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PROGRESS IN LOW TEMPERATURE PHYSICS IX

This Page Intentionally Left Blank

PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

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

VOLUME 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


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


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


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


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


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


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


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


[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)

Ch. 1, $11

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.

Ch. 1, $11


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


[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.)

Ch. 1, $21


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


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


[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

Ch. 1, $21


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.)

Ch. 1, 021


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


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


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


[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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


[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


[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


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


[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


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


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


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


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


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


[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


(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.)


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


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


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


[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


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


(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


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


[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


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.)


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


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


[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


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


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


[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


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


[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


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


[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


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


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)


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


[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


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


[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


-E

1.5 0

0

0

0 0

1.0

I

0

0

V \

E2

I

I

I

81

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


[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


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


[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


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


[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


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

I

I

I

I

I

I

I

I

I

c

h

c

I

0

I 10

1

20

t

I

I

30

(s)

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.)

91

[Ch. 1, $4

W.I. GLABERSON AND R.J.DONNELLY I .o

I

....

'.

7-2.01 -

I

I

I

I

1

h

1-

-

-3.0

:.-.

~

-.-:-

for

IfD’fMl

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)


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


[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


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


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-


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)


[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%.)


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

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]


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


[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


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?


[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


-

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


[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


(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


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)

1 I8


[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)

Ch. 1, $51


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


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)

122


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]


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


[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


125

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.)

126


[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

Ch. 1, $51


127

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


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

Ch. 1, $51


129

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


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,.

Ch. 1, SS]


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


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


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


[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 (> 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


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


[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


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%


[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


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-’.


[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


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


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


[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


[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


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


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


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


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


[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 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


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


[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


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


[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)


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


[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


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


[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


.

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


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


[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


-

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


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]


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


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


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


(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


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


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


-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


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


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


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.

-


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


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


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


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.

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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.)

268

S. HUNKLINGER A N D A.K. RAYCHAUDHURI

[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.

Ch. 3, 521

ANOMALIES IN GLASSES AT LOW TEMPERATURES

269

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.

770

S. HUNKLINGER AND A.K. RAYCHAUDHURI

[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


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


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.

Ch. 3, 021


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


[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)


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


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


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


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


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


[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).

Ch. 3, $21


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


[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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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,

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[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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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

[Ch. 3, 02

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

290


[Ch. 3, 42

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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29 1

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

'92


[Ch. 3, $2

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

294


[Ch. 3, 92

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:

796


[Ch. 3. $2

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

298


[Ch. 3, $3

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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299

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

Ch. 3, 831


301

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.

302

[Ch. 3. 53


1SOMHz

=;

TRANSVERSEWAVES

...-

LONGITUDINAL WAVES

I

0.5

2

1

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.)

I

-E U

l61

I

0

VITREOUS SILICA 1850MHz

0

-

t.

n

8 0

1800 Ni7eP22 MHz

0

Yz 11Q

I

u

0 0 0

'

i

z

E! 8 IQ 3

z

.

I

I

0

0

..-,

'

-

I

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)


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


[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


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


[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


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

-

12

--

^-__

1

1

1

T R A N S V E RS E WAV E S

E V

m

D -

w

e

0

Z

a I 0

=

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 \

i

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 , , , 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


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


(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

I

I

1

10 m

72

$ 1

r

Z 0

O

c U

6

3

c

A

A

a 0

0 0

0

1

0

CU60ZrL0

I

+ -I 3

0

0

01

h

01

,

I

1 TEMPERATURE i K )

I

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


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


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


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


{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-

Ch. 3, 041


319

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

320


[Ch. 3, $4

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-

327

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PULSE ELECTRIC, FIELD (Vlcm) Fig. 27. Amplitude of the spontaneous echo in Na-P-alumina as a function of pulse electric field strength. This measurement was carried out at a frequency of 590MHz and a temperature o f 18 mK. From the position of the maximum the electrical dipole moment of the tunneling svstems has been deduced. (From v. Schickfus and Strom. 1983.)

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). ''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.. 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' . \"

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