex rings nucleated by ions grow from Bngstrdms to microns in size, there is no possibility that the data could be consistent with a constant volume model. We have described two constant core radius models, one with surface tension, and one with a potential energy term owing to work done in

18

[Ch. 1, 92

W.I. GLABERSON AND R.J. DONNELLY

creating a hollow core against external pressure. Neither model is particularly relevant to helium 11. The latter, with a = /3 = 51 was recommended by Roberts and Donnelly (1970). The exact model chosen is a matter of taste unless one wishes to obtain absolute data. Most classical theories of vortices useful in t h e study of quantum vortices are appropriate to hollow vortices with no potential energy explicitly associated with the core, i.e., the free energy per unit length of the vortex is completely given by the core parameter. Such expressions are eq. (1.1) for vortex lines and eq. (1.5) for vortex rings. Thus we adopt a = 2, p = 1 as in eqs. (1.5) and (1.6) as perhaps the most conservative choice. Having made this choice we have re-analyzed Rayfield and Reif's experiment assuming K is known a priori, with the result a = (0.81 5 0.08) A. Rayfield (1968) first demonstrated that the vortex core parameter increases with both pressure and temperature. Shortly thereafter Steingart and Glaberson (1972) performed a series of highly accurate vortex core parameter experiments, using a time-of-flight technique. They were careful to allow for the effects of drag on the vortex core and ion, which is noticeable even at the lowest temperatures. Their results were interpreted on t h e Roberts-Donnelly (1970) recommendation: (Y = i, 0 = i. In view of our discussion above we can approximately correct the data to coincide with the model a = 2, /3 = 1 we are suggesting here. We do this by noting that the last term in eq. (2.2.3) being the intercept on Rayfield and Reif's analysis remains constant under different assumptions for a and p. It follows that the last term in square brackets in eq. (2.2.3) is also constant, which implies that a y = constant where (a + p)/2 = In y. Thus the data of Steingart and Glaberson can be approximately corrected by multiplying their core parameters by a factor of -0.61. The results of the Rayfield-Reif and Steingart-Glaberson experiments are shown in table 2.2.1. It is interesting to note that eq. (2.1.14) gives a = 0.48 8, assuming cr = 0.378 dyn/cm. All analyses give a vortex core parameter substantially smaller than the mean interatomic spacing 3.6 A. The pressure dependence of the relative core parameter a/a, was also measured by Steingart and Glaberson. Their results are summarized in table 2.2.2, where the error in a/a, is k0.03. It is interesting to note that the vortex core parameter increases with both temperature and pressure,

i,

Table 2.2.1 Temperature dependence of the vortex core parameter. ~~~~

T (K) 0.28

a(h)

0.35 0.40 0 . 8 1 ~ 0 . 0 80.7720.02 0.77+0.02

0.45 0.50 0.55 0.60 0.8O-tO.02 0.79k0.02 0.8220.02 0.84?0.02

Ch. 1, 021

19

VORTICES IN HELIUM I1 Table 2.2.2 Pressure dependence of the relative core parameter at T = 0.368 K.

P(atm) a/@

4.4 6.9 0 3.7 1.0 1.06 1.05 1.11

7.5 10.5 11.1 13.7 14.6 18.0 20.1 21.0 24.3 1.10 1.15 1.17 1.16 1.21 1.21 1.23 1.24 1.29

an observation which must be accounted for by any successful theory of core structure. Apart from an early estimate of a based on a vortex wave experiment by Hall (1%0), the results quoted above are all the existing direct data on t h e vortex core parameter. Vortex ring experiments cannot be extended to much higher temperatures because the drag on the rings becomes too large. There are experiments on measuring the healing length which we refer briefly t o at the end of section 2.5, but the interpretation of them in terms of a core parameter is model dependent.

2.3. GINZBURG-PITAEVSKII THEORY In the preceding sections, the vortex core is described as being “hollow” or “rotating”. These descriptions are clearly inadequate for dealing with the structure of an object whose size is of the order of the interparticle spacing. Indeed it is one of the mysteries of superfluid physics that simple phenomenological concepts work as well as they do on a microscopic scale. A first principles quantum mechanical description of the vortex core has thus far eluded theorists. In this and the next section we discuss two approaches which yield similar equations from quite different starting points. Ginzburg and Pitaevskii (1958) used a modified Ginzburg-Landau theory (1950) to study the structure of the vortex core in the vicinity of the lambda point. In this approach, the superfluid is described in terms of a complex order parameter P such that p, = rnlP1’ and u, = (h/rn)VY. The principal assumption is that the free energy per unit volume can be expanded in the form

Minimizing this free energy with respect to variations of P yields the Ginzburg-Landau equation

-(h2/2rn)~*~-aP+c~Pyl2P= 0, where it is assumed that a is proportional to (7’’ - T) and p

(2.3.2)

- constant.

20

W.1. GLABERSON AND R.J.DONNELLY

(Ch. 1, 12

As was true for superconductors, their model predicts a healing length that diverges as (T, - T)-In;in particular, a

- 4(T, - 7 y 2(A).

(2.3.3)

The variation of superfluid density near a wall was calculated and found to be given by the form (2.6.7) below. They obtained a solution which corresponds to a vortex filament in the center of a container of radius b whose energy is usually quoted as E = (p,K2/4T)ln(l.&b/a) =

(2.3.4)

( p , ~ ~ / 4 7 f ) [ h ( b/ a0.3781 ) ,

where the constant 0.378 represents the situation in t h e healing region near t h e vortex core. A principal difficulty with this approach is that the superfluid density is predicted to vary as (T, - 7’)whereas in fact it varies as (T, - T)2’3. Mamaladze (1967) simply incorporated into eq. (2.3.2) those values of a and /3 consistent with the observed temperature dependence of p, and t h e specific heat jump at T,. This leads to a vortex core radius given by LJ

- 3(TA- T)-”’(A)

(2.3.5)

and a maximum superfluid velocity consistent with superfluidity of (2.3.6)

r , = 5.76 x 103(q- T ) - ~cm/s ’~.

At high enough rotation speeds, the vortex cores begin to overlap thus

depressing the A-point. Mamaladze predic:s a A-point shift

AT, = - 5 . 4 ~1

(2.3.7)

0 ~ 9 ~ 3 ‘ 4 ,

where R is the rotation speed. N o reliable experimental evidence currently exists for a rotation-induced A-point shift. 2.4. GROSS-P~AEVSKII THEORY: MOTIONS

IN A

BOSECONDENSATE

The imperfect Bose condensate is governed by equations that were derived by Gross (1961, 1%3) and by Pitaevskii (1961). I n t h e Hartree approximation. t h e single-particle wavefunction 4 ( x , t ) for the N bosons of mass m that fill a volume V obey t h e nonlinear time-dependent

Ch. 1, 821

21

VORTICES IN HELIUM I1

Schrodinger equation

a*

-h2

ih -= -v’+ at 2m

+ w,J/IJ/I’,

(2.4.1)

where W, is the strength of the assumed &-function repulsive potential between bosons. If N is the total number of particles in V then (2.4.2) and the number current density is (2.4.3)

If E, is the average energy level per unit mass of a boson, we write (I = exp(-imE,r/h)P

(2.4.4)

so that by eq. (2.4.1)

a9

ih-=-

at

-hZ V z P + W,PJ?P)z-mE,P. 2m

(2.4.5)

Eq. (2.4.5) is analogous to eq. (2.3.2), although the physics behind the derivations is very different. It can be put into dimensioniess variables by the transformation (e.g., Jones and Roberts, 1982)

(2.4.6)

where pg = mE,/ Wo

(2.4.7)

giving (2.4.8)

72

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 82

The Madelung transformation enables us to give a hydrodynamical interpretation of eq. (2.4.8):

where R and S are real. Substituting into eq. (2.4.8) separating into real and imaginary parts, and introducing a fluid density p and fluid velocity u by p = R’,

(2.4.10)

cs,

(2.4.11)

u=

we find the equation of continuity of mass rlpldt t

v

(pu)= 0

(2.4.12)

and a generalization of Bernoulli’s equation

as/ar+ f u2t ;( p

-

1) - ; p -1/2v*p”2 = 0 ,

(2.4.13)

which involves a quantum potential. The actual equation of motion comes from taking the gradient of eq. (2.4.13)

(2.4.14) where the stress tensor

&, is (Hills and

Roberts, 1977b)

(2.4.15) where a constant tensor a6,) has been added in order that 2,, vanish at infinity where p = 1 . The transformation (2.4.6)defines a velocity of sound c given by c 2= E,

(2.4.16)

and a healing length which serves as a vortex core parameter for the condensate a

=

hI(2ni’E,.) I ”

=

~ 1 2 x TC 6 .

(2.4.17)

Ch. 1, $21

VORTICES IN HELIUM I1

23

The magnitude of a is 0.471 8, for c = 238 m/s, quite reasonable (about 40% low) compared to t h e data of table 2.2.1, i.e. -0.8A. At 25atm pressure c = 366 m/s and a = 0.31 8, compared to the value 18, from tables 2.2.1 and 2.2.2, or a factor of 3 t o o low. Scaling for vortex dynamics has a unit of mass p-a3, a unit of energy 2p,a3c2

= p,K3/8fi

7r3c

(2.4.18)

and a unit of impulse p , a 3 c ~ =p,~~/167r~c'.

(2.4.19)

The equations above yield a remarkable range of results of interest to our discussion of quantized vortices. The Bose condensate model applies at absolute zero and is a single fluid model represented by a compressible gas. The identification of results of t h e Bose condensate model with helium I1 has some problems. For example it is known experimentally that the density of the fluid circulating about a vortex in helium I1 is p, and not the density of t h e condensate which is -13% of the superfluid density [see Svensson (1983) for a recent review of the experimental situation]. The depletion of t h e condensate in the Bose condensate model is relatively small, whereas it is not small in helium 11. The comparison of results of the Bose condensate model with helium I1 must assume that the non-condensate fraction is dragged into motion by the condensate. Thus pmis identified directly with PS

Outside healing layers, the fluid is governed by Euler's equation for a barotropic fluid for which (2.4.20)

an equation of state which can hardly be directly applied to the properties of liquid helium. It therefore should be no surprise that the healing length (2.4.17) does not have the correct pressure dependence. One might suspect that the best agreement between condensate models and liquid helium would occur for the smallest density of the liquid, since that density will perhaps be closest to that appropriate to a gas. The dispersion relationship for infinitesimal sound waves corresponds with the usual Bogoliubov (1947) spectrum: phonons for the longest wavelengths and free particles for shorter wavelengths [see, for example, Jones and Roberts (1982) p. 26031. There are no rotons in the Bose condensate.

23

[Ch. 1, 92

W.I.GLABERSON AND R.J. DONNELLY

Vortex lines exist in the condensate: they are hollow, and are of constant radius when stretched so that differentiations such as in eq. (2.1.13) are at constant a. The solution for the vortex is the same as that discovered by Ginzburg and Pitaevskii (1958). The energy per unit length is

E = -P Z K 2

[In(b/a) + 0.381

(2.4.21)

471

a value independently confirmed by Roberts and Grant (1971). Vortex rings in the condensate were investigated first by Amit and Gross (1966). Roberts and Grant (1971) have made a numerical investigation obtaining for large rings

E Ll

= $p,~’R[In(8R/a)-1.6151,

(2.4.22)

K [In@Rla)- 0.6151 ,

(2.4.23)

47rR

P=~,KTR 2 ,

(2.4.24)

[as conjectured by Donnelly and Roberts (1969, $3)) which shows that excitations in t h e condensate obey Hamiltonian dynamics. The dipole moment p of these large rings, a quantity entering in quasi-particle interactions, is given by (Jones and Roberts, 1982) g = P/4n-p .

(2.4.25)

Examples of the use of dipole moments in discussing the interactions among elementary excitations appear in Donnelly et al. (1978) and Roberts et al. (1978). Just as in the theory of classical vortex rings, there are families of rings of increasing alR, which have been investigated by Jones and Roberts (1982). with some remarkable results. The authors have searched numerically for axisymmetric disturbances that preserve their form as they move through t h e condensate. A continuous family is obtained whose dispersion curve consists of two branches which we show in fig. 2.4.1 compared with the known spectrum of elementary excitations as determined by neutron scattering (Donnelly et al., 1981). The lower branch is (for large enough P ) a vortex ring of circulation

K

; as P + m, its

radius (3 = ( P / T K ) ’becomes ~ infinite and its forward velocity tends to zero. The upper branch lacks vorticity and is a rarefaction sound pulse

Ch. 1, 021

VORTICES IN HELIUM I1

25

30

-

20

Y

v

m

Y

\

w

10

0

0

I

2

P/k

3

4

(K')

30

-

20

Y

v

m

Y

\

W

10

0

Fig. 2.4.1. The axisymmetric solitary wave solutions obtained by Jones and Roberts (1982) compared to: (a) the phonon spectrum and the Bogoliubov spectrum. (b) the dispersion curve for helium I1 determined by Donnelly et al. (1981). The location of the large ring formulae eqs. (1.5) and (L.6)on the diagram is shown by a dashed line. The core parameter was taken to be 0.471 A, consistent with the Bose condensate calculations. The error in using the large ring formulae is evident.

26

[Ch. 1, $2

W.1. GLABERSON AND R.J. DONNELLY

that becomes increasingly one-dimensional as P + "; its velocity approaches c for large P. The velocity of any member of the family is shown, both numerically and analytically, to be dEldP, the derivative being taken along t h e family (Roberts, 1972). The results (in nondimensional units) are shown in table 2.4.1 and in fig. 2.4.1. It is interesting to note that the radius of the ring W vanishes on the lower branch at a point marked X. The cusp of the dispersion curve is at EM = 50.7, PM = 69.6 (E,,,/k = 6.4 K, PMlh= 0.34 A-' in dimensional units). The disturbances on the upper branch are rarefaction pulses which are the three-dimensional analog of the Tsuzuki soliton (Tsuzuki, 1971). The upper branch was also found by Iordanskii and Smirnov (1978), but no connection with the lower, vortex branch was reported. Small vortex rings have energies and momenta which deviate from simple formulae such as eqs. (1.5) and (1.6). For problems such as the theory of vortex nucleation in helium 11, these deviations may be crucially important, and naive applications of formulae for small alR are likely to be quantitatively wrong. We reproduce plots for t h e energy and impulse of small quantum rings in the Bose condensate as calculated by Jones and Roberts (1982) in fig. 2.4.2. The deviation from the thin ring formulae can easily be appreciated. Comparison with the classical formulae (see fig. 2.1.1) shows that in t h e Bose condensate quantum effects cause a departure from thin ring formulae far earlier in the Rla sequence than for classical rings. One motivation in studying small vortex rings has been the desire to understand what happens to vortex rings which shrink, and conversely to understand how vortex rings are nucleated. Large vortex rings moving through helium I1 under their self-induced velocity suffer collisions with rotons and phonons which dissipate energy. Since the energy of a ring is Table 2.4.1 Results of Jones and Roberts (1982) for axisymmetric disturbances in the condensate.

U 0.4

0.5 0.55 0.a 0.63 0.66 0.68 0.69

E

P

P

CE,

129.0 0.7 66.5 56.4 52.3 50.7 53.7 60.0

233.0 123.5 96.5 78.9 72.2 69.6

22.6 13.0 10.6 8.97 8.37 8.20 8.80 9.92

3.36 2.31 1.82 1.06

74.1

83.2

-

-

Ch. 1, 02)

27

VORTICES IN HELIUM 11

250

2 00 150

w 100

50

0

1

I

I

I

1

2

3

4

5

R/a

500

-

(b

I

400

-

-

300

-

-

200

-

-

100

-

-

a

0

I

0

I

2

3

4

_

5

R/a Fig. 2.4.2. Plots of (a) the energy E and (b) momentum P of the small quantum rings of Jones and Roberts (1982). The units are dimensionless. The dashed lines represent eqs. (2.4.22)and (2.4.24).Comparison of this figure with fig. 2.1.1 shows that small quantum rings deviate from the large Rla fomulae much sooner than classical rings. The circled points are the calculated results. (After Muirhead et al., 1984.)

proportional to its circumference, the ring shrinks, and moves faster. This process is described by mutual friction, and the rate of decay derived by Barenghi et al. (1983) is given by eq. (4.1.12) below. Onsager and Feynman had the idea that a roton might be the end state of a shrinking vortex ring. The forward motion of a vortex ring produces the Magnus

28

W.I. GLABERSON AND

R.J.DONNELLY

[Ch. 1, $2

force necessary to balance t h e tension tending to shrink a ring. Feynman reasoned that when the ring is small enough quantum effects balance the tension and the ring will be at rest, or nearly so, near the roton minimum of the dispersion curve (see fig. 2.4.1). Donnelly (1974) has written a history of the idea of the “ghost of a vanished vortex ring” for a conference honoring Onsager‘s 70th birthday. Donnelly and Roberts (197 1) noted that the data then available seemed to lead the large vortex ring sequence into the dispersion spectrum to t h e right of the roton minimum (see fig. 2.4.1), so that vortex rings could be considered to decay naturally into rotons. They observed further that in order for intrinsic nucleation theory to succeed [see Langer and Reppy (1970)], t h e population of small vortex rings in thermal equilibrium needs to be extremely large, comparable with the roton density. They then suggested that rotons were the population from which nucleating rings are created. The work of Jones and Roberts discussed here gives a remarkably different picture of small vortex rings. The consequences of these new ideas deserve much more attention. Grant (19771) has examined t h e problem of vortex waves on quantized vortex lines in a Bose condensate. This is a subject which we shall address in section 5.1 below. The resulting dispersion relationship for the “slow” branch is given by eq. (5.1.3) with Euler’s constant y (=0.5772) replaced by 0.692, and is thus very close to the value for classical hollow vortices. Gross (1966), Padmore and Fetter (1971), Grant and Roberts (1974) and Fetter (1976) take up the matter of charged and uncharged impurities in a Bose condensate which are relevant to the motions of negative and positive ions and 3He in liquid helium 11. While the contents of these papers are not directly relevant to the discussion of vortices themselves, they are relevant to the interaction of ions, 3He and vortices which are discussed in sections 2.7-2.9 below. The uncharged impurity can be used as a model for a single 3He atom dissolved in helium 11. If the hard core radius is b and b %- a, the healing length, then it is shown (Grant and Roberts, 1974) that the effective hydrodynamic radius is i7rpzbi,,, where be, = (b + a f i ) . Padmore and Fetter (1971) suggest that b = 1.83A, and since a = 0.471 A, be, = 2.50 A. Thus the hydrodynamic mass is 0.954m3and the sum of t h e physical and hydrodynamic masses is 1.95m3 compared to the experimental value -2.4m3.

When a hard impurity also carries a charge, the theory is amended to take into account the atomic polarizability a’which is done through t h e dimensionless parameter a = ( a ’ m *Z2e2alhZb3),where Ze is t h e ionic charge (in esu) and m * is the ionic mass, modified to allow for recoil effects. Roberts and Grant then find that be, is reduced to ( b + a f i 5aa/7). Pandmore and Fetter estimate the positive ion radius to be

Ch. 1, $21

VORTICES IN HELIUM I1

29

5.68 A. The experimental value is considered to be -5.8 A. The positive ion radius is observed to depend on the particular atomic species on its core (Johnson and Glaberson, 1974). The structure of the negative ion bubble has been examined by Roberts and Grant with a theory that treats E = ( ~ r n J l m ) ”as~ small and mJm as negligible, where me is the electron mass and 1 is the electron-Boson scattering length. The authors show that to leading order, the radius of the bubble is b = (7rmZuz/mep,)”5.The corrections required at the next order in the alb expansion are given, and it is shown that even when polarization effects are negligible, be, is less than b. The effective mass is still however close to fmp,b:,, although motion tends to expand the bubble and make it oblate. The value of b turns out to be 11.8A compared to the experimental value -17 .&.IR a further paper, Roberts calculated the normal modes of pulsation of the negative ion. This paper contains an introduction useful to anyone wishing to learn about the Bose condensate. The normal mode results are needed to calculate the mobility of the negative ion at low temperatures. The results of these calculations have not yet been exploited experimentally. 2.5. THEBOUND EXCITATION MODEL This model (Glaberson et al., 1968; Glaberson 1969) is an attempt to gain some microscopic understanding of the core of a vortex in real helium 11. In attempting to understand excitations near a vortex line, one can consider two topics. First, the vortex line at finite temperatures will be bombarded with collisions by rotons and phonons and hence will be in Brownian motion. We discuss this topic in section 5.5. Second, the circulation of the superfluid about the vortex (here considered at rest) will produce a shift in the roton energy given by the p us interaction [see, for example, Donnelly et al. (1967) section 121. Thus, if E ( P ) is the dispersion relation for elementary excitations, the Bose distribution is modified in a flow field to

-

(2.5.1) The normal fluid is considered to be at rest, on the average, near a vortex core and the superfluid has the velocity us= ~ / 2 ~ Values r. of II,become so large near a vortex core that the energy of a roton (with Landau parameters A, po, and p ) (2.5.2)

30

[Ch. 1. 02

W.I. GLABERSON AND R.J. DONNELLY

becomes depressed, and the number density N , increases until ultimately the Landau critical velocity w L is reached and rotons can then be created spontaneously. The situation is shown in fig. 2.5.1 where the quantities N,, A, and p , / p are plotted as a function of distance from the vortex core. A careful discussion of this problem was undertaken by Glaberson (1969) who considered the effects of roton-roton interactions which arise when N , grows, and the effects of the uncertainty principle when the roton is localized near the core. Neither of these effects qualitatively change t h e expected variation of roton density near t h e line. At distances shorter than that where the Landau critical velocity is reached, it is assumed that the core is normal, with properties perhaps like helium I. It was assumed that the total density is constant everywhere. One highly significant result is that the radius of the normal core increases with pressure. One might

I

I

I

I

I

I

I

I

I

1

1

1

1

1

1

1

L t I 1 1 1 l 1 . I tlt1Ill11

16 1412108 6 4 2

2 4 6 8101214 16

RG) Fig. 2.5.1. Behavior of roton density N,,roton energy gap A, and pJp near the core of a vortex line at T = 1.6 K on the bound excitation model (Glaberson et al., 1968).

Ch. 1, 921

VORTICES IN HELIUM I1

31

have assumed intuitively that this distance would scale with the total density, i.e., a p - and hence decrease with applied pressure. Glaberson’s results for N , as a function of temperature and of pressure are shown in fig. 2.5.2. At the lowest temperature the normal core has a radius of -2.5& somewhat larger than the core parameter -1 & , deduced by Rayfield and Reif (1W) The . radius of the normal core is seen to increase with temperature, and the region of excess roton density (the “tail”) spreads out with increasing temperature. Near T, the core radius diverges, and is given approximately by

-

a + 3.2/(T, - T)’” ( A ) .

(2.5.3)

It is generally believed that the coherence length 6 is a measure of the core parameter a and that 6 p i ’ , and hence diverges approximately as (T, - T)-u3near T,, in conflict with the results of the bound roton model. Efforts to measure 6 have been generally confined to the region near T,. A very nice summary of attempts to measure 6 is contained in table 111 of the paper by Ihas and Pobell (1974). Methods include light scattering, attenuation of first sound, reduction of the lambda transition in narrow channels, decay of persistent currents, third sound and heat

-

Fig. 2.5.2. (a) Behavior of the roton density N, near a vortex line as a function of temperature at P = 0. (After Glaberson, 1969.)

32

[Ch. 1, $2

W.I. GLABERSON AND R.J. DONNELLY

22t

L-

20

-P = O ohn

18

- -

-P = 5

16

R

-gg 7 7

0 x

14 14

12 12

m

-

I0

-

08

-

zL

i

I

T = 1.2”K

-

-

P = 6 ohn

- . . . ’P r P r n

\\\\r-

06 04

02 0

1

2

3

4

5

6

7

8

9

W

l

l

G

!

B

#

6

6

i

7

B

R(8) Fig. 2.5.2. (b) Behavior of the roton density N, near a vortex line as a function of pressure at 7 = 1.2 K . (After Glaberson. 1%9.)

transfer in films. The results indicate that 5 - l . 2 ~ - * ”(A), where (T, - V I T A .

E

=

2.6. THEHILLS-ROBERTS THEORY A number of experiments on helium I1 show evidence that the superfluid density is depleted near solid boundaries. In particular, experiments on the propagation of third sound in thin, unsaturated films of helium I1 show that t h e superfluid behaves as if it had a lower areal density than that given by the product of the superfluid density and the film thickness at the same temperature. Rudnick and his collaborators (Rudnick and Fraser, 1970: Scholtz et a]., 1974, as well as other groups) have associated this reduction in density with “healing”, the notion that the superfluid density decreases near a boundary. Hills and Roberts (1977a, b; 1978a, b) have advanced a two-fluid theory which incorporates healing as well as relaxation (the processes which prevent the superfluid fraction from changing instantaneously when the thermodynamic state is altered). Neither healing nor relaxation are incorporated in the Landau two fluid theory. The Hills-Roberts theory rests on accepted macroscopic balance laws for mass, momentum and energy together with a postulate for entropy growth. Their theory is entirely hydrodynamical, valid over the entire temperature and pressure range of helium 11. The superfluid density is

Ch. 1, 921

VORTICES IN HELIUM I1

33

regarded as an independent thermodynamic variable and the development allows for a constitutive dependence on superfluid density gradients. Boundary conditions on the superfluid density are not given a priori and some assumption needs to be made. Usually one assumes that p, vanishes at rigid boundaries and at vortex cores. The resulting theory becomes completely determinate once a free energy A ( p , T , p , ) is known. Irrespective of the details of A, however, a necessary condition for a “hydrostatic state” with a persistent superflow is that the helium be isothermal. Various idealized models for A are possible. For example, in the spirit of Ginzburg-Pitaevskii theory (section 2.3) Hills and Roberts (1978a) suggest that for small t = TA- T

where f (temporarily) denotes the bulk superfluid density. For the case of stagnant helium filling the half-space above a plane z = 0, and taking only the first two terms, the familiar hyperbolic tangent solution of Ginzburg and Pitaevskii (1958) is obtained for the superfluid density p, = f tanh2(z/D),

D = (h/m)(A,f)-”,

(2.6.2)

To construct A ( p , T, p,) over the whole density-temperature range Roberts et al. (1979) explored the idea that the state functions of the Landau theory depend not only on p and T, but also on w2, where w = u, - us [see, for example, Donnelly et al. (1%7), Roberts and Donnelly (1974)l. The normal fluid density p, can be obtained from the Helmholtz free energy F ( p , T, w 2 ) by differentiation p,/2p

=

-aF/aw2.

(2.6.3)

For sufficiently small T and density of excitations n ( p ) , FE can be accurately obtained by using the classical expression for the non-interacting Bose gas

where ~ ( p is) the dispersion relation for stagnant helium. For finite temperatures, Donnelly and Roberts (1977) have devised a way to obtain equilibrium state functions such as F ( p , T ) to within f = 0.1 K provided

34

(Ch. 1, 42

W.I. GLABERSON AND R.J. DONNELLY

the dependence of ~ ( p on ) T (and of course p ) is consistently incorporated. The Brooks-Donnelly tables (1977) contain a representation of ~ ( pand ) this can be used to numerically construct the energy function A of Hills and Roberts using a Legendre transformation A = FE- W’ 3FE/aw2.

(2.6.5)

This scheme fails in two areas, at low temperatures and near TA.The latter case was perhaps to be expected. Scaling shows that the appropriate potential is a Gibbs free energy G ( p , T,p> rather than A. Hills and Roberts (private communication) have expanded t h e Gibbs free energy in a power series near T, and have shown that eq. (2.6.1) is replaced by the truncated expansion

Unfortunately this model of the Gibbs free energy cannot be evaluated near T, by thermodynamic and neutron data - among other things neutron scattering linewidths are too large to be approximated as deltafunctions. This means that thermodynamic data cannot be obtained by simple statistical mechanical methods useful at lower temperatures [cf. Donnelly and Roberts (1977)l With a numerically constructed A, Roberts, Hills and Donnelly considered the static healing above a plane wall z = 0. The resulting equations are then [cf. Hills and Roberts (1978b) eqs. (3.7)-(3.10)].

where A is a free energy function, dj0 and po are the Gibbs free energy per unit mass and pressure at great distances from the wall. It is difficult to solve eqs. (2.6.7) and (2.6.8) simultaneously for general A(p, T, p,) since p, may range independently of p and T from 0 to f(p, T) and p itself may vary near the wall. The authors observe however, that t h e free energy A is dominated by the ground state free energy A&) which, according to Brooks and Donnelly (1977), is -15 J/g whereas the excitation part A, is at most 0.4 J/g. This suggests an expansion in small AJA, which shows that the total density p differs from po by order poAE/A, whereupon eqs. (2.6.7) and (2.6.8) give (h2/8mZp5)(dp~dz)2-p~E(po, ps) = -POAE(POl

where po is the value of p far from the wall.

f )

7

(2.6.9)

Ch. 1, 421

VORTICES IN HELIUM I1

35

Eq. (2.6.9) must be solved subject to t h e conditions p,(O)=O

on z = O ,

as f - ~ ,

p,+f

(2.6.10)

the latter of which has, in essence, been incorporated in eq. (2.6.9). One obtains

I

[A(fRZ)- A(f

z = (hZf/2pm2)”

dR ,

(2.6.11)

where the suffix has been suppressed on po, p6 has been replaced by fR2 and A,(p, T,pJ has been written A(fRZ),since po and T are constants. There are various definitions of “healing length” possible. Hills and Roberts adopt the idea of a “displacement thickness” 6, which is defined to be that distance for which fS is the superfluid mass (per unit area of wall) “displaced” from the wall through healing, so that the hypothetical density distribution

(2.6.12)

has the same net superfluid mass as the actual solution. For the density distribution (2.6.2), 6 = D. For general values of A the displacement thickness is 6=

(1 - RZ)dz

=

I,’

(hZf/2pm2)”

(1 - R 2 ) [ A ( f R 2-)A ( f ) ] - ’ ”dR (2.6.13)

and numerical integration produces the table 2.6.1. As we have remarked the low temperatures T 6 0 . 6 K had to be treated separately and the values shown were obtained by a careful asymptotic analysis which gave as T+O 6 -+ K

/ ~ w ~ ,

(2.6.14)

where w L = Alp, is t h e Landau critical velocity which can easily be evaluated using t h e Brooks-Donnelly tables. For the rectilinear vortex the governing equations are: (2.6.15)

36

W.I. GLABERSON AND

[Ch. 1, $2

R.J.DONNELLY

Table 2.6.1 Displacement lengths 8 at a plane wall (A)”. Temperature (K) 0.2

Density (g/cm’) ~

0.14520 0.14795 0.15070 0.15345 0.15620 0.15895 0.16 170 0.1W5 0.16720 0.16995 0 17270

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.370 2.41 I 2.472 2.565 2.625 2.710 2.7%

2.576 2.623 2.692 2.775

2.890 2.9% 3.023 3.108 3.20% 3.319 3.445 3.586 3.750 3.937

3.286 3.363 3.454 3.561 3.683 3.824 3.986 4.174 4.397 4.664 4.989

3.851 3.956 4.077 4.222 4.396 4.604 4.854 5.156 5.525 5.993 6.629

4.965 5.159 5.391 5.695 6.087 6.605 7.304 8.259 9.726 12.622

7.3” 7.829 8.535 9.552 11.18

~~

2.13) 2.162 2.207 2.261 2.320 2.380 2.430 2.502 2.568 2.641

2.241 2.279 2.331 2.m 1.294 2.365 2.463 2.429 2.535 2.493 2.607 2.560 2.681 2.630 2.763 2.709 2.852 2.165 2.200 2.247

-

-

2.886 2.983 3.095

2.868 2.968 3.072 3.184 3.306 3.668

-

-

’Courtesy of R.N. Hills.

(2.6.16) where y = 2 m 2 p / h f and pr= fR‘,subject to the conditions R(0) = 0, R + 1 as r + x. ‘The general method of solution is much as was outlined for the plane wall whereby the solution is regarded as a perturbation to t h e state p = po. The equations have to be solved numerically using a specific model for A. If eq. (2.6.1) is truncated to the first two terms then the Ciinzburg-Pitaevskii solution (2.3.3) is obtained for the vortex core parameter. In this theory the pressure near t h e core does not diverge as it does in t h e classical case and Hills-Roberts (1978b) determine the pressure variation for the simple model (2.6.1) truncated to two terms (see fig. 2.6.1). Recently Hills and Roberts have considered the system using the more general model for A in the range 0 G T S 2 K. To integrate the equations a collocation method based on cubic B-splines was used and the results are summarized in table 2.6.2. These values of a make t h e vortex formula (1.1) equal to the Gibbs free energy per unit length and eqs. (1.5) and (1.6) become t h e appropriate expressions for the Gibbs free energy for vortex rings. Roberts and Hills (private communication) have shown that it is not possible to derive a simple relationship between a and 6 except for T - 0 and T - , T, ; in the former case, a +0.436, and in t h e latter a -+0.4836.

Ch. 1, $21

VORTICES IN HELIUM I1

31

1.o

0.8

0.6 0.4

0.2 0

1

0

2

3

5

4

/a Fig. 2.6.1.The ratio of the superfluid density to the bulk density near a vortex line on the Hills-Roberts theory. The dashed line represents the quantity 2p(p, - P ) / A l f * as a function of r,a. It is clear that P the total stress normal to the axis decreases with decreasing r and reaches its minimum value of p.;- A1f2/2p at r = 0.

Table 2.6.2 Vortex core energy parameter a (AY. Temperature (K) Density

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1.091 1.110 1.140 1.175 1.214 1.255 1.297 1.341 1.388 1.443 -

1.205 1.229 1.263 1.304 1.345, 1.398 1.450 1.504 1.565 1.634 -

1.372 1.401 1.437 1.479 1.527 1.582 1.645 1.715 1.794

1.577 1.614 1.659 1.711 1.772 1.841 1.921 2.014 2.124 2.257 2.417

1.863 1.915 1.975 2.046 2.132 2.235 2.359 2.508 2.691 2.923 3.238

2.420 2.516 2.631 2.781 2.975 3.231 3.577 4.051

2.0

Wm3) ~~

0.14520 0.14795 0.15070 0.15345 0.15620 0.15895

0.16170 0.16445 0.16720 0.16995 0.17270

0.931 0.961 1.014 0.945 0.978 1.031 0.966 0.999 1.056 0.991 1.024 1.0% 1.017 1.052 1.118 1.043 1.082 1.152 1.070 1.113 1.186 1.098 1.143 1.222 1.124 1.175 1.261 1.158 1.211 1.303 -

'Courtesy of R.N. Hills.

1.886

-

4.776

6.215

-

3.851 3.842 4.191 4.695 5.506 -

-

38

W.I. GLABERSON A N D R.J. DONNELLY

[Ch. 1, $2

Thc vortex core parameter in table 2.6.2 can now be compared with the experimental data of section 2.2. Referring to the result for p = 0.14S2 glcm' approeriate to zero pressure, and interpolating to T = 0.3 K, we find u = 0.946 A, whjch is to be compared with the Rayfield-Reif result a = (0.81 f0.08) A. The agreement is satisfactory - perhaps remarkable -when one considers the difficulties in obtaining absolute magnitudes from experiment as described above in this section. The first row of table 2.6.2 shows that the core parameter a increases slowly with temperature, t h e increase being about 6% between 0.4 and 0.6K. The data of table 2.2.1 shows an increase of about 9% in ala, over the same range - thus the data appear to be rising somewhat slower than predicted but the absolute magnitudes and trend is satisfactory. The low temperature pressure dependence of the core parameter a can be deduced from the first row of table 2.6.2. ala, is predicted t o increase by .- 26'10 going from 0 to 20 atm at T = 0.4 K. The data of table 2.2.2 shows an observed increase of 24% going from 0 to 20.1 atm. Going from 0 to --15 atm ala, is predicted to increase by 19% and is observed to increase by 21% going from 0 to 14.6atm. On the whole, the predictions are in satisfactory accord with low temperature data obtained from vortex ring experiments and emphasize that a does not scale with the average interatomic spacing P - " ~ . We have remarked that vortex ring core determinations cannot be extended much above 0.6 K because of drag. We would still like to know the vortex core parameter all the way to T, for a variety of reasons, including nucleation theories. One method is to find an alternative way to measure 6. Roberts et a]. (1979) attempted to compare their calculations at p = 0 to the results of third sound experiments o n unsaturated films. Third sound in thin films travels at velocities considerably slower than would be inferred from t h e film thickness and t h e expression for the velocity of third sound. This has been interpreted as evidence for healing behavior near the substrate and possibly near the free surface. The authors showed that quantitative agreement with predicted displacement lengths was best assuming healing at both the substrate and free surfaces. Another technique is t h e measurement of fourth sound in porous media. When t h e powder size is sufficiently small, healing behavior influences the index of refraction of the porous medium which can be studied as a function of both temperature and pressure. Tam and Ahlers (1982) have made such a study for a number of fourth sound cavity packings and have interpreted their results in terms of a displacement length. They have also re-analyzed some data of Heiserrnan et al. (1976). Two sets of data for 0.05 pm and 0.009 pm data are shown in fig. 2.6.2. The data cover the range 1.4 to 1.8 K and 0 to 25 bar and are compared to

39

VORTICES IN HELIUM I1

Ch. 1, 221 10.0 O

T

v

8.6

I

IU

z

7.2

W _J

W

z

5.8

H

-I

a w r

4.4

3.0 0

5

10

15

20

25

PRESSURE ( b a r )

10.0 d = 0 . 0 0 9 prn

n

.a v

8.6

I IU

z

7.2

W J U

z

5.8

H _I

a W

4.4

I 3.0

0

5

10

15

20

25

PRESSURE ( b a r ) (b)

Fig. 2.6.2. Healing lengths calculated by Tam and Ahlers (1982) from the fourth sound velocity data of (a) Tam and Ahlers (1982) and @) Heiseman et al. (1976). The solid lines represent the displacement length calculation of R.N. Hills (table 2.4.1). (c) The temperature and pressure dependence of the displacement length found by converting t h e data of table 2.4.1 from density to pressure.

40

(Ch. 1, $2

W.I. GLABERSON AND R.J. DONNELLY

0

04

08

12

16

20

T(K) (c)

Fig. 2.6.2 (continued).

the results of table 2.4.1. The agreement with the calculations of Roberts et al. (1979) is seen to be quite good except possibly at the highest T and p where the healing length becomes large. At temperatures near TA it is seen from eq. (2.6.2) that the displacement length given by the Ginzburg-Pitaevskii theory S = D - f In. The general expectation [see, for example, Ahlers (1976)l is that coherence lengths and hence displacement lengths will scale as f near TA.The difficulty lies in the use of a Helmholtz free energy for A rather than a Gibbs free energy. The problem cannot be easily resolved without recourse to further experimental data. Instead, one can appeal to a different experiment to estimate G, in eq. (2.6.6). The Hills-Roberts theory (1978a, b) gives the condition for the onset of superflow in circular channels, as a minimum radius

- tiol,

uo= V 2

(2.6.17)

where jol = 2.40482 is the first zero of the Bessel function J,. [This result was also obtained by Mamaladze and Cheishvili (1966).] Here the dis-

Ch. 1, 021

41

VORTICES IN HELIUM I1

placement length S = 26 where S = D in eq. (2.6.2) in terms of A,. The use of the model (2.6.6) in place of (2.6.1) results in the replacement A, + &,G,E’~. Ihas and Pobell (1974) have used superleak second sound transducers to determine the onset of superfluidity in a variety of pore sizes. Expressing their results for onset temperature in terms of a reduced onset temperature E~ = (T, - To)/&,they found that the relationship E~

= [d x 108/(5.7 5 0.6)]-’~54z0~05

(2.6.18)

described their results over a range of pressures up to 30 bar (here d is the nominal pore diameter and included the sizes d = 0.1, 0.2, 0.4 and 0.6pm). If we approximate t h e exponent in eq. (2.6.18) by the Ihas-Pobell experiment suggests

-;,

do= 5 . 7 ~ A ; ~. ~

(2.6.19)

The combination of eq. (2.6.17) with eq. (2.6.19), using do = 2a,, and 6 = 26 gives the needed result for displacement length near T,:

s = 5.7 AE-’”/dTj0,

= 1.68

(2.6.20)

This result is pressure-dependent because T, (p) is pressure-dependent and the results of eq. (2.6.20) can be combined with the data of table 2.4.1 to produce approximate fits of S over the entire (T,p)-plane. The result (2.6.20) can be combined with the results discussed above for S in terms of G2 to evaluate that quantity as a function of pressure. To do that we need the bulk superfluid density f near T,, which has been discussed by Ahlers (1976). For present purposes it is accurate enough to approximate his recommended result [see Ahlers (1976) eq. (2.2.29)] by

where k ( p ) = 2.3% - 0.02883~. It is not difficult to raise objections to the Hills-Roberts theory, which in a continuum model being used on very small length scales. But the results are in remarkable accord with a wide variety of experiments and at the time of writing is the only predictive model of healing and core structure addressing the behavior of real liquid helium. BEHAVIOR OF IONS INTHE PRESENCE OF VORTICES 2.7. STOCHASTIC

Much of what we know about vortices is information obtained from experiments using ions as probes. Negative ions are electrons in relatively

42

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 02

-

large (R 16A) bubbles cut out of the liquid owing to the repulsive electron-helium atom interaction. Positive ions are smaller (-6 A) solid objects held together by electrostrictive forces. In the vicinity of a vortex line an ion does not experience any azimuthal drag from the superfluid, but it does experience t h e pressure gradient shown in fig. 1.1. The pressure gradient near a vortex is given by eq. (1.2) and hence a small impurity of volume V will experience an inward force due to the pressure gradient given by - Vdp/dr, and thus varies inversely as t h e cube of the distance from the vortex line. This estimate is modified if the flow about the impurity is properly allowed for. A discussion of t h e modification has been given by Donnelly et al. (1%7, section 24). A general discussion of t h e determination of t h e equation of motion of a sphere immersed in a superfluid containing an arbitrary configuration of quantized vortex lines is given by Painten et al. (1985). The equation of motion is in a form suitable for use with numerical computations of the way in which vortex lines interact with a moving sphere, and therefore is particularly useful in calculations of t h e behavior of ions and quantized vortices in various configurations. Clearly either species of ion (and indeed ’He impurities) will be pushed into a vortex core by such a large pressure gradient. Another way to appreciate t h e magnitude of the force is to consider the potential energies involved. When an ion is situated symmetrically on the core, it supplants a substantial volume of high velocity circulating superfluid: recall that the ionic radii are both considerably greater than the core size, which is of order 1 A. Since the ion displaces fluid with high kinetic energy, one refers to the resulting lowering of the energy of the system as a “substitution energy”. When an ion of radius R is on the vortex it can be shown that this classical substitution energy is given by (Donnelly et al., 1%7; Donnelly and Roberts, 1969) u(0) = -27rpS(h/m)’R[1 - (1 + a2/R’)” sihh-’(R/a)]

and off axis

(2.7.1)

u ( r )= -27rp,(h/m)2R[1-( r 2 / R Z - 1)1’2sin-’(R/r)]

*

for r a. The resulting potential energy for a negative ion is shown in fig. 2.7.la. The use of the concept of healing in eq. (2.7.1) is essential. Assuming a hollow core the potential of an ion exhibits unphysical behavior when the ion touches t h e core. The general expression u ( r ) was obtained using a healing model of the vortex core [Donnelly and Roberts (1969) p. 5321. We shall comment later on the applicability of eq. (2.7.1) in t h e case of a solvated 3He atom.

-

Ch. 1, 421

r/(a)R = 1 . 6 n m ; ~ = B B V / m n ; T = 1 . 6 ( K

(6)R = 0 . 7 8 n m ; I = 7 k V / a m ; T = < 1 K

Fig. 2.7.1. Potential energy wells for (a) the negative ion and (b) the positive ion. (After Parks and Donnelly, 1966.)

The very earliest rotating experiments with ions and quantized vortices (Careri et al., 1962) showed that negative ions are captured by quantized vortex lines and positive ions are not. These experiments were done at 1.37K. If a beam of negative ions is sent across a rotating bucket of helium 11, the ion current Z diminishes according to

d I / I = -nu dx

(2.7.2)

so that

I = I, exp(-2&rx/K),

(2.7.3)

where I, is the current in the ion beam for R = 0 and c is the capture diameter for a single vortex line. The magnitude of CT is ==lO-’cm and decreases with increasing field. Furthermore, above 1.7 K n o trapping is observed at all. We shall show that these magnitudes, which at first were very mysterious, have a simple explanation in stochastic theory. It was suggested by Donnelly (1%S) that the behavior of ions in liquid helium might be discussed in terms of Brownian motion in potential wells (Chandrasekhar, 1943). The potential wells are produced by vortices in the superfluid as indicated above, and the Brownian motion of the ions

w

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 82

(at temperatures well above 1 K), buffeted by rotons and phonons, is an effect produced by the normal fluid. The capture of ions, therefore, is pictured as a process of sedimentation of Brownian particles into a well. The escape of ions from vortices, which occurs at a temperature depending on the size of the ion, is modelled as t h e escape of Brownian particles over a potential barrier. The barrier is formed, in the case of ions and vortices, by the combined potential due to substitution energy and the steady electric field used in vortex experiments (see again fig. 2.7.1). It should be emphasized that the simple Brownian motion picture for ions in helium I1 will only be valid when the mean ion drift velocity is much smaller that the thermal velocity. It is not difficult to illustrate the basic processes and magnitudes involved in t h e ion capture phenomenon. To do so we study a very simple problem, at first sight unrelated to vortices. Consider a one-dimensional beam of ions of charge e in helium I1 moving along t h e x-axis in an electric field E and incident upon a collector at x = 0. It is easy to show [see Donnelly and Roberts (1969) p. 5241 that the density of ions in the beam for x < 0 must fall to zero at the collector in a diffusion-related distance of order k,T/e€, while maintaining constant particle flux at all negative values of x. At T = 1.5 K and E = 20 V/cm, this distance is surprisinglv large: of order 6.5x 10 ‘cm. In thc ion capture process the potential well illustrated in fig. 1.1 forms a sink for ions with a range of a few ingstroms. This means that the ion density is forced to zero as ions are captured and migrate up and down the lines under their mutual Coulomb repulsion. Again, the diff usion-related distance over which the ion density in the beam will fall to zero will be related to k,T/eE and is independent of t h e details of the ion-vortex interaction, and even the species of ion. It seems reasonable to suppose that this distance will determine the magnitude of t h e cross-section for ion capture. Detailed calculations by Donnelly and Roberts and experimental measurements confirm this simple reasoning (Careri et al.. 1%2; Springett et al., 1965; Tanner. 1966; Ostermeier and Glaberson, 1975a; Williams and Packard, 1978). McCauley and Onsager (1975a, b) have presented a more rigorous calculation i n which they explicitly included t h e vortex force in t h e theory. At temperatures below 1 K. t h e effective ion capture cross-section is observed to drop very rapidly (Ostermeier and Glaberson, 1974 and 1975a). The theoretical treatment discussed in t h e previous paragraph is based on the assumed relevance of the Smoluchowski equation in describing the capture process. This assumption is only valid when the force acting on t h e ion does not change appreciably over a diffusion length L,). This condition is obeyed as long as t h e ion remains a distance r,

Ch. 1, 421

VORTICES IN HELIUM I1

r B L D = (Miple)(2k, TIMi)”

45

(2.7.4)

from the vortex core. Here Mi and p are the ion mass and mobility respectively. If an ion finds itself therrnalized within a distance r, of t h e vortex, where r, is the radius within which the vortex potential is less than -k,T, the ion will be effectively trapped. The validity of the Smoluchowski equation is therefore ensured by having r, %- LD. This inequality is grossly violated at temperatures below -1 K. Ostermier and Glaberson (1974, 1975a) have performed a “Monte Carlo” calculation in which ions were permitted to move ballistically in the ion-vortex potential, suffering random collisions with rotons. Individual ion-roton scattering events were taken into account by assuming isotropic scattering and adjusting the frequency of collision to give the correct mobility. This approach yielded excellent agreement with both the temperature dependence and the electric field dependence of the capture cross-section. The thermal activation of ions out of their potential well and over the barriers of fig. 2.7.1 depends very much on the depth of the well, and this depth in turn depends on the substitution energy and hence the size of the ion through R and Rla. Again a simple one-dimensional solution illustrates the physics involved. The probability of escape of a Brownian particle over a potential barrier is given by the expression (Donnelly and Roberts, 1969)

where w A and wc are curvature parameters describing t h e characteristics of the well and the barrier, p is a diffusivity, U is the potential difference between well and barrier. The exponential term dominates t h e total probability and hence reflects the primary role of UIk,T as the most influential property of the escape process: U, of course depends strongly on the radius of the ion. In fact this characteristic of the escape enabled Parks and Donnelly (1966) to estimate the radii of the positive and negative ions from observations of the temperature above which ions cannot remain trapped for appreciable times (a temperature they call the “lifetime edge”). Shortly afterwards Springett and Donnelly (1966) used the same phenomenon to show how the radius of the negative ion changes with applied pressure. Other related investigations for both species of ion are reported by Douglass (1964), Ostermeier and Glaberson (1975b), Williams et al. (1975). See McCauley and Onsager (1975a, b) for a related and somewhat more rigorous theoretical treatment. Measurement of the escape of ions from vortex rings in very high electric fields (see fig. 2.7.1) (Cade, 1%5; Johnson and Glaberson, 1974)

4h

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1. $2

yield consistent results but, because the ion-vortex potential well is strongly distorted by the electric field, the data is more difficult to interpret. More serious is the use of the crude assumption that the vortex line remains perfectly stationary during the escape process, and the neglect of a proper quantum mechanical treatment of the problem for particles of atomic dimensions. We shall return to this latter problem in t h e case of trapped 'He below. The data which emerges from these measurements indicate the rough size of the trapped ion, and involve the core parameter a but do not provide a determination of t h e detailed core structure.

,Collector

Source /'

\Source Grid

\Source Collector

Icm

Fig. 2 . X . I . A schematic of the experimental cell (Ostermeier and Glaberson. 1976). The crosc-hatched areas indicate insulating surfaces. Ali other areas and grids are gold-plated stainless steel.

Ch. 1, 421

47

VORTICES IN HELIUM I1

2.8. THEMOBILITYOF IONS ALONG VORTICES

Ions trapped on vortex lines, pushed along the lines by an electric field, experience considerably more drag than is experienced by free ions moving through the bulk liquid. In a very simple sense, this implies that the vortex cores are more "normal" than the bulk. The experimental cell used by Ostermeier and Glaberson (1976) is shown in fig. 2.8.1. The vortex lines are charged with ions in a trapping region at the bottom of the rotating cell. A small amount of charge is then gated into the uniform electric field region and the time-of-flight to the collector at the top is determined. Fig. 2.8.2 is a plot of the inverse low field mobility as a function of inverse temperature for negative and positive ions. The data at temperatures above 0.8K are in good agreement with earlier results (Douglass, 1964; Glaberson et al., 1%8). The solid lines labelled 3 and 4 in the figure are the results of calculations I0.C

\\

0NGmlVE IONS OPOSlTlVE IONS

I

-

.c

N

E

< 0

0.1

(L,

lA

'

>

0 .

\

-

u

\

\

'2 OM

\

0

.

\

-

\

0

\ \ o \

0 0

\

\ I .o .o .o .o '0

0.001

I I

2 2

\

3.0 3.0

4 4

T-' (OK-') Fig. 2.8.2. Inverse i o n mobility in low electric fields, as a function of inverse temperature, for positive and negative ions. The lines are discussed in the text (Ostermeier and Glaberson, 1976).

48

(Ch. 1, 82

W.I. GLABERSON AND R.J. DONNELLY

of drag on t h e ion arising from the scattering of thermally excited vortex waves (Fetter and Iguchi, 1970). Curve 1 is a calculation of ion drag associated with the scattering of bound rotons (Glaberson, 1%9). At higher temperatures, there is an order of magnitude agreement with all of these approaches but there is serious disagreement at low temperatures. An addition of a small amount of 3He impurities to the system, restores agreement with the vortex wave theories at low temperature. It may be that, in the absence of 'He, the vortex line is not sufficiently damped so that some instability plays a role. The high electric field mobility data of Ostermeier and Glaberson (1975d, 1976) has an interesting feature which may be related to the vortex core structure. Fig. 2.8.3 shows the ion velocity as a function of electric field. The ion velocity saturates in high fields at some nearly temperature independent velocity that depends on the sign of the ion. The values are 1100 cm/s for the negative ion and 1600 cm/s for the positive ion. Making an explicit assumption regarding the superfluid distribution in the vortex core one can obtain the curvature of the harmonic potential that binds the ion to the vortex. The authors suggest that the frequencies corresponding to the harmonic potential well determine characteristic ion velocities beyond which the ions cannot easily be pushed. It is suggested that resonant generation of vortex waves, and therefore substantial drag, occurs when two conditions are satisfied:

-

-

3

13

o P O S I T I V E IONS A NEGATIVE IONS

-

I

1 oo

10'

1OE

E (V/cm) Fig. 2.8.3. The electric field dependence of trapped ion drift velocity at T = 0.415 K .

(Ostermeier and Glaberson, 1976.)

Ch. 1, $21

VORTICES IN HELIUM I1

w(k,)- kruL= Re@),

49

(2.8.1)

where w(k,) is the frequency of a vortex wave of wave vector k, and Re(R) is the real part of the natural frequency of the ion in its potential well. The first condition states that, at the limiting ion velocity uL, the frequency of a vortex wave of wave vector k , (in the frame of reference of the moving ion), is the same as the natural ion frequency. The second condition is that the group velocity of the vortex wave is the same as the ion velocity. These two conditions yield a unique ion velocity that depends on 0.Assuming that p,(r) in the vicinity of the core is that of a weakly interacting Bose gas, the equations yield values for uL, for the two ionic species, about a factor of three too large. The authors suggest a number of possible explanations for the discrepancy including uncertainties with respect to the calculation of the ion-vortex interaction potential. 2.9.

3 H E CONDENSATION ONTO VORTEX CORES

Rent and Fisher (1969) and Ohmi et al. (1969) predicted that, in 3He-4He solutions, 3He condensation or phase separation of the 3He rich phase into vortex cores should occur at sufficiently low temperatures. A number of experiments provided indirect evidence for the existence of such a condensation. Ostermeier and Glaberson (1975a), in measuring the capture cross-section of ions by vortex lines, observed no measureable trapping for the bare positive ion in a 1% solution. According to stochastic theory, significant capture should have occurred at temperatures below about 0.6 K, the positive ion thermal lifetime edge. It was suggested by the authors that their observations might be accounted for by the condensation of 3He onto the core with a resulting increase in the core parameter and thus a reduction of the thermal lifetime edge to below their lowest accessible temperature. A similar mechanism was also proposed by Williams et al. (1975) who observed a lack of positive ion trapping in a similar solution down to 0.1 K. Williams and Packard (1978) measured the thermal trapping lifetime of positive ions on vortex lines in dilute 3He-4He solutions. Their observations indicated a decreasing lifetime and decreasing ion-vortex binding energy with increasing 3He concentration, also consistent with the idea of 'He condensation onto the vortex cores. Ostermeier et ,a]. (1975), and Ostermeier and Glaberson (1976) obtained measurements of the mobility of ions trapped on vortex lines in

SO

[Ch. 1, 82

W.1. GLABERSON AND R.J. DONNELLY

various 3He-4He solutions at temperatures down to 0.3 K. Assuming that the vortex contribution to t h e drag experienced by the ion is simply additive to the contributions arising from bulk thermal excitations and 'He atoms. the important quantity is the difference between t h e inverse trapped ion mobility and t h e inverse free ion mobility measured under the same circumstances. Plots of this quantity as a function of inverse temperature for both positive and negative ions, are shown in fig. 2.9.1 for various fixed 3He concentrations. At temperatures above some concentration dependent critical temperature, the data fall along a common

1

I

c = 2.90%

I

I

0

I .o

2 .o

T-'

3.0

4

( O K - ' )

Fig. 2.9.1. The vortex contribution 10 the trapped ion inverse mobility as a function of inverse temperature. The dashed lines are approximate fits of the Fetter-lguchi thermal vortex wave theory to the higher temperature data. (a) negative ions (h) po.S I ~ I V Cions. (Ostenneier and Glaberson, 197%. 1976.) '

'

Ch. 1, $21

VORTICES IN HELIUM I1

51

c

2.o

C =0.072%

1.5

0.5

/I

-

1.0-

7

\

\ \

-

I.o

20

T-'

3.0

(OK-')

@)

Fig. 2.9.1 (continued).

curve, in good qualitative and reasonable quantitative agreement with the vortex wave drag theory of Fetter and Iguchi (1970). Comparable agreement with the data can be obtained using the other drag theories. The sharp increase in the vortex contribution to the ion drag, for both the negative and positive ions, is identified with the onset of 3He condensation onto the vortex core. Fig. 2.9.2 contains a plot of the ambient 3He concentration as a function of the critical temperature at which the vortex drag deviates from its universal high temperature behavior. The line in the figure is the result of a calculation which predicts the onset of condensation of 3He atoms onto the vortex cores, in excellent agreement with the data. In the presence of a vortex line, it is argued that the radial dependence of the 3He number density is given by

52

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 02

1

~

1.0

2.0

1.5

T-'

2.5

I ( O K -

Fig. 2.92. Ambient 'He concentration as a function of the critical temperature at which 'He condensation onto vortex lines OCCUTS. (Ostermeier and Glaberson, 1 9 7 5 ~ .1 9 6 . )

where n, is the ambient ,He number density and U(r, T) is an effective hydrodynamic potential related to the kinetic energy of the superfluid displaced by a 'He atom. 3He condensation is associated with the number density on the core n{O, T) becoming equal to the critical density measured at phase separation in bulk solutions. The quantity U(r, T) is obtained using t h e formalism of Parks and Donnelly (1%6) for t h e binding energy of a sphere of radius R to the vortex line. The effective hydrodynamic radius of an 'He atom, R,, is determined from its measured hydrodynamic mass m : m

'5 = m3+ (2/3)rrR:p,

(2.9.2)

where m3 is the bare 'He atomic mass. Numerous objections can be raised against this very simple macroscopic treatment of t h e problem, but the good agreement between the data and the calculation, with n o adjustable paramerers, suggesrs fhat the essential physical ideas are correct. Carrying this simple theory even further. Ostermeier and Glaberson

Ch. 1, 82)

53

VORTICES IN HELIUM I1

(1976) have attempted to explain their observed ion drag at temperatures below the critical temperature for 3He condensation. Fig. 2.9.3 is a plot of their measured vortex contribution to the ion drag as a function of reduced temperature below T,. In their analysis, they picture a phase separation of the 'He-4He solution into a 4He rich phase and a 3He rich phase about the vortex line. The radius of the 'He rich phase grows continuously from zero as the temperature is lowered below T,. Calculations of ion mobility, in the circumstances considered, are very involved (Huang and Dahm, 1976) and to a large extent speculative. Nevertheless, the calculated ion inverse mobilities, shown in fig. 2.9.4, are in satisfactory agreement with the data. The calculated 3He rich core radii, developed in the calculation, are shown in fig. 2.9.5. 2 .E

2 .c c

N

E

V \ V

A

V

1.5

0

a, in

8

> 8

v

-

1m

;t 1.c

.I

CLOSED SYMKILS: NEGATIVE IONS OPEN SYMBOLS :POSITIVE IONS I

(

1.04% 0.2 7% 0.072% 0.017%

V rl 0 0

0 0.5

A=

2.90%

m

V

08

0

A

0.5

1 I .o

I

1.5

Fig. 2.9.3. The condensed 'He contribution to ion drag along vortices, for positive and negative ions, as a function of reduced inverse temperature. (Ostermeier and Glaberson, 1975c, 1976.)

W.I. GLABERSON AND R.J. DONNELLY

54

1.5 -

[Ch. 1, $2

Positives

\

0

0.5

.o

I

1.5

(+- I) Fig. 2.9.4. The calculated condensed 3He contribution to ion drag along vortices, for positive and negative ions. as a function of reduced inverse temperature. (Osterrneier and Glaberson, ISnk, 1976.)

Since this work was completed several interesting and relevant developments have occurred. One is that in experiments on the nucleation of vortex rings by ions, it has been discovered that even very sma11 concentrations of 'He can have a large effect on the results (Bowley et al., 1980). A method of completely removing 'He, providing a useful reference measurement for any experiment with vortices and mixtures, has been developed by McClintock (1978). The binding energy for an 'He atom to a vortex line at SVP has been estimated as described above by Ostermeier and Glaberson (1976) to be about 3.0 K at 0.5 K. A Hartree-Fock approximation by Ohmi and Usui (1969) gave a value of about 3.5K in reasonable agreement with Ostermeier and Glaberson's estimate. A more sophisticated quantum mechanical calculation has been given by Senbetu (1978) who obtained the rather surprisingly small result -0.78 K. Muirhead et al. (1985) have argued that Senbetu's calculation fails to take account of two potentially important effects. One is that the binding energy of a single 'He atom in

Ch. 1, 921

VORTICES IN HELIUM I1

55

Fig. 2.9.5. The calculated 'He-rich vortex core radius as a function of reduced inverse temperature for variousambient 3He concentrations. (Ostermeier and Glaberson, 1975c, 1976.)

the bulk superfluid is pressure dependent. Fig. 1.1 shows that an 3He atom near the core of a vortex will be at a much lower pressure than in the bulk helium far from the core: the authors estimate this effect to be perhaps 8 bar. This calculation, strictly valid for external pressures above 8 bar, gives an estimated binding energy of -3.51 K. The wavefunction for the ground state of the 3He atom so obtained corresponds to a probability distribution peaked outside the vortex core and to an angular momentum of -1h. This orbital motion of the 3He atom could induce a motion of the vortex line in the form of a localized bending wave. This possibility was .not considered by either Senbetu or Muirhead et al., although the latter authors consider a classical analogue to the effect which suggests that the effect may be small. This interesting system desxves further attention.

56

W.I. GLAEBERSON AND R.J. DONNELLY

[Ch. 1, 83

3. Equilibrium vortex distributions Consider a situation in which there is a single straight vortex line along the axis of a cylindrical container of radius R. Assuming a hollow vortex core of radius a, the energy per unit length of the vortex is [eq. (1.1)]

E

=

I

:p,vi d2r = ( p S ~ * / 4 7In(R/a) r) ,

(3.1)

where psis the superfluid density. The angular momentum per unit length about the cylinder axis, i,is given by

For the cylinder rotating about its axis with an angular velocity 0, the free energy per unit length is given by

The free energy is therefore negative for 0 > 0,= ( ~ / 2 7 r RIn(R/a). ~) F is obviously zero in the absence of the vortex line. It follows that for low enough rotation speeds, the equilibrium state of the superfluid contained within t h e rotating cylinder is the vortex-free state, analogous to the Meissner state for a type two superconductor below Ifcl, whereas the one-vortex state is the equilibrium state for rotation speeds just above 0,. In doing this calculation, we have ignored the contribution t o the free energy from t h e entropy associated with excitations of the vortex line (see section 5.5). These may be important near the lambda point. Furthermore, one must be careful to distinguish between equilibrium configurations and the sometimes extremely metastable situation encountered in the laboratory. As the rotation speed is increased, the free energy minimum states correspond to increasing numbers of vortex lines distributed throughout t h e container. In t h e limit of very high rotation speed t h e equilibrium configuration approximates a regular triangular array (Tkachenko, 1966a) in a region of space not too far from the axis of the cylinder. Under these circumstances, t h e energy of the fluid can be written as E

=

1

ip,uf,, d2r =

ip,(U, + v,-J2 d2r

Ch. 1, 031

VORTICES IN HELIUM I1

57

For a uniform array of vortices, having an areal density n,, it is easy to show that

where 52, = (nv/2)w.It follows that

where Z = f psr2d2r is the classical moment of inertia per unit length of the fluid. The last two integrals can be identified with the angular momentum and energy associated with the vortices:

where E,, the energy per unit length of a vortex, is

and L,, the angular momentum per unit length of a vortex, is

and where the mean inter-line spacing is b = ( K / V % ~ ) ’Note ~ . that E, is not quite the same as the energy of a single vortex line given in eq. (3.1). The corresponding expression for the angular momentum of the fluid is L = IOV+

I

n,L, d2r.

(3.9)

The free energy is then

E

- 132 = ZLl,(tJ2,

- 0) + d2rnV{E,+ L,(Llv- a)]

(3.10)

and since for a large cylinder the second term is much smaller than the first, it follows that

a,=n.

(3.11)

On the average, therefore, the superfluid mimics solid body rotation at

58

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 03

the rotation speed of the container. This same result follows from the requirement that in equilibrium there can be no dissipation so that the vortices must move with the normal fluid velocity. There are interesting deviations from a regular triangular vortex array that warrant discussion. Near the cylinder walls there is a vortex-free strip having a thickness of the order of the mean vortex spacing. Stauffer and Fetter (1968) suggest that this comes about because of the competition that takes place between the tendency to mimic solid body rotation, and hence keep t h e vortex density uniform, and the minimum energy associated with each vortex. When values of the mutual friction coefficients B, B‘ and B” were first reported (see section 4.1) there was a substantial amount of scatter in the results from different laboratories. It is now recognized that there are at least two reasons for this problem. One is the proper treatment of the vortex-free strip, and the second is metastability. An early report on detection of the vortex-free strip was a negative ion absorption experiment in a cylindrical annulus by Northby and Donnelly (1970). Their technique was to measure the current Z delivered to the top of the rotating annulus by ions trapped on the vortex lines produced by rotation compared to t h e radially collected current Zo. If there is a vortex-free strip proportional to t h e interline spacing, then the current ratio will have the form (Z/Zo)/fI = A ( l - BfI-”),

(3.12)

where A and B depend on the trapping cross-section and the geometry. Northby and Donnelly arranged their collector to study only the vortexfree strip on the outer cylinder. Their data established that the form (3.12) is obeyed, and the results correspond to the absence of two rows of vortices at the outer cylinder instead of the expected single row. The first attempt to measure the vortex-free strip gave a value more than an order of magnitude larger than the theory (Tsakadze, 1964). Northby and Donnelly speculated that their results are consistent with the notion that the liquid may well have on average the equilibrium distribution of vortices, but that t h e lifetime of the vortices in the outer row is short compared to the time for an ion to migrate to t h e top collector (a matter of less than a minute). Second sound is also an important tool for studying the vortex-free strip. Here, of course, the wavelength is long enough so that t h e average properties of the vortex distribution are probed, although some spatial information can be obtained by using different modes (Bendt and Donnelly, 1%7; Bendt, 1%7). A careful study of the attenuation of second sound in rotating rectangular cavities has been reported by Mathieu et a].

Ch. 1, 831

VORTICES IN HELIUM I1

59

(1980). They are able to distinguish three vortex distributions for any given value of the rotation rate R: an equilibrium state with No vortices, and two limiting metastable states containing the minimum number Nl and maximum number N2 vortices. For R d lo-’ s-’ they find AN/No 1 where AN = N,- N,, and for R 3 1s-’ they find A N / N o S10%. In the higher velocity range AN can be related to a variation in thickness d of the vortex-free strip near the walls: d varies between external values d , and d, which appear to be independent of the boundary geometry. From free energy considerations, they infer that the equilibrium vortex-free width should be

-

do= (b/fi)[ln(b*/a)]’”,

(3.13)

where b* = e-3’4b and b2 = ~/27rR.They consider eq. (3.13) to be valid for any arbitrary-shaped cylinder with walls parallel to the axes of rotation. They report that the vortex distribution they achieve can be reduced to the equilibrium one with No and do by strong perturbations of the cryostat, or feeding large heat flux into the cavity. The results reported above make clear why values of B are likely to have been subject to scatter. The experimental measurement of B requires an investigation of the attenuation of second sound over a wide range of 0 and efforts to check against metastability. And of course the value of B depends upon the frequency of the second sound (see section 4.1). In 1966, Snyder and Putney investigated t h e angular dependence of mutual friction, confirming that B depends on sin’ 8, where 8 is the angle of tilt of the resonator with respect to the axis of rotation. The angulal dependence of mutual friction and the related problem of the B” coefficient is discussed in section 4.1. Mathieu et al. (1984) have carefully restudied this problem with a tilting rectangular cavity. They address the important question of the behavior of vortices near the boundaries when 8 ZO. They advance a model for this behavior based on the fact that vortices must touch a boundary normally (cf. fig. 3.1). They find that the curvature of the vortices takes place over a characteristic length do= 0.29 K’” mm. They also examine the mutual friction as a function of angle, reconfirming the Snyder and Putney result, and giving some evidence for a non-zero value of the mutual friction coefficient parallel to the axis of rotation. Their data are shown in fig. 3.2. The authors avoid interpreting their finite value of B” in terms of axial mutual friction since they found, as did Snyder and Linekin (1966), that the ratio B”/B can vary over a substantial range. The nature of B”, then, remains an open problem. An interesting effect comes about because of the incompatibility of

W.I. GLABERSON AND R.J. DONNELLY

(Ch. 1, $3

Fig. 3.1. Conjectural drawing of vortices in a tilted rectangular second sound resonator. (After Mathieu et al., 1984.)

0.3

0.2

0. I

0

0.1

0.2

0.3

0.4

0.5

sin2 (0)

Fig. 3.2. Measurements of B as function of sin 0 by Mathieu et al. (1984). The intercept marked B” is evidence for some axial mutual friction effect not yet completely understood.

Ch. 1, $31

61

VORTICES IN HELIUM I1

triangular symmetry and the circular symmetry of the rotation field. Campbell and Ziff (1978, 1979) have pointed out that significant distortions of the triangular lattice occur even at substantial distances from the outside edge of the array. They discuss the circular distortion in terms of a destabilizing velocity, that is the difference between the velocity induced at a vortex in a triangular array and that corresponding to solid body rotation. The destabilizing velocity typicaliy varies as (r/R)’ and does not seem to decrease as the vortex density is increased. Several of the detailed vortex distributions predicted by Campbell and Ziff are shown in fig. 3.3 for 18 vortex lines. Only one of these, 18, is the lowest free energy configuration, the other configurations being only local free energy minima (nos. 2, 3, and 7) or free energy saddle points (nos. 4, 5, and 6). Fig. 3.4 shows two predicted patterns for 217 vortices wherein

‘

.

6

. .. a

1

0

.. I

1

6

. . ... 0

.

1

8

3

0

. . .

* .

.2524 6 11

4

Q 1

.2032 I512

0

.3511 3 312

J - ( 6 I

..

. . .3521 6 12

.3524 3 312

.3563 6 12

.4773 5 13

Fig. 3.3. Seven stationary patterns of 18 vortices. Immediately below each pattern is the corresponding value of Afo-the free energy of the pattern relative to what it would have been as a component of an infinite array. Also indicated are the number of vortices in each circular ring in the patterns. (Campbell and Ziff, 1979.)

62

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 53

1.2725

1.1221

Fig. 3.4. IAwest free energy patterns for 217 vortices and their corresponding relative free energies. (Campbell and Ziff, 1979.)

PH INTENSIFIER

MONITOR

- CAMERA

EXPERIMENTAL CELL

2

mm4--L ‘-TRITIUM

SOURCE

Fig. 3.5. A block diagram of the apparatus. The cylindrical vessel is a 2 mm-diameter hole drilled in a stack of three carbon composition resistors. It is 22.7mm in height. Voltage differences applied across each resistor section produce axial electric fields for the rnanipulation of the ions. A tritiated titanium foil forms the bottom surface of the vessel and serves as the ion source. A 700 V potential difference is applied between the phosphor screen and the top of the vessel for acceleration of the electrons. The solenoid produces a magnetic field of 0.5T.which prevents defocusing of the accelerated electrons. (Yarrnchuk et al., 1979. 1982.)

Ch. 1, 431

VORTICES IN HELIUM I1

63

the circular distortion can be readily observed. In the calculations just discussed, the effects of the boundary are ignored and are presumed to be unimportant except very near the boundary. Williams and Packard (1974, 197$), Yarmchuk et al. (1979) and Yannchuck and Packard (1982) have obtained “photographs” of vortex distributions showing excellent qualitative agreement with the predicted distributions. The experimental arrangement is shown in fig. 3.5. Negative ions, which are simply electrons in relatively large bubbles, are trapped

Fig. 3.6. Two different 6vonex patterns observed. Below these are the predicted stable (S) and metastable (MS)patterns. ( Y m c h u k and Packard, 1982.)

W.I. GLABERSON AND R.J. DONNELLY

6‘4

[Ch. 1, 93

o n the vortex lines produced in a rotating cryostat, in a manner discussed earlier. The ions are then forced through the free surface of the liquid, where they become ordinary electrons, and are accelerated onto a phosphorescent screen. The spots of light produced on the screen correspond to the location of t h e vortices in the liquid, at least at t h e points where they intersect the free surface. Fig. 3.6 shows two different 6vortex patterns observed, along with the predicted patterns. The “S” pattern corresponds to an absolute minimum free energy whereas the “MS” pattern corresponds to a metastable configuration. The agreement is reasonably good but it should be mentioned that there are some noticeable discrepancies. The patterns are invariably slightly distorted from those predicted. Furthermore, whereas the interline spacing is frequently within -5% of its predicted value, some of the data yield values differing by as much as 30%. Many observations of the mean separation between the vortices in a two-line pattern yields values averaging about 10% smaller than predicted. It is suggested that vortex pinning at the bottom or side surfaces of the experimental cell may play a role. The vortices are then not perfectly rectilinear and some discrepancies of the sort observed are to be expected. Interesting oscillation modes observed in the vortex distributions will be discussed later. The form of the energy of a straight vortex line, eq. (3.1), has an important consequence for the thermodynamics of two-dimensional superfluid films as pointed out by Kosterlitz and Thouless (1W3). Ignoring contributions from three-dimensional excitations of the line, the entropy is just proportional to the logarithm of the total number of different independent positions that can be occupied by the line: S = k , fn(A/a2)= Zk, ln(R/a)+ constant,

(3.14)

where k , is Boltzrnann’s constant and A is the area of the system. The free energy (in a non-rotating film) is then F = E - Ts = ( L p S ~ * / 4In(R/a) r) - 2k,T In(R/a) ,

(3.15)

where L is the length of the line. It follows that it is free-energetically favorable for a line to be present when the temperature is above some critical temperature TKTgiven by (Nelson and Kosterlitz, 1977; Kosterlitz and Thouless, 1978) ksTKT= u , ~ ~ / 8 r , where

(3.16)

a; is the areal superfluid density in the film. At temperatures below

Ch. 1, 841

VORTICES IN HELIUM I1

65

TKTthere is some thermal distribution of oppositely oriented bound vortex pairs. Above T K T , vortex pairs will dissociate completely, leaving a finite number of free vortices. The spontaneous appearance of free vortex lines implies dissipation for any superflow (see section 4.2) and hence a breakdown of superfluidity. The transition from the normal state above TKTto the superfluid state below TKTis accompanied by a discontinuous jump in the effective superfluid density given by eq. (3.16). Of course, even below TKT, the presence of superflow will induce some vortex pair dissociation. Furthermore the presence of a large number of vortex pairs will modify the energy of any particular pair so that the vortex number and the superfluid density must be “renormalized”. Many of the detailed predictions of this theory have been confirmed. The predicted jump in the superfluid density at the superfluid onset has been observed in the torsional pendulum experiments of Bishop and Reppy (1978, 1980), the quartz microbalance experiments of Chester and Yang (1973), and in the third sound onset experiments of Rudnick (19778). A predicted free vortex contribution to the flow resistance in the film above TKThas been observed (Maps and Hallock, 1981; Agnolet et al., 1981). Hegde et al. (1982) and Maps and Hallock (1982) have observed the predicted non-linear resistance below TKTassociated with flow induced vortex pair breaking [see, however, Joseph and Gasparini (1982) for a contrary result from a film on a metal substrate].

-

4. Vortex dynamics steady state 4.1. MUTUAL FRICTION Isolated quantized vortices at low temperatures have well-understood dynamics (apart from the core effects which were the subject of section 2). At any finite temperature, however, vortices are acted on by other quasi-particles of the fluid: phonons, rotons and solvated 3He atoms. The most familiar result of such collisions is drag. We note in passing that collisions can set vortex lines into thermally induced vibrational states (Brownian motion) and, at least in principle, can thermally activate the production of more vortices or, in the case of thin films, produce a phase transition. The subject of this section is the drag forces on vortex lines which couple the two components and thus give rise to “mutual friction”. We will not pursue this subject in depth because a review article on the subject has recently been published (Barenghi et al., 1983). The term “mutual friction” first arose in the consideration of the effective heat transfer in,narrow channels under conditions of fairly strong

66

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, $4

turbulence in the superfluid. Gorter and Mellink (1949) proposed that the two fluid equations should be supplemented by a force F, to make the temperature gradient down narrow tubes proportional to the cube of the heat flux. Subsequently, the understanding of mutual friction was greatly enlarged by the work of Vinen and Hall on uniformly rotating helium I1 and the development of the dynamics of vortex lines in the presence of friction [see, for example, Hall (1%0), Vinen (1%1)]. The simplest form of the two fluid equations (neglecting certain second order terms and bulk viscosity and thermal conductivity) is ps dus/dt = - ( p J p ) V p

+ p , S V T - Fns,

(4.1.1)

The mutual friction terms Fnscan be written in different ways according to t h e problem at hand. For the case of superfluid turbulence, Fnshas the Gorter-Mellink form with mutual friction coefficient A

and is discussed in depth in a review by Tough (1982) in Volume VIII of this series. For the case of helium I1 rotating uniformly at angular velocity R one can write ~ n , = -(Bpnps/p>h

x q ) - B ’ ( p n p J p ) ( a x 419

(4.1.4)

where h is t h e unit vector a/lRI, q = (us- un), and t h e dimensionless coefficients B and B’ describe the dissipative and non-dissipative contributions to Fns.Implicit in this simple form for the mutual friction is the presumption that neither the orientation nor the distribution of t h e vortex lines is disturbed by the flow. This form, first advanced by Hall and Vinen, is based on results of experiments on second sound propagation in uniformly rotating helium 11. In particular, the mutual friction force modifies the propagation of second sound, the relevant wave equation becoming

where u2 i s t h e velocity of second sound in the absence of friction. This fact is used to measure the parameters B and B’.The term containing the parameter B gives rise to a contribution to the attenuation of second sound where the attenuation coefficient is given by

Ch. 1, 141

VORTICES IN HELIUM I1

a = BOJ2u’

67

(4.1.6)

and the term containing (2 - B’) gives rise to a measureable coupling between otherwise degenerate modes in a suitably designed resonator. The quantities B and B’ depend slightly on the frequency of the second sound and are complex (Miller et al., 1978; Mathieu and Simon, 1982). The reader is referred to a recent review by Barenghi et al. (1983) for details of these determinations. We give a list of recommended values of B and B’ in table 4.1.1 and display their temperature dependence for frequencies of the order of 1 kHz in fig. 4.1.1. While the values reproduced here are accurate enough for many purposes, there are problems, such as the study of vortex line turbulence, where the frequency dependence of B and B’ introduces a substantial error. The best procedure for obtaining accurate values of B and B’ would be to work from the microscopic scattering cross sections (see below) back to B and B‘ at the desired frequency; but there are substantial numerical difficulties in doing so. It would be useful to develop a practical procedure for obtaining accurate values of B and B’ at any frequency. One should note in addition that a method exists for coping with the special problem which arises in steady flow, when the frequency is zero. Values of B and B’ can be calculated by substituting a length appropriate for steady flow in place of the usual penetration depth for viscous waves [see Yarmchuck and Glaberson (1979) p. 429, Vinen (1957)l. Following Donnelly et al. (1%7), the wave equation for second sound (ignoring the bulk attenuation) can be generalized to include still another friction term B” [called B, by Snyder (1%3)] which would give rise to attenuation of second sound transmitted parallel to the rotation axis:

4 + (2- B ’ ) n X 4- Bd? X (aX 4)+ B”d?((n. 4 ) = u2V’(V*q). (4.1.7) For second sound transmitted in the z-direction, we let

where q,, 9, and q, are constants, u is the angular frequency of the second sound, k is the second sound wave number and a is a second sound attenuation coefficient (due to vortex lines only). We also let = R(sin 8, 0, cos e), so that 8 is the angle between the transmission (2-direction) and the vortex lines. Then, to second order in O/a, a = (O/2u2)(B sin’ 8

+ B”cos’ 8 ) .

(4.1.8)

68

W.I. GLABERSON AND R.J. DONNELLY Table 4.1.1 Mutual friction coefficients B and B'(Barenghi et at., 1983).

1.52

1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

1.40 1.35 1.29 1.24 1.19 1.14 1.10 1.06 1.02 0.99 0.98 0.98 1.01

0.61 0.53 0.45 0.38 0.31 0.25 0.19 0.15 1.10 0.07 0.05 0.04 0.04 0.05 0.04

2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10

1.02 1.04 1.05 1.07 1.10 1.13 1.16 1.21 1.26 1.33

0.04 0.04 0.03 0.02 0.01 0.00 -0.01 -0.03 -0.05 -0.08

2.11 2.12 2.13 2.14 2.15 2.16

1.42 1.53 1.69 1.90 2.21 2.67

-0.12 -0.17 -0.24 -0.36 -0.54 -0.83

2.161 2.162 2.163 2.164 2.165 2.166 2.167 2.168 2.169 2.170 2.171

2.73 2.80 2.88 2.99 3.12 3.28 3.49 3.75 4.13 4.72 5.93

-0.94 -1.00 - 1.07 -1.15 -1.25 -1.37 -1.51 -1.71 - 1.98 -2.40 -3.28

1.46

[Ch. 1, 94

Ch. 1, $41

VORTICES IN HELIUM I1

69

rn

3

0

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

T (K)

1.o

0.8 0.6 0.4 0

'

-I

0.2

0

- 0.2 1

Fig. 4.1.1. The behavior of B as a function of (a) temperature and @) reduced temperature, and the behavior of B' as a function of (c) temperature and (d) reduced temperature after a fit by Barenghi et al. (1983).

70

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 04

8

6

m

4

N

0' 0.8

I

I

1

1

I

I

1.C

1.2

1.4

1.6

1.8

2.0

T

2.2

(K)

(C)

I .o 0.8

m I

0.6

N v

0 I

m

0.4

0 _I

0.2

0

-4

10

I

o-~

1 0-z

(T,-T)

lo-'

1 oo

/T,

(d ) Fig. 4.1.1(continued).

Evidence for a finite value of B" has emerged from time to time. The most recent determination by Mathieu et al. (1984) is shown in fig. 3.2. The paper should be consulted for a discussion of this measurement. We now proceed to discuss the drag forces on vortex lines themselves.

Ch. 1, 841

VORTICES IN HELIUM I1

71

The Magnus force on a unit length of vortex line can be written

where uL is the velocity of the line in the laboratory system and us is the superflow velocity. The drag force o n unit length of line is fD

= -yo< x (2 x (u, - UL))+ y;2 x (U"--UL),

(4.1.10)

where ri is the unit vector along the core of the line and yo and yh are drag coefficients calculated from B and B' through eqs. (4.1.17) and (4.1.18). The gammas have units g cm-'s-'. Provided one can neglect the inertial forces on the line (an issue which we comment upon in section 5.1) the sum of forces on the line must vanish and we have fM+fD =

*

(4.1.11)

There are a number of useful applications of this equation, some of which are discussed by Barenghi et a]. (1983). We note here the useful relation for the decay of a vortex ring at finite temperatures. This is given by

where u, is the self-induced velocity of the vortex ring [see, for example, eq. (1.6)], us is the superflow velocity associated with sources other than the ring and where a further drag coefficient y is defined by (4.1.13)

We show representative values of the coefficients yo, y;, and y in table 4.1.2. Derivations of new relationships in mutual friction often require switching back and forth between various forms of the mutual friction coefficients. Two useful combinations in such calculations are a and a', defined as

The quantity y is then given by

72

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 94

Table 4.1.2 Drag coefficients yo, yi. and y (Barenghi et al., 1983).

'

1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95

5.08X 10 6.12 x lo-' 7.25 x 10.' 8.45 x lo-' 9.68x 10 1.09x 10.' 1.21 x lo-' 1.32 X 10.' 1.42 x 10 1.51 X 1.58~ 1.65 x 10-5 1.69 x 10.' 1.72X 10

2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10

1.71 x 1.70 x 1.69 x 1.67 x 1.66 x 1.63 X 1.60 x 1.57 X 1.52 X 1.47 X 1.40 x

10.'

2.11 2.12 2.13 2.14 2.15 2.16

1.31 x 1.20 x 1.06 x 8.76 x 6.56 X 3.91 x

10-5 10-5

2.161 2.162 2.163 2.164 2.165 2.166 2.167 2.168 2.169 2.170 2.171

3.62x 10 3.33x 10 3 . 0 3 ~10 2.73 x 10 -6 2.42 x 1 0 ~ b 2.11 x 1.78 x 1.45 x 10.' 1.11x lo-' 7.62 x 10.' 3.95 x

f,

'

'

4.61 x 1.13X 1.77 x 2.34 x lo-* 2.83x 10 f, 3.33 - 10.'

4.94 x lo-' 5.94 x 10-4 7.03 x 10-o 8.19 x 10.' 9.39 x 1 0 ' " 1.06 x 10.' 1.18 x lo-' 1.30 x 1.41 X 10 1.51 X 10' 1.60 x lo-' 1.68 x 10-~ 1.74 X lo-' 1.78 x 1O-s

4 . 1 7 ~10 4.65X 10 4.39x 10 4.93x 10 5.24 x 10.' 5.59x 10 5.97 x lo-* 6 . 3 9 ~10 6 . 8 4 ~10.' 7.31 x lo-& 7.81 x 10

1.81 x 10F' 1.81 x 10.' 1.82 x lo-' 1.82X 10 1.82 x 1.82 x lo-' 1.83 x lo-' 1.83 X 1.83 x lo-' 1.83 x 10-~ 1.83 x lo-'

8.29 x 8.71 x 10 8.99 x 10." 9.00 x W h 8 . 3 8 ~10 6.64 x lo-*

1.82 x 1.81 X 1.79 x 1.74 x 1.63 x 1.37 x

6.37 x 6 . 0 9 ~10.' 5.78 x lo..' 5.46x 10 5 . 1 2 ~10 4.74 x lorb 4.31 x lo-* 3.83 x 10 3.27 X 10.' 2.60 x 10-6 1.73 x lo-&

1.33 x 1.28 x 1.24X 1.19X 1.14X 1.09 x 1.03 x 9.58 x 8.74 x 7.66 x 6.08 x

- 1.83 X

1.92 X - 1.92 X -1.82X -~1.59 X -1.24X -7.63 X .-

-1.8s x

lo-* lo-'

lo..' 10

'

'

10 10 10.' 10.' 10.' 10.'

'

'

lo-' 10-.5

lo-$ 10.' 10P 10

'

10-5

10.' lo-' 10.' 10

'

'

'

'

'

' '

1Ws 10-~ 10"' 10"'

'

10. 10.' 10 10 10

' ' '

los 10.'

lo-* 10.' lo-*

Ch. 1, $41

Y = PsKa

VORTICES IN HELIUM I1

73

(4.1.16)

and the transformation for yo and yh from the observed B and B ‘ is given by (4.1.17)

(4.1.18)

A microscopic form of mutual friction is directly related to roton scattering from an element of line. The drag force on unit length of line due to scattering of excitations is:

where vR is the normal fluid velocity at the core of the line, and where the microscopic parameters D and D‘ are related to scattering lengths uHand U I by

V, being the average group velocity of rotons. Before completing the balance of forces with the Magnus force one must first note the existence of a subtle effect, first discussed by Iordanskii (1%5), which modifies the microscopic forces by the addition of a new force per unit length of line called the Iordanskii force f,:

The Iordanskii force is independent of the structure of the vortex core and arises principally because the normal component in the two fluid theory cannot be identified exactly with the fluid formed by the gas of excitations. The momentum density carried by the normal fluid is pnun, while that carried by the excitations is pn(en- 0,). Two derivations of the Iordanskii force are given by Barenghi et al. (1983). The balance of forces is now f M + f i +fa = 0: i.e.,

74

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 84

where the transverse friction constant D, is defined as Dt = D' - p , , ~ .

(4.1.23)

Values of D, D,, D'. a,, and aI are subject to considerable experimental uncertainties, and the interested reader should consult Barenghi et al. (1983) for recommended values of these quantities as well as interrelations among all the friction coefficients. 4.2. THERMOROTATION EFFECTS Anderson (1966) has shown that the motion of vortex lines through superfluid helium is accompanied by a chemical potential gradient in the fluid. This prediction is based upon a description of the superfluid in terms of a complex order parameter whose time rate of change of phase is proportional to the chemical potential. Anderson showed that the passage of a vortex line between two points in the fluid produces a change of 2 7 ~ in t h e order-parameter phase difference between the points. Applying this idea to a situation in which a large number of vortex lines move uniformly through the superfluid leads t o the result that, on a scale large compared to the vortex spacing, a chemical potential gradient will be present which is proportional to the line density and velocity and oriented in a direction perpendicular to t h e direction of motion. This very simple situation, in which an array of vortex lines is moved through the superfluid in a controlled manner, has been closely approximated in the series of experiments done by Yarmchuk and Glaberson (1978, 1979). Being based upon a microscopic model of vortex line dynamics, the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations [eqs. (4.1.1), (4.1.2) and (4.1.4)] (Hall and Vinen, 1956; Hall, 1958; Hall, 1960; Hall, 1963; Bekarevich and Khalatnikov, 1%1) allow t h e interpretation of experimental observations in terms of vortex line motion that reveals their relationship to the concept of phase slippage proposed by Anderson. In the following, the concept of phase slippage will be briefly reviewed, and the result will be applied to an idealized experiment in which the HVBK microscopic equations yield simple results for the vortex line motion. A more complete description of this experiment is then obtained by solving the HVBK macroscopic equations in the case where the flow is confined between paraliel plates. Anderson (1966), considered the superfluid to be described by a complex order parameter, the phase of which is related to macroscopic superfluid properties. In particular it was shown that (see also Roberts

Ch. 1, 641

VORTICES IN HELIUM I1

75

and Donnelly, 1974) -(film) d&dt = p ,

(4.2.1)

where 4 is the order parameter phase, m is the mass of a helium atom, and p is the chemical potential per unit mass. By requiring that the order parameter be single-valued and noting that its phase need only be defined to within an integer times 27r, the quantization condition is obtained:

V4 - dl = ~ N T ,

(4.2.2)

N is an integer and may be non-zero if the path of integration encloses a region of superfluid exclusion. Associating the superfluid velocity with the gradient of the order parameter phase

us= (fi/m)V4

(4.2.3)

we see that eq. (4.2.2) is a statement of the quantization of circulation. A vortex line passing between two points in the superfluid then leads to a phase “slippage” of 27r in the phase difference between the points. In the case where a large number of vortices move through the fluid, appropriate time derivatives may be defined in an average sense so that

where (dn/dt) is the average rate at which vortices cross a line connecting the two points of interest. The chemical potential gradient is perpendicular to the circulation of the line K = (h/m)& and to the velocity of the lines relative to the superfluid uL - us, so that

where uL is the number density of vortex lines. (Note that -p,Vp, is the Magnus force per unit volume on the superfluid.) In the absence of superfluid acceleration, all of the chemical potential gradient in the system is associated with vortex motion

V p / p - SV T = Vp, ,

(4.2.6)

whereas, if acceleration is allowed, the system can also respond to the chemical potential gradient by accelerating

76

W.I. GLABERSON AND R.J. DONNELLY

V p = V,U,

- dv,/dt.

(Ch. 1, 44

(4.2.7)

Note that eqs. (4.2.1) and (4.2.3) are valid locally (outside of vortex cores), whereas eq. (4.2.7) deals with an average over distances large compared with the intervortex line spacing. This relation simply expresses the fact that a chemical potential gradient gives rise to a time-varying phase gradient, which, in turn, can arise from either accelerating fluid (us CI: Vi) or moving vortices. In a system in steady state while rotating at frequency 0,this becomes Vk = V p v - 2 a x v , ,

(4.2.8)

where v, is now measured in the rotating coordinate system. In order to relate this last expression to measureable quantities, it remains to determine vL in terms of the experimentally controllable normal and superfluid velocities. From eqs. (4.1.9) and (4.1.10) we write

where on and us are the normal fluid velocities averaged over a region containing many vortex lines, K is the vortex circulation, k is the direction of local vorticity, and yo and yh are parameters related to roton collision diameters for parallel and perpendicular momentum transfer to the vortex lines. The right-hand side of eq. (4.2.9) is a general expression for the frictional drag experienced by the vortex line as a consequence of a transverse normal fluid flow, and eq. (4.2.9) states that the net force per unit length on a vortex line-Magnus force plus friction force-must be zero, i.e. the assumption of eq. (4.1.11). For the case of thermal counterflow in an infinite medium, v, and v , are given by

where it has been assumed that a uniform heat flux density given by pSTuo is applied in the f direction. Choosing i as t h e direction of ri we find that t h e component equations for the vortex line velocity are given by (4.2.11) where v L r i+ o L , j = uL. These equations can be reduced to expressions

Ch. 1, 14)

VORTICES IN HELIUM I1

77

for uLx and uLv separately:

(4.2.12)

Using eqs. (4.2.5) and (4.2.9) and these expressions for the vortex line velocity, we can calculate the contribution to t h e chemical potential gradient due to vortex line motion

where mL has been taken as ZLUK and R is the rotation speed. In order to simpIify this expression, the parameters yo and y i will be replaced by their equivalent forms in terms of the macroscopic mutual friction parameters B and B’. Using eqs. (4.1.17) and (4.1.18) we obtain the much simpler form

and eq. (4.2.8) yields

Naturally, the same result is obtained when the HVBK macroscopic equations are solved directly. The macroscopic equations, however, yield additional information. By including the effects on the normal fluid, one obtains the separate pressure and temperature gradient contributions to Vp. In the absence of vortex curvature, the equations of motion in the rotating coordinate system are

78

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 94

where F,, is the mutual friction force given by

and w = V x u s + 2L2, and 4 = w / ( w ( is a unit vector along the local vorticity. The equations have been written for a point on the axis of rotation, so that the centripetal acceleration is zero. The effect of this term is to produce a radial pressure gradient independent of applied heat flux and therefore it is not detected experimentally. The steady-state results for uniform counterflow transverse to the uniformly rotating superfluid are

Under experimentally accessible conditions, the magnitude of this temperature gradient does not exceed several microdegrees per centimeter. The pressure gradient obviously plays a minor role in infinite-medium t hermorotation. The results in eqs. (4.2.17) and (4.2.18) indicate roughly what is to be expected in a rotating counterflow experiment if the counterflow channel is very large. The effects of the channel surfaces on vortex line motion, however, are not included, and in order to deal with these effects, the macroscopic equations must include terms due to vortex curvature. Effects of curvature cannot be completely neglected in a channel of finite height because in such a system the normal fluid velocity is not constant throughout space and therefore will affect the lines differently at different points along their lengths. For a solution of the full hydrodynamic equations for flow in a channel of finite height and including the effects of vortex line pinning on the channel surfaces, the reader is referred to Yarmchuck and Glaberson (1979). The experimental arrangement used by Yarmchuk and Glaberson (1978, 1979), shown schematically in fig. 4.2.1, involved a glass channel of rectangular cross section, closed at one end with the other end open to a pumped helium bath. A resistive heater was placed near t h e closed end and the channel was rotated about a vertical axis perpendicular to the heat current. Temperature gradients parallel to the channel axis and perpendicular to it, as well as chemical potential gradients parallel to the channel axis were obtained as a function of heater power and rotation speed. The parallel component of t h e temperature gradient is shown in fig. 4.2.2 as a function of heater power at several rotation speeds at T = 1.3K. For heater powers below the critical power associated with the

Ch. 1, 841

79

VORTICES IN HELIUM I1

HEATER

Fig. 4.2.1. A schematic drawing of the counterflow channel (Yarmchuk and Glaberson, 1978, 1979). Of the three superconducting films, the center one is used to regulate the ambient bath temperature and the outer two are used in a bridge as temperature-difference detectors. The chemical potential detector is described in the references cited. I

I

I

I

*

c

Q (millrwatt) Fig. 4.2.2. A plot of the parallel component of t h e temperature gradient as a function of heater power for several rotation speeds. (Yarmchuk and Glaberson, 1978, 1979.)

80

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, $4

onset of turbulent counterflow, the temperature gradient is proportional to the heater power and increases with increasing rotation speed. Plots of the linear regime slopes versus rotation speed, for both parallel and perpendicular components of the temperature gradient, are shown in fig. 4.2.3. The solid lines are fits to the data using a full solution to the equations as outlined above, for a channel of finite height but infinite aspect ratio. The only fitting parameters are the mutual friction coefficients B and B' and the values obtained are in excellent agreement with those determined from second sound damping experiments. A plot of t h e chemical-potential gradient V p versus heater power Q for R = 0 and 0 = 10radfs is shown in fig. 4.2.4. Also shown for camparison is the temperature gradient. There exists a critical heater power Qc,,not observable in the temperature measurements, below which V p is zero. Q,, was associated with the power at which the vortex array "depins" and begins to move in response to the counterflow. Below Q,, the vortices are pinned to protuberances in the channel walls and accommodate the counterflow by deforming. If the boundary condition on V X us in the calculation is adjusted, so as to produce no longitudinal chemical-potential gradient, it is found that, indeed, little influence on VT

n(rodhec) Fig. 4.2.3. A plot of A T l / Q and AT,/Q in the linear regime as a function of rotation speed. The solid lines are discussed in the text ( Y m c h u k and Glaberson, 1978. 1979.)

Ch. 1, 841

VORTICES IN HELIUM I1

-E

1.5 0

0

0

0 0

1.0

I

0

0

V \

E2

I

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0

0 0

v

0

0

0

t=

D

0

0.5

0

0

0 0

I.o

I.5

2.o

Q (mwatt) Fig. 4.2.4. The chemical-potential gradient and temperature gradient as a function of hearer power for R = 0 and R = 10 rad/s. (Yarmchuk and Glaberson, 1W8, 1979.)

is predicted-the pressure gradient does, of course, become large. A systematic study of vortex pinning is discussed in the next section. 4.3.

VORTEX PINNING

A significant motivation for the investigation of vortex dynamics is the possibility that such dynamics might play the role in the spin down behavior of pulsars - rotating neutron stars. What is observed (Lohsen, 1972, 1975; Manchester et al., 1975, 1978; Reichley and Downs, 1%9, 1971) in the Crab and Vela pulsars (as well as others) are catastrophic events - glitches - in which the pulsing frequency suddenly increases. Immediately following a glitch, t h e deceleration rate is increased, relaxing to the preglitch rate with a long relaxation time. There have been numerous attempts at explaining the glitch phenomena and, in most of them (e.g. Packard, 1972; Ruderman, 1969, 1976; Baym et al., 1969; Baym and Pines, 1971; Anderson et al., 1978; Krasnov, 1977; Campbell, 1979) the role played by quantized vortices is crucial. Alpar et al. (1981) have been quite successful in explaining the bulk of the observed data. T h e y picture the neutron star (or at least those components of the star that are relevant to its observed dynamical behavior) as consisting of several distinct components which may rotate at different frequencies (see

82

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 84

fig. 4.3.1). The first consists of t h e charged components including t h e star crust, electrons, superfluid protons and a possible interior normal core. Throughout this matrix, there is neutron superfluid which is threaded by a uniform array of quantized vortices. The vortices interact with the crust nuclei in a manner that depends on t h e relative magnitudes of the neutron superfluid energy gap and the energy gap of t h e neutrons within the (large) nuclei. The crust therefore contains two neutron superfluid regions - a pinning region where the vortex lines are strongly pinned to the crust nuclei and a non-pinning region where the lines avoid the nuclei and can flow relatively easily. There is, of course, also t h e core neutron superfluid inside the crust which as can be shown, equilibrates rapidly with the crust. The system is then described in terms of a two-component model consisting of (a) the pinned superfluid within t h e crust and (b) everything else. The giant glitches are represented as events in which unpinning of vortices takes place in a weakly pinned transition layer. This of course leads to a sudden transfer of angular momentum to the crust and a consequent speed-up. Subsequent to the glitch, the “driving force” responsible for vortex creep is relieved, thereby decoupling some of t h e crustal superfluid from t h e crust so that t h e effective moment of inertia is reduced and the deceleration rate increases. The depinning event itself is

SUPERFLUIO NEUTRONS

SUPERFLUIO PROTONS NORMAL e-

Vy 7 x lO’%J

0 - 3

Fig. 4.3.I . Sketch of the structure of a neutron star. The magnitudes of the various radii and densities can vary 50% depending on estimates of the parameters in the equation o f state. (Anderson et al., 1982.)

Ch. 1, $41

83

VORTICES IN HELIUM I1

triggered by a buildup to some critical value of the differential rotation rate between the crust and the pinned superfluid. Macro-glitches in the Crab pulsar are pictured as being triggered by some external event, the younger and hotter pulsar having a faster steady state vortex creep rate so that creep alone can relieve the differential rotation stress without spontaneous discontinuous unpinning events. Hegde and Glaberson (1980) performed a series of experiments in order to systematically investigate the pinning of vortices to surface protuberances. It had been generally believed [e.g. Tsakadze (1978)] that vortex pinning could be described in terms of a viscous-like flow of vortices along surfaces. The observation described in the previous section, that no chemical potential gradient at all could be observed for a vortex array subject to a counterflow intensity less than some critical value, suggests that static pinning is important. The experimental arrangement was similar to that of Yarmchuk and Glaberson (1979). A glass channel of large aspect ratio was rotated about

.

r

--8 V

ff

'I3

'*

a " 2

0 Q6

I .o

14 .

1.8 2.2 2.6 ( rad/sec)'I2

3.0

Fig. 4.3.2. A plot of the critical heat flux for the onset of vortex motion as a function of the square root of the rotation speed, at the temperature T = 1.30K.(Hegde and Glaberson, 1980.)

84

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 94

an axis perpendicular to the channel axis and to t h e large-area walls. The channel was closed at one end, near which a resistive heater was placed, and open at the other end to a pumped liquid helium bath. A small hole in one of the channel walls near the heater allowed a connection to a chemical potential detector. The absence of a chemical potential gradient was taken as an indication of the absence of vortex motion. The large-area channel walls, which are perpendicular to the vortices in the absence of thermal counterflow, were of various kinds: (a) cleaved mica, (b) polished plate glass, (c) glass coated with a dense distribution of either 1 km, 10 Fm or 100 km diameter glass microspheres. The areal density of the microspheres was less than that of close-packed spheres but

(a) Fig. 4.3.3. (a) A plot of the critical heat flux, at a fixed rotation speed of fl = 6 rad/s as a function of temperature. (b) A plot of the depinning critical heat flux y:,, equal to t h e difference between the critical heat flux in “rough” surfaced channels and ”smooth’. surfaced channels, as a function of temperature, for a rotation speed of R = 6 radis. The solid line is t h e result of a calculation based on the HVBK equations and assuming the boundary condition that t h e vorticitv is parallel to the surface at the surface. The dashed line is t h e result of an isolated line calculation using a similar boundary condition. (Hegde and Glaberson. 1980.)

Ch. 1, 041

VORTICES IN HELIUM I1

85

much larger (except in the case of the 100 Fm diameter spheres) than that of the vortices. A plot of the critical heat flux in the channel, qcl,versus the square root of the rotation speed O”, is shown in fig. 4.3.2 foc the temperature T = 1.3K. qcl clearly becomes independent of for all channel surfaces at high rotation speed. Of course, as the rotation speed is reduced t o zero, qcl becomes equal to the critial heat flux corresponding to the transition to turbulence. qcl is plotted against temperature at a fixed high rotation speed of O = 6rad/s in fig. 4.3.3. The data evidently divides into two distinct groups - “rough” surface channels and “smooth” surface channels - within each of which, they are completely coincident. That the data for cleaved mica and for bare polished glass are completely coincident, in spite of the fact that their surface characteristics on a microscopic scale are very different, suggests that qcclin these systems has nothing at all to do with surface roughness. The authors suggest that there are two distinct mechanisms involved in determining qcl, only one of which is associated with surface roughness, and that the two mechanisms produce additive contributions to qc,. In fig. 4.3.3 we show a plot of q:,, the difference between qcl for the “rough” channels and averaged values of qc, for the “smooth” channels, as a function of

86

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, $4

temperature. The solid line is a result of a calculation to be discussed below. This result, having no adjustable parameters, is clearly in excellent agreement with the data. It was assumed that all of the “rough” systems studied involve perfectly pinned vortex lines. By this was meant that the lines will remain pinned as long as they can terminate on their pins at an angle, with respect to the surface, greater than zero. Yarrnchuk and Glaberson solved the linearized Hall-Vinen-Bekharevich-Khalatnikov (HVBK) hydrodynamic equations for counterflow in a rotating channel of infinite aspect ratio (no side walls) and infinite length. This solution is clearly inadequate in our situation where the local vorticity can, and indeed will, greatly exceed 2 0 . The full nonlinear equations were numerically solved by Hegde et al., and the calculated value of the heat flux at which the vorticity at the channel surface is parallel to the surface is shown as the solid line in fig. 4.3.3. This is given approximately by the expression

2 upST “I

=

(4.3.1)

d [G : + @/p, - G,)2]’’2’

where p is the fluid density, p r t h e superfluid density, S the specific entropy, T the temperature, and d the channel height; G, and G, are related to the mutual friction coefficients by

G ,=

1 - a’ a 2 +(1-a‘)*’

G,=

a a2+(1-d)Z’

-

(4.3.2)

where as before a = Bpn/2p,a’ = B‘pn/2p,u (~1477)In(b/a), and b is the interline spacing. Because B, and particularly B’, are not known very accurately, there is -10% uncertainty in t h e theoretical values of 9cl. The HVBK equations ought to be valid in situations where all characteristic lengths in the problem are much greater than the inter-vortex line spacing, this requirement is well satisfied in the interior of the channel. Near the channel surfaces, however, the radii of curvature of the vortex lines go to zero faster than t h e inter-line spacing and t h e HVBK equations are not strictly valid. In this region, t h e behavior of isolated vortex lines would be a better representation of t h e situation. It turns out that isolated-line behavior is qualitatively similar to the average hydrodynamic behavior so that these considerations are probably not very important. To obtain a feeling f o r isolated-line behavior, Hegde and Glaberson (1980) solved the (nonlinear) equation [see eq. (4.2.9)]

Ch. 1, $41

VORnCES IN HELIUM I1

87

where us and u, are superfluid and normal fluid velocities, R is a unit vector along K, and y and yi are constants simply related to B and B’, as discussed in the previous section. This equation states that the net force on a stationary vortex line, Magnus force plus friction force, must be zero. u, is taken as that for Poiseuille flow and us= u,+ urn, where u, = -@,/p, (i.e., “plug” flow) and urn is the Arms and Hama self-induced velocity given by eq. (1.9). The values of the calculated heat flux for which the vortices terminate parallel t o the surface are shown as the dashed curve in fig. 4.3.3. Although this curve agrees about as well with the data as the HVBK curve, the interactions between the vortices are indeed large and the effects of the image in the surface have been ignored so that the good agreement is probably fortuitous. The nature of the critical heat flux in the “smooth” channels is not understood. The HVBK equations for a channel of finite width and length have not been solved and the highly speculative possibility remains that the vortices are capable of distributing themselves so as to produce no vortex motion, below some critical heat flux, even in the absence of pinning. In the presence of pinning, once 9T1is exceeded the vortices would be capable of redistributing themselves so as to maintain immobility of the vortices in the steady state. Only when the heat flux exceeds t h e sum of 9T1 and qco, t h e critical heat flux in t h e smooth channels, would steady-state vortex motion occur. The qualitative difference between the temperature dependence of qcl and qrl in fig. 4.3.3 lends support to the assumption that the mechanisms involved produce additive contributions to qcl. The picture of vortex pinning developed here is that, at least in a situation where pinning sites are dense compared to the vortex distribution, the vortices remain pinned as long as they conceivably can. Onset of depinning is characterized in terms of a macroscopic boundary condition -the curl of the (average) superfluid velocity field is parallel to the surface at the surface. The details of the interaction between the vortex line and its pinning site do not appear to be important here, but might be important in situations where the vortex distribution is dense compared to that of the pins or where the distance between vortices is not much smaller than the channel height. A calculation of the interaction between an isolated vortex line and a hemispherical pinning site was done by Schwarz (1981). The evolution of a vortex line terminating on a pinning site and subject to a transverse superflow, was followed numerically. The analysis is the same as that used by Hegde and Glaberson in their single-line calculation, except in two respects. Hegde et a]. solved directly for the stationary line shape whereas Schwarz determined the time evolution of the shape. The line, as expec-

88

W.I. GLABERSON A N D R.J. DONNELLY

[Ch. 1, 94

ted, “spirals” into its stationary configuration when pinned. More importantly, Schwarz solved the boundary value problem so that the effects of the image vortex in the surface were explicitly included. The calculation confirmed the picture of depinning suggested above. Below some critical superflow velocity a stationary line shape exists. As the superflow velocity is increased, the line comes into the “pin” at increasingly smaller angles with respect to the plane surface. Finally, the line passes too close to the plane surface and the effect of the image vortex in that surface dominates. A stationary shape no longer exists and the line “breaks” away from its pin. Fig. 4.3.4 shows various calculated line shapes. Fig. 4.3.4a shows the stationary line shape at a superflow velocity just below the critical velocity and figs. 4.3.4b-d show the time evolution of a vortex line under slightly supercritical conditions. This treatment of depinning is closely related to the extrinsic critical velocity model proposed by Glaberson and Donnelly (1966) but, for the reasons discussed earlier, is not particularly well suited for a quantitative analysis of a situation involving dense vortex arrays.

i ___

..

..&

.

J

Fig. 43.4. Behavior of a pinned vortex as the supertluid velocity is increased from slightly below the depinning critical velocity to slightly above i t . Parameters for this calculation are D = 10‘’cm, h = IO-‘cm. and (I = 0.1. The stationary configuration (a) corresponding to vs = 0.64cm/s becomes unstable when vI is increased to 0.67 cm/s, the vortex reconnecting t o the plane and moving off as in (b) to (d). (Schwarz, 1981.)

Ch. 1, $41

4.4.

VORTTCES IN HELIUM I1

89

SPIN-UPAND THE VORTEXSURFACE INTERACTION

Perhaps the most straightforward experimental test of any dynamicai model of rotating superfluid helium is the classic spin-up problem in which a freely rotating bucket of superfluid is impulsively spun-up and allowed to relax back to solid body rotation. The transient behavior of the container as it transfers angular momentum to the fluid not only probes the nature of the internal fluid dynamics but also the interaction of the fluid with the walls of the container. The investigation of the spin-up of helium I1 is not new. In the early seventies Tsakadze and Tsakadze (1972, 1973a, b, 1975) performed several spin-up experiments, both below and above TA.They observed exponential-like decay in helium I and in helium 11, the major difference between the two being that helium I1 relaxation was slightly quicker. When they roughened the container inner walls in order to investigate their interaction with the superfluid they observed much shorter relaxations times (- f~,,~,,) which were independent of temperature through TA.This curious result was thought t o be a consequence of normal fluid turbulence (Alpar, 1978). More recently Campbell and Krasnov (1982) have attempted to explain the results of some early experiments of Reppy and Lane (1961, 1%5) and Reppy et al. (1960) in which helium I1 was spun-up from rest. In these experiments it was found that after an initial normal fluid relaxation the superfluid component induced a relaxation characterized by Oc=A(l-

1 + B + C e-"T

(4.4.1)

(0, = container angular velocity), qualitatively very different from the typical exponential decay of a classical fluid. Campbell and Krasnov have produced a model of these experiments which incorporates a viscous vortex-boundary interaction in which the vortex drag force is simply proportional to the relative vortex-surface velocity. By varying the strength of the interaction they were able to fit Reppy and Lane's data quite well, giving further evidence t o the widely held assumption that a moving vortex line is indeed subject to a viscous force at the fluidboundary interface. The quality of their model's predictions does not, however, make a compelling case for a viscous interaction. They only considered spin-up from rest in which necessarily large fractional changes in vorticity occur. Clearly, in such cases vortex nucleation at the outer walls of the container becomes an important, if not the primary, superfluid relaxation mechanism.

yo

[Ch. 1. $4

W.I. GLABERSON AND R.J.DONNELLY

Some recent experimental evidence for a “static-friction’’ type of vortex-boundary interaction, in which the vortices appear to move along a surface exerting a force equal t o the vortex line tension, was presented by A d a m et al. (1985). A schematic diagram of their experimental apparatus is shown in fig. 4.4.1. The helium cell consisted of a hollow lead coated magnesium cylinder, A, 5 cm high and 2.8 cm in radius. The cylinder contained a set of thin aluminum disks and spacers which formed 8 cylindrical cells, B, each with a typical height to radius ratio of 0.12. The cylinder was sealed with a magnesium cap into which a small hole had been drilled, thus minimizing film flow out of the container during a run. After submerging t h e container in He I1 for a sufficient time for it to fill through the cap hole, the inner jar, C, was emptied via a fountain pump, D. The container was levitated by means of a superconducting magnet, E, surrounding its base. Axial stability was provided by a second superconducting magnet, F, positioned over the top portion of the container. The container was accelerated by a non-contacting induction motor consisting of a thin copper sleeve, G, surrounded by four superconducting drive coils, H. (a)

(b)

Fig. 4.4.1. Schematic diagram of the experimental apparatus. (b) Depicts the arrangement of photo-detector marks. used in monitoring the container’s angular velocity, on the top of the cell. (Adams et al.. 1985.)

Ch. 1, 041

VORTICES IN HELIUM I1

91

The experimental procedure involved impulsively spinning-up (and + do. spinning-down) the container of He I1 from 0, = ooto 0, = w 1 = o,, The container’s angular velocity, was then monitored as a function of time. The experiments were performed at T = 2.1 K and T = 1.3 K with both smooth disks and roughened disks, coated with #20 Aluminum Oxide Powder. The character of the T = 1.3 K relaxations depended dramatically on the disk surface roughness. Fig. 4.4.2 shows a typical response of t h e smooth-surfaced cell to an impulsive torque and figs. 4.4.3 and 4.4.4 are typical responses for the rough-surfaced cell. Note the exponential-like behavior of the former and the nearly linear response of the latter. Substantial departure from exponential behavior was characteristic of all the low temperature rough disk relaxations for which oo> 1.5 rad/s. The remarkable linearity of the decay curve, observed to be symmetric with respect to spin-up and spin-down, indicates that the superfluid applied a strong, relatively constant, internal torque throughout most of the relaxation. The data are interpreted in terms of a vortex-boundary force which is taken to be the sum of a “static” and a viscous boundary force, I .o

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t

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Fig. 4.4.2. The angular velocity of the smooth disk cell after an impulsive spin-up torque (00 = 8.54 rad/s, Aw = 0.283 rad/s). The solid line represents the prediction of the model for 6 = 0.005 and f, = 0.0. (Adams et al., 1985.)

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t (s) Fig. 4.4.3. The angular velocity of the rough disk cell after an impulsive spin-up torque (m= 2.21 rad/s. AW = 0.283 rad/s). The solid curve is the prediction for 6 = 0.005 and f, = 0.53 x 10-6dyn/cm. The dashed cume is the prediction for 6 = 0.005 and f,,= 0.0. (Adams et al., 1985.) 4

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I

I

I

I

I

I

1

I

I

I

I

3

u)

\ L

N

'0 -

-3 2

I

c

v

I

; 0 I

20

30

t

40

tsf

Fig. 4.4.4. The angular velocity of the rough disk cell after an impulsive spin-down torque (wg = 3.35 rad/s, d w = -0.283 rad/s). The solid curve is the prediction for 6 = 0.005 and f, = 0.53 X lo-* dyn/cm. (Adams et d.,1985.)

Ch. 1, 041 Fb

= -(fD =

93

VORTICES IN HELIUM I1 + f k l ) l V ~ - Vb

-.fpQLb-

6P,K(VL-

vb>

for

IfD’fMl


for

IfD’fhd

’fp

9

(4.4.2)

where fp is the minimum force per unit length required to break loose a pinned vortex, fM is the Magnus force on a unit length of line eq. (4.1.9), fD is the drag force on a unit length of line [eq. (4.1.10)], f is the dimensionless viscous interaction coefficient, V, and vb are the vortex line and boundary velocities and is a unit vector in the direction of V,- Vb. Though fp and f are treated as adjustable parameters, fpL is assumed to be of the order

vu

PsK T L = -ln(b/a)

4lT

,

(4.4.3)

where TLis the line “tension” of an isolated vortex, a is the vortex core parameter, b is the inter-vortex line spacing and L is the vortex line length. This boundary force is, of course, in addition to the mutual friction and Magnus forces acting on the vortex lines throughout the fluid. Simulations of the experiments were made in which f and fp were independently adjusted to fit the low temperature data. The exponentiallike behavior of the smooth disk data suggests that viscous drag was the dominant superfluid-container interaction. The smooth disk decays were therefore fit by setting fp = 0 and varying f. This is equivalent to the approach taken by Campbell and Krasnov. Shown in fig. 4.4.2 as the solid line is the numerical fit. The fact that the data falls along a somewhat straighter line than that of the fit is probably a consequence of not having perfectly smooth disks. Reasonable fits were obtained with f ranging from 0.005 to 0.007 for all of the smooth disk data, f increasing with decreasing rotation speed. As will be discussed, these values of 6 are consistent with treating the drag as arising from mutual friction in the Ekman layer. The rough disk relaxations were fit by simply “turning on” the “static” interaction. It was assumed that f was independent of surface roughness, thus leaving fp as the only adjustable parameter. Excellent fits, such as shown as the solid lines in figs. 4.4.3 and 4.4.4, to all of the “rough” disk data for which w,, > 1.5 rad/s were obtained with f,,= 0.53 X dyn/cm and were relatively insensitive to the value of 6. This value of fp corresponds to a maximum line “tension”, assuming a vortex line cannot apply a force to the boundary greater than its “tension”, of

TL= ;Lfp = 1.59 x lO-’dyn

(4.4.4)

W.I. GLABERSON AND R.J. DONNELLY

91

[Ch. 1, 04

in good agreement with the theoretical value,

=

1.5 X lo-' dyn

.

(4.4.5)

Note that the approach taken suggests that observation of the spin-up rate of the superfluid yields a reasonably direct measure of the quantum of circulation. At T = 2.1 K, where p,/p = 0.88, exponential-like behavior that was independent of disk roughness was observed. Simulations of these relaxations yielded poor results - predicting decay times much longer that what was observed. It seems likely that the poor quality of the fits at these higher temperatures was due to normal fluid secondary flow. In the model of pinning presented, a vortex line is absolutely immobilized on a surface protuberance until external forces stretch it to the point where it bends parallel to the surface at the surface. At this depinning threshold the vortex line can apply a maximum force, antiparallel to the sum of the forces acting upon it, equal to its energy per unit length. Thus, fp has a well defined theoretical value [eqs. (4.4.4) and (4.4.5)] which scales with temperature as ps. An experimental verification of this temperature dependence is needed t o guarantee that the agreement between simulation and theory was not fortuitous. A possible explanation for the necessity of including a viscous interaction in the model lies in the inadequacy of the normal fluid equations solved. By neglecting the axial and radial components of the normal fluid velocity, Adams et a]. have explicitly assumed that the normal fluid relaxes via viscous diffusion. However, it is well known that secondary flow is the primary relaxation mechanism in all contained (Newtonian) spin-up flows (Greenspan, 1968). Secondary flow is characterized by a quasi-steady Ekman layer at each disk surface.through which fluid is pumped radially by centrifugal action. Adams et al. believe that it is this viscous layer, unaccounted for in the simulations, that requires the ad hoc addition of a viscous vortex-boundary interaction to their model. For a typical T = 1.3K spin-up experiment in which w, = 3 radls, r] = 16 X P, and p, = 0.007 g/cm3, the thickness of the Ekman layer on each disk is,

(4.4.6)

The totaI normal fluid friction force on a vortex line due to its motion

Ch. 1, $4)

VORTICES IN HELXUM 11

95

through the top and bottom Ekman layers (assuming that in the layers V,= Vb)is,

where y o is a mutual friction coefficient. After equating FE to the viscous interaction of the model and solving for f , they obtain

(4.4.8) a value surprisingly close to that used in the simulations. Eq. (4.4.8)is also qualitatively consistent with the observed wo dependence of f . It thus appears that the viscous vortex-boundary interaction is indeed associated with the mutual friction exerted by the vortices moving through the Ekman layers. At low temperatures the normal fluid friction parameter, yoyo, is approximated by

which when plugged into the expression for f gives, (4.4.10)

It f does indeed represent vortex drag through normal fluid Ekman layers it should scale with temperature as pin at low temperatures. The value of f used to fit the data is four orders of magnitude smaller than typical values used by Campbell and Krasnov. It can easily be shown that at low temperatures the smooth disk (f, = 0) relaxation time, T ~ ,is proportional to (6' + l ) / f , so that ~ ~= 0.005) ( 6 = 7s(f = 200). Although the smooth data could be fit with either f = 0.005 or f = 200, the rough data restricted f to small values. As a final note, we point out that the vortex boundary interaction is likely to have a profound effect on turbulent flow through channels, particularly when the channel walls are rough [see, for example, Yamauchi and Yamada (1985)l. 4.5. VORTEX DYNAMICS IN THIN FILMS The static theory (Kosterlitz and Thouless, 1973, 1978) of phase tran-

96

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 94

sitions in two dimensional systems is fairly well established. The dynamical theory (Ambegaokar et al., 1978, 1980) is considerably more speculative and vortex diffusivity plays a crucial role in the dynamical theory. We will briefly review the theoretical treatment. The theory is constructed in terms of two parameters: K, the reduced stiffness constant, and y, the vortex excitation probability. These quantities are renormalized by thermally excited bound vortex pairs. Theory predicts that knowledge of K and y at any given length scale r can be used to predict behavior at other lengths through the Kosterlitz recursion relations: (d/dl)(K-') = 47r'y"

(4.5.1)

dy/dl= (2 - 7rK)y,

(4.5.2)

where I is a scale parameter defined as 1 = In(r/a) and a is the vortex core size. The connection with experimental quantities is

K = ~~a,/471~k,T.

(4.5.3)

where usis t h e areal superfluid density, k, is Boltzmann's constant and T is the temperature. Eqs. (4.5.1) and (4.5.2) contain a fixed point which determines TKTand is given by (Nelson and Kosterlitz, 1977) lim K ( 1 ) = 2/71

(4.5.4)

I-.=

at y = 0. [This is t h e same as eq. (3.16).] In order to observe predictions of this static theory, the characteristic time of the measurement must be long enough for the system to relax under the influence of an external current. For measurements carried out at a frequency w, the dynamics of vortexpair polarization fixes an effective I given as

I, = 0.5 In(l4D/wa),

(4.5.5)

where D is the vortex diffusivity. The physical situation then corresponds to the result of iterating the recursion relations, not to 1 = (below T,,), but to 1 = f,. . The principal consequences are then a small effective frec vortex density for T < TKTand non-vanishing of usfor T > TKT.Ambegaokar et al. (1978, 1980) demonstrated heuristically that, unlike most and transport coefficients at critical points, D should remain finite at TKT, D should be of furthermore argue on dimensional grounds that, at TKT, order (hlm ). Huber (1980) derived this result somewhat more rigorously.

Ch. 1, $41

VORTICES IN HELIUM I1

97

The experimental measurements of CT, near TKTby Bishop and Reppy (1978, 1980) are consistent with a value of I, = 12 corresponding to D 20(fi/m). Measurements (Fiory et al., 1983) of the flux flow resistance in thin film superconductors at TKTyield D h/2m, where m, is the electron mass. The first direct observation of vortex dynamics and determination of the vortex diffusivity in thin superfluid films was by Kim and Glaberson (1984a). Their experimental arrangement involved the use of a high Q third sound resonant cavity, similar in design to that of Rutledge et al. (1978). Two thin circular polished amorphous quartz discs were welded together at their edges. Helium was allowed to diffuse through the discs at room temperature. At low temperatures, the helium condenses into films coating the inner quartz surfaces and constitutes a third sound resonant cavity. The third sound cavity was mounted in their rotating cryostat and was rotated about an axis perpendicular to the film surfaces. The quality factor and resonance frequency of various modes were then monitored as a function of rotation speed and temperature. At temperatures not too far below TKT, it was possible to observe the damping of the third sound resonance associated with the motion of the rotation-induced vortices. In this temperature regime, the resonance width was observed to be strictly linear in the rotation speed (and therefore in the induced vortex density). We note that in superconducting films the flux flow resistance is linear in the applied magnetic field only at TKT.Below TKT,non-linear behavior at low fields, presumably associated with the non-infinite value of the effective penetration depth, is observed (Fiory et al., 1983). It is not surprising that such effects are not observed in helium. Following the treatment of Ambegaokar et al. (1980) they write the velocity of a vortex line subject to superflow past itself as

-

-

UL =

2.RDha, 2 x o,+(l- q u , , mK,T

~

(4.5.6)

where 0,the vortex diffusivity and C are related to the phenomenological drag coefficients describing interactions with the substrate and with thermal excitations analogous to the mutual friction coefficients of the HVBK equations in the bulk, and us is the superfluid velocity. Vortex flow gives rise to superflow decay [eqs. (4.2.3) and (4.2.4)]: do, dt

-=-

-2~hn ixu,, m

(4.5.7)

98

[Ch. 1, $4

W.I. GLABERSON AND R.J. DONNELLY

where n = Om/& is the vortex density and R is the rotation speed. It follows that du,/dt

3~

-DRu, + (term Ito us).

(4.5.8)

The vortex diffusivity can then be directly extracted from the contribution of the rotation-induced vortices to the third sound damping and is given by the expression (4.5.9)

where Aw is the excess resonance (full) width associated with rotation. In this discussion, it was assumed that the contribution to the sound damping from rotation-induced vortices is simply additive to the contributions from thermal conduction in the substrate and gas, polarization of bound vortices and flow-induced broken vortex pairs. It turns o u t that a convenient way of representing t h e data is as a function of T/us.The diffusivity appears to collapse to a universal curve and can be reasonably well represented, for T not too low, by the relation (see fig. 4.5.1)

0

20

40

60

80

100

Fig. 4.5.1. Vortex diffusivity in thin film plotted as a function of (T/u,).The solid line is the The arrow indicates the location of the Kosterlitzrelation D = 0.17(l/m)-’(k~T/u,)Z. Thouless transition. (Kim and Glaberson, 1 W a . )

Ch. 1, 841

D

L-

VORTICES IN HELIUM I1

0.17(h/m)-3(k,T/a,)2

99

(4.5.10)

and has the value 0.4hlm at TKT’ At low temperatures [ T/a,< O.3(T/as),,],the diffusivity falls off more rapidly than at higher temperatures. The fall off occurs at the upper limit of T/u, where the authors observe vortex creep into and out of the film with the characteristic time behavior observed in persistent current decay experiments (Ekholm and Hallock, 1979, 1980; Browne and Doniach, 1982). At still lower values of T/a,the vortices are strongly pinned and substantial persistent currents could be achieved. Where necessary to make sure that the rotation-induced vortex density was indeed its equilibrium value, L?m/.rrh,and that there was no persistent current present which could in principle result in vortex pair-breaking and therefore extra sound damping - the cell was cooled to the measuring temperature from some high temperature, above TKT,while rotating. In practice, this procedure made no perceptible difference. Petschek and Zippelius (1981) have carried out a calculation of the variation in the vortex diffusivity near TKTdue to the existence of bound vortex pairs. They predict that the interaction of free vortices with the small vortex pairs should lead to a small decrease in the diff usivity comparable in magnitude to the increase in the dielectric constant. It is clear from the data, which show a much more rapid variation with the opposite sign, that either this prediction is wrong or else the data reflect a rapid variation of the bareunrenormalized - diffusivity. Assuming the latter, experimental verification of the relatively small universal change predicted by Petschek and Zippelius would require a much more extensive investigation than has been carried out. The contribution of rotation-induced vorticity to third sound damping is linear in the rotation speed only for relatively low sound amplitudes. For higher sound amplitudes, free vortices arising from the breaking of otherwise bound vortex pairs by the superflow also contribute to the damping. Treating the resonance as though it were Lorentzian and taking the resonance width as a measure of the effective free vortex density, that density is found to be reasonably well represented by the relation

(4.5.11) This is interpreted in terms of a “law of mass action”, in which vortices in a bound pair can dissociate in the presence of superflow by diffusing over a free energy barrier. The effect of rotation is to decrease the barrier height for one sign of vortex and to increase it for the other so that the product of nupand ndom(where nfree= nUp+ ndoW)is fixed. At still higher

1(X)

W.1. GLABERSON AND R.J. DONNELLY

[Ch. 1, 94

sound amplitudes, rotation produces a peculiar and unexplained chaotic time dependence of the third sound resonance width. Persistent current decay, in the temperature/film thickness regime where it can be observed, is presumably associated with vortex nucleation and/or creep. The decay rate is determined as follows (Kim and Glaberson, 1984b). The cell is cooled through the superfluid transition temperature to a target temperature in some state of rotation; the state of rotation is changed quickly and the cell is kept at the target temperature for some specified delay time; the cell is then cooled to a low temperature where the effective persistent current remaining is determined. Fig. 4.5.2 is an isochronal map of the decay behavior, that is, a plot of t h e effective persistent current remaining in the cell after a delay of 30min as a function of the target temperature. Because the sound cavity is sealed, the film thickness decreases as the temperature increases. The thickness in helium atomic layers is indicated at the top of the figure. The circles correspond to vortices entering the film (the cell is accelerated from rest to 8.4rad/s at the target temperature) and the triangles correspond to vortices leaving the film (the cell is cooled to t h e target temperature while rotating and then brought to rest). At relatively low temperatures, t h e decay is logarithmic in time as is characteristic of vortex hopping from pinning site to pinning site (Browne and Doniach, 1982). For thinner films, particularly for decays associated with vortices entering the film, non-logarithmic time behavior is observed. 5.9

8.5 C

Z W I-

'

.

8

4.3

VORTICES O "ENTERING"CELL O I

TEMP

(OK)

Fig. 4.5.2. Persistent current decay behavior in films. (Kim and Glaberson, I%.)

VORTICES IN HELIUM I1

Ch. 1, 051

101

The very clear difference between the vortex inflowing and outflowing decay behavior in fig. 4.5.2 suggest that in the latter the decay rate is limited by vortex hopping whereas in the former the decay rate is dominated by a much slower nucleation rate. We note that the decay behavior is, in many respects, qualitatively similar to that observed by Ekholm and Hallock (1979, 1980). By deliberately introducing vortices into the system by rotation, it may have been possible to disentangle vortex nucleation from vortex creep. One serious problem with the suggestion that the vortex inflow decay is limited only by nucleation is the implication that t h e vortex distribution is then always reasonably uniform in this situation. It follows that at a given temperature the decay rate should only be a function of the persistent current. It is found, however, that a persistent current prepared by accelerating from some non-zero rotation speed in which the cell was cooled down, decays much more rapidly than one prepared from a cell accelerated from rest that has been allowed to decay to the same effective persistent current. More work is clearly required for a satisfactory understanding of vortex motion in films. We point out a somewhat curious effect. Vortex pinning is observed to become important as the film thickness is increased. Intuitively, one might have expected the contrary: sensitivity to surface irregularities should be diminished in thicker films. We speculate that this effect is associated with a competition between van der Waals forces, tending to keep the film thickness and hence vortex potential uniform, and surface tension which would tend to decrease the film surface area and give rise to a modulated vortex potential for a microscopically rough surface. For the very thin films in which diffusivity is observable, the van der Waals force dominates and the vortex potential is relatively uniform. Another possibility is that, as the film thickens, it ceases to properly wet the surface, and the film begins to be characterized by a distribution of small droplets. A more intriguing speculation is that, as the film thickness is increased, t h e vortices gradually freeze into a regular array [see Fisher (1980)l and therefore become more and more sensitive to the presence of isolated pinning sites. 5. Vortex dynamics - waves

5.1. ISOLATED VORTEX

LINES

In a situation in which a vortex line is deformed into a helix, the deformation propagates as a wave, as discussed by Lord Kelvin (Thornson, 1880). In order to obtain an intuitive understanding of these Kelvin

102

W.1. GLABERSON

AND R.J. DONNELLY

[Ch. 1, $5

waves and their dispersion relation, a simple analysis is in order. We consider, in particular, a helical deformation of wave vector k and amplitude d, where d G k- ’ (see fig. 5.1.1). According to Helmholtz’ theorem, an element of vortex line always moves with the velocity of the superflow at the line in the absence of dissipation. In t h e case of the deformed line considered, this velocity can be separated into a component induced by “local” line elements and a component induced by the rest of t h e line. Here, “local” means within a distance from t h e point considered which is much larger than a core radius but much smaller than the wavelength of the deformation. Ignoring the non-local contribution, the line moves with Arms-Hama velocity [eq. (I .9)]where L is reasonably taken as being of order k - I . When d << k - ’ , this velocity is perpendicular to t h e undisturbed line and to the displacement vector from the undisturbed line to the point considered. Each vortex line element therefore executes motion about the undisturbed line in a circle of radius d and with a frequency w = v/d. The radius R is given approximately by R l / d k 2 s o that

-

i

I

2n / k

I Fig. 5.1.1. A vortex line deformed into a helix. The amplitude of the deformation is d and its wavenumber is k. R is the radius of curvature of the line at some point.

Ch. 1, 451

103

VORTICES IN HELIUM I1

It is evident from the analysis that vortex waves are intrinsically polarized, always executing motion about the undisturbed line in a sense opposite to the sense of the superfluid velocity field. The correct expression, assuming a thin hollow vortex core, was given by Lord Kelvin (Thomson, 1880):

where K, is a modified Bessel function of order n. The same formula was derived by Pocklington (1895) for waves on a hollow ring with n = kR the mode number and k = 27r/A, A being the wavelength. In both cases the core is considered to be thin compared to other dimensions. The dispersion formula (5.1.2) shows that there are two branches, a fast (positive) wave w + and a slow (negative) one w - , corresponding to + and - respectively in eq. (5.1.2), see fig. 5.1.2. The formula generally used in the literature of helium vortices, is an approximation to w - for long wavelengths. On expanding the Bessel functions in eq. (5.1.2), we have t h e following expression for w - for ka 4 1: Kk2 47r

w - = - -[ln(2/ka) - 71,

Kk2 [In(l/ka) - 0.1161 , 47r

-

- --

(5.1.3) I

-

-

N

-

2

oJ+

T=1.4 K O a=l .49 A

-

0 X

7

1 U 0)

m

0

+I

3

W-

-1

0

I

I

1

2 ka

Fig. 5.1.2. The dispersion curves for helical vortex waves.

3

104

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, §S

where y is Euler’s constant 0.5772. . . . Similar formulae have been shown to apply to vortex waves in an imperfect Bose gas by Pitaevskii (1%1) and by Grant (1971). The positive branch W + does not follow the self-induced local velocity discussed above, because it owes its existence to a non-zero effective mass in the core. To see this, let us consider a rectilinear vortex line with core of effective density p rotating at angular velocity w at a distance A from the undisturbed position. By equating the centrifugal force per unit length p ~ w ’ u ’ A to the Magnus force per unit length ~ K u Aone , obtains w = ~/?ra* which is the limit of w + from eq. (5.1.2) for &a < 1 (see fig. 5.1.3). There has been considerable progress in our knowledge of classical vortices in recent years. Maxworthy et al. (1985) have made a detailed comparison of classical and quantum vortices. Classical vortices can support helical waves of the type shown in fig. 5.1.1. When such waves reflect from a boundary two (slow) waves of opposite polarization combine to form a plane standing wave called a Kelvin wave which rotates with angular velocity w given by eq. (5.1.3) in the long wave limit. There are also axisymmetric waves for which the core diameter varies as the wave propagates. It has been shown that such waves propagate at a long-wave speed approximately equal to the maximum swirl velocity within the core, almost independent of the detailed vortex structure (Maxworthy et a]., 1983). For the quantum vortex this velocity is of the order of 160 mls, assuming a core diameter of 2 A. It is not known what

J

<>

0.2

3 0

I 0

I

2

1

3

ka

Fig. 5.1.3.The dispersion curve for the slow branch compared to the long wave approximarion (dashed line). The error in using the approximation beyond ka 4 1 is evident.

Ch. 1, S5)

105

VORTICES IN HELIUM I1

significance such waves have for quantum vortices. There is at present no known experimental evidence for such axisymmetric waves. For classical vortices axisymmetric and helical wave trains can be produced with suitable excitation. Constant excitation, and certainly an impulsive one, can produce isolated groups of waves (solitary wave packets) in the helical case and isolated core expansions (solitary axisymmetric waves) (see fig. 5.1.4). The latter were predicted theoretically by Benjamin (1%2,1%7) and have been observed in viscous vortices by Pritchard (1970) and Maxworthy and Hopfinger (1984). Solitary wave packets or kinks have been predicted for a dispersion relationship of the form given by eq. (5.1.1), plus a cubic non-linearity by Hasimoto (1972) and Kida (1981). The complex wave function for these disturbances is the non-linear Schrodinger equation. Evolving solitary kink waves have been observed in classical vortices (Maxworthy and Hopfinger, 1984), but there is as yet no conclusive evidence that they exist on quantum vortices. What evidence exists is assessed by Maxworthy et al. (1985) and has to do with the interpretation of ion experiments such as we have described in section 2.8 above. Given the easy excitation of solitary kink waves in classical systems it would be surprising if such disturbances were not observed on quantized vortices. Hall (1958) was the first to demonstrate that the vortex system in rotating He I1 could sustain oscillatory modes. His pioneering investigation involved placing a stack of thin mica plates, closely spaced, parallel to each other, and suspended by a long metal torsion fiber, in a

1

0.4

I

I

I

I

0

I

1

/-

a=l .49 A

0.1 .-

n -

I

-

I

I

I

I

I

0.5

1.0

1.5

2.0

2.5

ka

Fig. 5.1.4. The group velocity for helical waves.

3.0

106

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, $5

rotating He4 cryostat. After rotating the system to the desired speed, Hall allowed time for equilibrium, and he then perturbed the system so that an oscillation of t h e disks set in. The period of oscillation was determined as a function of rotation speed. Several different disk spacings were employed in separate experimental runs, ranging from 0.02 to 0.7cm. The disks were left smooth for some runs, while for others they were coated with 50 Fm grains to increase the surface roughness. Hall obtained resonant behavior of his system when using t h e roughened disks, with 0 values on t h e order of unity. A modified version of this experiment was performed, in which only one disk was used. The vortices ran from the disk to t h e free surface of helium some distance above. T h e behavior of

the period of the free oscillations of the disk as a function of distance below the surface was studied. A similar experiment was performed by Andronikashvili and Tsakadze (1960) on the damping of a disk in rotating He 11. Nadirashvili and Tsakadze (1968) performed another such experiment in which the logarithmic damping decrement was measured under various conditions. In all of this work, the disk oscillation periods were quite low so that the corresponding wavelengths were large compared to the distance between the vortex lines. As we shall discuss later, this implies that vortex-vortex interactions were important in determining t h e dynamical behavior and t h e experiments were not a direct probe of simple Kelvin waves. Halley and Cheung (1%9) and more recently Halley and Ostermeier (1977) have suggested that an rf electric field, transverse to a vortex line charged with ions moving along the line, would couple strongly to vortex waves under suitable conditions. They argued that it is reasonable to expect strong coupling, i.e. "resonant" generation of vortex waves, when the following two conditions are satisfied: cod =

~ ( k- )kvi,,

t:,,, = d u ( k ) / d k .

. (5.1.4)

Note that these conditions are very similar to t h e conditions discussed with respect to the limiting trapped ion velocity in section 2.8. The first condition is simply that the vortex-wave frequency and sense of rotation, in the frame of reference of t h e moving ion, is the same as t h e rf frequency and sense of rotation. The second condition ensures that any energy pumped i n t o t h e vortex wave, remains in the vicinity of t h e ion. These two conditions determine a characteristic ion velocity which depends on the rf frequency. If the longitudinal dc electric field, driving the ions along the vortex line, is measured as a function of ion velocity, an

Ch. 1, 051

VORTICES IN HELIUM I1

107

anomaly should be observed at t h e characteristic velocity. The vortex waves produced near “resonant” conditions clearly have phase velocities in the same direction as the ion velocity so that t h e ion drag should be enhanced. The only propagating vortex-wave modes at reasonable frequency are those which are polarized in a sense opposite to the circulation sense of the vortex. Because the vortex-wave group velocity is larger than its phase velocity, the sign of wd is opposite to that of w ( k ) and conditions (5.1.4) can only be satisfied simultaneously for an rf electric field polarization in the same sense as the vortex circulation. The ion-velocity anomaly should therefore be observed for only one rf field polarization for a given sense of rotation of the apparatus. Ashton and Glaberson (1979) were able to observe the predicted effect. The experiment involves the use of a rotating ’He refrigerator in which a sample of 4He could be cooled to 0.3 K while rotating at 10 rad/s. This rotation speed yields a uniform distribution of vortex lines oriented along the rotation axis with a density of 2 X 104cm-’. A schematic of the experimental cell is shown in fig. 5.1.5 and except for the rf electrodes, is ROTATlON AXIS COLLECTOR

/ SIDE VIEW

H B -/

-

GATING GRIDS

I--I 1 ---

n

n

U

U

RF ELECTRODES TOP VIEW

U

Fig. 5.1.5. A schematic drawing of the experimental cell for observing vortex waves. (Ashton and Glaberson, 1979.)

108

[Ch. 1, $5

W.I.GLABERSON A N D R.J. DONNELLY

similar to those used previously for ion mobility studies. In order to properly observe the predicted anomaly, it was necessary to ensure that both the longitudinal dc electric field and t h e transverse rf electric field were reasonably uniform. This was accomplished by having t h e drift field defined by four stacks of electrodes, each o f which consisted of eight electrodes stepped in dc potential. Rf potentials were ac coupled to the electrodes and applied in a circularly polarized mode. Circular polarizat i o n not o n l y helped distinguish real from spurious effects (because of the intrinsic polarization of t h e vortex waves), but also helped achieve rf-field uniformity over the cross-section of the drift region. A plot o f ion velocity versus dc electric field, for t h e cases of no rf field and for the rf field polarized in the clockwise and counterclockwise senses, is shown in fig. 5.1.6 (in this case the i o n cell was rotated in the counterclockwise sense). Note that the ion velocity is anisotropic with respect to rf field polarization. Reversing t h e apparatus-rotation sense, and hence the vortex-circulation sense, reproduces the data with the roles ~.

!

8.0 I

7.0

G3

-

1

u

a,

>

5.0

1

t

t-

4'01 3.0

1 t I

2.0

ERF GV/CIT

,

I.o

2.0

ED^

3.0

4.0

5.0

(V/cm)

Fig. 5.1.6. A plot of ion velocity as a function o f dc electric field f o r n o rf field and for the rf field polarized in the clockwise and counterclockwise senses. (Ashton and Glaberson. 1979.)

Ch. 1, §5]

VORTICES IN HELIUM I1

109

of the two rf-field polarizations reversed. An anomalous kink and plateau in the velocity versus dc electric field curve, for counterclockwise rf-field polarization, is observed near the characteristic velocity determined by eqs. (5.1.3). A simultaneous solution of eqs. (5.1.3) yields uion= 3.0 m/s for the rf frequency used. The small discrepancy between this value and the plateau velocity observed can be explained in terms of the ac field inhomogeneity - the ions move much faster near t h e top and bottom of the drift region where the ac-field amplitude is small. At the characteristic velocity, the wavelength of the reasonantly generated vortex waves (2000 A) is two orders of magnitude larger than the ion radius and three orders of magnitude larger than t h e vortex-core radius. The kink is not perfectly sharp, of course, because of finite vortex-wave damping as well as residual field inhomogeneities. This observation confirms the most important prediction of Halley and Ostermeier and constitutes a measurement of the vortex-wave dispersion relation at rf frequencies. 5.2. COLLECTIVE EFFECTS-

INFTNITE VORTEX ARRAYS

Kelvin waves propagate along isolated vortex lines with a dispersion [eq. (5.1.3)] w ( k ) = (Kk2/4'7T)[hl(l/kU)f 0.1161 .

(5.2.1)

It is important to realize the effect of moving into a frame of reference rotating with angular velocity R, in the same sense as the velocity field of the vortex line. In this situation the vortex will appear to rotate faster, at an angular velocity given by w = R,+ (~k~/4'7~)[ln(l/ka)+ 0.1161.

(5.2.2)

We now consider the effect on the spectrum of allowing for a uniform distribution of vortices. It is obvious that this extension of the theory is necessary; a vortex line moves with the local net velocity at its core, no matter what the source (but see section 2.2!), so that it will respond to t h e fields generated by other vortices in addition to any self-induced field. Thus a complicated interaction can occur between vortices that in general should not be neglected, although in certain limits the behavior is relatively simple. This problem was solved in an approximate fashion by Rajagopal (1964). He made several simplifying assumptions: (I) There is a mean spacing between vortices in a uniform, but not necessarily ordered, array, given by a parameter b.

110

W.I. GLABERSON AND R.J.DONNELLY

[Ch. 1, $5

(2) There are no vortices within a radius b of t h e particular vortex under consideration. Beyond this the vortex density is constant and non-zero. (3) An integral over these velocity fields is sufficiently accurate so that a summation over discrete vortices is not required. He then found t h e angular velocity of this central vortex’s oscillation in a frame rotating at fl,,to be

where (5.2.4) and K,, is a modified Bessel function. The J integral results from the approximate summation of t h e velocity contributions from all of the other vortices. Two limits are of interest here and are concerned with t h e extent that the vortices influence one another; i.e.. the degree to which the modes are collective. If the mean spacing b becomes very large relative to t h e wavelength A ( = 2 7 r / k 1. then the vortex oscillations become independent and the frequency must approach the Kelvin result. Indeed, as k b - + r , .I I . giving -+

w =

R,, t (,k2/47r)[In(l/ku)+ 0.1161

(5.2.5)

which is just Kelvin‘s result in a frame rotating at speed 0,. Conversely, in the collective limit, as k h - + 0 . J - + O in such a way that w --

?R,,+(Kk2/47r)In(b/n).

(5.2.6)

The situation is as follows: w =

F(f2,))fIl) * V k 2.

(5.2.7)

where 1 ’ is a weak function of k and where F(f2,) is 2 for a dense array of vortices (relative to the wavelength) and goes smoothly to 1 in the limit of independent oscillations. In general. o f course, the value should be wmewhere in between, It i h appropriate to notice the correspondence limit o f the dispersion relation just found. For a uniform. but not necessarily ordered, array.

Ch. 1, 551

VORTICES IN HELIUM I1

111

b (l/n)”*= ( K / ~ R , ) ”Since ’ . K = h/m, if we let h -0, then b - 0 , which implies that for any k , F(R,)-2. Also, Y-0, since Y x k , so that we are left with the result = 2R,, which is just the dispersion relation for classical inertial waves with wave vector along 2. The classical result for general k is

” = 2R,l-2.

(5.2.8)

k is perpendicular to 2, these modes do not propagate. The HVBK macroscopic two-fluid equations deal with velocity fields which are averaged over distances large compared to 6. Quite naturally, these equations yield a dispersion relation for oscillation modes, eq. (5.2.6) characteristic only of the extreme collective limit. On the other hand, the macroscopic equations are well suited to a determination of the interaction between the normal fluid and the vortex system and the consequent wave damping. Rajagopal’s calculation was important in pointing o u t the need for considering collective effects in the vortex wave spectrum. However, an assumption made in his work was that t h e vortices oscillate in phase, which eliminated the important possibility that vortex waves could exist with wave vectors oriented perpendicular to the vortices themselves. Such modes arise naturally from lattice stability calculations, for stability of a vortex lattice is just t h e condition required for the existence of oscillatory behavior of t h e vortices, neglecting bending. Tkachenko (1966a, b, 1969) was the first to attempt a stability calculation of this kind for an infinite array of classical vortices; he found that triangular lattices were stable, and that the normal modes of such a lattice (plane waves) consisted of elliptical motions of the vortices about their equilibrium positions. Other studies of stability yield similar results. It should be emphasized, however, that these results pertain only for an infinite system. When a finite set of vortices is considered, it is found that t h e triangular lattice is often not the lowest free-energy configuration. The effect this might have on t h e spectrum will be discussed later. Because as we shall see, the lattice sums involved in the calculation of t h e dispersion relation are complicated, it is instructive to attempt a simple derivation of an approximate expression. Consider a triangular array o f rectilinear vortices aligned along t h e z axis. Assume t h e presence of a long wavelength plane Tkachenko standing wave mode having nodal lines parallel t o the y axis (see fig. 5.2.1). Focus attention o n vortices that are near an antinodal line. Each vortex executes elliptical motion about its equilibrium position, where t h e ellipse has semiminor and semimajor axes S,(x) and S,(x) respectively. The frequency with which the ellipse is If

1I?

W.1. GLABERSON AND R.J. DONNELLY

[Ch. 1, 95 I

I I

1 I

I

I

I

I

I I I I

tI I I

Fig. 5.2.1. A Tkachenko standing wave mode. The dots are the undisturbed triangular vortex lattice points. h0 and &O are the semimajor and semiminor axes of the elliptical paths executed by the vortices.

traversed is clearly o f order

(5.2.9) where uL,,,= and uT,,,= are the maximum longitudinal and transverse components of t h e vortex velocity. We begin with a calculation of uT.,,=. This is the velocity that t h e vortex has when the displacements of all of the vortices are purely longitudinal. Under these circumstances S,(x) = 0 and S,(x) = S,, cos(lx) where I is t h e wavevector of the wave. Corresponding to these displacements is a vortex density oscillation along t h e x axis d n ( x ) = n ( x )- ti,, - lS,,n,

sin(/xx),

(5.2.10)

where no = ~ L ? is / Kthe equilibrium density. It is only the excess vortex density, An@), that is important here since, in t h e rotating coordinate system, a uniform vortex distribution induces no velocity at a vortex position. It is a trivial matter to show that a column of vortices having a density A lines/cm, each line having a circulation K , induces a velocity (parallel to the column) u ? A K / ~ at all points in the fluid not too close to the vortices. Fortunately, t h e net vortex density change for any region between adjacent nodal lines is zero. We need therefore only be concerned with the excess vortices within a distance (7d2)I from t h e vortex o f interest. It follows almost immediately that

-

Ch. 1, $51 VT. m a

VORTICES IN HELIUM I1

-

113

(5.2.11 )

.

K ~ O ~ L O

Now consider the situation when the displacements of all of the vortices are purely transverse to the wavevector: S L ( x ) = O and 6 , ( x ) = 6Tocos(Zx). There is no vortex density oscillation and no transverse component of the vortex velocity. A column of vortices induces a velocity component perpendicular to itself which oscillates as a function of displacement along t h e column with an amplitude that falls off exponentially with distance from the column. The characteristic decay length is of order b, the inter-line spacing. It follows that we can obtain a reasonable estimate for the longitudinal component of the velocity of a vortex by considering only the effects of vortex columns immediately adjacent to the one containing the vortex of interest. A short calculation will show that a displacement E from a symmetry position along the vortex column and at a distance b from the column, the longitudinal velocity (i.e. the velocity along the wave vector or perpendicular to the column) is of order K E / b 2 . For a long wavelength, E (lb)’S,, so that

-

Note that, although all of the vortices in a particular column have the same longitudinal velocity, the average longitudinal superfluid velocity must be zero if the fluid is assumed incompressible. In a Tkachenko wave, the transverse vortex velocity is close to t h e average transverse superfluid velocity, whereas the longitudinal vortex velocity is very different from the average transverse superfluid velocity. Combining eqs. (5.2.1 1) and (5.2.12) we have (5.2.13) so that w ( l )= CTl where CT- ~ n h ” -( 0 ~ ) Furthermore, ”~. the eccentricity of the elliptical motion is given by

As can be seen, the vortices rotate about their equilibrium positions in a sense opposite to that of the superflow about each of the lines. It is now necessary to consider the fact that in laboratory systems it is difficult, if not impossible, to attain t h e conditions required for t h e application of the previous analysis - the reason being that vortices are easily bent. If the vortices are contained in a finite chamber, they will

1 I4

W.1. GLABERSON AND R.J. DONNELLY

[Ch. 1, $5

attempt to hydrodynamically clamp or “pin” to the walls, so that any motion of a vortex requires, if it is to remain firmly attached, that the vortex line bend. This in turn induces a velocity at its core that will result in the excitation of a wave whose wave vector lies along the axis of the undisturbed vortex. Thus a theory is required that combines both the Tkachenko and the Kelvin contributions to t h e dynamics. The combined effects have been studied by Williams and Fetter (1977) and Sonin (1976). The two results are essentially identical, and we will draw extensively from both papers. The starting point is simply the Biot-Savart formula, eq. (1.8). summed over all the vortices except the one under consideration. If we work in a frame rotating at speed 0, and recognize that t h e velocity induced at site i will result in a small displacement u,(z,, I ) of the vortex away from its equilibrium position ri, then

(5.2.15) where rl! _= r, - r,, RY, = rjl + ( z ,- z , ) i , and u:, = (3u,/dz. In order for the equilibrium configuration to rotate at t h e same speed as the frame, we , forces the impose the condition on t h e density of vortices n = 2 0 / ~which first term to drop out. We assume that the vortices are of length L, impose periodic boundary conditions in the z direction, and require translational invariance in the z direction and discrete translational invariance in t h e x-y plane. We can then expand the 14, in plane waves,

where IV is the total number of vortices in the system, k = 2mi/L, in = 0, * l , 22.. . . , and 1 is a reciprocal lattice vector in the first Brillouin zone. This is substituted i n t o eq. (5.2.15), t h e integrals are performed, and t h e Fourier-transformed equations become, after invoking t h e assumed triangular symmetry of the lattice,

Ch. 1, $51

VORTICES IN HELIUM I1

(auk//af)y= =

7 -6)(uk/)r-ff(Ut/)y,

2i[vy(uk/)x- vr(uk/)yl

115

(5.2.18) (5.2.19)

9

where t h e Greek letters represent the following lattice sums:

Kk2 0;= Ro+-C' K o ( k r l ) , 47r

Kk2

2' [1- exp(if

T =-

47r

*

5 )]Ko(kr,) ,

(5.2.21)

I

Kk2

6 = -2' [I 47r

(5.2.20)

I

- exp(if

- q ) ]y ; - x; K2(kr,1

Kk2 2x Y f f = - 2' [ 1 - exp(i1- rl )] K2(kr, ) 47r I

yz

c'[1 41T

Kk2

- exp(i1- 5 )]

X

f

(5.2.23)

(5.2.24)

Y 2' [ 1 - exp(il- r l ) ]-L K,(kr, ) .

(5.2.25)

I

5

vy = -

47r

(5.2.22)

K,(kr, ) ,

vx = -

Kk2

9

r:

I

11

I

Self-induced motion is accounted for by modifying the first lattice sum as follows:

R,

+

Kk2

= 0; -[In(2/ka)- y ]

47r

(5.2.26)

with y = 0.5772. . . (Euler's constant), assuming a hollow core [eq. (5.1.3)]. A long-wavelength approximation to the spectrum can be obtained as follows. We change the coordinate system so that the x and y components transform to components in the direction of I and transverse to it. The resulting equations, with somewhat different lattice sums than those above, are 8% -exp(iQ -R,) = u, =

at

[(a, + p)u, + au,] exp(iQ - R,) ,

(5.2.27)

116

[Ch. 1. PS

W.1. GLABERSON AND R.J.DONNELLY dU,

-exp(iQ dl

- R,) = uf

=

[-(L?,

-

y)u,

-

-

(141exp(iQ R,) ,

(5.2.28)

where Q is the total wave vector and R,is t h e three-dimensional position vector of the vortex line. In the approximation I b 4 1 , and assuming h = [ ~ / ( f l ~ V ~in) a) “triangular * lattice, the lattice sums reduce to a

==o. kz- I?

(5.2.29) Kl’

Kk2

In (r/a) + -, k 2 + I Z 47r 167r

p -= R, --

+ --

(5.2.30)

(5.2.31) where r is the smaller of the two lengths b and I l k . These sums neglect t h e effects o f t h e nature of the core. Solving for t h e eigenvalues of eqs. (5.2.27) and (5.2.28), we have w:=(n+p)(n-y)-(Y

2

.

(5.2.32)

Eq. (5.2.32) is greatly simplified in two limits. If I + O , w becomes XIl,+ ( ~ k ’ / 4 7 rln(r/a), ) thus recovering the Kelvin result. A second limit is that of k + O , the case of pure Tkachenko waves. Then

This result follows when 2 0 0 * ~1’/167r.i.e.. when there is a dense lattice . this we can see that pure Tkachenko and long wavelength 2 ~ / 1 From waves are non-dispersive and travel at a speed

Returning to the original equations. we find that t h e vortices in such a mode will move on elliptical paths, with t h e major axis perpendicular to 1. As 111 decreases, the eccentricity increases, so that for very long wavelengths t h e motion reduces t o pure transverse displacements of t h e line relative to 1. The ratio of maximum transverse displacement to maximum longitudinal displacement is

Ch. 1, 051

VORTICES IN HELIUM I1

117

(5.2.35) A final comment is in order regarding pure Tkachenko modes. The correspondence limit is nonexistent, as the velocity and frequency of the modes are proportional to 6.Thus Tkachenko modes have no classical analog, in contrast to the case of Kelvin modes, which reduce to inertial waves. Tkachekno waves thus depend crucially on the existence of macroscopic quantization conditions. It is of interest to write down a set of macroscopic hydrodynamic equations, in which an appropriate averaging is done over a length scale much larger than the interline spacing, from which one can obtain the normal modes of the system. The Hall-Vinen-Bekharevich-Khalatnikov equations were such an early attempt. We reproduce them here including a term associated with the bending of the vortex lines, as viewed in a coordinate system rotating with angular velocity 0.

do Pr Ps p , 2= --vp + ~ , S V T- 2p,n x us+ - ~ ( l xnr 1)'

'dt

2

p

dun

Pnx=-pV

p

+ B 'Ps- oPn 2P

-

p,SVT

- 2 p n 0 x u,

+ fi V ( ( 0 X rl') 2

(5.2.36)

(5.2.37) x { u s - u,+ Y V x d},

where as before B and B' are the mutual friction coefficients, o = V X us+ 2 0 , d is a unit vector along w, and v = ( ~ / 4 " r In(b/a) ) where b is of the order of the interline spacing, a is the core parameter, and K is the vortex circulation. For a modern account of such equations the reader is referred to an extensive article by Hills and Roberts (1977a). To these can be addded equations expressing t h e incompressibility of the superfluid and normal fluid components v.os=o,

v.u,=o.

(5.2.38)

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W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 55

These last two equations are justified since, at t h e relevant frequencies, the wavelengths of first and second sound are much larger than any other lengths of interest. In the superfluid acceleration equation, the terms 2pJ2 x u, and fp,V(lfl X rl’) are the Coriolis and centripetal forces associated with the transformation t o the rotating coordinate system. The term important for o u r consideration is t h e vortex bending term p s v ( o .V)&. It comes from assuming a contribution to the energy of the system simply proportional to the length of vortex line present. It is straightforward to linearize the equations and determine plane wave solutions of wavevector k. In the absence of mutual friction, the superfluid and normal fluid equations decouple and the normal modes correspond to ordinary damped inertial waves in the normal fluid obeying w ( k ) = ? 2 R ( k , / k ) + i(v/pn)k2,

(5.2.39)

where k’ = kS+ k t + k : and a = Ri and to mixed inertial-vortex line waves in the superfluid having the dispersion relation w ( k ) = 2 ( k Z / k ) [ ( 2 L+ ?

vkf)(2R+ ~ k ’ ) ] ’ ’ ~

(5.2.40)

For propagation along the rotation axis, the dispersion relation corresponds to the long wavelength limit in Rajagopal’s analysis. It is these modes which were presumably observed in t h e early vortex wave experiments. As is true for classical inertial waves, these modes do not propagate in a direction perpendicular to t h e axis o f rotation. The HVBK equations do not generate Tkachenko modes because, as stated, it was assumed that the energy of the system depends only o n the vortex line density and not on vortex lattice deformations. As we have seen, the vortex lines d o not, in general, move with the average superfluid velocity so that a complete description of the system must involve an additional equation of motion for the vortex lattice deformation field. Volovik and Dotsenko Jr. (1980) and Dzyaloshinskii and Volovik (1980) have developed powerful Poisson bracket techniques for dealing with t h e dynamics of defects in condensed systems and have used those techniques to derive linearized macroscopic equations for a rotating superfluid which yield Tkachenko modes. A more physically intuitive and more easily generalized approach was recently developed by Baym and Chandler (1983). We shall follow their approach closely. It is assumed that, in equilibrium, the vortices form a regular two dimensional lattice rt: = ia + Jp

(52.41)

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VORTICES IN HELIUM I1

119

where a and p are the fundamental translation vectors of the lattice. In non-equilibrium situations, the vortices will be displaced from their equilibrium positions by a two dimensional deformation vector E.. = 11

r.. - r?.. 11 11

(5.2.42)

Note that E~~is a two dimensional object so that for rotation about t h e z axis, qj has only x and y components. i., is not the velocity of a line element - i t is the projection of that velocity in the x-y plane. We now average the superfluid velocity and the lattice deformation over a region of space large compared to t h e interline spacing but small compared to any other macroscopic lengths. We are left with macroscopic fields us(r,t ) and E ( r , t ) . o = V x us, the macroscopic vorticity in the fluid, has a direction parallel to that of t h e vortex lines and has a magnitude proportional to the two dimensional vortex density in a plane perpendicular to that direction. It follows that us and E are related to each other through a simple equation of continuity. Let n,(r, t ) be the number density of vortex lines passing through a plane parallel to t h e x-y plane. Of course, this is just proportional to the z-component of w : q ( r ,t)=

(5.2.43)

Kn,(r, t ) .

The equation of continuity is

(5.2.44) which simply states that the vortex density in some region can change only if vorticity flows into or away from that region. Corresponding, but more complicated, equations can be written for the vortex density in t h e x, z and y, z planes. These equations can be combined into a single vector equation

(5.2.45) This, in turn, can be integrated to yield

-+ at

w x E(r,t ) =

-V4(r, t ) ,

(5.2.46)

where energy conservation requires that +(r, t ) = p ( r , t ) + u f ( r , t)/2 and where p is the chemical potential. Finally, this can be rewritten as

120

W.I. GLABERSON AND R.J.DONNELLY

dt

+ ( u , . V)u, + w

x (i. - us)= - v p

.

[Ch. 1, § S

(5.2.47)

In order to determine the momentum conservation equation, it is first necessary to discuss t h e (kinetic) energy associated with the elastic vortex deformation. Restricting consideration first to two-dimensional behavior, it can be shown that the energy associated with deformation of a triangular lattice is

(5.2.48) The momentum conservation equation is then

(5.2.49) where the stress tensor is

(5.2.50) and where p is t h e pressure and y;; is given by

SE,,

Y:; = Writing j

P K R o d&, 6 ( d & , / f i r , ) 8 (firk = pus, the

3+ ( v ,

*

fiEk

+ -- 34,

--

fir,

(5.2.51) fir,

momentum conservation equation becomes

me I V ) v ,= - v p - ,

(5,2.52)

P,

i it

where t h e elastic forcing term is Ue,=

PKR,[2V(V. E ) - v 2 4 8

(5.2.53)

Comparison o f this equation with eq. (5.2.47) shows that the elastic force -ue,is what drives the line displacement u s - P. We n o w include the effect of line bending o n the stress tensor. For small line deflections, t h e excess length of line associated with the bending is given by

Ch. 1, 051

VORnCES IN HELIUM I1

121

(5.2.54) where I, is the undeflected line length. It follows immediately that the energy associated with this bending is given by

(5.2.55) where Y = (pr~*/47r) In(b/a) is the energy per unit length associated with the line. This leads t o an additional term in the stress tensor

(5.2.56) and therefore t o an additional force term

)a--(

0,Y

ub = -az

a&

(5.2.57)

which is the same form as the vortex bending term in the H V B K equations. In the rotating coordinate system, assuming incompressibility of the superfluid, the linearized equations become

v

(5.2.58)

us = 0 ,

avs + 2.a x

F

(5.2.59)

= -vp ,

(31

where V, refers to t h e components of the gradient in the x-y plane only. Certain extreme cases are easily dealt with. For wave motion propagating along the rotation axis, t h e equations reduce t o the HVBK equations (without dissipation, of course) and the corresponding mixed Kelvinvortex waves are obtained. For propagation perpendicular to the rotation axis, the vortex bending term plays no role and the equations reduce to d2

- (V x at2

K

F),

+ (-v2+ 1677

a

20,) --(V.

F )=

0,

(5.2.61)

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W.I. GLABERSON AND R.J. DONNELLY

d

[Ch. 1, $5

(5.2.62)

The solutions, in the long wavelength limit, are Tkachenko waves with a dispersion relation

“$(I)

=

(KQ,/~T)I~.

(5.2.63)

For propagation at arbitrary angles with respect to t h e rotation axes, the dispersion relation determined using t h e lattice sum approach is obtained. Baym and Chandler pointed out an interesting consequence of the inequivalence of the average superfluid velocity field u,(r, t ) and the vortex lattice deformation velocity field E(r, I ) . In a frame of reference at rest with respect to the superfluid at some point ( u s= 0), the energy per unit length of a vortex line is not precisely equal to t h e quantity v. In addition there is a small term which can be written in t h e form f m * i 2 where t h e vortex effective mass m * is approximately t h e mass of superfluid excluded by the vortex core. There is a corresponding small contribution to the mass current. The effect of these terms is to allow an additional normal mode of the system in which t h e vortices rotate about their equilibrium positions in the same sense as that of the superflow about t h e vortices and opposite to that of Tkachenko modes. This mode is not likely to be directly observable because of its very high frequency: w K/U‘ 10l2s-I. This “inertial” mode is not confined to vortex arrays, but exists also in the presence of a single isolated vortex line, having the dispersion w + as discussed in section 5.1. A series of experiments was performed by Tsakadze and Tsakadze (1973, 1975) and Tsakadze (1976, 1978) in order to observe Tkachenko oscillations. An experimental cell was employed that allowed for free rotation of various shapes of buckets, including a spherical one so as to simulate a neutron star interior. The buckets were filled with superfluid and suspended magnetically o n a long rod. The vessels were driven to high rotation speeds and their period of rotation was monitored. Oscillations of the period were observed at frequencies that were not simply related to expected Tkachenko mode behavior. Following Sonin (1976), they analyzed their data in terms of an empirical viscous vortex slip coefficient, making a rather long extrapolation to the relevant experimental regime, and obtained results which were consistent with Tkachenko waves. The results were, however, not conclusive in establishing their existence o r in verifying details of t h e theory. Tkachenko waves have probably been obseived by Andereck et al. (1980, 1982). Their experimental cell is shown in fig. 5.2.2. A stack of

-

-

Ch. 1, SS]

VORTICES IN HELIUM I1

123

1

e

L

Fig. 5.2.2. The experimental cell used to observe Tkachenko waves by Andereck et al. (1980, 1982). Subassemblies shown are: (a) mounting frame, (b) detection capacitor plates, (c) noise isolation cylinder, (d) disk and spacer stack, (e) epoxy on brass sheath, (f) torsion fiber, (g) epoxy, (h) magnet, and (i) driving coils.

Macor disks was suspended on a stainless-steel torsion fiber from a relatively massive platform which was itself suspended on a fiber from the cryostat. The disks were 0.0127 cm thick, 3.05 cm in diameter and were separated from each other by spacers ranging in thickness from 0.02 cm to 0.27cm. The entire assembly was immersed in helium and mounted in a rotating 'He refrigerator. The disks were driven into torsional oscillation magnetically and the response was detected electrostatically. The response was monitored using phase-sensitive detection with the phase in quadrature with respect to the drive so that the empty disk response was suppressed. As the drive frequency was swept through a vortex resonance, the effective complex moment of inertia of the disk assembly was altered and response such as shown in fig. 5.2.3 was observed. At relatively high temperatures, the quality factor of the resonance was found to decrease with increasing temperature, presumably because of mutual friction, but was temperature independent below 1.3K, where it is believed to be limited by disk spacing inhomogeneity. In order to ensure that the vortices are pinned at the disk surfaces, the disks were coated with a layer of 10 pm-diameter glass beads. Although

124

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 05

I .72Hz

koA

AMPL.

.077Hz

I

I

Fig. 5.2.3. Vortex resonance plot. (Andereck and Glaberson, 1982.)

I

'

1

I

I

1

1

I

I

I

1

1

1

35

d= .0206cm

0

30 25

W

ca 20 a

a

u

15

3 10

5

'0

1

2

3

4

5

6

7

8

9

10

II

Q(RAD/ SEC) Fig. 5.2.4. Resonance frequencies as a function of rotation speed for various disk spacings. The solid lines are predictions based on the density of states picture discussed in t h e text. The dashed line is the prediction of Rajagopal for a disk spacing of 0.076 cm in the absence of a Tkachenko contribution. The shaded area represents the region explored in Hall's oscillating disk experiments. (Andereck et al., 1980, 1982.)

Ch. 1, $51

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this affected the large-amplitude response, no significant effect on the “zero”-amplitude resonance frequencies was observed. The authors observed hysteretic behavior with respect to drive amplitude changes after a change of rotation speed. The hysteresis disappeared after t h e disks were driven at sufficiently large amplitude. This suggests that t h e vortices were indeed strongly pinned at the surfaces, which is consistent with the observations of Yarmchuk and Glaberson (1979) and of Hegde and Glaberson (1980), but at variance with the presumptions of Hall (1958) and Tsakadze (1976, 1978). In fig. 5.2.4 we show the measured resonance frequencies for various disk spacings as a function of rotation speed. Also shown for comparison, as the dashed line, is the predicted resonance frequency for the lowest purely longitudinal (Kelvin) mode appropriate for the 0.076 cm spacing. Consider a plane-wave vortex displacement of wave vector q = k + f, where k is the component of the wave vector parallel to the rotation axis and f its perpendicular component. In the experimental situation, the value of k is fixed by the disk spacing, k = .rr/disk-spacing.As pointed out by Williams and Fetter (1977), the vortex oscillation dispersion relation, eq. (5.2.32), has the interesting property that for fixed k the frequency first decreases and then increases as 1 is increased from zero, as shown in fig. 5.2.5. This, of course, yields a peak in the vortex-wave density of states for some 1 which depends on the values of k and R. The authors assert that the vortex-wave resonances observed are associated with these 22

-

I

I

I

------.

L A T T I C E SUMS LONG WAVELENGTH RPPROXIMATION

18

V W

m

\

\

2

14

w v 10

6 0

200

100

Q

300

4 00

(cm-‘1

Fig. 5.2.5. TkachenkeKelvin mixed mode dispersion relations. (Andereck et al., 1980, 1982.)

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W.I. GLABERSON AND R.J.DONNELLY

[Ch. 1, 05

particular values of 1. The frequency interval between standing-wave modes corresponding to successive possible values of 1 in the cell is much smaller than t h e inhomogeneity-induced spread of t h e resonance frequencies for a particular I, so that many modes are necessarily excited. The observed response is then a convolution of the density of states and the line width associated with disk-spacing inhomogeneity. Minimizing w ( k , 1 ) with respect to 1 at fixed k yields values of 1 such that lb 1, so that the long-wavelength limit, eq. (5.2.32), was not strictly reached in t h e experiment and a detailed sum over the vortex-line lattice was performed to obtain the correct dispersion relation. The detailed calculation yielded results qualitatively similar to that of t h e continuum calculation (see fig. 5.2.5). Furthermore, since IR 200 B 1, where R is the disk radius, the cylindrical geometry of the experiment has negligible influence on the resonance frequencies. The predicted values of the resonance frequencies, with n o adjustable parameters, are shown as the solid lines in fig. 5.2.4. There can be n o doubt that the data is properly accounted for. Several points concerning the experiment should be mentioned. Resonances were o n l y observed when a cylindrical sheath was placed outside the stack of disks. Whether the effect of the sheath was to minimize the influence of side to side mechanical vibrations or to fundamentally change t h e resonance cavity characteristics was not clear. The theoretical resonance condition does not involve the question of the coupling of the mechanical system to the vortex oscillations. This coupling remains something of a mystery. Finally, the experiment does not appear to be sensitive to a density of states peak associated with the Brillouin zone edge of the triangular lattice. Perhaps the global circular distortion of the lattice, discussed in section 3, smears out this peak. It should also be pointed out that Sonin (1983) has recently suggested an alternative interpretation of the experimental data which does not involve Tkachenko waves at all. This interpretation, in terms of the normal modes inherent in the HVBK equations, yields reasonable agreement with the data. It is apparent that a resolution of this disagreement and a truly unambiguous demonstration of Tkachenko wave behavior will require additional experimental work. (See note added in proof, p. 142.)

-

-

5.3. COLLECTIVE EFFECTS-FINITE

VORTEX ARRAYS

Thus far we have been concerned with the normal modes in systems of infinite extent. Campbell (1981) has carried out extensive calculations of the transverse normal modes of finite rectilinear vortex arrays, extending the early work of Thomson (1883) and of Havelock (1931). The free

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energy of N vortices contained within a circular boundary of radius R,in a frame of reference rotating with angular frequency 0,is written in the form

(5.3.1) where f,,= prc2/47r and zj= xi + iyj is the complex position of vortex j . The second term on the right comes from the effect of the boundary and is left out in an unbounded fluid. The last term plays no role in the situation considered here, where the total number of vortices is fixed. Minimizing this quantity with respect to the vortex positions yields the equilibrium quasi-triangular arrays discussed earlier. The free energy plays the role of a stream function for the vortex velocity. In a situation where dissipation dominates the vortex dynamics, the vortices always move antiparallel to the gradient of the free energy. In the opposite limit, no dissipation at all, the vortices move perpendicular to the gradient with a velocity proportional to its magnitude. We shall mostly consider this latter situation. The equations of motion thus generated are a 2N X 2N matrix equation. The eigenvalues and eigenvectors of the matrix are the normal modes of the system. The vortex trajectories for various normal modes and for certain values of N in an unbounded fluid are shown in fig. 5.3.1. As expected, the vortices execute elliptical motion about their equilibrium positions in a sense opposite to that of the vortex circulation. There are, of course, 2N normal modes for a given stable (or metastable) configuration, only N of which are independent. Some of these are common to all configurations: a rotation mode having zero frequency, a breathing mode whose amplitude vanishes in the limit of zero dissipation, and a displacement mode. The rotation mode is simply a rotation of the array as a unit about the center of symmetry and obviously has no restoring force associated with it. The breathing mode - a simple expansion or contraction of the array - corresponds to a relaxation of the system back to its equilibrium configuration, without oscillation, in a time that depends on the mutual friction coefficients. Ignoring the dragging along of the normal fluid by the vortices, the vortex density relaxation time is given by T

where

- ( 2 a cos e sin ~ a ),- '

(5.3.2)

128

[Ch. 1, 85

W.I. GLABERSON AND R.J. DONNELLY

0.922Q918

16

-- ..__--0.9993647

0.9993647

19

a 9990495

1.0000000

Fig. 5.3.1. Some of the vortex normal modes for an equilibrium distribution of 19 vortices. Only half of each ellipse is shown to better illustrate the correlation of vortex motion. Below each mode is its corresponding oscillation frequency in units of ufl. 0 is taken as zero. (Campbell, 1981.)

and 8 = tan-" Y d ( P s K - Y 3 1 I

(5.3.4)

and where y and y: are the vortex drag coefficients defined by eq. (4.1.10). Perhaps the most interesting mode is the displacement modes in which the array is displaced as a unit, at some instant of time, along some direction. In the absence of a boundary, this mode is equivalent to a shift of the center of symmetry from the axis of rotation of t h e frame of

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reference, and has the frequency 0.The presence of a boundary decreases the frequency of this mode. The frequency for N not too large, is then approximately the same as that of a single vortex of circulation N at t h e center of the container:

(5.3.5) - 0- N K / ~ I T R ~ . the continuum limit, 0 - N K / ~ I T Rthis ’ , mode softens considerably w

In but, because of the vortex free region near the boundary, remains finite. This then corresponds to an rn = 1 “edge wave” discussed by Campbell and Krasnov (1981). Another important feature of finite vortex arrays in the absence of a boundary is the rigorous absence of angular momentum from any oscillation mode. In general, relatively long wavelength phenomena predicted by the continuum calculations are confirmed by the detailed finite, but large, vortex array calculations. An important exception involves situations where the perturbation wavelength is comparable to the size of t h e system. Williams and Fetter (1977) have solved their continuum equations for the normal modes of a vortex distribution in a cylindrical container. They utilize the tractable but physically unresonable boundary condition that the vortices have no radial velocity at the boundary. For the axisymmetric case, the vortex displacements are elliptical with an amplitude that varies as a Bessel function of the distance from the axis. Campbell has simulated this mode by considering large finite arrays of vortex lines, constraining the outer ring of lines to be fixed in position. He finds that the lowest normal modes of the systems have a frequency about a factor of two smaller than predicted by the continuum calculation. Campbell suggests that the global circular distortion, discussed in section 3, is responsible for this effect. The built-in dislocations render the array considerably softer, for very long wavelength perturbations, than is the case in a perfect triangular array. Furthermore, this effect does not seem to depend on the number of vortices in the system. The Tkachenko wave experiments of Andereck et af. (1980) involve transverse wavelengths much smaller than the size of the system and are therefore not affected by these considerations. Yarmchuk and Packard (1982), using their vortex photography technique, have observed oscillations in vortex arrays containing small numbers of vortices. Detailed measurements were made of the oscillation periods and damping constants for two-, three- and four-vortex arrays. None of the observed oscillation behavior could be directly associated with any of the normal modes predicted by Campbell (1981). For example, two vortices were observed to execute damped oscillatory azimuthal

130

[Ch. 1, 35

W.I. GLABERSON AND R.J. DONNELLY

motion, that is oscillation in the angle formed by a line connecting the two vortices and some reference line passing through t h e center position. There is clearly no restoring force for rectilinear vortices perturbed in this manner. It is suggested that the vortices are in fact not rectilinear, being pinned to either the bottom or side surfaces of the experimental cell. The experiment is, of course, only sensitive to positions of the vortices at the free surface. They calculated t h e response of small vortex arrays (N = 2, 3. 4) to three-dimensional perturbations, and obtained a simple expression for the frequency of azimuthal modes:

(5.3.6) where N is the number of vortices and k is t h e longitudinal wavevector. Assuming that the vortices were pinned at the bottom surface of the cell, this expression yields frequencies 2-5 times smaller than observed. A number of possible explanations of t h e discrepancies were put forward, the most likely being that the vortices were pinned at t h e sides of the cell rather than at t h e bottom. In t h e case of three-vortex arrays, it was found that t h e vortex orbits, after subtracting out t h e azimuthal oscillation and t h e instrumentally induced oscillation of the center position, were similar to those for displacement waves as predicted by Kelvin and Havelock. As mentioned earlier, this displacement mode corresponds to a displacement of the array, as a unit, off the axis of the cylindrical cell, and has a frequency very close to the rotation frequency. 5.4. A

VORTEX INSTABILITY

Glaberson et a]. (1974) and Ostermeier and Glaberson (1975~)have pointed out the existence of a simple hydrodynamic instability involving vortex lines. The instability arises in the presence of counterflow along the lines. Consider the linearized HVBK equations [eq. (5.2.36)].Taking f2 = Rf and introducing two scalar potentials, +l(r,t ) and &(r, f ) for the gradient terms, solutions to the equations are sought, having the form

-

cb, = dl0exp[i(k r + w l ) ] ,

-

us= U~ exp[i(k r

+ wt)],

-

(b2 = 4, exp[i(k r + w t ) ] , u, =

-

U,,i + u,,, exp[i(k r + w r ) ] ,

(5.4.1)

In the absence of mutual friction, the normal modes have the form given in eqs. (5.2.39) and (5.2.40) except that, in eq. (5.2.39), w is replaced by o + U,k,.

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VORTICES IN HELIUM I1

131

The state of marginal stability, when the mutual friction force is included, is determined by the condition Im(w) = 0. This determines t h e critical value of the axial normal fluid velocity for a mode having wavevector k :

u

0. c

=

X"(2w

+ Yk5)(20 + Y k ' ) ] ' R .

(5.4.2)

The critical frequency is given by eq. (5.2.40) so that the condition for instability is: a mode of wavevector k is marginally stable when the projection of the normal fluid velocity onto that wavevector is equal to the phase velocity of that mode. The simplest modes to consider are those propagating along the vortex lines. Generalizing to off axis modes, for which one should in principle include Tkachenko effects, probably does not qualitatively affect t h e calculation. The stability condition for longitudal modes is similar to a Landau condition in which the critical velocity is given by

uo,c= (wlk),,

= 2(20v)'R.

(5.4.3)

For normal fluid velocities larger than this value, there exist infinitesimal helical deformations of the vortex lines which grow exponentially in time. We now present a simplified derivation of t h e critical velocity for axial normal flow along an isolated vortex line. This derivation, although not completely rigorous, lends considerable insight into the nature of the instability. Consider a vortex line oriented along the z axis and deformed into a helix of wave number k and infinitesimal amplitude 6. A unit vector along the vortex line is given by iL(z= ) -k6[sin(kz)f

+ k8[cos(kz)]j + f

(5.4.4)

and the superfluid velocity at the line is given approximately by u,(z) = vk26([sin(kz)]i- [cos(kz)]j + k ~ i }

(5.4.5)

where Y = (~/47r)In(l/ka). The normal fluid is assumed to be in solidbody rotation at frequency R about the z axis and at the same time translating along the z axis with velocity V,: u,(z) = -RS[sin(kz)]i

+ 08[cos(kz)]j + Voi.

(5.4.6)

Writing the velocity of a vortex line element as uL, the Magnus force per unit length on the line is [eq. (4.1.9)]

132

W.I. GLABERSON AND R.J. DONNELLY

jM = psu x

(11,

- u,) = P ~ K B ,X

(uL- v,) .

[Ch. 1, 85

(5.4.7)

The drag force per unit length experienced by a vortex line is [eq. (4.1.lo)]

The motion of the linc is determined by requiring that the net force o n each line element vanishes [eq. (4.1.1l)] fD+fM

=

0.

(5.4.9)

The helical deformation of the line will either grow or decay, depending on whether the radial component of t h e line velocity is positive or negative. Solving eq. (5.4.9) for this radial component and setting it to zero yields a "critical" value for U,:

U",c= (l/k)(R + " k 2 ) .

(5.4.10)

This expression is essentially the same as that derived from the HVBK equations, for a wave vector along t h e rotation axis, except that 2 0 is replaced by 0.As pointed out before, this reflects the difference between isolated line dispersion ( w = R + vk') and extreme collectivization of the helical waves (w = 2R + vk'). In an experiment reported by Cheng et al. (1973), thermal counterflow was impressed along the axis of rotation in rotating helium. The attenuation of a transverse negative-ion beam-due to trapping o f the ions on vortex lines-was found to decrease significantly as a result of the counterflow. These results probably can be explained in terms of the vortex-array instability. In a thermal-counterflow experiment (assuming Poiseuille flow for the normal fluid), one can estimate the heat current necessary to begin disrupting the vortex array by 4,

- fpSST(2Rv)'",

(5.4.11)

where p, is the superfluid density, S is the specific entropy, and T is the temperature. The points in fig. 5.4.1 are the measured heat currents at which 20% of the recoverable ion beam was restored and the solid line is a plot of eq. (5.4.11)* where R is taken as 2.5 rad/s. There is good qualitative and fair quantitative agreement between t h e theoretical and experimental results. A more direct observation of the instability has recently been obtained

Ch. 1, $51

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1.2

1.3

I.4

1.5

133

1.6

T(K) Fig. 5.4.1. Critical heat current for the onset of the vortex instability as a function of temperature. The points are taken from smoothed data of Cheng et al. (1973),and the solid line is a plot of the theoretical critical heat current from equation (5.4.11). (Giaberson et al., 1974.)

by Swanson et al. (1983). They measured t h e attenuation of second sound propagating in a direction perpendicular to an array of vortex lines induced by rotation. In the presence of thermal counterflow along the lines an onset of excess attenuation was observed at critical counterflow velocities in very good agreement with the predicted values. 5.5. THERMALLY INDUCED VORTEX

WAVES

We remarked in section 5.1 that the dispersion relation (5.1.2) applies to vortex rings with the substitution n = kR. An interesting question is to find the population of vortex waves found on vortex rings in equilibrium with the remainder of the liquid at a temperature T. Such an investigation has been carried out by Barenghi et al. (1985) with rather surprising results. They find that near the lambda transition the free energy to create certain sizes of rings with waves seems to vanish so that spontaneous production of such rings should occur, destroying the superfluidity of

134

W.I. GLABERSON AND R.J. DONNELLY

[Ch. 1, 5.5

helium 11. T h e authors refer to this phenomenon as a "free energy catastrophe", and speculate on its cause. Let us consider a sample of helium I1 at temperature T when no vortex rings are present. Then the free energy is Fl = F, + F,,

(5.5.1)

,

where F, is a constant and F,, i s the free energy of the excitations. In t h e case in which a vortex ring of given size and orientation is present we have F, = Ft,+ F,,

+ E +- F, ,

(5 5 2 )

where the free energy of the ring F consists of the energy E given by eq. (1.5) and the contribution F-, of t h e thermally excited waves. W e assume that F,, is unaffected by the presence of t h e vortex ring. If the quantity 46,- F 2 - F l = E + F w < 0

(5.5.3)

then the system lowers its energy by creating vortices spontaneously. W e are led therefore to studying the quantity AF, = E + F, as a function of temperature and ring radius. T h e free energy of the vortex waves is calculated by quantizing the oscillations of the ring and evaluating the partition function using the Pocklington dispersion curve numerically, applying a Debye cutoff in the usual way. T h e surprising result is indeed that, for a given ring size R, there is a temperature (
-

Ch. 1 , $51

VORTICES IN HELIUM I1

135

bov spectrum in a uniform condensate - phonons for small k , free particles for large k ) is the spectrum of vortex waves. This suggests that for real liquid helium the vortex waves should be regarded as a particular form of elementary excitation bound to the vortex. The result is that the free energy cannot be computed by a method which does not take into account t h e modification of the excitation spectrum of the liquid produced by the presence of vortex lines.

5.6.THEEFFECT

OF MUTUAL FRICTION ON VORTEX WAVES

One of the calculations carried out by Barenghi et al. (1985)is of interest here: namely the effect of mutual friction on vortex waves. Consider the vortex ring as a rectilinear vortex of length 27rR lying along t h e z-axis. We denote displacements of the vortex in the x and y directions by ,$(z), ~ ( 2 )The . balance of forces on a massless vortex filament is, as discussed in section 4.1, fD +f h 4=09

(5.6.1)

where fM is the Magnus force per unit length

and fD is the friction force per unit length

where ui is the self induced velocity of the line and uL is its velocity in the laboratory system. We assume that there are no background velocities in either the normal fluid or the superfluid. A vortex has an energy per unit length given by eq. (1.1) and therefore a tension. If the vortex is bent the tension gives rise to a restoring force, acting at right angles to the vortex and in the plane in which the vortex is bent. The motion is such (see the discussion in section 5.1) that this restoring force is balanced by a Magnus force acting on the moving line. Thus the term - p s x~ ui can be replaced by a tension force To.From [eqs. (5.6.1)-(5.6.3)]we have

(5.6.4)

136

W.I. GLABERSON AND R.J. DONNELLY

Assuming a wave-like dependence like exp[i(wf- kz)] for both we have iw( y ; - P , K )-~(ioy, iw(y6 - P

[Ch. 1 , 05

4 and 7

+ Tok2)[= 0 .

+ (iwyo+ Tok2)7= 0 ,

(5.6.5)

, K ) ~

which we can solve for w : 2iy0T0k2+[-4yZ,Tik4+4TZ,k4(yi+ (~A-p,k)~)] w =

(5.6.6)

Z [ Y ; + ( 7 ; - PSk)’1

Therefore the real and imaginary parts of w are

_PsK

YOPSK

(5.6.7) PrK

where the mutual friction coefficient y, defined in section 4.1, is given by (5.6.9)

and the coefficients a and a’ are defined by eqs. (4.1.14) and (4.1.15). W e know that in the limit of long wavelengths we have for the negative branch [see eq. (5.1.3)] w =

K~~L/~IT,

(5.6.10)

where L is a slowly varying logarithmic term. From eqs. (5.6.7) and (5.6.10) we identify in this limit T , = PsK‘ L / ~ ? T ,

(5.6.1 1)

which is the energy per unit length eq. (1.1). Eqs. (5.6.7) and (5.6.8) can therefore be written as Re(@)= w-(1 - a’)

(5.6.12)

Irn(w) = w - a .

(5.6.13)

and

Ch. 11

VORTICES IN HELIUM I1

137

The effects of mutual friction are therefore a shift in the frequency of the wave [given by eq. (5.6.12)] together with a damping [given by eq. (5.6.13)]. The authors show that, although the damping is large at t h e higher temperatures, the resulting uncertainty in the energy of the vortex wave is less than its total energy unless the reduced temperature, E = (T, - T)/T,, is less than about 2 x Furthermore, the relative shift in frequency or relative shift in energy, is less than 20% unless E is less than 1.3 x Acknowledgements We are grateful to our colleagues and students who have helped us gather and understand this large body of material. We are particularly indebted to R.N. Hills and P.H. Roberts who have made useful suggestions for improvements and have allowed us to quote results in advance of publication, and to W.F. Vinen and T. Maxworthy for comments on an early draft. Our research is supported by the Low Temperature Physics Program of the National Science Foundation. References Adams, P.W., M. Cieplak and W.I. Glaberson (1985) Phys. Rev. 32,171. Agnolet. G., S.L. Teitel and J.D. Reppy (1981) Phys. Rev. Lett. 47, 1537. Ahlers, G. (1976) in: The Physics of Liquid and Solid Helium: Part I, eds. K.H. Benneman and J.G. Ketterson (Wiley, New York). Alpar, M.A. (1978) J. Low Temp. Phys. 31, 803. Alpar, M.A., P.W. Anderson, D. Pines and J. Shaham (1981) Ap. J. 249, L29. Ambegaokar, V., B.I. Halperin, D.R. Nelson and E.D. Siggia (1978) Phys. Rev. Lett. 39, 1201. Ambegaokar, V., B.I. Halperin, D.R. Nelson and E.D. Siggia (1980) Phys. Rev. B21, 1806. Amit, D. and E.P. Gross (1966) Phys. Rev. 145, 130. Andereck, C.D. and W.I. Glaberson (1982) J. Low Temp. Phys. 48, 257. Andereck, C.D., J. Chalupa and W.I. Glaberson (1980) Phys. Rev. Lett. 44, 33. Anderson, P.W. (1966) Rev. Mod.Phys. 38,298. Anderson, P.W., D. Pines, M. Ruderman and J. Shaham (1978) J. Low Temp. Phys. 30,839. Anderson, P.W., M.A. Alpar, D. Pines and J. Shaham (1982) Phil Mag. &A, 227. Andronikashvili, E.L. and Yu.G. Mamaladze (1967) in: Progress in Low Temperature Physics, Vol. V, ed. C.J. Gorter worth-Holland, Amsterdam) Ch. 111. Andronikashvili, E.L. and D.S. Tsakadze (1960) Sov. Phys. JETP 10, 227. Arms, R.J. and F.R. Hama (1%5) Phys. Fluids 8, 553. Ashton, R.A. and W.I. Glaberson (1979) Phys. Rev. Lett. 42, 1062. Awschalom, D.D. and K.W. Schwarz (1984) Phys. Rev. Lett. 52,49. Barenghi, C.F., R.J. Donnelly and W.F. Vinen (1983) J. Low Temp. Phys. 52, 181. Barenghi. C.F., R.J. Donnelly and W.F. Vinen (1985) Phys. Fluids 28, 498. Baym, G. and E. Chandler (1983) J. Low Temp. Phys. 50, 57.

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Baym, G. and D. Pines (1971) Ann. Phys. 66, 816. Baym. G.. C.J. Pethick, D. Pines and M. Ruderman (1%9) Nature 224, 872. Bekharevich. I.L. and I.M. Khalatnikov (I%]) Sov. Phys. JETP 13,643. Bendt, P.J. (1%7) Phys. Rev. 164, 262. Bendt, P.J. and R.J. Donnelly (1967) Phys. Rev. Lett. 19, 214. Benjamin, T.B. (1%2) J. Fluid Mech. 14.593. Benjamin, T.B. (1967) J. fluid Mech. 28, 65. Bishop, D.J. and J.D. Reppy (1978) Phys. Rev. Lett. 40, 1727. Bishop. D.J. and J.D. Reppy (1980) Phys. Rev. 822,5171. Bogoliubov, N. (1947) J. Phys. USSR 11, 23. Bowley, R.M., P.V.E. McClintock, F.E. Moss and P.C.E. Stamp (1980) Phys. Rev. Lett. 44, 161. Brooks. J.S. and R.J. Donnelly (1977) J. Phys. Chem. Ref. Data 6 , 51. Browne, D.A. and S. Doniach (1982) Phys. Rev. B25, 136. Cade, A.G. (1%5) Phys. Rev. Lett. 15. 238. Campbell, L.J. (1979) Phys. Rev. Lett. 43, 1336. Campbell, L.J. (1981) Phys. Rev. A U , 514. Campbell L.J. and Yu.K. Krasnov (1981) Phys. Lett. MA,75. Campbell, L.J. and Yu. K. Krasnov (1982) J. Low Temp. Phys. 49, 377. Campbell. L.J. and R.M. Zit7 (1978) Los Alamos Scientific Laboratory Report No. LA-7384MS. Campbell. L.J. and R.M. ZiR (1979) Phys. Rev. BU), 1886. Careri, G., W.D. McCormick and F. Scaramuui (1%2) Phys. Lett. 1. 61. Chandler. E. and G . Baym, 1985, preprint received. Chandrasekhar. S. (1943) Rev. Mod. Phys. 15, 1. Cheng. D.K.. M.W. Cromar and R.J. Donnelly (1973) Phys. Rev. L e t t . 31. 433. Chester, M.and L.C. Yang (1973) Phys. Rev. Lett. 31. 1377. de Bruyn Ouboter, R.. K.W. Taconis and W.M. van Alphen (1%7) in: Progress in Low Temperature Physics, Vol. V, ed. G.J. Gorter (North-Holland, Amsterdam) Ch. 11. Donnelly, R.J. (1965) Phys. Rev. Lett. 14,39. Donnelly, R.J. (1974) in: Quantum Statistical Mechanics in the Natural Sciences, Vol. 4, eds. S.L. Mntz and S.M.Widmayer (Plenum, New York) p. 359. Donnelly, R.J. and P.H. Roberts (1969) Proc. Rov. Soc. A312, 519. Donnelly. R.J. and P.H. Roberts (1977) J. Low Temp. Phys. 27, 687. Donnelly, R.J., W.I. Glaberson and P.E. Parks (1%7) Experimental Superfluidity (University of Chicago Press, Chicago). Donnelly, R.J., R.W. Walden and P.H. Roberts (1978) J. Low Temp. Phys. 31,375. Donnelly, R.J.. J.A. Donnelly and R.N. fills (1981) J. Low Temp. Phys. 44. 471. Douglass, R.L. (1964) Phys. Rev. Lett. 13,791. Dyson, F.W. (1893) Phil. Trans. Roy. Soc. A185, 43, 1041. Dzyaloshinskii, I.E. and G.E. Volovik (1980) Ann. Phys. 125, 67. Ekholm. D.T. and R.B. Hallock (1979) Phys. Rev. Lett. 42, 422. Ekholm, D.T. and R.B. Hallock (1980) Phys. Rev. Lett. B21,3902. Fetter, A.L. (1971) Phys. Rev. Lett. 27,986. Fetter. A.L. (1976) i n : The Physics of Liquid and Solid Helium, Part I, Ch. 3, eds. K.H. Benncrnan and J.B. Ketterson (Wiley, New York). Fetter. A.L. and I. Iguchi (1970) Phys. Rev. A2, 2067. Feynman, R.P. (1955) in: Progress in Low Temperature Physics, Vol. I, ed. C.J. Gorter (North-Holland, Amsterdam) Ch. 2. Finry, A.T.. A.F. Hebard and W.I. Glaberson (1983) Phys. Rev. B28,2-36,

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Note added in proof We have received a preprint of an article by E. Chandler and G . Baym (1985) which extends thcir earlier work by incorporating t h e effects of the normal fluid on the Tkachenko wavc

dirpcrsion relation. They determine a temperature dependence for the resonant frequencies in the experiments of Andereck and Glaberson (1982) in good agreement with the experimental results.

CHAPTER 2

THE HYDRODYNAMICS OF SUPERFLUID 'He BY

H.E. HALL and J.R. HOOK Schuster Laboratory, University of Manchester, Manchester M13 9PL, UK

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

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The thermodynamic basis of hydrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 2.2. The A-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Hydrodynamics in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The interaction between flow and textures in 'He-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The Ho circulation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The stability of superflow in 'He-A and the stability of t h e i parallel . . . . . . . . . ......................................... to us texture . . . . . . . . . . . . . . ....... 3.3. Helical textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Bevond helical textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The effect of container walls on textures generated by superflow . . . . . . . . . . . . . 3.6. The effect of textural singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Superflow in 'He-A and 'He-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Persistent current experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Critical velocity for the onset of dissipation in 'He-B . . . . . . . . . . . . . . . . . . . . . . . 4.3. Higher critical velocities and the depairing critical current . ........................... 4.4. Magnitude of flow dissipation in 'He-B . . . . 4.5. The possibility of dissipationless flow in 'He-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. 4.6. A simple model of orbital dissipation in 'He-A . . . . .................. 4.7. Observed dissipation in A-phase toroidal flow . . . . . 4.8. Observed dissipation in A-phase driven flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Uniformly rotating ?He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. A-phase vortex textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. B-phase vortex structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Vortex induced flow dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Measurement of thermodynamic and hydrodynamic parameters . . . . . . . . . . . . . . . . . 6.1. Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Normal fluid densitv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Second viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146

146 150

159 171 171 174 176 182 191 1%

200 202 205 216 218 219 222 227 236 23f1 245 248 248

251 253 256 257 259

1. Introduction

In this article we shall build on the foundations laid by the review of spin and orbital dynamics in Vol. VIIa by Brinkman and Cross (1W8). In particular we refer the reader to that article and to the earlier review by Leggett (1W5)for a full explanation of the structure of the order parameter in the various superfluid phases of 3He. This article will be somewhat more limited in scope than the earlier one in that we shall restrict ourselves to orbital hydrodynamics, though we shall attempt to cover experiment and theory equally. Spin dynamics is deserving of a separate review, as is the subject of ultrasonics. There has been much very interesting work in the latter field in the last few years, but it is basically concerned with non-hydrodynamic order parameter modes, and is therefore much more directly linked to microscopic theory than the topics we shall discuss in this article. Hydrodynamic equations may be derived either directly from microscopic theory or phenomenologically from thermodynamics and the conservation laws. Brinkman and Cross gave a very clear exposition based on simple physical arguments relating to the structure of t h e order parameter and appeal to microscopic theory. This heuristic approach gives a good intuitive feel for what is going on, but lacks reliability in dealing with subtleties arising from the interaction of various aspects of a problem with many degrees of freedom. We therefore feel that the time has come for a review of the thermodynamic approach. In principle, this is rigorous and should provide both a touchstone for the correctness of microscopic theories and a limitation on the amount that needs to be calculated microscopically, as elementary classical thermodynamics does for statistical mechanics. In practice, historically, each application of the thermodynamic method to a new class of system has revealed defects in what was previously thought to be a general and rigorous procedure, and superfluid 3He has proved no exception to this rule. We therefore discuss as carefully as we can in section 2 the thermodynamic foundations of hydrodynamics and their application to the superfluid phases of 3He. In this section, as elsewhere in this article, the rather uneven weighting of different topics reflects the level of activity over the last few years. The question of angular momentum in the A-phase has been the greatest source of controversy, but the other phases have enhanced our understanding of the concept of a spontaneously broken relative symmetry (Liu, 1982).

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fCh. 2, 52

As far as finding actual solutions of the hydrodynamic equations is concerned, by far the greatest effort, both analytic and numerical, has gone into the interaction of superflow and textures in the A-phase that is a unique consequence of the broken refative gauge-orbit symmetry. This we review in section 3. Proceeding from experiments that might be done to experiments that have been done, we give in section 4 a general discussion of flow experiments and their interpretation, and in section 5 we review the mainly very recent work on uniformly rotating superfluid 3He. We conclude in section 6 with a review of progress since 1978 in the determination of the thermodynamic and transport coefficients that appear in the hydrodynamic equations. This article was first completed in July 1983, but has been revised to take account of major developments to September 1984, most notably the observation of persistent currents.

2. The thermodynamic basis of hydrodynamics 2.1. GENERAL PRINCIPLES Originating with Khalatnikov’s (1965) derivation of the two-fluid equations for ‘He 11, there has over the years evolved a so-called standard procedure for deriving the equations of motion of ordered systems from thermodynamics, the conservation laws, and the structure of the broken symmetry variables only. For a recent review of the principles, oriented towards superfluid 3He, see t h e article by Liu (1982). For irreversible thermodynamics, see the book by de Groot and Mazur (1962). There are differences of detail between t h e procedures used by various authors. but broadly the argument proceeds in three stages: (i) Write down a generalized thermodynamic identity, including t h e order parameter variables arising from spontaneously broken symmetry a s additional independent thermodynamic variables. (ii) Take the time derivative of t h e thermodynamic identity, and for all variables except the entropy replace time derivatives by appropriate space derivatives obtained from t h e conservation laws. Manipulate the resulting equation into a continuity equation for entropy, containing an explicit Galilean invariant entropy production which is a sum of products of fluxes and forces. ( i i i ) Write down the most general possible linear relation between fuxes and forces, consistent with t h e symmetry of t h e problem and the Onsager reciprocal relations, that will give a positive definite entropy

Ch. 2, 921

HYDRODYNAMIC'S OF SUPERFLUID 'He

147

production. It is at this stage that arbitrary phenomenological coefficients appear that must be calculated by microscopic theory. Crucial to the above procedure are the concepts of global thermal equilibrium and the lowest order deviation from it, lacal thermal equilibrium. Although hydrodynamic equations are always non-linear, and conspicuously so for 3He, in a deeper sense local hydrodynamics is a linear theory: we consider only the lowest order deviation from global thermal equilibrium, and only linear relations between fluxes and forces. The superficial non-linearity of t h e equations of motion comes from the internal structure of the fluxes and forces. Although the standard procedure is in principle well defined and (apart from minor problems we shall mention later) unique, there has in fact been considerable disagreement about the hydrodynamics of 3He, as there was about liquid crystals (de Gennes, 1974). The problems have arisen in the first part of the procedure, the formulation of a local thermodynamic identity, and have come partly from doubts about what are the lowest order deviations from global equilibrium, as emphasized by Liu (1982), and partly from problems in making the transition from extensive variables to the corresponding densities. We shall therefore go through this part of the procedure, pointing o u t where the difficulties lie. Even in global thermal equilibrium hydrodynamic variables such as the density p and entropy density s are not necessarily uniform; consider, for example, a fluid in a gravitational field, or a uniformly rotating fluid. It the thermodynamically conjugate variables that are always uniform in global equilibrium: temperature T, chemical potential p, angular velocity 0 (=;curl u"); we use u" for the fluid velocity, equal to that of the system with which it is in equilibrium, because it is this velocity that becomes the normal velocity for a superfluid. The basic assumption of local thermal equilibrium is that, for variations that are sufficiently slow in space and time, we can take the local energy density as the same function of T, p , un, 0 as in global equilibrium. The lowest order gradients of these variables appear in the dissipation function R (or entropy production R / T ) which is a positive definite quadratic form in VT, V,l [ = i ( a , u q + d,u:)], aIO,etc. Note that V p is absent from R because the mass current is identical with the momentum density. Note also that this lowest order hydrodynamic expansion is not identical with a gradient expansion, which is what is usually done in microscopic theories. In particular note that the antisymmetnc part of the velocity gradient tensor can exist in global is equilibrium, and is therefore zero order, but that the symmetric part first order. It would be straightforward to decide what is a lowest order deviation from global equilibrium if we could work entirely in terms of the

is

v,

148

H.E. HALL AND J.R. HOOK

[Ch. 2, $2

conjugate variables (T,p, 0, . . .). But in order to find t h e specific form of R we must first write down a local thermodynamic identity, for which we need to know the local energy density as a function of its natural independent variables e ( p , s, . . .). Since p, s need not be uniform in global equilibrium t h e question arises as to whether V p , Vs should properly be included in t h e list of arguments on which e depends. There is no evidence of any such dependence in simple fluids, but it cannot be ruled out a priori. This thought, together with a specific suggestion for a V p term in 'He-A to which we shall return later, has led to some heroic efforts (Combescot, 1981; Pleiner and Brand, 1981) to include all possible dependences of the energy density on gradients in the derivation of the hydrodynamics. A glance at ref. [42] of Pleiner and Brand should suffice to convince most people that the resulting hydrodynamics is to be avoided if at all possible. Fortunately, one can be reasonably confident that most of the additional terms go beyond a hydrodynamic approximation; we must await positive evidence that particular terms produce observable effects. The other problem concerns the proper treatment of the mechanical work terms arising from motion. For the whole system in global equilibrium we can write: dEmOllOn= u-dP+f2.dL,

(1)

where u is t h e centre of mass velocity, f2 t h e angular velocity, P the total momentum and L the total angular momentum. The linear momentum is trivially related to a momentum density

I

P = gd3r, but problems arise with the angular momentum. To allow for a possible angular momentum density (of aligned Cooper pairs, for example) not included in g we write

The ambiguity arises from a point made by Martin et al. (1972); one can define a new current glo,= g + curl 1"

(4)

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149

(compare Amperian currents in magnetism) and give an equivalent description by using g,, instead of g in eqs. (2) and (3), omitting the explicit 1" term. In this sense the angular momentum density I" is redundant. Incidentally, g,, is what is normally calculated by microscopic theory. However, the motivation for retaining an explicit 1' in eq. (3) is that it gives a contribution to L varying as R2 in a cylindrical system of radius R, whereas t h e r x g term gives a contribution varying as R". In other words, 1" integrates to an Lo which is a proper extensive variable. On the other hand, the equivalent current ;curl lo makes no contribution to the extensive variable P,and does not look like the current density associated with an extensive variable, since for uniform 1' it is entirely superficial. The picture represented by eqs. (2) and (3) thus lends itself to a clear classification of variables into extensive and intensive, and avoids the occurrence of other types of dependence on the size of the system. We now define

and readily obtain

I

-

dEmorion = d3r (u" dg +

dlO);

(6)

the integrand is the motional contribution to t h e energy density. Note that although 0" and J2 are globally reIated by 12 = ;curl on,locally they are independent variables. Note also that only 1" is conjugate to J2; the rest of the angular momentum is accounted for by g. We are now in a position to determine the local thermodynamic identity and t h e pressure. The thermodynamic identity for the whole system may be written as (excluding magnetic terms for simplicity)

d E = T d S - p d V + p d M + u . d P + J2 - d L + X d y ,

(7)

where we have used y to stand for one or more order parameter variables (intensive in all cases we know) and X for its extensive conjugate. We now consider a change in which the energy density changes by d e and the volume changes locally by d V, so that d E = l v d 3 rd e + e d V and similarly for the other extensive variables. Eq. (7) then becomes

1

so

H.E. HALL AND J.R. HOOK

[Ch. 2, $2

d3r (de - T d s - p d p - v " - d g - R - d l O - x dy)

+ (P

-

-

TS+ p - pp - 0" g - f2 lo)d V = 0 .

(9)

The second term is zero by the Euler relation for the energy in volume dV, so that the pressure is

the first term in eq. (9) must then be zero for arbitrary volume V, yielding the local thermodynamic identity

Note that only independent variables that are extensive in eq. (7) make a contribution to the pressure in eq. (10). Had we used the current density g,, and omitted 1' we would have obtained a coordinate dependent contribution to the pressure

We are reluctant to believe either that t h e pressure depends on the viewpoint adopted, or that this rather singular pressure distribution is correct, and would conjecture that eq. (12) is balanced by a redistribution of t h e energy density e in the g,,, picture; but we have not attempted to verify this conjecture. The viewpoint embodied in eqs. (2) and (3) seems less prone to subtle error, and we adopt it henceforth. 2.2. THEA-PHASE

In this section we consider the A-phase as specified by the orbital part of the order parameter, rir + iri, with f = x ri. The effect of the spin degrees of freedom in the dipole locked regime d I( f is merely to modify certain constants in the bending energy; more general situations will be considered in section 3. In the lowest order deviation from global equilibrium we expect t h e Free energy density to contain bending energy term< quadratic in the spatial gradients of the order parameter. In the local normal fluid rest frame the most general form is clearly

where

a,, is t h e tensor introduced by

Mermin and H o (1976) to specify

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HYDRODYNAMICS OF SUPERFLUID 'He

151

spatial gradients of the orientation of the orbital triad and rjik, is a uniaxial tensor with respect to the orbital axis l", even in l" and symmetric in the suffix pairs ( i j ) and ( k l ) . By use of the explicit form

eq. (13) may be written as

with us= ( u s - v " ) , which is in the notation of Hu and Saslow (1977) except that we have introduced the notation f = (h02rn), so that all coefficients have the dimensions of density, and used the suffixes s, t and b to remind ourselves of the association with splay, twist and bend. In terms of the variables u s and l" the thermodynamic identity (11) becomes

since we have not yet introduced an angular momentum density lo into the picture; A' = g,, - pun by Galilean invariance. Following Hu and Saslow we obtain A' = ( a f : / W ) and hence

The foregoing straightforward picture was upset by the discovery from microscopic theory (Mermin and Muzikar, 1980; Nagai, 1980; Combescot and Dombre, 1980; Combescot, 1980) of an additional term in the current

where according to Combescot (1980) and Nagai (1981)

if Fermi liquid corrections are included. Similar spin currents in a field gradient have been found by Muzikar (1980). Mermin and Muzikar argued persuasively that eq. (18) represents part of the contribution from

152

H.E. HALL AND J.R. HOOK

[Ch. 2, 82

an intrinsic angular momentum density; this we now consider. Taking a ( T ) as a function of temperature and pressure only we write eq. (18) as -:fx Vp, so that the total current now becomes g,,, = psus+ pnun- p , i [ i - (us- un)l + $.xrI(pLi)

+ ( C - fp,) curl i- c,i[i, curl 4.

(20)

To try to extract an angular momentum density P from eq. (20) we follow Volovik and Mineev (1981) and evaluate t h e change in angular momentum SL in a volume SV when everything in eq. (20), including the coefficients. is allowed to vary within SV but not on the boundary of SV o r outside it. We have

and find

+ terms in ( p s ,p,,, p o ) involving r explicitly + terms in [ ( C- Sp,), c,]involving r explicitly = &p, + (2C- C o ) d +other terms .

(22a) (22b)

As pointed out by Volovik and Mineev, eq. (22) means that the change in local angular momentum density is not a total differential. We may identify the first term in eq. (22a) as Sf', but this still leaves the change in local angular momentum density represented by the second term. Unfortunately. the last two terms in t h e current (20) give rise both to this local term and to non-local terms involving t h e moment arm r ; we therefore really have no option but t o leave these terms in the current, despite their contribution to the local angular momentum density. Accordingly we set 1" = p,.f and redefine the current as

we note for future reference that ~ ( x rg) = ( 2 -~pL - c,)sP

+ non-local

terms .

(24)

We now consider t h e relationship of the currents (23) or (20) implied by eq.

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153

(18) to the bending energy (15). To obtain eq. (20) phenomenologically requires the ad hoc addition of

to eq. (15). This is disquieting, because eq. (25) is not a term included in eq. (13). It was the apparent necessity of eq. (25) that motivated the inclusion of terms in gradients of the conserved variables, as well as those in gradients of the order parameter variables, in the free energy density of superfluid 3He, as we have mentioned in section 2.1. However, by use of the Mermin-Ho relation for curl v s we have the identity

The last term in eq. (26) is a surface term that integrates to zero over the sample and corresponds to a redistribution of the free energy density over space associated with a differently defined current density; we may ignore it. The term -pL$.d2 has just the form of the energy of an angular momentum density pL#in equilibrium with a rotating vessel, and exists independently of any gradients, even in global equilibrium. The remaining terms give a free energy of t h e canonical form (15) derivable from eq. (13), but with modified values of t h e constants C and K,. So modified, this free energy yields by differentiation the current (23). We thus see an additional advantage to the explicit separation of an angular momentum density lo, beyond the arguments given in section 2.1: it enables us to retain a bending energy involving gradients of the order parameter variables only. It is now straightforward to write down the thermodynamic identity and derive t h e hydrodynamics following Miyake et al. (1981). We use the conservation laws for mass, energy and momentum, and introduce as yet undetermined quantities j4and X to represent the rates of change of phase and of -i^ respectively. To obtain explicit forms for the conjugate variables from the bending energy, we use the Galilean transformation properties of the energy to obtain

-

d(e - Ts - g v" - 1"

- a)= d[fo - i p ( ~ " ) ~ ] = -S

d T + /L dp - g * dun+ A" do'

- 1". d R

+ 4, d t + +,, d(a,{),

(27)

154

[Ch. 2, 92

H.E. HALL AND J.R. HOOK

Note that the bracket on the left side of eq. (27) is t h e free energy that is a minimum in equilibrium for prescribed U ” and R. From eq. (27) we see that 4, contains a term - p L c f , 4f, is independent of R, and the molecular field P,= dt - d,&, - [ I x (A’-V)Q, includes a term -2C0,. Consequently, by combining the definition of pressure (10) with the thermodynamic identity we find that in global equilibrium

+ textural

terms not explicitly dependent on

a.

(28)

Similarly, by expressing d p as a function of the independent variables in eq. (27) and making use of Maxwell relations we find that also in global equilibrium u: dp = p(dp,/dp), d(&

0)

+ textural terms not explicitly dependent on R.

(29)

Eqs. (28) and (29) represent the observable consequences of the angular momentum density pLi. The pressure differences are quite small, -lo-’ Pa for 1 rad/s, so that a static experiment requires some improvement in present techniques; an ac experiment, using alternating f 2 to excite a fourth sound resonance in a suitable cell, and thus obtain an amplification Q, may be more feasible. Note that it is important that the geometry should be such that the texture is clamped to prevent it being distorted by (for example, a thin slab). Nagai (1980, 1981) and Combescot (1980), as well as Miyake et al. (1981), obtain the result (29), except that Nagai and Combescot do not distinguish (dpL/8p)Rfrom pJp ; this can probably be traced to an implicit assumption that the Yosida function Y ( T ) is independent of densitywhich i t is not, because of the pressure dependence of T,. Combescot (1980) and Nagai (1981) also obtain the pressure difference, eq. (28), but Combescot (1981) does not. Nagai (1981) and Combescot (1981) use different definitions of the pressure, but in fact get the same result for the contribution to the stress tensor proportional to a,, ; and this agrees with Combescot’s definition of pressure, and thus does not give eq. (28). These problems with the pressure can perhaps be traced to the difficulties arising from the use of g,,, that we discussed in section 2.1. In fact, the relation between eqs. (28) and (29) is rather general; compare the dependence of pressure and density on ( u s - u”)’ in 4He I1 [Khalatnikov (1965) p. 60).

Ch. 2, $21

HYDRODYNAMICS OF SUPERFLUID 'He

155

We are in broad agreement with Miyake et al. (1981) about the entropy production and hydrodynamic equations, and therefore refer the reader to them for details. Before discussing the results and their experimental implications there are a few more general comments we should make: (i) We do not see the necessity of Miyake et al.'s distinction between d2 and ;curlu". Globally icurlu" certainly relaxes towards the angular velocity a of the container, but locally un and ;curl un are all there is to define the velocity and angular velocity of the thermal equilibrium frame. (ii) Strictly, P should also include an induced angular momentum I * proportional to d2 (Volovik, 1979) which is actually necessary for the stability of thermal equilibrium (Liu, 1982). Though important in principle, there is no evidence that this term is large enough to produce any observable effect, and we therefore ignore it for simplicity. (iii) Miyake et al. are at some pains to choose reactive coefficients so that their results agree with a modification of Hu and Saslow's work by one of us (Hall, unpublished preprint). Since that much-quoted preprint was flawed by its failure to use the thermodynamics discussed in section 2.1 (hence, no doubt, its rejection, after a struggle, by the editors of Physical Review Letters), it seems a mistaken effort to try to agree with it in detail; indeed, the conclusions so reached are not plausible. The dissipation function now follows, with the use of one additional continuity equation for the angular momentum density 1' 81; -+ at

&.. a . Ilk kI

+ airji= 0 ,

in which T~~is a tensor to be determined. The result, in the notation of Hu and Saslow (1W7) with #= (h02rn) as before is R = - ( q i / T )diT -

-p

-

a)div A'

- u" os-

v,

- (a-. Jl - ps.. Jl - v;g; - A;v: - fpljaj4) - (Tji -

+ [x; - (i)"

- Ejk1fpkiij)aiaj v){ - & j j k $ O k ] p;.

- 19V;

(31)

Two features of eq. (31) are noteworthy: the appearance of d2 in the second term and E x d2 in the last; and the natural separation of the symmetric velocity gradient tensor V,. Both these effects are the consequence of including the rotational term in the thermodynamics, as is the appearance of the penultimate term in aiOj; the latter, however, seems unlikely to have important consequences. Strictly eq. (31) is not unique,

IS6

H.E. HALL AND J.R.HOOK

[Ch. 2, 82

because of the possibility of adding suitable derivatives of an arbitrary function to the various brackets [compare the discussion of 4He I1 by Putterman (1974)l; however, eq. (31) becomes unique if we allow the second (erm to be fixed by a microscopically derived Josephson equation. The L - d2 in t h e second term is the gauge wheel effect introduced by Liu and Cross (1979); it and the 4 x 0 in the last term are both manifestations of the idea of H o and Mermin (1980) that the order parameter rotates with the container in global equilibrium. It is very satisfactory that these terms should appear naturally in the structure of the fluxes rather than have to be added via reactive coefficients, as in the treatment of Hu and Saslow. Despite appearances, the d2 term in the Josephson equation cannot produce superfluid acceleration if d2 is uniform; textural effects act via t h e Mermin-Ho relation and the I)"- u s term to prevent this, as is carefully discussed by Ho and Mermin (1980). For this reason the proposal of I h and Cross (1979) to detect the gauge wheel effect involved a fourth sound resonator consisting of two vessels oscillating torsionally in antiphase and connected by a superleak, in order to The pressure difference, eq. (28), due to intrinsic secure a non-uniform 0. angular momentum is of t h e same form as the gauge wheel term (they cancel each other at T = 0), but can produce effects from textural gradients in uniform rotation, a possibility that was overlooked by Ho and Mermin because they did not consider explicit dependence of ,u on 4.0. This makes an experiment to detect intrinsic angular momentum via eq. (28) simpler to realize than a gauge wheel experiment, though similar in concept, because the fourth sound resonator can be a rigid body in this case. It has been customary to take the second factor of each term in eq. (31) as a thermodynamic force and the preceding bracket as the conjugate flux. However, we notice that the bracket in the last term of eq. (31) is closely related to a quantity of clear physical significance, the angular velocity of rotation of the i vector relative to the normal fluid background, defined by 1

ai

w ' = I x (-+ at

(un.V)i+

i x a).

We shall see that if we take u'rather than ly as a thermodynamic force we shall obtain the orbital equation of motion in a form that is directly comparable with microscopic theory. We therefore rewrite the dissipation function as

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HYDRODYNAMICS OF SUPERFLUID 'He

157

where we have used a tilde to give an obvious shorthand notation for the various brackets in eq. (31). By making use of the conservation of angular momentum and the Onsager relations we then find the allowed reactive fluxes:

where we have used the superscript 0s to denote a traceless symmetric tensor, and 8; = - fi{. Note that the antisymmetric stress tensor (34f) is totally determined by the conservation of angular momentum. Interestingly, we can get yet more mileage out of angular momentum conservation by taking the vector product of r with the statement of linear momentum conservation to yield

which with eq. (34f) gives

a

ai

at

at

- ( r X g ) = ( A -pL)-+other

terms.

Comparison of eqs. (36) with (24) now enables us to identify

so that the reactive dynamics of i is determined by constants in the bending energy, as one might expect.

158

H.E. HALL AND J.R. HOOK

[Ch. 2, 62

We shall not bore the reader, and ourselves, by writing down all the dissipative fluxes, but it is worthwhile to display the full orbital equation of motion, t h e dissipative version of eq. (34d)with eq. (37). It is

This is identical with the equation discussed by Brinkman and Cross (1978) except for the reactive term in [for which see Nagai (1980)l and the inertial term in (2C- Co); Cross's (1977) Lo is (h/2m)(2C- Co). Volovik and Mineev (1981) argue that this is of order at T = 0, but we can see from Cross (1975) that it is expected to be appreciable at intermediate temperatures because of Fermi liquid corrections. Taking orbital viscosities from Wheatley (1978) we find that (h/2m)(2C - Co)/pLI should reach a value of order 0.1 near the A-B transition at high pressures. Eastop et al. (1984) have proposed a specific experimental arrangement to detect this small inertial effect in the presence of the large orbital viscosity. There are thus three different angular momentum experiments that should be done in 'He-A.: (i) Measurement of the pressure difference between f parallel and antiparallel to d2, as discussed under eq. (28); this should give a result characteristic of an angular momentum density pL$ (ii) Measurement of the inertia associated with a local reorientation of i;this should give the inertia characteristic of angular momentum density (2C - C 0 ) i (iii) A gyroscopic measurement of the ground state angular momentum of a thin lenticular slab; integration of ( r X gta)throughout the volume, or integration of ( r x g ) and addition of the angular momentum density p& shows that this should give a total angular momentum density 2Ct to an accuracy of t h e order of the angle 8 at the side of the lens. The first two of these predictions may still be regarded as controversial; the third, we believe, is not. It is interesting t o note that in the limit of a Bose condensed gas of diatomic molecules, considered by Mermin and Muzikar (1980) and by Volovik and Mineev (1981), 2 C = pL and C, = 0, so that all three experiments would give the same answer. The most striking indication that 'He-A contains Cooper pairs rather than diatomic molecules is therefore the absence of reactive orbit waves. With the wisdom of hindsight one could say that the inertia associated with a local change in the direction of i is obvious from eq. (22b). Despite our success in producing a consistent macroscopic hydro-

yk

Ch. 2, 821

HYDRODYNAMICS OF SUPERFLUID 3He

159

dynamics, it is still not certain that complete agreement with microscopic theory has been achieved. For example, Nagai (1981) has several other V p terms in the bending energy for which he has microscopically calculated coefficients. However, these coefficients are such that the contribution of the extra terms t o the energy is comparable with ordinary elastic compressional energy only when the length scale is as short as @/mu,), which is of the order of atomic dimensions; we therefore think that these terms should be regarded as beyond a hydrodynamic approximation. 2.3. HYDRODYNAMICS IN A MAGNETIC FIELD

In a strictly zero magnetic field the hydrodynamics of the B-phase is formally identical with that of 'He 11. There are large quantitative differences, which result in second sound being replaced by two diffusive modes (Hall, 1981), and these are important in the interpretation of flow experiments, as we shall see in section 4. But the characteristic signature of the B-phase, spontaneously broken spin-orbit symmetry, reveals itself only when we compare the responses to rotations and to magnetic fields (Liu and Cross, 1978). It is therefore convenient to preface our discussion of t h e B-phase, and of t h e A,-phase which exists only in the presence of a magnetic field, with some general remarks about hydrodynamics in a magnetic field. This will give us a unified starting point from which to discuss the somewhat discordant theoretical work on the hydrodynamics of these phases. 2.3.1. Quasi-hydrodynamic variables: magnetic work In a strictly hydrodynamic theory the degrees of freedom of a many body system are divided into microscopic and hydrodynamic variables. The hydrodynamic variables are defined as those which relax to thermal equilibrium with an arbitrarily long time scale in the limit as the wavenumber of the disturbance tends to zero. The microscopic variables, on the other hand, have a finite relaxation time in the limit of zero wavenumber, and this time is normally of the order of magnitude of a microscopic collision interval. An awkward intermediate case arises when a degree of freedom has a finite relaxation time in the limit of zero wavenumber, but this relaxation time is nevertheless quite long on a normal laboratory time scale; a typical example of such behavior is t h e NMR relaxation times TI and T,. It is clearly desirable that the hydrodynamics of superfluid 'He should if possible be formulated in such a way as to include the Leggett equations of spin dynamics; indeed we should like to include all phenomena on a

160

[Ch. 2, 82

H.E. HALL AND J.R. HOOK

time scale long compared with hlA. To do this we must include the magnetization density M as a quasi-hydrodynamic variable, with a phenomenological relaxation time introduced into the appropriate conservation law. The concept of a quasi-hydrodynamic variable means that we assume that all magnetic degrees of freedom other than the magnitude of the total magnetization density relax on a microscopic time scale. With this assumption we can write the magnetic contribution to the free energy density as

where H is the externally applied field and M = XH in complete thermal equilibrium. Since M is in general not in equilibrium with H, M and H are independent variables in eq. (39) and we have dfmag=( M x-H)*dM-M.dH

where we have introduced the spin density S and the so-called spin angular velocity o = (i?f/aS),,. w is t h e Larmor precession rate in the extra field that would be required to produce the actual magnetization; it is clearly zero in complete equilibrium. Commonly in derivations of hydrodynamics the second term in eq. (40) is omitted, corresponding to the assumption, not usually stated explicitly, that the external field is kept constant. The difference between the work terms, eq. (40), for spin angular momentum and those discussed in section 2.1 for orbital angular momentum arises from the fact that S is a quasi-hydrodynamic variable. It is the second term in eq. (40) that is analogous to - l o d R in eq. (27); it should therefore be included in the thermodynamics to give a complete account of spin angular momentum. If the intrinsic angular momentum is entirely due to spin y d H in eq. (40) is replaced by d(d2 + $7) when dependence on R is included; this gives rise to normal gyromagnetic effects.

-

2.32. Order parameter symmetries and Josephson equations In a magnetic field symmetry of the order parameter is reduced by the fact that different components of the energy gap become unequal. The symmetries of the various possible superfluid phases and the relationship

Ch. 2, $21

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161

between them are most clearly seen if we write the order parameter in such a way as to bring out the relationship to eigenstates S, = 0, 2 1 and L, = 0, *1 of the spin and orbital angular momentum of the pairs. To do this we make use of the orthogonal triad of unit vectors in orbit space rii, ri and f = m X ri that we have already used to discuss the A-phase in section 2.2, and introduce a corresponding triad d, B and = d x B in spin space; this latter triad corresponds to the notation of Saslow and Hu (1981). The existence of a definite relationship between these two vector triads is the spontaneously broken spin-orbit symmetry that is fundamental to the superfluid states of 'He, and it is the nuclear dipolar coupling that selects a particular relative rotation R as being energetically favorable.* In the B-phase relative spin-orbit symmetry is the only spontaneously broken symmetry in addition to the broken gauge symmetry characteristic of all superfluids. The order parameter is now specified by i ( k ) = A&k) where the unit vector k^ [Leggetts' (1975) n J specifies the location of the pair on the Fermi surface. For the A and A, phases we have

which for A , = A reduces to the result for the A-phase in zero field

The important broken continuous symmetry for these phases is relative gauge-orbit symmetry: a rotation of rii and ri about i by 6; is equivalent to an overall phase change 4 = -6;. In the A,-phase only one spin is paired (A = 0, say) and a similar effect is produced by a rotation of d and 8 about f;the broken continuous symmetry is relative gauge-spin-orbit symmetry. We also note the discrete symmetries of the A-phases. For any A, and A, there is a single special direction f in orbit space, so they are uniaxially nematic. The A-phase in zero field and the A,-phase have unique special directions in spin space (d and respectively), and are therefore uniaxially antiferromagnetic. The A phase in a high magnetic field (A, # A, # 0) has both these special directions and thus becomes biaxially antiferromagnetic. The notation we have used for these discrete symmetries is the conventional crystallographic one, which relates most * Note that we refer both these triads, and the spin and orbital angular momenta, to a common coordinate system; R is the rotation that takes the orbital vector triad into the spin vector triad. It is more usual in the literature to use R to rotate the coordinatesystem and to use different coordinate systems for spin and orbital angular momenta.

162

[Ch. 2, 52

H.E. HALL AND J.R. HOOK

directly to t h e physical properties. Liu (1981, 1982) has used the word biaxial rather differently, to mean that all three rotational symmetries are broken, whether or not different directions are crystallographically equivalent. The various A-phase symmetries are summarized in table 1. For the B, planar and polar phases the order parameter is, apart from an overall phase factor,

which for A , = A, zero field

=A,

reduces to the familiar result for the B phase in

Note that the planar phase is obtained from eq. (43) with A = A , A, = 0, and the polar phase with A , = A, = 0. It is interesting to note that the planar phase differs from the A-phase only in that L, is of the opposite sign to S,, rather than being the same for both spin states. In weak coupling BCS theory with the dipolar coupling neglected the A and planar phases are degenerate; up and down spin states are uncoupled and the relative orientation of t h e orbital angular momenta of the two spin states is undetermined. This degeneracy is removed by spin fluctuation feedback (Leggett, 1975). The broken continuous symmetry for t h e B phase is relative rotation of the spin and orbit axes about any direction; for example, rotations about a a n d about 13r are equivalent. It is isotropic in small fields but becomes uniaxial in both spin and orbit space when A,, A, and A, are unequal. These symmetries are also summarized in table 1. Table 1 Order parameter symmetries

Phase

Spin symmetry (antifenomagnetic)

Orbit symmetry (nematic)

A A in field

uniaxial biaxial

d

AI

uniaxial

1

i i uniaxial i

B B in field

isotropic uniaxial f

isotropic uniaxial i

uniaxial uniaxial

)

Equivalent rot at ions

6.07 4,o;. 0;

Ch. 2, 021

163

HYDRODYNAMICS OF SUPERFLUID 3He

With the order parameter separated out into a coherent sum of terms in eqs. (41) and (43) we may easily identify a phase variable associated with each spin state and note the effect of rotations in spin and orbit space on each phase variable. We can then use the gauge wheel argument of Liu and Cross (1979) to write down Josephson equations for each spin state; thus eq. (41) yields for the A phases *

.a+f.o= - - 2m p, h

*

2m h

ddt -+

1

dd, -+ dr

1 . a-j.0 = -- p ,

dr

in which dldt is the convective time derivative slat+ (u, venient to define mean and relative phases by

(45)

*V).It

is con-

which may be associated with mass and spin superfluid velocities defined by us= (hI2rn)Vd = (h/2rn)miVni,

(48)

uq = (h/2m)V+,, = (h/2rn)daVea. The corresponding Josephson equations are dd --$I dt

- .a

d j., -d + dt

2m h

= --p

(49)

=0.

Eq. (49) is the Josephson equation for the A-phase in zero field (Liu, 1976; Liu and Cross, 1V9) and eq. (50) is t h e f component of the Leggett equation for Just as the Mermin-Ho relation expresses curlu, in terms of the f texture, so will curl uv be related to the f texture. In the A, phase eq. (45) [or perhaps eq. (a)]is the only relevant Josephson equation and ut = (us+ uq) is t h e only relevant superfluid velocity. From eq. (43) we obtain the corresponding Josephson equations for the B phase

4k).

I64

H.E. HALL AND J.R. HOOK

[Ch. 2, 52

(51)

(52)

d4,

1

p i 1

dr

- R -f m

0 =

2m h

--p

(53)

We notice that if 4 and 45p are defined by eq. (47) we have the additional relation 4,, = 4, so that there are o n l y two independent phase variables and we have d& _ dr

2/72

--p’

(54)

the only difference from the A-phase equations is that the i . 0 term appears in the equation for 4sprather than 4. We should conclude this section with two cautionary remarks. First, we have so far neglected spin-lattice relaxation; when it is included there should be relaxation terms in eqs. (50) and (55). Second, we have said nothing about t h e explicit form of t h e chemical potential p. We have already seen in section 2.2 that p depends explicitly on i-fl in the A-phase, so that the experimental implications of eq. (49) are not what they at first seemed: the gauge wheel effect is modified by intrinsic angular momentum. We should therefore be alert for a possible explicit dependence of p on w ) as well as on a), in t h e B-phase as well as in the A-phase.

(p-

( / a

2.3.3. The A , phase and the A, transition Several authors have tried to find the behaviour at t h e A, transition and in the A,-phase by taking an appropriate limit of t h e A-phase hydrodynamics in a large field. There is now general agreement about the wavelike normal modes, but questions of texture, particularly the behaviour of the f vector, are more controversial. We therefore start by considering the straightforward situation of a uniform texture with f parallel to t h e applied field and i perpendicular to f, and use linearised equations. This is t h e 3-fluid model of Liu (1980) and Gurgenishvili and

Ch. 2, 821

HYDRODYNAMICS OF SUPERFLUID 3He

165

Kharadze (1980) and relates directly to our discussion in section 2.3.2. Intuitively we expect the mass and spin supercurrents in the normal fluid rest frame to be given by

in which pt and p, should be tensors with principal values parallel and perpendicular to Eqs. (56) and (57) may be expressed in terms of us and 0 , by

with p, = ps = (pt + p, ) and py = ( p , - p, ); if Fermi liquid corrections are included (Leggett, 1975) we no longer have ps = p, except near T,, so t h e more general model of Liu (1980) is required. To complete the dynamics we need to combine the Josephson equations, (49) and (50), with the conservation laws of mass, momentum, entropy and spin density; the last of these is X

= 7q

at

Y

Ap

9

(60)

where S is the longitudinal spin density and R,, the longitudinal resonance frequency; the right hand side of eq. (60) is the dipolar torque. With these equations it is found that in the A-phase not too close to the A, transition the normal modes are first sound, a longitudinal spin wave with dispersion relation 0 2 ( q )=

n;f + c : q 2 ,

where

and second sound with velocity

166

[Ch. 2, $2

H.E. HALL AND J.R. HOOK

As the A? transition is approached from below Rl,+O and the modes (61) and (63) mix; at T., and in the A,-phase we are left with a single mode, a combined spin-temperature wave with velocity given by (Liu, 1979)

Qualitatively, these predictions are easy to understand. In t h e A,-phase only one spin state is condensed so that magnetization and superfluid density are essentially identical. The wave mode associated with this therefore has both magnetic and thermal restoring forces; numerically, the former are by far the larger. As soon as t h e second spin state becomes superfluid, magnetization and superfluid density are no longer constrained in this way, and we have two possible modes. Of course, as we have mentioned already, when dissipation is included the low frequency mode n o longer propagates, and second sound is replaced by two diffusive modes. The behavior predicted above has been nicely verified in the experiment of Corruccini and Osheroff (1980), illustrated in fig. 1. They used oscillating superleak (Nuclepore) transducers to generate and detect oscillations of superfluid density and half way along the propagation path I

-..

---

--1

Fig. 1 . Velocity of the combined spin-temperature wave in the A, phase as a function of temperature. The inset shows the experimental cell. Gold-plated 5 km Nuclepore membranes, not shown, are stretched across both end plates to generate and detect the wave. The NMR coil monitors the magnetization change associated with the wave. The propagation length is 1 cm. (Corruccini and Osheroff, 1980.)

Ch. 2, $2)

HYDRODYNAMICS OF SUPERFLUID ’He

167

they had a continuous wave NMR coil to detect changes in magnetization. When the transmitter was pulsed in the A, phase appropriately timed responses were seen in both types of transducer, but all signals disappeared less than 1 p K below the A, transition. This is as expected: below the A, transition the superleak transducers will no longer function as a “spin filter”, and a propagating second sound mode is not expected. In principle, comparison of the NMR and superleak signals should identify the spin state that is superfluid in the A, phase; in practice doubts about the response of the superleak transducers make the identification of spin parallel to the field as superfluid a tentative one. A hydrodynamics going beyond this simple 3-fluid model has been attempted by several authors. The most extensive is probably that of Pleiner and Brand (1981). They include gradients of the conserved variables in the thermodynamics and the complexity this introduces forces them to confine their attention to reactive hydrodynamics in the strictly hydrodynamic regime. They do obtain, for example, the Muzikar (1980) i x VH term in the spin current, but we believe that this is more properly obtained from angular momentum considerations than from gradients of conserved variables. More seriously, they choose somewhat inconvenient phase variables which have complicated spin rotation and Galilean transformation properties. Thus, instead of our mean phase they use the overall phase of the order parameter, which is not simply related to the physically transparent 4t and 4, when A , # A,. The work of Saslow and Hu (1981) is more readily connected to experiment, but they treat f as an independent hydrodynamic variable and we are inclined to agree with Dombre and Combescot (1982) that this is incorrect. From eq. (41) f is the local spin quantization axis; it is well defined only if A t f A,, and then the condensate magnetization is necessarily along .f. This has no direct implications for the total magnetization (Leggett, 1975), but in the spirit of eq. (39) and the spin relaxation theory of Leggett and Takagi (1977) we expect the condensate magnetization and the total magnetization to equilibrate with each other in a time of the order of the quasiparticle collision time. In a consistent hydrodynamic picture we should therefore take parallel to the total spin density S. But although f and S are parallel it is useful to preserve the distinction between them: we have argued in section 2.1 that gradients of conserved variables, such as VS, should not be included in the thermodynamics; on the other hand f is an order parameter variable and we do expect textural energies quadratic in Vf. A rather careful formulation of the total order parameter distortion energy for the A-phase in a magnetic field is therefore necessary, based OR the order parameter, eq. (41). This has been attempted for the Ginzburg-Landau region by Saslow and Hu (1981) (their eq. A3).

168

H.E. HALL A N D J.R. HOOK

(Ch. 2, 52

2.34. The B-phase The 3-fluid model that we discussed at the beginning of section 2.3.3 is readily extended to the B-phase. If we associate densities P , , potp , with t h e three condensate spin states we readily find that eqs. (58) and (59) remain true with P, = Pt + Po + P, Psp

= Pt i- P,

'

Liu and Cross (1978) have given a more complete hydrodynamics for the B-phase in small magnetic fields. They introduce a new hydrodynamic variable q to describe relative rotations of orbit and spin space; our 4, is essentially the f component of their q, though our vector triads are not defined in t h e limit of zero field which they consider. From an equation equivalent to eq. (55) they obtain the result that under conditions where 1 and f are both parallel to the applied alternating field (which is expected on energetic grounds) the system responds only t o the vector (a+ y H ) . This equivalence of rotation and magnetic fields gives the usual Barnett and Einstein-De Haas effects. Unfortunately, the smallness of laboratory rotation rates relative to typical Larmor frequencies means that the predicted effects are very small. This simple picture of gyromagnetic effects has been modified by Combescot's (1980) discovery from microscopic calculations of the total current density that there is an intrinsic orbital angular momentum density associated with magnetization. Specifically, if we write S = S' Combescot finds an orbital angular momentum density L = -p,Sf where p, decreases from 1 at T = 0 to zero at T,. We must therefore subtract a term ( L+ S)* f 2 from the free energy density, so that the spin dependent part of the free energy density becomes

in which t h e last two terms are the contribution of orbital and spin angular momentum respectively. The spin angular velocity is therefore

where

i = 1-R. Setting o = 0 in thermal equilibrium gives t h e anomalous

Ch. 2, $21

HYDRODYNAMICS OF SUPERFLUID 3He

169

Barnett effect found by Combescot; p, = 0 gives the normal Barnett effect. With eq. (67) the Josephson-Leggett equation (55) becomes sp= d4

dt

(1 - p$.

n - Y 2 S + f - ( y H + a), ~

X

so that the f - D term disappears at T = 0. Combescot (1980) does not obtain the last term in eq. (a),and neither does Liu (1982) unless we interpret his SS as t h e deviation from that produced by a normal Barnett effect. Note that in thermal equilibrium equation (68) gives (d+,Jdt)= i . 0 ; this is the rotation of the order parameter with the container discussed for the A-phase by Ho and Mermin (1980). Combescot suggests an interesting experiment to test the anomalous Barnett effect. Consider ’He-B in a thin slab geometry so that the axis of the rotation R is fixed normal to the slab. If the slab is now rotated about an axis in its plane eq. (67) predicts a magnetization kL?/y)(l- p, cos 0,) parallel to the axis of rotation and a magnetization (,yO/y)P, sin 0, perpendicular to the axis of rotation but in the plane of the slab, where cos e, = -a.1 Volovik and Mineev (1983), [see also Mineev and Volovik (1984)l have queried Combescot’s gyromagnetic orbital angular momentum on the grounds that his calculation method was not accurate enough to determine the current as -~j?,curl(R.S). They maintain that the current is in fact - ~ ~ , E ~ ~ and $ ~ argue ~ V ~that S ~this is a surface current that should not be associated with a bulk angular momentum density. However, Volovik and Mineev’s current is quite analogous to t h e f X V p current in the A-phase that we discussed in section 2.2. If we treat the free energy associated with it in the same way as eq. (26) we have the identity

where 0’ = $urlos. For Combescot’s current the first term on the right is absent. But for either current we have the last term, which is the surface term discussed by Volovik and Mineev, and the second term, which is of the form - L - Q associated with a bulk angular momentum density if there are no vortices, so that as= 0 everywhere. The absence of vortices may therefore be necessary for the success of Combescot’s proposed experiment. The only attempt at a full hydrodynamics of the B-phase in a magnetic field is the work of Pleiner and Brand (1981, 1983), and unfortunately this

170

H.E. HALL AND J.R. HOOK

[Ch. 2, 52

is subject t o similar criticisms to the A-phase work we discussed in section 2.3.3. The work of Liu and Cross (1978) is for the zero field limit and consequently their variable q treats all directions of relative rotation of the spin and orbit coordinates equivalently. This limiting case seems to us to conceal some of the physics. Any non-zero magnetization is sufficient to define a unique direction in spin space f = S/lSl, and consequently a unique direction i = R in orbit space. Rotations about f in spin space or about f in orbit space change &, and hence usp,whereas rotations of f or f are essentially textural in nature. There is therefore a need for a B-phase hydrodynamics that takes account of this distinction.

i-

2.3.5. Future work It will be apparent from t h e preceding discussion that we are of the opinion that a completely satisfactory hydrodynamics in a magnetic field has not yet been given for either A- or B-phase. Such a hydrodynamics will clearly be necessary to interpret the experiments on superfluid 'He in high magnetic fields that are likely to be done in the next few years; we suspect it is also necessary in order to obtain, as a low field limit, a spin dynamics that correctly generalizes the Leggett equations to inhomogeneous situations. It appears from the history of the past few years that the problems and controversies have arisen not from difficulties in carrying out the standard procedure but from difficulties in correctly formulating the equilibrium thermostatics. We therefore list below t h e steps that we think are important, in the light of our preceding discussion, in formulating the equilibrium thermostatics. (i) Include spin and orbital angular momentum densities in t h e thermodynamic identity, as discussed in section 2.1. Guidance from microscopic theory is probably necessary here. (ii) Identify correctly the independent hydrodynamic variables. Thus, we have already argued that f and S are not independent. On the other hand, even if, for example, f and i are strongly pinned to a particular orientation by a high magnetic field they should still be treated as hydrodynamic variables in order to obtain correct low field behaviour; the strong pinning is an energetically determined restriction on the solutions of t h e hydrodynamic equations in particular circumstances, not a restriction to be inserted into the derivation. (iii) Write down the complete thermodynamic identity, with gradients of order parameter variables as independent variables but not gradients of conserved variables. The latter restriction, as we have pointed out in section 2.1, is to some extent a practical one and not rigorous; it is nevertheless important. We saw in section 2.2 that if step (i) above is

Ch. 2, 031

HYDRODYNAMICS OF SUPERFLUID 'He

171

carried out there is no convincing reason to include gradients of conserved variables; since the hydrodynamic equations we shall obtain will in any case be very complicated, it is vital for comprehension to avoid needless complication. (iv) Write down the most general free energy density allowed by symmetry and quadratic in order parameter gradients. (v) Calculate the matrix of thermodynamic derivatives and check Maxwell relations and stability conditions. The last step, for example, shows that t h e induced angular momentum density i* (section 2.2) must exist; its inclusion in the hydrodynamics, which we have not carried through, may be related to the orbital susceptibility of Leggett and Takagi (1978) and Volovik and Mineev (1981).

3. The interaction between flow and textures in 'He-A

In this section we will discuss the textures of f that occur in the presence of superflow and also the possibility of superflow dissipation through orbital motion, a possibility unique to 3He-A amongst known superfluids. We summarize the extensive theoretical work that has taken place on these topics in recent years. The only experiments discussed in section 3 are ones in which the effect of flow on texture has been directly observed. Discussion of the possible role of orbital motion in flow dissipation experiments is postponed until section 4. 3.1. THEHo

CIRCULATION THEOREM

The link between superflow and textures was first expressed quantitatively by Mermin and Ho (1976) in the relation that now bears their names ~

h A

1

1

.- V ~. ~ S=t - I

r 1 2 m

.(viix v,i) .

This has since been shown (Liu and Cross, 1978) to be a consequence of the non-commutativity of three dimensional rotations which may be expressed in the form

where V and S are first order differential operators, such as Vj or a/& and d o is an infinitesimal rotation. A more physically transparent, and there-

172

H.E. HALL AND J.R. HOOK

[Ch. 2, $3

fore more useful, result is the integral form of eq. (70) derived by Ho (1978a) for the circulation around a closed curve C,

h q-dr=--[u(D)+2m1], 2 i?l

(71)

where n is an integer and a ( D ) has the following geometrical interpretation. Let the direction of [ at any point on C be represented by a point on the surface of a unit sphere, constructed by drawing the vector i from t h e centre of the sphere. Then the texture on C is mapped into a closed curve D on the surface of the unit sphere. a ( D )is the net area on t h e surface of the unit sphere to t h e left of D when D is traversed in the sense corresponding to the direction of integration along C. Eq. (71) was actually proved by Ho for a simply connected region containing n o line singularities, in which case n has to be an even integer. We believe, however, that eq. (71) is more generally true; an odd value of n requires either a toroidal vessel with C encircling the hole, or an odd number of line singularities threading C.If a pure disgyration crosses C there is a change in a ( D ) by 27r, just balancing t h e change of n by one unit, and the circulation is not changed. If the texture is known throughout a simply connected region of fluid, eq. (71) may be used to calculate the absolute circulation by expanding a contour of zero length, and hence zero circulation, until it reaches C.A more generally useful application of eq. (71) is to calculate changes in circulation when the behaviour of the texture is known only in the neighbourhood of C; such changes may occur either through time dependence of the texture on C or through motion of C. It can be seen from eq. (71) that a change in texture on C such that D sweeps out the complete area (47r) of the unit sphere once will cause a change in circulation of two quanta. An example of the use of eq. (71) is to calculate the superfluid circulation around a torus along a surface contour C, at latitude 0 as shown in fig. 2a, for t h e case where the texture is t h e singularity free Mermin and Ho (1976) texture shown in fig. 2b. For the contour C,,,?the texture is represented on the unit sphere by a vanishingly small loop U,,, at the south pole. so that eq. (71) becomes

nh

u s -d r = 2m

n in eq. (72) is just the number of complete rotations of the order paramctcr ti + iri about i as C,, is traversed once. The texture o n t h e

Ch. 2, 431

HYDRODYNAMICS OF SUPERFLUID 'He

173

(C) Fig. 2. (a) Definition of the surface contour C, on the torus. (b) The Mermin-Ho texture. (c) The texture of i on C, is represented by the circle Do on the sphere.

contour C, is represented by a loop Do on the surface of the sphere which is a line of latitude in the southern hemisphere of angle d as shown in fig. 2c. The shaded area on the surface of the sphere enclosed by D, is 27r(1- sin 0 ) and therefore using eqs. (71) and (72) we may write t h e circulation around C, as

h q - d r = -((n C# 2m

f

+ 1- sin d ) ,

(73)

174

H.E. HALL AND J.R. HOOK

a result first obtained by circulation around each quanta around C3,,n.For predicts a circulation of con tour.

[Ch. 2, 93

Mermin (1977). Hence there are n + 1 quanta of of the geodesic contours Co and C,,and n + 2 t h e contour in the centre of the channel, eq. (71) n + 1 quanta if i is taken t o be parallel to this

3.2. THESTABILITY OF SUPERFLOW IN 'He-A

AND THE STABILITY OF THE

i

PARALLEL TO Us TEXTURE

The change in circulation associated with a time-dependent texture suggests the possibility of superflow decay through orbital motion, the kinetic energy of t h e flow being dissipated by means of the Cross-Anderson (1975) orbital viscosity. That orbital motion can also lead to dissipation in situations where us is maintained at a constant value and may be seen by considering the superfluid equation of motion, which in t h e absence of normal fluid motion may be written as

If au,l& is zero on average then a finite chemical potential gradient will be required to cancel the final term and work must therefore be done to maintain a constant u, in t h e presence of orbital motion. The integration of eq. (74) along a contour C in the fluid joining points A and R gives

where duldr is the rate of sweeping out of area o n t h e surface of the unit sphere by t h e curve D which represents the texture on C. For a closed curve C for which A and B are the same point, p B - p A must vanish and eq. (75) then becomes the time derivative of eq. (71) and in this form describes the decay of superflow in a toroidal geometry by orbital motion. For a situation where t h e normal fluid velocity is spatially uniform the equation of motion, eq. (38), of i in the normal fluid rest frame may be written in t h e concise form (Brinkman and Cross, 1978)

i+-+-v)=o. ai (3t

2m

We have ignored t h e inertial term in eq. (38) and t h e free energy density t o be used in evaluating the molecular field ly is

Ch. 2, 031

175

HYDRODYNAMICS OF SUPERFLUID ’He

a

where we have added the dipole energy and bending energy of to eq. (15). We continue the practice of using8 to represent hG2m. We note that the coefficients C and K4 do not appear in Y so that the modifications to f represented by eq. (26) do not affect the dynamics of i in the limit in which we are interested. The bending energy of d becomes simpler in two limits. In the dipole-locked limit where is parallel to f the effect of these terms is to increase the bending energy of f according to: Ks-’Ks+Psp~t

l(l-’KtfPspL.

Kb-’Kb+Pd.

In the limit of spatially uniform d, such as might occur in a completely vanishes. For the dipole-unlocked situation, the bending energy of remainder of section 3, except where we explicitly state otherwise, we will assume that one of these limits is appropriate. If for the moment we ignore the effect of the walls of the flow channel, then we can see that the question of the stability of superflow around a torus is closely linked to the question of whether the gradient free energy f is such as to favour a spatially uniform texture with f parallel to 0,. If this is the case then the effect of superflow on an initially uniform f texture will be to rotate f into the us direction, and the spatial variation in the flow direction and temporal variation of f that are necessary from eq. (74) to cause superflow decay will not occur. The complications that arise because of the existence of the walls of the flow channel are discussed in section 3.5. Until 1977, the stability of the fIIu, texture was accepted almost without question (Leggett, 1975) because it was favoured by both the superfluid density anisotropy terms and the f bending energy terms in f . In 1977, Hook and Hall (unpublished) obtained the result that the f 11 us texture was unstable in the Ginzburg-Landau limit. Unfortunately their stability analysis was based on incorrect equations. Their work, however, stimulated a flurry of theoretical activity and Bhattacharyya et a]. (1977) and Cross and Liu (1978) derived the correct condition

a

(Co + ;P$

Pol<,

(78)

for t h e stability of the illus texture, where psll= ps- po. The physical

176

H.E. HALL AND J.R. HOOK

[Ch. 2, 93

interpretation of this result is that t h e terms in f involving the anisotropy of t h e superfluid density and the bending of i are such as to favour the ill u, texture, and the coefficients of these terms appear o n the right hand r l ~ to side of eq. (78). whereas the cross term - - C o ( u S - ~ ( ~ . c utends destabilize the texture so that C, appears on the left hand side. For T near to T, and for the case of dipole-locking

so that inequality (78) is just satisfied. In t h e absence of dipole-locking

and the inequality is not satisfied. As pointed out by Bhattacharyya et al. it is remarkable that if bulk superflow in 'He-A is stable then it may owe its stability t o the magnetic interaction between the 'He nuclei. At lower temperatures the decrease in po may cause instability even for the dipole-locked case but it is doubtful whether this will occur above the A-B transition temperature i n zero magnetic field (Fetter, 1979; Schopohl and Tewordt, 1980). Our discussion of superflow decay has considered only the term in the supercurrent involving us. It is also necessary to consider t h e term arising from curl i If the us supercurrent decays through orbital motion it is not obvious that curl f currents of similar magnitude will not be generated. We return to this problem in sections 3.3 and 3.5. Throughout section 3 we will assume that it is possible to model toroidal flow by applying periodic boundary conditions to linear flow. This assumption has been generally made, but Bailin and Love (1978) suggested that a real toroidal geometry may be significantly different. They find in a geometry of cylindrical symmetry with us parallel to that t h e ill us texture is never stable. Subsequent wor4 by Harris (1980) shows that this problem is only important when the superfluid circulation is of order one quantum. For larger circulations the texture corresponds to ill u, to a good approximation.

6

Fetter (1978) has shown that a helical texture of i arises in the presence of superflow if po is reduced below the value pu: which makes eq. (78) an equality. f makes a constant small angle B to the flow direction but spirals about this direction with a pitch P as shown in fig. 3. The helix remains stationary in the frame of reference in which the normal fluid is at rest

Ch. 2, $3)

HYDRODYNAMICS OF SUPERFLUID 3He

I Fig. 3. A helical texture of

P i with opening

177

I angle 0 and pitch P.

and in order to reduce the -C‘,(i*u,)(i-curl~) term in the gradient energy the helix must have a left (right) handed sense about us for i initially parallel (antiparallel) to us. The transition to the helical state is second order with 6 growing as (pot - p0)”* as po is reduced below pot. The pitch of the helix at the critical point ( p o= pot) is

P=

Kb

h-

(c,+ ipsl,)2mu,

~

h 1.2 mu,

for T near T,

The formation of a helical texture from the i 1) us texture does represent superflow decay. The area swept o u t on t h e unit sphere is 2741 - cos 6 ) for every turn of the helix and therefore, using eq. (71), us changes by

Au, = -(1- cos 6)h/2mP.

(82)

The change in mass flow j is more difficult to calculate because allowance has to be made for both the anisotropy of the superfluid density and the appearance of curl f currents. The change in the component of j parallel to us is

Aj

= po sin’

6uo+ (p,- po cos2O)Au, - cos 6 sin26Coh/2mP,

(83)

where uo is t h e value of us before helix formation. The final term is the contribution of the curl i current and it can be seen that for this situation it is in the opposite direction to the initial mass flow. Takagi (1978) suggested that the application of a magnetic field parallel to us should destabilize the us texture in the dipole-locked limit because of its orienting effect on the vector. The required field is

where A x is the anisotropy in t h e magnetic susceptibility. Lin-Liu et al. (1978) have shown that instability of the us texture should also occur in

17X

H.E. HALL AND J.R. HOOK

[Ch. 2, 83

flows large enough that the characteristic scale of spatial variation of becomes less than the dipole-locking length

i

Dipole-unlocking then occurs as the cost in energy of bending d on such a short length scale exceeds the energy gained from the dipolar interaction. In a flow us, i varies on a length scale of order hlmu, determined by competition between flow alignment and bending energy terms in the gradient free energy. Hence the critical velocity of dipole-unlocking is of order h/rn(,, the actual value in the Ginzburg-Landau region for H = 0 being 0, =

0.413h/m(,

2

1 mm s-’ .

(85)

Helical textures arise when the illu, texture is destabilized in either of the ways described in the previous paragraph. In the dipole-unlocked case both and d form helices of the same pitch but at different angles to the flow direction. Lin-Liu et al. (1979) and Vollhardt et al. (1979) have investigated the helical textures that arise for T near T, when both a superflow and a magnetic field are present. Fig. 4 shows the region of field and flow over which t h e small-angled helical texture exists. The axes of the figures are the dimensionless magnetic field ( H = H i )

c

and the dimensionless mass flow

Note that our ID is related to the (l of Vollhardt et al. by 6, = V%$. Vollhardt et al. give the pitch P and angle ,y of the helix as functions of p and h and also give the NMR frequencies for the helical textures. Rromley (1980) reports that the anisotropy of the spin susceptibility was incorrectly ignored in the NMR analysis and presents corrected frequencies. Rrornley finds that o n l y one of t h e two longitudinal and one o f t h e four transverse resonances have appreciable spectral weight. Vollhardt et al. (1981) p o i n t out that Bromley’s calculations are for t h e case where t h e pitch remains constant after the helix forms rather than changing to

Ch. 2, 031

HYDRODYNAMICS OF SUPERFLUID 'He

179

h Fig. 4. Stability region for helical textures in the presence of a supertlow us and magneric field (Ilu.) for T near T, (Vollhardt et al., 1979). p and h are dimensionless mass current density and magnetic field as defined by eqs. (86) and (87). In region I the //la, texture is stable. In region I1 helical textures are stable with i and d making a small angle to 0,; the broken curves indicate the optimal pitch (P= 27&/k) of the helix. In region I11 helical textures are unstable. I n region IV helical textures are stable with / and d almost perpendicular to H if the assumption is made that the pitch of a helix remains constant after its formation (Fetter and Williams, 1981; see section 3.4). A uniform texture with ClldlH is stable within a region similar to IV. (Vollhardt, 1979; Fetter, 1981; Dow, 1984; see also section 4.8.)

minimize the energy. They present corrected frequencies for a helix in which the pitch is allowed to change. As can be seen from fig. 4, the small-angled helical texture does not exist over a wide range of the h , p plane. We discuss in the following section the textures that arise when the helical texture becomes unstable. On the zero field axis the helical texture becomes unstable at p = 1.13. At this point the pitch P is 6.65, and the angles of the i and helices are 35" and 13.1" respectively. These figures correspond to decay of v, and the component of j , parallel to us by 14% and 7% respectively as can be seen using eqs. (82) and (83). There is as yet n o unambiguous experimental observation of a helical texture. Kleinberg (1979) has suggested an interpretation of some of his observations of ultrasonic attenuation in the presence of a heat flow in terms of the formation of helical textures. The superflow arises because the heat is transported in part by t h e counterflow mechankm. The sound attenuation is measured transverse to the heat flow. Fig. 5 shows the measured attenuation as a function of a gradually increasing magnetic

180

[Ch. 2. 03

H.E. HALL AND J.R. HOOK

0239

'-T/T; H,

13 ~

~ O U S S

Theory

05

I0 H/H,

Fig. 5. Sound attenuation at 15 MHz and 1 - T/T, = 0.0239 observed for increasing magnetic field by Kleinberg (1979). The heat flux was 1 nW cm-* and the dashed line was calculated f o r a helical texture of the type described by Lin-Liu et al. (1979). H / H c is the same as the dimensionless field h of Lin-Liu et at.

field applied parallel to the heat flow. The initial constant attenuation is interpreted as arising from the ill us texture. The onset of a textural change is clearly distinguishable. The dashed line indicates a theoretical calculation of t h e attenuation that would be expected for a transition to the helical state predicted by Lin-Liu et al. (1979) and agreement between experiment and theory is good; there is evcn a suggestion in the experimental curve of another transition at a higher field, which is in the right place to be interpreted as arising from instability of t h e helical texture. However this type of behaviour was not consistently observed, results obtained under apparently identical conditions often looking quite different. It is also true that the results may be explained equally as well by a theory in which the i texture remains spatially uniform. Kleinberg's observation that the observed threshold fields for the onset of the helical texture are too high for T near to T, may result from problems in calculating the field H' needed to convert actual magnetic fields into dimensionless ones. According to eq. (86). H' should be independent o f T near T, but the experimental data used by Kleinberg gives a value of H' which decreases strongly near T,.

Ch. 2, $31

181

HYDRODYNAMICS OF SUPERFLUID 'He

Bates et a]. (1983) have begun to investigate experimentally the phase diagram of fig. 4. They use ultrasonic attenuation at 30 MHz to study the texture within a rectangular channel through which fluid is driven by a bellows arrangement. Two channels of cross section 3.05 X 1.77 mm2 and 1.35x 2.07 mm2 were used, but the results obtained so far do not depend on the channel size. By applying a steadily increasing magnetic field

0

.90 T,

0

.

.94 Tc

*o

O

0

0

," &@

ooog

0 OO

O -

Q)

0

.

0

c:

m U

i

2

0

ti,

.9E

(Oe)

(C'

4

2 H,

4

(Oe)

Fig. 6. Phase boundary for the uniform texture as observed by Bates et al. (1984). (a), @) and (c) show data at three temperatures at 28.8 bar in flow channels with cross-sectional areas 0.028 cm2 (open circles) and 0.054cm2 (filled circles). Smoothed data is shown for all three temperatures in (d) by dashed curves. Curve (i) is calculated for 0.94Tcfor weak-coupling values of the free energy coefficients. Curve (ii) has K b increased by 10%.

I82

H.E. HALL AND J.R. HOOK

[Ch. 2. 53

parallel to a uniform flow through t h e channel they have determined the limits of stability of the ill u, texture shown in fig. 6. Comparison with fig. 4 shows that the phase boundary has the correct shape but the illus texture persists over a larger region of the us-H plane than is predicted by fig. 4, and in particular to much larger flow velocities. As can be seen from fig. 6d the phase boundary is very sensitive to the parameters entering the free energy, eq. (77). Two factors which could affect the theoretical boundary are strong-coupling corrections to these coefficients and depairing effects. In connection with this latter possibility we note that t h e maximum mass current used by Bates et al. was about 6% of the depairing critical current of Kleinert (1980). It is possible that the observed transition at large flows is a change in the t y p e o f the time dependent texture and not the instability of the us texture. Such a transition has been predicted by Dow (1984) (see section 3.4). There are two additional noteworthy features of the results of Bates et al. There were often large fluctuations in the texture before the flow was applied. These were probably driven by the flow associated with residual heat leaks and are thus similar to the persistent orbital motions observed by Paulson et al. (1976) and Krusius et al. (1978) that are discussed in the following section. When flow was introduced into the channel through a superleak consisting of 830 OOO glass capillaries of 2 Fm diameter t h e results obtained differed from those shown in fig. 6 and suggested the presence in zero magnetic field of a spatially non-uniform texture in t h e channel.

3.3. BEYOND HELICAL

TEXTURES

The textures that occur when the helical texture becomes unstable have not yet been completelj elucidated, although they are crucial to any consideration of superflow dissipation. Fetter and Williams (1981) argue persuasively that when the helical texture is made unstable by an increasing magnetic field the resulting behaviour depends o n whether t h e situation is one of toroidal flow in which the superflow is free to decay or one in which the superflow is driven such as thermal counterflow. I n neither case is there a nearby time-independent texture into which the system can evolve. Fetter and Williams assume that as the texture evolves i t remains approximately helical so that insight i n t o t h e behaviour can be obtained by considering only the energies of helical textures. The toroidal geometry is simulated by considering one dimensional flow with periodic boundary conditions, the period being much greater than h/tncs. Onc important consequence of restricting discussion to helical textures with

Ch. 2, 931

HYDRODYNAMICS OF SUPERRUID ’He

183

periodic boundary conditions is that once a helix is formed its pitch remains constant since there is a large energy barrier between helices of different winding number. The only possibility for change occurs when the opening angle 8 of the helix is 0 or 7r where the energy barriers go to zero. Periodic boundary conditions are also used in the driven flow case, although here, as realized by Fetter (1982), the consequent constancy of pitch is a more unrealistic assumption, since there are likely to be distant walls perpendicular to the flow at which 8 will vanish. Fetter and Williams find that, if this assumption is made, helical textures become stable again in higher fields as shown on fig. 4. For these helices i and d are almost perpendicular to the field. The difference between toroidal and driven flow arises because of the different dependence of us on 8 in the two cases. For toroidal flow the dependence is given by eq. (82). For driven flow we consider the case of thermal counterflow where v, is constant and the total mass flow is zero; us is then given by us = u,

+

-pun

+ (C,,h/2rn~)cos e sin’ e p, - po COS* e

(88)

P is the pitch of the left-handed helix formed when the illu, texture becomes unstable with increasing magnetic field. Note that for the heat flow case a helical texture will precess in the laboratory frame since it is only stationary in a frame of reference where v,, = 0. In the dipole-locked limit to which our further discussion in this section is limited, P is given by eq. (81) even in a finite field. For this pitch and at the critical field H, at which the small-angled helical texture becomes unstable, the energy of a helical texture as a function of 8 is shown in figs. 7c and 7e for the toroidal geometry and for heat flow respectively. Both curves are qualitatively the same and their form suggests that 0 will continue to increase to

e = T. At this point the possibility exists for a change in pitch and the subsequent behaviour in the two cases is then likely to differ because of the very different values of us. For heat flow eq. (88) predicts that us has the same value as that prior to helix formation and, as i is now antiparallel to us, the most likely occurrence is the formation of a right-angled helix of pitch similar to that given by eq. (81). This helix is unstable as indicated by energy diagram 7f and is likely to evolve to 8 = 0 where the whole process repeats itself. Time-dependent solutions are likely to occur therefore in the case of driven heat flow. That the suggested time dependence is an oscillating 8 at constant pitch is we believe a misleading result, arising only because the pitch was assumed to

18.1

[Ch. 2, 83

H.E. HALL AND J.R. HOOK

L

0

712

T

TEXTURES-

ol

L

0

0

712

n?

r

0

712

T

H,

Fig. 7. Evolution of the free energy density of helical textures with increasing H. f and f are the free energies for toroidal flow and heat flow respectively. HO and H I are the fields at which the IIIu, and small angled helical textures become unstable. The dot on the curve indicates the behaviour of the system. (Fetter and Williams, 1981.)

remain constant. We shall see below that t h e time dependence can involve precession of the texture about the flow direction with 0 remaining almost time independent. For the toroidal geometry on t h e other hand eqs. (81) and (82) predict us, = UJ 1- (2Cn+ p d ) / K , ] = -0.24,

for T near T, ,

(89)

where u6) is the value of us prior to helix formation. The most likely helix to form under these circumstances of both reversed us and reversed i is a left-handed helix with pitch close to that given by (81) with us= /us,l. The energy diagram for a helix of this pitch is shown in fig. 7d and has a deep minimum near 8 = 77/2, t h e reduction in 1uS1 at fixed H has brought the system i n t o the stable large-angled helix region IV of the phase diagram of fig. 4. The likely end result is therefore a time-independent wideangled helix with a small reversed superflow u f , which from eq. (82) is of order

Ch. 2, 031

185

HYDRODYNAMICS OF SUPERFLUID 3He

Many features of these conclusions are confirmed for the heat flow case by Hook and Hall (1W9) who solved numerically the equation of motion of i,eq. (76), for the case of heat flow perpendicular to two parallel planes situated at z = 0 and z = d. From an initial texture which consisted of a helix with pitch given by eq. (81) and a small opening angle 6 varying with position as sin(?rz/d) in order to satisfy the boundary condition that i should be perpendicular to the planes, the textures shown on the phase diagram of fig. 8 were obtained in the dipole-locked limit for T near T,. H * and d * are the dimensionless magnetic field (applied perpendicular to the planes) and plane separation defined by

d* = 2mw,d/h.

(92)

where w,, is the value of the counterflow Iu, - u,l at the plane surfaces due to the heat flow. The situation considered by Fetter and Williams corresponds to the large d * limit, where, with increasing H, Hook and Hall find final textures which are: the f l l u , texture, a small-angled helical texture, periodic but anharmonic time-dependent textures that are discussed further below, and a wide-angled helical texture with i almost perpendicular to H over most of the region between the planes. H*

1

TIME DEPENDENT

1.0

8.0 0

I

10

I

20

I

I

30

40 di

Fig. 8. Textures which arise in a heat flow between two parallel planes when a magnetic field is applied perpendicular to the planes (Hook and Hail, 1979). The initial texture had 8 close to zero everywhere. The dimensionless magnetic field H' and plane spacing d' are defined by eqs. (91) and (92).

186

H.E. HALL A N D J.R. HOOK

[Ch. 2, 83

The calculations of Hook and Hall illustrate a very important feature of textural problems in t h e A phase, that the texture can depend on the previous history of the system, a feature which is also evident in experimental work (see section 4.7). For a given heat flow and magnetic field there is often more than one possible final texture, the choice of initial texture determining which of these actually arises. For example for H = 0 and large d * , in addition to t h e illo, texture, there are also time-dependent final textures. In the simplest of these the angle 8 that i makes to the heat flow is almost time independent but changes with position from 0 at z = 0 to approximately 7r at z = d/2 and then back to 0 again at z = d. This solution is referred to as t h e 0-7r-O solution, and more complicated solutions of the form kr-0-r-0, etc., also exist. The existence of the 0-7r-O texture, together with the demonstration by Fetter and Williams that i is likely to rotate through a large angle with respect to the heat flow when the small-angle helix becomes unstable, suggests that &7r domain walls may play an important role in superflow dissipation. Hook and Hall investigate the properties of such walls by looking for textures which satisfy 8 = 0 at z = 0 and 8 = 7r at z = d. As well as t h e simple domain wall (0-.rr solution) they find for sufficiently large d * more complicated solutions of the form 0-7r-O-~, etc. containing larger numbers of domain walls. The 0-7r domain wall precesses about the flow direction at a uniform angular frequency w . The significance of this precession may be seen by considering the representation of the texture on the surface of the unit sphere (fig. 9). As the wall precesses, the area o n the unit sphere is swept o u t at a rate 2 0 . There is, therefore, from eq. (75) a chemical potential difference A p = hw/m

(93)

associated with t h e wall. The frequency of precession w is shown in fig. 10 as a function of l/d*. The dimensionless frequency w * is related to w by

where p, is the orbital viscosity. In real units we have

If we suppose that superflow dissipation for driven flow is associated with the formation of an array of domain walls then we might expect the

Ch. 2, 331

HYDRODYNAMICS OF SUPERFLUID 'He

187

e=o

t

e=n Fig. 9. Trajectory of the 0-7r domain wall on the unit sphere. The arrows indicate the direction of precession for a heat flow which is vertically upwards. (Hook and Hall, 1979.)

*

w

1.0-

.5 -

0

I

0.05

I

I

015

0.1

l/d* Fig. 10. Precession frequency as a function of lld' for 0 - 7 r domain walls (curve A) and the 0-7r-O texture (curve B). (Hook and Hall, 1979.)

1X8

H.E. HALL AND J.R. HOOK

[Ch. 2, 93

separation of the walls to be of order filmv,. The chemical potential gradient associated with such an array is of order

where w e have used eqs. (93) and (95). This result, which we use extensively in section 4 in our discussion of measurements of flow dissipation in 'He-A. has also been obtained by Volovik (1978). This result is confirmed by the numerical calculations of Hook and Hall. Their 0-7r-O texture does indeed look like two 0-7r domain walls back-to-back, the two walk precessing in opposite directions at a frequency close to that of a single 0-7r domain wall of the same width (see fig. 10). The chemical potential difference associated with t h e 0-7r-O texture is therefore approximately double that for a single domain wall [eq. (93)).If 0-7r domain walls result when the small-angled helix becomes unstable in driven flow, then the time-dependent textures of fig. 8 might be expected to be of this form. This is almost the case, except that, because of the magnetic field, the texture is closer to 0-7r/2-O than t o 0-7r-O for most of the time. The opposite precession of the two ends in this case then leads to the texture acquiring an ever increasing twist; if the texture is treated as an approximately helical one, then the number of turns of the helix gradually increases. The twist is removed periodically by an increase in 8 towards 7r in t h e middle of t h e channel. The passage of 8 through 7r results in a decrease of twist by 27r as one turn of the helix disappears. Such a process also occurs for t h e 0-7r-4) texture, although here the value of 8 in t h e middle of the channel is close to 7r for most of the time. Experimental evidence for the existence of 0-7r-O textures and 0-.rr domain walls comes from the persistent orbital motions observed by Paulson et al. (1976) and Krusius et al. (1978) respectively. The orbital motions observed by Paulson et a]., had a period of order 2 4 p 4 , v f , close to t h e value predicted for 0-7r-O textures in fig. 10. Fig. 11 shows that reasonable agreement is obtained between the period of the small amplitude oscillations observed by Krusius et al. and the precessional period of 0-7r solutions. In both cases only semiquantitative agreement is to be expected because of the complexity of the experimental geometry. The predictions of Fetter and Williams for toroidal flow have been investigated numerically by Dow (1984). He applied periodic boundary conditions to linear flow to simulate toroidal geometry, but in other respects his calculations are similar to those of Hook and Hall. In the dipole-locked limit, a helical texture driven unstable by increasing magnetic field remained approximately helical until the opening angle was

Ch. 2, 931

HYDRODYNAMICS OF SUPERFLUID ’He

189

Period (minutes) 5

1 0.05 0.06 007 0.08 i- TIT,

0.04

Fig. 11. Measured period of the small amplitude oscillations observed by Krusius et al. (1978) compared with theoretical values for a 0-T domain wall. “Superoscillations” occurred at temperatures higher than that indicated by the vertical arrow. Curves A and B assume different values for d and p,/pd in calculating theoretical values of w . (Hook and Hall, 1979.)

close to T,at which stage the helix lost turns through local increases of the opening angle to T.The consequent reduction in mass flow resulted in the system reaching the stable large angle helix region of the phase diagram of fig. 4; the final mass flow was not reversed as suggested by Fetter and Williams. In some cases the final texture was spatially uniform with f and d perpendicular to H, i.e. a helical texture of infinite pitch. It is possible that the final textures in these calculations are dipolelocked helices of finite pitch only because dipole-locking is assumed. Dow has extended his calculations to allow dipole-unlocking for the cases of both driven and toroidal flow. In the latter case the stationary spacedependent textures arising from superflow collapse are dipole-unlocked and contain solitons (Maki, 1979), the nature of the soliton depending on the magnetic field and residual mass flow. In small fields, h Q 1 in the dimensionless units of eq. (%), d remains nearly parallel to the flow throughout the collapse and the final state consists of an array of stationary domain walls of the form O-?z-&z--etc. in t h e i texture only. In large fields, h >> 1, d remains largely uniform and perpendicular to H and

190

H.E. HALL A N D J.R. HOOK

[Ch. 2, 13

the final state has an array of almost pure i solitons, in each of which a 27r rotation of i is involved. For the case of heat flow perpendicular to parallel planes at which 8 = 0, Dow finds that the stationary textures in region IV of fig. 4 also contain solitons. For flows less than p 1= 0.5 in the dimensionless units of eq. (87) t h e solitons resemble t h e composite solitons of Maki and Kumar (1978). At larger flows the solitons resemble the pure i soliton of Vollhardt and Maki (1979a), with d perpendicular to t h e field (see fig. 32 in section 4.8). At a dimensionless mass flow p 2 = 0.9 the texture becomes time dependent. Vollhardt and Maki also predict the change in soliton type but as they consider only stationary textures they do not see the transition to a time-dependent state. They do however find that the uniform texture with i and d perpendicular to H becomes unstable against the formation of pure i solitons at a mass flow very close to p 2 . Dow’s calculations suggest that solitons may appear at much smaller mass flows than this. Dow has also investigated the time dependent dipole-unlocked textures that arise in t h e presence of a heat flow. In zero H and for flows that exceed that at which the helical texture becomes unstable, he finds precessing O-.ir4-7r-etc. domain walls of 1 only; d remains parallel to the How. In large fields the “pure f” soliton texture becomes time dependent. d remains perpendicular to the field and at any point i precesses about the d direction. Dow (private communication) has suggested that the transition from the low field to the high field behaviour could be t h e transition observed at large flow rates in the experiments of Bates et al. (1984) discussed in t h e previous section. For all the time-dependent textures discussed above the essentially dimensional argument leading to eq. (97) should apply and we will use this and similar arguments in section 4 in our discussion of A-phase flow dissipation measurements. A shortcoming of t h e work discussed in this section is the restriction to textures which vary only in the flow direction. That this should be the case is by no means obvious even for fluid of infinite extent in the other two dimensions. In practice the texture will certainly have to vary in the direction perpendicular to the flow in order to satisfy boundary conditions at the container walls (see following section). Calculating one dimensional textures is sufficiently difficult in general that the three dimensional problem is likely to be completely intractable. It may be necessary to resort to methods which do not seek to solve for the texture in detail but which attempt to describe the system using variables averaged over finite regions of space and time. Dimensional arguments can often be used to obtain space and time scales for variation

Ch. 2, §3]

HYDRODYNAMICS OF SUPERFLUID 'He

191

of t h e averaged variables. Vinen (1957) uses an approach of this kind to discuss flow of He-I1 in the presence of a large vortex density. Such ideas are applied to 3He-A in deriving eq. (97) and in section 4.Volovik (1980) describes an approach of this kind to the case of textures varying only in one dimension. He proposes a closed system of equations, the solution of which produces results similar to those discussed in this section. Hu (1979a) has discussed the topological implications of the structure of the A-phase order parameter for persistent current decay.

3.5. THE EFFECT OF CONTAINER WALLS ON THE TEXTURES GENERATED BY SUPERFLOW

U p to this point we have ignored the effect of the container walls parallel to the flow in discussing the textures that arise in the presence of superflow. We have considered textures with spatial variation only in the flow direction. Since the natural scale of spatial variation of f in a flow us is h/2mv,, we must at least expect the channel geometry to be the dominant influence on the texture for v S 4 h / 2 m dwhere d is &hechannel size. We begin by considering a toroidal geometry with a singularity-free texture. As shown by Mermin (1977) a torus is topologically the only type of container where a singularity-free texture is allowed. An example of such a texture is the Mermin and Ho texture of fig. 2b. We have already calculated in section 3.1 the circulation along a contour on the surface of the torus for this texture. Since f is fixed on the surface, the circulation along a surface contour must remain constant. We see immediately that the existence of container walls may have a profound influence on superflow decay. For small superflow, v, < h/2md, the bending of f will be on a length scale of order the channel size d. As the number of quanta of circulation is increased the region with f approximately parallel to t ) , in the centre of the channel will grow until for o,Bh/2rnd it wiIl occupy most of the channel as shown in fig. 12. We have assumed d %- 5 , so that the situation of fig. 12 is one of dipole-locking for which the ill i), texture is stable. A further increase in u, to the critical value of eq. (85) will cause dipoleunlocking to occur and we may then expect to obtain a small-angled helical texture in the central region of the channel of the type described in section 3.3. It is not difficult to see how this texture may be matched to a constant surface texture and therefore how a state of partially collapsed superflow in the central region may be matched to the fixed surface circulation. A further increase in the surface circulation should eventually give rise

192

H.E. HALL AND J.R. HOOK

[Ch. 2, $3

Fig. 12. Mermin-Ho texture in a cylinder of diameter d for the case of a large superflow, o, hRmd. Note however that we assume os < f t R m f ~so that dipole-locking occurs and the illu, texture is stable.

*

to catastrophic superflow collapse in the central region of the type discussed in the previous section. The orbital motion associated with this collapse can be expected to give rise to a complicated transitional texture between t h e region of collapsed superflow and t h e walls. The need for this transitional texture becomes clear if eq. (71) is applied for a contour C that is gradually moved from the wall to the region of collapsed superflow. To obtain the required change in circulation the texture must change in such a way that the appropriate area is swept out on the surface of the unit sphere. Ho (1978a) has considered this problem in some detail. He concludes that the superflow collapse may be viewed as arising from the formation of coreless vortex rings in the channel. The rings grow, eventually producing an array on the channel walls as shown in fig. 13. The simplest kind of coreless vortex (Anderson and Toulouse, 1977), as shown in fig. 14, has cylindrical symmetry and is non-singular at the centre. Although

Fig. 13. An array of coreless vortex rings on the surface of the channel provides an interface between the region of collapsed superflow in the interior and the unchanged superflow at the surface. The cross sections of six rings are shown and the arrows indicate the direction of circulation around the vortex which occupies the region o f dimensions W X S shown. (Ho, 19778a.)

Ch. 2, 831

HYDRODYNAMICS OF SUPERFLUID 'He

193

I I Fig. 14. The Anderson-Toulouse (1977) coreless vortex. The texture has cylindrical symmetry about the dashed line.

the coreless vortex needed to cause superflow decay in a torus is necessarily more complicated in order to satisfy the boundary conditions, its essential properties are the same. The circulation around a coreless vortex along a contour of large radius is two quanta (hlm) as may readily be seen from fig. 14 by gradually shrinking such a contour to one of vanishingly small radius. In this process the line representing the texture on the surface of the unit sphere sweeps out the area of the sphere once. The circulation around the initial contour then follows from eq. (71), because the circulation around the final contour must be zero. The formation and expansion of a coreless vortex ring causes the superflow in the central region of the channel to decay by two quanta. The situation is analogous to the decay of persistent currents in superfluid 4He by the formation of vortex rings. An important difference is that the coreless vortex ring cannot annihilate itself at the wall because of the fixed i texture there. Ho gives an explicit form for a suitable coreless vortex texture in which the vortex ring forms by continuous deformation from an initially uniform texture. The texture is again uniform in the central region of the channel after collapse has occurred. Thus the collapse need not give rise to large curl f currents in the central region of the channel. In order to reduce the circulation to zero in the centre of the channel, the spacing S (see fig. 13) of the vortices along the channel walls must be S = h/mu, where us is the superfluid velocity at the walls. The width W of the vortex layer on the surface is of order hfmv, so that W and S are of similar magnitude. The gradient free energy associated with the coreless

104

H.E. HALL A N D J.R. HOOK

[Ch. 2, 03

vortex array is of order 7rd27rRWip,u:; in this there will be contributions of the same order of magnitude from curl f currents in the neighbourhood of the walls and the bending of f as from us supercurrents. R is the radius of the torus and d the diameter of the flow channel. It is interesting to compare this with the kinetic energy of the superflow prior to its collapse which is of order 2 7 r R ~ 7 r d 2 f p S to v ~ show that superflow collapse is energetically favourable if us> h/rnd, that is if the natural scale of spatial variation of i for a flow v, is smaller than the channel size. That the decay may not occur until us reaches a higher value us texture in of order film(, is associated with the metastability of the the dipole-locked limit. Until t h e velocity reaches the higher value there is likely to be an energy barrier against the formation of coreless vortex rings. Note however that a critical velocity of order h / m d might be expected for the creation of cored vortices (see section 4.2), and these could allow superflow collapse at this reduced velocity. For the case of driven flow through a channel of size d with a gradually increasing flow velocity, the situation will initially be the same as that for toroidal flow. An initial Mermin-Ho texture should gradually distort until for us B h/2rnd (but us< h/2mSD) the texture is like that of fig. 12. At t:, = h/2rn{, dipole-unlocking should occur and a small-angle helical texture arises. The difference to the toroidal case arises when the flow becomes large enough for the helical texture to become unstable. For driven flow we expect time-dependent textures to arise in the centre of the channel as discussed in the previous section. The matching of these to t h e constant surface texture has been discussed by Ho (1978b). He shows that a time-dependent texture of the form of the 0-7r4 texture of Hook and Hall (see previous section) will lead to the continuous production of coreless vortex rings near the surface, a conclusion that is not altogether surprising in view of the discussion of toroidal flow above. If the texture is not to be immobilized by entanglement arising from an accumulation of vortex rings then some mechanism must be found for removing the vortex rings. If the flow channel is open at the ends then Ho suggests that the vortex rings will flow downstream out of the channel. If the ends of the flow channel are closed, however, then surface singularities in the texture are required to annihilate the vortex rings. i n the following section we describe the way in which a moving surface point singularity (boojum) may be used to match a 0-7r domain wall in a flow channel to the boundary texture. The stationary surface singularities proposed by Ho are however more complicated than this. We have so far restricted our discussion of the effect of container walls to the case where the channel size d % t D . In narrow channels in the absence of flow the texture of lowest energy is no longer the Mermin-Ho

Ch. 2, 931

HYDRODYNAMICS OF SUPERFLUID 'He

Fig. 15. The radial disgyration of

195

i appropriate to cylinders of small radius.

texture but is the radial disgyration texture of fig. 15 (Maki and Tsuneto, 1977; Bucholtz and Fetter, 1977). The effect of flow on this texture has been discussed by Bruinsma and Maki (1979) who state that vortex ring formation in such a small geometry may have such a large activation energy that it is unlikely to occur. In the following section we will describe, however, a mechanism for superflow dissipation in small channels. . .. We conclude this section by discussing briefly t h e effect of the container wails on superflow between two infinite parallel planes. The application of periodic boundary conditions to this situation simulates flow in the annulus between two concentric cylinders. In the absence of flow the texture will be as shown in fig. 16a. Hu (1979a) and Ham and Hu (1980) find that for T near T, this texture persists up to a critical velocity v, = 6 d i f 4 r n d

Fig. 16. Texture of

(98)

i between parallel planes for: (a) zero flow, (b) large flow parallel to the planes.

1%

H.E. HALL AND J.R. HOOK

[Ch. 2, 93

in the dipole-locked limit where the plane spacing d is much greater than ID, At this velocity a second order textural transition occurs and, with increasing us, i acquires a component not only in the flow direction but also, contrary to the earlier assumptions of de Gennes and Rainer (1974) and Fetter (1976), in the direction perpendicular to the flow and parallel to the planes. This departure for a planar texture occurs because of the - C , ( ~ - u , )(i-curl i) term in the gradient free energy. As the how increases further the situation increasingly becomes one in which [ is parallel to u, over most of the channel (fig. 16b). Eventually dipoleunlocking should occur and the subsequent behaviour should be similar to that for the Mermin-Ho texture discussed above. For a narrow plane spacing d e tD where dipolar forces are unimportant, the initial transition from t h e texture of fig. 16a is a first order one which occurs at a critical velocity. t), =

V?.lrh/4rnd.

(9)

The appearance of L'3 rather than fi as in eq. (98) reflects the smaller bending energy coefficient K b in the dipole-unlocked limit. Hu (1979b) suggests that the application of a magnetic field perpendicular to a slab of 'He-A contained between two parallel planes with superflow parallel to the planes might in certain circumstances lead to a state with superflow varying periodically in space, thereby resembling the convection patterns developed beyond the Rayleigh-Bernard instability in a normal liquid.

3.6. THEEFFECT OF TEXTURAL

SINGULARITIES

The possible role of surface singularities in removing the coreless vorticity generated by driven superflow has already been mentioned in section 3.5. In this section we will concentrate most of our attention on the type of point surface singularity that has become known as a boojum (Mermin, 1981). These singularities may play an important role in superflow dissipation in 'He-A and their properties have been widely discussed. The boojum first turned up in the proposed texture of fig. 17a, which is likely to have the lowest energy for a spherical container. A singularity free texture is not possible in a simply connected geometry (Mermin, 1977) and the texture in fig. 17a has a point singularity on the surface of the sphere at which the direction of i is reversed. Fig. 1% shows that a boojum also arises when the Anderson and Toulouse coreless vortex of fig. 14 meets a plane surface. The point singularity o n the upper surface is a hyperbolic boojum, that on the lower

Ch. 2, $31

HYDRODYNAMICS OF SUPERFLUID 'He

197

Fig. 17. (a) Boojum texture in a sphere. (b) Boojums arising where an Anderson-Toulouse vortex meets a solid surface.

surface is a circular boojum (Mermin, 1977). The circulation around a contour of large radius surrounding the coreless vortex is two quanta as explained in section 3.5.If the contour is translated vertically to either the upper or lower surface and then shrunk to a vanishingly small radius surrounding the boojum, then the texture on all the contours is represented by a single point on the unit sphere at the south pole. According to eq. (71) therefore the circulation around a contour of small radius surrounding a boojum is two quanta. This property of boojums explains their importance in the discussion of superflow decay. The circuIation for a surface contour is constant in the absence of singularities but will change by two quanta every time the contour is crossed by a boojum. The texture shown in fig. 18a for a torus was proposed by Mermin (1977) and contains two boojums. Motion of these as shown in the inset of the figure will cause superflow around the torus to decay. In the case of driven superflow the motion of boojums over the surface provides a mechanism for creating a finite chemical potential gradient on the surface at fixed superfluid velocity. The texture on a path along the centre of the channel in fig. 18a looks like the 0-7r-O texture of Hook and Hall (see section 3.4). We see that a 0-T domain wall together with a boojum serves the purpose of matching two oppositely directed Mermin-Ho textures in a flow channel. The

I58

H.E. HALL AND J.R.HOOK

[Ch. 2, $3

Fig. 18. (a) Two boojum texture in a torus. The boojum at the top is a circular one, that at the bottom has been called a “semi-hyperbolic” one by Mermin (1977) because it looks like a circular boojum in the plane perpendicular to the paper. The inset shows the motion of the boojurns necessary to cause superflow decay. (b) Semi-hyperbolic boojum and 0-n domain wall in a tube with a large superflow.

stability of the 0-r-0 texture for sufficiently large flows suggests that the two boojums i n fig. 18a may not annihilate each other. Indeed, the discussion of section 3.4 suggests that a large superflow may lead to the creation of more domain walls with their associated boojums. It is interesting to discuss briefly the dynamics of a 0-n- domain wall/boojum combination in a tube. Consider just the semi-hyperbolic boojum at the bottom of fig. 18a. Provided that t h e domain wall is well separated from another wall, the energy of this texture is invariant under

Ch. 2, 831

HYDRODYNAMICS OF SUPERFLUID ’He

199

rotation. The critical flow velocity required to make the wall precess and thereby cause dissipation is therefore zero. For a small flow velocity the bending of the texture will be on a length scale d and we may expect the terms linear in I), in the gradient free energy to dominate the motion of i. Dimensionally the equation of motion of I is then

leading to a precession frequency for the wall of order

By Comparison with eq. (%) we see that the effect of lateral confinement on a domain wall is similar to that of longitudinal confinement. At large flow velocities the texture will look like that in fig. 18b and it is likely that the precession rate will be that for an unconfined domain wall [eq. (95)). We have presupposed that the domain wall and boojum remain locked together in the precession. The motion of a boojum in the presence of superflow has not yet been calculated. In a real tube it is possible that the boojum may be pinned by the surface geometry. Should the domain wall and boojum become unlocked then the more complicated stationary surface singularities described by Ho (1978b) will have to be formed to remove the excess coreless vorticity generated by the precessing domain wall. It is possible that the “super-oscillations” observed by Krusius et al. (1978) were the result of the motion of textures near to the surface. The above discussion is appropriate to the limit d %- tD.In the small channel limit, as already mentioned in the previous section, the texture in the absence of flow is likely to be the radial disgyration of fig. 15. Thuneberg and Kurkijarvi (1981) have proposed that at the ends of the flow channel the singularity might become a cored vortex (fig. 19). Rotation of the vortices about the channel axis would then lead to superflow dissipation. Pinning of the vortices by surface irregularities of size D would lead to a critical current for superflow of

where R is the channel radius and &(T) the temperature-dependent coherence length. For R = 1 pm, d = 100nm, & =30 nm this equation gives u, = 6.3 mm s-’.

H.E. HALL A N D J.R. HOOK

[Ch. 2, I4

Fig. 19. Radial disgyration in a narrow channel. The disgyration is assumed to become a cored vortex ar the ends. Rotation of the vortices about the channel axis causes superflow dissipation. m u n e b e r g and Kurkijarvi, 1981.)

4. Superflow in 'He-A and 'He-B

The existeoce of a superfluid fraction in both phases is clearly demonstrated by their ability to propagate a fourth sound mode and by the existence of an irrotational component in torsional oscillator measurements. The extent and nature of the dissipation in the flow of this fraction are matters of great experimental and theoretical interest. There are considerable experimental problems in making meaningful measurements, in particular the need to ensure that the observed dissipation is associated with pure superflow in a well located and defined geometry. Failure to overcome such problems in some earlier experiments led Einstein and Packard (1982) to suggest that dissipation intrinsic to the superfluid component was present in 'He-B even at the lowest flow velocities, a suggestion that led directly to a joint Berkeley-Helsinki experiment to search for persistent currents in the rotating cryostat in Helsinki. The subsequent observation of persistent currents in 3He-B at Cornell (Gammel et al., 1984) and Helsinki (Pekola et al., 1984a) convincingly demonstrated that dissipationless flow of 3He-B is possible at low velocities. Persistent currents have not yet been directly observed in the A phase and it can be seen from our discussion in the previous section that we might expect the A phase to behave quite differently. For this reason we discuss the two phases separately in this section. We will attempt to fit the many different observations for both phases i n t o a coherent but as yet incomplete picture and to investigate t h e extent to which existing theory is able to account for this picture. It is perhaps worth remembering that

Table 2 Details of flow experiments Experiment referred to in text as:

References

Pressure range (phases studied)

Geometric and other details

Pekola et al. (1984a) Pekola et al. (1984b) Pekola and Simola (1984) Gammel et al. (1984) Gammel and Reppy (1984)

0-29.3 bar (A and B phases)

3 m m radius toroidal flow channel contained 20 pm plastic powder-packing fraction 13%

15 and 29 bar (A and B phases)

Cornell torsional oscillator

Parpia and Reppy (1979) Crooker et al. (1981) Crooker (1983)

Helsinki diaphragm driven flow

Manninen and Pekola (1982) Manninen and Pekola (1983)

20-28 bar (18 pm orifice-A and B phases; others B phase only) C27.4 bar (A and B phases)

rectangular cross section flow channel contained 100 pm (9.5 Fm) particles with 15% (30%)packing fraction in earlier (later) experiments t h e three toroidal flow channels studied contained orifices of diameter 18 pm, 5 pm and 2 pm

sussex

Dahm et al. (1980) Hutchins (1981) Hutchins et al. (1981a) Hutchins et al. (1981b) Ling (1984) Ling et al. (1984) Brewer (1983) Eisenstein et al. (1980) Eisenstein (1980) Eisenstein and Packard (1982) Gay et al. (1981) Gay et al. (1983)

Helsinki persistent current Cornell persistent current

Berkeley

Manchester

Bell Laboratories

Paalanen and Osheroff (1980a, b)

4.9-33 bar (A and B phases)

0 bar (B phase) 29.3 bar (A Phase) 29.5 bar (A phase)

flow channels of length 10 pm, diameter 0.8 pm in Nuclepore and 1.5 X mm2 filter. Total channel area 6.4 X for B and A phase measurements respectively diaphragm driven flow through channel of length 9.16 m m and rectangular cross section 2.86 mm x 48 pm

0 bar U-tube flow through four circular channels of radii 102, 126,177, and 227 pm and lengths 5.02,5.02,10.04and 10.04m m respectively toroidal flow channels of rectangular cross section. First cell had 25 channels 49 pm X 0.76 mm. Second cell had 75 channels 17 pm x 0.76 mm. flow channel of rectangular cross section 5 mm X 0.5 mm length 10mm

21?2

[Ch. 2, 84

H.E. HALL AND J.R. HOOK

the problem of superflow dissipation in superfluid 4He is still not fully understood despite over forty years of study. We will rely heavily on some of the ideas that have been proposed for 4He in discussing t h e 'He problem, particularly when describing the B phase results. To avoid unnecessary repetition we give in table 2 a list of the experiments that have been performed together with references and important details. Throughout section 4 we shall refer to the experiments by the title given in the first column of the table and leave the reader to consult the table or the original references for further details.

4.1. PERSISTENT CURRENT EXPERIMENTS

Both the Cornell and Helsinki persistent current experiments work on t h e same principle. A toroidal Row channel is mounted horizontally in such a way that it is capable of torsional oscillations about two mutually perpendicular axes in its plane - the 8 and axes in fig. 20. We denote the resonant frequencies of the two modes by w,, and OJ+. A persistent angular momentum L of t h e fluid in the channel produces a coupling between the modes which can be detected by driving the 8 oscillation at frequency u, and measuring the resulting amplitude of oscillation about the axis. The torque about the 4 axis due to L is io,OL where 19= fI,exp(io,r) is the angular displacement about the 8 axis. The resulting angular displacement 4, exp(iw,r) about the 4 axis has an amplitude

+

+

where I, is t h e moment of inertia about the 4 axis and Q, is the quality

Fig. 20. Schematic diagram of geometry of persistent current experiments.

Ch. 2, 841

HYDRODYNAMICS OF SUPERFLUID 'He

203

factor of the 4 resonance. The angular displacement is detected capacitively and calibration of the apparatus is achieved by measuring the angular displacement about the 4 axis &exp(io,t) generated by the Coriolis force when the whole apparatus is rotated at angular velocity R G bOcand a simple calculation yields about a vertical axis. Usually

where I, is the moment of inertia about the vertical axis. Thus if bOcand 4oLare measured for the same 8, then from eqs. (103) and (104)

and L is determined in terms of readily measurable quantities. An alternative method of determining L would be to measure the amplitude of vibration about the 4 axis caused by driving the 8 oscillation at 0,. Both methods were used successfully at Cornell although the latter has the disadvantage that the total motion of the cell is much bigger. The design of the Cornell experiment was such that the splitting w, - wowas small, thus increasing the value of 0, for a given drive and hence enhancing the sensitivity. In the Helsinki experiment u4 and u8were very different; this gives a lower sensitivity but is less subject to complications arising from changes in the normal modes of vibration. The first observation of a persistent current was by the Cornell group but the Helsinki observations are in some respects simpler and more detailed and we shall therefore discuss these first. The torus contained 2 0 k m powder to a packing fraction of about 13%, which locked the normal fluid to the motion of the channel. The presence of t h e powder required that, even for pure potential superflow, the superfluid was dragged along by the channel to an extent describable by a dragging factor

where GS is the mean superfluid velocity. The experimental procedure was to measure t h e amplitude bar(+) with the cryostat at rest following rotation at a preparation angular velocity (Ip for a period of about 1 min, and then to measure +oL(-) with the cryostat at rest following a 1 min rotation at LtPin the opposite sense. The difference

A 4 L = $&(+I-

&(-)I

(107)

204

H.E. HALL AND J.R. HOOK

[Ch. 2, 44

was used in eq. (105) to deduce a value for L. The use of a differential method overcame the problem of uncertainty in t h e value of & corresponding to L = 0. For small values of Op.n o persistent currents were observed suggesting irrotational superflow throughout t h e above measurement procedure. Above a critical value 0:of a,, the frozen-in angular momentum L was proportional to - Op) and for Op> 2O,, L saturated at a value L,. These results are consistent with the simple idea that there is a critical velocity IF,- u,J = u, which cannot be exceeded but that at subcritical velocities the superfluid behaves in an apparently irrotational fashion. Since the saturation angular momentum L, and the critical rotation rate fl, correspond to flow at this relative velocity, there should be a relationship between these quantities of t h e form

(a,

where R is the mean radius of the torus and M t h e total mass of fluid within it. This equation was used to establish an experimental value for x. There was n o perceptible decrease in t h e frozen-in angular momentum for periods up to 4.8h, implying an effective viscosity for superflow at least 13 orders of magnitude smaller than the normal liquid value at the same temperature. When the temperature was changed t h e value o f L varied in proportion t o p,/p indicating a temperature independent superfluid velocity as might be expected. We postpone our main discussion of t h e critical velocity u, until the following section but we note here the existence at certain pressures of discontinuities in c, as a function of temperature. The locus in the p, T plane of these discontinuities is shown in fig. 21. Also shown is a line indicating points at which a discontinuity occurred in NMR frequency in experiments on rotating 3He in a simply connected container [Pekola et A. (1984b). see also section 5.21. This behaviour has been interpreted as being due to a change in vortex core structure. In regions I and 111 of fig. 21 t h e core structure is assumed to be t h e same and u, is independent of magnetic field. In region I 1 u, increased strongly with increasing field. The -'vortex core" transition is first order with a finite latent heat, t h e magnitude of which we discuss further in section 5. In t h e Cornell experiments, persistent currents were also generated by rotation at a preparation angular velocity 0,. As in t h e Helsinki experiments t h e observed L saturated for large values of OP.In earlier experiments in which the flow channel was filled with 100 p m particles a finite L wiis observed for the smallest f2, used whereas in later experiments in

Ch. 2, 44)

HYDRODYNAMICS O F SUPERFLUID 3He

LO

I

I

205

I

SOL1D

30

L

:: 20

Y

a 10

0

1.5

1

2 T (mK)

2.5

3

Fig. 21. Transitions observed on the phase diagram of 'He: in NMR experiments on rotating 'He-B (open circles); in persistent current measurements at H = 0 (solid circles) and H = 40 G (crosses). (Pekola et al., 1984b.)

which 10 pm particles were used there was a critical rotation rate for the appearance of a finite L. This difference probably arises because the critical velocity is higher in the small pores for the reasons discussed in the following section.

4.2. CRITICAL VELOCITY

FOR THE ONSET OF DISSIPATION IN

3He-B

The persistent current experiments discussed in the previous section demonstrate that dissipation in superflow of 3He-B at low velocities is vanishingly small. In the absence of a coupling between the centre of mass and internal motion of the Cooper pair in 3 He-B, the onset of dissipation is usually interpreted as arising from the motion of singular vortex lines as in superfluid 4He. By calculating the flow velocity at which it becomes possible for vortex rings to be created in a state of pure superflow with conservation of energy and momentum one arrives at a critical velocity of the form (Feynman, 1955) ah

/

d

\

206

H.E. HALL AND J.R. HOOK

[Ch. 2, 84

where a and p are dimensionless numbers of order unity, d is the size of t h e flow channel and (( T ) the temperature-dependent coherence length that measures the size of the vortex core. For a flow channel of complex geometry, d would be expected to be the smallest dimension, but a and p are likely to depend on the geometry. Vinen (1957) proposed equations to describe the behaviour of a vortex tangle in superfluid 4He. These enable calculation of the total length L of t h e vortex line per unit volume and hence of t h e mutual friction force per unit volume betwcen normal and superfluid

Here H is a temperature-dependent dimensionless coefficient of order unity. Although Vinen's theory was introduced initially for an infinite volume and contained no provision for a critical velocity, one was subsequently introduced by the device of excluding vortex production processes within a distance of t h e ordcr of t h c interline spacing L I" from the channel walls.The vortex tangle is then only self-sustaining if L"'d > 2, i.e. if the interline spacing is less than the channel sizc. The corresponding critical velocity is

where cr is likely to depend o n temperature and geometry. At u,, L rises discontinuously to a value close to the equilibrium value for an infinite medium. Despite t h e differences in derivation the similarity in form of eqs. (109) and (11 1) is striking. Many of the features of the Vinen theory have been reproduced using more rigorous arguments by Schwarz (1978). Further discussion of these theories and their application to 'He can be found in t h e review article of Tough (1982). We merely note here for future reference that Schwarz predicts that the value of L to be inserted in eq. (110) to calculate t h e mutual friction force for infinite homogeneous turbulence is given approximately by

where a ( 'I' i h) a dimensionless temperature-dependent coefiicient, u = I D , u , ~and r,, 1 cm s-'. We suspcct that q,may well be an artefact of the numerical methods employed by Schwarz t o solve his equations. since 5

Ch. 2, 041

HYDRODYNAMICS OF SUPERFLUID ’He

207

the application of dimensional arguments to the equations themselves appears to rule out the appearance of a characteristic velocity in a system without a fixed length scale. Another critical velocity which is relevant in 4He is that calculated by Langer and Fisher (1967) for the production of vortex rings by thermal excitation

c(k) 3

”‘

P S

167rkBTIn(hAvJ2m@,) ’

where C is a number of order unity, @, is the minimum observable value of du,/dt in a particular experiment, A the area of the flow channel and vo a characteristic frequency for processes per unit volume on an atomic scale. Although Soda and Arai (1981) have suggested that this could also be relevant in 3He we calculate it to be lO(1- VT,) m s-l, which exceeds the depairing critical velocity (see following section). In the Helsinki persistent current experiments, as shown in fig. 22, u, was almost temperature independent for T < 0.9Tc for the low pressure phase of fig. 21. The weak pressure dependence was such that u, varied from 4.5hlmd at 3 bar to 6.7hlmd at 23 bar, for d = 20 Fm. This pressure dependence is consistent with

1.9h md

u, = -ln(d/19~,),

(113j

where 6, = huF/(dcBTc)is the zero temperature weak-coupling coherence length. Eq. (113) is just eq. (109) with {(T) replaced by 6,. It is not clear why eq. (109) should give the pressure dependence correctly but predict a temperature dependence which is much stronger than that observed. For T > O.9Tc, u, dropped towards zero. The temperature-dependent u, close to T, seems too small to be associated with depairing and at present this feature is not understood. The effect of flow channel size on u, has been investigated in the Cornell torsional oscillator experiments. The toroidal flow channel contained an aperture of small diameter. Above a critical amplitude of oscillation 0, the quality factor of the oscillator decreased and t h e period began to increase, indicating an increase in dissipation associated with a greater fraction of the fluid mass becoming coupled to the oscillator. From 0, it is possible to deduce a u, for flow through t h e aperture. This is not entirely straightforward since, even for a perfectly irrotational superfluid, a fraction of the superfluid mass is coupled to the oscillator by the presence of the constriction. Accurate determination of this fraction is

208

[Ch. 2, 84

H.E. HALL A N D J.R. HOOK

lo> vc cms'

i

3

O

0

.

0

0.4

1

L

0.6

D

0.8

1.o

Fig. 22. Critical velocity for the onset of dissipation for the B phase in: Cornell torsional oscillator experiments, V 2 k m orifice, 0 5 p n orifice, A 18 pm orifice; Helsinki persistent current experiments at 12 bar; !JCornell persistent current experiments at 29 bar; 0 earlier Sussex experiments at 29.5 bar; 0 later Sussex experiments at 16.5 bar; x Berkeley experiments in 126pm and 2 2 7 ~ mtubes. Also shown are critical velocities for the A phase from: A Cornell torsional oscdlator 18 Frn orifice: Bell Laboratories experiments -dashed line. Solid lines indicate (1 - T/Tc)IRdependence.

difficult because of the effect of superfluid compressibility (Carless et al., 1983). Values of u, calculated by ignoring this effect are shown in fig. 22. L',tends to decrease near T, but assumes a more constant value at lower temperatures. Because t h e compressibility effect is larger at higher temperatures w e consider only the low temperature value of u,. For the 18 pm, 5 p n and 2 bm channels we take u, to be 0.5, 1.7 and 3.8 cm s-' respectively. These are given to within about 10% by u, = 4fi/md,a result very similar to that from the Helsinki persistent current experiment. We note that although the Cornell torsional oscillator measurements were made at pressures between 20.5 and 22.8 bar there was n o indication in them of the phase transition shown in fig. 21. The critical velocity observed in the earlier Cornell persistent current experiment is shown in fig. 22. This was also of order 4h/md (with d = 100 pm) at t h e lowest temperatures at 29 bar. There appears to be

Ch. 2, $41

HYDRODYNAMICS OF SUPERFLUID 3He

209

more temperature dependence of u, than in the Helsinki persistent current experiments, but the large experimental scatter prevents any firm conclusions concerning this. The data would be consistent with a temperature dependence as strong as 0, a (1- TIT,). In t h e later Cornell persistent current experiments with 9.5 pm powder in the flow channel a temperature-independent critical velocity of 3 mm sfl was observed. This is 1.4hlmd for d = 9.5 pm. The 100 pm geometry of the earlier Cornell persistent current experiments is comparable in size to the flow channel in the Berkeley and Sussex experiments. In t h e earlier Sussex experiments the oscillatory flow following a step function change in the equilibrium position of the diaphragm was studied. At 29.6 bar the quality factor of the oscillation increased markedly below an almost temperature independent velocity of 0.6rnm s-’ as shown in fig. 22. This is equal to 1.4hlrnd for d = 48 pm, the smallest dimension of the flow channel. This critical velocity was not observed in later Sussex experiments in which the electrostatic force on the diaphragm was ramped linearly in time, i.e. V2 a r where V is the applied voltage, although this may be because of the lower sensitivity of the later experiments. For small ramp rates the position of the diaphragm at any instant was undetectably different from that it would have occupied had the voltage been constant at its instantaneous value, indicating flow of the liquid with a small, or perhaps zero, pressure difference. At larger ramp rates for low pressures, the diaphragm lagged behind its “equilibrium” position but usually with a capacitance C still changing linearly with time. Discussions with the Sussex group have convinced us that their published current densities for dissipative flow are incorrect because proper allowance was not made for the fact that the electrostatic force acted only over approximately one quarter of the diaphragm. For dissipationless flow the Sussex group correctly used the relationship between equilibrium capacitance C, and voltage dC,/dt

=

E

d( V2)/dt

(114)

and calculated t h e current from

where E and y are geometrical constants of t h e apparatus. But they also calculated currents from eq. (115) when a pressure gradient was present, whereas the correct equation in this case is

210

[Ch. 2, §4

H.E. HALL AND J.R.HOOK

which reduces to eq. (115) only if the geometric constant cy = 1. From the geometry of the Sussex diaphragm we estimate a = 0.81. Eq. (116) predicts that dC/dr varies with ramp rate even when j remains constant. Fig. 23 shows a plot of the constant value of dCIdr during a ramp against the ramp rate for a series of results at TIT, = 0.983 at 15.5 bar. The dashed line has been drawn through the “dissipationless” region at small dV2/df and thus represents eq. (114). The dotted lines have been drawn with a slope smaller than that of the dashed line by a factor equal to our estimated value of 1 - a, and points parallel to this line would, according to eq. (116) represent flow at t h e same current density. The points for large dV*/dt are approximately on the upper dotted line and the corresponding “saturation” current density is obtained by applying the Sussex recipe, eq. (115), at the intercept P, of the upper dotted line with the extrapolation of the dissipationless line. The “saturation” current is therefore only slightly higher than the “dissipation onset” current j,, calculated by applying eq. (115) at the point P, at which dissipation first occurs. The points indicated by squares on fig. 23 show the value of dCldr after

4,

‘I

K ( dt

I

I

0

/

j

1

I

b 0

I

I

31I

O 0

oi.

/

0.5

d

1.0

0

J

1.5

2.0

dt Fig. 23. The circles show the constant rate of change of capacitance C observed during a ramp of the voltage V on the diaphragm as a function of the ramp rate for the later Sussex experiments. The squares show the value ofdCldT after the ramp had finished. See text for explanation of the dashed, dotted and continuous curves.

Ch. 2, 041

HYDRODYNAMICS OF SUPERFLUID 'He

211

t h e ramp had finished. The intersection on the dC/dt axis of the continuous line through these points with the dotted line through PI indicates that these points represent flow at a rate close to j c l . There is thus evidence for hysteresis in the flow in that currents greater than jcl could be obtained while the driving force was increasing whereas the flow dropped to jclfor decreasing driving force. We will discuss only the values of jclpublished by the Sussex group since their original analysis is correct for currents up to j c l . At low pressures the measured jcl varied with temperature as (1T/TJUzindicating a u, of the form A ( l - T/Tc)IRas shown in fig. 22 for a pressure of 16.5 bar. A increased from about 2 cm s-l at 4.9 bar to about 6 cm s-l at 20 bar. In the 0 bar Berkeley experiment, as shown in fig. 22, a tube-size independent u, of order 1.1(1- T/Tc)*ncms-l was observed at which the dissipation increased and became non-linear (force no longer proportional to velocity). It can be seen that this value is consistent with the Sussex observations. At the polycritical pressure the temperature dependence of jc, in the Sussex experiments changed abruptly to (17',/Tc)2indicating a critical velocity of the form A ( l - T/Tc) with A approximately 10 cm s-'. No reason for this abrupt change in behaviour has been advanced although it is tempting to associate it with the phase transition observed in t h e Helsinki persistent current experiments (fig. 21). There is experimental evidence therefore in all but the Berkeley experiments for a critical velocity of order ahlrnd with d the minimum dimension of the flow channel and a a number of order unity. There is evidence from the Helsinki persistent current experiments that the pressure dependence (but not the temperature dependence) of a is given correctly by eq. (109). There have not yet been a sufficient number of experiments in well characterized geometries for the geometry dependence of a to be determined. The failure to observe a critical velocity of order hlrnd in the Berkeley and later Sussex experiments is probably due to lack of sensitivity in these experiments; the higher critical velocity that was seen in these experiments could indicate a transition to a vortex state in which the dissipation increases more rapidly with flow velocity. Such transitions have been observed in flow of superfluid 4He (Tough, 1982). We have not discussed the Helsinki diaphragm driven flow experiments in this section since, for the small size (0.8 pm) of flow channel involved, a critical velocity of order hlrnd would exceed the depairing velocity. These experiments are therefore discussed in t h e following section.

4.3. HIGHER CRITICAL VELOCITIES

AND THE DEPAIRING CRITICAL CURRENT

In the Berkeley and the earlier Sussex experiments t h e mass current

212

H.E. HALL AND J.R.HOOK

[Ch. 2, 84

density was observed to saturate at the largest pressure differences. These saturation currents varied with temperature as j , = A (1 - T/T,)’”

(117)

as did the only critical current observed in the Helsinki diaphragm driven flow experiments. The values of A for all these currents are shown in fig. 24 as functions of pressure. Because of their similarity in magnitude and temperature dependence, these currents have often been identified with each other. The temperature dependence has led many authors to propose an explanation of the currents in terms of the depairing critical current. Before discussing

Fig. 24. Values of the prefactor A in j c = A ( l - T/T<)”*for: V, the Berkeley saturation the saturation current j,2 in the earlier current j c 2 (average value for three larger tubes); 0. Sussex experiments; 0. the dissipation on.set current in the Helsinki experiments. The continuous theoretical curve is the weak-coupling depairing current: the dashed curve is obtained hy multiplying the weak-coupling result by the 0.48 factor appropriate to the restricted geometry of the Helsinki experiments.

Ch. 2, 541

HYDRODYNAMICS OF SUPERFLUID ’He

213

the correctness of this identification, we give a brief description of the theory of the depairing current. There is a theoretical upper limit to the superfluid mass current density in a Fermi superfluid, which arises when the kinetic energy of the flow becomes comparable to the condensation energy. It then becomes energetically favourable for a transition to the normal state to occur and this phenomenon is known as depairing. The transition is most easily calculated for a simple S-wave superfluid in the Ginzburg-Landau regime. The free energy density when a spatially uniform superfluid mass current density j, is imposed on the fluid may be written [de Gennes (1966) p. 1821

where we have assumed that p, is spatially constant and denoted by pd the equilibrium value of p, in the absence of flow.The first term in g is the condensation energy, the second the kinetic energy of superflow and the final term gives the correct thermodynamic potential for a situation where j , rather than u,(=j,/p,) is an independent variable. Minirnising g with respect to p, gives t h e equilibrium condition j , = (-2a)lnp’paf(1-f)”’,

(119)

where f = p,/pd. Fig. 25 shows a plot of j , against f from this equation and

7 213

13

f Fig. 25. Supercurrent versus reduced superfluid density, f = p/p&, for an S-wave superfluid in the Ginzburg-Landau limit. The symbols are explained in the text.

214

H.E. HALL AND J.R. HOOK

[Ch. 2, 34

it can be seen that the maximum value of j , is

which arises for f = $ and us= (-2a/3)ln. Solutions of eq. (119) with f < are unstable since dzg/dpi < 0. The prediction therefore is that, as j , is increased from zero, p, falls continuously according to eq. (119) until f = at which point a first order transition to the normal state occurs. For a P-wave superfluid the calculation of j, is more difficult because of the complexity of the order parameter, the distortion of which by the flow is, not unexpectedly, anisotropic. The first calculation for 'He-B was made by Fetter (1975) using a generalized Ginzburg-Landau free energy approach. Subsequently Vollhardt et a]. (1980) and Kleinert (1980) have given calculations for the weak-coupling limit that are valid at all temperatures. The calculations include Fermi liquid corrections which affect the values of u, and p, at the transition, although t h e current psu, is unaffected. Above TIT, = 0.6, j, is well represented by (1 - T/Tc)3'2 .

The theoretical prediction therefore is that the pressure difference should jump discontinuously at the transition from zero to the value required to drive the normal fluid at this critical flow rate. The coefficient of ( 1 T/Tc)3nin eq. (121) is plotted on fig. 24 and it can be seen that the measured values are smaller by amounts which exceed the experimental errors. It is unlikely that strong-coupling corrections can account for this discrepancy. A trivial strong-coupling coirection (Serene and Rainer, 1978) can be made by multiplying j , in eq. (121) by AC,,,,,/AC,, where JC' is the specific heat discontinuity at T,. This procedure increases t h e theoretical depairing current if the measurements of AC of Alvesalo et a]. (1981) are used. Non-trivial corrections to j , have not yet been calculated in entirety. Fetter's (1975) calculation includes strong-coupling corrections to the condensation energy but uses weak-coupling expressions for the current density and the corresponding gradient terms in the energy. If the strong-coupling parameters of Sauls and Serene (1981) are used in this calculation, the predicted corrections are small. Fetter's theory predicts that t h e fluid makes a transition to the A phase before the depairing current is reached; the critical current for the transition is however close to the depairing current except near the polycritical point where it goes rapidly to zero.

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HYDRODYNAMICS OF SUPERFLUID 'He

215

It is likely therefore that the discrepancy between experiment and theory arises for some other reason and we investigate this possibility by examining the extent to which the assumptions made in the theoretical calculation are valid and consistent with the experimental observations. An important assumption is that in the presence of a current density the order parameter remains time and space independent (apart from the phase gradient associated with the superflow). It is also assumed that the order parameter remains B-phase like -more specifically, that it is of the form

A ( k ) = eiy[A,(k h)d+ A,(k.A)i?

+ A,,(&-i)f],

for Row in the i direction [cf. eqs. (43) and (&)I. The anisotropic gap distortion is accounted for by the different amplitudes A, and A,. With these restrictions the depairing calculation finds a local free energy minimum but there is no guarantee either that it is an absolute minimum or that space and time dependent perturbations of the order parameter will not occur. Fetter's demonstration that a transition to the A phase may occur below the B phase depairing current is an example of a possible failure of the assumptions. More importantly, the existence of dissipation below the upper critical currents in both the Berkeley and Sussex experiments indicates that space and time dependent states were present below these currents. In none of the experiments was it possible to observe a transition to t h e normal state since it was impossible to apply the enormous pressure gradient required to cause the normal fluid to flow at the critical rate. Instead in the Helsinki diaphragm driven flow experiments, currents in excess of the critical values were produced by applying larger pressure gradients; quite contrary to the behaviour expected from the depairing theory. The increased current was much too large to be explained by normal flow in the finite pressure gradient. It is possible that the apparent saturation of current in the Berkeley and earlier Sussex experiments arose because the dissipation increased so rapidly with flow that increases in pressure difference produced no measurable increase in current. Such behaviour might be associated with a change in t h e nature of t h e vortex state similar to those observed in superfluid 4He (Tough, 1982). The critical current in the Helsinki experiment is more likely to be associated with depairing than the others since here an explanation of the discrepancy with the theoretical value of eq. (121) is available. K.W. Jacobsen and H. Smith (unpublished) predicted that the very small size of t h e flow channels in this experiment should reduce the depairing current by a factor 0.48. Using this factor gives the dashed curve o n fig. 24 which fits the Helsinki data well.

216

[Ch. 2, 84

H.E. HALL A N D J.R. HOOK

4.4. MAGNITUDE OF FLOW DISSIPATION IN 3He-B In the Berkeley and earlier Sussex experiments dissipation was observed down to the lowest velocities studied. The suggestion by Eisenstein and Packard (1982) that t h e dissipation was intrinsic to the superfluid component was refuted by the subsequent observation of persistent currents (see section 4.1). Brand and Cross (1982) have proposed a mechanism for the dissipation observed at low flow velocities in t h e Berkeley experiments. Although only superfluid passes through the flow channel, it is the total liquid density p which flows at the free surface in the reservoirs that form t h e two arms of the U-tube. Brand and Cross use two-fluid equations, together with the boundary conditions at the free surface that u, = us= velocity of surface, to show that the region where conversion to pure superflow takes place is within a distance of order ( p 2 ( J ~ c l l ) ’ nfrom d the surface; here d is the width of t h e annular region that forms the reservoirs, l3is the second viscosity and qc,, an effective first viscosity defined by Brand and Cross. The pressure difference driving t h e superfluid through the tube is obtained by subtracting the pressure difference 2p13(qell)”xld across the two conversion regions (one in each reservoir) from the driving pressure head 2pgx, where 2 x is the difference in level between the two reservoirs. The equation of motion is therefore x

+ 2LX + w 2 x = 0 ,

(123)

where ( 124a)

and W’

(124b)

= 2p,galplA,

A is the area of cross section of the annulus and 1 and a are the length and cross-sectional area of the flow channel. Fig. 26 shows that theoretical values of L (continuous curves) are larger than experimental values by a factor less than two - reasonable agreement bearing in mind the degree of approximation in the theoretical calculation. We have used theoretical values of (Wolfle, 1978) and experimental values of JI (Carless et al., 1983) in the calculations. We note that the agreement, particularly at low temperature, is significantly

c3

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HYDRODYNAMICS OF SUPERFLUID 3He

217

Fig. 26. Values of the damping coefficient L in the Berkeley experiments for the 102 krn 0, 126 Bm 0, 177 pm 0, and 227 pm A tubes. The continuous curves are obtained from eq. (124a) for the tubes shown. The dashed curve is for the 102 pm tube and allows for finite mean free path effects by replacing d in eq. (124a) by d + 24.

improved if allowance is made for normal fluid slip within the reservoirs. If we use an effective value of d in eq. (124a) of d + 21, where 1, is the viscous mean free path, then we obtain the dashed curve of fig. 26 for the 102 pm diameter tube. Wolfle (private communication) has pointed out however that the mean free path at the lowest temperatures is too long for such a first order slip correlation to be strictly valid. In an attempt to see whether there is a similar explanation for the dissipation observed at low flow velocities in the earlier Sussex experiments, we have estimated the dissipation occumhg in a gap of mean width d between a moving diaphragm and a parallel stationary surface. Solving the two fluid equations for this situation gives a Q factor for the Sussex experiments of order

where 1 and a are the length and area of the flow channel and A is the area of the diaphragm. The observed Q’s are much smaller than those predicted by eq. (125) for T near T, and they increase strongly at low T whereas eq. (125) predicts no strong T dependence. The observed Q factors depended on whether the cell was warming or cooling, implying

218

[Ch. 2, 44

H.E. HALL AND J.R. HOOK

that remanant vorticity provides a more likely explanation than the dissipation represented by eq. (125). As far as we are aware there has been little or no attempt to explain the dissipation Occurring above the critical current discussed in section 4.2. If t h e dissipation is due to a vortex tangle then we might expect the resulting pressure gradient to be given by the mutual friction force of eq. (110). The calculation of the effect of such a pressure gradient is complicated and is different for different experiments. We feel that such calculations would be worthwhile, but merely note here that eq. (110) with a value for L given by eq. (112) gives a pressure difference of the correct order of magnitude to explain the dissipation observed in both the Berkeley and the later Sussex experiments. The predicted pressure gradient for the Helsinki diaphragm experiments is however much too small. This lends further weight to the interpretation of these experiments in terms of depairing critical currents. 4.5. THEPOSSIBILITY OF DISSIPATIONLESS FLOW

IN

'He-A

In addition to dissipation from motion of singular vortices and depairing we expect in 'He-A to find dissipation due to orbital motion, a possibility we have discussed in some detail in section 3 and which includes dissipation resulting from motion of coreless vortices. In a large open volume of liquid the (meta) stability of the i ] l v , texture for velocities up to 1 mm s-' [eq. (85)] could prevent Occurrence of orbital dissipation at smaller velocities. In an open volume however the critical velocity for dissipation due to singular vortices may be lower than this (see discussion for 'He-B in section 4.2). Persistent currents were not detected for the A phase in either the Cornell or the Helsinki experiments, t h e limit of resolution in both cases being about 1 mm s-I. The tortuous geometry of these experiments implies the existence of many order parameter singularities and in these circumstances the stability of the illu, texture in an open volume may not be relevant to the existence or otherwise of persistent currents. Indirect evidence for persistent currents in the A phase was obtained in some as yet unpublished experiments of Hook, Crooker and Zimmermann in which the Cornell rotating cryostat was used in an attempt to create persistent currents in the toroidal flow channel (cross section 1 mm x 5 mm) of a torsional oscillator. The intention was to detect t h e current through its effect o n t h e i texture within t h e channel and hence on t h e damping and period of the oscillator because of the anisotropy of the normal fluid density and viscosity. The interpretation of the results is not yet completely clear but, at 29.34bar and for T/Tc<0.9, rotation

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219

rates equivalent to flow velocities less than 0.2 mm s-l produced textural changes which persisted for periods of five minutes, the longest period for which they were studied. Flows at larger velocities (up to about 1mm s-') appeared to decay on a time scale of order tens of seconds. Truscott (1979) has proposed that the difference between the two measurements of NMR line shifts in 2 b m capillaries of Gould and Lee (1978) and Saunders et al. (1978) might be due to the existence of persistent currents in the latter experiments. The flow velocity required to explain the discrepancy would however need to be close to the depairing velocity.

4.6. A SIMPLE MODEL OF ORBITAL DISSIPATION IN 3He-A Since an exact treatment, which would involve solving coupled equations for i a n d 0, in three dimensions, is likely to prove intractable we will present in this section a simple model that can be used to describe flow dissipation through orbital motion. Although this was initially proposed by Gay et al. (1983) to account for the results obtained in the Manchester experiments, we will see in sections 4.7 and 4.8 that with appropriate generalizations it can be applied to describe with some success the other A phase experiments that have been performed. The starting point is the picture of Ho (1978a) of superflow collapse in a toroidal channel which has been discussed in section 3.5. In the absence of surface singularities a layer of coreless vortices with thickness of order )i/m]u,- o,I is required to screen the constant superfluid velocity at the surface from the region of collapsed superflow in the interior. This situation is reminiscent of the Meissner effect in superconductivity and a London equation A2V2g,= g,

is proposed to describe it. g, is the relative superflow g - p, which must be as small as possible in the interior if the energy is to be a minimum. The complicated supercurrent [eq. (17)] in 3He-A is represented by a single parameter fis defined by

and the penetration depth A is taken to be crh/2mlfis- u,] with a an adjustable parameter of order unity. As the state of collapsed superflow is achieved by orbital motion, the rate at which g, can adjust itself to satisfy eq. (126) is controlled by an

220

H.E. HALL AND J.R. HOOK

orbital relaxation time equation of t h e form

7.

[Ch. 2, 84

It is therefore proposed that us should satisfy an

r dt7, AZV2(5,- u,) = (5, - 0 , ) + -. dt r can be estimated from the equation of motion, eq. (76), of In the limit that the scale of spatial variation of i is of order A, the terms in eq. (76) arising from flow and bending energies have the same order of magnitude so that dimensionally this equation becomes

Hence we take an orbital relaxation time

where /3 is another adjustable parameter of order unity. Inserting values for A and r in eq. (128) gives

In the absence of surface singularities the boundary condition to be applied is that 5, is constant at t h e surface. In t h e presence of n surface singularities per unit length of flow channel, precessing around the channel at an angular velocity w (see section 3.6) the superfluid velocity at the surface should relax according to

The value of w , has been discussed in section 3.6 and depends on whether the precession is dominated by motion of the bulk or surface texture. If the former is true then w , 7 - ' . Eq. (131) can be pictured as arising from the superfluid acceleration eq. (74). The first term on the right hand side arises from the h i - (CX V i i ) / 2 m term in eq. (74) and for a steady superflow leads to a chemical potential gradient of the form predicted by eq. (97) for dissipation arising from precessing domain walls. This identification with eq. (74) enables us to generalize eq. (131) to the case of driven flow. By adding the V p term from eq. (74) and taking v p = v p l p we obtain

-

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HYDRODYNAMICS OF SUPERFLUID 3He

($)V2("'

-

u,)

=

PPi av, PP, +-- -v p .

(iiS-

PSI

at

22 1

(133)

PSl

In driven flow I&- 0,) takes its maximum value u, in the centre of the channel and in calculating the flow through the channel there are two situations in which it may be possible to ignore the variation of gS- u, transverse t o the flow as represented by the left hand side of eq. (133). Firstly if h/2mvm, G d then the region near the wall in which this term is important is too small to effect the flow. Secondly if the variation of CS with time is sufficiently slow or the density of mobile surface singularities sufficiently high then the boundary condition, eq. (132), may allow us at the surface to relax quickly enough that 0, remains uniform in the direction transverse to t h e flow. Under these circumstances the dissipation should be described by

au, V P -=--at P

P S l

(ii, - UJ3 .

PP,

This equation has also been proposed by Volovik (1980). In a confined geometry of dimension d such that d G h/2mIijS- uJmU we might expect different behaviour. The picture of screening by a layer of coreless vortices on the surface is clearly no longer appropriate. We use eq. (74) as a starting point and assume that the scale of spatial variation of i is d so that orbital motion will be driven by the terms in eq. (76) linear in us to obtain

[cf. eqs. (96)and (loo)].We see that the effect of the confined geometry is to speed up the orbital motion. A dimensional estimate for the final term in eq. (74) is therefore

h i 2m

i ( I x vil)

h2 -m2

P S I ( U S- 0,)

d2pi

leading to an equation for 0, -a%---

at

VP p

h2 Ps,(U,- 0,) P'm2 d2pi '

where p' is an adjustable parameter of order unity. Although we have not

222

[Ch. 2, 84

H.E. HALL A N D J.R. HOOK

explicitly considered the variation transverse to t h e flow direction it is interesting to observe that a term of the form of the final term in eq. (136) has a similar effect to the term on t h e left hand side of eq. (131). If the time-dependent texture consists of precessing domain walls matched to the surface by boojums precessing at the same rate (see section 3.6) then we may take fijPto be constant over the cross section of the flow channel.

4.7. OBSERVED DISSIPATION IN A-PHASE TOROIDAL FLOW Dissipation in ac flow of 'He-A has been observed in the toroidal flow channels of torsional oscillators at Cornell and Manchester. In ac flow it is possible that dissipation due to singular vortices may be inhibited because t h e period of the oscillator is shorter than a nucleation time. That this may be the case in the Manchester experiments is suggested by measurements made in the B-phase under the same conditions in which the dissipation could be explained entirely by normal fluid viscosity. To observe orbital dissipation it is necessary that the orbital relaxation time T [eq. (130)j should satisfy O T < 1 where w is the angular frequency of the oscillator. In earlier torsion pendulum experiments at Manchester (Main et al., 1976) with a flow channel of larger dimension this condition was not satisfied and orbital motion was not observed. Instead the stationary i texture was determined by the torques on i in eq. (76) that are either independent of us- u, such as those arising from bending of l o r quadratic in us- on such as those arising from the anisotropy of the superfluid density. For small amplitudes of oscillation t h e former dominated and was largely perpendicular to the flow. For larger amplitudes the importance of the latter increased and i became more parallel to the flow. With increasing oscillation amplitude the measured superfluid density decreased thereforc from pSl to p4, (Main et al.. 1977). In later experiments at Manchester (table 2) the smaller channel size reduced the orbital relaxation time in t h e experimentally accessible amplitude range so that W T 1 and additional dissipation was observed. This was quantified by measuring its contribution 6vw to the bandwidth of the oscillator. The dissipation was accompanied by collapse of superflow, a fraction of the superfluid component becoming coupled to t h e motion of the oscillator. The superflow collapse was quantified by measuring the shift $vs in resonant frequency from its value at T,. In general Sv, decreased as the velocity amplitude of oscillation o0 increased indicating irrotational behavior by a smaller fraction of the superfluid component. T h e dissipation was non-linear; Svw depended strongly on uo and had a maximum value close t o the amplitude where W T - 1, an indication that orbital motion was involved.

c

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HYDRODYNAMICS OF SUPERFLUID 'He

223

Perhaps the most striking feature of the dissipation was its strong history dependence. At a given value of measuring amplitude and temperature 6v, could vary by a factor of more than ten, depending on the previous history of the fluid. This is illustrated in fig. 27 which shows Sv, as a function of 1 - VT, as the pendulum warmed up to T, at a measuring amplitude vo = 1.55 mm s-'. The different curves correspond to different values of the oscillation amplitude uo(Tc)on the initial cool down through T,. The dissipation increases monotonically with increasing vo(T,). Eq. (131) together with boundary condition (132) was used by the Manchester group in an attempt to account for their observations. The history dependence was introduced by postulating that the density of surface singularities in eq. (132) would depend on previous history. If this interpretation is correct then fig. 27 shows that n increases with uo(Tc). Other experiments suggested that n could be increased by a short burst of high amplitude oscillation below T, but decreasing the amplitude did not change n. On warm up at a measuring amplitude uo > uo(T,)there was a tendency near T, for n to increase to the value it would have had if uo had been equal to uo(Tc). The profound influence of vo(Tc)on n is to be expected since the energy barrier for the formation of singularities vanishes at T,. Better agreement between the experimental results and the model predictions was obtained by assuming that the motion of surface sin-

I----

1.5 -

N

x E

-

3

1.0-

7

a

0.5-

Il-T/Tr 1

Fig. 27. The effect of the oscillation amplitude while cooling through T,, w(Tc)on the damping measured during warm-up at a measuring amplitude of 1.55 mm s-' for the 17 km cell in the Manchester experiments. (Gay et al., 1983.)

224

(Ch. 2, 44

H.E. HALL AND J.R. HOOK

X

I

X

Fig. 28. Amplitude dependence in the Manchester experiments of the additional bandwidth 6vw and frequency shift 6v, at TIT, = 0.9 compared with model calculations for the 17 Krn cell [(a) and (b)) and t h e 49 p n cell [(c) and (d)]: 0, m(T,)= urn=; X , og(T,)= UO; 0, u0(T,)= 0. Full curves are calculated for up= 0.030,,, broken cuxves for uc = 0.03~0 and dotted curves for uc = 0. Both 6vwand 6v, are normalized by dividing by the frequency shift &(O) at small amplitude (Gay et at., 1983).

Ch. 2, $41

HYDRODYNAMICS OF SUPERFLUID 'He

I

1

225

I

1

0

I I

OO

2

1

4 v, [ m m i l ]

I

1

6

8

Fig. 28 (continued).

gularities was determined by the bulk rather than the surface textures, i.e. by taking o,= T-' in eq. (132). A comparison of the experimental results with the predictions is shown in fig. 28. Sv, and Sv, are shown as functions of measuring amplitude uo at T/T, = 0.9 for both 49 p m and 17 p,m cells. Experimental results for three different values of u,(T,) are shown:

226

H.E. HALL A N D J.R. HOOK

[Ch. 2, $4

the maximum amplitude used in the measurements; 7.75mms-' and 8.49mm s-' for t h e 49pm and 17pm cells respectively. (ii) u,,(T,)= uo. (iii) uo(T,)= 0 . Since the results for (i) are likely to be for states with the same value of n, values of a, p and u, (=nh/2rn) were chosen to give as good a fit as possible to these results. Values of a' = 5.0, pp,/pSL= 5 x lo-' cm2s-' ( = p = 1.6) and v, = yu,, with y = 0.03 give the continuous theoretical curves. The small value of y indicates that the density of surface singularities was much less than the density of coreless vortices required to give complete superflow collapse (=y = 1). The other theoretical curves are calculated for the above values of a and p and for u, = 0 . 0 3 (dashed ~~ curve) and v, = 0 (dotted curve). The difference between the three sets of calculated curves, like the difference between the three series of experimental points, is larger for the 17pm cell than for the 49 pm cell, thus indicating the greater importance of the surface in the narrower channel. The tendency for experimental values of Sv, to fall below the calculated values may be due to the partial alignment of parallel to the flow which is not properly allowed for in the theoretical model. The experimental curves show that for uo > 6 m m s-* a highly dissipative state independent of past history is reached. For uo(Tc)= 0, comparison with the theoretical curves suggests that surface singularities are formed below T, with a density increasing approximately linearly with uo when uo exceeds 2 mm s-'. This velocity is close to the dipole unlocking velocity [eq. (SS)] and it is possible that creation of surface singularities is associated with this. The series of experiments with uo(T,)= uo seem to correspond to a value of u, between yo, and yu,,. All the observed history dependence can be explained qualitatively by postulating that the density of surface singularities may be increased by high amplitude oscillation but otherwise remains constant except close to T,. Thus the simple model proposed by the Manchester group gives a reasonable semi-quantitative description of their observations. Cornell torsional oscillator experiments o n the A-phase were performed with the 18 pm aperture cell only. As mentioned in section 4.3 the presence of the constriction forces a fraction of the superfluid mass to couple to the oscillator so that this experiment represents a cross between toroidal and driven flow. The dissipation behaved in a similar way to that in the B-phase, with a well defined critical amplitude of oscillation 8, at which the quality factor started to decrease. In contrast to the B-phase, the resonant frequency decreased continuously from zero amplitude with no apparent change in behaviour at 0,; the changes in frequency below 8, (i) vo(T,)= u,,,

u,,

=

c

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HYDRODYNAMICS OF SUPERFLUID 3He

227

could result from alignment of i by flow. Fig. 22 shows the critical velocity uc that is deduced from 8, if superfluid compressibility effects are ignored and presumably if the fluid is assumed to be isotropic. u, appears to extrapolate to a finite value of about 1.5 mm s-' at T,. Critical velocities of this order may be obtained from several mechanisms. The velocity at which the scale hlmu, of spatial variation of f is equal to the aperture diameter is 1.1mm s-'. The discussion of section 3.5 suggests that textural changes will occur at a velocity of this order. The measured v, is also close to the dipole-unlocking velocity [eq. (85)] at which the onset of orbital dissipation in dc superflow in an infinite medium might be expected. There is some evidence in the Manchester experiments described above for a critical velocity of order 2 mm s-'; see for example the open circles on fig. 28a. This velocity was associated with the creation of surface singularities by flow. The requirement w r < 1 for orbital motion gives a characteristic velocity u (wp,/p,,)'' = lO(1- T/T,)'"mm s-l which is slightly higher than the Cornell critical velocity, suggesting that the amplitude of motion of i at the oscillator frequency is likely to be small. The relevance of these tentative suggestions could be examined by applying the simple model discussed in the previous section to the Cornell geometry. Until something akin to this has been done it will be impossible to say conclusively whether orbital motion can explain these experiments.

-

4.8. OBSERVED DISSIPATION IN A-PHASE DRIVEN FLOW Dissipation occumng in A-phase driven flow has been studied in three very different geometries at Sussex, Helsinki and Bell Laboratories (see table 2). Although there are qualitative and quantitative differences between results for the A and B phases in the Sussex and Helsinki experiments, one should be wary of concluding immediately that these differences are due to the existence of orbital dissipation in the A phase; t h e Bell Laboratories experiments were performed only in the A phase so such a comparison is not possible. There is good evidence for orbital dissipation in all three experiments but there are still unexplained features of the results. It is worth pointing out that dissipation due to cored vortices as represented by eqs. (110) and (112) can be comparable in magnitude to orbital dissipation. Results for the earlier Sussex experiments are shown in fig. 29 in which the volume of fluid displaced by the diaphragm following a step function change in the applied voltage is plotted as a function of time. Curves for four different sizes of voltage step are shown and they show the dia-

H.E. HALL AND J.R. HOOK

[Ch. 2, 84

Fig. 29. Response of the diaphragm in the Sussex experiments to step functions of different height in the applied voltage at TIT, = 0.86.The dashed curves were calculated from eq. (139). The dimensionless quantities V’ and t‘ are defined in the text.

phragm moving to its new equilibrium position. It is clear that, in contrast to the B-phase, the flow has not been saturated. Eq. (134) gives a good explanation of these results. We put u, = 0 since the normal fluid is clamped and we assume that the pressure difference across t h e diaphragm is related to the volume displaced d V by p , - p 2 = y A V. Using conservation of mass in the form ap,ii, = p dA V/df together with eq. (134) gives the following equation of motion of AV

where w o = ( p p y / p l ) ” ’ is t h e natural oscillation frequency and ps is the effective superfluid density for flow through the flow channel which has area a and length 1. It is convenient to write eq. (137) in dimensionless units V and t’ defined by

in terms o f which it becomes

Ch. 2, 041

HYDRODYNAMICS OF SUPERFLUID ’He

229

+AV’=O When the dissipation is large the second derivative can be ignored and the resulting equation readily integrated to give

where ti is the constant of integration and is physically the time for t h e diaphragm to reach equilibrium. Theoretical curves obtained from eq. (139) have been plotted on fig. 29 and agreement with experiment is good. The theoretical curves have been constrained to go through the origin. The dimensionless units have been identified by using wo = 105 s-’. Comparison of the dimensionless and real volumes enables a value of p t o be deduced. Using p,/psl = 10-4(1- T/Tc)’nm2s-l and &/p = 0.3(1- T/TC) gives p = 1.8, a value that indicates that the dissipation does arise from orbital motion and which is very similar to the value p = 1.6 obtained for the Manchester experiments described in the previous section. Since the model used does not take into account the effects of the channel walls we must assume the presence of a sufficient density of surface singularities to allow the superfluid velocity at the surface to relax freely. At lower flow rates than those shown in fig. 29, the diaphragm 0scillated about its equilibrium position. Oscillatory behaviour is predicted by eq. (138) for small AV’, and it is possible that this equation, with the possible addition of a linear damping term to allow for processes such as those observed in the B-phase (see section 4.4), might account for the entire transient. The effect of the non-linear term can be expected to become unimportant for dV’/dt’< 1, which in real units corresponds to w07 > 1, that is to a flow velocity

(2) 1R

us< w:n

0.5(1- T/TC)”mm s-l

.

A change in the nature of the dissipation was indeed observed at a velocity of this order although detailed comparison with the experiments is complicated by a strong hysteresis effect in the experiments; results obtained while cooling were different from those during warming. Amongst possible explanations of this hysteresis are a history dependent surface singularity density, as in the Manchester experiments described previously, and the effect of residual heat flows within the cell. It is also possible that the observed change in dissipation may be associated with a textural change. According to the discussion of section 3 the precessing

730

[Ch. 2, 54

H.E. HALL AND J.R. HOOK

domain wall array or similar time-dependent texture responsible for the dissipation could disappear at a flow velocity us h/md = 0.44 mm s-l for d=48km. The above model sheds light on two other features of t h e Sussex experiments. The initial mass current for a particular voltage step is predicted to vary as (1 - T/Tc)4’3which is somewhere between the temperature dependences observed during warming and cooling. The initial at mass current for a voltage step 6E is predicted to vary as (6E)1’3 constant temperature, close to the observed dependence. We feel that a detailed comparison of the Sussex results with the model would be worthwhile. In later Sussex experiments in which the force on the diaphragm was ramped linearly in time. a temperature-independent critical velocity for the onset of observable dissipation of order 0.5-1 cm s-’ was observed. This seems larger than any critical velocity associated with orbital dissipation. Calculations based on eq. (134) suggest that these later experiments should have been sensitive enough to see orbital dissipation occurring at flow rates above 0.2 cm s-’. The method used in the Helsinki diaphragm driven flow experiments was similar to that in the later Sussex experiments. Fig. 30 shows a plot of t h e average pressure difference Ap during the voltage ramp against the average mass current density j for various reduced temperatures at 24.6 bar. There is a qualitative difference between the results at the four lower temperatures and the rest in that at higher temperatures the graphs have two distinct linear regions with two critical currents jl and j 2 whereas at lower temperatures only one linear region is observed. At the four lowest temperatures the fluid outside the flow channels was in the B phase whereas that inside the tubes was in the A phase; this was possible because of the depression of the A-B transition in the channels to lower temperatures by the restricted geometry. The magnitude and temperature dependence of the higher critical suggests its identification with the critical current for current j 2 for T > TBA T < TBA,an indentification which is confirmed by comparison of the magnitude of the dissipation dApldj above these currents. The observed temperature dependence of j 2 is (1 - T/Tc)3’2 which suggests depairing as a possible explanation. The weak-coupling limit value for the A phase depairing current, corresponding to eq. (121) for the B phase, is

-

5

(1 - T/TC)’R

(Kleinert, 1980). This is about a factor four higher than jz. Some reduction

23 1

Ch. 2, 841

20

n L

a n 3.

v

> 10 La

a

0 1

0

0.05 J, (kg/m2s)

I

0.10

Fig. 30. Ap,,. as a function of J, in the Helsinki experiments at 24.6 bar. The reduced temperature for each set of points is shown on the figure. (Manninen and Pekola, 1983.)

in the theoretical current due to the restricted geometry is to be expected as for the B phase but probably not by such a large factor. As for the B phase, strong coupling corrections to eq. (141) are likely to increase the discrepancy between theory and experiment. An interpretation of the observed phenomena in terms of orbital dissipation may also be given as follows. The difference in behaviour for T C TBAand T > TBAsuggests that the ends of the channels are involved in the dissipation occurring between jl and j2. The similarity in behaviour above j2 at all temperatures implies that the dissipation there is occurring mainly within the interior of the channels. Manninen and Pekola (1982) suggest that the dissipation between j l and j , may be associated with vortex precession at the ends of the channels as suggested by Thuneberg and Kurkijiirvi (1981) (see section 3.7); presumably the vortices are not present if the fluid outside the tube is in the B phase. This explanation can be combined with one involving orbital dissipation if the rate of precession of the vortices is governed by the rate of motion of an associated orbital texture. Because of the restricted geometry the textural dynamics will be described by eq. (136) rather than eq. (134). We would expect therefore a

232

H.E. HALL AND J.R. HOOK

[Ch. 2, $4

pressure gradient of the form

where we have inserted ul to allow for the existence of the critical current j , . We envisage j l as arising from textural pinning. If we suppose that between j l and j 2 the pressure gradient only appears in regions of width of order d at each end of each channel then we obtain a pressure difference

This has the correct velocity dependence and predicts a slope

where P, is t h e effective superfluid density for flow in t h e channels. Inserting p , / p = 0.3(1- TITc),pIJpsl= 10-4(1- TlT,)'' cm2 s-' and d = 0.8 p m gives a value for p' of 1.5 when theoretical and experimental slopes are compared. If we assume that j 2 represents a current at which the time-dependent texture enters the flow channel, then we might expect ddpldj, to increase by a factor of order L/2d at j = j z where L = 10 pm is the length of the channels. This predicts an enhancement by a factor 6 whereas the observed increase was by a factor 7. The quality of this agreement must be regarded as coincidental since many factors of order unity have been ignored, but it does suggest strongly that our interpretation is valid. According to eq. (102) the pinning of vortices by surface irregularities o f size D should lead to a critical current j , of the form

This equation with a = 0.5, D/t(0)= 16 and p , / p = 0.3(1 - t ) predicts t h e magnitude and temperature dependence of j , very well. It is however slightly worrying that t h e size of the irregularity should be comparable to the channel diameter. It is tempting to associate j 2 with the instability of t h e

Ch. 2, 441

233

HYDRODYNAMICS OF SUPERFLUID ’He

radial disgyration texture in a cylinder which occurs at a flow velocity u, = 3.31h/md = 8 cm s-’

for R

= 0.4

Krn

(14.6)

(Bruinsma and Maki, 1979). Unfortunately this has the wrong temperature dependence and also predicts too high a critical velocity if bulk values for the superfluid density are used to deduce flow velocities from the measured jz’s. Taking into account the depression of the superfluid density by the flow and the restricted geometry might improve the agreement. In the Bell Laboratories bellows driven flow experiment a millipore filter at one end of the flow channel was used to “filter” the flow. The effect of the filter is shown in fig. 31 where the pressure difference Ap

vs (rnmls) Fig. 31. The main figure shows a typical set of data for the Bell Laboratories experiment. The arrows indicate the time order ofthe points. The unfiltered critical velocity (down) is always less than the reproducible filtered critical velocity (up). The inset shows an unusual set of data points which show a plateau in pressure. (Paalanen and Osherofi, 1980a.)

234

H.E. HALL AND J.R. HOOK

[Ch. 2, $4

across the flow channel is plotted as a function of flow velocity us through i t . The critical velocity u,, defined by the extrapolation of the linear part of the curve to zero Ap, was higher for flow entering the channel through the filter than for unfiltered flow. Furthermore u, (filtered) was reproducible and approximately equal to 0.55 mm s-I at all temperatures as indicated by the dashed line on fig. 22 whereas u, (unfiltered) varied from zero up to u, (filtered). us was not directly measured but was deduced from the measured mass flow density by dividing by ps1; this was the appropriate eigenvalue of the superfluid density tensor for small flows because of the strong magnetic field applied parallel to the flow. The transverse NMR frequency varied linearly with (us-- u,) as the flow velocity increased above its critical value. As yet there is no theoretical model that can explain all the observations. In trying to find such a model one should perhaps be wary of interpreting u, as a critical velocity for the onset of dissipation because of the observed curvature of the Ap versus u, curve at small Ap. However, as mentioned in section 3.4, Vollhardt and Maki (1979b) have shown that a uniform texture with parallel to d and both perpendicular to a strong magnetic field, should become unstable against the formation of domain walls (solitons) when t h e flow alignment energy becomes comparable to t h e dipole-locking energy, 1.e. when us (AD/pd,)"'. The precise value is 0.45(AD/psL)"= 0.4 mm s-' at 29.5 bar, close to the measured u,; note that the value given by Vollhardt and Maki (1979b) which exceeds this by a factor o f two is erroneous (Paalanen and Osheroff, 1980b; Vollhardt, 1979). The t y p e of domain wall proposed is illustrated in fig. 32. d is independent of position and, because the susceptibility anisotropy energy is much bigger than the dipole energy, it remains perpendicular to If. i rotates through T about an axis perpendicular to both d and H from a direction parallel to d to one anti-parallel to d. Dow (1984) has shown that this soliton texture becomes tirne-dependent at a velocity similar to that derived by Maki and Vollhardt for its formation from the uniform texture. The time dependence is necessary for orbital dissipation to occur.

c

-

Fig. 32. Type of domain wall which should occur in the presence of flow parallel to a strong magnetic field. The direction of d is indicated by dotted arrows, that of i by continuous arrows. (Vollhardt and Maki, 1979a.)

Ch. 2, 84)

HYDRODYNAMICS OF SUPERFLUID ’He

235

We might expect the pressure difference arising from the textural motion to be given by eq. (134) as

where L(= 1cm) is the length of the flow channel. This correctly predicts t h e order of magnitude of the observed dissipation but does not predict t h e observed linear dependence of d p on us at large us. The linear dependence would arise if the scale of spatial variation of 1 was determined by a flow independent length such as tDrather than by hlmv,. More worrying is the failure of eq. (147) to predict the observed temperature dependence of adplav,. Eq. (147) predicts a temperature dependence (1- T/Tc)-‘” whereas the observed dependence was (17-JT,)? A possible explanation of this discrepancy has been given by Cross (1983). He suggests that the correct temperature dependence could arise if the dissipation were due to the motion of an array of non-singular vortices at a velocity uL determined by balancing the Magnus force p , x~(v, - vJ with t h e force d , p , ~ ( t-) ~0,) + dip$ X (q- 0,) arising from motion of the array relative to the normal fluid. Here K is a vector of magnitude equal to the circulation of a vortex and direction parallel to the vortex, and d, and d , are dimensionless coefficients of order unity. This motion gives rise to a pressure gradient in the flow direction

where n is the areal density of vortices. The coefficients dlland d , have been calculated by Hall (1984) who finds

If dll%d , then eq. (148) gives a result similar to eq. (147) if n is taken to be (h/muJ2. If d, 4 d , - 1 then

which has the (1 - T/TC)’”temperature dependence observed in the experiments. Since d , is proportional to (1 - T/TC)” and d , - 1 is independent of temperature the condition for eq. (148) to reduce to eq.

236

H.E. HALL AND J.R. HOOK

[Ch. 2, $5

(150) is certainly valid sufficiently close to T,. Direct evaluation gives d,,< d l - 1 for T/Tc>0.97 so that Cross’s suggestion is probably not the complete solution to the problem. Perhaps dissipation due to cored vortices is important in this experiment.

5. Uniformly rotating 3He

The objective of experiments on uniformly rotating 3He is to study the vortex textures that are produced in rotating equilibrium. Minimization of the free energy F - L R together with the idea (Ho and Mermin, 1980) that the order parameter is stationary in the rotating frame in equilibrium, apart from a trivial uniform rate of change of phase, leads to the expectation that most of a sample of rotating 3He will contain vortices at an areal density of exactly ( 2 0 / K ) , where K is the appropriate circulation quantum. The effect of vortex creation energy on the free energy balance is that regions of t h e sample near boundaries are free of vortices. Experiments on these vortex textures have become possible as the result of the construction of a nuclear demagnetization cryostat that can be rotated at speeds of the order of 1 rad/s (Hakonen et al., 1981). The experiments to date have used NMR to probe the texture in a cylindrical volume about 5 mm diameter and several cm long. Since we have n o pretensions to cover spin dynamics in this article, we shall in sections 5.1 and 5.2 simply quote the results of calculations of NMR frequencies in order to assess the evidence for the various vortex structures that have been proposed. For further details and extensive references o n the field. w e refer t h e reader to the review paper of Hakonen et al. (1983a). More recently, rotating millikelvin cryostats have been used to perform flow experiments. We have already discussed the persistent current experiments (Gammel et al., 1984; Pekola et al., 1984a) in section 4, since the use of rotation to set up the persistent current is incidental to what is primarily a flow experiment; but in this section we will mention aspects related to vortices. We shall also, in section 5.3, discuss the experiments o n vortex-induced flow dissipation (Hall et al.. 1984) and their impl i cat i o n s .

-

5.1. A-PHASE VORTEX TEXTURES

The connection between flow and textures made explicit in eq. (71) means that continuously distributed vorticity is possible in 3He-A; we therefore expect t o avoid t h e singularity i n us and logarithmic vortex energy characteristic of ‘He 11. The first suggested structure for rotating ?He-A

Ch. 2, $51

HYDRODYNAMICS OF SUPERFLUID 'He

237

was proposed by Volovik and Kopnin (1977), who pointed out that two-quantum Anderson-Toulouse vortices could be packed together to form a lattice, probably triangular. There is no single quantum analytic vortex that will pack together to form a lattice, but Fujita et a]. (1978) showed that a lattice could be constructed by mixing two types of single quantum analytic vortex, with four quanta per unit cell of the structure. They also showed that this type of lattice had a lower free energy in the Ginzburg-Landau regime in zero magnetic field; this is the dipole-locked case, and the scale of the texture is that of the separation between vortices. However, the experiments (Hakonen et al., 1982a,b) were done in magnetic fields of 284 or 142G, usually along the rotation axis. This is ample to produce dipole unlocking on a scale SD 10 Fm, much less than a typical vortex spacing; we therefore now consider the textures appropriate to this situation. Maki (1983a) has shown that the lattices proposed by Fujita et a]. are modified so that (i and i lie more nearly in a plane perpendicular to H. The patterns of i are shown schematically in fig. 33(a) for the lattice of circular and hyperbolic vortices and in fig. 33(b)

-

0

00 0

(a)

Ib)

Fig. 33. Schematic diagrams of the i texture for (a) the circular-hyperbolic vortex lattice and (b) the radial-hyperbolic vortex lattice. In an axial magnetic field each texture is largely in the plane of the paper but escapes upwards at the centres of the hyperbolic vortices and downwards at the centres of the radial or circular vortices, for clockwise rotation; for anticlockwise rotation the direction of escape is reversed.

338

H.E. HALL AND J.R.HOOK

[Ch. 2, 95

for the lattice of radial and hyperbolic vortices. If i were everywhere in the plane of the paper there would be singularities and no circulation, but “escape in the third dimension”, downwards at each circular or radial singularity and upwards at each hyperbolic singularity removes the singularity and creates one quantum of clockwise circulation (note that there is an error in Maki’s fig. 2). The i vector escapes in the third dimension on a scale f,, and the d vector only on the shorter length scale 6” = tDHa/H,where H, 27 G . Maki shows that the circular-hyperbolic configuration, fig. 33(a), is energetically preferable in the temperature range of the experiments. The distortion of t h e Anderson-Toulouse vortex by an axial magnetic field is more drastic. Seppala and Volovik (1983) suggest that d is almost uniform and perpendicular to H,t h u s breaking the rotational symmetry, and that follows d except in a core region of size cD Eq. (71) shows that to secure two quanta of circulation i must assume all possible orientations somewhere within this core region; various possibilities are illustrated in fig. 34. By following the sense of rotation of i along the paths C, and C, it can be seen that the configuration %(a) is a hyperbolic (H) and circular (Cj vortex pair; similarly 34(b) is a hyperbolic and radial (R) vortex pair. Other possibilities, in which the position of the two vortices are interchanged, are shown in figs. 34(c) and (d). Note that the dotted circles in fig. 34 mark the boundary between regions with f parallel to d and f antiparallel to d. The modified Anderson-Toulouse two-quantum vortex can therefore be considered as a tube of domain wall of diameter of the order of tD;it is the tension in this domain wall that binds together the pair of single quantum vortices. Seppala and Volovik also consider the possibility of a singular vortex with one quantum of circulation. This is equivalent to allowing t h e i configuration on the dotted circle in fig. 34 t o continue to the origin. It is interesting to note that a lattice like fig. 33(a) can be constructed from figs. 34(aj and (c), and similarly a lattice like fig. 33(b) can be constructed from figs. 34(b) and (d). The essential difference is that in fig. 33 the angle between i and (or - d ) is always less than W, there are no domain walls. and there is a long range bending of f and d in t h e plane Perpendicular to H. on the scale of the separation between vortices. In fig. 34, o n the other hand. d is almost uniform. as is i between the vortex pairs. and t h e vortices are bound together in pairs by the domain walls t h a t this arrangement entails. Calculations by Fetter (198%) indicate that ;i vortex pair texture as in fig. 34 becomes stable above a field of the order of t h e dipole-unlocking field Ho. Experimentally, Hakonen et a]. (1982b) found that t h e main transverse N M R line was somewhat broadened by rotation, and a broad satellite of

-

c

a

Ch. 2, SS]

-+ ,*'

.

2

--

--+

.......0;;.....

,

,.

I

I

239

HYDRODYNAMICS OF SUPERFLUID 'He

.

J

+ '\\[I \

I I I

c I I

?H

I

\

c

c

J

......o...." H

c-

2

J t

c

1.......

.......

J t

c-

Fig. 34. Modified Anderson-Toulouse i textures in an axial magnetic field. They may be considered as bound vortex pairs- hyperbolic (H) and circular (C) in (a) and (c), and hyperbolic and radial (R) in (b) and (d). The dotted circles are domain walls between i parallel and antiparallel to d.

about 5% t h e area of t h e main peak appeared at a lower frequency, as shown in fig. 35. They also found that a weak satellite of irreproducible intensity in the initial non-rotating state was removed by a period of rotation; this they associated with domain walls (solitons) in the initial state. The resonance frequency is characterized by a parameter R, where the observed frequency f is given by

or approximately

HYDRODYNAMICS OF SUPERFLUID 'He

4 n=

z

0 I-

(ch.2, 85

1 2 1 radh

Y

U

W

n

n

a 0 cn

z

C

m

I

U

w I

a 5 z

c U.

0 zp -6

1,

-2 f

2

0

- 1, (kHz1

Fig. 35. Transverse NMR absorption spectra of supercooled 'He-A at 1 - T/7; = 0.267; 920 kHz is the Larmor frequency. Frequencies are measured relative to f A (1924 kHz), the frequency of the main peak in 'He-A. Fig. 35a shows the R-dependent broadening of the main peak. Fig. 35b is a close-up of fig. 3Sa and shows a soliton peak at 0 = 0 and a vortex peak at R = 1.21 rad s-'. The intensity of t h e soliton peak vaned from one experiment to another and with time; the soliton peak had already decayed in the case of the lower spectrum of fig. 3Sb. (Hakonen et al.. 1983a.)

fo =

ft

where fo is the normal Larrnor frequency and the A-phase longitudinal resonance frequency. Values of R, obtained at the two magnetic fields used are shown in fig. 36, together with values calculated for circular and hyperbolic vortices by Maki (1983b), who predicts that a doublet structure might be resolved at lower temperatures. However, Seppala and Volovik (1983) claim that vortices of the type shown in figs. 34(b) and (d) are also

Ch. 2, 951

24 1

H.E. HALL AND J.R. HOOK 10

oe

06

OL E”

10

I

I

I

0

0.1

02

1

I

08

06

04 04

1-T/ T,

Fig. 36. R,[see eq. (152)] as a function of 1 - T/T, at HO= 284 Oe for the vortex and soliton peaks (Hakonen et al., 1Y83a). For upper figure cryostat rotation was started after cooling through T,, whereas for lower figure A-phase was entered with cryostat rotating. A, soliton peak; the dashed line is for the twist composite soliton (Maki and Kumar, 1978). Other symbols are for vortex peak. 0 . 0 . 6 r a d s-’; 0, 1.04 rad s-‘; V,1.21 rad s X, 1.43 rad s-I. +, 1.32 rad s-’ and HO= 142 Oe. Continuous theoretical lines for circular (C) and hyperbolic (H) vortices are from Maki (1983b).

‘;

consistent with the present experimental data, though their value R : = 0.5 in the Ginzburg-Landau limit seems a little low. Singular vortices would give a value of R, much closer to unity, as well as a stronger signal, and are definitely excluded by the data. A convincing decision between t h e textures illustrated in figs. 33(a) and 34 therefore requires a probe that focuses on the major difference between them. Fig. 33 is characterized by a slowly varying texture of i and d between the vortices, on the scale of the vortex separation, whereas the vortices of fig. 34 are separated by regions of uniform texture, breaking the rotational symmetry. This suggests the use of ultrasonic attenuation to probe the symmetry perpendicular to the rotation axis. We may remark that the broadening of

232

H.E. HALL AND J.R. HOOK

[Ch. 2, 85

the main NMR peak is perhaps suggestive of the long range bending shown in fig. 33(a). An intriguing phenomenon, which however may not help to decide between different vortex structures, is the removal of solitons (domain walls) by a period of rotation. It seems to us plausible that, whatever the usual vortex lattice, any domain wall initially present will on rotation be incorporated in the structures of the type shown in fig. 34, and will subsequently be swept out of the system when rotation is stopped. While i t may be difficult to decide between different configurations of continuously distributed vorticity, the observation that rotational equilibrium is attained more rapidly in the A-phase than the B-phase is strongly suggestive of the easier nucleation that one would expect for any structure with continuously distributed vorticity.

5.2. B-PHASE VORTEX

STRU~RES

Since the phase variable is decoupled from the rest of the order parameter in the B-phase, we expect singular vortices as in 4He I1 as t h e only possibility. However, the longer coherence length and more complicated order parameter in 'He means that we may expect a more interesting vortex core structure, at least part of which is some sort of spatially varying anisotropic superfluid state. The key element in the observation o f such vortices by NMR is the existence of t h e flare-out texture in a cylindrical container of B-phase (Brinkman et al., 1974; Spencer and Ihas, 1982) even in the absence of rotation. This texture has a characteristic length of the order of millimetres, comparable to the size of the container, and acts as a trapping potential for spin waves, so that multiple resonances are seen, fig. 37. It can be seen from fig. 37 that the splitting of these spin wave modes is increased by rotation. The basis of this effect is t h e tendency of a superflow to orient the n texture [Brinkman and Cross (1978) eq. (71)j. Since the vortex spacing is very much less than t h e length scale on which bending of n occurs, t h e texture remains locally smooth to a very good approximation, but is globally distorted by the mean value of ( u s - 0")' in the neighbourhood of the vortices. Maki and Nakahara (1983) have shown that this effect can account for the observations, except that the characteristic velocity scale is rather too small; the effect of rotation is almost an order of magnitude larger than expected. Salomaa and Volovik (1983a) suggest that this discrepancy arises from orientation of n associated specifically with the vortex core rather than the long range velocity field. The experiments (Bunkov et at., 1983) do indeed provide some qualitative evidence for effects associated specifically with t h e vortex cores. Values of the spin wave splitting

Ch. 2, 051

HYDRODYNAMICS OF SUPERFLUID 3He

-

243

1 kHz

R=0

I

FREQUENCY

R

3

0 6 radls

-

Fig. 37. NMR absorption spectra for 'He-B at 1 - T/T,= 0.45 at rest and at 0 = 0.6 rad s-' (Hakonen et al., 1983a). The main peak is close to the Larmor frequency and the "background" curve qualitatively corresponds to the flare-out texture, with the spectrum calculated using the local oscillator model (Brinkman et al., 1974). A series of almost evenly spaced satellites is superimposed.

normalized by the B-phase equivalent of eq. (152), are shown for various angular velocities in fig. 38; the most conspicuous feature of these results is that the effect of rotation changes by a factor of about 1.5 at (1T/T,) 0.4. It is difficult to think of any effect of the long range velocity field that could produce such a discontinuity, a change in structure of the vortex core seems more likely, though Sonin (1984) has considered instability of the uniform n texture near a vortex as a possible alternative explanation. For the experiments of fig. 38 rotation was stopped and restarted at each temperature; a variation of this procedure is illustrated in fig. 39. Fig. 39(a) shows stopstart results as before; fig. 39(b) shows a warming run taken with continuous rotation; and fig. 39(c) a run in which rotation was stopped and restarted at (1 - T/TJ = 0.38. It is clear that there is a range of temperatures in which the low temperature state of the vortex lattice can be superheated, and it therefore seems reasonable to suggest a first order transition in the vortex core structure. This phase transition, and an associated latent heat, has also been observed in persistent current experiments (Pekola et al., 1984b). The transition manifests itself as a change in the maximum persistent current that can be created, i.e. in the critical velocity, as we discussed in section 4.1; the

-

244

[Ch. 2, 55

H.E. HALL A N D J.R. HOOK

0 010

0 008

i=i

2 a

0.OOL

0.002

0

0.2

0.3

0.4

0.5

1-TI T, Fig. 38. Frequency splitting of the satellite peaks in 3He-B. suitably scaled, as a function of I - T/T, at Ho = 284 Oe. Solid lines are guides to the eye. Arrows indicate the direction o f tcmpcrature drift. (Hakonen et al., 1983a.)

transition line in the ( p , T) plane was shown in fig. 21. The observed latent heat is surprisingly large; if we take a value o f the order of the A-B transition latent heat over the volume of the vortex core as reasonable, the experiments suggest a very high residual vorticity in t h e persistent currents of order ( u J d ) , where d is the size of t h e packed powder. This vorticity is about 1o(K) times greater than any angular velocity used in the experiments. It suggests that in the tortuous geometry used, the persistent current is associated either with a very dense array of trapped vortices, or with a surface change in the order parameter that shows a similar phase transition to the vortex cores in an open volume of ‘He-B. A further intriguing effect has been observed (Hakonen et al., 1983b) that is particularly noticeable below this phase transition. It is found that there is a contribution to the spin wave splitting that is odd in both 0 and

Ch. 2, §5]

HYDRODYNAMICS OF SUPERFLUID 'He

001 5

I

I

a)

-

I

I

I

245

1

b)

0010

-

0

c N

-m>d

2

9'

u-

Y

P

a 0005

/ 03

0.5

0.3

05

03

05

l-T/T, Fig. 39. Scaled frequency splitting of the satellite peaks in 'He-B as a function of 1 - T/Tb for 0 = 1.33 rad s-'. (a) Rotation was successively started and stopped at 15 min intervahl this produced a stable lineshape. (b) Continuous rotation; no discontinuity at 1 - T/T, = 0.4, (c) Cryostat rotated continuously from a low temperature to 1- T/T,= 0.38 where motiMl was stopped; after a new start the jump in Af appeared and the higher branch was thereaftd followed in a start/stop experiment. Arrows indicate increasing time. (Hakonen et al., 1983h)

H, suggesting a hitherto unsuspected term in t h e free energy of t h e form (dl R - H ) . Since S = x H / y the second term in eq. (69) is of t h h form if there are no vortices so that Os= 0 everywhere. However, with an equilibrium array of vortices, as in the experiments, the average value df asis 0.Consequently, with the bending scale of the n texture much larger than the vortex spacing, this bulk term makes almost n o con. tribution [though not, we believe, for the reason given by Volovik and Mineev (1983) and Hakonen et a].]. Hakonen et al. suggest instead d; contribution of this type to the tree energy from spontaneous mag. netization of the vortex core; this suggestion is made more plausible bf the marked effect of the vortex core transition on the magnitude of t h e effect odd in and H. Salomaa and Volovik (1983b) have showri theoretically how a B-phase vortex with a ferromagnetic core can arise, and Fetter (1985b) has shown that reasonable strong coupling parameter9 give such a core at high pressures.

-

5.3. VORTEX-INDUCED FLOW

DISSIPATION

It has recently been shown that the dissipation in an oscillatory superflow

246

[Ch. 2, $5

H.E. HALL AND J.R. HOOK

experiment can be profoundly modified by a superposed rotation (Hall et al., 1984). The geometry of the flow cell used is shown in fig. 40; the experimental 3He is essentially in the form of a slab 100 pm thick folded over onto itself so that the end regions are separated only by a 12pm Kapton diaphragm, displacement of which is used both to drive and detect flow oscillations. This geometry was originally designed to test the feasibility of measuring the pressure difference associated with intrinsic angular momentum given by eq. (a) but , it became apparent in the first exploratory experiments that the most marked effect of a superposed rotation was to quench flow oscillations: a Q of 40 to 200 (depending o n amplitude) in the non-rotating state was reduced to about 4 by rotation at 0.5 rad/s. This excess dissipation was observed when t h e cell was mounted with the axis f2 vertical, so that most of the slab was perpendicular to t h e axis of rotation, but not when it was mounted with the axis X vertical. The orientation dependence makes it very likely that the excess dissipation is associated with vortex lines, since no vortices would be expected in a 100 p.m slab when t h e rotation axis lies in its plane. The experiments are therefore interpreted in the same way as the analogous second sound experiments in 4He (Hall and Vinen, 1956; Hillel, 1981; Hillel and Vinen, 1983), in terms of a mutual friction between normal fluid and superfluid. The force o n unit volume of superfluid is written as n

Fig. 40. Schematic diagram of the apparatus used to study the effect o f rotation on oscillatory superflow. Flow oscillations are strongly damped by rotation about the axis R, but not by rotation about the axis X.

Ch. 2, $51

HYDRODYNAMICS OF SUPERFLUID 'He

247

where the suffix I denotes a component perpendicular to the rotation axis. To obtain this force we note that the superfluid is accelerated by transport of vortices [see the discussion by Anderson (1966) for 4He or 3 He-B, or eq. (74) for 3He-Al as if it were acted on by a Magnus force

per unit length of line, where uL is the velocity of the vortex line and K is the circulation quantum. There is also an interaction between the normal fluid and the vortex (related to the scattering of excitations in 4He) which we parameterize as fn

=

d,,P,K(

UL

- on),

+ d, P ~ KX (uL

- 0,)

*

(155)

We can now obtain the mutual friction parameters by setting fn+ fs= 0 and F, = (2O/K)fs.For the dissipative force, which concerns us here, the result is

So far reliable measurements in 3He have been made only close to T,. Typically, for (1 - T/TJ 0.07, (Bp,/p) is -5 in 'He-B at 20 bar and -3 in 3He-A at 29 bar. These rather large values, which tend to increase as T, is approached, are intriguing, especially when considered in conjunction with the well established critical divergence of B at TA in 4He (Mathieu et al., 1976). For eq. (156) shows that large values of (BpJp) can result only if both d, and Id,- 11 are substantially less than one. The reason for this behaviour in 4He was for long not understood, but recently Onuki (1983) and Pitaevskii (1977) have proposed explanations based on Ginzburg-Pitaevskii theory. However, in 3He-A the absence of a singular vortex core means that the interaction between normal fluid and vortex should be describable in purely hydrodynamic terms. Hall (1985) gives an argument relating eq. (155) to eq. (34f) and finds

-

where y is a numerical factor depending on the vortex texture (Cross,

2.18

H.E. HALL AND J.R. HOOK

(Ch. 2, 56

1983; Kopnin, 1978). Thus

By taking the orbital viscosity as known, Hall finds experimentally Id,-- 11 = 0.11 kO.02 at (1 - T/T,)=0.01. Since we expect A - 0 and pL= i p s [eq. (19)] near T,, this suggests that the superfluid density entering the Magnus effect may be a little less than pSI=ps; this is quite possible, depending o n the vortex texture. Apart from such niceties, it is gratifying that our treatment of intrinsic angular momentum in section 2.2 can be carried through to this unforeseen connection with experiment. Further experiments are needed, not only for a full theoretical interpretation, but because a direct experimental measurement of the dissipation associated with vorticity can help in the interpretation of other flow experiments. Measurements at lower temperatures may be best done with a torsion pendulum, as suggested by Sonin (1981), since the dissipation in t h e non-rotating state is likely to be lower. For a complete interpretation it will be necessary to measure B' as well as B, by doing a fourth sound analogue of the second sound experiments in 4He (Mathieu et a]., 1976).

6. Measurement of thermodynamic and hydrodynamic parameters

The hydrodynamic theory involves many phenomenological parameters. Within t h e spirit of the theory these may be regarded as quantities which must be measured. Alternatively they may be calculated from microscopic theory. Very few have been measured to date; rather more have been calculated microscopically. In this section we describe recent measurements of some of t h e more important parameters; for earlier measurements we refer the reader to the review articles of Wheatley (1975, 1978). We will compare the measurements with theoretical predictions, but will not give details of t h e theories themselves. For a review of the microscopic theory of transport parameters t h e reader is recommended to read Wiilflr and Einzel (1984). A microscopic calculation of the many phenomenological parameters that enter t h e hydrodynamics of the A phase has been given by Nagai (1979, 1980). 6.1.

SPECIFIC HEAT

The specific heat of the superfluid phases has been measured recently by Alvesalo et a]. (1979, 1980, 1981) and by Zeise et a]. (1981). There is

Ch. 2, $61

249

HYDRODYNAMICS OF SUPERFLUID 3He

controversy surrounding these measurements because the values obtained for the normal phase in both cases are about 30% below those obtained by Abel et al. (1966). In addition, recent normal state measurements by Greywall (1983), Mitchell et al. (1984) and Mayberry et al. (1983) are nearer to those of Abel et al. If the superfluid state values are normalized by dividing by the specific heat C, in the normal state at T, then it can be hoped that any systematic error contributing to the above discrepancy will disappear. Fig. 41 shows measured values for the specific heat discontinuity AC/CN at T, as a function of pressure. Also shown are the discontinuities obtained by using

(ACIC,),

=

1.43

(AC/CN), = 1.43

516 (1 + A&,/3) ’

1 1+ O.6(Ap1,+ A&/3)’

where the strong coupling parameters A&5, A&, and APMs are as defined and calculated by Sauls and Serene (1981).

1 /6 A

, 0

0

B

9’

/’

1

0

OA

1. o l

0

I

I

I

10

20

30

P Bar Fig. 41. Normalized specific heat discontinuity at T, as a function of pressure as measured by Alvesalo et al. (circles) and Zeise et al. (triangles). Open symbols are for the B phase, closed symbols are for the A phase. The continuous and dashed theoretical curves are for the B and A phases respectively as explained in the text.

250

[Ch. 2, 06

H.E.HALL A N D J.R. HOOK

The temperature dependence at four pressures of the specific heat measured by Alvesalo et al. is shown in fig. 42. The dashed curves are obtained from the BCS theory with a renormalized energy gap; the renormalization factor at each pressure was adjusted t o give the measured value of ACIC,. The continuous lines, which fit the data rather better, are obtained from the weak-coupling-plus model of Serene and Rainer (1979). In this theory the dependence of C/C, on T/T, depends only on one parameter which may be taken as AC/CN. The fit to this theory suggests that any systematic errors in C may indeed have been eliminated by division by C,. The fit also provides a method for calculating C/C, for use with the hydrodynamic equations; t h e actual value of C i s of course still uncertain because of the doubts about C,. The entropy can be obtained if required by integrating the specific heat.

.3.01. -

1

P = 18.1 bar

A C l C , = 1.70

'i t t

P : 12 9

2.'

A C l C , = 1.64 1 1_ -I-

0.4

0.6

I

1

0.8

P : 3.0 bar A C / C , = 1.43

I

1.01

0.L

0.6

0.8

1.0

Fig. 43. Temperature dependence of the specific heat at four pressures. Theoretical curves are explained in the text. (Alvesalo et al., 1981.)

Ch. 2, 561

HYDRODYNAMICS OF SUPERFLUID 3He

25 1

6.2. NORMAL FLUID DENSITY Measurements of the normal fluid density in the B phase have been made using a torsional oscillator by Archie et al. (1981) and Saunders et al. (1981). The lower temperatures obtained in t h e latter measurements enabled a more accurate determination of the low temperature limiting value of t h e resonant frequency and hence there is less uncertainty in the resulting values of pn. Discussion of these results is facilitated if Fermi liquid corrections (Leggett, 1975) are removed from the measured normal fluid density by evaluating a stripped normal fluid fraction

In the absence of strong coupling effects p:/p should equal Yea, t h e Yosida function 1 ”

Y(dIkT)= 2

dx sech2i[x2+(d/kT)’]’’,

0

evaluated for an energy gap equal to the BCS value. Fig. 43 shows a plot of Yea - p:/p against TIT, for various pressures. The upper and lower sets of points were obtained by using values of Fl from Wheatley (1975) and Alvesalo et al. (1981) respectively. There is uncertainty in the measured values at low temperatures because of the possible need for finite mean free path corrections to the hydrodynamic theory. The quasiparticle mean free path 1 becomes comparable to the channel width d = 95 p.m of the torsional oscillator at a reduced temperature varying from about 0.6 at 0 bar to 0.3 at 29 bar. At lower temperatures than these 1 becomes comparable to the viscous penetration depth 6 = (q/pnu)”. A first order slip correction appropriate to the limit 14 d,6 may be made by introducing a slip length as discussed in the following section. Such a procedure suggests that finite 1 corrections to the normal fluid density are probably small, but this has not yet been verified by a calculation valid for arbitrary 1. The continuous curves on fig. 43 show values of Yea- Y ( K - ” A , , / ~ T ) for the various gap scaling factors K - indicated. ~ ~ The experimental points obtained using the F, values of Wheatley are close to the curve for K-”= 1.1 whereas those obtained using the Fl values of Alvesalo et al. do not correspond to a simple gap renormalization. The dashed curve shows the prediction of the weak-coupling-plus model of Serene and Rainer (1983) for a pressure of 20 bar in which both trivial and non-trivial

252

[Ch. 2, 06

H.E. HALL AND J.R. HOOK

-0.051 , 0.3 0.4 1

1 I

0.5

I

0.6

0.7

0.8

0.9

I 1.0

Fig. 43. Difference between Y m and the stripped normal fluid fraction p:/p obtained from the B phase torsional oscillator experiments of Saunders et al. (1981). Upper and lower sets of points were obtained using values of Fl from Wheatley (1975) and Alvesalo et al. (1981) respectively. 0. 4.98bar; B, 9.%bar; A. 15 bar; A, 20.5 bar; x , 24.13bar; 0, 29.2 bar. Theoretical curves are as explained in the text.

strong-coupling corrections have been included. The predictions of this model fit the experimental values obtained using M e a t l e y FI values rather better than those obtained using Helsinki F, values. Reasonable agreement is also obtained if Fl values of Greywall (1983) are used. It is unlikely that this agreement will hold at lower pressures where strongcoupling corrections are likely to be much smaller whereas the experimental results show very little pressure dependence. As far as we are aware strong coupling corrections to the superfluid density at other pressures have not yet been calculated. Fig. 43 suggests that the normal fluid density may reasonably be calculated when required from the formula

Ch. 2, $61

HYDRODYNAMICS OF SUPERFLUID ’He

253

using the F, values of Wheatley and a Yosida function evaluated for an energy gap enhanced by a factor 1.1 from its BCS value. This procedure has been used in this review article to calculate values of p,. 6.3. SHEARVISCOSITY

Recent measurements of t h e shear viscosity q of the B phase have been made using a variety of techniques. Archie et a]. (1981) report vibrating wire and torsional oscillator measurements. Values of q from later Cornell measurements, in which a torsional oscillator and an oscillating sphere were used, have been quoted by Ono et al. (1982). Eska et al. (1980, 1983) deduced values for q from first sound attenuation. Carless et al. (1983) used a vibrating wire technique. All these experiments have fundamental problems of interpretation at low temperatures where the quasiparticle mean free path 1 increases as exp(A/kT) and eventually exceeds the dimension d of the apparatus or t h e viscous penetration depth S = ( ~ / p , ~ )at” the frequency w at which the measurement is made. The condition 1 6 can alternatively be written or 1 where r is t h e quasiparticle relaxation time. Strictly these problems represent a breakdown of the hydrodynamic theory, but provided l / S and Nd are not too large they may be overcome by applying to the hydrodynamic equations a slip boundary condition of the following form

-

-

on the tangential component unt of u, at a surface. Here u is the tangential velocity of t h e surface and the value of the slip length has been calculated by Hbjgaard Jensen et al. (1980). 5 is of the same order as 1 and becomes frequency dependent when o r is finite. Slip corrections are smallest for the spherical viscometer since the diameter of the sphere (2 cm) is sufficiently large that Ild Q 1 throughout the experimental temperature range and at the experimental frequency of 1800 Hz,1 becomes comparable to S only at the very lowest temperatures achieved at the lowest pressures. Application of a first order slip correction as described above suggests that finite mean free path corrections are small in this experiment throughout the temperature range in which measurements were made. Fig. 44 shows the viscosity measured in the spherical viscometer (closed symbols) at pressures of 5 bar (triangles), 20 bar (circles) and 30 bar (squares). Data from the Cornell torsional pendulum at the same pres-

[Ch. 2, 86

H.E. HALL AND J.R. HOOK

2S4

V

I

i 1 .02 OL 1

01

1

06

I

I

I

T/T,

Fig. 44. Experimental values of shear viscosity for 'He-B from torsional oscillator (open symbols) and spherical viscometer (closed symbols) measurements at Cornell University. Triangles. circles and squares are for 5 bar, 20 bar and 30 bar. Theoretical curves are from Einzel (1984).

sures (open symbols) are also shown. T h e theoretical curves were obtained by Einzel (1984) using a separable kernel approximation to the collision operator. The theoretical curves are very sensitive to the energy gap A and t h e scattering parameter A2 which is also an important factor in determining the normal state viscosity. A, values of Pfitzner and Wolfle (1983) have been used in calculating the theoretical curves in fig. 44.The value of A was chosen to vary smoothly with temperature in such a way as to give the measured ACIC, of Alvesalo et al. (1981) at T, and t h e low temperature limiting values of A deduced from the analysis of Bioyet et al. (1979) of spin relaxation experiments; this energy gap differs significantly at low temperatures from that predicted by the weak-coupling-plus theory of Serene and Rainer (1983). The viscosity values calculated by Ono et al. (1982) using a variational solution of the Boltzmann

Ch. 2, 061

HYDRODYNAMICS OF SUPERFLUID ’He

255

equation do not differ significantly from those calculated using the separable kernel approximation. Whilst there is good agreement between experiment and theory at higher temperatures, there is a marked disagreement at lower temperatures. The theoretical curves go through a minimum with decreasing temperature whereas t h e experimental curves decrease monotonically; indeed the rate of decrease becomes larger at lower temperatures. This latter feature, which has been described as “the low temperature droop” by Ono et al., is more marked for the torsional oscillator data than for t h e spherical viscometer data. The low temperature droop cannot be explained by a simple slip correction of the type described above because such corrections are very small for the spherical viscometer. The problem is unlikely to arise because of error in the calculation of the quasiparticle scattering time, since the theory correctly predicts t h e relaxation time which determines the damping of order parameter collective modes (Einzel, 1984). Recently Einzel et al. (1984) have suggested that the droop may be caused by Andreev reflection of the quasiparticles by the spatially dependent B phase order parameter near a solid surface. This occurs when the quasiparticle mean free path becomes long compared t o the scale of spatial variation of the order parameter, and prevents excitations from transferring drift momentum to the surface. Their calculations show that this effect can remove the discrepancy between experiment and theory for the torsional oscillator data, but calculations have not yet been performed for the spherical viscometer. The viscosity values obtained from the vibrating wire viscometer measurements of Carless et al. (1983) are shown in fig. 45. These authors propose a method for making finite mean free path corrections to vibrating wire experiments which should work for arbitrary 1. Experimental evidence for the validity of this method is provided by the vibrating wire measurements in 3He-‘He mixtures of Guenault et al. (1983). The theoretical curves on fig. 45 are identical to those on fig. 44. Both t h e temperature and pressure dependence of the viscosity is predicted well by the theory at high temperatures but there is a discrepancy at lower temperatures similar to that found in the torsional oscillator and spherical viscometer measurements. Values of 7 were obtained from the first-sound attenuation data of Eska et al. (1982) using finite mean free path corrections calculated by Nagai and Wolfle (1981) and Hpjgaard Jensen et a]. (1980). The reduced viscosity agrees with theoretical values at high temperature but also exhibits a “droop” at low temperatures. The effect of AndrCev reflection on vibrating wire and sound attenuation measurements has not yet been calculated.

-

H.E. HALL AND J.R. HOOK

256

[Ch. 2, 96

0.2

01

t

0.1

1

0.2

I

i

i

.,

0.L

I

0.6

I

I

'O

I

TIT,

1

Fig. 45. Experimental values of shear viscosity for 'He-B from vibrating wire measurements of Carless et at. (1983). V, 0.1 bar; 2.1 bar; A, 9.89bar; 0,19.89bar; 0.29.34 bar. Theoretical cumes are from Einzel (1984).

6.4. SECOND wscosm Values for the second viscosity l3 at low pressures have recently been deduced from the B phase vibrating wire experiments of Carless et al. (1983). This was possible because of the realization by Hall (1981) that it was necessary to consider the compressibility of the superfluid fraction when analysing vibrating wire measurements in superfluid 3He. In obtaining the experimental values of l3shown on fig. 46,it was necessary to assume a form for finite mean free path effects and this introduced some uncertainty into the values obtained. The theoretical curve was obtained from the theory of Wolfle and Einzel (1978) in the way described by Carless et al. and is almost identical to the curve obtained by Einzel (1984) using a separable kernel approximation to the collision operator. The experimental values of f ; are consistent with the theoretical cal-

Ch. 2, 86)

257

HYDRODYNAMICS OF SUPERFLUID 3He

O'

i7

0.8 TITC

0.9

11

Fig. 46. Second viscosity of 3He-B determined from vibrating wire measurements of Carless et al. (1983) at 1.28 bar. Theoretical curve is explained in the text.

culation for the whole temperature and pressure range (T/T,> 0.65, < 5 bar) for which it was possible to deduce experimental values. The theoretical value of 5; was also used successfully in predicting the dissipation at low flow velocities in the Berkeley experiments discussed in section 4.4. It is significant that in the analysis of this experiment and in the vibrating wire measurements, it was necessary to use a boundary condition on the normal component of v, at the surfaces of the fluid to obtain a solution of the two fluid equations. In both cases the boundary condition that was successfully used was to equate the normal component of u, to the normal component of the velocity of the surface. This boundary condition is not at all obvious and has been little discussed in the literature (see Carless et al. for a brief discussion); a theoretical justification of this boundary condition would be very desirable. p

6.5. THERMAL CONDUCIIVITY

Recent measurements of the thermal conductivity of the B phase at

258

[Ch. 2, 96

H.E. HALL AND J.R. HOOK

20 bar have been reported by Wellard (1982). The measurements were made by observing the propagation of a heat pulse along a tube of diameter 4mm. The results were analysed by solving the two fluid equations of motion. Experimental values of 17 (Carless et al., 1983) and theoretical values of [3 (Wolfle and Einzel, 1978) were used in this analysis. Near T, the heat flow is predominantly due to counterflow of the normal and superfluid fractions but at low temperatures the thermal conductivity term is dominant. The counterflow contribution appeared to be limited by a critical velocity for superflow of order 4mms-'. The values obtained for K are shown in fig. 47. They are subject to error at all temperatures because of uncertainty in the value of the specific heat capacity of the liquid. In obtaining the results on fig. 47, the reduced specific heat capacity C/C, was assumed to follow a BCS curve as a function of T/T, for an energy gap renormalized by a factor of 1.1 to give the observed discontinuity (Alvesalo et al., 1981) in C at the transition temperature. The value used for CJT,) was 0.024 J g-' K-'. There is additional uncertainty in the values of K near to T, because the thermal conductivity contribution to the heat flow is much smaller than the counterflow contribution. The theoretical curves on fig. 47 are as calculated by Einzel (1984) using a separable kernel approximation to the collision operator for the various values of the scattering parameter A , indicated. The curves are insensitive to energy gap renormalization. Near T, the experimental results are consistent with t h e predicted value of A , of Pfitzner and Wolfle of 1.33 but

1

I-

0 4 1

I

1

I

/I

I

1

I

I

1

/

'I

I

Fig. 47. Thermal conductivity of 3He-B at 20 bar as measured by Wellard (1982).Theoretical curves are explained in the text.

Ch. 21

HYDRODYNAMICS OF SUPERFLUID 'He

259

at low temperatures they fall below the theoretical curve for this value of A, in a manner reminiscent of the drop in the low temperature viscosity (section 6.3). The experimental results at low temperatures also indicate a A 1 value greater than that (1.52 < A, < 1.92) predicted by the variational calculation of Hara (1981). The reason for this discrepancy is not known. Acknowledgements

We are grateful to Mario Liu for extended discussions on the foundations of hydrodynamics during a visit by him to Manchester (supported by SERC Grant GR/B/99125). Much of this article was written while HEH was on leave at Cornell University, and we would like to acknowledge the hospitality and stimulating conversations of the Cornell low temperature group; especially Dave Lee for forcing HEH to give some lectures and David Mermin for t h e stimulus provided by his lectures on what lies beyond hydrodynamics. JRH would like to acknowledge the hospitality of and useful conversations with the low temperature groups in Garching and Julich during a visit to Germany in 1982. The Helsinki group, especially Matti Krusius, Jukka Pekola and Martti Salomaa, and the Sussex group have been generous with pre-publication information about their work. We have benefited greatly from conversations and correspondence with Helmut Brand, Douglas Brewer, Michael Cross, Dietrich Einzel, Jason Ho, Chia-Ren Hu, Tony Leggett, Kazumi Maki, Yoshi Ono, Dierk Rainer, Wayne Saslow and Dieter Vollhardt. We would like to thank Judy Burkhart, Jandy Walker, Christine Renshaw, Catherine Formby and Ian Callaghan for producing the typescript and figures. References Abel, W.R.,A.C. Anderson. W.C. Black and J.C. Wheatley (1966) Phys. Rev. 147,111. Alvesalo, T.A., T. Haavasoja, P.C. Main, M.T. Manninen, J. Ray and L.M.M. Rehn (1979) Phys. Rev. Lett. 43, 1509. Alvesalo, T.A., T. Haavasoja, M.T. Manninen and A.T. Soinne (1980) Phys. Rev. Lett. 44, 1076. Alvesalo, T.A., T. Haavasoja and M.T. Manninen (1981) J. Low Temp. Phys. 45, 373. Anderson, P.W. (1966) in: Quantum Fluids, ed. D.F. Brewer (North-Holland, Amsterdam) p. 146. Anderson, P.W. and G. Toulouse (1977) Phys. Rev. Lett. 38,508. Archie, C.N., T.A. Alvesalo, J.D. Reppy and R.C. Richardson (1981) J. Low Temp. Phys. 42, 295. Bailin, D. and A. Love (1978) J. Phys. C11, L909.

260

H.E. HALL AND J.R. HOOK

(Ch. 2

Bates. D.M., S.N. Ytterbce, C.M. Could and H.M. Bozler (1984) Phys. Rev. Lett. 53, 1574. Bhattacharyya, P., T.-L. Ho and N.D. Mermin (1977) Phys. Rev. Lett. 39, 1290. Bloyet, D., E. Varoquaux, C. Vibet, 0. Avenel, P.M. Berglund and R. Combescot (1979) Phys. Rev. Lett. 42, 1158. Brand, H. and M.C. Cross (1982) Phys. Rev. Lett. 49, 1959. Brewer, D.F. (1983) in: Quantum Fluids and Solids, eds. E.D. A d a m and G.G. Ihas (AIP, New York) p. 3%. Brinkman, W.F. and M.C. Cross (1978) in: Progress in Low Temperature Physics, Vol. VIIa, ed. D.F. Brewer (North-Holland, Amsterdam) p. 105. Brinkman, W.F., H. Smith, D.D. Osheroff and E.I. Blount (1974) Phys. Rev. Lett. 33,624. Bromley, D.J. (1980) Phys. Rev. B21, 2754. Bruinsma, R. and K. Maki (1979) J. Low Temp. Phys. 37,607. Rucholtz, L.J. and A.L. Fetter (1977) Phys. Rev. B15,5225. Bunkov, Yu.M.,P.J. Hakonen and M. Krusius (1983) in: Quantum Fluids and Solids. eds. E.D. A d a m and G.G. Ihas (AIP, New York) p. 194. Carless. D.C., H.E. Hall and J.R. Hook (1983) J . Low Temp. Phys. 50,605. Combescot, R. (1980) Phys. Lett. 78A,85. Combescot, R. (1981) J. Phys. C14, 1619. Combescot, R. and T. Dombre (1980) Phys. Lett. 76A,293. Corruccini. L.R.and D.D. Osheroff (1980) Phys. Rev. Lett. 45, 2029. Crooker, B.C. (1983) PhD thesis (Comell University). Crooker, B.C.. B. Hebral and J.D. Reppy (1981) Physica lOSB+C,795. Cross. M.C.(1975) J. Low Temp. Phys. 21,525. Cross, M.C. (1977) J. Low Temp. Phys. 26, 165. Cross, M.C. (1983) in: Quantum Fluids and Solids. eds. E.D. A d a m and G.G. Ihas (AIP. New York) p. 325. Cross, M.C. and P.W. Anderson (1975) in: Proc. 14th Int. Conf. on Low Temperature Physics - LT14, Vol. 1. eds. M. Krusius and M. Vuorio, p. 29. Cross, M.C. and M. Liu (1978) J. Phys. C11,1795. Dahm. A.J.. D.S. Betts, D.F. Brewer, J. Hutchins, J. Saunders and W.S. Truscott (1980) Phys. Rev. Lett. 45, 1411. de Gennes, P.G. (1966) Superconductivity of Metals and Alloys (Benjamin, New York). de Gennes, P.G. (1974) The Physics of Liquid Crystals (Oxford University Press). de Gennes. P.G. and D. Rainer (1974) Phys. Lett. 46A.429. de Groot, S.R. and P. Mazur (1962) Non-Equilibrium Thermodynamics (North-Holland, Amsterdam). Dombre, T. and R. Combescot (1982) J. Phys. C1S. 692.5. Dow. R.C.M. (1984) PhD thesis (Lancaster University). Eastop. A.D.. H.E. Hall and J.R. Hook (1984) in: Roc. 17th Int. C o d . on Low Temperature Physics-LT-17, eds. U. Eckern. A. Schmid. W. Weber and H. Wiihl (NorthHolland, Amsterdam) p. 37. Einzel. D. (1984) J. Low Temp. Phys. 54. 427. Einzel. D.. P. Wolfle. H. HBjgaard Jensen and H. Smith (1984) Phys. Rev. Lett. 52. 1705. Eisenstcin, J.P. (1980) PhD thesis (Berkeley). Eisenstein. J.P. and R.E. Packard (1982) Phys. Rev. Lett. 49,564. Eisenstein, J.P., G.W. Swift and R.E. Packard (1980) Phys. Rev. Lett. 45, 1569. Eska, G., K. Neumaier. W. Schoepe, K. Uhlig, W. Wiedemann and P. Wolfle (1980) Phys. Rev. Lett. 44. 1337. Eska, G.. K. Neurnaier, W. Schoepe, K. Uhlig and W. Wiedemann (1983) Phys. Rev. B27, 55%.

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261

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Tough. J.T. (1982) in: Progress in Low Temperature Physics, Vol. VIII. ed. D.F. Brewer (North-Holland, Amsterdam) p. 133. Truscott, W.S. (1979) Phys. Lett. 74A. 80. Vinen, W.F. (1957) Proc. Roy. SOC. A240, 114. 128; A242, 493. Vinen. W.F.(1958) Roc.Rov. Soc. A243, 400. Vollhardt. D. (IY7Y) PhD thesis (University of Hamourg). Vollhardt, D. and K. Maki (1979a) Phys. Rev. B#), %3. Vollhardt. D. and K. Maki (1979b) Phys. Lett. 72A,21. Vollhardt. D..Y.R. Lin-Liu and K. Maki (1979) J. Low Temp. Phys. 37,627. Vollhardt, D.. K. Maki and N. Schopohl (1980) J. L o w Temp. Phys. 39,79. Vollhardt. D.. Y.R. Lin-Liu and K. Maki (1981) J. L o w Temp. Phys. 43,189. Volovik. G.E. (1978) Pis'ma Zh. Eksp. Teor. Fiz. 27. 605 (JETPLett. 27, 573, 1978). Volovik. G.E. (1979) Sov. Sci. Rev. A l . 23. Volovik, G.E. (1980) Zh. Eksp. Teor. Fiz. 79, 309 (Sov. Phys. JETP 52. 1561. Volovik, G.E. and N.B. Kopnin (1977) JEW Lett. 25, 22. Volovik. G.E. and V.P. Mineev (1981) Zh. Eksp. Teor. Fiz. 81, 989 (Sov. Phys. J E W 54, 5241. Volovik, G.E. and V.P. Mmeev (1983) Zh. Eksp. Teor. Fiz. 86(5). Wellard, N.V. (1982) PhD thesis (University of Manchester). Wheatley, J.C. (1975) Rev. Mod. Phys. 47, 415. Wheatley, J.C. (1978) in: Progress in L o w Temperature Physics, Vol. VIIa, ed. D.F. Brewer (North-Holland, Amsterdam) p. 1. WiilRe, P. (1978) in: Progress in b w Temperature Physics, Vol. Vlla, ed. D.F. Brewer (North-Holland, Amsterdam) p. 191. Wolfle, P. and D. Einzel (1978) J. L o w Temp. Phys. 32, 19, 39. Zeise, E.K., J. Saunders, A.I. Ahonen, C.N. Archie and R.C. Bchardson (1981) Physica 108B+C. 1213.

Note added in proof M. Liu (Phys. Rev. Lett. 55,441)has questioned the identification A = 2 C - C, [eq. (37)] o n the grounds that it cannot be deduced rigorously by equating the right-hand sides of eqs. (36)and (24). The argument leading t o eq. (22b) provides some direct evidence for identifying 2C- C, with the inertia associated with local reorientation of 1. hut this approach can be criticised hecause it ignores possible collisional contributions to the inertia. The experimental evidence to date (D.N. Paulson, M. Krusius and J.C. Wheatley, 1976. Phys. Rev. L e t t . 36, 1322) suggests only that A is small. Ah/?rnp, S O . 1 .

CHAPTER 3

THERMAL AND ELASTIC ANOMALIES IN GLASSES AT LOW TEMPERATURES BY

S. HUNKLINGER* and A.K. RAYCHAUDHURI** Max-Planck -Instirut fur Festkorperforschung, Heisenbergstrasse I , 7000 Stuttgart 80,FRG

Present address: Institut fur Angewandte Physik der Universitat Heidelberg. 6900 Heidelberg, Federal Republic of Germany. * * Present address: Department of Physics, Indian Institute of Science, Bangalore 560012, India.

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

Contents 1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Insulating glasses and the tunneling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Low temperature specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Ultrasonic e x p e r i m e n t s . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Tunneling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 2.4. Consequences of the tunneling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Ultrasonics revised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. "Transverse relaxation" of tunneling states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Thermal expansion . . . . . . . . . ....................................... 3. Metallic glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Specific heat and thermal conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultrasonic properties of normal conducting metallic glasses . . . . . . . . . . . . . . . . . 3.3. IJltrasonic properties of superconducting metallic glasses . . . . . . . . . . . . . . . . . . . 4. "Glassy properties" of disordered crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Ionic conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Orientationally disordered crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Radiation damaged crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.Two-phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Ongin of the tunneling systems - theoretical attempts . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Connection hetween low temperature anomalies and the glass transition . . . . . . . . . teniperaturc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 269 269 273 280 284 287

291 295 298 298 301 311 318 319 323 325 327 329 332 338 339

1. Introduction

It is more than a decade since Zeller and Pohl (1971) presented clear and unambiguous evidence that below 1K the thermal properties of amorphous insulating solids differ markedly from their crystalline counterparts. The specific heat of amorphous solids (see fig. 1) is much larger than that of crystals, whereas their thermal conductivity is considerably lower (see fig. 2). In pure and defect-free dielectric crystals both quantities are proportional to T 3 at temperatures below 1 K. In amorphous solids, however, t h e specific heat is almost linear and the thermal conductivity varies almost quadratically with temperature. Previous to this pioneering work there was already evidence that amorphous solids show unusual behaviour at low temperatures (Fisher et al., 1968; Hornung et al., 1%9;

Fig. 1. Specific heat as a function of temperature of vitreous silica Suprasil W ((1.5 ppm OH content) and Suprasil (1200 ppm OH). The Debye specific heat is indicated by a dashed line. (After Lasjaunias et al., 1975.)

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[Ch. 3, 51

TEMPERATURE f K) Fig. 2. Thermal conductivity as a function of temperature of vitreous silica and crystalline quartz (after Zeller and Pohl, 1971). The amorphous solid exhibits a characteristic plateau at a temperature where a maximum is observed in crystalline materials.

Heinicke et al., 1971). It was, however, t h e work of Zeller and Pohl, which stimulated a wide variety of experimental and theoretical investigations studying these low energy excitations manifesting themselves below 1 K. This review attempts to present a comprehensive survey of t h e various investigations done since then. We try to make the basic concepts clear rather than give a bibliographical account. Nevertheless most of the major contributions in this field will be covered. In order to keep the Article of a manageable size, we have restricted ourselves to experiments done below 1 0 K . The review is divided into seven sections. After presenting the basic observations in insulating glasses and t h e models required to explain them (section 2), t h e recent works on metallic glasses (section 3) and disordered crystals showing glassy properties (section 4) will be discussed. A brief survey of the theoretical suggestions trying to explain the origin o f low energy excitations is presented in section 5. Finally, in section 6 we report on recent observations indicating a connection between low temperature anomalies and the glass transition temperature.

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2. Insulating glasses and the tunneling model 2.1.

L O W TEMPERATURE SPECIFIC HEAT

At low temperatures the only excitations contributing to the specific heat of pure dielectric crystals are acoustic phonons of long wavelength. In this temperature range solids can be treated as an elastic continuum. For T < 0/100, where 8 is the Debye temperature, the specific heat can be calculated from the measured values of the elastic constants using Debye’s theory, which predicts a T3-dependence. In contrast, the specific heat of amorphous, or more generally speaking disordered solids cannot be deduced from the Debye theory, though t h e wavelength yf t h e dominant phonons at this temperature is of the order of 1000A and thus is large, compared to t h e scale of atomic disorder. The different behaviour observed in the specific heat of crystalline SiO, (quartz) and amorphous SiO, (vitreous silica) (Zeller and Pohl, 1971; Lasjaunias et al., 1975) becomes obvious from fig. 1. The specific heat of crystalline quartz is given by C = 0.55T3p.J/gK and is thus slightly smaller than t h e Debye specific heat of vitreous silica which is represented by a dashed line in fig. 1. The specific heat of the amorphous modification is considerably higher, although t h e elastic constants differ only slightly. We subtract t h e phonon contribution C, calculated from Debye’s theory from the measured specific heat C, defining in this way the “excess” specific heat C, = C - C,,, which is characteristic of the amorphous state. This additional heat capacity C, can be approximated by

C,= a,T”’ + a , T 3 .

(1)

The exponents are 8, = 0.22 and S,, = 0.3 for Suprasil and Suprasil W, respectively (Lasjaunias et al., 1975). Since 6 is always small, specific heat well below 1 K is often said to be “linear”. The importance of the different terms for the specific heat is clearly demonstrated by their contribution at 1 K. For vitreous silica Suprasil I the numerical values are 1.65, 1.0 and 0.8 pJ/gK for t h e linear, t h e “excess” cubic and t h e Debye term, respectively. From the measured specific heat one can easily deduce the density of states n ( E ) of the excitations with energy E present in the material*. If the specific heat varies with power law TITm, then n(E) E m .Therefore t h e “excess” density of states n ( E ) can be expressed as

n(E)= A&’ + A , E 2 ,

(2)

For convenience we often express energy inconsistently in units of temperature, i.e. we do not multiply with Boltzmann’s constant k.

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[Ch. 3, 92

where A, and A2 are constants characteristic of t h e particular amorphous solid. A deviation from perfect linearity of the specific heat at the lowest temperature could indicate that the density of states is not completely uniform. But this conclusion is only valid if the internal relaxation times are much shorter than the time constants determined by the procedure of measurement. In a subsequent section we will see that t h e finite value of S is indeed caused by such relaxational effects. Surprisingly, the excess specific heat is not only observed in specific, but rather in all amorphous solids, irrespective of their chemical composition and structure. In fig. 3 we show as examples t h e specific heat of polymethylmethacrylate (PMMA) (Stephens et a]., 1972), of the semiconducting element a-Se (Stephens, 1976), and of the metallic glass PdZr (Graebner et a]., 1977). In metallic glasses t h e "excess" specific heat is usually masked by the contribution of free electrons. However, in superconducting glasses like PdZr the contribution of electrons vanishes at temperatures well below t h e superconducting transition temperature and again the "excess linear" specific heat is observed in these materials.

F

'

a

/

/

,

0 1305

,

,I

1

,- _ _ -Se

1

L

1

L

-

01 02 05 1 TEMPERATURE I K )

Fig. 3. Specific heat versu\ temperature o f different amorphous solids: PMMA (after Stephens et al.. 1072). a-Se (after Stephens. 1976). superconducting metallic glass Zr,oPda (after Graebner et al., IW7) and Na-P-AI201 (after Anthony and Anderson, 1977).

Ch. 3, $21

ANOMALIES IN GLASSES AT LOW TEMPERATURES

271

In fig. 3 the specific heat of the crystalline fast ionic conductor Na-pA1,0, is shown (Anthony and Anderson, 1977). The curve shows clearly that the existence of an enhanced low temperature specific heat is not restricted to amorphous solids only (see section 4). It has to be mentioned that in contrast to impurities in crystals, impurities in glasses only have a minor influence. As can be seen from fig. 1, the specific heat of vitreous silica increase only slightly with the impurity content. On the other hand the density of states of the additional excitations does seem to depend on the chemical nature of the amorphous solid. In random structures with a high local rigidity, i.e. in covalently bonded amorphous solids with higher coordination numbers, the density of states is smaller than in two-fold coordinated silicate glasses or polymers. This has been demonstrated by measurements of the specific heat of a-As and a-Ge. In a-As (Loponen et al., 1982), which is three-fold coordinated, the value of the constant a , is reduced by roughly a factor of three, compared to that of vitreous silica. In a-Ge (v. Lohneysen and Schink, 1982) with its four-fold coordinated covalent bonds, no characteristic excess states could be detected. Similar conclusions can be drawn from studies of the velocity of sound (see section 2.2). In first experiments on a-Si and a-Ge (v. Haumeder et al., 1980; Bhatia and Hunklinger, 1983)n o evidence for excess states could be found in oxygen free films. Measurements with higher sensitivity (Duquesne and Bellessa, 1983; Tokumoto et al., 1984) have demonstrated that such states are indeed present, although their density of states is an order of magnitude smaller than in vitreous silica. Brillouin scattering experiments (Heinicke et al., 1971; Vacher et al., 1980) and measurements with tunnel junctions up to 400 GHz (Rothenfusser et al., 1984)have shown that in the long wavelength limit the phonon concept also holds for amorphous materials. Consequently it may be concluded that Debye’s theory is applicable to the phonon contribution to the specific heat and the “excess” specific heat C, can be defined. This theory is not valid at higher temperatures because of phonon dispersion and the possible existence of localized model. Therefore we are not even able to define an excess specific heat at higher temperatures. Nevertheless we want to compare the specific heat of vitreous silica with that of their crystalline counterparts also at elevated temperatures. Most crystals show a deviation from the Debye T3-law for T > @/loo. The extent of deviation and the temperature where it occurs depend on the particular properties of the material, namely its phonon dispersion curve. For the discussion of this question the plot C/T3versus T2is more appropriate. In fig. 4 we show such data for vitreous silica, quartz and cristobalite (Bilir and Phillips, 1975). All these substances have the same chemical composition but differ in structure. Vitreous silica, t h e amor-

272

[Ch. 3, 02

S. HUNKLINGER AND A.K. RAYCHAUDHURI I

I

f

I

1

CRISTOEALITE

I

0

I

100

1

200

I

300

1

LOO

1 500

T ~ I K ~ I Fig. 4. Specific heat of the three modifications of S i G quartz, vitreous silica and cristobalite plotted as CIT' against T2. (From Bilir and Phillips, 1975.)

phous phase, has the least density ( p = 2.20 g/cm3) and an open structure. It has a maximum in C / T 3around 10 K t h e magnitude of which is about three times higher than that of crystalline quartz ( p = 2.65 g/crn'). Cristobalite is another crystalline modification of silica with a relatively low density ( p = 2.32 g/cm3). Surprisingly, t h e specific heat of vitreous silica and that of cristobalite are more similar than those of quartz and cristobalite. On the other hand, at low temperature the specific heats o f both crystalline phases approach t h e Debye value whereas t h e amorphous phase exhibits t h e "excess" specific heat discussed above. The maximum in C/T' for crystalline solids is due to contributions from phonons with 3 small group velocity and hence a high density of states. Thus t h e specific heat of cristobalite indicates that there the (transverse) acoustic phonons with a large wave vector have extremely low energies (Bilir and Phillips, 1975). This idea is supported by neutron scattering experiments. It is tempting to extend this explanation to vitreous silica, although at high phonon frequencies the plane wave picture breaks down in amorphous solids. Very recent studies of vitreous silica by inelastic neutron scattering (Buchenau et al., 1984) indicate that at vibrational frequencies well below 1 THz acoustic phonons contribute only a small part to t h e observed scattering intensity. The additional vibrational modes have been ascribed t o the coupled rotation o f SiO, tetrahedra and are highly localized. So far it is not clear whether these

Ch. 3, 921

ANOMALIES IN GLASSES AT LOW TEMPERATURES

273

states are related to those introduced by the fracton concept (Derrida et al., 1984) in order to describe the dynamic properties of amorphous polymers. An alternative explanation (Karpov and Parshin, 1983) is based on t h e occurrence of strong anharmonicities in the local atomic potential due to fluctuations in the structural parameters. This leads to singularities in the energy density of states of t h e anharmonic oscillators associated with these local instabilities.

2.2. ULTRASONIC EXPERIMENTS It has been said previously that the thermal conductivity of amorphous solids is considerably reduced compared to that of their crystalline counterparts (see fig. 2), thus indicating that the phonon mean free path is much shorter in glasses. A similar type of behaviour is also found in doped alkali halides (Narayanamurti and Pohl, 1970). where the motion of dopant molecules gives rise to additional excitations. There t h e short mean free path of phonons originates in resonant scattering from these excitations. In glasses, phonon scattering has been extensively investigated by thermal conductivity measurements and examination of ultrasonic properties. These investigations have established that the excitations seen in specific heat, or at least a fraction of them, interact strongly with phonons, i.e. they intimately couple to lattice distortions. They are localized in space and exhibit a two-level type nature with a broad distribution of the level splitting. In the following we discuss first the acoustic measurements and then return to the thermal transport studies conducted o n glasses at low temperature. Since our aim here is to bring out the essential physics of the dynamics of these low energy excitations, we make references to experiments only with this in mind. For more details we refer to previous review articles (Hunklinger and Arnold, 1976; W.A. Phillips, 1981). In fig. 5 we show the typical result of a microwave acoustic absorption measurement (Hunklinger et al., 1973; Hunklinger, 1977). Below 1 K the attenuation increases with decreasing temperature if t h e acoustic intensity is very low, but decreases at higher intensities. This observation, known as saturation of the attenuation (Hunklinger, 1972; Golding et al., 1973), is a very valuable observation because it limits t h e number of models which are able to describe t h e low temperature anomalies. Without going into detail w e state that the low energy excitations are best described in terms of two-level systems (TLS). We postpone t h e discussion of their microscopic origin and consider first their influence on the propagation of sound by a phenomenological approach. As we will see in the following sections the acoustic and dielectric properties can be described using a constant density of states n ( E )= no.

274

S. HUNKLINGER A N D A.K. R A Y C H A U D H U R I

-

[Ch. 3, 92

V I TRE3U 8 SILICA

Fig. S, Temperature dependence of the ultrasonic absorption in vitreous silica for longitudinal waves at I G M . At higher intensities a continuous decrease with decreasing temperature is observed, whereas at low intensities the ahsorption rises again below 0.7 K. For comparison the ahsorption of a quartz crystal is also included. (From Hunklinger. 1977.)

This means we may put 6 = 0 and A?= 0 in eq. (2). We will raise the question of the correct density of states again in later sections. At low temperatures the most likely interaction between a phonon and a TLS is the resonant scattering where the energy splitting E coincides with t h e phonon energy hw. It leads to an absorption* ares[Hunklinger and Arnold (1976); a careful treatment of the saturation effect is given in Graebner et at. (1983)l: %s

=

n,M' ~

hw tanh __ pu3 (1 + l/&)''* 2kT . WT

(3)

The deformation potential M describes the average transition probability between the two levels under the influence of the strain field of a sound * For simplicity we do n o t write down indices defining longitudinal or transverse polarization explicitly except in cases where they are absolutely necessary. Furthermore we neglect t h e tensorial character o f the elastic strain field as well as that of the deformation potentials.

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275

wave. u stands for the sound velocity, p for the mass density, and w for the circular frequency. The resonant absorption of a sound wave is proportional to the difference in population of the lower and upper levels which we denote by AN. If the acoustic intensity I is well below a critical value I,, the resonant process does not change AN noticeably. Hence AN is proportional to tanh(hw/2kT), the population difference in thermal equilibrium. As the acoustic intensity increases, the upper level becomes more and more populated. AN starts decreasing from its thermal equilibrium value and the attenuation coefficient decreases. In eq. (3) this fact is taken into account through the factor (1 + I/IC)-ln.The acoustic intensities applied in the experiment shown in fig. 5 lie above and below the critical value I,. In the limit I 41, the absorption becomes am 0: w tanh(h42kT). Measurements at small intensities and down to very low temperatures have verified this relationship (Golding et al., 1976a). The dependence of a m on temperature and intensity is characteristic of the existence of only two levels and proves that no third level of comparable energy splitting exists. When the intensity of the sound wave is rather high (I% Ic), we finally reach a situation where AN = O and the attenuation ams approaches zero. This is called saturation and reflects the dynamic equilibrium between excitation and recombination processes. The critical intensity 1, is proportional to t h e bandwidth Aw within which the TLS are excited and to the ratio between the recombination rate T;' and t h e probability for excitation. This can formally be expressed as h 'pv' I, = -Aw . 2M2T,

(4)

The relaxation time T, is the time within which TLS return to thermal equilibrium after a perturbation of their population. It is the same as the spin-lattice relaxation time if we use the spin analogy (Hunklinger and Arnold, 1976). In dielectrics this time is determined by the strength of the interaction between TLS and phonons. At low temperatures, where the direct or one-phonon process is dominant, the relaxation rate of a TLS with energy splitting E is given by (Jackie, 1972):

The indices 1 and t indicate the polarization of the interacting phonon branches. For any phonon-assisted process t h e third power is the weakest energy dependence. Higher order phonon processes will give relax-

27h

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, 52

ation rates with much stronger energy dependence and will be inconsequential at lower temperatures. In metallic glasses the expression for 7-y’ is modified since free electrons also contribute to t h e recombination process. We will discuss this point in detail in section 3. The width AO in eq. (4) is determined by the relaxation time T2 describing the interaction among t h e TLS (see section 2.6). T2 can also be called the transverse or spin-spin relaxation time, if we again apply the spin an a logy. The relaxation of the TLS also gives rise to a second absorption process (Jackie, 1972). A sound wave travelling through an amorphous solid modulates the level splitting of the TLS by an amount of SE and thus disturbs their thermal equilibrium. Because of the finite value of T I the response of the TLS is delayed with respect to the strain of t h e sound wave. resulting in energy dissipation and hence attenuation. This attenuation is given by (Jackle et al., 1976):

E w2T, d E sech’ 2kT 1+ w2T: The deformation potential D is defined by 6E = D * e, where c is t h e strain field of the applied sound wave. This process of sound attenuation, commonly known as the “relaxation process”, is non-resonant in character in contrast to t h e process described previously. Therefore, all t h e TLS which can be excited thermally ( E S 3 k T ) take part in the process, i.e. one has to integrate over all energy splittings. The integration of eq. (6) can be carried out analytically in t h e two limiting cases w T , S 1 iind wT, G 1. In t h e case of very low temperatures or high frequencies the condition wT, 1 holds and eq. (6) results in an attenuation

Due to the non-resonant character, the relaxation absorption cannot be saturated in general and can also be observed at high acoustic intensities (see also section 3.2.2). Thus the absorption at higher power levels is due to this process. As expected, t h e strong increase of absorption shown in fig. 5 is proportional to w ” T 3 in agreement with eq. (7). This behaviour has been observed not o n l y in vitreous silica, but in many other disordered dielectrics as well. According to theory, t h e exponent in t h e

Ch. 3, 42)

ANOMALIES IN GLASSES AT LOW TEMPERATURES

277

temperature dependence of the absorption a,, must be equal to the exponent of the energy dependence of the relaxation rate [see eqs. (5)-(7)]. Thus the measurements strongly support the assumption that t h e relaxation time of the TLS is due to the direct TLS-phonon interaction. The other limit ~ 7 ' 4 will 1 be discussed in a following section, after having introduced the Tunneling Model. An important quantity in acoustic experiments is the variation of the sound velocity with temperature or frequency. It is related to the absorption via the Kramers-Kronig relation. Thus both absorption processes contribute to t h e variation of the velocity. Provided that n(E)= no= constant and hw < kT, the resonant interaction leads to a logarithmic (PichC et temperature variation of the velocity Av/u = [ u ( T ) - v(To)]/u(To) al., 1974; Hunklinger and Arnold, 1976):

where To is an arbitrary reference temperature. It is interesting to note that the contribution of the resonant process to the sound velocity is independent of the measuring frequency, whereas a, depends strongly on frequency. The validity of eq. (8) has been verified in a number of insulating glasses. In fig, 6 we present data on vitreous silica (PichC et al.,

0 SUPRASIL

n

I

m QUARTZ CRYSTAL

6

>

02 0.3

0.5 1 2 TEMPERATURE [ K )

3

5

Fig. 6. Relative variation of longitudinal sound velocity Avlu = [ o ( T ) - v ( T ~ ) ] / u ( T in~ ) vitreous silica against temperature. Full squares show for comparison the constant velocity in quartz crystals. (From Hunklinger, 1977.)

27 8

ICh. 3, $2

S. HUNKLINGER AND A.K. RAYCHAUDHURI

1974; Hunklinger, 1977). Clearly at the lowest temperature the sound velocity varies logarithmically with temperature. The contribution of the relaxation process is negligible compared to that of the resonant process as long as wT,s= 1. At high enough temperatures, however, when wT,= 1, the relaxation process becomes dominant. Since its contribution is negative, the sound velocity starts to decrease again (see fig. 6). At frequencies in the MHz-range t h e maximum is observed around a few Kelvin. From the slope of the logarithmic rise at low temperatures the coupling parameter n&f2 can be easily, and rather precisely, determined. Therefore, accurate measurements of the sound velocity not only demonstrate the presence of low energy excitations, but also give a measure of their "strength". Without a detailed discussion we want to point out that there is a close relationship between acoustic and dielectric low temperature properties. If the TLS couple with electric fields, the behaviour of t h e dielectric properties is expected to be analogous to that of the acoustic properties. As an example we show the saturation of the dielectric absorption at 10 GHz in vitreous silica Suprasil I (v. Schickfus and Hunklinger, 1974) (see fig. 7). Clearly the dielectric absorption decreases with increasing electromagnetic intensity. This and other experiments demonstrate that TLS

,---l'o-3

i5

INTENSITY

SUPHASIL 1 I1200 p p m OH1

- IWlCrn21

lOGtl7

c

I

/

r/

k 0

w

_1

! n

5 '-

178

A-

02

I

05

410-5

/I I

I

/

1 2 TEMPERATURE ( K

I

I

5

10

1

Fig. 7. Temperature dependence of the dielectric absorption in vitreous silica Suprasil 1 (1200 ppm OH) at 10 GHz for different microwave intensities. The dashed line indicates the T'-contribution of the relaxation process. (After v . Schickfus and Hunklinger, 1974.)

Ch. 3, $21

ANOMALIES IN GLASSES AT LOW TEMPERATURES

279

also possess an electrical dipole moment. Furthermore a temperature variation of the dielectric constant has been observed, which is analogous to the variation of the sound velocity. Here we do not discuss this aspect but refer to a review article published previously (Hunklinger and v. Schickfus, 1981). Let us now briefly consider thermal conductivity A. From simple kinetic theory one obtains for dielectric crystals: A

= :C,vl.

(9)

At low temperatures the mean free path 1 of thermal phonons in pure dielectric crystals is limited by the size of the sample. Therefore, the thermal conductivity is proportional to C, and is consequently proportional to T3. As shown in fig. 2 for vitreous silica, the thermal conductivity of disordered solids also behaves anomalously at low temperature. It is roughly proportional to T2 and its magnitude depends only slightly on chemical composition. Assuming that t h e TLS are localized in space and do not contribute to the transport of heat, only phonons have to be considered in dielectrics. Because eq. (3) is also valid for thermal phonons, we may put I = 0 and tanh EI2kT = 1 resulting in I-' = arax w. Since the energy of the dominant thermal phonons is proportional to T, we obtain 1 7'-' and consequently A = T2. In a more careful calculation, integrating over all frequencies and adding both phonon polarizations, one finds:

The coupling parameters n&4: and n f l : can be determined from measurements of the resonant absorption or from the velocity of sound [see eqs. (3) and (S)]. It is worthwhile to note that the calculated thermal conductivity using coupling parameters from velocity measurements agrees very well with the experimental value at the lowest temperatures (Hunklinger and PichC, 1975). It should be pointed out that one expects a perfect T2-dependence of the conductivity, if the density of states is constant. However, for all t h e dielectric glasses investigated so far, it has been observed that A = TC where 5 lies between 1.75 and 1.95 (Raychaudhuri et al., 1980). This deviation could arise from a non-uniformity of the density of states, but in this case a corresponding deviation from the perfect logarithmic temperature variation of the velocity of sound should appear. Although some evidence for such a correlation has been reported (Golding et al., 1976b).

180

[Ch. 3, §2

S. HUNKLINGER AND A.K. RAYCHAUDHURI

the low value of 6 is not fully understood. As shown in fig. 2, for vitreous silica the thermal conductivity of amorphous solids reaches a plateau above a few Kelvin. This observation clearly indicates that the mean free path of the dominant thermal phonons drops drastically with temperature. Many attempts have been made to explain this phenomenon (Anderson, 1981) but we feel that so far no unambiguous answer exists.

2.3. TUNNELING MODEL Since the anomalous properties of amorphous solids are observed down to very low temperatures, we are confronted with the question, what is there in a solid that can give rise to such small energy splittings? For example in crystalline alkali halides such small energies originate in the tunneling of impurity atoms or molecules like Li’, CN- or O H (Narayanamurti and PohI, 1970). Therefore it has been suggested that the TLS in glasses are caused by the tunneling motion of groups of atoms. This is the origin of t h e so-called “Tunneling Model”. (W.A. Phillips, 1972; Anderson et al., 1972). In the regular lattice of a crystal all atoms or molecules occupy a well defined position, allowing only one possible configuration. In contrast the random network of an amorphous solid can be realized as a large number of different configurations. Therefore the basic assumption of t h e “Tunneling Model” seems to be quite natural, namely that certain atoms or clusters of atoms can occupy at least two different positions or configurations of almost equal potential energy (see fig. 8). For a formal description we may introduce “particles” of still unknown microscopic nature moving in double-well potentials. In an isolated well such a particle has a series of

I

i I

\

\

i

\

! ~

t-

d

-_j

Fig. 8. Double well potential with barrier height V, asymmetry energy A and distance d . M2/2 i s the ground-state energy of the tunneling particle with mass m .

Ch. 3, $21

ANOMALIES IN GLASSES AT LOW TEMPERATURES

28 1

vibrational states separated by an energy hR which is of the order of the Debye energy. At low temperatures we are only interested in the ground states with the wave functions $L and ll/R for the particle located either in the “left” or “right” well, respectively. The separation in the energy of the two minima is often referred to as the “asymmetry” A (see fig. 8). The energy splitting due to the tunneling of the “particle” between t h e two configurations can be approximated by A, = hR e-A. The tunneling parameter A = d(2mV/h2)” reflects the overlap of t h e wave functions ll/L and (LR. Here d is the separation between t h e two wells (in any configuration space), m the effective mass of the tunneling particle and V the barrier between the two minima. In the basis (&, ll/J the Hamiltonian of a single tunneling system is given by

A 2 -A,

-A

Because of the tunneling, (jlL and $R are not true eigenstates. In the orthogonal basis with the eigenstates $+and I+- the Hamiltonian reads

H,

=-

where

E = (Az +

(13)

Thus in the basis ($+, $-) we get a TLS with energy splitting E. Because of the random amorphous network the parameters of the tunneling systems (henceforth we will denote them by TS) do not have well defined values but are expected to vary over a wide range. In the Tunneling Model the quantities A and A are assumed to be independent of each other and to have a constant distribution P(A,A)dAdh=FdAdA,

(14)

where F is a constant. It must be emphasized here that a constant distribution of A and A is one of t h e most important assumptions o f the Tunneling Model, which agrees amazingly well with most experimental results. Recently, a more concrete microscopic approach has been published (Karpov et at., 1983), which is based on an explicit model for t h e double-well potential. Although this description is able to specify important features of the TS. we will stick in o u r further discussion to the original Tunneling Model, because it is already capable of explaining most observations, despite its simplicity.

282

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, P2

For further discussion it is more convenient to rewrite eq. (14) using new variables E and u = A,/E -

P ( E , u ) d E du

=

P

,R

u(1- u )

d E du.

This function is energy independent and is shown in fig. 9. At the boundaries of the allowed interval, i.e. at u = 0 and u = 1 the distribution function tends to infinity. In order to keep the number of TS finite, some sort of a cut-off has to be introduced for u + 0, which will be discussed in the following section. The spectral distribution P(E, u) may be split into two parts Pf and P,, representing those TS, which have preferentially large and small values of u, respectively (Hunklinger, 1984):

P ( E , u ) =P , + P , = P '

U

((1- U Z ) ' R

+

U

where P' is again a constant. Variations of the additional parameter W shift the relative weight of the two branches, but for the moment we put P' = p and W = 1. Of course the procedure we have just described is not the

U=

AOIE

Fig. 9. Distribution function P(E, u ) against u. The dashed lines represent P, and P. for W = 1 (see text).

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ANOMALIES IN GLASSES AT LOW TEMPERATURES

283

only way to modify the spectral distribution and more sophisticated attempts have been proposed (Golding et al., 1978; Doussineau et a]., 1980). The major drawback in all these attempts, however, is the fact that new free parameters have to be introduced without knowing the underlying physical meaning. TS couple to their environment via strain and electric fields. Since both mechanisms can be described in the same way, we will only discuss interaction via strain fields. The interaction can be accompanied by a simultaneous transition of the tunneling particle from one well to the other. This so-called phonon-assisted tunneling leads to a variation of A,,. If the position of the particle is not changed during interaction, then the asymmetry A is varied. These two processes can formally be described by the introduction of the deformation potentials y4 and y A . As already pointed out we neglect the tensorial character of the relevant quantities and define

Thus the Hamiltonian of interaction can be written as:

Usually it is assumed that yA % y4, meaning that strain fields mainly couple to the asymmetry A . At first glance this is surprising because geometrical parameters of the TS determine A , in an exponential way. Nevertheless coupling constants associated with the variation of the geometry are expected to be rather small, namely of the order of the energy splitting itself (W.A. Phillips, 1973; Hunklinger and Arnold, 1976). However, it should be mentioned that based on observations on tunneling states in doped crystals, it has also been argued that y , may even be of the order of 300 K (Case et al., 1972; Fisher and Klein, 1980). A more effective way of varying the energy splitting could be a variation of the local environment of the TS. A deformation may enhance or reduce the irregularity of the structure in the neighbourhood, thus leading to a rise or decrease of the asymmetry A . Consequently the resulting deformation potential can exhibit both signs. In addition larger values are possible and we will see that deformation potentials of the order of 1 eV or lo4K are necessary to explain the acoustic properties (Golding and Graebner, 1976; Hunklinger and Arnold, 1976; Golding et al., 1979). Recently, the experimentally observed magnitude of y has also been estimated theoretically (Karpov et al., 1983). Because of these arguments we

284

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, $2

will neglect yJ, in our further discussion and use only one coupling constant y = yJ. The interaction Hamiltonian of eq. (18) thus simplifies to

In t h e basis ($+, &) this becomes

2.4.

CONSEQUENCES OF THE TUNNELING MODEL

In this section we will discuss the differences in the theoretical predictions based on a constant distribution of TLS and on t h e Tunneling Model. We rewrite the total Hamiltonian from eqs. (13) and (20)

where

D

=

2yAIE and M

=

-

yA,/E.

(22)

The coupling constants D and M correspond to those used previously for the TLS. They were assumed to be constant and independent. In the Tunneling Model only the coupling constant y exists, but the effective constants D and M now depend on the overlap of the wavefunctions GL and t,bR, i.e. on the ratio of A / E and d,/E. For a description of the acoustic response of the TS, we may use eqs. (3)-(8), but we have to carry out an additional integration. If we apply the distribution function P ( E , u ) this integration has to be performed with respect to u. In the following we will discuss this procedure for the most important cases. First we consider t h e density of states which was assumed to be constant in the TLS picture. Integration of eq. (15) results in divergence of n(E)because of the strong rise of P ( E , u) for small values of u. This divergence can be avoided either by modifying P(E, u + 0) or by introducing a minimum value for u. Not knowing which approach is correct, we introduce a minimum value urnin.Thus we obtain

2

n(E)=FIn-=

urn"

2E

Fin-. 4,mi"

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ANOMALIES IN GLASSES AT LOW TEMPERATURES

285

As we will see below, dynamic measurements of the specific heat (Zimmermann and Weber, 1981a) indicate that u,,, must be smaller than In these experiments the energy splitting of the relevant TS corresponds to 1K. Consequently the minimum value of the tunnel splitting do,^,, must be smaller than 1 FK. Eq. (23) leads to a constant density of states no, if do,^,, is proportional to E, i.e. if urninis a constant. Otherwise t h e density of states of t h e TS will depend weakly on energy, resulting in a deviation of the specific heat from its linearity. We will see that this small effect would be masked completely by another phenomenon, even if it existed. The main difference between the TLS picture and the Tunneling Model lies in the fact that in the latter the relaxation time Tl [see eqs. (5) and (22)] shows a distribution even if the energy splitting E is kept constant. For do= E the relaxation time has a minimum value T l , , and with the decreasing value of Ao/E the relaxation time TI increases and tends to infinity. The wide distribution has interesting consequences: the experimentally observable density of states becomes a function of the time r of measurement. If the experiment is performed over longer and longer time scales, more and more TS will have a chance to couple to t h e perturbation produced in the experiment. This fact is formally expressed through the following relation:

4t P(E, t) = - In -. 2 T1,m P

As a consequence the specific heat should also become time-dependent: 2 4t 7 T C(T, t ) = -Pk2T In -. 12 TI,m

This interesting aspect, known as “time dependent specific heat”, has been studied by different groups. In the first experiments (Goubau and Tait, 1975) the heat diffusion across a thin glass plate was measured, but no satisfactory agreement with theory could be achieved. In more recent experiments (Loponen et al., 1980; Meissner and Spitzmann, 1981; Loponen et al., 1982; Knaak and Meissner, 1984) thin plates of glass were uniformly heated and the “diffusion” of phonons into the “bath of TS” was registered. In this way a time dependence of the specific heat could be observed, covering a time scale of 10 ~s to 100 ms. For the “linear” term reasonable agreement was found with eq. (25), demonstrating that a broad distribution of relaxation times does indeed exist. However, these experiments do not present an answer to the question,

286

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, $2

why the heat diffusion across thin glass plates (Goubau and Tait, 1975) cannot be understood within the framework of the Tunneling Model? Much longer relaxation times can be studied by “thermal relaxation” experiments (Zimmermann and Weber, 1981a; Loponen et al., 1982). At temperatures around 1 K relaxation times as long as 104s have been reported for vitreous silica. O n t h e other hand the minimum relaxation time T,,,, is rather short: it is known from ultrasonic experiments that T,.,, is generally of t h e order of 1 ns for TS having an energy splitting corresponding to 1 K. Therefore we may state that the relaxation times span at least 13 orders of magnitude. As mentioned before, the tunnel splitting determines the relaxation time and according to eqs. ( 5 ) and (22), the relation T,,,/T, = (d,,/E)’= u2 holds. Thus we may use the “thermal = relaxation” measurements to give the above mentioned lower limit urnin Ao,,,/E = lo-‘. In contrast to the behaviour of the “linear term” no time dependence is observed for the excess cubic term of t h e specific heat (Loponen et al., 1980; Meissner and Spitzmann, 1981; Knaak and Meissner, 1984). This qualitatively different behaviour clearly demonstrates that t h e underlying excitations are of different nature. We will see that a similar conclusion can be drawn from acoustic experiments and we will consider this question once again in section 2.5. Surprisingly this fascinating result has not yet attracted more attention. In section 2.1 we discussed the specific heat and the density of states without specifying the time scale. We simply assumed that the specific heat is a well-defined quantity. But we have just seen that the relaxation times can be extremely long, whereas measuring times are usually of the order of 10 s. This means that the experimental data shown in figs. 1 and 3 do not represent the total specific heat. We have to use eq. (25) and a constant value of P will lead to a temperature dependence of the specific heat with a slightly increasing gradient. Putting in plausible numbers for the time of observation t and T,.,, obtained from acoustic measurements, good agreement with the experimentally determined specific heat is achieved (Zimmermann and Weber, 1981b). In eq. (1) a factor S was introduced to describe t h e deviation from the linear temperature dependence. Of course this factor can only be an approximation and its value should depend o n the ratio t/T,,,. However, in order to compare theory and experiment quantitatively one would have to take into account that the time of measurement I depends on the diffusivity of t h e sample and its therefore temperature dependent itself and not properly defined in many long-time specific heat measurements.

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287

2.5. ULTRASONICS REVISED

In the sections 2.1 and 2.2 we introduced the concept of TLS and discussed how the dynamics of TLS show up in elastic experiments. We are now asking for the changes which must be introduced into the description of the acoustic properties, if we take into account the distribution of coupling parameters as proposed by the Tunneling Model. A minor alteration is required for the resonant interaction: in eqs. (3), (4), (5), (8) and (10) we replace no by p and M by y. This means that both concepts are equivalent and the experiment will not be able to distinguish between the two. The situation is not that simple in the case of relaxation. In eq. (6) we have to introduce the distribution function and carry out a second integration. As long as wT1.,,b 1 holds, only a slight change is necessary: the second integration leads to a factor 3 in the denominator of eq. (7), i.e. we have to replace t h e factor 32 by 96. Qualitative differences are expected at higher temperatures or lower frequencies when we enter the regime wT1,,,4 1. This is due to the fact that the main contribution to the relaxation absorption as well as to the accompanying velocity change is caused by relaxing states fulfilling the condition wTl = 1. Since in the TLS picture the coupling is constant, all existing TLS having an energy splitting comparable to kT relax so fast that wT, G 1 and their contribution is depressed. In the Tunneling Model there exists a distribution of the coupling constant and one can always find systems with wTl = 1 as soon as the condition wT1,,,< 1 is reached. Thus the relaxation absorption becomes temperature independent for wT,,,,6 1: Py2

%I=

-$,(

7 T u

mil -c-. 2v

The quantity C is identical to the one introduced in eq. (8). Since it is a direct consequence of the distribution assumed in the Tunneling Model, this simple result deserves some comment. It only contains the coupling factor py’ and no information on the mechanism of relaxation. The value of py2 can be obtained from the resonant absorption or the dispersion of sound velocity. Thus in the Tunneling Model, once we have determined py2 (or C) from the resonant process, the absorption coefficient in the plateau region is automatically determined. Any deviation from this expected behaviour would therefore imply a deviation of the distribution function P(E, u ) assumed in the Tunneling Model. In fig. 10 we show the normalized attenuation or internal friction

288

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S. HUNKLINGER A N D A.K. R A Y C H A U D H U R I

0

; 0

0

0.001 0.01

A

a-SiO, 3.17 kHz PMMA

90MHz

0.1 1 TEMPERATURE (K)

10

Fig. 10. Internal friction 0.' = ado of polymethylmethaaylate and vitreous silica as a function of temperature. At higher temperatures a clear plateau is observed in both cases. (After Federle and Hunklinger, 1982 and Raychaudhuri and Hunklinger, 1984.)

0.' = a v / w for the polymer PMMA (Federle and Hunklinger, 1982) and for vitreous silica* (Raychaudhuri and Hunklinger, 1984). In both cases we can clearly distinguish between the two regimes wT,,,,S- 1 and W T , , ,1.~ At the lowest temperatures the absorption is proportional to w 0 T 3as expected for relaxation of TS via thermal phonons [eq. (8)]. At higher temperatures a plateau as predicted by eq. (26) is observed. The presence of a plateau shows that in both materials the constant distribution of eq. (14) is a good approximation although the microscopic structures of these amorphous solids are fundamentally different. Similar results have been obtained for other amorphous solids (see, for example, Ng and Sladek, 1975; Jacqmin et al., 1983). In all cases the observed plateau is not an ideal one. This could indicate that minor deviations occur from the distribution assumed in the Tunneling Model. On the other hand it is also possible that a more sophisticated treatment of the relaxation process is necessary. For example, t h e effect of environmental vibrations o n the tunneling process (Heurov and Trakhtenberg, 1982) and L o w frequency acoustic measurements are usually carried out by applying a vibrating reed technique. In this type of experiment Young's modulus determines the acoustic behaviour. The resulting vibration is closely related t o the motion in a longitudinal wave. Since the difference between the two is not significant for our considerations, we will neglect it in our further discussion.

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the influence of finite phonon relaxation times (Laikhtman, 1984)have been studied. Both calculations lead to a modification of eq. (26), but no critical comparison with existing experimental results has been carried out so far. In the regime wTt,,,4 1 the temperature variation of the velocity of sound also reflects the relaxation mechanism. As long as wT,,,,%-1 the relaxation process gives only a negligible contribution to this variation compared to that of the resonant process [eq. (8)]. But for wTl.,, Q 1 we obtain :

provided that the TS relax via the direct process [eq. (5)]. Here Tois again an arbitrary reference temperature. A logarithmic temperature variation is therefore deduced for both the resonant and the relaxation interaction. Since they are of opposite signs, the total variation is given by

Av

-Av

I OI-

+-A v

= -zCIn-. I T Irel,ph

TO

Thus with increasing temperature we expect a logarithmic decrease of the velocity as soon as @TI,,, Q 1. The temperature variation in A u will show a maximum at a temperature where the cross-over between the two regimes occurs. This result is remarkable since strongly coupled TS contribute mainly to the resonant process and weakly coupled TS to the relaxation process. By measuring the velocity of sound over a wide enough temperature or frequency range, one can probe the distribution of the tunneling parameters and the validity of eq. (14) can be checked. Eq. (28) is only valid when TS relax via the one-phonon process. When other processes make dominant contributions to the relaxation process, the behaviour of Au will be different. If the relaxation rate is still proportional t o u2= Ai/Ez and shows a power law dependence on energy E, the logarithmic dependence of A u will persist, though with a different prefactor which reflects this power law dependence. One example is the presence of free electrons discussed in section 3. Another possibility is the occurrence of many-phonon processes at higher temperatures. For example, for Raman processes x T7)one would expect A V , , , ~ = 7 -zAu, (Doussineau et al., 1980), but no direct experimental evidence for this process exists in disordered solids so far. Until very recently all attempts t o verify eq. (28) experimentally failed in dielectric glasses. At radio frequencies oT,.,,approaches unity around a few Kelvin, so that a logarithmic decrease of the sound velocity can be

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expected at helium temperatures. Surprisingly enough a linear temperature variation was observed in this temperature range (Bellessa, 1978). So far there is no satisfactory explanation for this. The linear variation could be caused by a deviation from the constant density of states, by the relaxation of TS via higher order processes or by completely new phenomena of unknown nature. To verify the predictions of the Tunneling Model unambiguously, experiments have to be carried out at much lower frequencies, so that the condition @TI.,, = 1 is reached well below 1 K. Fig. 11 shows the results of a recent experiment carried out at 1kHz, using a vibrating reed technique (Raychaudhuri and Hunklinger, 1982a). In this case the condition w ? ’ , , = 1 is fulfilled around 50mK. Clearly the velocity increases logarithmically due to the resonant interaction, passes a maximum and decreases again logarithmically. In full agreement with theory, the prefactor of the decrease is only half that of the increase. Above 1 K, however, the decrease is much steeper, thus indicating why high frequency measurements d o not exhibit the expected logarithmic decrease. This study shows that the distribution of t h e tunneling parameter [eq. (14)] is in excellent agreement with the experimental observations up to a temperature of 1 K. In fig. 10 we have already shown the absorption at low frequencies. The existence of a plateau at higher temperatures indicates that the density of states is constant and free of irregularities around 1 K. This seeming discrepancy between the velocity and the absorption measurement w

c3

z a

l-

a

TEMPERATURE ( K ) Fig. 11. Temperature dependence of the sound velocity of a silica based microscope cover glass at 1028Hz. The dashed line indicates the logarithmic decrease of the velocity with temperature. Its slope is exactly -i of the logarithmic rise represented by the full line. A clear deviation from the expected behaviour is observed above 1.5 K. (From Raychaudhuri and Hunklinger, 1982.)

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demonstrates that above 1K a new relaxation mechanism, probably higher order phonon processes becomes relevant. Furthermore it is worthwhile pointing out the fact that in these dynamical experiments the assumption of a constant spectral density is sufficient. The excitations contributing to the excess cubic term in the specific heat remain invisible. As in t h e experiments on the short time specific heat, the constant and the quadratic part in the density of states [eq. (2)] behave completely different, thus implying that their origins are different as well. RELAXATION’’ 2.6. “TRANSVERSE

OF TUNNELING STATES

So far we have only discussed the interaction of the TS with external elastic fields. But there also exists an interaction between the TS themselves which is probably of elastic origin. This additional aspect is taken into account in the description of the dynamical behaviour of the TS using the Bloch-equations (see, for example, Joffrin and Levelut, 1975; Hunklinger and Arnold, 1976).As in the case of spin systems in a magnetic field we have to distinguish between T, and T,, the longitudinal and the transverse relaxation time respectively. The longitudinal relaxation time TI we have already introduced in our discussion. T2reflects the phase memory of the TS and is determined by the interaction between the TS. It should be mentioned that a completely different description of TI and T2 has been proposed (Kagan and Maksimov, 1980), in which the kinetics of a single tunneling particle are already characterized by two relaxation times. In this article we use the conventional description but do not introduce the Bloch formalism and its application to the TS since this has already been done in previous papers (Hunklinger and Arnold, 1976;W.A. Phillips, 1981). We will discuss some interesting aspects but not present a formal description. In section 2.3 we saw that the difference in the populations of the two states determines the attenuation of a weak acoustic or electric wave while an intense pulse leads to an equipartition of the two states. In this case a “hole” of width d w [see eq. (4)] is burnt into the spectral distribution at the energy corresponding to the applied frequency. If now a second, weaker probing pulse of slightly different frequency is sent through the sample, its attenuation will reflect the population difference at that frequency. Thus a measurement of the attenuation as a function of frequency difference between the two pulses traces out the spectral character of the hole created by the intense pulse (Arnold and Hunklinger, 1975; Bachellerie et al., 1977). The first experiment of this kind used 1ps acoustic pulses of 750 MHz propagating simultaneously through the glass sample (Amold and Hunklinger, 1975). Saturation can still be seen by the probing pulse if its frequency is displaced by as much as 50MI-k at 0.5K (see fig. 12). In view of the relatively small lifetime

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FREQUENCY OF SATURATING PULSE (MHz) Fig. 12. Resonant attenuation of a weak probing pulse of fixed frequency (738MHz)as a function o f the frequency of a strong saturating pulse (acoustic intensity J = 5 x 1 0 'W/cm2, pulse duration 1 ps). (After Arnold and Hunklinger. 1975.)

broadening (T,,,,is expected to be of t h e order of 10 ~ s and ) the small frequency uncertainty of t h e applied pulses (Af- 1 MHz) this result is rather astonishing. Furthermore it has been observed that t h e width of t h e hole depends linearly on temperature and increases with the time of measurement. This remarkable effect is ascribed to temporal fluctuations of the level splitting of the TS at resonance. At finite temperatures TS with an energy splitting comparable or smaller than kT continuously absorb or emit thermal phonons giving rise to strain fluctuation in the environment. Because of the coupling of the energy splitting of TS to strain these fluctuations will be accompanied by energy changes of the neighbouring TS. Therefore TS having an energy close to the phonon energy of t h e intense pulse will temporarily be in resonance with the sound wave, although at T = 0 an energy difference would exist. As a consequence TS can be excited within a wider energy range and a broad hole is observed in the attenuation of the weak probing pulse. The excursion of the energy splitting of a given TS will depend on time, since t h e contributions of the neighbouring TS add statistically. Thus the time dependence of the width of the burned hole ( h o l d et al., 1978) reflects t h e distribution of the relaxation times T , and consequently the distribution of the tunneling parameters. After very long times t h e hole width will reach its largest value when all the TS in the neighbourhood

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have changed their state. With increasing temperature the number of thermally activated TS rises and results in an increase of the observed hole width (Arnold and Hunklinger, 1975; Golding and Graebner, 1981). The mechanism of hole broadening is called “spectral diffusion” by analogy with the mechanism known from magnetic resonance experiments. It has been discussed in detail (Black and Halperin, 1977) and agreement between theory and experiment has been obtained. In principle such a two-pulse technique should also be suitable to study the relaxation time T I .Again an intense first pulse is applied to saturate the TS around the centre frequency of the pulse and the attenuation of a weak probing pulse is measured. But now the probing pulse has the same frequency and is delayed with respect to the intense pulse. By varying the delay one can monitor the recovery of the population difference. The first experiments of this type have been performed at relatively high temperatures, namely between 0.3 K and 1 K (Golding et al., 1973; Hunklinger and Arnold, 1976). However, as discussed above, spectral diffusion changes the spectrum of the excited states and saturation recovery studies result in too short values for T I . At very low temperatures around 100 mK the contribution of spectral diffusion to saturation recovery becomes negligible and TI can be measured by this experiment. In vitreous silica values of roughly 70 p,s were found at 100 mK and 690 MHz (Golding and Graebner, 1981). Saturation recovery develops in a nonexponential way, it slows down with increasing time separation between the two pulses. This fact manifests the distribution of relaxation times, since at large delay times those TS are seen which have a small effective coupling &,/E [see eq. (22)j. With decreasing temperature, both relaxation times TI and T2become longer and longer. Well below 100 mK also the transverse relaxation time T2may exceed the duration of applied acoustic or electromagnetic pulses. Under this condition TS are excited coherently, i.e. a macrosopic or elastic polarization is generated. A variety of new phenomena are based on this coherence. We only mention polarization decay (“free induction decay”), generation of echoes (spontaneous, stimulated and rotary echoes), population inversion and self-induced transparency. Since a comprehensive review on these phenomena has been published recently (Golding and Graebner, 1981), we will only briefly discuss results obtained by echo-experiments as an example. The occurrence of spontaneous echoes has been predicted (Kopvillem, 1977) and they have in fact been generated in glasses by acoustic (Golding and Graebner, 1976) and electric (Bernard et al., 1978; v. Schickfus et al., 1978) means below 50mK. They allow the relaxation times T, and T2to be determined directly, as well as the coupling parameter y (from acoustic echoes) or the electric dipole moment p (from electric echoes). The decay of the

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spontaneous echoes is t h e most direct measurement of T, at temperatures below 100mK. A typical value is T,= 15 ps at 20mK and 690MHz in vitreous silica (Graebner and Golding, 1979). The theoretical explanation of t h e echo decay (Black and Halperin, 1977; Hu and Walker, 1977) is based on the occurrence of spectral diffusion as in the case of holeburning experiments. Observed decay times agree with theory in magnitude as well as in temperature dependence. Discrepancies from theory are found in measurements at temperatures between 4mK and 20mK (Piche. 1978) and in the shape ot t h e echo decay (Golding and Graebner, 1981). In principle T I can be determined from the decay of t h e stimulated echo (Graebner and Golding, 1979), but spectral diffusion makes a correct analysis of the data difficult. It is more promising to use the decay of the population inversion to obtain reliable values for T I . This has been demonstrated in studies of electrically generated echoes in vitreous silica (Golding et al., 1979). In these experiments the minimal longitudinal relaxation time, TI.,, was found at a frequency of 720MHz and at T = 19 mK. As expected from eq. (5), TI l / T and for y I a value of 1.5 e V can be deduced. There is another interesting parameter of the TS which can be measured in echo experiments. Variation of the applied electric or elastic field strength results in a change of the amplitude of the spontaneous echo. A maximum occurs and from its position the electrical or mechanical coupling constant can be determined. For vitreous silica Suprasil W an average electrical dipole moment of 0.6 D was found, where 1 D = 1 X lo-’* esu (Golding et a]., 1979). Interestingly an additional maximum was observed in vitreous silica Suprasil I containing roughly 1200ppm OH impurities. From this maximum a dipole moment of 3.7 D and from TI a coupling constant of only y , = 0.9 eV was deduced. Obviously there exist two different types of TS in vitreous silica. One species is called “intrinsic” and is present in all different types of vitreous silica, whereas the second one is “induced” by the OH-impurities. It is, however, unclear whether additional TS are created by impurities or whether they simply “mark” ordinary TS by enhancing their dipole moment and weakening their coupling to the network. A similar conclusion has also been drawn from the amplitude of the backward wave phonon echoes in glasses, whose amplitude exhibited a linear dependence on the concentration of hydroxyl ions (Shiren et al., 1977). An interesting influence of irradiation with band-gap photons on the spontaneous electric echo has been observed for the amorphous semiconductor As& (Fox et al., 1982). O n irradiation the amplitude of the echo decreases and simultaneously the dephasing time T2 rises. These changes are accompanied by an increase of the electrical dipole moment

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of the states responsible for the echo from 1.8 D to 2.5 D. Reirradiation with light in the near infrared restores the echoes. Without going into a detailed discussion, we want to state that these studies indicate that in a-As$, the TS are somehow linked to the localized electronic states present in the band-gap. Finally we want to mention briefly the influence of TS on optical properties. In these studies optically active molecules are embedded in amorphous matrixes (see, for example, Friedrich and Haarer, 1984). Because of the strain sensitivity of the electronic levels an inhomogeneous broadening of the optical absorption line is caused by the varying local environment. Measurements via fluorescence line-narrowing and optical hole-burning have shown that the homogeneous linewidth is also considerably broadened. According to recent measurements on organic glasses the width of the burnt hole increases approximately proportional to at temperatures between 0.4 K and 20 K (Thijssen et al., 1983). The same temperature variation has been found for the optical dephasing rate of photon echoes in a Nd3'-doped glass fibre in the temperature range 0.1 K to 1 K (Hegarty et al., 1983). As in the acoustic or dielectric case, line broadening or optical dephasing is likely to be caused by spectral diffusion.Thus an analogous treatment is possible. The only difference is that the TS being at resonance with the sound wave or microwave field have to be replaced by the optically excited molecules. Such a theoretical treatment of the temperature dependence of the homogeneous linewidth leads to perfect agreement between experiment and theory (Hunklinger and Schmidt, 1984). 2.7. THERMAL EXPANSION For crystals thermal expansion is caused by the anharmonicity of the atomic potentials. Like other thermal properties, thermal expansion of glasses is anomalous at low temperatures: the magnitude as well as temperature dependence differ from those of crystals (Barron et ai., 1980). We are going to discuss this phenomenon briefly although it is not well understood. Since measurements of the thermal expansion below 1 K are extremely difficult, only a few experiments have been carried out. We will only discuss these experiments, because extrapolation of the data obtained above 1.5 K to lower temperatures leads to controversial results. The linear expansion coefficient p is intimately connected with the Griineisen paramerer r:

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where C is the specific heat and B the elastic bulk modulus. In crystals r and B are virtually constant at low temperatures, resulting in p a T 3 because of t h e temperature dependence of the specific heat. In contrast thermal expansion of amorphous solids varies linearly with temperature and is well approximated by

p

=

6, T t b,T3 ,

(30)

where b, and b, are material dependent constants. In fig. 13 the linear expansion coefficient of several amorphous solids is shown for temperatures below 1 K (Ackermann et al., 1984). To demonstrate the validity of eq. (30) we have plotted PIT as a function of T 2 .The similarity between eqs. (1) and (30)immediately suggests that fl is related to the specific heat anomaly of glasses (W.A. Phillips, 1973; Papoular, 1972). Therefore we may define two different Griineisen parameters r, and r3 which are connected with the linear and t h e cubic term, respectively. I', is thought to be caused by the anharmonicity of the TS. r3reflects the

a

LT W

a

x W I-

-15;

I -

I

02

01 06 08 ( T EMP E RAT R EI (K

u

10

I

Fig. 13. Thermal expansion of various amorphous solids, divided by T and plotted versus T2(after Ackennann et al., 1984). x -Epoxy SC5,O- AS&. A - PMMA, I-vitreous silica Spectrosil B (1200 ppm OH), 0- vitreous silica Spectrosil W F (20 ppm OH).

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contribution of the ordinary phonons and the “excess” excitations leading to the a,T3-term of the specific heat in eq. (1). Here we want to mention that thermal expansion at low temperatures has also been measured for the metallic glass PdSiCu by two different groups. Unfortunately tneir results differ drastically: rl was found to be 1.6 (Ackermann et al., 1984) and -350 (Kaspers et al., 1983). Measurements of via the thermoelastic effect (Tietje et al., 1984) result in values of rlclose to zero. It is not clear whether this large discrepancy can be attributed to sample preparation. As in the case of phonons, the Griineisen parameter of TS relates elastic energy changes with volume changes:

r

r = -d(ln

E)/d(ln V) .

(31)

Using eq. (17) and the definition of the energy splitting we may write for a given tunneling state

Since d, do and E are of the order of 1 K (or even smaller), we would is of the order of yA or y . From acoustic experiments we expect that know that y = lo4 K but such a high va ue is not observed. The largest absolute value of the Griineisen parameter of dielectric glasses is found for vitreous silica Spectrosil WF, namely r, = -65 (Ackermann et al., 1984). As pointed out in section 2.3, y can be positive or negative and it is likely that the contribution of the TS is averaged out for a macroscopic sample. Of course, it could happen that this cancellation is not complete and that TS with a certain sign of y predominate (W.A. Phillips, 1973; Ackermann et al., 1984). On the other hand the elastic coupling of t h e TS via variation of the geometry (see section 2.3) should also contribute to the Gruneisen parameter. It is, however, very unlikely that this effect could be strong enough (W.A. Phillips, 1973; Hunklinger and Arnold, 1976) to account for the value rlin the case of vitreous silica. Very recently the thermal expansion has been discussed using the specific Tunneling Model (Karpov et al., 1983) mentioned in section 2.7. According to these calculations (Galperin et al., 1984a), TS with A < A, give a negative contribution to the thermal expansion coefficient, whereas the contribution is positive if A > A,. Since the relaxation time Tl depends on the value of A,/E, a glass is expected to contract initially on heating, but to begin to expand as soon as those TS become dominant for which A > A , . So far experimental evidence for this expected exotic behaviour does not exist. Obviously, thermal expansion of glasses below 1 K remains an unsolved question.

r,

7

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Finally we want to point out that r, of amorphous solids does not differ markedly from those of crystals. In general t h e elastic value of the Griineisen parameter is approached. An exception is vitreous silica where r, falls significantly below this value. Nevertheless the qualitatively different behaviour of rl and f, indicates once again that the “excess” excitations seen in the linear and the cubic term of the specific heat of amorphous solids behave in a completely different manner and are therefore probably also different in their microscopic nature.

3. Metallic glasses Metallic glasses, as the name suggests, have two aspects. On one hand they have a random structure and are therefore expected to exhibit low temperature properties similar to ordinary glasses. O n the other hand they are metallic, and it is only natural to expect them to have certain properties which owe their origins to the presence of conduction electrons. In crystalline metals electrons play an important role for the thermal and elastic properties such as specific heat, thermal conductivity or sound absorption. As we will see, the influence of electrons on thermal properties of metallic glasses is not different from that on crystalline alloys. However, the dynamical properties, as revealed through ultrasonic or audiofrequency measurements, are clearly affected by the presence of electrons. 3.1. SPECIFIC HEAT

AND THERMAL CONDUCI~VITY

The specific heat of metallic glasses below 1K contains the lattice or Debye contribution (C, a T’), the electronic contribution (C, a T) and the TS contribution (C, a T). It is, however, not clear whether an “excess” cubic term [see eq. (l)]also exists in metallic glasses (Golding et a]., 1972; Samwer and v. Lahneysen, 1982). In Zr-based metallic glasses an additional term (C,,= T-*) has been observed (Lasjaunias et al., 1979; Lasjaunias and Ravex, 1983) at very low temperatures ( T < 0.1 K). It is correlated with the magnitude of C, and has been attributed to the quadrupolar nuclear contribution of the 91Zrnuclei. The electronic contribution is more or less the same in glassy and crystalline metals: measurements after recrystallization (Lasjaunias and Ravex, 1983) have shown that noncrystallinity has an almost negligible effect on the density of electrons at the Fermi surface and consequently on the specific heat. Of particular interest to us is the TS contribution which is also linear in temperature. In normal conducting metallic glasses the linear term due t o electrons C e B C,. As a result it is difficult to identify C, from measurements on normal conducting glasses.

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First unambiguous evidence of a linear term assigned to the amorphous state came from specific heat measurements on ZrJd,', (Graebner et a]., 1977) which becomes superconducting at T, = 2.53 K. The data are shown in fig. 3 along with those of other disordered solids. At T 4 T,, when the electronic contribution is frozen out, a linear term has been observed whose magnitude is very similar to that of vitreous silica. Subsequent measurements on other superconducting metallic glasses confirmed this observation (see, for example, v. Lijhneysen, 1981;Ravex et al., 1981; Kampf et a]., 1981; Samwer and v. Lahneysen, 1982). There is, however, a slight difference in the behaviour of insulating and conducting glasses. If we approximate the contribution of the TS at very low temperatures by C, = ulT1" [see eq. (l)], we find S > 0 for insulating glasses, whereas for most metallic glasses 6 is close to zero or even negative. Measurements of the low temperature thermal conductivity on amorphous metals provided first evidence that low energy excitations, as found in insulating glasses are also present in amorphous metals (Matey and Anderson, 1977). As in crystalline metals and alloys both electrons and phonons take part in the heat conduction process. If A, is the conductivity due to electrons alone and A p is that due to phonons, the total conductivity is A =A,+A,.

(33)

In amorphous metals, the mean free path of electrons is extremely short, namely of the order of a few atomic distances and is mainly limited by the scattering from structural disorder. From the Wiedemann-Franz law A , = L u T the contribution of the electrons can be estimated. Here L = i.rr2(k/e)2is the Lorenz number and u the electrical conductivity which is typically 3 X lo5- lo6 (0m)-' for metallic glasses. The electronic contribution to the thermal conductivity is therefore expected to be roughly A, 10-4T(W/cm K). At low temperatures the measured thermal conductivity A is larger (see fig. 14) and the phonon contribution A p can be deduced with reasonable accuracy. In metallic glasses the mean free path of phonons is limited by interaction with TS and electrons. Therefore the total scattering rate T-' is a sum of the individual rates T ; and T ; ~caused by the interaction with TS and electrons, respectively. In section 2.2 we have already discussed the attenuation due to TS: 74 = vl-' a T for thermal phonons [see eqs. (3) and (lo)]. The electronic contribution 7;' to the phonon scattering is given by (Pippard, 1965) ~1

-1

T,

0:

/,T2

(Ch. 3, $3

TEMPERATURE ( K ) Fig. 14. Thermal conductivity versus temperature of the superconducting metallic glasses ZnoPdU (Graebner et al., 1977) and Zr&a together with various other amorphous solids. (From Raychaudhuri and Hasegawa, 1980.)

This relation is valid in the limit /,q 4 1, where I, is the mean free path of the conduction electrons and q the wavevector of t h e interacting phonon. Based on the free electron model the magnitude of 7,' can be calculated (Morton, 1977; Raychaudhuri and Hasegawa, 1980). From such an estimate we know that electrons dominate scattering at higher temperatures. On cooling their contribution becomes comparable with that of TS in the plateau region of the thermal conductivity. Well below 1 K only the scattering process due to TS has to be considered. As for dielectric glasses, a T2-dependence is expected [see eq. (lo)] which compares well with t h e observed temperature variation, for which A p 3~ T' (6 = 1.8-1.9) is reported (Matey and Anderson. 1977; v . Ltihneysen, 1981). In the plateau region, i.e. between 1 K and 10 K. the two scattering rates are comparable and it is difficult to distinguish between both processes. In superconducting metallic glasses the contribution of the electrons can clearly be separated. Above the superconducting transition temperature T,, cq. (34) holds a s in normal conducting metallic glasses. Below Tc, 7,' decreases o n cooling as t h e electrons freeze out and d o n o t participatc in the

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74

scattering process anymore. Because is expected to vary smoothly at q , the drastic change in the temperature dependence of 7;’ gives rise to a minimum in the thermal conductivity at T,. In a number of superconducting metallic glasses such a minimum has been found (Graebner et al., 1977; v. Lijhneysen, 1981; Ravex et al., 1981). As an example we show data on Zr,Pd, (Graebner et al., 1977) and Zr&e, (Raychaudhuri and Hasegawa, 1980) in fig. 14. At low temperatures the thermal conductivity is found to be proportional to indicating that TS are the dominant scattering centres as in insulating glasses. At T, a clear minimum is observed due to the strong variation of the electronic contribution to phonon scattering. In conclusion, electrons play an important role for the thermal properties of metallic glasses. Their contribution to specific heat and thermal resistivity is well described by simple theories developed for ordinary crystalline alloys with a very short electron mean free path.

3.2. ULTRASONIC PROPERTIES OF NORMAL CONDUCTING

METALLIC GLASSES

The thermal properties of metallic glasses seem to indicate that free electrons do not influence the behaviour of the TS. It seems that the presence of conduction electrons neither modifies the spectral density of the TS nor changes their coupling to phonons. Both statements appear to be quite natural and as a consequence the unsaturated resonant sound absorption as well as the sound dispersion should remain unchanged when going from dielectric to.metallic glasses. I n particular one would expect that below 1 K the sound velocity increases logarithmically with temperatures as predicted by eq. (8). As shown in fig. 15 for NiP (Bellessa et al., 1977), such a temperature variation has indeed been observed giving the first unambiguous evidence for the presence of TS in metallic glasses. However, we will see in the following that this straightforward interpretation is not completely correct because free electrons have a pronounced influence on the velocity of sound. The difference in the properties of metallic and insulating glasses become obvious in ultrasonic measurements which probe the dynamics of the TS. To be more specific, any property explicitly involving the relaxation time of TS will show a change when going from non-conducting to conducting glasses. In particular measurements of the ultrasonic attenuation revealed that the relaxation absorption [eq. (6)] as well as the critical intensity Ieq. (4)] required to saturate the resonant absorption are both significantly different from those found in dielectric glasses. In fig. 16 we show “typical data” on the temperature dependence of the relaxation absorption arc,of the metallic glass NiP (Doussineau et al., 1977) along with that of an insulating glass, both obtained at similar frequencies.

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1SOMHz

=;

TRANSVERSEWAVES

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TEMPERATURE ( K ) Fig. 15. Relative variation of the sound velocity versus temperature in NiglP,g for both, longitudinal and transverse sound waves. (After Bellessa et al., 1977.)

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Fig. 16. Variation of the ultrasonic attenuation against temperature o f vitreous silica and NiP at similar frequencies. In both cases an acoustic intensity of about 0.1 mW/cm' was applied. (From Doussineau et al.. 1977.)

Ch. 3, $3)

ANOMALIES IN GLASSES AT LOW TEMPERATURES

303

Three features we want to point out. First, the attenuation of the metallic glass remains relatively high down to the lowest temperature indicating that the relaxation rate of the TS in metallic glasses is much higher than in dielectric glasses. Secondly, the low temperature rise is much less steep than in the insulating glass. Thirdly, the “plateau” is by n o means perfect. More direct evidence for such high relaxation rates comes from saturation experiments (Golding et al., 1978; Doussineau et al., 1978). In fig. 17 we show t h e intensity dependence of the attenuation in PdSiCu. Obviously the critical intensity Z, to saturate the resonant absorption is much higher than in insulating glasses. Since Z, TI‘Ti1[see eq. (4)], this measurement demonstrates clearly that relaxation rates in PdSiCu are very high, i.e. relaxation times are extremely short. This conclusion is supported by the fact that no saturation recovery effects (see section 2.6) could be observed down to temperatures of 10 mK (Golding et al., 1978). Therefore it has been estimated that even at that temperature T, is shorter than 2511s for TS with energy splitting corresponding to 1GHz. In contrast, in vitreous silica under the same condition the TS need 250 p s to relax. Unlike the casc for dielectric glasses the value of T2will not be determined by spectral diffusion but by the short lifetime TI and we may put T , = T2.Therefore TI can also be estimated from the

w

l-tQ

0

1

I

1o-L

1

ACOUSTIC INTENSITY

I

1u2 (W/cm2)

Fig. 17. Change of the resonant ultrasonic attenuation as a function of the applied acoustic intensity. (From Doussineau et al., 1978.) Note that this experiment was carried out at 62mK, whereas the lowest temperature attained in the experiment shown in fig. 5 was only 400 mK.

304

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, 93

influence of t h e acoustic intensity on t h e temperature variation of t h e velocity of sound (CordiC and Bellessa, 1981).From such a measurement on PdSiCu the relaxation rate T, was estimated t o be only 2 ns at 10 m K for TS with an energy splitting corresponding to 500 MHz. The above discussion has shown that 1’s in metallic glasses have extremely short relaxation times and the main difference between metallic and dielectric glasses arises from this fact. In the following we will show how such a short relaxation time is caused by t h e electron-TS interaction.

.Q.1. Interaction between condiiction electrons and tunneling systems

Previously we have discussed t h e coupling of TS to local strains. In a similar way the interaction between TS and conduction electrons can be described (Golding et al., 1978; Black, 1981).This interaction is thought to be analogous to the electron-phonon coupling, where electrons are scattered because they see t h e variations of the electronic potential which are associated with thermal motion of t h e ions. In amorphous solids the local occupation of the valence electron states may be changed due to the motion of the tunneling particles. As a result the potential seen by the conduction electrons also varies with time. In a first approximation t h e conduction electrons can be considered to be plane waves and the Hamiltonian of the interaction between TS and electrons can be expressed in a form similar to that of t h e elastic interaction; as in that case we only take into account the coupling via a variation of t h e asymmetry energy A. The interaction accompanied by a change in the position of the tunneling particle (electronassisted tunneling) is neglected since the relevant coupling parameter is expected to be smaller by a factor exp(-A) than that responsible for the asymmetry variation. As a consequence the scattering process becomes angular independent, resulting in a great simplification of t h e theoretical calculations. Thus we may write (Black et al., 1979; Black, 1981)

where N is the number of atoms in the sample. The operators c ; and c ~ + ~ create and destroy electrons in the states k and k + 9, respectively. Similar to the elastic coupling constants D and M [see eq. (22)], the coupling parameters Vl and V , are proportional to A / E and A J E . Although they also depend on 9, we consider yl and V, as mean values,

Ch. 3, $31

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305

which have been appropriately averaged over the Fermi surface. For a TS the simplest method of relaxation is the direct transition between the eigenstates P+ and P-.These transitions caused by the conduction electrons can be described by analogy with the Korringa relaxation of nuclear magnetic moments

T - - m 1 2 ( A ) 2 E c oEt h ~ 4h E

‘,‘

Here we have put J5V, = qA,/E, where J5 is the electronic density of states per atom at the Fermi level. It has been pointed out (Black, 1981) that two serious questions arise in connection with these relations. Firstly, the lowest-order perturbation theory is only permitted if higher-order terms become successively smaller. This seems to be guaranteed, since an expansion in the dimensionless coupling parameter q represcnts these terms. From experiments (Golding et al., 1978; Arnold et al., 1Y81; Weiss et al., 1981) it is known that q is smaller than 1, but values as large as 0.85 (Weiss et al., 1981) have been reported. Recently, it has been pointed out that this approach might be oversimplified (Zawadowski, 1980), and that in fact both scattering processes have to be taken into account, namely the electron-TS interaction with and without change of the position of the tunneling particle. Because of the relatively strong interaction between TS and electrons it could be that “bound states” are formed where the motion of the tunneling particle and of the conduction electron screening cloud around the TS are strongly correlated (Kondo 1976; Vladar and Zawadowski 1983). In fact, very recent acoustic experiments under high magnetic fields (Neckel et al., 1985) seem to indicate that such a more elaborate theory is necessary. Since its significance is not yet completely clear, we will stick to the simple approach because it provides an explanation for many of the observed features. The second question arises from the short electronic lifetime which by far exceeds Elh in the case of metallic glasses. Although the problem is not completely solved, it is assumed (Black, 1981) that eq. (36) remains valid as long as the size of the TS does not exceed the electronic mean free path considerably.

3.2.2. Comparison between theory and experiment Before we discuss the influence of the TS-eiectron interaction on the relaxation process, we want to consider resonant absorption further. In

306

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, 93

principle eq. (3) is only valid for lightly damped TS. In metallic glasses, however, the TS at resonance with the sound wave are generally heavily damped: since in most of these experiments E G k T , eq. (36) leads to 7i.: > E/fi,i.e. t h e lifetime broadening exceeds the level splitting. However, it has been pointed out (Thomas, 1983) that even under such circumstances eq. (3) is probably still correct. Let us now discuss the relaxation absorption in greater detail. To simplify this discussion, we only take into account the relaxation of the TS via interaction with free electrons. As in t h e case of phonon relaxation we , ~TI,,,again distinguish between the two regimes wT,,,,* 1 and W T , , , 1. represents the minimum relaxation time which we obtain here from eq. (36) by putting A,/E = 1. For wT,,,,% 1, i.e. for high frequencies and/or low temperatures one finds (Doussineau and Robin, 1980; Black, 1981)

For wT1.,,G 1 , i.e. for low frequencies and/or high temperatures one obtains (Jackle, 1972: Black and Fulde, 1979; Doussineau et al., 1980)

How do these results compare with those obtained for relaxation via the TS-phonon interaction? The temperature dependence of the absorption in the regime wTl.mS- 1 reflects the energy dependence of the relaxation = T 3in insulating glasses whereas are,,e rate. According to eq. (7) T is predicted by eq. (37). In this regime the relaxation process has only a negligible influence on the velocity of sound for both relaxation mechanisms. For wT,,,, 1 the absorption is independent of temperature irrespective of the way of relaxation, and its magnitude depends only on t h e strain coupling constant C, defined in eq. (26). However, the contribution of the relaxation process to the change of the sound velocity reflects the relaxation mechanism. For both, phonon and electron relaxation the sound velocity change varies logarithmically with temperature. But t h e prefactor is --3C/2 for phonons and - C/2 for electrons [see eqs. (27) and (a)]. In fig. 18 w e show the attenuation data for NIP taken at three

Ch. 3, $31

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307

frequencies (Doussineau and Robin, 1980). Below 0.3 K the attenuation is linear in temperature and frequency independent as expected from eq. (37) since wT1.,,9 1. At higher temperatures the attenuation becomes less temperature dependent, but a true temperature independent region as expected from eq. (39) is not reached. The regime w T 1 , , , 4 1 can best be studied by low frequency experiments. In fig. 19 we show t h e results of such a measurement carried out at about 1 kHz (Raychaudhuri and Hunklinger, 1984). Although at this frequency the condition oT1,,,4 1 holds over t h e whole temperature range, the attenuation increases with temperature. Up to 5 K only a weak rise is observed and eq. (39) may be said to be obeyed approximately. In principle an energy dependence of t h e coupling constant C could explain this deviation from a “plateau”. In this case it would be sufficient if either or y exhibited this energy dependence, but so far no further evidence for such a modification of the original assumptions of the Tunneling Model exists. The variation of the sound velocity of a typical metallic glass has already been shown in fig. 15. The experimental curve looks very similar to that of insulating glasses (see fig. 6). First the velocity rises logarithmically with temperature, then it passes through a maximum and falls

-

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1

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E V

m

D -

w

e

0

Z

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=

El k

I

Q

3

Z W II-

4

o L90 MHz

Q

+ 275 MHz

a 0 I

0.5

-L.1

1.5

TEMPERATURE I K ) Fig. 18. Change of the ultrasonic attenuation of NiP versus temperature for three frequencies. The curves are shifted in such a way that coincidence is obtained at 0.1 K. (From Doussineau and Robin, 1980.)

308

[Ch. 3. 43

S. HUNKLINGER A N D A . K . RAYC‘HAUDHURI

2 7 I I

,.--

8

1030Hz

-00’

01

10 13 TEMPERATURE (K \

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103

Fig. 19. L o w frequency measurement of the internal friction @ ’ = au/w of the metallic glass PdSiCu against temperature. (From Raychaudhuri and Hunklinger, 1984.)

again. However, this apparent similarity is somewhat misleading because there are two important differences between t h e insulating and t h e metallic glasses. Firstly, the logarithmic rise found for insulating glasses is due to t h e contribution of the resonant process alone. According to eq. (8) the slope is determined by C = & ‘ / p d . In metallic glasses t h e relaxation process contributes too, because the crossover from t h e regime wT,,,, :’ 1 to w T , , , < 1 occurs at a much lower temperature due to t h e fast relaxation of the 7 s via free electrons. We may write for. the total \uriation of the sound velocity in the rcpime w T , , , <. I for normal conducting met;tllic glasses

The slope o f the logarithmic rise is only half of what is expected from the resonant interaction itself. At much lower temperatures, when t h e regime t,,l’l,,,,+ 1 is entered, the contribution of t h e relaxation process should freeze out [see eq. (38)].Only the contribution of the resonant interaction remains and t h e slope of t h e logarithmic rise is given by t h e constant C. So far t h e crossover from one regime to another has not clearly been detected in normal conducting metallic glasses, because measurements in

Ch. 3, 93)

ANOMALIES IN GLASSES AT LOW TEMPERATURES

309

these materials have not been carried out down to low enough temperatures and high enough frequencies. The second difference between insulating and metallic glasses has to do with the maximum in the velocity. In insulating glasses this maximum appears when the increase due to the resonant interaction is just compensated by the contribution of the relaxation process. This condition is fulfilled close to the crossover temperature T,,, where oTl,,,= 1. Thus we may put T,,,= TM,the temperature of the velocity maximum. In dielectric glasses Ti,M= T - 3 , since the TS are relaxed by phonons. Therefore TM should follow the relation u T i 3= constant, whose validity has been demonstrated in many experiments. As an example we may compare figs, 6 and 11, where measurements on silica based glasses at 90MHz and 1 kHz are shown. In metallic glasses the relation wTl,,,= 1 is fulfilled at much lower temperatures. Therefore the maximum around a few K does not indicate that the crossover temperature T, is reached. It marks t h e crossover from the electron to the phonon dominated relaxation process (Raychaudhuri and Hunklinger, 1984). The main contribution to relaxation dispersion comes from TS having an energy splitting E = 2 k T . Therefore the relaxation rate is either proportional to F3 or to T, depending whether phonons or electrons are dominant. Consequently electrons will be responsible for relaxation at low temperatures but at higher temperatures phonons will take over. Using typical parameters, we find that this transition should take place at around a few K. Below this temperature eq. (41), which predicts a positive value of the slope, is appropriate. Above the crossover temperature relaxation by phonons dominates and eq. (28) holds, and the velocity decreases with temperature. Thus the position of the maximum should be independent of the frequency of measurement. This conclusion is nicely verified by data on PdSiCu taken at about 1GHz (Golding et al., 1978) and about 1 kHz (Raychaudhuri and Hunklinger, 1982a). In fig. 20 it can clearly be seen that by varying the measuring frequency by six orders of magnitude, the maximum hardly shiftsfrom 2 K to 1.5 K. In a similar situation in dielectric glasses the maximum will shift from 2 K to 20 mK. In a series of papers (see, for example, Araki et al., 1980; Park et al., 1981) it has been shown that the acoustic attenuation of metallic glasses like PdSiCu in the temperature range 0.3 K to 2 K can become intensity dependent even for frequencies as low as 10MHz. Under these experimental conditions the attenuation usually arises from the relaxation process only and saturation is not expected. Subsequently a similar effect was observed in the quenched alloy ZrNb which behaves as a glass (see section 4.4). It was found that the intensity dependences are only observable as long as the.relaxation absorption is present (Wang et al., 1982).

310

ICh. 3, 83

S. HUNKLINGER A N D A.K. RAYCHAUDHURI

s W

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1

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u

b

W

b

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01

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1 TEMPERATURE ( K )

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4

1J *

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10

Fig. 20. Temperature variation of the sound velocity of PdSiCu at low and high frequency. Data at 1030Hz were taken using a vibrating reed technique (Raychaudhun and Hunklinger. 1982a) whereas data at 960 MHz were obtained by conventional ultrasonic technique (Golding et al.. 1978). (From Raychaudhuri and Hunklinger, 1984.)

This indicates that the origin of the saturation effect lies in the relaxation process and not in the resonant interaction. The “standard” Tunneling Model turned o u t to be sufficient to explain the high frequency investigations in the range 400-1100 MHz and temperatures down to 50 mK (Doussineau, 1981). The seeming discrepancy between high and low frequency measurements was settled by the observation of two distinct stages of saturation (Hikata et al., 1982). The first stage seen at high frequencies is due to saturation via the resonant interaction as we have discussed so far. The second stage occurs at low frequencies and high amplitudes and is in fact due to an intensity dependence of the relaxation absorption. Two explanations for this surprising effect have been reported so far. The first one (Arnold et al., 1982a) takes account of the fact that up until now the effect has only been observed in metallic systems, where the relaxation times of the TS are extremely short. Therefore it is assumed that a hole is burnt into the spectral distribution of the occupation of the two states by the intense sound wave (see section 2.6) whose width can be of the order of kTM. Thus the occupation number of those TS contributing most to the relaxation process can be changed and the absorption becomes intensity dependent In the,second explanation (Galperin et al., 1984b) the strong

Ch. 3, 831

ANOMALIES IN GLASSES AT LOW TEMPERATURES

31 1

modulation of the level splitting of the TS due to the intense sound wave is taken into account. Non-linearities are expected as soon as the amplitude of the modulation exceeds kT. The critical intensity estimated in this way compares favourably with the measured values. Finally we want to comment on the behaviour of sound velocity above a few Kelvin. As in insulating glasses the velocity in this region decreases roughly linearly with temperature (Bellessa et al., 1977) in contrast to the logarithmic variation expected from theory. The reason for this behaviour is still unknown, but it is common to both metallic and insulating glasses. In summary, we find that the basic theory of TS-electron interaction (Black, 1981) briefly discussed in section 3.2.1 can account for most of the observed features. An important remaining question is the weak temperature dependence of the attenuation in the regime wT,,,,4 1.

3.3. ULTRASONIC PROPERTIES

OF SUPERCONDUCI-ING METALLIC GLASSES

So far we have considered the ultrasonic properties of normal conducting metallic glasses. However, the best way to see the effect of free electrons on the relaxation of TS is by modifying their density through the superconducting transition. Below T, t h e electrons will gradually form Cooper pairs which do not take part in relaxing the TS. Thus the elastic properties of metallic glasses for T 4 T, will be similar to those of an insulating glass. The application of a high magnetic field destroys the superconducting state and one can directly study the influence of electrons on the properties of TS. Recently very interesting experiments have been carried out on superconducting metallic glasses. They elucidate in an appealing way the complex situation of the relaxation of TS in superconductors. 3.3.1. Relaxation of tunneling systems in the superconducting state We will first present the basic ideas of the T k l e c t r o n interaction in superconducting materials (Black and Fulde, 1979; Black, 1981). At the transition temperature T, the electronic density of states is modified at the Fermi level because of a gap d,(T) opening up. This change has also consequences for the Korringa-like relaxation process occurring between TS and free electrons. The resulting relaxation rate Ti,5is then given by (Black and Fulde, 1979)

317,

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[Ch. 3, §3

Here ck is the energy of the electron with the wavevector k, f ( ~ ) = [exp(&/kT)+11-l is the occupation number and N(E)=I E I ( E ~ is the density of states if I E ~> A,. This expression differs strongly from the Korringa law for the NMR Ti' in superconductors because of the coherence factor 1 - A : / E ~ E Obviously ~.. the relation for normal conducting metallic glasses is recovered if the energy gap A, tends to zero. In general no analytic expression can be found for except for TS with very small energy splittings. For E G A , , eq. (42) reduces t o the simple expression (Black and FuIde, 1979)

Due to t h e exponential factor this relaxation rate will decrease rapidly below T,. Ultimately, at very low temperatures phonons will govern the relaxation again, i.e. T;,:< Ti,',. In fig. 21 we have plotted the relaxation rates of hypothetical metallic glasses having transition temperatures of 0.5 K and 3 K. In addition the rates are presented for the case when the superconductivity is suppressed by a high magnetic field. In the calculation we have put E = kT in order to show t h e properties of those TS which are the main contributors to the relaxation process. A,/E as well as all other parameters was kept constant. For both materials it was assumed that at 1.5 K the relaxation rate caused by free electrons in the normal conducting state is equal to the rate due to phonons, i.e. TT.L(l.5K) = TT,'&lSK). If we consider the rate in the superconducting state, due to the interaction with quasiparticles alone, we see that t h e rate increases exponentially at small temperatures. Close to T, t h e energy splitting of the TS reaches 24,. Now Cooper pairs can be broken by the TS and an additional channel of relaxation opens up. It turns out, however, that this process only gives a minor contribution to the relaxation process. First we consider t h e glass with a high transition temperature. In this case the phonons dominate over the whole temperature range. In particular no sharp change in the relaxation rate of the TS occurs in t h e vicinity of T,. Consequently attenuation as well as velocity does not undergo any change at T,. If, however, the glass is turned into a normal-conducting glass by a magnetic field, relaxation due to electrons also comes into play below 1 K. When the transition temperature is low, the relaxation rate of t h e TS undergoes various changes. At the lowest temperature the phonons dominate, but above a certain temperature relaxation via quasiparticles becomes more and more prominent and t h e relaxation rate rises drastically. Above T, relaxation via normal electrons

Ch. 3, 031

ANOMALIES IN GLASSES AT LOW TEMPERATURES

313

T E MPE RATURE ( K )

TEMPERATURE ( K ) Fig. 21. Relaxation rate of TS in two hypothetical metallic glasses with a transition temperature of 0.5K and 3 K . Note that the scales are different for the two figures. The contribution of the quasiparticles ( T < T,)and electrons ( T > T,)is shown by Curve 1 (- . - . -). The relaxation rate due to phonons is represented by Curve 2 (---) and the sum of both by the full line (Curve 3). If the superconductivity is suppressed by a high magnetic field, free electrons give rise to relaxation also below T, and their rate is given by Curve 4 (-. . . - . . .). The total relaxation rate (due to electrons and phonons) for T < T, is shown by Curve 5 ( . . . . . ).

takes place and at even higher temperatures phonons finally dominate again. In summary, the relaxational behaviour of TS in superconducting metallic glasses depends crucially on three things: (a) the relative strength of the electron-TS and phonon-TS interactions (we have artificially put Ti,: = TT,',,at 1.5 K),

314

[Ch. 3, $3

S. HUNKLINGER AND A.K. RAYCHAUDHURI

(b) t h e transition temperature T, and (c) t h e presence or absence of magnetic fields. In addition experimental observations will also depend on the measuring frequency, because it determines the time scale.

3.3.2. Comparison between theory and experiment We will first discuss experimental results obtained by ultrasonic measurements at relatively high frequencies. In fig. 22 we show the attenuation of the superconducting glasses CuZr (Arnold et at., 1982b) and PdZr (Weiss et al., 1980). Their transition temperatures are 0.4 K and 2.6 K. respectively. In CuZr the attenuation is temperature independent , ~1. By cooling through T, the relaxation rate drops above T,, since w T , G very rapidly and t h e attenuation decreases, since the regime w T , . , > 1 is entered. In contrast to CuZr the attenuation of PdZr does not drop at T, h u t a t lower temperatures. There are two reasons for this different behaviour. On one hand phonons already contribute t o relaxation

I

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I

1

10 m

72

$ 1

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6

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c

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A

a 0

0 0

0

1

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CU60ZrL0

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0

0

01

h

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,

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1 TEMPERATURE i K )

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10

Fig. 22. Ultrasonic attenuation in two superconducting metallic glasses versus temperature. The transition temperature T, = 0.4 K for CuZr and T, = 2.6 K for PdZr are indicated by m o w s . The measurements were carried out at 745 MHz and 740 h4Hz in the case of CuZr (Arnold et al., 1982b) (full dots) and PdZr (Weiss et al., 1980)(open circles), respectively. By application of a magnetic field of 8 T the superconductivity of PdZr was suppressed, resulting in a plateau-like attenuation (full triangles) down to the lowest attained temperature. Note t h e different behaviour of the two solids at the transition temperature. (After Arnold et al., 1982b and Weiss et al., 1980.)

Ch. 3, 031

ANOMALIES IN GLASSES AT LOW TEMPERATURES

315

because of the relatively high T, value. On the other hand the electronTS coupling is much stronger in PdZr than in CuZr. Whereas r) = 0.4 has been deduced for CuZr (Amold et al., 1982c), the high value of r ) = 0.85 is necessary to fit the theory to the experimental results on PdZr (Weiss et al., 1981). Such a high coupling constant means that thermally excited < 1 to as far quasiparticles cause such high relaxation rates that oT,,,, down as 1K. The same figure also shows the attenuation of PdZr under a magnetic field. Applying a magnetic field of ST, the material was kept normal-conducting. Therefore in this experiment wT,,,, G 1 down to the lowest attained temperature and the attenuation remains in the plateau region. The high frequency measurements of the sound velocity are presented in fig. 23 (Arnold et al., 1982c; Weiss et al., 1982). In both cases at the lowest temperature the velocity rises logarithmically with temperature due to the resonant interaction. According to eq. (8) the elastic coupling constant C determines the slope. As T, is approached, thermally excited quasiparticles emerge. Because of the low transition temperature of CuZr, phonons do not contribute and relaxation is entirely caused by electrons (see fig. 21). Thus the relaxation rate rises drastically with temperature because of the increasing number of quasiparticles and leads to a decrease of the velocity before T, is reached. Above the transition 1

"

a a

a

I

a 0

0 0 ~

15

01

02

05

1

2

TEMPERATURE I K )

Fig. 23. Variation of the sound velocity against temperature in the superconductingglasses CuZr (Arnold et al., 1982) and PdZr (Weiss et al., 1982) measured with 797 MHz transvem and 720MHz longitudinal waves, respectively. Note the different behaviour at the superconducting transition temperature Tc.(After Arnold et al., 1%%, and Weiss et al., 1982.)

316

[Ch. 3, $3

S. HUNKLINGER AND A.K. RAYCHAUDHURI

temperature the material is normal conducting and the velocity varies logarithmically again, however with the slope C/2 [see eq. (41)]. Therefore a pronounced kink is observed at T, in agreement with theoretical predictions. In PdZr the maximum is also due to the onset of t h e relaxation process, but now both quasiparticles as well as phonons contribute. The position of the maximum coincides with the transition of the absorption from t h e steep rise to the plateau behaviour. As i n CuZr a kink is found close to T,, but the feature is only very weak. This indicates that phonons already dominate relaxation at that temperature, so that electrons can only give a minor contribution to the relaxation process. Above T, the velocity decrease is entirely due to phonon relaxation. Let us now consider the experimental results obtained from these materials at low frequencies. Using a vibrating reed technique, the measuring frequency is nearly six orders of magnitude lower. In fig. 24 the internal friction 0-' is shown for CuZr (Raychaudhuri and Hunklinger. 1982b) and PdZr (Raychaudhiri and Hunklinger, 1984). In both cases 0-' rises at t h e lowest temperatures, but with a different slope. In PdZr t h e acoustic properties are determined by phonons only because of the relatively high transition temperatures (see fig. 21). Therefore the acoustic behaviour at the lowest temperatures resembles very much that of dielectric glasses (see fig. lo). In CuZr the

0

001

0.1 1 TEMPERATURE ( K )

'

10

Fig. 24. Internal friction Q- of CuZr (at 1560 Hz) and PdZr (at 1180 Hz) as a function of temperature. Close to T, the slope of the attenuation changes its sign. (After Raychaudhuri and Hunklinger, 1984.)

Ch. 3, 831

ANOMALIES IN GLASSES AT LOW TEMPERATURES

317

rise of Q-' is much steeper, indicating that not only phonons but also quasipanicles contribute to the strong increase of the relaxation rate. Above 0.1 K a plateau-like behaviour of the internal friction is observed in both cases. No great change is seen at T, because in both materials the condition wT1.,,< 1 is already reached below this temperature. Surprisingly a kink in Q-' is observed close to the transition temperature for PdZr as well as for CuZr. Although this feature is quantitatively not very important, it may be qualitatively significant. According to theory [see eqs. (26) and (39)] we expect Q-' to be constant in the regime wTlS,,-e 1 irrespective of the mechanism causing the relaxation. We believe that this discontinuity in the slope of Q-' is due to the presence of electrons, which somehow modify the distribution function P(E, u ) or the elastic coupling constant y. As mentioned before, theoretical considerations (Kondo, 1976; Zawadowski, 1980; Vladi~rand Zawadowski, 1983) as well as recent acoustic experiments under high magnetic fields (Neckel et al., 1985) support this idea. In fig. 25 the variation of the sound velocity of CuZr and PdZr (Raychaudhun and Hunklinger, 1982b, 1984) is shown for low frequency measurements. At the lowest temperatures the velocity rises logarithmically as in all glasses, due to the resonant interaction of TS and phonons.

.

U

I

TC

CU60 ZrLO Pd30Zr70

I

001

I-3-

01 1 TEMPERATURE ( KI

1

Fig. 25. Low frequency measurements of the variation of the sound velocity for CuZr (at 156OHz)and PdZr (at 118oHz). Note the different behaviour of CuZr at the superconducting transition temperature for low and high frequencies (see also fig. 23). (After Raychaudhuri and Hunklinger, 1984.)

318

S. HUNKLINGER AND A.K. RAYCHAUDHURI

{Ch. 3, 94

For both materials the observed slopes are considerably higher than those reported from high frequency measurements (Arnold et al., 1982c; Weiss et al., 1982). This result is a striking contradiction to theoretical expectations [see eq. ( S ) ] . It could be explained by assuming a frequency dependence of t h e elastic coupling constant y, But so far no theoretical or further experimental evidence for such a behaviour exists. As mentioned in the discussion of the internal friction, the behaviour of PdZr at t h e lowest temperatures is determined by the interaction of the TS with phonons only. In particular after having passed the maximum at about 70 mK the velocity decreases logarithmically with a slope which is half of the slope of t h e logarithmic rise seen at lower temperature. This behaviour is analogous to what has been observed in insulators (see fig. 11). Above 0.2 K the velocity starts to decrease with increasing rate until 1 K is reached, then the rate slows down considerably until T,. This non-logarithmic variation is attributed to the contribution of quasiparticles to the relaxation process. Although this explanation is qualitatively consistent with the theoretical predictions outlined above, t h e quantitative agreement is by no means satisfactory. Above the transition temperature the velocity decreases further and is attributed to phonon relaxation, because the contribution of electrons can be neglected in this temperature range. In CuZr the decrease of the velocity following the maximum is nonlogarithmic, indicating that relaxation of the TS takes place preferentially via interaction with quasiparticles. Whereas the high frequency data are in agreement with the theoretical prediction (Arnold et al.. 1982c), it is not possible to give a quantitative description of the velocity decrease at low frequencies. Above T, the velocity seems to exhibit a plateau again in striking contradiction tn what has been observed at higher frequencies, where the velocity rises iogarithmically ag?in. In summary it is found that the elastic behaviour of amorphous superconductors is rather complex, due to relaxation of TS by interaction with electrons. Using t h e theory outlined in t h e previous section, the dynamic properties can he understood qualitatively. A relatively good agreement between experiment and theory is found at ultrasonic frequencies. Pronounced deviations, however. become obvious in low frequency measurements of t h e internal friction and the velocity o f sound. These experiments seem to indicate that either the density of states or t h e elastic coupling o f the TS is modified by the presence of electrons. 4. “Glassy properties” of disordered crystals The Tunneling Model of amorphous solids owes its origin to t h e well-

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studied problem of tunneling of impurities in crystalline solids (Narayanamurti and Pohl, 1970). Unlike in glasses these tunneling entities exhibit well-defined values for the tunnel splitting, the energy and the relaxation rates. If, however, the impurity content is increased, a linear excess specific heat is observed in many cases, without a pronounced change of the thermal conductivity. Obviously these tunneling states are only weakly coupled to the lattice, unlike those TS which are found in amorphous solids. The aim of recent studies of the low temperature properties of disordered solids was mostly to understand the microscopic origin of the TS in glasses and not just to study tunneling of well-defined units in welldefined crystals. So an assessment is necessary as to what extent these investigations really teach us about the origin of the low temperature anomalies in glasses. The various crystalline solids in which glassy behaviour of thermal, acoustic and dielectric properties has been found, can broadly be classified into four categories: (a) ionic conductors and other non-stoichiometric compounds, (b) orientationally disordered crystals, (c) radiation damaged crystals and (d) two-phase systems. We will briefly discuss representative examples here.

4.1. IONICCONDUCTORS p-alumina is an ionic conductor the low temperature properties of which have been studied extensively during the last few years (see, for example, Strom, 1983). Therefore our discussion of ionic conductors will mainly be devoted to this material. p-alumina is a two-dimensional fast ionic conductor containing various metallic ions like Li, Na, K or Ag moving in conductioa planes. These planes are separated by rigid spinel blocks of about 8 A thickness. There is an excess of cations with respect to stoichiometry and a corresponding number of charge balancing oxygen ions, i.e. the crystal has a non-stoichiometric composition. This makes the conduction plane highly disordered. As we will see, investigations carried out at low temperatures show that a broad spectrum of low energy excitations arises from this disorder. First evidence for anomalous low temperature properties came from measurements of the specific heat. As already mentioned in section 2.1 and shown in fig. 3, compared to the usual Debye value of crystalline dielectrics t h e specific heat exhibits an excess, which varies almost linearly with temperature (McWhan et al., 1972; Anthony and Anderson, 1977). In addition low temperature thermal conductivity was found to depend quadratically on temperature (Anthony and Anderson, 1976). Unlike the case of the low temperature properties of glasses, t h e size of t h e cations has a strong influence on t h e magnitude of the thermal quantities. Both

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specific heat and thermal resistivity decrease in the sequence Li, Na and K. This means that the density of states of t h e low energy excitations decreases with an increasing cation radius. The sequence, however, is broken by Ag-P-alumina, which has a specific heat close to that of Li-P-alumina but a thermal conductivity similar to that of Na-P-alumina. The apparent similarity in the dynamics of these low energy excitations and those of glasses first became evident from dielectric measurements in the microwave range (Strom et al., 1978). There it was demonstrated that the dielectric constant and dielectric absorption of Na-P-alumina behave in exactly t h e same manner as they do in vitreous silica. Even the magnitude of t h e observed effects turned out to be Comparable. This means that the Tunneling Model is also an appropriate approach to describe the low temperature properties of p-alumina. This idea is supported by the detailed analysis of dielectric data obtained in a wide frequency range (Anthony and Anderson, 1979; Strom et al., 1982). Recently it was found that a change of the Na-ion concentration does not alter t h e magnitude of the temperature variation of the dielectric constant (Dobbs et a]., 1983). Similar observations have been made for real glasses, where a variation of t h e chemical composition has little influence on the cryogenic anomalies. The similarity between the properties of glasses and those of ionic conductors is nicely demonstrated by ultrasonic measurements (Doussineau et a]., 1980). In fig. 26 we present the relaxation absorption of Na-P-alumina in the ultrasonic range. At low temperatures (uT,.,,* 1) the absorption rises proportionally to u0T3[see eq. (7)]. At higher temperatures ( w T , , ,1)~the characteristic plateau [a a 07"'" according to eq. (26)] is found. At first glance p-alumina seems to be a good example to prove the validity of the Tunneling Model for disordered materials. However, one also finds deviations, which are generally present in glasses as well but become more prominent in the case of p-alumina. Firstly the original distribution function as given by eq. (15) does not lead to a quantitative fit of the resonant and t h e relaxation processes simultaneously. In other words, the value of C = P y 2 / p o 2deduced from the logarithmic increase of the sound velocity does not agree with the value of C obtained from the fit of the data of the relaxation absorption (Doussineau et al., 1980). A similar observation was made in the dielectric measurements (Strom et al., 1982). Agreement can be obtained by modifying the distribution function. For example, we can use eq. (16) instead of eq. (15) and find a weighting factor W - 3. Unfortunately agreement is not reached by using a single value of W for sound waves of different polarization (Doussineau et al., 1980). A possible explanation could be that elastic waves of different polarization do not couple to exactly the same type of TS. The second problem is related with the

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coupling of the TS to the crystalline lattice. It follows from the analysis of the dielectric and acoustic measurements that the relaxation time T, is considerably larger than that found in silica based glasses (Strom et a]., 1978, 1982; Anthony and Anderson, 1978). This result, however, is in striking contradiction to values deduced from coherent echo phenomena in Na-P-alumina (v. Schickfus and Strom, 1983). The observed echo decay rates are several orders of magnitude faster than expected from fitting the Tunneling Model to the dielectric data obtained in the incoherent regime. In addition the temperature dependence of the echo decay is considerably weaker than expected from spectral diffusion (see section 2.6). It may be that in the case of Na-P-alumina not spectral diffusion but direct exchange of energy between like defects ("spin-spin flip processes") is dominant (Continentino, 1980). We have already briefly discussed in section 2.6 that the dipole moment associated with the TS can be directly determined by echo experiments. In fig. 27 the amplitude of the spontaneous echo is shown as a function of the electric field strength of the applied pulse. From t h e position of the maximum at dipole moment p = 4.8 D is derived. Taking local field cor-

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rection via the Clausius-Mossotti equation roughly into account, one obtains a bare dipole moment po = 0.5 D. This result is of interest for the construction o f TS models on a microscopic basis. For example it has been suggested (McWhan et a]., 1972). that it is the tunneling motion of Na'-ions from a so-called mid-oxygen site to another, which gives rise to TS. In that case the resulting bare dipole moment would be, 13 D, because the scparation between two mid-oxygen sites is 2.63 A . I n addition t h e tunneling rate would also be 100 low (Walker and Anderson. 1984). A small dipole moment would therefore mean charge compensation of the mobile atomic species by oxygen atoms. I t seems to be likely that the tunneling unit is n o t a single cation but a group of cations and oxygen atoms. which are changing positions in a correlated manner (Wolf, 1979). The existence of such regions is also corroborated by other experiments and i t seems that there exists even some kind of a "glass transition" a t about 100 K involving these associated regions (Strom, 1983). ''<;lass)" propcrties have been found in several other fast ionic conductors (see. fur example. frieur and Ciplys, 1981) but not in all. For example the "one-dimensional" ionic conductor Hollandite does not exhihit ii linear specific heat but rather a Schottky anomaly (v. Liihneysen et al.. 1981). In addition thermal conductivity experiments indicate that these excitations are only weakly coupled to phonons. This raises the

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question whether the existence of a wide distribution of tunneling parameters in crystalline systems depends on the dimensionality. Other crystalline systems which fall into the class of non-stoichiometric solids, also show “glassy” properties. As examples we want to mention yttriumboron YB, (with n = 61.7 or 66) (Raychaudhuri et a].. 1980: Slack et al., 1971), yttria-stabilized zirconia ((Zr02),-x(Y203)x) (Walker and Anaerson, 1984) and “Nasicon” (Nal+,ZrZ-(1,3),SixP.7_,012-(2,3)r) (Gmelin and Villar, 1981). As in the case of p-alumina it is not clear whether the tunneling particle is a single atom or a larger cluster. In contrast to p-alumina, Li,N is an ionic conductor which has nominally stoichiometric composition. First evidence of glass-like properties came from acoustic and dielectric experiments (Baumann et a]., 1980). It was suggested that TS arise from the tunneling of H’ impurities whose charge is compensated by Lit defects. Surprisingly the elastic coupling constant C as determined from the logarithmic rise of the sound velocity does not depend on the hydrogen concentration (Baumann et a]., 1981). Subsequently the glass-like behaviour was also confirmed by thermal measurements (Guckelsberger and de Go&, 1980; Gmelin and Guckelsberger, 1981; Ackermann et a]., 1981). However, it was found that the magnitude of t h e observed thermal or dielectric effects is much higher in polycrystalline material than in single crystals.

4.2. ORIENTATIONALLY DISORDERED CRYSTALS A further possibility to create a “glassy” behaviour of crystals is the introduction of orientational disorder. In this case atoms or molecules occupy regular lattice sites, but disorder exists in orientation and/or molecular conformation. The first member of this family was quenched cyclohexanol, which exhibits thermal and dielectric anomalies (Bonjour et al., 1981; Calemczuk et al., 1984). Recently the thermal properties of the very interesting system (KBr),-,(CN), has been investigated at low temperatures (De Yoreo et al., 1983; Moy et al., 1984). Fig. 28 shows the specific heat for different values of the molar ratio x. As expected the Debye value is found for pure KBr. The admixture of small amounts of KCN leads to a substitution of Br- ions by CN- ions, which are known to possess low energy quasirotational tunneling and librational states (Narayanamurti and Pohl, 1970). Consequently, the specific heat is drastically increased. However, as a big surprise, the specific heat decreases again with a higher KCN concentration. For x = 0.25 it is almost identical with that of PMMA and does not change anymore with higher concentration. There are two important observations we want to emphasize in particular. Firstly,

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the magnitude of the specific heat depends on t h e time of measurements, as already discussed for glasses in section 2.4. Secondly, the “orientational” glass also exhibits an enhanced T3-term, which is comparable to the Debye specific heat (see section 2.1). Furthermore the thermal conductivity also shows at low temperatures t h e typical behaviour of amorphous solids. Thus the situation is completely analogous to that in glasses. Here again the question of the physical nature of the excitations arises. One might try to understand the amorphous-like behaviour of the thermal low temperature properties by assuming that the motion of all but a small number of CH- ions is frozen in at low temperatures. In this case t h e mobile ions would have to have a coupling to phonons more than one order of magnitude stronger than that of ions at low concentration in order to explain the reduced thermal conductivity. This seems rather unlikely. Therefore it has been proposed that the orientational glass does not freeze into an absolute ground state configuration but into one with a local energy minimum. The transition from one local minimum to another

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thus gives rise to collective excitations. It is tempting to assume that only a small number of ions takes part in such a motion, in accordance with the ideas developed for the TS in glasses. Since one can control the puzzling low energy excitations by varying x in the orientational glass, one may have a better chance of understanding their nature.

4.3. RADIATION DAMAGED CRYSTALS Another class of crystalline materials with low temperature anomalies is formed by radiation damaged solids. It is not a new fact that these materials possess excitations additional to those present in unirradiated solids. A long time ago it was observed that the thermal conductivity of neutron irradiated a-quartz approaches that of glasses with gradually increasing irradiation dose (Berman, 1951). At the highest dose the conductivity peak of a-quartz is converted into a plateau similar to that of glass. Upon subsequent annealing the conductivity increases and returns nearly to that of unirradiated quartz. Measurements of the specific heat produced further evidence for the glass-like behaviour of neutron irradiated quartz (Westrum, 1956). The first dynamic experiment at low temperatures on irradiated quartz was the measurement of the ultrasonic attenuation at 9.4 GHz (Laermans, 1979). The sample irradiated with a dose of 8 x lof8n/cm2 with a neutron energy larger than 0.3 MeV exhibited an attenuation which could be partially saturated at higher acoustic intensities. A T3-increase was observed between 2.5 K and 4.2 K for the nonsaturable part of the attenuation. Clearly, these observations can be explained by the interaction of the sound wave with the TS present in the irradiated crystal. Subsequently the acoustic coupling y1 of longitudinal waves was deduced from phonon-echo experiments (Golding and Graebner, 1980). The value y1 = 1.2eV was further evidence that the TS in irradiated quartz and in vitreous silica are of the same nature. Recently the thermal properties of neutron irradiated quartz have also been studied below 1 K (Gardner and Anderson, 1981; de Go& et al., 1981; Saint-Paul et al., 1982). As an example we show the thermal conductivity for different irradiation doses in fig. 29 (de G&r et al., 1981, 1985). Clearly with increasing dose the thermal conductivity decreases and comes very close to that of glasses. Measurements of both specific heat and thermal conductivity support the existence of a broad distribution of TS similar to that found in glasses. Furthermore, states due to impurities and of unknown character have been observed (Gardner and Anderson, 1981; Saint-Paul et a]., 1982).

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The occurrence of these states depends on radiation dose and neutron energy. For example, it has been observed that the magnitude of t h e acoustic absorption increases with the irradiation dose (Laermans and Esteves, 1984). In general the density of states increases until a radiation dose of 3 x 10" n/cm' is reached and t h e n decreases. This reversion is also seen in t h e change of other physical quantities like mass density and paramagnetic susceptibility (Laermans, 1985). That dose roughly corresponds t o one where all atoms should have suffered a displacement. Careful studies of diffuse X-ray scattering of fast neutron irradiated quartz have shoyn that amorphous regions are generated with roughly a diameter of 2 0 A (Grasse et al.. 1981). The size of these regions hardly changes with the dose but their number increases. Thus these measurements indicate that TS are formed within "amorphous islands", behaving like small

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inclusions of vitreous silica. The dimension of these regions also gives an upper limit for t h e spatial extension o f t h e TS. ‘The search for glassy properties has also been carried out in y- and electron-irradiated quartz. In both cases excess excitations are created, but their nature seems to be different from the TS found in glasses. The excitations caused by y-irradiation are also tunneling states, but they have a well-defined tunnel splitting and are only weakly coupled to the lattice (see, for example, Saint-Paul and Nava, 1984).They are not caused by ionic motion but are due to the tunneling of holes present in the neighbourhood of A-impurities. Measurements of the thermal conductivity of electron-irradiated quartz give some evidence for the existence of a broad distribution of low energy excitations (Laermans et al., 1980). It seems that their microscopic nature is different from the TS observed in glasses but no clear evidence for one or the other possibility exists so far. Of course the question arises whether by irradiation other crystals can be made to show glassy properties. The answer is negative for MgO crystals (Gardner and Anderson, 1981), but some evidence was found in thermal conductivity for minerals heavily damaged by a-irradiation originating from incorporated thorium and uranium (Raychaudhuri et al., 1980). SYSTEMS 4.4. TWO-PHASE

The fourth kind of crystals which show glassy properties are “two-phase materials” where a small portion of one phase is quenched in the matrix of another. ZrNb is an example for such a system (Lou, 1976). It was found that Zr-Nb (20%) quenched to room temperature shows a linear specific heat and a T*-dependence of the thermal conductivity. On annealing the magnitude of the linear specific heat is decreased and the thermal conductivity increases. Subsequent investigations of the ultrasonic properties of Zr-Nb (20%) have confirmed the existence of TS (Thomas et al., 1980). In fig. 30 we show the velocity change in this material as a function of temperature for both the superconducting and normal state (Weiss et al.. 1981). Below 1 K two remarkable features are found, which can be quantitatively explained by the theory presented in section 3.3.1. The absolute value of the velocity as well as the slope of the velocity rise, with temperature depends on the state of t h e sample, normal or superconducting. This is a consequence of the electronic relaxation process whose contribution to the velocity is negligible in the superconducting state but relatively strong in the normal state. Since Zr-Nb (20%) becomes superconducting at T, = 8.6 K, i.e. well above 1 K,

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phonons already dominate the relaxation process at that temperature and no specific feature is observed at the transition temperature, which is in agreement with expectation (see fig. 21). Let us briefly discuss the origin of the TS present in ZrNb alloys quenched from a bcc single crystal P-phase to room temperature. The P-phoase matrix contains roughly 10" isolated particles per cm3, -3-S A in size consisting of a metastable athermal w-phase. Structural transformation between the two phases is possible without diffusion. This can be accomplished by moving two atoms only by 0.5 A along the ( 1 11) direction (Lou. 1976). Neutron- and diffuse X-ray scattering, Mossbauer studies and dark field electron microscopy have established that in fact a dynamic fluctuation between t h e 0- and the w-phase occurs. These local structural fluctuations have been suggested to be the origin of the TS. Therefore ZrNb seems to be a system where the microscopic origin of t h e TS is positively known. However, at least one question remains: how does a distribution of the parameters A and A of the TS come about in this well-defined system'? Having the astonishing properties of (KBr),_,(KCN), of t h e previous section in mind, one might argue that also in ZrNb more than

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two atoms need to move in order to form a TS. It seems possible that in this material t h e collective motion of clusters is also the origin of the TS. There are other “two-phase’’ systems with ‘‘glassy” dielectric and thermal properties. For example the polycrystalline materials Pb,(MnTa)O, (also called PMT) and Pb,(MgNb,)O, (PMN), which have a perovskite structure and are relaxational ferroelectrics (Raychaudhuri and Pohl, 1982). T o explain the ferroelectric and dielectric behaviour of PMN, a two-phase model has been suggested where local fluctuations and configurational changes (similar to ZrNb) take place (Holste et a]., 1976).At low temperatures tunneling between these configurations may give rise to t h e glass-like behaviour. In conclusion one can say that crystals can exhibit cryogenic anomalies analogous to those of amorphous solids. This phenomenon originates from the fact that to some extent disorder is also possible in crystals. In contrast to expectation in these materials too the nature of the tunneling particle cannot be easily predicted. The important lesson one draws, from studies of t h e glassy properties of crystalline systems, is that t h e single particle tunneling (as in systems like KCI : OH) cannot be the origin of the glassy anomalies. It is the “correlated” motion of more than one atom between configurations of nearly the same energy which gives rise to the existence of the TS.

5. Origin of the tunneling systems - theoretical attempts Recently several theoretical attempts have been made to explain the occurrence of TS in glasses. Some of these theories also include a model of t h e glass transition. However, at the present stage none of t h e theories can be considered complete. As far as t h e low temperature anomalies are concerned, they restrict themselves to arguments of plausibility and speculations. Thus no concrete quantitative estimates can be drawn from them. T h e y do not answer threc very fundamental questions: (a) Why are TS present in the amorphous network at all? (b) Why does the asymmetry A and, even more important, the tunneling parameter A have such a wide and uniform distribution? (c) What gives rise to such a high coupling to the lattice with a deformation potential y = 1 eV? In our opinion a satisfactory theory should be able to make quantitative estimates and give an answer to these questions. In view of the mentioned

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shortcomings we will restrain from presenting a detailed account of the theories. We will only briefly mention their basic concepts. The glass forming process must have a direct effect on the number and properties of t h e TS. To combine glass transition and low temperature is therefore a challenging problem. In one of the theoretical approaches the free volume theory of the glass transition (Cohen and Grest, 1979) has been extended to include the low temperature properties of glasses too (Cohen and Grest, 1980, 1981). According to this theory “free volume” gets frozen in on cooling the melt through the glass transition temperature T’. This free volume can be distributed without expenditure of energy. Molecular diffusion can take place when the accumulation of free volume leads t o formation of ephemeral voids having a size equal t o that of the diffusing species. As t h e glass is cooled to low temperature, these ephemeral voids become long-lived vacancies. The assumption is now that these voids are the direct cause of the TS. Any of the neighbouring atoms can move i n t o the void by multi-particle tunneling along a suitable one-dimensional path. By applying concepts of percolation a rough estimate of the total number of such tunneling centres has been obtained. which is i n reasonable agreement with experimental observation. According t o theory the energy scale of t h e tunneling barriers is set by the maximum curvature of the free energy surface determined by kTg itself. This leads to a correlation between the spectral density and T, given by P ‘Tg’ (Cohen and Grest. 1980, 1981).In the following section we will see that experiments partially support but also contradict this prediction. Recently in a series of papers another theory of the glassy state was dcveloped (J.C. Phillips. 1979, 1081a). which also proposes the existencc of tunneling centers (J.C. Phillips, 19Xlb). ’This theory is mainly applicable to chalcogenide glasses. where the bond constraints can be defined with less ambiguity than in other glasses. The basic approach is of topological nature. The short range order of glasses is determined by t h e condition of mechanical stability which simultaneously minimizes t h e configurational entropy. The condition of mechanical stability, which exhausts all degrees of freedom, is given by N , = Nd. Here N , is t h e number of interatomic force constraints per atom and Nd the dimension o f the formed network. Such a treatment leads to problems in the magnitude of t h e configurational entropy, which are resolved by the introduction of “medium range order”. The idea is that glass transition is associated with the formation o f clusters whose morphology avoids periodicity. They mainly determine the kinetics when t h e supercooled liquid freezes i n t o t h e glassy state. In individual clusters. which may still be partially polymerized t o other clusters, chemical ordering is imperfect and local fluctuations in stoi-J

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chiometry occur. In A,B,-, alloys there are clusters existing which are locally rich in A or B atoms and contain an excess of A-A or B-B bonds. These clusters are like fragments of the crystalline structure bordered by A or B dimers, for which evidence has been obtained through high resolution Raman spectroscopy. On a much larger scale t h e number of A and B atom-rich clusters preserves the overall stoichiometry. Generally the clusters are held together through nonbonding layerlayer interaction, but occasionally also cross-linking atoms will appear. These atoms are identified with the tunneling centers. In addition an enlargement of the T3-term in t h e specific heat of glasses is made plausible because softening of internal surface phonons is expected. However, n o quantitative estimates have been derived from this theory so far. An alternative and mathematically elegant theory has been developed, using t h e concept of gauge fields to explain the origin of the tunneling centers (see, for example, Duffy and Rivier, 1981). Here the glass is treated as an elastic continuum formed by a continuous random network. This network consists of rings with a varying number of atoms (Rivier, 1979, 1983). Odd-numbered rings, however, cannot exist in isolation. They are threaded by continuous lines into necklaces, which are either closed or end at the surface of the sample. These odd lines are the only topologically stable defects of the network. They can be characterized by a core, which is a puncture in the elastic continuum. This means that matter lies in a non-simply connected space and it has been shown that there exist two degenerate states of t h e same elastic energy per puncture loop. Tunneling between these states is possible leading to a general, microscopic and universal explanation for the tunneling centers. At this stage it is difficult to get any quantitative estimates from this theory. It seems, however, that annealing experiments are appropriate to test its relevancy. A completely different approach to explain the low temperature specific heat (Zeyher and Dandoloff, 1981) is based on the recently developed theory of the hydrodynamics of disordered solids (Cohen et al., 1976). Basically eight hydrodynamic modes exist in solids consisting of only one kind of atoms. Six of them are propagating and two are diffusive. One of the diffusive modes corresponds to vacancy diffusion, which is present in almost all otherwise perfect crystals. For a glass it is difficult to define a vacancy (or the equilibrium density of the lattice sites). One may adopt an operational definition and have ”configurational rearrangement” for glasses instead of vacancy diffusion. In this case the hydrodynamic mode can give rise to a low temperature specific heat C 0: T”’. The constant of proportionality depends on t h e transport

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coefficient associated with this extra mode. The question whether this specific heat is large enough to make a significant contribution to the total specific heat cannot be answered because t h e prefactor is still an unknown quantity. Attempts have been made to study ionic motion in simulated glasses to understand the nature of the tunneling centers (Brawer, 1981). The molecular dynamic study of simulated BeF, glass has shown tohat in addition to ordinary vibrations with a displacement of roughly 0.2 A, two other kinds of atomic motion exist. In both motions bond breaking is involved predominantly at defect sites. One kind of motion is due to large scale configurational changes and leads to changes in the network’s bond topology. This motion leads to microscopic diffusion and freezes out at the glass transition temperatu5e. The other kind of motion involves ionic motion on a scale of 0.5 to 2 A leading to the breaking and reforming of bonds. which give rise to localized structural alternations. This motion persists down to temperatures well below the glass transition and thus represents “frozen in” defects. In some cases the local structural alternations are quasiperiodic. This result indicates t h e existence of at least two metastable equilibrium positions being qualitatively equivalent t o t h e double-well potential of the Tunneling Model. For vitreous silica a concrete model for t h e low energy excitations has been proposed (Ferrari and Russo, 1983). By analogy with chalcogenide glasses the existence of “intimate valence alternation pair” (IVAP) defects has been postulated for vitreous silica. The nature of these two-center defects is electronic in origin and consequently differs fundamentally from the TS we have discussed so far. Nevertheless these defects are found to display a two-level-like energy spectrum and to exhibit properties simiiar to those known from t h e “classical” TS. The appealing feature is here that calculations are based on a concrete defect structure. On the other hand there also remain serious open questions. First, such a type of defect is not common to all amorphous solids. In particular it is difficult to imagine that similar defects also exist in metallic glasses. where cryogenic anomalies are also observed. Secondly the IVAP-originated two-level system cannot explain t h e long-time tail of t h e time dependent specific heat (see section 2.4) (Russo and Ferrari, 1984). Additional excitations have to be introduced which are assumed to be of t h e same nature as t h e TS discussed so far.

6. Connection between low temperature anomalies and the glass transition temperature

So far we have seen that there are two important quantities determining

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the low temperature anomalies, namely the density of states of t h e TS and their coupling to lattice, electric fields or free electrons. The magnitude of these quantities can only be deduced from experiments. The crucial question of the whole field is now to understand the microscopic origin of the TS and to find a way to deduce the density of states and the coupling parameters. Solving this puzzle is a twofold problem. On the one hand it is a theoretical problem and model calculations are needed. On the other hand it is also an experimental problem. Empirical rules have to be established between these quantities and other glass properties so that more information can be provided and a theoretical model can be developed. The question therefore is: Which properties of a glass should be changed to affect the low temperature properties? In this section we will present the results of some recent investigations which can be considered a first step in such a direction. We have stated before that the spectral density P does not vary too much when various glasses are considered. However, recent experiments have shown that in glasses with similar composition the specific heat changes when the glass transition temperature is varied (Raychaudhuri and Pohl, 1980, 1982). From the data (see fig. 31) the correlation P Ti' is deduced, which has also been found in ultrasonic experiments (Doussineau et al., 1983). Before the measurements were carried out, a similar LOO

2r

25

375

350

325

300

2 75

I

I

I

I

1

3.0 Tg' [

35 K'1

Fig. 31. Variation of the density of states of nitrate glasses with the glass transition temperature TI. The composition of the glasses was: O-[C~(NO&&O-~ZNQ)W and IC~(NO&]&KNO&U doped with water: A-undoped, 0 - (1.1 x ld'cm-'), 0-(2.4 x Id'cm '), V - (3.3 x I d ' cm-') and A - (3.8 x Id'cm-'). (From Raychaudhuri and Pohl, 1982.)

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[Ch. 3, 06

conclusion had already been drawn from t h e analysis of t h e thermal conductivity of a variety of amorphous solids (Reynolds Jr, 1978). This correlation between the density of states and t h e glass transition temperature is rather fascinating and is also supported by theoretical arguments (Cohen and Crest, 1980). It establishes a relation between a phenomenon seen below 1 K and another occumng at a much higher temperature, e.g. at 1400K in the case of vitreous silica. It clearly demonstrates that the low temperature anomalies are not an isolated phenomenon, but should be understood with the fact in mind that a glass is an entity in its own right and not just an extrapolation of heavily disordered crystalline systems. Theoretical attempts to explain the cryogenic anomalies have to take the glass forming process into account. The question arising now is: How general is t h e relation F T i ' ? Does it fit the data of all glasses irrespective of their structure and chemical composition? Very recent studies of t h e electrolyte glass LiCl. nH,O with variable composition show that the glass transition temperature can have a much stronger influence on the density of states (Reichert et al., 1985). On the other hand it is on the whole difficult to find a systematic variation of t h e spectral density F with T, if we also include metallic glasses in our considerations. Definitely a more complex relation is needed if amorphous solids of completely different structures are considered, as already pointed o u t (Raychaudhuri and Pohl, 1982). Nevertheless t h e above relation reflects at least t h e general trend that an increase of T, is associated with a decrease of p. How can this behaviour be understood? A simple way of dealing with the problem is to consider T, as the determining factor for the energy scale of the problem. In this case, if a fixed number N of configurational states were distributed in some form over the energy range kT,, the coefficient would be proportional to NIkT, and the inverse scaling of P with T, could be understood. It may be said that T, plays the same role for the excess specific heat of a glass as the Debye temperature plays for t h e lattice specific heat of crystals. But again nature demonstrates that this explanation might be too simple. I n fig. 32 we show the internal friction of vitreous silica joined together from measurements of different authors (Marx and Silverstone, 1YS6: Scott and MacCrone, 1970: Kaychandhuri and Hunklinger, 1984) and of the electrolyte glass L i C I . 7 H 2 0 (Kasper and Riihring, 1984). The temperature scale is normalized to t h e glass transition temperature, which differs by as much as a factor of ten. It is roughly 1500 K in vitreous silica whereas in LiCI.7H20 i t is only 140K. For both glasses a plateau is observed in the attenuation. In the case of the electrolyte glass the presence of a perfect plateau region becomes much more obvious if the temperature is plotted on a linear scale. The elastic coupling constant y of I X I . 7H,O has lint yet been determined, but velocity measurements (Kasper and Kiihring,

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0

y

335

LiCI.7Hz0 SiOz

lo-?

lo-'

REDUCED T E M P E R A T U R E ( T I T 9

Fig. 32. Internal friction of vitreous silica and electrolyte glass LiCl .7H20 against the reduced temperature TIT,. The plateau region is indicated by dashed lines. The data on vitreous silica are put together from low frequency measurements in the range 3.2 k H z to 37 kHz (Marx and Silverstone, 1956; Scott and MacCrone, 1970; Raychaudhuri and Hunklinger, 1984). The measurement on LiCl ' 7 H 2 0 has been carried out at 17 MHz (Kasper and RBhring, 1984). For clarity only a few data points are drawn in the plateau region. The plateaus are observed in different temperature intervals because of the different frequency of measurement and the reduction of the temperature scale.

1984) indicate that it is not too different from that of vitreous silica. Since the height of the plateau is determined by &z/puz, we may conclude that with decreasing T, the density of states p indeed increases as we have discussed before. The main point we want to make here, is however the following. If the energy of the TS will exhibit a plateau up to temperatures close to kTg, internal friction caused by the TS will exhibit a plateau up to temperatures close to T,. At still higher temperatures the absorption increases in all glasses because of the onset of the viscoelastic damping in the vicinity of T,. The expected behaviour is clearly seen for LiCl. 7Hz0, but not for vitreous silica. The simplest explanation for the deviation in vitreous silica would be that the density of states of the TS does not extend beyond energies

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[Ch. 3, 06

corresponding to 100K. Another explanation would be, that a new mechanism shifts the contribution of the TS expected to be found at higher temperatures to lower temperatures (Morr, 1985). Further experimental and theoretical investigations are necessary to clarify this point. It has been proposed that annealing of metallic glasses near the crystallization temperature should also have a strong influence o n the low temperature properties (Banville and Harris, 1980). In fact, annealing has significant effects on electrical and superconducting properties of these glasses (see, for example, Esquinazi et al., 1981). In agreement with the above mentioned idea, a strong increase of t h e thermal conductivity is also observed on annealing (Matey and Anderson, 1977; Ravex et al., 1981; 1984; Esquinazi et al., 1982; Grondey et al., 1983; Cotts et al., 1983; Lasjaunias et al., 1984). Similarly the slope of the logarithmic rise of the sound velocity is reduced (Matey and Anderson, 1977; Cibuzar et al., 1984). In fig. 33 we show the influence of t h e heat treatment o n the thermal conductivity of Zr,,Cr, for two stages of annealing (Esquinazi and de la 93,

7

1

r,

a

x

IL0L 5i

2 5 TEMPERATURE ( K )

Fig. 33. Phonon thermal conductivity of ZrmCua as a function of temperature. From these data the contribution of the electrons has already been subtracted. The conductivity of the quenched sample is shown by full circles. After keeping the sample for 28 days at room temperature the conductivity indicated by crosses is observed. Finally after annealing for 20 h at 250°C the data represented by open circles are found. (From Esquinazi and de la Cruz, 1984.)

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Cruz, 1984). Clearly the conductivity already rises with annealing at room temperature. The change in the conductivity above the transition temperature is probably also influenced by the TS, but is not understood so far. The increase well below T, as well as t h e change in the velocity (Matey and Anderson, 1977; Cibuzar et al., 1984) could easily be explained by a reduction of the density of states p. This idea can be verified by specific heat measurements. Indeed on annealing a strong decrease is found for sputtered Zr-based metallic glasses (Ravex et al., 1981, 1984; Lasjannias et al., 1984). For the liquid-quenched metallic glasses Zr,,Cu, and Zr,,Cu,, however, n o variation of the specific heat could be detected although a strong rise was seen in thermal conductivity (Grondey et al., 1983). This surprising result can be caused either by a reduction of t h e elastic coupling parameter y or by a change of the distribution function P ( A , A), e.g. by an increase of the weighting factor W in eq. (16). In both cases the original prediction of a decrease of the number of TS (Banville and Harris, 1980) is not fulfilled. The different behaviour of sputtered and liquid-quenched samples indicates that their microscopic structure is different. It could be that quenched samples possess a higher degree of order which is approached by the sputtered sample through annealing. Two experiments have been reported where reversible effects have been observed. Annealing of PdSiCu at 570K (for 50min) results in an irreversible increase of the phonon contribution to the thermal conductivity of about 25% (Cotts et al., 1983). Subsequent heat treatment at 520 K (for 20 min) causes a further but reversible increase by somewhat less than 10%. Renewed annealing at 570 K (for 15 min) reduces the conductivity to the value measured after the first treatment. A similar reversible, but opposite effect is seen in vitreous silica (Rusing and v. Lohneysen, 1984). Annealing at 1600 K (for 6 h) gives rise to an increase of the conductivity by about 10%. Further annealing at 1170K (for 50h) reduces the conductivity to the original value. In view of these controversial results an explanation is n o t possible at present. Two important questions remain open. First of all, is the number of TS constant? We suspect that chemistry enters into this problem. Whether annealing changes t h e number of TS is not clear yet. Is the energy scale of the TS limited by kT,? Again no direct answer is available. It seems that both questions deserve a more fundamental theoretical treatment than so far accomplished by the free volume theory. In conclusion we want to point out that a correlation between the low temperature anomalies and the glass transition temperature definitely exists. Of course this correlation does not solve the problem of the origin of the TS, but it might present a clue as well as new features to the problem.

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7. Conclusion

It is always difficult to conclude something when there is n o final answer to the main problem. Such is the case here. For this reason we will point out the major problems remaining in addition to what is already understood. The studies done so far have shown that we understand in principle the dynamics of the low energy excitations. The Tunneling Model seems to provide a good phenomenological basis on which various observations can be explained. This is definitely the main achievement of the past decade of extensive investigations. The major problem is of course the microscopic nature of the tunneling entity. Not much can be said about this, although there are certain ideas existing. First, it is not a simple defect in the sense of a dangling or broken bond. For the existence of tunneling systems, however, the rigid constraints of bonding have to be relaxed in some region. In a way we may say that some softer regions must be present in glasses. Second, tunneling systems do not arise from a single particle tunneling as in impurity doped alkali halides, since otherwise chemistry would play an important role. Probably they involve the co-operative motion of several atoms. However, a tunneling system will not consist of too many atoms. Five might be a typical number, c.g. in silica glasses it could be a Si0,tetrahedron. We hope that in the not too distant future more definitive statements can be made about them. An interesting aspect is the distribution of t h e parameters of the tunneling systems, in particular of the tunneling parameter A, giving rise to such a wide distribution of the relaxation rates (Klein et al., 1978; Klein, 1985). In the Tunneling Model a uniform distribution of the asymmetry energy A and the tunneling parameter A is assumed. Due to random strain splitting a uniform distribution of A can be imagined. But for A, which is determined by several parameters of the system, it is difficult to find an argument for the occurrence of such a uniform distribution. From experiments we know that this is indeed the case although no satisfactory explanation exists. The consequence of this distribution of A is a distribution of relaxation times extending over at least 13 decades. Another system in nature with such a broad spectrum of relaxation rates is hard to find. Looking at the trend of investigations done so far, we feel that t h e time has come to study t h e low temperature anomalies with a much broader perspective and not in isolation, as a problem in itself. It is a property of the glassy state in general and a connection should exist between low temperature anomalies and other properties of glasses. It is hoped that

ANOMALIES IN GLASSES AT LOW TEMPERATURES

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such an approach will lead to new ideas and directions and maybe to further surprises. We will have to go a long way before the final words about this intricate problem can be written. Acknowledgments

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S. HUNKLINGER A N D A.K. R A Y C H A U D H U R I

Kiisper. G . . and V. Riihring (1983) in: Proc. 17th Int. Conf. on Low Temperature Physics, LT17. cds. U . Eckern. A . Schmid. W. Weber and H. Wuhl (North-Holland, Amsterdam) p . 3x5. Kaspers. W.. R . Pott, D.M. Herlach and H . v. Liihneysen (1983) Phys. Rev. Lett. 50, 433. Klein. M.W. (1985) Phys. Rev. 831, 1114. Klein. M.W.. B. Fischer. A.C. Anderson and P.J. Anthony (1978) Phys. Rev. B18. 5887. Knaak. W.. and M. Meissner (1984) in: Phonon Scattering in Condensed Matter. Solid-state Sciences 51. eds. W. Eisenmenger. K. Lassmann and S. Dijttinger (Springer. Berlin) p. 416. Kondo. J . (1976) Physica 84B. 40, 207. Kopvillem. CJ.Kh. (1977) Sov. Phys. J . 20. 892. Laermans. C. (1979) Phys. Rev. Lett. 42. 250. 1.aermans. C. (1985) in: Structure and Bonding in Non-Crystalline Solids, eds. G. Walrafen and A. Revesz (Plenum Press, New York) in press. Laermans. C , and V. Esteves (1983) in: Phonon Scattering in Condensed Matter, Solid-state Sciences 51. eds. W. Eisenmenger, K. Lassmann and S. Dottinger (Springer, Berlin) p. 407. Laermans. C . . A . M . de Goer and M. Lwatelli (1980) Phys. Lett. 80A. 331. Laikhtman. H . D . (1984) Phys. Rev. B29. 3601. Lasjaunias. J.C.. and A. Ravex (1983) J. Phys. F13. L101. L.‘ibjaunias. :. J . C . , A . Ravex. M. Vandorpe and S. Hunklinger (1975) Solid State Commun. 17. 1045.

Lasjaunias. J . C . . A . Kavex and D. Thoulouze (1979) J . Phys. F9, 803. Lasjaunias. J . C . . A . Ravex. 0. Laborde and 0. Bethoux (1984) Physica 1 2 6 B + C . 126. Loponen. M.T.. R.C. Dynes. V. Narayanamurti and 1.P. Garno (1980) Phys. Rev. Lett. 45. 457.

Loponen. M.T.. R.C. Dynes. V. Narayanamurti and J.P. Garno (1982) Phys. Rev. B25.4310. L o u . L.F. (1976) Solid State Commun. 19, 335. Marx. J.W.. and J . M . Silverstone (1956) J . Appl. Phys. 24,81. Matey. J . R . . and A . C . Anderson (1977) J . Non-Cryst. Solids 23, 129; Phys. Rev. B16. 3406. McWhan. D.B.. C.M. Varma, F.L.S. Hsu and J.P. Remeika (1977) Phys. Rev. B15, 553. Meissner. M . , and K. Spitzmann (1981) Phys. Rev. Lett. 46,265. Morr. W. (1985) Thesis (Univ. of Heidelberg). Morton. N . (1977) Cryogenics 17, 335. Moy. I).. J . N . Dohbs and A . C . Anderson (1984) Phys. Rev. B29. 7160. Narayanamurti. V.. and R.O. Pohl (1970) Rev. Mod. Phys. 42, 201. Neckel. H . . P. Esquinazi. G. Weiss and S. Hunklinger (1985) submitted. Ng. D.. and R.J. Sladek (1975) Phys. Rev. B l l . 4017. Papuular. M . (1972) J . Phys. C5. 1943. Park. G . . A . Hikata and C. Elbaum (1981) J . Non-Cryst. Solids 45. 93. Phillips, J.C. (1979) J. Noncryst. Solids 34. 153. Phillips. J.C. (198la) J . Noncryst. Solids 43. 37; 44. 17. Phillips. J.C. (1981b) Phys. Rev. 824, 1744. Phillips, W.A. (1972) J . Low Temp. Phys. 7. 351. Phillips. W.A. (1973) J. Low Temp. Phys. 11, 757. Phillips. W.A.. ed. (1981) Amorphous Solids-Low Temperature Properties, Topics in Current Physics. Vol. 24 (Springer, Berlin). Piche, L. (1978) J . Physique 39. C6-1545. Piche. L., R. Maynard, S . Hunklinger and J . Jackle (1974) Phys. Rev. Lett. 32. 1426. Pippard, A . B . ( 1965) The Dynamics of Conduction Electrons (Gordon and Breach, New York) Prieur. J.Y.. and D. Ciplys (1981) Physica 107B. 181.

ANOMALIES IN GLASSES AT LOW TEMPERATURES

343

Ravex, A,, J.C. Lasjaunias and 0. BCthoux (1981) Solid State Commun. 40, 853. Ravex, A., J.C. Lasjaunias and 0. BCthoux (1984) J. Physique F14, 329. Raychaudhuri, A.K., and R. Hasegawa (1980) Phys. Rev. 821, 479. Raychaudhuri, A.K., and S. Hunklinger (1982a) J. Physique 43, C9-485. Raychaudhuri, A.K., and S. Hunklinger (1982b) Solid State Commun. 45, 703. Raychaudhuri, A.K., and S. Hunklinger (1984) Z. Phys. B57, 113. Raychaudhuri, A.K., and R.O. Pohl (1980) Solid State Commun. 37, 105. Raychaudhuri, A.K., and R.O. Pohl (1982) Phys. Rev. B25, 310. Raychaudhuri, A.K., J.M. Peech and R.O. Pohl(l980) in: Phonon Scattering in Condensed Matter, ed. H.J. Maris (Plenum Press, New York) p. 45. Reichert, U., M. Schmidt and S. Hunklinger (1985) submitted. Reynolds Jr, C.L. (1978) J. Noncryst. Solids 30,371. Rivier, N. (1979) Phil. Mag. AM, 859. Rivier, N. (1983) in: Topological Disorder in Condensed Matter, eds. F. Yonezawa and T. Ninomiya (Springer, Berlin) p. 13. Rosenberg, H.M. (1985) Phys. Rev. Lett. 54, 704. Rothenfusser, M., W. Dietsche and H. Kinder (1984) in: Phonon Scattering in Condensed Matter, Solid-state Sciences 51, eds. W. Eisenmenger, K. Lassmann and S. Dottinger (Springer, Berlin) p. 419. Riising, H., and H. v. Lohneysen (1984) in: Proc. 17th Int. Conf. on Low Temperature Physics, LT17, eds. U. Eckern, A. Schmid, W. Weber and H. Wiihl (North-Holland, Amsterdam) p. 253. Russo, G., and L. Ferrari (1984) Phil. Mag. B49, 311. Saint-Paul, M., and R. Nava (1984) Ferroelectrics 51, 193. Saint-Paul, M., J.C. Lasjaunias and M. Locatelli (1982) J. Physique C15, 2375. Samwer, K., and H. v. Lohneysen (1982) Phys. Rev. B26, 107. Scott, W.W., and R.K. MacCrone (1970) Phys. Rev. B1, 3515. Shiren, N.S., W. Arnold and T.G. Kazyaka (1977) Phys. Rev. Lett. 39, 239. Slack, G.A., D. Oliver and F.H. Horn (1971) Phys. Rev. B4, 1714. Stephens, R.B. (1976) Phys. Rev. B13, 852. Stephens, R.B., G.S. Cielosyk and G.L. Salinger (1972) Phys. Lett. %A, 215. Strom, U. (1983) Solid State Ionics 8, 255. Strom, U., M. v. Schickfus and S.Hunklinger (1978) Phys. Rev. Lett. 41, 910. Strom, U., M. v. Schickfus and S. Hunklinger (1982) Phys. Rev. B25, 2405. Thijssen, H.P.H., S. Volker, M. Schmidt and H. Port (1983) Chem. Phys. Lett. 94, 537. Thomas, N. (1983) Phil. Mag. B48, 297. Thomas, N., W. Arnold, G. Weiss and H. v. Lohneysen (1980) Solid State Commun. 33,523. Tietje, H., M. v. Schickfus, E. Gmelin and H.J. Giintherodt (1984) in: Proc. 17th Int. Conf. on Low Temperature Physics, LT17, eds. U. Eckern, A. Schmid, W. Weber and H. Wiihl (North-Holland, Amsterdam) p. 365. Tokumota, H., H. Kajimura, S. Yamasaki and K. Tanaka (1984) in: Proc. 17th Int. Conf. on Low Temperature Physics, LT17, eds. U. Eckern, A. Schmid, W. Weber and H. Wiihl (North-Holland, Amsterdam) p. 385. v. Haumeder, M., U. Strom and S. Hunklinger (1980) Phys. Rev. Lett. 44, 84. v. Lohneysen, H. (1981) Phys. Rep. 79, 161. v. Lohneysen, H., and H.J. Schink (1982) Phys. Rev. Lett. 48, 1121. v. Lohneysen, H., H.J. Schink, W. Arnold, H.U. Beyeler, L. Pietronero and S. Strassler (1981) Phys. Rev. Lett. 46,1213. v. Schickfus, M., and S. Hunklinger (1977) Phys. Lett. @A, 144. v. Schickfus, M., and U. Strom (1983) Phys. Rev. B28, 1068.

334

S. HUNKLINGER A N D A.K. R A Y C H A U D H U R I

v . Schickfus. M.. B. Golding. W. Arnold and S. Hunklinger (1978) J. Physique 39, C6-959. Vacher. R . H. Sussner and S. Hunklinger (1980) Phys. Rev. B21, 5850. Vladar. K.,and A. Zawadowski (1983) Phys. Rev. B28. 1564. 1582, 1596. Walker. F.J.. and A.C. Anderson (1984) Phys. Rev. B29, 5881. Wang. J . 1 ... G . Weiss and S. Hunklinger (1982) J . Physique 43, C9-533. Weiss. G . . W. Arnold, K . Dransfeld and H.J. Giintherodt (1980) Solid State Commun. 33,

111.

Weiss. G . . S . Hunklinger and H . v. Lohneysen (1981) Phys. Lett. 85A, 84. Weiss. G . . S. Hunklinger and H. v. Lijhneysen (1982) Physica 109+ llOB. 19%. Westrum. E.F. (1956) in: Proc. IVth Int. Congress on Glass. Paris (North-Holland. Amsterdam) p. 396. Wolf. D J . (1979) 1. Phys. Chem. Solids 40. 757. Zawadowski. A . (1980) Phys. Rev. Lett. 45. 211. Zcller. R.C., and R . O . Pohl (1971) Phys. Rev. B4, 2029. Zeyher. R.. and R . Dandoloff (1981) Physica 1088. 1265. Zimmermann, J.. and G. Weber (1981a) Phys. Rev. Lett. 46. 661. Zimmermann. J.. and G . Weber (1981b) Phys. Lett. %A. 32..

AUTHOR INDEX Abel. W.R. 249.259 Ackermann, D.A. 296,297,323,339 Adams. P. W. 90.9 I. 92,137 Agnolet, G. 65, 137 Ahlers, G. 40.41, 137 Ahlers. G.. see Tam, W.Y. 38, 39, 142 Ahonen, A.I., see Zeise, E.K. 243. 248,264 Allen. L.C.. see Graebner, J.E. 274,341 AIpdr. M.A. 81.89,137 Alpar. M.A.. see Anderson, P.W. 82, I37 Alvesalo. T.A. 214,248,249,250,251,252, 254,258,259 Alvesalo. T.A.. .see Archie, C.N. 251,253, 259 Ambegaokar. V. 96,97,137 Amit. D. 24. 137 Andereck. C.D. 122, 123, 124, 125, 129, 137 Anderson, A.C. 280.339 Anderson, A.C.. see Abel. W.R. 249,259 Anderson. A.C., see Ackermann, D.A. 296, 297,323,339 Anderson, A.C.. see Anthony, P.J. 270, 271.319,320.321,339 Anderson. A.C.. see Cotts. E.J. 336, 337, 340 Anderson. A.C.. see Dobbs, J.N. 320,340 Anderson, A.C.. see Gardner, J.W. 325, 327,341 Anderson. A.C., see Klein, M.W. 338,342 Anderson, A.C., see Matey, J.R. 299, 300, 336.337,342 Anderson, A.C., see Moy. D. 323,342 Anderson, A.C., see Walker. F.J. 322, 323, 344 Anderson, P.W. 74,81, 82, 192, 193,247, 280,137.259.339 Anderson. P.W.. see Alpar, M.A. 81.137 Anderson. P.W.. see Cross, M.C. 174,260 Andronikashvili. E.L. 5, 106, 137

Andronikashvili, E.L., see Hakonen, P.J. 237,261 Anthony, P.J. 270,271, 319, 320, 321,339 Anthony, P.J., see Klein, M.W. 338. 342 Arai, T., see Soda, T. 206,263 Araki, H. 309,339 Archie, C.N. 251,253,259 Archie, C.N., see Zeise, E.K. 243,248,264 Arms, R.J. 10, I37 Arnold, W. 291, 292,293. 305, 310, 314, 315,318,339 Arnold, W., see Hunklinger, S. 273, 274, 215,271,283,291,293,297,341 Arnold, W., see Jackle, J. 276,341 Arnold, W., see Shiren, N.S. 294,343 Arnold, W., see Thomas, N. 327,343 Arnold, W., see v. LBhneysen, H. 322,343 Arnold, W., see v. Schickfus, M. 293,344 Arnold, W., see Weiss, G. 314,344 Asthon, R.A. 107, 108,137 Avenel, O., see Bloyet, D. 254,260 Awschalom, D.D. 8.137 Bachellerie, A. 29 I , 339 Bagley, B.G., see Golding, B. 298.341 Bagley, M., see Gay, R. 201,219,223,224, 26 I Bailin, D. 176,259 Band, W.T., see Main, P.C. 222,262 Banville, M. 336, 337,339 Barenghi, C.F. 27,65.67,68,69,71,72,73, 74, 133, 135, 137 Barenghi, C.F., see Maxworthy, T. 104, 105, I40 Barenghi, C.F., see Swanson, C.E. 133,142 Barron, T.H.K. 295,339 Bates, D.M. 181, 182,190,260 Baumann, T. 323,339 Baym, G. 81,118,138 Bekharevich, I.L. 74, 138

345

346

AUTHOR INDEX

Bellessrr, G . 290. 301. 302. 31 I . 33Y Bellessa. Ci.. see CordiO. P. 304.340 Bellessa. G . ..YW Doussineau. P. 301. 302. 340

Bellessa. C . , .jet*Duquesne. J.Y. 271.340 Bendt. P.J. 58. 1% Benjamin. T.B. 105. 13X Berglund, P.M.. srr Bloyet. D. 254. 260 Herman. R . 32.5. 339 Bernard. L. 293. 339 Bkthoux. 0...see Doussineau. P 301. 302. 340

Bethoux. 0 , sec Lasjaunias. J.C.

336, 337,

342

Bkthoux. 0..sw Ravex. A. 299. 301. 336. 737.343 Belts. D.S.. scv Dahm. A.J.

201. 260 Berts. D.S.. (rc Hutchins. J.D. 201, 262 Bets. D.S.. ,see h n g . Ren-zhi 201. 262 Brits. D.S.. .w Saunders. J. 219, 263 Reyeler. H.U.. ice v. Lohneysen. H. 322. ?43 Bhatia. K . L . 271. 339 Bhattacharyya. P. 17.5. 176. 260 Bilir. N . 171. 272.33Y Billmann. A . . . w A r n o l d .W. 314, 315. 318. 339 Bishop. D.J. 65. 97. /3X Black. J.I.. 293. 294. 304. 305. 306. 31 I . 3 12. 339. 3 4 ) Black. J.L.. .set' Golding. B. 283. 303. 304. 305. 309. 310. 34/ Black. W.C.. .we Abel. W.R. 249. 259 Blount. E.1.. .we Brinkman. W.F. 242. 243. 760

Bloyet. [I. 254. 260 Bogoliubo\. N . 23. 13X Honjour. E. 323. 340 Bonjour. E . . .SCC Jacqmin, L. 2X8.341 Bowley. R.M. 54. 138 Bozlcr. 11.M.. we Bates. D.M. 18 I. 182. 190. 260

Brand. H. 216. 260 Brand. H.. S O Y Pleiner. H. 148. 167. 169. 263 Brawer. S A. 332.340 Brewer. I1.F. 201. 260

Brewer. D.F.. see Dahm, A.J. 201,260 Brewer, D.F.. see Hutchins, J.D. 201,262 Brewer, D.F., see Ling, R e n - h i 201, 262 Brewer, D.F.. see Saunders. J. 219.263 Brinkman. W.F. 145. 158. 174. 242.243, 260 Brodale. G.E., see Fisher. R.A. 267.3417 Brodale. G.E.. see Hornung, E.W. 267,341 Broer, M.M.. see Hegarty, J. 295. 341 Bromley. D.J. 178. 260 Brooks, J.S. 34, 138 Browne. D.A. 99. 100. 138 Bruinsma, R. 195, 233,260 Buchenau, U . 272,340 Bucholtz, L.J. 195, 260 Buckel, W., see Kampf. G. 299.341 Bunkov. Yu.M. 242.260 Bunkov. Yu.M.,see Hakonen, P.J. 240. 24 I , 243.244.245,26 I Cade. A.G. 45, 138 Calemcz.uk. R. 323,340 Calemczuk. R., see Bonjour. E. 323,340 Calemczuk. R.. see Jacqmin. L. 288.341 Campbell. L.J. 61.62, 81. 89, 126. 128, 129, 138

Careri. G . 43.44. 138 Carless. D.C. 208,216,253.255.256.257. 258.260

Case. C.R. 283,340 Chalupa. 1.. sec Andereck, C.D.

122, 123,

124. 125, 129. 137

Chandler. E., see Baym, G. 118. 138 Chandrasekhar, S. 43,138 Cheishvili. O.D., see Mamaladze. Yu.G. 40.140

Chen. H.S.. see Graebner. J.E. 270.299, 300.301.341

Cheng, D.K. 132, 133.138 Chester, M. 65, 138 Cheung. A.,see IIalley, J.W. 106. 13Y Cihuzar, G . 336.337,340 Cibuzar. G.. see Hikata, A. 310,341 Cielosyk. G.S., see Stephens, R.B. 270. 343 Cieplak. M.. see Adams, P.W. 90.91.92. 137

Ciplys. 11..see Prieur. J.Y.

322. 342

AUTHOR INDEX Cohen, C. 331.340 Cohen. M.H. 330, 334,340 Collins. J.G., see Barron. T.H.K. 295,339 Combescot, R. 148, 151, 154. 168, 169,260 Combescot, R., see Bloyet, D. 254,260 Combescot. R., see Dombre. T. 167.260 Continentino. M.A. 321,340 Cordie, P. 304,340 Corruccini, L.R. 166,260 Cotts. E.J. 336. 337,340 Cotts, E.J.. see Ackermann, D.A. 296, 297, 339 Cromar, M.W.. see Cheng, D.K. 132,133, I38 Crooker, B.C. 201.260 Crookcr. B.C., see Hook, J.R. 218,261 Cross. M.C. 158. 174, 175,235,236,247, 260 Cross. M.C.. see Brand, H. 216,260 Cross, M.C.. see Brinkman. W.F. 145, 158. 174.242.260 Cross, M.C.,see Liu. M. 156, 159, 163, 168. 170, 171,262 Dahm. A.J. 201,260 Dahm. A.J.. see Huang. W. 53, 139 Dahm. A.J., see Hutchins, J.D. 201,262 Dandoloff, R.. see Zeyher, R. 331,344 De Bruyn Ouboter. R. 5.138 De Gennes, P.G. 147. 196.213,260 De Goer. A.M. 325,326,340 De G d r . A.M., see Guckelsberger, K. 323, 34 I De Goer. A.M.. see Laermans, C. 327,342 De Groot. S.R. 146,260 De la Cruz, F., see Esquinazi, P. 336, 337, 340 De la Cruz. M.E., see Esquinazi, P. 336, 340 De Yoreo, J.J. 323, 324,340 DeConde. K.. see Williams, G.A. 45,49, 142 Depatie, D., see Reppy, J.D. 89, 141 Derrida, B. 273,340 Devismes, N.. see de Goer, A.M. 325, 326, 340 Dianoux. A.J.. see Buchenau. U. 272.340

347

Dietsche, W., see Rothenfusser. M. 271, 343 Dobbs, J.N. 320,340 Dobbs, J.N., see Ackermann, D.A. 296, 297,339 Dobbs, J.N.. see Moy, D. 323,342 Dombre, T. 167,260 Dombre, T.,seeCombescot. R. 151,260 Doniach, S., see Browne, D.A. 99, 100,138 Donnelly, J.A., see Donnelly, R.J. 24, 25, 138 Donnelly, R.J. 24.25, 28,29,33,34,42,43, 44,45,67.138 Donnelly, R.J., see Barenghi, C.F. 27,65, 67,68,69,71,72,73,74, 133, 135, 137 Donnelly, R.J., see Bendt, P.J. 58, 138 Donnelly, R.J., see Brooks, J.S. 34. 138 Donnelly, R.J., see Cheng, D.K. 132, 133, I38 Donnelly, R.J., see Glaberson, W.I. 29, 30. 47,88,139 Donnelly, R.J., see Maxworthy, T. 104, 105,140 Donnelly, R.J., see Muirhead, C. 10. 16. 27,54,140 Donnelly, R.J., see Northby, J.A. 58, 140 Donnelly, R.J., see Parks, P.E. 43,45, 52, 141 Donnelly, R.J., see Roberts, P.H. 11, 14, 18, 24, 33,38,40, 74,75, 141 Donnelly, R.J., see Springett, B.E. 44, 45. 141 Donnelly, R.J.. see Swanson, C.E. 133,142 Dotsenko Jr, V.S., see Volovik, G.E. 118. 142 Douglass, R.L. 45,47,138 Doussineau, P. 283,289,301, 302, 306, 307, 310,320,321,333,340 Doussineau, P.,see Arnold. W. 305,3 10. 314,315,318,339 Doussineau, P., see Bachellerie, A. 291.339 Doussineau, P., see Bellessa, G. 301, 302. 311,339 Dow, R.C.M. 179, 182, 188, 189, 190,234. 260 Downs, G., see Reichley, P. 81, 141 Dransfeld, K.,see Calemczuk, R. 323,340

AUTHOR INDEX

34s

Lhiisfcld. K.. Y W Golding. 8. 293. 294, 34 I Dransl'eld. K.. . v w Heinicke. W . 268, 271. 34 I Dransfeld. K . . . v w Iiunklingrr. S. 273. 341 11ransfeld. K . . ,\et Weiss. '3. 314. 344 Dull'?. I>.M. 331. 340 Iluquesne. J,Y'. 271. 341 DSnes. R.C.. .roc Loponcn. M.T. 271. 285. 2Xh.34'

306.340

Dyson. F W. 15. 138 Dqaloshinskii. LIZ.. 118. 138 Ilastop. A . D . 158. '60 Einiel. D. 254. 255. 256. 258. 260 Eiiizcl. D.. Y W Wiilflc. P. 248. 256. 258, 264 Eisenstein. J P. LOO. 201. 716. 260 Ekholrn. 11.1. 99. 101. 138 Elhaum. (... Y W Araki. I i . 309. 33Y lilbaum. C.. scc C'ihuzar. G. 336. 337.340 Elhaurn. (... v w llikata, A. 310. 341 Elhaurn. C., .see Park. G . 309.342 Iiskii. G 253. 2.55. -760 Esquinazl. P. 336. 337. 340 Esquinaii. P.. wc Neckel. H. 305. 317. 342 l..sle\ec. L'.. .wc Laermans, C. 326. 342 F'araJ. E.. .we Mitchell. K

Fisher, X.X. 101. 13Y Fleming. P.I>.,reeCohen, C. 331. 340 Fleurov, V.N. 288.340 Foglc. W.E.. see Mayberry. M.C. 249.262 Fox. D.L. 294.340 Fraenkel, L.E. 12, 14. 15. 139 Frdscr, J.C.. see Rudnick, 1. 32. 141 Frcnois. C.. see Arnold. W. 305.33Y Frcnois. C.. see Doussincau. P. 283. 289,

249. 263

Federle. (2. 28X. 340 Fcrreri. 1. 332. 340 Ferrari. L., .SLY Russo. G . 332. 343 Fetter. A.L.. 5. 2X.4X. 51. 176. 179. 182. 183. 1x4. 1x6. 18X. 189. 196. 214. 238, 245. 1-18. 26 I Fetter, A.L.. .wt Bucholtz. L..J. 195. 260 Fetter. A.L.. w e Padmore. T.C. 28. 141 Fetter. A.L.. st^' Stauffer. D. 58. I41 1-etter. A . L . . svc Williams. M.R. 114. 125. ILL). 14: Feynman. R.P. 5. 14. 205. 138.261 Fiory. A.T. 97. 13X 1,'ischcr. H.. .we Klein. M.W. 338. 342 1,'isher. 8 . 7x3. 340 I'isher. 1 2 . . .WP Rent, L S . 49. 141 Fi.ihcr. M.E.. .wc Langer. J.S. 206. 262 Fisher. R.A. 267. 340 Fiber. K.A.. .we Hornung. E.W. 267. 341

Friedrich, J. 295, 340 Fujita. T. 237,261 Fulde, P.. see Black, J.L. 306, 31 I , 312. 33Y Galperin. Yu.M. 297, 310,340 Gammel. P.L. 200, 201. 236.261 Gammel, P.L., see Hall, H.E. 236. 246. 261 Gamota. G . 15, 139 Gardner. J.W. 325. 327. 341 Garibashvili. D.I., see Hakoncn, P.J. 236, 237. 261

Garno, J.P.. see Loponcn. M.T. 271. 285. 286.342

Gasparini. F.M.. see Joseph, R.A. 65. I3Y Gay. R. 201,219. 223,224, 261 Gianque. W.F., see Fisher, R.A. 267,340 Gianque, W.F., see Hornung. E.W. 267, 34 I Gibbs. J.H.. see Cohen, C. 33 I . 340 Ginzburg. V.L. 19, 24, 33. 139 Glaberson, W.I. 29. 30. 31. 32.47.48, 88, 130. 133. 13Y

Glaherson. W.I., see Adams, P. W. 90. 9 1. 92, 137

Glaberson, W.I.. see Andereck. C.D. 122. 123. 124, 125, 129. 137 Glaberson. W.I., see Asthon. R.A. 107, 108. 137 Glaberson. W.I.. see Donnelly. R.J. 29, 33, 42.67. 138 Glaberson, W.I., see Fiory. A.T. 97, 138 Glaberson. W.I., see Ilegde. S.G. 65. 83. 84. 86, 125, 13Y Glaberson, W.I.. see Johnson, W.W. 29, 45.13Y

Glaberson, W.I.. see Kim, M . 97.98, 100. 140

AUTHOR INDEX Glaberson, W.I.. see Ostermeier, R.M. 44, 45,46.47,48,49, 50. 52. 53. 54, 55. 130, 140 Glaberson. W.I., see Steingart. M. 18, 141 Glaberson, W.I.. see Yarmchuk, E.J. 62, 67. 74, 78. 79, 80. 81,83, 125, 142 Gmelin. E. 323,341 Gmelin. E.. see Tietje, H. 297.343 Golding. B. 273,275,219.283.293.294, 298.303, 304,305,309, 310,325,341 Golding, B.. see Fox, D.L. 294,340 Golding. B.. see Graebner, J.E. 270,274, 294,299. 300.301.341 Golding, B.. see Grasse, D. 326.341 Golding, B.. see Hegarty, J. 295,341 Golding. B., see v. Schickfus, M. 293,344 Gordery. R.A., see Hegde, S.G. 65,139 Gordon, M.V.J.. see Yarmchuk, E.J. 62, 63, 14-7 Gorter. C.J. 5,66. 139 Goubau. W.M. 285,286,341 Could. C.M. 219.261 Could. C.M., see Bates. D.M. 181, 182. 190.260 Graebner. J.E. 270,274,294,299,300,301. 34 I Graebner. J.E., see Golding, B. 273,275, 279.283,293.294.303,304,305,309,310, 325.341 Grant. J. 28, 104, 139 Grant. J., see Roberts, P.H. 24. 29, 141 Grasse, D. 326,341 Greenspan. H.P. 94,139 Grest, G.S., see Cohen. M.H. 330, 334,340 Greywall. D.S. 249, 252.261 Grondey. S . 336.337,341 Gross, E.P. 20.28,139 Gross, E.P.. see Amit, D. 24, 137 Gross, W.M.. see Manchester, R.N. 8 I , I40 Guckelsberger. K. 323,341 Guckelsberger. K., see Gmelin. E. 323,341 Guenault, A.M. 255,261 Guntherodt, H.J., see Tietje, H. 297,343 Giintherodt, H.J.,see Weiss. G. 314, 344 Gurevich. V.L.. see Galperin. Yu.M. 297, 310.340

349

Gurgenishvili, G.E. 164, 165.261 GyorlTy, B.L., see Black, J.L. 304,340 Hadrer, D.. see Friedrich, J. 295,340 Haavasoja, T.. see Alvesalo, T.A. 214. 248, 249.250,251,252,254.258,259 Haemmerle, W.H.. see Fox, D.L. 294,340 Hakonen, P.J. 236.237.238,240.241.243. 244,245,261 Hakonen, P.J.. see Bunkov. Yu.M. 242, 260 Hakonen, P.J., see Pekola. J.P. 201, 204, 205,243.263 Hall, H.E. 19,66, 74, 105, 125, 155, 159, 235,236,246,247.248.256, 139.261 Hall, H.E., see Carless, D.C. 208,216, 253, 255,256,257,258,260 Hall, H.E., see Eastop, A.D. 158,260 Hall, H.E., see Gammel, P.L. 200, 201, 236, 26 I Hall. H.E., see Gay, R. 201,219,223. 224, 26 I Hall, H.E., see Hook, J.R. 175, 185, 186. 187, 188, 189, 194, 197.261 Hall, H.E., see Main, P.C. 222,262 Hall, H.E., see Mitchell, R. 249, 263 Halley, J.W. 106, 139 Hallock, R.B., see Ekholm, D.T. 99, 101, 138 Hallock, R.B., see Maps, J. 65,140 Halperin, B.I., see Ambegaokar, V. 96,97, 137 Halperin, B.I.. see Anderson, P.W. 280, 339 Halperin, B.I., see Black, J.L. 293,294,340 Halperin, B.I., see Golding, B. 273,341 Ham, T.E. 195,261 Hama, F.R., see Arms, R.J. 10, 137 Hamilton, P.A., see Manchester, R.N. 81. 140 Hara, J. 259,261 Hard, J., see Ono. Y.A. 253,255,263 Harris, C.G. 176,261 Harris, R., see Banville, M. 336,337,339 Hasegawa, A. 15.139 Hasegawa, A,, see Gamota, G . 15, 139 Hasegawa. R., see Raychaudhuri, A.K.

350

AUTHOR INDEX

300. 30 I . 34.J I4;twnuk~.F 1 . J . 105. 13Y lfavelt>ck. T . H 126, 13Y Haqes. h... v w I M h . J.N. 320. 340 Ilrbard, A . F . v w Fiory. A.T. 97, 138 tlzhral. H.. Y W C'rookcr. f3.C. 201. 260 I-fegarlb. J. 295.341 tlsgdc. S . G 6.5, 33. Y?. 86. 125. 139 Heinickc. W 268. 271. ? 4 / Heiserman. J. 38. 39. /3Y I lerlach. D.M.. wc Kaspers. W. 297. 342 I l l C k \ . W.M. I I . 13Y Hikara. A . 310%3 4 / Hikata. A , . . s w Araki. t l . 309. 339 Hlkuta. A,. wo Cibuiar. G . 336. 337.340 tlthaia. A,. .wc Park. G . 309. 342 Ijillcl. A . 246. 261 llills. K.N. 22. 32. 33. 34. 35. 36. 37. 39. 40. 117. 139 Hills. R.N.. we Donnelly. R.J. 24, 25. 138 Hills. R.N.. wc Rohc.rts. P.H. 33. 38. 40. 141 I1o.T.-1.. 156. 169. 172. 192. 194. 199. 219.

236.261

Hu, T.-L...set Bhattacharyya. P. 175. 176, 160 Ho.T.-L...scv Mcrmin. N.D. 150. 171. 172. 9 7 -6-

I-lojgaard Jenscn. H. 253. 255,26/ llojgaiird Jcnscn. 14 . . v w Einzel, D. 255. -760 liolstc. J.(.. 329. 34I tlook. J . R . 175. 185. 186. 187. 188. 189. 194. 197.218. 261 l i o o k . J R.. .rerC'arless. D.C. 208. 216. 253. 255. 256. 257. 258. 260 Hook. J R.,wc Eastop. A.D. 158. 260 tlook. J.R ...s ecC;ay. R . 201. 219. 223, 224. 16 I Hook. J.R.. w e Main. P.C. 222. 262 Iinok, J.R.. wc Mitchell. R . 249. 263 llopfinger. E.J . .we Maxwwrthy. T. 104. 105. 140

I lorn. EH.. .w Slack. G.A. 323. 343 Hornung. E.W. 267. 341 Hornung. E.W ..w Fisher, R.A. 267. 340 Hsu. F.S.L.. .wc Golding. B. 298,34/

Hsu. F.S.L., see Graebner, J.E. 270. 299. 300.301.34/

Hsu. F.S.L.. see McWhan. D.B. 319, 322. 342

Hu, C.-R. 151. 155, 191, 195. 196.262 H u . C .-R...ser€iam. T.E. 195,26/ Hu,C.-R..seeSaslow, W.M. 161. 167.263 Hu. P. 294. 341 Huang, W. 53.139 Hubex. D.L. 96, 139 Hulin. J.P., see Heiserman. J. 38. 39, 13Y tlunklinger, S. 273. 274, 275. 277. 278. 279. 282.283.291.293.

295. 297.34/

Hunklinger. S., see Arnold, W. 291, 292, 293.339

Hunklinger. S.. see Baumann. T. 323.33'2 Hunklinger, S., see Bhatia, K.L. 271,339 Hunklinger, S.. see Federle, G . 288,340 Hunklinger, S.. see Golding. B. 293. 294. 34 I Hunklinger. S., see Jackle, J. 276.34/ flunklinger, S., see Lasjaunias. J.C. 267. 269,342

Hunklinger, S.. see Neckel. H. 305. 3 17. 342

Hunklinger. S.. see PiehC, L. 277, 278. 342 Hunklinger, S., see Raychaudhuri, A.K. 288.290, 307.308. 309, 310,316,317, 334, 335.343 Hunklinger. S., see Reichert, U. 334.343 flunklinger, S., see Strom, U . 320. 321,343 Hunklinger. S., sre v. Haumeder, M. 271. 343 Hunklinger, S., see v. Schickfus. M . 27X, 293.343,344 Hunklinger. S., see Vacher, R. 271.344 Hunklinger, S., see Wang. J.L. 309.344 Hunklinger. S.,.see Weiss, G. 305, 315. 318. 327. 328,344 Hutchins. J., see Dahm, A.J. 201. 260 Hutchins. J.D. 201,262

Ignatev. F.N., see Karpov, V.G. 297.341 Iguchi. 1.. .see Fetter, A.L. Ihas. G.G. 31,41,/3Y

281,283.

48, 51, /38

Ihas, G.G.. see Spencer. G.F. 242,263

AUTHOR INDEX Ikkala, O.T.. see Hakonen, P.J. 236, 237, 238, 261 Iordanskii, S.V. 26,73.139 Islander. S.T.,see Hakonen, P.J. 236. 237, 238, -761 Jickle. J. 275. 276. 306,341 Jackle. J., see Baumann, T. 323,339 Jickle. J.. see Black, J.L. 304,340 Jickle. J.. see Piche, L. 277,278,342 Jacobsen, K.W. 215.262 Jacqmin. L. 288,341 Jacqmin, L., see Calemczuk, R. 323.340 JolTrin. J. 291,341 JoRiin, J.. see Bernard, L. 293,339 Johnson, W.W. 29,45,139 Johnson, W. W., see Glaberson, W.I. 130, 133,139 Jones, C.A. 21,23,24. 25, 26, 27. 139 Joseph. R.A. 65,139 Kagdn. Yu. 291.341 Kajimurd, H.. see Tokumota. H. 271,343 Kampf. G. 299,341 Kane, A.B., see Golding. B. 279, 283,303, 304.305,309,310,341 Kane. A.B., see Graebner, J.E. 274,341 Karpov. V.G. 273.281.283.297,341 Kasper. G. 334,335.342 Kaspers, W. 297.342 Kazyaka, T.G., see Shiren, N.S. 294,343 Keith, V., see Guenault. A.M. 255.261 Kennedy, C.J.. see Guenault, A.M. 255, 26 I Khalatnikov, I.M. 146, 154,262 Khalatnikov, I.M., see Bekharevich. I.L. 74. 138 Kharadze, G.A., see Gurgenishvili. G.E. 164, 165.261 Kida. S. 105. 139 Kiewiet, C.W.. see Main, P.C. 222,262 Kim. M. 97,98. 100,140 Kinder, H., see Rothenfusser, M. 271,343 Klein. M.W. 338,342 Klein, M.W., see Fisher, B. 283,340 Kleinberg, R.L. 179. 180,262 Kleinert. H. 182. 214, 230, 262

35 I

Klinger, M.I., see Karpov, V.G. 28 I , 283, 297,341 Knaak, W. 285,286.342 see Grasse, D. 326.341 Kocar, 0.. Kondo, J. 305,327,342 Kopnin, N.B., see Volovik. G.E. 237. 264 Kopnin, N.V. 248,262 Kopvillem, U.Kh. 293,342 Kosterlitz, J.M. 5,64,95. 140 Kosterlitz, J.M., see Nelson, D.R. 64. 96, 140

Krasnov, Yu.K. 81, 140 Krasnov, Yu.K., seeCampbel1. L.J. 89, 129,138 Krusius, M. 182, 188, 189. 199,262 Krusius, M., see Bunkov, Yu.M. 242,260 Krusius. M.. see Hakonen, P.J. 240,241, 243,244,245,261 Krusius, M., see Paulson, D.N. 182. 188, 263 Krusius. M., see Pekola, J.P. 201, 204, 205. 243.263 Kumar, P.. see Maki, K. 190, 241,262 Kurkijarvi, J., see Thuneberg, E.V. 199, 200,23 1,264 Laborde, O., see Lasjaunias, J.C. 336, 337. 342 Laermans, C. 326,327,342 Laermans, C., see de Goer, A.M. 325, 326, 340 Lagnier, R., see Bonjour, E. 323,340 Laikhtman, B.D. 289.342 Lamb, H. 12,140 Landau, L.D., see Ginzburg, V.L. 19, 139 Lane, C.T., see Reppy, J.D. 89,141 Langer, J.S. 5, 28, 206, 140,262 Lasjaunias, J.C. 267, 269,298, 336, 337, 342 Lasjaunias, J.C., see Ravex, A. 299, 301, 336,337,343 Lasjaunias, J.C., see Saint-Paul, M. 325, 343 Lawless, W.N., see Ackermann, D.A. 323, 339 Lawless, W.N., see Holste, J.C. 329,341 Lee, D.M.,seeGould, C.M. 219,261

352

AUTHOR INDEX

Ltggctt.A.J. 145. 161. 162. 165. 171. 175. 251.32 l-eisurt. R . G . wc Doussineau, P. 283. 289. 306. 3411

262

Lcvclut. A...w(~Arnold.W. 305. 310. 314. 315. 7 I X . 339 Levelut. A,. .sw Bachcllerie. A. 291, 33Y I.e\elut. A , . .wc Bcllessa. G . 301. 302. 31 I . 339 I-e\,elut. A , . scc Douwneau. P. 283. 289. 3U 1. 302. 306. 340 I.cvclut. A.. .WL, Joffrin. J . 29 I , 341 Lin-Liu. Y.R. 177. 178. 180.261

Lin-1.i~.Y.R.. . W L ~Vollhardt. D .

242,262

Maki. K., see Bruinsma. R. 195,233.260 Maki. K.. see Lin-Liu, Y.R. 177. 178. 180.

178. 179.

264

Linekin. D.M.. scc Snyder. H.A. 59. 141 Ling. Ren-zhi 201.262 Liu. M. 145. 146. 147. 155. 156. 159, 162. 163. 164. 165. 166. 168. 169, 170. 171.262 Liu. M...sw Cross. M.C. 175. 260 120cdlelli. M.. .sw dc Goer. A.M. 325. 326. 340 1.ocatelli. M...sec Jacqmin. L. 288. 341 Locatelli. M.. SCC Laermans. <'. 327. 342 L.ocatelli. M.. .wc Saint-Paul, M . 325, 343

Maki, K.. we Vollhardt, D. 178, 179. 190. 2 14.234.264

Maksimov, L.A.. see Kagan, Yu. 291,341 Mamaladze, Y u.G. 20, 40.140 Mamaladze, Yu.G.. see Andronikashvili. E.L. 5 . 137 Mamniashvili. G.. see Pekola, J.P. 201. 204,205.243.263

Manchester. R.N. 81. 140 Manninen. M.T. 201,231,262 Manninen. M.T., see Atvesdlo. T.A. 214. 248,249.250,251.

252.254.258. 259

Maps. J. 65,140 Marechal. J.C.. see Mathieu. P. 58. 59. 140 Markkula. T.K.. .see Hakonen. P.J. 236, 237.261

Marlin. P.C. 148.262 Martinon. C.. see Arnold. W. 292,33Y Marx, J.W. 334, 335. 342 Matecki, M., see Doussineau, P. 333.340 Matey, J.R. 299,300. 336,337.342 Mathieu. P. 58, 59, 60.67. 70. 247, 248.

Lnhsen. t. X I . 140 Loponen. M.T. 271. 285. 286.342 Lou. L.F:. 317. 728, 342 L.ounasmaa. O.V.. see Hakonen, P.J. 236. 237.31 Lounasmaa. 0 . V . . .we Pekola. J.P. 200. 201. 204. 205.236. 243.263 I.ove. A.. we Bailin. Ll. 176. 25Y 1.ugt. H.J. 17. 140 1.ynaIl. L i t . . see Miller. R.J. 67, 140

Maxworthy, T. 104. 105.140 Mayberry. M.C. 249.262 Maynard. J., see Heiserman. J. 38, 39, 13Y Maynard. R..see Piche. L. 277.278.342 Mazur, P.. see de Groot. S.R. 146.260 McCauley, J.L. 44,45. 140 McClintock, P.V.E. 54, 140 McClintock. P.V.E.. see Bowley. R.M. 54,

fvlaack Bisgaard. T.. we Hojgaard Jensen.

McCormick. W.D.. see Careri. G. 43.44.

140,262

138

ii. 253. 255.261 MacC'hesriey. J.B.. sec Hegarty. J. 295. 341 LIacCrone. R.K ..we Scott. W.W. 334. 3 3 5 . 342 MacDonald. W.M.. SCP Ackermann. D.A. '-96. 297, 339 Main. P.C. 212. 262 Main. P.C.. wc Alvesalo. T.A. 24X. 249. 259 'viahi. K 189. 190. 195. 237. 238, 240. 241.

I38

McLean. E.D.. see Scholtz. J.H. 32. 141 McLean. K.O.. see Case, C.R. 283.340 McMillan. W.L.. see Rutledge, J.E. 97. 141 McWhan. D.B. 319,322.342 Mehl, J.B.. see Miller, R.J. 67, 140 Meissner. M. 285. 286.342 Meissner. M., see De Yoreo. J.J. 323. 324. 340

Meissner, M.. see Knaak, W. 285, 286.342

AUTJ-1OR INDEX

353

Mellink, J.H., see Gorter, C.J. 66, 139 Mermin.N.D. 150, 151. 158, 171, 172, 174. 191. 196, 197, 198,262,263 Mermin. N.D.. see Bhattacharyya, P. 175, 176,260 Mermin, N.D.. see H0.T.-L. 156, 169,261 Miller. R.J. 67. 140 Mineev. V.P. 169,263 Mineev, V.P., see Hakonen. P.J. 240, 241, 243.244.245.261 Mineev, V.P.. seeVolovik, G.E. 152, 158, 169. I7 1,245.264 Mitchell, R. 249. 263 Miyake, K. 153. 154, 155.263 Mochel, J.M.. see Rutledge, J.E. 97, 141 Morr. W. 336.342 Morton, N. 300,342 Mory, M., see Maxworthy. T. 104. 140 Moss, F.E., see Bowley, R.M. 54, 138 Moss. S.C., see Grasse, D. 326,341 Moy, D. 323,342 Moy, D., see Ackermann, D.A. 323,339 Muirhead, C. 10. 16.27.54,140 Muirhead, C.M.. see Paintin. J. 42, 141 Muzikar, P. 151, 167,263 Muzikar. P., see Mermin, N.D. 151, 158, 263

Ohmi, T. 49. 54. 140 Ohmi, T., see Fujita, T. 237,261 Oliver, D., see Slack, G.A. 323,343 Ono, Y.A. 253,255,263 Onsager, L. 5 , 17,140 Onsager, L.. see McCauley, J.L. 44.45. 140 Onuki, A. 247,263 Orbach. R., see Derrida, B. 273.340 Osheroff, D.D., see Brinkrnan, W.F. 242, 243,260 Osheroff, D.D., see Cormccini, L.R. 166. 260 Osheroff, D.D., see Paalanen, M.A. 201. 233,234,263 Ostermeier. R.M. 44,45,46,47. 48,49, 50, 52,53,54,55, 130, 140 Ostermeier, R.M., see Glaberson, W.I. 130. 133,139 Ostermeier, R.M., see Halley, J.W. 106, 139

Nadirashvili, N.S. 106, 140 Nagai. K. 151, 154. 158, 159,248,255,263 Nagai, K., see H~jgaardJensen, H. 253. 255,261 Nagai, K.. see Ono, Y.A. 253,255,263 Nakahara, M., see Fujita. T. 237,261 Nakahara, M.. see Maki, K. 242,262 Narayanamurti, V. 273,280. 319,323,342 Narayanamurti. V.. see Loponen, M.T. 27 I , 285,286.342 Nava, R.. see Hunklinger, S. 273,341 Nava, R., see Saint-Paul, M. 327,343 Neckel, H. 305,317,342 Nelson, D.R. 64.96. 140 Nelson, D.R., see Ambegaokar, V. 96,97, 137 Neurnaier, K.. see Eska, G. 253,255,260 Newton, L.M.. see Manchester, R.N. 81, 140

Paalanen, M.A. 201,233,234,263 Packard, R.E. 81,141 Packard, R.E., see Eisenstein, J.P. 200,201, 216,260 Packard, R.E., see Pekola, J.P. 200,201, 204,205,236,243,263 Packard, R.E., see Williams, G.A. 44.45. 49,63,142 Packard, R.E., see Yarmchuk, E.J. 62.63, 129, 142 Padmore, T.C. 28,141 Paintin, J. 42, 141 Papoular, M. 296,342 Park, G. 309,342 Park, G., see Araki, H. 309,339 Parks, P.E. 43.45, 52, 141 Parks, P.E., see Donnelly, R.J. 29, 33,42, 67, 138 Parodi, P., see Martin, P.C. 148,262

Ng, D. 288,342 Norbury, J. 15,140 Northby, J.A. 58,140 Niicker, N., see Buchenau, U. 272,340 Nummila, K.K., see Pekola, J.P. 200, 201, 204,205,236,243,263

354

AUTHOR lNDEX

Parpia. J.. .see Saunders. J. 2.51. 252. 263 Parpia. J.M. 201. 263 Parshin. D . A . . .set’ Galperin. Yu.M. 297. 3 10. 340 Parshin. D.A.. .rep Karpov, V.G. 273. 341 Paulson. D.N. 182. 188. 263 Paulson. D.N..we Krusius. M. 182. 188. 189. 199.262 Prech. J.M.. st’c Raychaudhuri. A.K. 279. 3 3 . 327.343 PeisI. I I . . .wc Grasse. L). 326. 341 I’ekola. J.P. 200. 201. 204. 205. 236. 243. 3 3 Pekola. J.P.. s w Manninen. M.T. 201. 231. 262 Pershan. P.S.. .we Martin. P.C. 148. 262 Peshkov. V.P. 5. / I / Pethick. C’.J... s i ~Baym. G. 81. 138 Petschek. K.G. 99. I41 Pfitzner. M. 2.54. 258.263 Phillips. J.C. 773. 330.342 Phillips. N.E.. . s i v Mayberry. M.C. 249. 262 Phillips, W. A . 2x0. 283. 291. 296. 297. 342 Phillips. W.A.. .see Rilir. N. 271. 272. 339 Pichi.. 1.. 277. 278. 294. 342 Pichi. L... .we Bernard. L. 293.339 Piche. L.. see Hunklinger. S. 279. 341 Pichk. l... .see Jickle. J. 276. 341 Pickett. G.R.. s w Guenault. A.M. 2.55. 261 Pietronrro. L.. .we v . Lohneysen. H. 322. 34.1 Pines, D.. ree Alpar. M.A. X I . 137 Pines. D.. .ser Anderson. P.W. 81. 82. 137 Pines. D.. .see Bayrn. G. 81. /3X Pippard. A.B. 299.342 l’itaev>kii. L.P. 20. 104. 134. 247. /4/. 263 Pit;tevskii. L.P.. .swGinzburg. V.L. 19, 24. .7 .3 . /‘I

PI.q d.i ’. s . R.. szc Mathieu. P. 59.60. 70. 140 Pleiner. f l . 148. 167. 169. 263 Pobell. F.. s w Ihas. G . G . 31. 41. 13Y Pocklington. H . C . 103. 141 Pohl. R.O.. .we De Yoreo. J.J. 323. 324. 340 Pohl. R.O.. see Narayanarnurti, V. 273. 2x0. 7 19. 323. 342

Pohl. R.O.. see Raychaudhuri. A.K. 279, 323.327.329, 333, 334.343 Pohl. R.O., see Zeller. R.C. 267, 268, 269, 344 Poon. S.J., see Cotts, E.J. 336, 337. 340 Port. It.. see Thijssen. H.P.tl. 295. 343 Potl, R.. see Kaspers, W . 297, 342 Potter, R.C.. see Ackermann. D.A. 323, 33Y Prieur. J.Y. 322.342 Prieur. J.Y ..see Doussineau. P. 283. 289, 306,340 Pritchard. W.G. 105. 141 Putney, Z.. see Snyder, H.A. 59, 141 Putterman, S.J. 156.263 Rainer. D.. see de Gennes, P.G. 196. 260 Rainer, D.. see Serene. J.W. 214, 250, 251, 254.263 Rajagopal. E.S. 109. 14/ Ravex. A. 299,301, 336,337,343 Ravex. A,, see Lasjaunias, J.C. 267, 269. 298,336, 337,342 Ray. J., see Alvesalo. T.A. 248. 249,ZSY Raychaudhuri, A.K. 279,288.290.300. 301. 307. 308.309.310,316,317,323,327, 329, 333. 334, 335,343 Rayfield. G.W. 9. 17. 18. 31. 141 Rehn, M .M.. see Alvesalo, T.A. 248,249. 259 Reichert, U. 334.343 Reichley. P. 81. 141 Reif, F., .see Rayfield, G .W. 9. 17. 18, 3 I , 141 Remeika. J.P.. see McWhan. D.B. 319, 322.342 Rent, L.S. 49, 141 Reppy, J.D. 89,141 Reppy, J.D.. see Agnolet, G. 65, 137 Reppy. J.D., see Archie, C.N. 251.253.259 Reppy, J.D.. see Bishop, D.J. 65, 97, 138 Reppy. J.D.. see Crooker, B.C. 201.260 Reppy, J .D., see Garnmel, P.L. 200, 201, 236, 26 I Reppy. J.D.. see Hall, H.E. 236. 246, 261 Reppy. J.D., see Langer. J.S. 5 , 28, 140 Reppy. J.D.. see Parpia, J.M. 201,263

AUTHOR INDEX Reppy. J.D., see Saunders, J. 25 I , 252,263 Reynolds Jr, C.L. 334,343 Richardson, R.C.. see Archie, C.N. 251, 253,259 Richardson, R.C., see Saunders. J. 251, 252.263 Richardson. R.C.. see Zeise, E.K. 243,248, 264 Ridner. A,. see Esquinazi, P. 336,340 Rivier, N. 331.343 Rivier. N., see Duffy. D.M. 331.340 Roberts. P.H. 11, 14, 18. 24, 26. 29. 33. 38, 40. 74, 75. 141 Roberts. P.H.. see Donnelly, R.J. 24, 28, 34.42.44.45, 138 Roberts, P.H., see Grant. J. 28, I39 Roberts, P.H., see Hills. R.N. 22, 32. 33, 34, 35, 36.40. 117, 139 Roberts. P.H..seeJones. C.A. 21.23, 24, 25. 26. 27. 139 Robin. A,, see Doussineau, P. 306. 307, 320. 32 1,340 Rohring. V.. see Kasper. G. 334, 335.342 Rothenfusser. M. 271,343 Roubeau. P.. see Hakonen, P.J. 236. 237, 26 I Rowe, J.M.. see De Yoreo, J.J. 323, 324, 340 Ruderman. M. 81. 141 Ruderman, M., see Anderson, P.W. 8 1, 137 Ruderman, M.. see Baym. G. 81, 138 Rudnick. I . 32. 65, 141 Rudnick, I.. see Heiserman, J. 38, 39, 139 Rudnick, I., srr Scholtz. J.H. 32. 141 Rush, J.J.. see De Yoreo. J.J. 323, 324,340 Rusing, H. 337,343 Russo. G . 332.343 Russo. G . ,.see Ferrari. L. 332,340 Rutledge. J.E. Y7, 141 Saint-Paul. M . 325, 327.343 Salce, B.. s e Bonjour. ~ E. 323. 340 Salce. B.. see de Goiir. A.M. 325, 326,340 Salinger. G.L.. .see Stephens, R.B. 270,343 Saloheimo. K.. .see Hakonen. P.J. 236. 237. 26 I Salomaa, M.M. 242,245.263

355

Salomaa, M.M., see Hakonen, P.J. 240, 241,243,244,245.261 Samara. G.A., see Holste, J.C. 329,341 Samwer, K. 298,299,343 Samwer, K., see Grondey, S. 336, 337.341 Sandiford. D.J., see Gay, R. 201,219. 223, 224,261 Sandiford, D.J., see Main, P.C. 222,262 Saslow, W.M. 161, 167,263 Saslow, W.M.,seeHu, C.-R. 151, 155,262 Sauls, J.A. 214,249,263 Saunders, J. 219,251,252,263 Saunders, J., see Dahm, A.J. 201,260 Saunders, J.. see Zeise, E.K. 243, 248.264 Scaramuni. F., see Careri, G . 43.44, 138 Schink, H.J., see Grondey, S. 336, 337.341 Schink, H.J., see v. Lohneysen. €1. 271, 322,343 Schmidt, M.,see Hunklinger. S. 295,341 Schmidt, M.. see Reichert. U. 334.343 Schmidt, M., see Thijssen, H.P.H. 295.343 Schoepe, W., see Eska, G . 253,255,260 Scholtz, J.H. 32. 141 Schon, W., see Doussineau, P. 333.340 Schopohl, N. 176,263 Schopohl, N., see Vollhardt, D. 214,264 Schuhmacher, G., see Bernard. L. 293.339 Schutz, R.J., see Golding, B. 273,275. 293, 341 Schutz, R.J., see Grdebner, J.E. 270, 299, 300,301.341 Schwarz, K.W. 87,88,206,141,263 Schwarz, K.W., see Awschalom, D.D. 8. 137 Scott, W.W. 334,335,343 Selisky, H., see Kampf, G. 299,341 Senebetu, L. 54. 141 Seppala, H.K. 238,240,263 Serene, J.W. 214,250,251,254,263 Serene, J.W., seeSauls, J.A. 214, 249, 263 Serra, A.. see Mathieu, P. 247, 248,262 Shaham, J., see Alpar, M.A. 8 I, 137 Shaham, J., see Anderson, P.W. 81.82. 137 Shiren, N.S. 294,343 Siggia, E.D., see Ambegaokar, V. 96.97. 137 Silverstone, J.M., see Marx. J.W. 334. 335,

356

AUTHOR INDEX

342 Simola, J.T., see Hakonen, P.J. 240,241, 243,244,245,261 Simola, J.T., see Pekola, J.P. 200,201,204, 205,236,243,263 Simon, Y., see Mathieu, P. 58, 59,60,67, 70,247,248, 140,262 Simpson, J.R.. see Hegarty, J. 295,341 Singh, G.P., see Calemczuk, R. 323,340 Slack, G.A. 323,343 Sladek, R.J., see Ng, D. 288,342 Smirnov, A.V., see lordanskii, S.V. 26,139 Smith, H., see Brinkman, W.F. 242,243, 260 Smith, H., see Einzel, D. 255,260 Smith, H., see H ~ j g a a r dJensen, H. 253, 255,261 Smith, H., see Jacobsen, K.W. 215,262 Snyder, H.A. 59,67, 141 Soda, T. 206,263 Soinne, A.T., see Alvesalo, T.A. 248,249, 259 Sonin, E.B. 114, 122, 126,243,248, 141, 263 Spencer, G.F. 242,263 Spitzmann, K., see Meissner, M. 285,286, 342 Springett, B.E. 44,45, 141 Stamp, P.C.E., see Bowley, R.M. 54, 138 Stauffer, D. 58, 141 Stein. S., see Hunklinger, S. 273,341 Steingart, M. 18, 141 Stephens. R.B. 270,343 Striissler. S., see v. Lohneysen, H. 322,343 Strayer. D.M., see Glaberson, W.I. 29, 30, 47, 139 Strom. U. 319,320,321,322,343 Strom, U.. .see v. Haumeder, M. 271,343 Strom. U.. see v. Schickfus, M. 321, 322, 343 Susman. S., see De Yoreo. J.J. 323,324, 340 Sussner. H., see Vacher. R. 271,344 Svensson. E.C. 23,142 Swanson. C.E. 133, 142 Swenson. C.A., see Case, C.R. 283.340 Swift. G.W.. see Eisenstein, J.P. 201. 260

Swithenby, S.J., see Saunders, J.

219,263

Ta, T.T., see Bachellerie, A. 291,339 Taconis, K.W., see de Bruyn Ouboter, R. 5,138 Tait, R.H., see Goubau, W.M. 285,286, 341 Takagi, H., see Miyake, K. 153, 154, 155, 263 Takagi, S. 177,264 Takagi, S.,see Leggett, A.J. 162, 171,262 Tam, W.Y. 38,39,142 Tanaka, K., see Tokumota, H. 271,343 Tanner, D.J. 44,142 Tanner, D.J., see Springett, B.E. 44,141 Teitel, S.L., see Agnolet, G. 65, 137 Tewordt, L., see Schopohl, N. 176,263 Thijssen, H.P.H. 295,343 Thomas, N. 306,327,343 Thomson, J.J. 15, 126, 142 Thomson, W. 15, 102, 103,142 Thouless, D.J., see Kosterlitz, J.M. 5, 64, 95,140 Thoulouze, D., see Lasjaunias, J.C. 298, 342 Thuneberg, E.V. 199,200,231,264 Tietje, H. 297,343 Tkachenko, V.K. 56,111,142 Tokumota, H. 271,343 Tough, J.T. 5,66,206,211,215,142,264 Toulouse, G., see Anderson, P.W. 192, 193, 259 Trakhtenberg, L.I., see Fleurov, V.N. 288, 340 Truscott, W.S. 219,264 Truscott, W.S., see Dahm, A.J. 201,260 Truscott, W.S., see Hutchins, J.D. 201,262 Truscott, W.S., see Saunders, J. 219,263 Tsai, C.Y., see Widnall, S.E. 15, 142 Tsakadze, D.S. 58, 89, 122, 142 Tsakadze, D.S., see Andronikashvili, E.L. 106,137 Tsakadze, D.S., see Nadirashvili, N.S. 106, 140

Tsakadze. J.S., see Hakonen, P.J. 236.237, 261 Tsakadze, S.D. 83. 125, 142

AUTHOR INDEX Tsakadze. S.D., see Tsakadze, D.S. 89. 122,142 Tsuneto. T.. see Fujita, T. 237.261 Tsuneto, T.. see Maki, K. 195,262 Tsuneto, T.. see Ohmi. T. 49. 140 Tsuzuki, T. 26. 142 Uhlig, K.. see Eska, G. 253. 255,260 Usui, T.. .see Miyake, K. 153. 154, 155,263 Usui. T.. see Ohmi. T. 49, 54. 140 V. Haumeder. M. 271,343 V. Lhhneysen, 11. 271,299.300,301, 322, 343 V. Liihneysen, H.. see Grondey, S. 336, 337.341 V. Lohneysen. H.. see Kaspers. W. 297, 342 V. Liihneysen, I{., see Rusing. H. 337.343 V. Lohncysen, H., .see Samwer, K. 298. 299.343 V. Lohneysen, H.. see Thomas, N. 327,343 V. Lohneysen, H.. see Weiss, G. 305, 3 15. 3 18, 327.328.344 V. Schickfus. M. 278, 293, 321, 322,343, 344 V. Schickfus. M.. see Baumann, T. 323,339 V. Schickfus. M., see Golding. B. 293.294, 34 I V. Schickfus. M., .see Hunklinger, S. 279. 34 I V. Schickfus, M.. see Strom, U. 320, 321, 343 V. Schickfus, M.. see Tietje, H. 297,343 Vacher, R. 271,344 Vacher. R.. see Calemczuk, R. 323,340 Van Alphen. W.M.. see de Bruyn Ouboter, R. 5. I38 Vandorpe. M., see Lasjaunias. J.C. 267. 269.342 Varma. C., .see Anderson, P.W. 280, 339 Varma. C.M.. see Gamota, G . 15. 139 Varma. C.M., see Hasegawa, A. 15. 139 Varma, C.M.. see McWhan, D.B. 319. 322. 342 Varoquaux. E.. .see Bloyet. D. 254,260 Vibet, C.. .see Bloyet. D. 254,260

357

Villar, R., see Gmelin, E. 323,341 Vinen, W.F. 5,66,67, I9 I , 206, 142.264 Vinen, W.F.. see Barenghi, C.F. 27, 65, 67, 68.69, 71,72,73, 74, 133, 135, 137 Vinen. W.F., see Hall, H.E. 74,246, 139. 26 I Vinen, W.F.. see Hillel, A. 246,261 Vinen, W.F.. see Muirhead, C. 10. 16, 27, 54.140 Vinen, W.F., see Paintin, J. 42, 141 Vladir, K. 305.3 17,344 Volker, S., see Thijssen, H.P.H. 295.343 Vollhardt, D. 178, 179, 190,214. 234. 264 Vollhardt, D., see Lin-Liu, Y .R. 177. 178, 180,262 Vo1ovik.G.E. 118. 152, 155, 158. 169, 171, 188, 191,221,237,245, 142,264 Volovik. G.E., see Dzyaloshinskii, I.E. I 18. 138 Volovik, G.E., see Hakonen. P.J. 236. 237. 240,241,243,244,245.261 Volovik, G.E., see Mineev, V.P. 169.263 Volovik, G.E., see Pekola, J.P. 201,204, 205.243,263 Volovik, G.E.. see Salomaa, M.M. 242, 245.263 Volovik, G.E., see Seppala, H.K. 238, 240, 263 Walden, R.W., see Donnelly, R.J. 24, 138 Walden, R.W., see Roberts, P.H. 24. 141 Walker, F.J. 322, 323,344 Walker, F.J.. see Ackermann, D.A. 296. 297,339 Walker, L.R., see HU,P. 294,341 Wang, J.L. 309,344 Weber, G., see Zimmermann. J. 285.286, 344 Weiss, G. 305, 314, 315, 318. 327, 328,344 Weiss, G., see Neckel. H. 305. 317, 342 Weiss, G., see Thomas, N. 327,343 Weiss, G., see Wang, J.L. 309, 344 Wellard, N.V. 258,264 Westrum, E.F. 325.344 Wheatley, J.C. 158, 248. 251.252. 264 Wheatley. J.C., see Abel, W.K. 249. 259 Wheatley, J.C.. see Krusius. M. 182. 188,

AUTHOR INDEX

IS8

..

189. l99..?6.? M'hcatle!. 1.C' . ICY' Paulaon. D.N. 182. 188. 33 l\'hiic. ( i . K . w Barron. T H . K . 29.5, 3.10 h h i t c . ( i K.. wc('asc. C . R . 283. 340 Widnall. S.F. 15. I42 Wicdcniann. W.. .wc Eska. G. 253. 255. 260 Wildes. D.<;.. wc Saunders. J . 2.51. 252. -'63 William\. ( i , f \ , 44.45. 49. 63. 142 Wi\li:iiiis. M . K . 114. 125. 129. I42 \i'illiarn>. h1.R . wt' Fetter, A L . 179. 182. 1x3. 1x4, 186. I X X . 189. \Vinterling. ( i . . , w t ' tlcinicke. W . 268. 271. 34 I \\'oil. I)..I. 3 2 2 . ZJJ b ' d f l c . P 2 16. 2 I 7. 248. 2.56. 2%. 264 Wiilflc. P.. \f'" Lilllcl. D. 2.5s. 260 w;,lllc. I' . \"<' E.;ka. <;. 253. 261) Wciltle. P.. UY tiojSa;trd Jensen. H . 2.53. '55.261 \+'llIRe. I'.. T"f' N,lgal. K . 2 5 5 . 26.1 N'cilflc.P.. w~Plitmcr. M. 254. 258. 263

xi

Yarnada. K . . .we Yamaguchi. J . 95. 142 Yamapuchi. J . 95. 142 Y;irnasaki. S.. uv Tokurnota. H. 271.343 Yang. L.C.. .rce Chester. M . 6 5 . / 3 8 Yarrnchuk. E . J . 62. 63. 67, 74. 78. 79. 80. 81. X3. 12.5. 129. 142 I'arrnchuk. E.J.. .w Ostermeier. R . M . 49. 14(1

Yttcrboc. S.N.. .set Bates. D . M .

1x1. 182.

190. 3/50

I'u. K.-W.. we Drrrida. R.

273, 340

Zawiidowski. A . 305. 317,344 Zawadowski. A , . .we Vladiir. K . 344 Zeise. E . K . 343. 248. 264

305. 317.

Zeller. U.C. 267. 268. 769.344 Zcyhcr. K . 331.344 ZiK. R.hl.. v w Camphell. L.J. 61. 62. 138 Zirnmermann. J . 28.5. 2x6. 344 Zirnmemiann. J . . .YW Hook. J.R. 218. -761 Zippclius. A ,. .wc Petschek. R . G . 99. 1 4 /

SUBJECT INDEX

A, phase of 3He 161-167 Absorption, dielectric 278f, 320 Absorption, ultrasonic of dielectric glasses 273-277, X7-289 of ionic conductors 320f of metallic glasses 301-304, 306-311 of radiation damaged crystals 325 of superconducting glasses 314-318 Andreev reflection 255 Angular momentum, conservation of 157 intrinsic 148-158, 247f due to magnetization 168f due to rotation 155, 171 Annealing effects 3361 t3 parameter for vortices 17, 58, 59, 66-68, 71, 73, 77, 78, 86. 87, 89, 95, 117 B' parameter 58. 66-68, 71, 73. 77, 78, 86, 87, 117 B" parameter 58. 59, 67 B, parameter 67 Boojums 194, 196199. 220-226, 229 Bose condensate 20, 23, 26. 28. 29 Bound excitation model 29 Chemical potential gradient 74-78, 80. 83, 84 Coherent effects 293-295, 321f Collective eHccts 109, 111, 126 Core parameter 6, 17-19, 22, 31, 36, 38, 46,49, 93. 117 Coupling constant elastic 283f. 294 electric 294, 321f Critical intensity 275, 303 Critical velocities, A phase o f 'He 17S182, 185. 190f. 194ff. 199. 218f. 2261. 230-234 H phase of 'He 204-215. 243f. 258 Deformation potential 274, 283. 294. 297f Density of states 269f, 284f Dielectric glasses 269-298 Dielectric properties of dielectric glasses 278f of disordered conductors 320. 329 359

Dipole coupling 150, 175179, 185. 188ff. 191, 194196, 237f critical velocity 178, 191, 218, 2261 length 175f, 191, 194, 237f Dipole moment of tunnelling systems 294, 321f Direct process 275f. 289 Disgyrations 172. 199f. 233 Disordered crystals 318-329 Distribution of tunnelling parameters 281283 Domain walls 186-190, 194, 197-199, 220. 222. 234, 238f, 242 Double well potential 280f Drag coefficient 71, 72. 97, 128 Drag force 65. 70, 71, 73. 89, 93, 132

Echoes acoustic 29S295 backward wave 294 electric 293f, 321f Echo decay 294 Energy splitting of tunnelling systems Entropy production 146ff. 155f Excess specific heat 269, 271

281

Fermi liquid corrections 151, 158. 165. 214. 251ff Flow dissipation A phase of 3He 174179. 186-199, 21% 2-36 B phase of 'He 205-219 rotation induced in 'He 245-248

parameter in 4 H e vortices 18. 28. 36, 71-73. 76, 77, 87. 103. 11.5-117. 17-0, 121, 128. 132. 136 yo 71-73, 76, 77, 86. 95, 128. 132, 135. 136 7; 71-73, 76, 77. 86. 95, 128. 135, 136 Gauge whecl 156, 163 Ginzburg-Landau theory 175. 178. 213f, 237, 241 Ginzburg-Pitaevskii theory 19. 33 y

?hU

SUBJECT INDEX

Glass transition temperature inHuence on low temperature propertie5 330. 33-L3.78 (iradient free energv 148. 1.%1S4. 167. 169. 171. 174-178. 182-184, 1%. 2 1 3 .

2371 Gros\-Pitaevskii theory 20 Griincisen parameter 29-%297 Gryomagnetic eliects 160. lh8f Hamiltonian of tunnelling systems 281 of interaction 283f. 304 Hamilton's equation 1 I . 12. 14 Healing 19. 20. 22. 23. 28. 32, 34, 35. 4 0 4 2 Ilclical textures 17b-I85. 191 Hill.;-Koherts theory 32. 40. 41 History depcndent cHects 211. 217f. 27.V 216. 229f Ho circulation thcorcm 171-174. 177. 191193. 197~736. 238 Holc hurning acoustic 29 1 - 3 3 optical 295 HVBK equation5 Xh. 87. 97, 118. 121. 126. 130. 137 Impulw

7, Y. 11-13. 15. 23. 26 5. 9. 10. 17. 18. 28. 29. 41-47. 48. 49. SI. 53. 58. 63. 105-109. 132 Ionic conductors 319-323 Iordanskii force 73

Models for tunneling systems 329-332 Mutual friction in helium 11 27. 58. 59. 65, 66. 68,71. 73. 77,78. 80. 86. 93, 95. 97. 117, 118, 123, 127. 131. 135-137 Mutual friction in superHuid 'He 206, 218. 227. 23Sf. 247f Negative ion in helium I 1 9, 28. 29, 4143, 45, 48, 50. 58. 63. 132 NMR in rotating 'He 23624.5 Orbital dynamics 153. 15&158. 174, 182IW, 218-222, 2348 Orbital relaxation time 220H. 225. 227, 229 Order parameter in 'He 135. 139f. 156. 1WlbJ. 167f. 170. 214f. 232-244 Orientational disordered systems 323-325 Persistent currents 20(k205, 218f stability of 174-176. 191-194. 19filW. 218 Persistent orbital motions 182, 188 Phase memory 291 Phonon assisted tunnelling 283 Photon echo 20.5 Positive ion in helium I 1 9, 28. 29, 42. 43, 37-5 1

Ion

.loseph\on equations Kelvin wavc

156. l63tf

102. 104, 106. 109

Landau parameter5 2SIH Leggctt equations 159, lh3f I-inear specific heat 269 1.ogarithmic velocity change 277f. 289f Madclung trandormation 22 Magnu\ force 27. ZX. 71. 73. 7.5.76, 87.93. 104. 131. 1.75

Mass supercurrent dcnsity In 'He l e t 7 A ph;t\e 1518. IhS. 177H, 182. 219. 230233 H ph:iw IhX. 2(IY-2I I dcpairing 1x2, 207. 21 1-215, 230f Metallic glasses 298-318

Radiation damaged crystals 325-327 Relaxation process 2761, 287-289.3 0 3 Relaxation time i n glasses longitudinal in dielectric glasses 275f in metallic glasses 30.3-30s i n superconducting glasses 31 1-314 transverse in dielectric glasses 276. 291-295 in metallic glasses M3f due t o the direct process 2751. 289 due to the Raman process 289 Resonant ahsorption 274276 Saturation of the resonant process 27.3-275. 303 o f the relaxation process 309 -311 Saturation recovery 293, 303 Second sound in helium I 1 41. 58, 59. 66, 67.80. 118. 133 Slip in 'He 2S1, 253, 2SSf Small vortex rings in helium I1 26. 28

SUBJECT INDEX Solitary wave 104 Solitons in 'He 189f, 2.34, 239. 242 Sound in 'He, first 165, 255 fourth 154, 156, 200, 248 second 159, 165ff. 246. 248 zero 145, 179-182, 241 Specific heat of 'He 214. 248ff, 254 Specific heat of glasses of dielectric glasses 269-273, 285f of four-fold co-ordinated solids 271 of ionic conductors 270, 319f, 3221 of metallic glasses 2 9 3 of orientational disordered systems 323f of radiation damaged crystals 3 2 3 of superconducting glasses 270. 295 of two-phase systems 327 time dependence 2851 Spectral diffusion 292f Spin supercurrent density 151, 163. 165I68 Spin-up 6, 89-91, 94 Spin waves 165ff. 242-245 Spontaneously broken relative symmetry 145, 159, 161f Standard procedure in hydrodynamic theory 146f. 170 Strong coupling 162. 182, 214. 231. 245, 249-252. 254 Superconducting glasses 270, 295, 2991, 311-318 Superfluid density of 'tte 165, 168. 251R

Thermal conductivity o f dielectric glasses 279f of ionic conductors 319f. 323 of metallic glasses 2981 of orientational disordered systems 323 of radiation damaged crystals 325f of superconducting glasses 2991 of supertluid 'He 257H of two-phase systems 327 Thermal equilibrium. local and global in 'He 147f. 150. 153. 156 Th cr m a I e x pa n si o n 29.5-298 Thermal relaxation in glasses 286 Thermodynamic forces and fluxes 146f. 156H Thermodynamic identity 146-1.54. 170

36 1

Thermodynamics choice of variables 146-155. 159-160, 167-1 7 1 calculation of work terms 148f, 15% of rotating systems 148ff Thermorotation effects 74 Thin films of helium 11 5. 6, 8, 38. 6.5, 95, 97, 101 Tkachenko wave 113. 116. 122, 126, 129 Tunnelling Model of glasses 280-283 Tunnelling particle 280f Tunnelling systems 280-339 energy splitting 281 Gruneisen parameter 2Y7 Hamiltonian 281. 2831 induced 294 intrinsic 294 theoretical models 329-332 Two-level system in glasses 27S280 Two-phase systems 327-329 Ultrasonic properties of glasses see Absorption. velocity change Uniform distribution in glasses 277. 309 Velocity maximum in glasses 277. u)9 Velocity change in dielectric glasses 277f. 289-291 in metallic glasses 301f. 306-311 in superconducting glasses 315318 two-phase systems 327f Vibrational modes in glasses 2721 Viscosity first 216f. 251, 253-256 second 216f. 256f Vortex diffusivity 96-99 Vortex distribution 6. 56, 58. 59, 63. 64,87. 101, 112, 129 Vortcx-free strip 58 Vortex instability 130 Vortex pinning 6, 64,81, 83. 87. 1 0 1 Vortex ring 9, 11, 12. 13, 1.5. 17. 18. 19. 24, 26, 28, 36. 38. 45, 54. 71. 13S-135 Vortex wave 6, I Y . 28, 48. 49, 51, 101, 102, 104, 10&109, 111. 118. 121. 12.5. 126. 1.34. 135, 137 Vortices in 'He A phase 192-197. 199, 218-222, 231f. 235242, 245-248 B phase 205ff, 211, 215, 218. 242-248 H phase, core structure of 204, 242-245

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