Progress in Low Temperature Physics, Vol. 12

PROGRESS I N LOW TEMPERATURE PHYSICS

XI1

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PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

D.F. BREWER Professor of Experimental Physics University of Sussex, Brighton

VOLUME XI1

1989

NORTH-HOLLAND AMSTERDAM. OXFORD. NEW YORK . TOKYO

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(Revised for vol. 12) Gorter, C.J. (Cornelius Jacobus) Progress in low temperature physics. (Series in Physics) Vol. 12 edited by D.F. Brewer. Vol. 12 has imprint: Amsterdam; New York: North-Holland; New York, N.Y.: Sole distributors for the U.S.A. and Canada: Elsevier Science Pub. Co. has imprint: Amsterdam, North-Holland Pub. CO.; New York, lnterscience Publishers. 1 . Low Temperatures. 1. Brewer, D.F. (Douglas Forbes) 11. Title. 111. Series. 536.56 55-14533 QC278.G6 PRINTED IN THE NETHERLANDS

PREFACE

It may seem eccentric to start the twelfth volume of Progress in Low Temperature Physics with a chapter on high-temperature superconductivity, however distinguished its author. When the discovery of high- T, superconductivity (defined by Ginzburg in Chapter 1 as Tc> 30-40 K) in rather complicated compounds was announced, many low-temperature physicists must have decided that its development was now in the hands of the expert materials scientists, and that their attention was better placed elsewhere. Nevertheless, if this were to mark a sharp decline of low-temperature physicists’ interest in superconductivity, it seemed appropriate to have some sort of obituary in the form of, as Ginzburg quotes from my letter to him, “an article of more general interest giving an historical perspective of the various speculations in the past on the possibility of such (HTS) superconductors, and the possible mechanisms for the superconductivity in the recently discovered materials.” Professor Ginzburg responded with la chapter of characteristic style - and with the short response time characteristic of Soviet authors to this series - which is well worth reading for more than its scientific interest. Other chapters illustrate the wide range of physics which are more usual low-temperature topics. Study of spin polarized quantum systems was much stimulated by a conference with this title held at Aussois in 1980. A considerable amount of work on ’He has been done in France, and the chapter on spin polarized ’He gas by Betts, Laloe and Leduc is a collaboration between authors from Paris and Sussex. It follows two other related articles, on spin polarized atomic hydrogen in Volume X, and on 3He-4He solutions in Volume XI. Chapter 3 is another article on helium, by Nakayama, dealing with Kapitza thermal boundary-resistance at, mainly, millikelvin temperatures. Although this is another characteristic, although not exclusively, low-temperature phenomenon, which is technically very important in the problem of thermal contact, it has not yet received attention previously in this series. The increasing availability of millikelvin temperatures emphasizes not only its practical importance but also the growing interest in the detailed mechanisms of the interaction of 3He quasiparticles with solid surfaces, a topic which was also treated in Volume XI. The subject of charge density waves discussed by Gruner in Chapter 4 is one of much current interest in metal physics. As such, it is one which many would assert is not really a topic for a low temperature physics review. The arid philosophical argument about what low temperature physics is, if V

vi

PREFACE

anything, has not deterred me from including the chapter here. It follows previous articles on metal physics in Volumes X and XI. The final chapter, from Helsinki, on multi-SQUID devices and their applications, represents somewhat of a departure in subject matter for this series. Previous volumes have concentrated almost exclusively on the physics of low temperatures, partly because application in the liquid-helium temperature range are not extensive, apart from superconducting magnets. The advent of the Josephson effect changed that, and the economic and social climate is inclined to favour applications: IUPAP in its sponsored conferences, for example, specifically encourages the inclusion of applications in the programme. To return to high-temperature superconductivity: as experimental information and theory have progressed, it seems to be emerging that the physical mechanism may be quite different from the usual BCS superconductivity, and that measurements at quite low temperatures (tens of millikelvins) may be very important in determining what it is. So there are still some interesting and rewarding things to be done by low-temperature experimental physicists. Not quite salue, perhaps, but at least not quite vale, and I think I can promise at least one more contribution to the series in the future. As usual, I am very grateful to colleagues for discussions about topics of current interest in the physics of low temperatures, to the authors who gave their valuable time to writing, and to the publishers, especially Peter de ChPtel and Anita de Waard, for their help. To future authors - if they are there - I should like to extend my anticipatory thanks, and to remind them that inopi benejcium bis dar, qui dot celerirer. Sussex, 1989

D.F. Brewer

CONTENTS VOLUME XI1 Preface ..................................................... Contents .................................................... Contents of previous volumes ...................................

Ch. 1 . High-temperature superconductivity: some remarks. V.L.Ginzburg..........................................

V

vii xi

1

1. Introduction ................................................... uestions of priority .................................. 2. Significance 3. Experimental studies of superconductivity . .. 4 . Microscopic theory of superconductivity ................................

5 . Critical temperature in the BCS theory

......

6. Superconductivity mechanism in the "neon" s 7. Superconductivity mechanism in the "nitrogen" superconductors . Ways for

raising T,

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

...................................... 8. Exciton mechanism of HTS . .. 9 . Nature of HTS in metal oxides ............................................ 10. Concluding remarks ..... References ........................................

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

21 26 32 41 41

Ch. 2. Properties of strongly spin-polarized 'He gas. D.S. Befts. F.Laloeand M . Leduc .................................. Introductory remarks ........................................................ 1. The0ry .................................................................. 1.1. Introduction .............................................. I ............................................. 1.3. Characteristic lengths ................................................ 1.4. Transport properties ..................................... 1.5. Classical transport theory ............................................. 1.6. Quantum mechanical transport theory 1.7. Thermal conductivity ................................................ 1.8. Viscosity ..................................... 1.9. Spin diffusion ....................................................... 1.10. Equation of state .................................................... 2. Polarisation methods ...................................................... 2.1. Optical polarisation of 'He nuclei ..................................... 2.2. Relaxation processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Experiments on spin rotation effects and spin waves ..................... 3.2. Experiments on thermal conductivity . . . . . . vii

47 47 47 51 52 54 56 59 66 67 70 73 78 78 89 93 93 98

...

CONTENTS

VIII

3.3. Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Liquid-gas equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ... , _ . .. ...... ........

106 109 111

Ch. 3. Kapitza thermal boundary resistance and interactions of helium quasiparticles with surfaces, Tsuneyoshi Nakayama . . . . . . . . . . 115 1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Kapitza thermal boundary resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition of the Kapitza resistance R , . . . . 2.2. General expression for the Kapitza resistanc 2.3. Anomalous behaviour of the observed Kapitza resistances 2.3.1. The liquid 'He-sintered powder interface at mK tem 2.3.2. The liquid He-bulk solid interface above about 3. Fenni liquid theory of the Kapitza resistance.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Heat transfer due to zero-sound excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Emission of zero-sound from a small particle into liquid 'He 3.1.2. Heat flux from a spherical particle at temperature 7 . .. . . . . . 3.1.2.1. Energy current into liquid ' H e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2. Kapitza conductance t o zero-sound excitations . . ..... 3.2. Energy transfer due to inelastic scattering of single quasiparticles interface . ,.............. .................. ..... 4. Anomalous Kapitza resistance between sintered powder and liquid 'He at mK temperatures ......................... .................................. 4.1. Heat exchanger using submicrometer metal particles . . . . . 4.2. Soft phonon-modes in sintered powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer due to the effect of soft phonon-modes . 5. The magnetic channel of heat transfer between sintered PO 5.1. Surface characteristics of submicron metal particles and surface magnetic .. impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 5.2. Heat transfer due to magnetic coupling at mK temperatures uid 'He-4He mixtures and sintered

............................... ............ .................................... 6.2. Magnetic channel . . . . .

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

117 118 118 120 124 124 126 128 128 128 130 130 132 135 140 140 141 144 149 149 152 158 159 162 165

7.1. Kapitza conductance k, and phonon transmission coefficient across the

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

165 168 171 175 176

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

179 186 187 189

7.2. High-frequency phonon scattering at the liquid He-solid interface . . . . . . . . . . 7.2.1. Cause of diffuse scattering at the surface without liquid He . . . . . . . . . . 7.2.2. Specular versus diffuse scattering of bulk phonons . . . . . . . . . . . . . . . . . . 7.2.3. Diffuse signals in the time-of-flight reflection signals. . . . . . . . . . . . . . . . . 7.2.4. Reduction of the diffuse signal at the solid surface in contact with 8. S u m m a r y . . . ...................................................... ........... ... Appendix . . . . . . . . . . . . . . . . . . . . . References . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS

ix

Ch. 4. Current oscillations and interference efects in driven charge density wave condensates. G. Griiner ......................

195

1. Introduction ............................................................. 2. Basic notions and observations in charge density wave dynamics ............... 2.1. The charge density wave ground state and model compounds .............. 2.2. The dynamics of the collective mode .................................... 2.3. Frequency and field dependent transport ................................ 3 . Current oscillations ....................................................... 3.1. General features ............ ..................................... 3.2. Current-frequency relation ............................................. 3.3. Size effects and fluctuation phenomena .................................. 3.4. Broad band noise .................................................... 4 . Models of charge density wave dynamics .................................... 4.1. The classical particle model ............................................ 4.2. Models including the internal degrees of freedom ........................ 4.3. The tunneling model .................................................. 5. Interference phenomena ................................................... 5.1. Harmonic mode locking ............................................... .................................... 5.2. Subharmonic mode locking ... 5.3. Nonsinusoidal and pulse drives ........................................ 5.4. Fluctuations and coherence enhancement ................................ 6 . Conclusions .............................................. List of review papers ........................................................ References .................................................................

197 201 201 205 211 217 217 220 223 226 227 227 232 237 239 239 246 255 259 262 265 266

Ch. 5. Multi-SQUID devices and their applications. Risto Ilmoniemi andJuhha Knuutila .....................................

271

1. Introduction ............................................................. 2 . SQUIDs ................................................................. 2.1. Single-junction (rf) SQUIDs ........................................... 2.1.1. General ........................................................ 2.1.2. Rf SQUID in the hysteretic mode ................................. 2.1.3. Discussion ...................................................... 2.2. Double-junction (dc) SQUIDS ......................................... 2.2.1. Operation ...................................................... 2.2.2. Problems with practical devices ................................... 2.2.3. The state of the art .............................................. 2.3. Electronics ........ ........................................ 3. Applications: biomagnetism ................................................ 3.1. Measurement techniq .............................. 3.1.1. Magnetically shielded rooms ..................................... 3.1.2. Gradiometers ................................................... 3.2. Neuromagnetism ..................................................... 3.2.1. Origin of neuromagnetic fields .................................... 3.2.2. Spontaneous activity ............................................

273 273 274 274 276 279 280 280 284 286 289 292 293 293 295 296 297 298

X

CONTENTS

3.2.3. Evoked fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Clinical aspects of MEG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Cardiac studies . . . . . .................................... 3.4. Other biomagnetic ap .............................. 3.5. Multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Optimization of multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . 3.5.2. Existing multichannel systems ....................... ........... 3.5.3. Planar gradiometer arrays . . . . ..................... 3.5.4. Use of multichannel magnetometers . . . . . 4 . Other multi-SQUID applications . . . . . . . . . . . . . . . . . . . . . . . .......... 4.1. Geomagnetism .................... 4.2. Physical experiments . . . . . . . . . . . . . . 4.2.1. Accelerometers and displacem 4.2.2. Monopole detectors ....................... ........... 5 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 302 303 304 305 310 319 323 326 326 328 328 329 332 333

Author Index

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

341

Subject Index

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

361

CONTENTS OF PREVIOUS VOLUMES

Volumes I-VI, edited by C.J. Gorter Volume I (1955)

I.

The two fluid model for superconductors and helium 11, C.J. Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 Application of quantum mechanics to liquid helium, 11. R.P. Feynman ................................... 17-53 111. Rayleigh disks in liquid helium 11, J.R. Pellam.. . . . . . 54-63 IV. Oscillating disks and rotating cylinders in liquid helium 11, A.C. Hollis Hallett . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-77 V. The low temperature properties of helium three, E.F. Hammel ........................................ 78-107 VI. Liquid mixtures of helium three and four, J.M.Beenakker and K.W. Taconis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-137 The magnetic threshold curve of superconductors, B. VII. Serin ........................................... 138-1 50 VIII. The effect of pressure and of stress on superconductivity, 151-158 C.F. Squire ...................................... Kinetics of the phase transition in superconductors, T.E. IX. Faber and A.B. Pippard . . . . . . . . . . . . . . . . . . . . . . . . . . 159- 183 Heat conduction in superconductors, K. Mendelssohn 184-201 X. The electronic specific heat in metals, J.G.Daunt . . . . 202-223 XI. Paramagnetic crystals in use for low temperature XII. research, A.H. Cooke.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 224-244 XIII. Antiferromagnetic crystals, N.J. Poulis and C.J. Gorter 245-272 x IV. Adiabatic demagnetization, D. de Klerk and M.J. Steenland ............................................ 272-335 xv. Theoretical remarks on ferromagnetism at low temperatures, L. NCel . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 336-344 XVI. Experimental research on ferromagnetism at very low temperatures, L. Weil.. . . . . . . . . . . . .. . . . .. . . .. . . . . . 345-354 XVII. Velocity and absorption of sound in condensed gases, A. van Itterbeek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355-380 XVIII. Transport phenomena in gases at low temperatures, J. 381-406 de Boer ......................................... xi

nii

CONTENTS OF PREVIOUS VOLUMES

Volume I1 (1957) I. 11.

Ill. IV. V. VI. VII.

VIII. IX. X. XI. XII. XIII. XIV.

Quantum effects and exchange effects on the thermodynamic properties of liquid helium, J. de B o e r . . . . . . . . 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- 186 187-225 226-265 266-291 292-337 338-367 368-394 395-430 43 1-464

Volume 111 (1961) 1. 11. 111.

IV. V. VI. VII.

VI11. 1x. X.

Vortex lines in liquid helium 11, W.F. Vinen.. . . . . . . . Helium ions in liquid helium 11, G. Careri . . . . . . . . . . The nature of the 301-transition in liquid helium, M.J Buckingham and W.M. Fairbank. . . . . . . . . . . . . . . . . . . Liquid and solid 'He, E.R. Grilly and E.F. Hammel 'He 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 IT, W.J. Huiskamp and H.A. Tolhoek . . . . . . . . . . . . . . . . . . . . . . Solid state masers, N. Bloembergen . . . . . . . . . . . . . . . . The equation of state and the transport properties of the hydrogenic molecules, J.J.M. Beenakker . . . . . . . . .

1-57 58-79 80-1 12 113-152 153-169 170-287 288-332 333-395 396-429 430-453

CONTENTS OF PREVIOUS VOLUMES

XI.

Some solid-gas equilibria at low temperatures, Z. Dokoupil .......................................

xiii

454-480

Volume IV (1964)

I. 11.

111.

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. NCel, R. Pauthenet and B. Dreyfus ......................................... Dynamic polarization of nuclear targets, A. Abragam and M. Borghini ................................. Thermal expansion of solids, J.G. Collins and G.K. White .......................................... The 1962 3He scale of temperatures, T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde . . . . . . .

1-37

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

Volume V (1967) I.

11.

111.

I v. V. VI.

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 Magnetic breakdown in metals, R.W. Stark and L.M. Falicov .........................................

1-43

44-78

79- 160

161- 180 181-234 235-286

xiv

VII.

CONTENTS OF PREVIOUS VOLUMES

Thermodynamic properties of fluid mixtures, J.J.M. Beenakker and H.F.P. Knaap ......................

287-322

Volume VI (1970) I. 11. 111.

IV.

v. v1. VII. VIII. IX.

X.

Intrinsic critical velocities in superfluid helium, J.S. Langer and J.D. Reppy ........................... Third sound, K.R. Atkins and 1. Rudnick . . . . . . . . . . . Experimental properties of pure He3 and dilute solutions of He3 in superfluid He4 at very low temperatures. Application to dilution refrigeration, J.C. Wheatley . . . . . . . . Pressure effects in superconductors, R.I. Boughton, J.L. Olsen and C. P a l m y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconductivity in semiconductors and semi-metals, J.K. Hulm, M. Ashkin, D.W. Deis and C.K. Jones . . . Superconducting point contacts weakly connecting two superconductors, R. de Bruyn Ouboter and A.Th.A.M. de Waele ....................................... Superconductivity above the transition temperature, R.E. Glover 111 .................................. Critical behaviour in magnetic crystals, R.F. Wielinga Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities, G.R. Khutsishvili The international practical temperature scale of 1968, M. Durieux .....................................

1-35 37-76

77-161 163-203 205-242

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

CONTENTS OF PREVIOUS VOLUMES

xv

Volumes Vll-XI, edited by D.F. Brewer

Volume VII (1978) 1.

2.

3. 4. 5. 6.

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

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

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

Volume VIII (1982) 1. 2. 3. 4.

Solitons in low temperature physics, K. Maki.. ...... 1-66 Quantum crystals, A.F. Andreev ................... 67-132 Superfluid turbulence, J.T. Tough . . . . . . . . . . . . . . . . . . 133-220 Recent progress in nuclear cooling, K. Andres and O.V. Lounasmaa ..................................... 221-288 Volume IX (1985)

1.

2. 3.

Structure, distributions and dynamics of vortices in helium 11, W.I. Glaberson and R.J. Donnelly ........ The hydrodynamics of superfluid 'He, H.E. Hall and J.R. Hook ....................................... Thermal and elastic anomalies in glasses at low temperatures, S. Hunklinger and A.K. Raychaudhuri ....

1-142 143-264 265-344

xvi

CONTENTS OF PREVIOUS VOLUMES

Volume X (1986) 1.

2. 3. 4.

Vortices in rotating superfluid 'He, A.L. Fetter.. . . . . . Charge motion in solid helium, A.J. Dahm .......... Spin-polarized atomic hydrogen, I.F. Silvera and J.T.M. Walraven ....................................... Principles of ab initio calculations of superconducting transition temperatures, D. Rainer . . . . . . . . . . . . . . . . .

1-72 73-137 139-370 37 1-424

Volume X I (1987) 1.

2. 3. 4.

5.

Spin-polarized 'He-4He solutions, A.E. Meyerovich Long mean free paths in quantum fluids, H. Smith . . . The surface of helium crystals, S.G. Lipson and E. Polturak ........................................ Neutron scattering by 4He and 3He, E.C. Svensson and V.F. Sears ....................................... Characteristic features of heavy-electron materials, H.R. Ott . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-73 75-125 127-1 88 189-2 14 2 15-289

CHAPTER 1

HIGH-TEMPERATURE SUPERCONDUCTIVITY: SOME REMARKS* BY

V.L. GINZBURG P. N. Lebedev Institute of Physics, USSR Academy of Sciences, Moscow, USSR

* Submitted

to “Progress in Low-Temperature Physics” in November 1987.

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elsevier Science Publishers 9.V., 1989

Contents I . Introduction . . . . . . . . . ................................ 2. Significance of HTS. Questions of priority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental studies o ............... 4. Microscopic theory of superconductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Critical temperature in the BCS theory . . . . . . . . ....... 6. Superconductivity mechanism in the ”neon” sup ................. 7 . Superconductivity mechanism in the “nitrogen” superconductors. Ways for raising

r, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. Exciton mechanism of HTS . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 9. Nature of HTS in metal o x i d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . emarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

2

3 3 7 12 15 19

21 26 32

1. Introduction

Reproducible high-temperature superconductors were obtained in 1986-87, as late as 75 years after the discovery of the superconductivity effect. We apply the term “high-temperature superconductors” to those superconductors for which the critical temperature T, exceeds 30-40 K and, what is especially important, is higher than the boiling point of nitrogen under the = 77.4 K. normal pressure, TLIN2 Less than a year has passed (at the time of writing) since the discovery of the “nitrogen” superconductors with T c 380-90 K. So, it is quite natural that the situation in the field of high-temperature superconductivity (HTS) remains, on the whole, not entirely clear, particularly concerning the detailed mechanism of superconductivity. In these circumstances it would be too early to write a review of HTS. The editor of this series thought, however, that the discovery of HTS should be reflected in this volume and he suggested tFat I should write “an article of more general interest giving an historical perspective of the various speculations in the past on the possibility of such (HTS) superconductors, and the possible mechanisms for the superconductivity in the recently discovered materials”. It is precisely this task that I have set for myself. It seems reasonable here to give also some information well known to the specialists for the convenience of the nonspecialist readers. It proved to be rather difficult to write such an article in the style typically employed for scientific papers and reviews. At any rate, what I was writing proved to be some sort of an essay and therefore I decided not to avoid using the first person singular pronouns, and to make digressions concerned with matters of priority and so on. The suitability of such a style in the present publication can, of course, be questioned but in my opinion there are no reasons to try to unify the style of all reviews. As an illustration, note that the Annual Review of Astronomy and Astrophysics (Annual Rev. Inc. California, USA) publishes even quasi-biographical notes by scientists starting from 1975.

2. Significance of HTS. Questions of priority

The discovery of HTS produced an exceptional interest and attracted intense attention in the scientific community and the general public. This discovery is unquestionably highly important for physics and for solid-state physics 3

4

V.L. GINZBURG

specifically. It is also unquestionable that discoveries of no lesser and even greater scientific significance have been made in physics in the last two or three decades. They have not produced such a strong response, though. There is hardly any doubt that the extreme attention drawn by HTS is caused primarily by the understanding of the potential uses of HTS in technology, medicine and so on. Until now the ever-expanding use of superconductors necessarily involved cooling with liquid helium. But there is a relative shortage of helium and it is expensive. For instance, in the USSR a liter of liquid helium costs about 10 roubles. Helium is extracted from natural gas the resources of which are limited and it can be assumed that in a few decades the shortage of helium will become acute. On the other hand, nitrogen is a main component of the atmosphere and it is very easy to separate and liquefy. A liter of liquid nitrogen is cheaper than that of helium by a factor of several hundreds. In addition, nitrogen is a more effective coolant than helium since its heat of evaporation is larger by an order of magnitude. Finally, since helium is so expensive, in most installations the gaseous helium is not discharged into the atmosphere but is recovered, reliquefied and used again. This requirement, in addition to other complications, makes the cryogenic apparatus for helium much more complex and expensive than for nitrogen. Moreover, its reliability is therefore diminished. An accident involving just a discharge of helium into the atmosphere can be catastrophic or, at least, result in the stoppage of the installation involved in the accident. Nitrogen, of course, will be released into the atmosphere and therefore the installations with the nitrogen superconductors will be very much simpler to operate. Thus, the applications of superconductors cooled by liquid nitrogen will increase the economic and the technological benefits by a factor of hundreds if not thousands. At the same time, an improvement in the efficiency by a few percent or even less is regarded as a considerable advance in electrical and heat engineering. It must be added that, according to some indications, a critical temperature T, as high as 200-300 K can be achieved for some superconductors. Such materials are still unstable, though, and it is too early to discuss their application. If, however, materials with T, 200-300 K can be made stable and easy enough to obtain, then the range of HTS applications will be extended even further. Apan from the above reasons, which explain the exceptional attention drawn to the discovery of HTS,it must also be noted that the new materials proved to be quite easy to manufacture - metaphorically speaking they can be cooked in any kitchen. This is why the nitrogen superconductors are being manufactured by schoolchildren less than a year after their discovery (see, for instance, Grant 1987).

-

HIGH-TEMPERATURE SUPERCONDUCTIVITY

5

I m u l d like to note that I had been working on the HTS problem since 1964, always believed in the feasibility of producing HTS, and always promoted the studies in this field. It was precisely the ease with which they have been produced that proved to be unexpected even for me (I believed that after the first success it would take a relatively long time to reproduce the results and, so to say, to master the HTS effect). On the whole, the significance of HTS is clear enough, though I hope that the above remarks were not superfluous. It seems worthwhile to note also the following facts. Firstly, the practical application of the HTS is not something trivial and self-evident, it will require considerable efforts. A wide variety of potential uses may be named now, new ones will be suggested but, in any case, much work is to be done before new devices are designed and deployed. Secondly, it would be quite wrong to think that the helium superconductors (or, to be more precise, the use of the superconductors cooled with liquid helium) are becoming or will become redundant. In addition to purely physical experiments, the low temperatures are needed in various apparatus for reducing the thermal noise (in the classical domain, that is for hw < ksT, the thermal noise for instance in conductors is proportional to the temperature T; here w is the frequency, for instance, the frequency of the circuit being used). The discovery of the HTS, therefore, will by no means detract from the significance of the low-temperature physics though it will of course strongly influence this science as a whole. Since the discovery, analysis and future application of the HTS have a vast, though potential, economic and, hence social effect, it must affect the work style and general attitudes in physics and technology. For this and other reasons, that will be clear from the following discussion, I would like to make a few brief remarks on questions of priority. The questions of priority are an old concern of the scientific community. Suffice it to recall the considerable attention paid by Newton to this subject (see, for instance, Westfall 1982). The history of science tells us of the great efforts wasted in fruitless priority disputes and the pain they brought to their participants. In our time such disputes are, of course, different in form but hardly in substance. A good illustration is given by the endless discussion of the contributions made by Logentz, PoincarC and Einstein to the special relativity theory (I have even written a paper entitled “How and who did create the relativity theory?” which has appeared in several Russian publications and in which I also discuss the general problem of priority; see the collection of papers, Ginzburg 1985). A general discussion of the subject would be out of place here but I would like to tell about an incident that took place during the 18th International Conference of Low-Temperature Physics (LT-18). A “Special symposium on high T, superconductors” was held at a plenary session of the

6

V.L. GINZBURG

conference on August 21, 1987. Five reports were to be delivered at the symposium, but I was not expected to participate. A day before it one of the organizers of the LT-18 had suggested, quite unexpectedly for me, that 1 make a brief report, too. I can only guess at the reasons for this suggestion, but it was obviously well-intended. Therefore I agreed, though I was in a quandary - indeed, what could 1 say in a few minutes after the reports of which 1 did not have any prior knowledge. Bearing in mind the tremendous excitement generated in the HTS studies I thought it would be worthwhile and opportune to remind the audience about the need to be cautious in raising priority questions. With this purpose I demonstrated a transparency with the following statements: Priority questions are dirty business. Priority mania or supersensitivity is illness. I commented briefly on these statements whose meaning seemed quite clear to me - as a person with fifty years' experience of work in physics and astrophysics I recommended not to attach excessive significance to priority questions. Incidentally, I made similar remarks in my introductory report delivered on August 3, 1987 to the 20th International Cosmic Ray Conference held in Moscow. As far as I could see, my statements and comments were then recognized exactly as 1 expected - as a piece of advice presented in a half-joking manner. Unfortunately, the response to my address at the LT-18 was different. A few persons told me (and perhaps many thought but did not tell me!) that they understood it as a complaint or a reproach. Humans are, of course, complicated creatures. It is not rare that people say something while thinking just the opposite. Apparently, some people suspected that while calling for restraint in the questions of priority I was, in fact, pursuing some personal ambitions. If I had expected that kind of misunderstanding I would not, of course, have made this address at all or, at least, would have worded it differently. I think it would be shameful to speak at a conference about one's claims or complaints. I can only make the assurance that I was misunderstood and that I do not have any priority claims or complaints. My work has been generally recognized. In fact, I consider myself to be quite lucky in this respect, since most of my papers and books have been published only in Russian or in translation with small circulation. In addition, I could not attend many international conferences (for instance, apart from the LT-18 I could attend only the LT-10 which was held in Moscow in 1966 though my first paper on physics of low temperatures was published in 1944; see Ginzburg 1944). I hope that the above remarks are relevant and no further comments are needed. 1 would like only to emphasize that when I cite below my papers

HIGH-TEMPERATURE SUPERCONDUCTIVITY

7

and even quote from them I do it exclusively for the sake of information. Such information can prove useful to those interested in the history and sociology of physics and science in general. As for me, I would prefer not to resume discussion of such questions (and, perhaps, I shall have no more chances for doing that) so I decided to use the.occasion of writing this paper for this purpose.

3. Experimental studies of superconductivity*

Let me begin with a brief account of the history of experimental studies on superconductivity. The liquefaction of helium by Kamerlingh Onnes in 1908 was the first milestone in this direction. By the way, up to 1923 liquid helium was produced only at Leiden. In 1911, Kamerlingh Onnes discovered superconductivity in mercury ( T , = 4.1 K). In 1913, the superconductivity of lead ( T, = 7.2 K) was discovered. It was observed in 1914 that superconductivity was destroyed by a magnetic field, the values of the critical field H , ( T ) being not very large (for example, in the case of mercury even at T + 0, when H , has its maximum value, it is about 400 Oe). Naturally, the critical current I, was also found to be small. Thus, immediately after the discovery of superconductivity, two obstacles appeared in the path of its practical application (the first superconducting magnet was constructed by Kamerlingh Onnes in 1914). The first obstacle was the temperature barrier (low values of T,), and the second was the magnetic and current barriers (relatively weak fields H , and currents I,). The magnetic and current barriers were overcome by the discovery in the 1930s of alloys which are now called superconductors of the second type (see Berlincourt 1987). For such materials, superconductivity vanishes only in a field Hc2(T ) which can be enormously high. For example, Hc2(0) 200 kOe for Nb,Sn. As a result (as late as the 1960s) strong superconducting magnets were constructed. For high-temperature superconductors discovered recently, the field Hc2 is generally still stronger. As regards the critical current I,, for high values of Hc2 it is mainly determined by the way materials are treated, that is, by technology. Therefore, there are apparently no serious reasons to fear that there may be any difficulties in principle in overcoming the magnetic and current barriers for high-temperature superconductors. Table 1 shows how the critical temperature T, was raised over the years up to 1973. I would also like to draw attention to some values of boiling

-

* In this section and some other sections I have used some parts of my introductory to the

HTS conference in Trieste delivered on July 5, 1987 (see Ginzburg

1987a).

report

V.L. GlNZBURG

8

Table 1 Material

Year of discovery of its superconductivity

Critical temperature

r, (K) 4.1 1.2 9.2 18.1 20 to 21 20.3 23.2-23.9

191 1 1913 1930 1954 1966 191 1 1973

points and melting points under atmospheric pressure (see table 2). Of course, these facts are well known but they are relevant to our subject. Now let us discuss the attempts to overcome the temperature barrier made since 1973 when the superconductor Nb,Ge with T,= 24 K was discovered. In connection with the recent advances we must note the work on BaPb,_,Bi,03 ceramics carried out since 1974 (Sleight et al. 1975). The highest temperature T,= 13 K for such ceramics is obtained for x =0.25. These and other ceramics had attracted considerable attention even before the latest discoveries as evidenced by more than 200 entries in the reference list of the review by Gabovich and Moisseyev (1986). The reproducible superconducting materials with high temperatures T, discovered in 1986-87 belong precisely to this class of metal-oxide ceramics. We shall discuss the metal oxides below but here we must note that high-temperature superconductors were perhaps produced in 1978 and later though they were “nonreproducible”. These include CuCl (Brandt et al. 1978, Chu et al. 1978, Lekowitz et al. 1979, Vezzoli and Bera 1981), CdS (Brown et al. 1980), Nb-Si films (Ogushi et al. 1983) and some other materials (the latest results for CuCI and CdS are discussed by Lefkowitz

Table 2 Substance Boiling temperature, Tb( K ) under atmospheric pressure Melting temperature T,,,( K ) under atmospheric pressure

He

H,

Ne

N,

0,

H,O

4.2

20.3

27.2

17.4

90.2

373.16

14.0

24.5

63.3

54.1

213.16

HIGH-TEMPERATURE SUPERCONDUCTIVITY

9

et al. 1987, Homan and MacCrone 1987, and Collins 1987). It is often argued (at least by Soviet workers) that the above reports of strong diamag= netic anomalies at temperature above and even much higher than Tb.N2 77.4K are erroneous, that is, they involve no HTS. But the experiments were carried out by various laboratories, particularly for CuCI, and, as far as I know, no errors have been found. Obviously, if the superconducting phases had been observed they were unstable and the conditions required for producing specimens with ‘‘anomalies?’ remain unclear. There is no rigorous proof, therefore, that HTS was observed in CuCl and some other materials before 1986. It would be instructive to analyze the causes for the lack of clarity in this matter. As for me, I never saw any reasons to believe that all the reported diamagnetic anomalies were erroneous. The detection of a large (in magnitude) diamagnetic susceptibility by itself in the absence of a similar sharp anomaly of resistance does not rule out the existence of grains of the superconducting phase that do not form a continuous electric circuit. In principle, superdiamagnetism not associated with superconductivity can also exist (see Ginzburg et al. 1984 and the references cited there). It is more probable, however, that it was precisely the HTS that was observed in those experiments. In view of the latest results, one can suggest, for instance, that the CuCl specimens contained copper oxides. Incidentally, A.P. Rusakov told me that in his later (after 1978) experiments with CuCl he was especially careful about preventing oxidation and perhaps that was why no strong diamagnetic anomalies were then detected. In addition, a large number of experiments have been reported in which quite different (metal oxide) specimens exhibited “nonreproducible” superconductivity-type anomalies at temperatures as high as room temperature (see below). In short, I believe that, quite probably, HTS of the “nonreproducible” kind has been observed, at least after 1978. Incidentally, in materials science nonreproducibility is not a strange phenomenon; it has been known, for instance, for some semiconductors. It was only in 1986 that Bednon. and Muller (1986) reported reproducible superconductivity with T,- 30-40 K for the La-Ba-Cu-0 ceramics. This report was entitled, though, “Possible high T, superconductivity in the Ba-La-Cu-0 system” and, indeed, it lacked the evidence that the resistance R = 0. But, in fact, it was the first reproducible “neon” superconductivity, = 27.2 K. that is, for which T, was higher than the boiling point of neon, Tb,Ne This work was, undoubtedly, of a great importance. I am not acquainted with the details of the subsequent studies in the HTS field carried out in Switzerland, Japan and USA which can be found in the reviews that will be published in Part 3 of the LT-18 Proceedings (the reports at the abovementioned symposium). Further details can be found in the review by Golovashkin (1987). However, I shall mention some milestones on the road

I0

V.L. GINZBURG

to the HTS development. For instance, Cava et al. (1987) reported a fairly sharp and undoubtedly superconducting transition with T, = 36.2 K for the La, $r0 *CuO, ceramics. The “nitrogen” superconductivity with T, > Tb,N2 -- 77.4 K was later obtained by replacing ,,La with ,,Y and some other elements. For the system Y-Ba-Cu-0 Wu et al. (1987) produced superconductivity with T,,, = 93 K (under atmospheric pressure) and T,, = 80 K (at T,,, the resistivity R starts to decrease sharply and at T,, we have R = 0 within the accuracy of the measurements). Similar results were obtained soon afterwards by groups of experimenters in many countries (the first work performed in the USSR is reviewed by Golovashkin 1987). Interestingly, the “neon” superconducting ceramics had been manufactured several years before their superconductivity was discovered. For example, Shapligin et al. ( 1979) produced the above-mentioned La, $r0 ,CuO, ceramics. Shapligin et al. even studied the resistivity of these ceramics but not in liquid helium. Naturally, they did not discover their superconductivity ( I believe that these ceramics did not include any “nitrogen” superconductors). In recent tests these old specimens proved to be superconducting and therefore they experienced little or no ageing. According to my information, similar ceramics had been manufactured before 1986 in Japan and France though they had not been examined for superconductivity. One should learn a lesson from that. Clearly, all new materials with metallic or semiconductor properties must be “tested for superconductivity” in liquid helium, and in successful cases T, must be measured. A wide variety of the “nitrogen” superconductors are now known. A typical one is the YBa,Cu,O,-, system. It has T,=93 K, for instance, for y = 0.1. When y is higher, for example, y = 0.8, the system does not exhibit superconductivity. In this system yttrium can be replaced with a number of other elements without a significant reduction in T,. The “nitrogen” superconductors are generally rather easily obtainable (I mean ceramics rather than single crystals) and, as mentioned above, they have even been manufactured by school pupils (Grant 1987). An intense search of the HTS materials with even higher critical temperatures is going on. Various criteria for “true” or “genuine” superconductivity have been put forward. Under ideal conditions the resistivity R of a genuine superconductor must, of course, vanish and the full Meissner effect (the expulsion of the magnetic field from the interior of the superconductor) must occur in a weak magnetic field. The suggested requirement that the effect be stable in time does not seem necessary. This condition is, of course, significant (perhaps, decisive) from the viewpoint of applications. In terms of physics, however, the metastable phases are by no means inferior to the stable ones

HIGH-TEMPERATURE SUPERCONDUCTIVITY

11

and, in fact, the concept of stability itself is, to a certain extent, arbitrary (for instance, diamond can be regarded as a metastable phase). As for the requirement of reproducibility, it is highly significant for proving the very existence of superconductivity. Let us assume, for the sake of argument, however, that we have accidentally produced a stable material with R = 0 which exhibits the Meissner effect. Obviously, it is a superconductor even if we cannot manage to produce it once again for a long time. The suspicious cases thus involve a combination of instability with nonreproducibility. In any case, at the time of writing (October 1987) I know of no “genuinely” superconducting material whose critical temperature T, has been proved to be much higher than 100 K. Some data on such “potential superconductors” with T c 3 100-150 K were reported to the LT-18 Conference (see Proceedings LT-18). It should also be mentioned that dgushi et al. (1987) reported T,, = 315 K and T,, = 255 K for La-Sr-Nb-0 films. These specimens also exhibited some diamagnetism and a dependence of T, on the current. Huang et al. ( 1987) reported evidence suggesting that specimens of the EuBaZCu306+, system contained inclusions of superconducting phase with T,- 230 K; see also the review by Golovashkin (1987). (A sign of the exceptional importance of HTS is the appearance of the newsletter “High T, update” that is published fortnightly in the USA. The main item in the newsletter is a list of preprints which contained about a hundred titles in the issue No. 9 published on September 1, 1987. At least 2500 papers on HTS will be published a year if the publication rate is kept at this level, and perhaps more since I d o not know the scope of coverage by the newsletter. The newsletter contains also sections for news and other topics. The issue No. 9 reported the production of stable and reproducible specimens of the system Y( BaSr)-Cu-0 with T, = 240 K, although apparently it is not the temperature T,, . Another group of experimenters have also produced specimens with T, = 260 K.) From the viewpoint of the theory, at least at present, there are no limitations preventing the existence of superconductivity at room temperature and even at higher temperatures (see below). We cannot safely predict, however, what critical temperatures will be obtained, especially, .for practicable materials. An intense search is underway and we shall probably know soon the possible limits for raising T, for metal oxides. At any rate, I shall not be surprised at all if in the very near future easily obtainable “room-temperature’’ superconductors are manufactured. (In view of the recent advances it is instructive to recall the papers which stated quite firmly that the highest possible value for T, was 25-30K; Matthias 1970, 1971.)

I?

V.L. GINZBURG

To conclude this section, let us discuss briefly the relevant terminology. Before 1986 the term “high-temperature superconductor” was often applied even to the materials with Tc> 10 K, for instance, to Nb,Sn with T, = 18.1 K. In my papers the term HTS was applied only to superconductors with a critical temperature near the boiling point for the liquid air, that is, T,> 80-100 K (see, for instance, Ginzburg 1968,1970,1972). At present, ceramics with T,> 30 K are typically referred to as HTS materials. A term is, of course, a matter of convention and it does not have a special significance. In my opinion, however, it would be more appropriate to apply the term HTS to materials with T, > T,,.,: = 77.4 K. This suggestion is supported by the fact that the temperatures T,-20-40 K were long regarded as being obtainable even for the “conventional” (phonon) mechanism of superconductivity (see, for instance, Ginzburg 1968, Ginzburg and Kirzhnits 1982). In order to prevent confusion, however, in this paper I apply the term HTS to all new “neon” and “nitrogen” superconductors, that is, the materials = 27.2 K. This approach is supported by the fact that all with T, > Th.Ne such materials known now are metal oxides and clearly have something in common (apart from the composition, the common features are the layered structure of the new materials and the presence of oxygen in them).

4. Microscopic theory of superconductivity A microscopic theory of superconductivity, which can be called reliable or fairly comprehensive, was first put forward in 1957, i.e. 46 years after the discovery of superconductivity. Of course, I mean the theory formulated by Bardeen, Cooper and Schrieffer (Bardeen et al. 1957). However, this theory did not grow out of nothing: a lot of work had preceded it (see, for instance, Bardeen 1956). Incidentally, I have participated in discussions of the following question on a number of occasions: what prevents us from creating a new theory (in fact, the discussions concerned problems other than superconductivity)? Do we lack additional facts or information, or are we waiting for the advent of a genius, a “new Einstein”? It is impossible to provide an unequivocal answer to this rhetorical question, since all depends on the prevailing situation. But as far as superconductivity is concerned, the following can be stated unambiguously: both Einstein and Bohr, and many other leading scientists of a lower calibre, were interested in the problem of superconductivity and made a number of useful comments (Einstein 1922, Hoddoson et al. 1987). However, they were not able to formulate the microscopic theory of superconductivity. It seems to me that, in the first place, such a

HIGH-TEMPERATURE SUPERCON1)UCTIVITY

13

theory had to wait until the discovery of superfluidity of He 11, which was finalized only in 1938.* Then it became generally clear that superconductivity was the superfluidity of the electron liquid in metals. This concept, treated in the context of Landau's theory of superfluidity, helped the understanding of some aspects of the problem (Landau 1941, Ginzburg 1944, 1946) though not at once. The matter was complicated by the opinion (even Landau held it for a time) that superfluidity of helium 11 had no relation to the Bose-Einstein condensation whose possibility and even inevitability under certain conditions was suggested by Einstein (1925). The relation of superfiuidity to the Bose-Einstein condensation or, at least, to the Bose statistics of the 4He atoms became obvious only after 1948 when liquid 'He whose properties sharply differed from those of liquid 4He had been produced (liquid 'He is known to lack a lambda point in the temperature range T > 0.1 K while for 4He we have TA=2.17 K). Many scientists, however, had believed in such a relation even before liquid 'He was produced (London 1954). One would think that then efforts had to be made to understand how electrons in 3 metal could produce a Bose system, for instance, by binding electrons into pairs with zero spin. In fact; however, the concept of such pairing was by no means trivial. It was very hard to imagine two electrons forming a bound state because normally they repel each other. Perhaps I do not have a complete picture but I know only one paper published before 1954 that mentioned a relation between superconductivity with formation of pairs and their Bose-Einstein condensation. It was written by Ogg (1946) who believed he had discovered superconductivity in metal-amonia solutions. As far as I know he was mistaken though I do not know the details of the experiment. However, Ogg suggested that electrons were bound in pairs which underwent a Bose-Einstein condensation. In 1952 I analyzed the * In science, discoveries are made in different forms and in quite different ways. The discovery of superconductivity and superfluidity is a very good illustration of that. For instance, superconductivity was discovered in 1911 with confidence and practically by a stroke by Kamerlingh Onnes. In the same year Kamerlingh Onnes observed some strange phenomena in liquid helium at 7'-2.2 K (nonmonotonic dependence of the density on temperature) and in 1922 he discovered an anomalously fast flow of the helium films. In 1928 the concept of two phases -helium I and helium 11- was put forward by Keesom and Wolfke and in 1932 Keesom and Clausius obtained a clear lambda curve for the specific heat of liquid helium at the lambda transition. In 1936 W. Keesom and A. Keesom discovered superhigh thermal conductivity of helium I 1 and finally in 1938 Kapitza, and Allen and Misener independently discovered superfluidity of helium I I - its frictionless flow through narrow slits and capillaries. A complete enough understanding of superfluidity was achieved by 1941 in the works of Kapitza and Landau. Therefore, we can say that it took thirty years to discover the effect of superfluidity. To save space, no references are given here, especially as a full enough list was given by London (1954).

V.L. GINZBURG

14

properties of a charged Bose gas and noted that the Meisnner effect had to occur in it (Ginzburg 1952). In this paper, however, I never mentioned the possibility of the formation of such a Bose gas from electrons and apparently I did not care for the concept at the time. In fact, in the paper on the tY-theory of superconductivity (this is how 1 refer to the macroscopic theory put forward in 1950 by Landau and me (Ginzburg and Landau 1950); we wrote when replacing -ihV with (-ihV - ( e / c ) A ) that “ e is the charge for which there are no reasons to believe that it differs from the electron charge”. I had some doubts about that even at that time ( I believed it would be better to take some effective charge e * ) but 1 could not find any convincing arguments in favour of a difference between e and e*. Later, when I analyzed experimental data I saw that the agreement of the Y-theory and experimental data could be improved by introducing the effective charge e * = ( 2 - 3 ) e (Ginzburg 1955). But Landau objected against this suggestion (Ginzburg 1955) since he noted that such an effective charge could depend on the pressure, temperature, composition and so on, and therefore it had to be regarded as a function of coordinates. But then the gauge invariance is violated. I failed to overcome this difficulty and to introduce the charge e*. Now the solution of the problem seems quite clear - any universal charge e * = ne where R is an integer is quite admissible from the viewpoint of gauge invariance and, in fact, in superconductors including the newlydiscovered ones we have e* = 2e (Gough et al. 1987). Is it not curious (it seems to be sheer blindness now) that such a conclusion was not drawn by me, by Landau, or by anybody else? I have spoken about this problem at length also in order to draw attention to the contribution made by Schafroth who was killed in a plane crash in 1959. Like me, he started by considering a charged Bose gas (Schafroth 1954a) and then put forward the idea of pairing in the most clear and definite form, even suggesting that the electron-phonon interaction could be responsible for pairing (Schafroth 1954b). Schafroth et al. (1957) developed their theory but left the question about the nature and mechanism of pairing unanswered. The pair size was assumed to be small (of atomic dimensions) and the critical temperature T, was determined to a first approximation by the formula for the temperature of Bose-Einstein condensation of an ideal gas: T, =

3.31 h’n”’ m*k,

where m* is the mass of the pair, m =9.1 x 10 28 g is the mass of a free electron, and n is the number density of pairs with zero spin. In recent years, and especially now when new superconducting materials have been obtained, Schafroth’s model with “tiny” pairs is being widely

HIGH-TEMPERATURE SUPERCONDUCTIVITY

15

discussed, and we shall return to it later. Thus, it is obvious that the generally accepted belief that the introduction of pairing in the theory of superconductivity is due to Cooper (1956) is entirely unfounded. The merit of Cooper’s work and, in the first place, of the subsequent BCS theory (Bardeen et al. 1957) lies in the real demonstration of pair formation (“large”, or Cooper, pairs) and of their collective “condensation” even under a weak attraction between electrons near the Fermi surface of the electron gas. I must emphasize that the above brief discussion was somewhat one-sided (perhaps even biased). An essential difference between superconductivity and superfluidity is, of course, the occurrence of the Meissner effect in superconductors that was discovered in 1933. A similarity can, of course, be found between the effect of rotation on the helium I1 and the effect of the external magnetic field on a superconductor but it is a different matter. The significant aspect is that superconductivity does not reduce to infinite conductivity and at equilibrium the external magnetic field cannot penetrate into the interior of a superconductor (we are discussing, of course, Type I superconductors for H < H, or Type I1 superconductors for H < H c , ) . In this connection, a significant contribution to our understanding of superconductivity (apart from an even irrespective of the similarity to superfluidity) was made by the concept of the macroscopic quantum state and the rigidity 6f the wave function in this state (see London 1950). The concept of the energy gap in the excitation spectrum was also promoted in this way (see London 1950, Ginzburg 1944, 1946, Bardeen 1956,1963). In the BCS model the Bose-Einstein condensation is somewhat indefinite in character because the size of the pairs is much larger than the distance between them. Owing to the above factors, the similarity to superfluidity apparently made no contribution to the formulation and presentation of the BCS theory (Bardeen et al. 1957). But now we understand that the BCS model with weak coupling is one limiting case while the Schafroth model (with localized pairs) is another limiting case. To be more precise, this fact was realized long ago but somehow the prevailing feeling was that all real superconductors are described by the BCS theory. Now it is clear that there are reasons not to limit the discussion to the BCSscheme and to consider also the general case (see below).

5. Critical temperature in the BCS theory The above considerations do not, of course, detract in any way from the significance of the BCS model and this approach on the whole. In fact I believe it quite possible and even probable that new superconductors with

V.L. GINZBURG

16

T,> 30 K are described basically by the BCS model. We shall continue this discussion below and now we shall base our treatment precisely on the BCS theory. The well-known formula of the BCS theory T, = 8 exp( - 1/ Aerr)

(2)

and, what is more important, the entire BCS theory were strikingly successful. Indeed, according to this formula, the value of T, is determined by only two parameters, viz. k B 8 , which is the region of attraction between electrons near the Fermi surface, and Aefr = N ( 0 ) V, which is a dimensionless parameter characterizing the intensity of this attraction. In the original version of the BCS theory, herr=N ( 0 )V < 1, where N ( 0 ) is the density of states on the Fermi surface and V is the matrix element of the interaction energy. It is not surprising that the BCS theory is indeed the real microscopic theory of superconductivity for the chosen model. However, the fact that this model gives a fairly accurate description of real superconductors with a weak coupling (Ae,,-* 1) can by no means be considered obvious. As an example, we can consider the BCS formula connecting T, with the energy gap 2 4 ( 0 ) in the superconductor spectrum for T = O , which has been confirmed experimentally:

2A(O) =3.52kBT,.

(3)

This relation might not be satisfied for the simplest model. Of course, the BCS theory was developed and refined (in the USSR, the contributions made by N.N. Bogoliubov, L.P. Gor'kov and G.M. Eliashberg are worth mentioning). The reasons behind the astounding success of the BCS model are discussed in Ginzburg and Kirzhnits (1982). In fact, the success of the BCS model is largely explained by the fact that it is a mean field theory. On the other hand, such a theory is, to a certain extent (formally only in the vicinity of Tc),equivalent to the Landau theory of second-order phase transitions, that is, the V-theory of superconductivity in this case, which also includes only two parameters in the simplest case. Besides, it is precisely for the conventional superconductors that the mean field theory is generally suitable owing to the smallness of the fluctuations near T,. The smallness of the fluctuations in its turn is due to the large correlation length to (Ginzburg 1960, White and Geballe 1979). Let us now go over to the calculation of the critical temperature and its possible maximum value. For the phonon mechanism of superconductivity, i.e. when the attraction between electrons is due to their interaction with the lattice vibrations, or phonons, the temperature 8 0,) in formula (2) is the Debye temperature. For the normally-accepted values @,,s500 K

-

HIGH-TEMPERATURE SUPERCONDUCTIVITY

17

and Aerr=S 1/3, we obtain Tc=S500 e-3 = 25 K. In general, this explains the fact that T c s 10-30 K. In a more realistic theory, and for weak coupling (under the condition I ) , we have

where A and p are dimensionless coupling constants for the phonon (exciton) and Coulomb interaction respectively, and 0,= EF/k , is the temperature corresponding to the Fermi energy EF of the metal. Further, in the commonly used approximation (homogeneity, isotropy, weak coupling) we obtain p - A = 4re2N(O)

where ~ ( wk,) is the longitudinal permittivity of the material, and the angle brackets indicate averaging over the momenta hk. It has been suggested that the requirement of stability of a metal leads to the condition ~ ( 0 k, ) > 0. If this be so, then A s p in accordance with eq. ( 5 ) , and superconductivity would be possible only because p* < p. This would indicate a low upper limit for T, (Cohen and Anderson 1972). Such a conclusion, however, is incorrect since the stability requirement actually has the form l/c(O, k ) s 1 (where k # 0), the values of ~ ( 0 k, ) < 0, and hence A > p are admissible. This important result was obtained by Kirzhnits (1976, 1987) and is associated with the following. The response function obeying the dispersion relations is not E but rather I / a As a matter of fact, the electric field E = ( I / E ) D where , D is the induction satisfying the equation div D = 477pext.The external charge pextcan be controlled, and hence it is D that plays the role of the “cause”, while the field E is the “consequence”. We cannot arbitrarily change E, and hence the function E in the formula D = EE need not obey the dispersion relations. To be more precise, the function ~ ( 0k, + 0) for small values of k --* 0 is also a response function, and hence the necessary condition for the stability of the system is c(0, k -+ 0) a 0. But in the theory of superconductivity, we are interested in large values of the wave vector k Usually, k is of the order of k , , and hence we should impose only the requirement l/e(O, k ) s 1. As mentioned above, this is in agreement with the condition ~ ( 0 k, ) < 0. What is more, it is just the values ~ ( 0k, ) < O that are realized in most metals (Dolgov et al. 1981, Dolgov and Maksimov 1982). Usually, p* 0.1 (the parameter p =S OS), and in some cases A s 3 (for example, in the PbBi alloys, A s 2.6; see Dolgov and Maksimov 1982, Wong and Wu 1986).

-

V.L. GINZBURG

I8

For intermediate and strong coupling when her[> 1, formula (2) under condition ( 4 ) is obviously inapplicable. For this case, a number of formulas of type ( 2 ) have been proposed with, say, Acfr=(A - p * ) / ( l + A ) .

(6)

As an illustration, note that for p* = 0.1 and A = 1 or A = 3, the temperature T, = 0.1 10 and T, = 0 . 2 5 0 , respectively, in accordance with (2) and (6). In other words, we obtain the values T,= 30 K and T,==75 K for 0 = Or,==300 K.Many other similar formulas for T,, obtained on the basis of Eliashberg's equations (Ginzburg and Kirzhnits 1982, Dolgov and Maksimov 1982, Wong and Wu 1986) and having a semi-empirical or model character, lead to high values of T, for appropriately chosen parameters. For example, the following formula has been proposed by Wong and Wu (1986) (we present it for p * = O . l ) :

0 T, ->( 1 +0.53A ) exp[ - 1.25/( A -0.1 1 )], 5.42

(7)

where

A =2

I

dw

a 2 ( w )F ( w ) w

(see the above papers for notation). According to ( 7 ) , Tc=0.078,, and T, = 0.31 8,, for A = 1 and_ A = 3 respectively. For strong coupling, when A b 10, we have T,- &/A (here 6 = h G / k , , where (I, is a certain mean frequency in the phonon spectrum; see Allen and Dynes (1979, Ginzburg and Kirzhnits (1982, Chapter 4). In any case, there are no restrictions in principle imposed on the possible values of T, by the conventional theory of superconductivity. Thus, it is clear that large ("high-temperature") values of T,- 100 K can be obtained for A =s3 and 0 O,, 8 0,) if at the same time eD3 300 K. The only question is whether such values of O,, and A can be realized for the phonon mechanism of superconductivity. For each type of material, there exists a maximum value of T, since the parameters A and 0,) are not independent. Thus, an increase in the value of the electron-phonon coupling constant A leads to a decrease in the phonon frequencies (due to the screening by conduction electrons and other reasons). This effect can be approximately described by the formula (see Dolgov and Maksimov (1982) in this connect ion ) :

- - -

HIGH-TEMPERATURE SUPERCONDUCTIVITY

19

where woo is the bare - not renormalized - (“pseudoatomic”) phonon frequency and C is a constant. As a result, we obtain the maximum value of T, which is proportional to woo. Superconductors obtained before 1986 were characterized by parameters leading to T C s25 K. Hence, and in view of the rough estimates presented above, it was usually concluded that T, < 30-40 K for the phonon mechanism of superconductivity (see, for example, Ginzburg and Kirzhnits (1982, Chapter 1). It was always stipulated, however, that this conclusion is not general. For example, it was mentioned that O,, 3000 K for metallic hydrogen and some (though not all) calculations lead to the values T,100-300 K (see Ginzburg and Kirzhnits (1982), Dolgov and Maksimov (1982) and the literature cited therein). Many metals are known to have high values of OD(for example, O,(Al) = @,(Ti) = 430 K, O,(Os) = 500 K, OD(Mo) =450 K, and @,(Be) = 1400 K). The high value of 8, for Be even led to the suggestion that a temperature T,- 100 K can be attained for beryllium under certain conditions. The presence of light atoms H and C in organic superconductors raised hopes (Budzin and Bulaevskii 1984) that the high phonon frequencies in this case might result in large values of T,. So far, HTS of such types have not been produced, but the possibility cannot be ruled out.

-

6. Superconductivity mechanism in the “neon” superconductors

Naturally, it is especially important at present to explain the mechanism of superconductivity in recently discovered materials, viz. metal oxide ceramics. Let me start with the alloy La,.8Sro.,Cu04and similar materials with T,=36 K. According to Cava et al. (1987) and Kwok et al. (1987), superconductivity in this case may be caused by the usual phonon mechanism described above. The high value of T, is attributed to the high frequency wo of vibrations in the “subsystem” formed by a Cu ion surrounded by a distorted octahedron of oxygen ions. A comparison with the superconducting metal oxide ceramic BaPb,-,Bi,03 (for this alloy, T, = 13 K for x = 0.25) for nearly identical values of OD leads to T, = 36 K under the assumption that the density of states N ( 0 ) near the Fermi surface is 2-3 times higher for La,.8Sr0.ZC~04. Kwok et al. (1987) assumed that for the alloy La,.,,Sro,,,Cu04 the value of N ( 0 )is 3-5 times higher than for BaPb0.,Bio.,O, in accordance with the measurements of the field H c 2 . Taking A = 1.2 for Putting A = 3 by BaPbo.75Bio.z503, we get A =3.1-3.9 for Lal.85Sro.15Cu04. way of an example and using formula (7) with an identical value of OD,. we obtain Tc(La,.,,Sro.,5Cu04)= 3.2Tc(BaPb0.75Bi0.2503) = 42 K. The same conclusion is drawn by Weber (1987) from quantitative analysis based on

V.L. GINZBIJRG

20

calculations (Mattheiss 1987) of the electron structure for the tetragonal crystal LalCu04 (such computations have been carried out also by Mazin et al. (1987)).According to Mazin et al., the values Tc=30-35 K for A = 1.5 are quite natural for the system Laz ,(Ba, Sr),CuO,. The light oxygen atom vibrating at a high frequency wo plays an important role in this case. A distinguishing feature of the metal oxide ceramics is that they have high values of wgowing to the presence (associated with light oxygen atoms) of rigid optical modes of lattice vibrations (phonons) which are strongly coupled with conduction electrons. In turn, this is due to a significant contribution of ionic-covalent bond in these compounds. About 20 valence electrons per unit cell contribute to the formation of this bond (the metallic bond is formed by just 1 to 2 conduction electrons which exert a rather weak influence on the lattice). Thus, it can be assumed that there is a tendency towards the optimization of the contributions from ionic and metallic bonds in metal oxide ceramics (see Ginzburg and Kirzhnits (1982, p. 170). The above discussion does not mean, of course, that the nature of superconductivity in the La-Sr-Cu-0 system has been understood in full. The nature of superconductivity in BaPb, ,Bi,O, is also a matter of controversy (see Gabovich and Moisseyev 1986). As noted above, however, the La-Sr(Ba)-Cu-0 system may have the conventional phonon mechanism of superconductivity. This i; also suggested by the results on the isotopic effect in the La, 8sSr,,,5Cu0, ceramics (Batlogg et al. 1987b, Faltens et al. 1987). The observed isotopic effect (the change in T,) was caused by the replacement of I6O with '"0.The isotopic effect is usually described by the where M is the mass of the isotope. For a singlerelation T,- M component metal the parameter a if superconductivity is caused by electron-phonon coupling and anharmonicity is ignored (as noted above, in the case of the phonon superconductivity mechanism 0 in the BCS formula is the temperature 9,)which is proportional to the characteristic frequency wg of lattice vibrations that is proportional to M -I"). The parameter Q has a different value for a complex (multicomponent) system and when anharmonicity is taken into account. According to Batlogg et al. ( 1987b) for the La, ssSr, ,5Cu04ceramic a = 0.16k0.02. Faltens et al. (1987) reported values of Q varying from 0.1 to 0.37 for various specimens. When anharmonicity is ignored the theory yields values of a that exceed experimental results (apparently these values are considerably higher than 0.16). I t would be interesting to analyze the isotopic effect for BaPb,Bi,-,O,. At any rate, a clear manifestation of the isotope effect suggests that the phonon mechanism plays a significant role for the La-Sr-Cu-0 system. On the other hand, the smallness of the observed values of a in comparison with the predicted values can be explained both by anharmonicity and by the 'I,

=:

HIGH-TEMPERATURE SUPERCONDUCTIVITY

21

effect of another superconductivity mechanism, for instance, the exciton mechanism that will be discussed below.

7. Superconductivity mechanism in the “nitrogen” superconductors. Ways for raising T,

We must state at the beginning that the mechanism of superconductivity in the “nitrogen” superconductors (and, partially, in the “neon” superconductors as discussed above) is still unclear. This is explained by the lack of reliable experimental data, particularly for single crystals. In the light of the above estimates we can admit logically that the temperatures T, 100 K can be explained in the framework of the conventional phonon mechanism. As an illustration, note that (7) with A = 3.5 and T, = 250 K yields eD = 680 K and this value is by no means excessive. If the experimental result for Q D , the electronic component of the specific heat and other parameters confirmed such a possibility then we would obtain, roughly speaking, a “trivial solution” for the high-temperature superconductivity in the cases under consideration. Indeed, the limit of Tc=S40 K for the phonon mechanism has never been substantiated since it was derived in a rather arbitrary way from rough estimates and few available experimental results. Now we, from this point of view, have produced materials with such parameters OD and A as lead to high temperatures T,. This conclusion can be reliably refuted, probably, only when we obtain sufficiently detailed results on the phonon spectra of the “nitrogen” superconductors and other data needed for calculating T, (though we assume here that we are dealing with BCS superconductivity). The widely held opinion based on incomplete data, and intuition suggest, however, that the “nitrogen” superconductors are unlikely to have the phonon mechanism of superconductivity. Note, first of all, the lack of a marked isotope effect (see, however, below) in the “nitrogen” superconductors of the type of YBa2Cu307and EuBa,Cu,O, (Batlogg et a]. 1987a, Bourne et a]. 1987). The replacement of I6O with I8O was found to produce no noticeable change in T, (the measurements gave a = 0.0 f 0.027 and 0.0 f 0.02). As noted above, this result by itself does not rule out the phonon mechanism (superconductivity in Ru and Zr is assumed to be due to the phonon mechanism but for them a =O; for PdH and PdD we even have a the thermal mean square velocity, (J the distance of closest approach, and rcOl1 the mean time between collisions equal to 6.2 x lO-’’/pT’/’ s ( p in g cm-’; T in kelvin). Numerical substitution leads to the approximate result 74 = 115 T ’ l 2 / ps ( p in g cm-’; T in kelvin). Figure 2.10 shows the results of more precise theory compared with experimental data and we see that for example at T = 3 K and n = loi8atoms (corresponding to p = 5 x g ~ m - ~rd ) ,is about 4 x lo7s which is over one year. We conclude that this mechanism can safely be neglected for present purposes. Schearer and Walters ( 1965) have considered the effects of field gradients, and made measurements on ’He gas at T = 3 0 0 K, and more recently a

I

I

I

2

I

I

I

I

10 12 14 16 18 20 T IK Fig. 2.10. The quantity p ~ versus , temperature, where p is the density of the polarised gaseous >He and T~ is the relaxation time due to the magnetic dipole coupling between the nuclei (from Chapman (1975)). According to eq. (2.6) this would vary as T”’,but a more thorough treatment (shown as a dashed line) gives results which agree well with experiment and confirm that T, is very high. 4

6

I

8

STRONGLY SPIN-POLARISED ’He GAS

91

generally applicable theory has been given by Lefkvre-Seguin et al. (1982). Gradient effects are important because any field inhomogeneity encountered by a randomly moving atom may be pictured as a randomly varying field applied to the same but motionless atom. A Fourier decomposition of this fluctuating field will in general yield a component lying in a direction perpendicular to that of the main field Bo and oscillating at the nuclear Larmor frequency w o . This field component will then induce transitions between the two magnetic levels *4 of the nucleus and contribute directly to the nuclear relaxation rate. In general this is a rather complicated matter, with T~ depending on Bo, the field gradient G, and at least three characteristic ~ to 3/[8r112(’.’’[~~]] times including the time between collisions T , ~ (equal where n is the number density), the diffusion time across the cell TD (approximately equal to 0.62r2nm3R“”’[Uk]/keT for a sphere of radius r), and l / w o where wo (the Larmor frequency) is equal to yBo with y being the nuclear gyromagnetic ratio. Lefhre-Seguin et al. (1982) have addressed the theory of this and obtained a general formula for T ~ .In one limit, W O T Ds= 1 , this reduces to a form given earlier by Schearer and Walters (1965). Although the formula in this limit is not appropriate for experiments of interest in this chapter, we mention it briefly here:

where G is the magnetic field gradient (part of which will be due to the magnet itself and therefore proportional to Bo while the other part may be an ambient gradient in the laboratory), ( U 2 )is the thermal mean square velocity proportional to T. The numerical form of (2.7) becomes “=

B~ ~ ; + 6 . 6 3x 1 0 1 5 p 2 ~ 2.28 x 101’pT3’2 ’

(5)

where Bo is in gauss and G is in gauss cm-’. Work by Schearer and Walters ( 1965) showed satisfactory agreement between this theory and experiment ~, Bo=lOG, and G = at 300K. If we take p = 5 ~ 1 0 - ~ g c r n -T=300K, 5x G cm-’ as representative values, eq. (2.8) gives T~ = 3 x lo* s (about ten years!). Equation (2.8) is not appropriate to the conditions in all of the experiments discussed in this chapter, but it is generally true to say that suitable design can ensure that field gradients are less destructive of polarisation than other mechanisms discussed below. Timsit et al. (1971) have given a helpful discussion of the physics of adsorption (for 7J. and absorption processes (for T,,,). In practice these are the most important mechanisms causing depolarisation and Timsit et al. (1971) identify the dipolar coupling between a ’He nucleus and a flipping magnetic dipole in the glass as the most important of all.

D.S. BEITS ET AL.

92

Their arguments lead, subject to some model assumptions, to the following expressions. We have omitted their arguments here and simplified their notation.

where N and N a d are the numbers of free and adsorbed ’He atoms, respectively, in the container (taken to be a sphere of diameter d ) , fFe is the atom fraction of iron in the glass (iron being the commonest paramagnetic impurity in glass), and T ; ~is the intrinsic relaxation time of a ’He nucleus near a paramagnetic site on the wall surface. The number Nad depends on temperature according to (2.10) where, in addition to quantities already defined, u:d is the lattice vibration frequency of an adsorbed atom, and Qadis the (positive) binding energy of an adsorbed atom. The thermal mean speed ( U )is, of course, proportional to TI”. Hence, from (2.9) and (2.10), (2.11) For Tab, the arguments of Timsit et a[. (1971) lead to expressions which parallel eqs. (2.9) and (2.10), (2.12) where /3 is the number of nearest neighbours for a ’He atom absorbed in the glass, v,h is the jump frequency of a ’He atom diffusing through the glass, and N a b is the number of ’He atoms dissolved in a surface layer of thickness (Ar) equal to the mean diffusion jump distance. This number is given by =

6 NkBT(A r ) S d

(2.13)

where S is the (temperature-dependent) solubility of ’He in glass, and (2.14)

STRONGLY SPIN-POLARISED ’He GAS

93

where Doexp(-Qab/k,T) is the diffusion coefficient of 3He in glass expressed in terms of the constant Do and the (positive) activation energy Qab.Hence from (2.11), (2.12) and (2.13), (2.15) The forms of Tad and T~~ are not particularly transparent but it is of interest to note that they have different signs in the exponentials. Thus at sufficiently high temperatures Tab will be the smaller, whereas at sufficiently low temperatures Tad will be the smaller. Substitution of suitable numerical values (given by Timsit et al. (1971)) suggests that at room temperature Tad and Tab are of comparable magnitude, depending of course on the type of glass used. It is sensible therefore to assume that Tad is the more important of the two in the helium range of temperatures. Substituting values into eq. (2.11 ) yields for a sphere of 6 cm diameter, (2.16) Let us assume that fFe is 1 in lo4. Then at 300 K this expression is equal to 7.8 x lo6 s. As temperatures fall from, say, 20 K to 4 K, Tad falls sharply s. The lesson is that, even from 1.4 x 10’ s (a day and a half) to 2.5 x allowing for approximations and uncertainties, it would be impossible to retain polarisation for enough time to do an experiment were it not for the expedient of coating the glass with some material which acts as a barrier between the spins and any paramagnetic impurities. A very thorough study of the beneficial effects of hydrogen and some other coatings has been made by Lefhe-Seguin et al. (1985) and it was found for example that in a spherical cell of diameter 3 cm, the relaxation time was as high as 1000 min at about 3.7 K, falling to 10 min at about 2.3 K. Times of this order allow a range of experiments to be performed, although at the lower temperatures some haste is necessary.

3. Experiments 3.1. EXPERIMENTS ON

SPIN ROTATION EFFECTS A N D S P I N WAVES

The theory of these effects has been discussed in sections 1.6 and 1.9 and experiments designed to measure the quantity p given by eq. (1.70) have been performed by Nacher et al. (1984) and Tastevin et al. (1985). Nacher et al. (1984) first used a method in which spin waves were excited by applying a magnetic field gradient over the sample, conditions being such

D.S. BETIS ET AL.

94

that the atoms are able to explore all the volume of the container before depolarisation is brought about by the field gradient (the condition for motional averaging is discussed by Abragam (1961) and by Slichter (1980)). This situation is only possible when the walls of the container have a very weak relaxation effect. The effects of spin waves can then be simply and completely described (see Lefkvre-Seguin et al. (1982)) in terms of the longitudinal relaxation time T,, the transverse relaxation time T,, and a frequency shift Aw of the precession of M, the transverse component of M. The origin of the shift can be understood from the following simple argument which is valid when (M,l is small compared to the longitudinal component Mo which is closely equal to M. Now the shift depends on the field gradient in the sample, and would be strictly zero if M were uniform (because of angular momentum conservation). The role of the variation FBo of the longitudinal component of the magnetic field across the sample is precisely to create a gradient of spin polarisation, that is, to couple the total transverse magnetisation M, of the sample to a spin wave. But this wave oscillates with a slightly different frequency and is also damped at a rate l / r D . The situation is then simply that of two coupled oscillators with different frequencies and damping, and one can show from an elementary theory of coupled oscillators that the equation of motion for M , acquires a damping rate 1/T2 and a frequency shift Aw given by

(3.2) where for the purposes of the experiments carried out by Nacher et al. (1984) it was unnecessary to know the magnitude of the coefficient a which occurs in both these equations. These experiments made use of the ratio of (3.2) to (3.1): T2Aw = @ M,

(3.3)

or, equivalently, A 4 = PM,

(3.4)

where A 4 was the phase shift accumulated during a time T 2 . Nacher et al. ( 1984) set out to measure p = Aq5/ M as a function of temperature. A general sketch of the experimental arrangement is shown in fig. 3.1 and many of the features of polarised 'He gas experiments discussed in various sections above are in evidence here. The 'He gas is contained in a sealed 'Pyrex cell, consisting of two different parts connected by a 3 mm inner diameter tube: a 5 cm inner diameter by 5 cm long cylindrical container where the atoms are submitted to a discharge and optical pumping at room

STRONGLY SPIN-POLARISED 'He GAS

I

95

kb C

d II

BO

Fig. 3.1. Double cell used by Nacher et al. (1984) for the detection of spin rotation effects and spin waves in polarised 'He. (a) Circularly polarised laser beam. (b) Upper 'He container submitted to optical pumping at 300 K. (c) Thermal shielding at high temperature. (d) Insulating foam. (e) Helium bath at 4.2 K. (f) Cylinder of brass coated with superconducting lead. (g) Thermal shielding at low temperature. (h) NMR pick-up coils (other sets of coils for tilting the spins and varying the field gradients are not shown). (i) 'He sample. (j) Pumped liquid 4He bath at adjustable temperature. (P) Exit to vacuum pump.

temperature, and a 13 mm inner diameter spherical bulb where the NMR experiments are done at low temperatures. Gaseous diffusion inside the connecting tube transfers polarisation from the upper cylinder to the lower bulb. With this experimental arrangement, optical pumping can be done at a temperature where it is much more effective than at a few kelvin. The source of light for optical pumping is a colour centre laser operating at A = 1.08 p,m. The sealed cell contains a mixture of 3He and molecular hydrogen, which freezes on the cold parts of the inside walls of the cell. This provides a coating which, as explained above, strongly reduces the nuclear relaxation rate. The pressure of the 3He gas was p = 0.3 Torr corresponding to (eq. (2.4)) n , = 1OI6 atoms cmP3 and n, = 10" atoms at high and low temperatures respectively. Thermal shields are used to control the temperature gradient along the tube. Their purpose was to restrict the thermal gradient to within a small region, in order to reduce the area of

96

D.S. BET73 ET AL.

surfaces at intermediate temperatures (cold enough to relax 'He nuclear spins but not cold enough to be covered by a few protective layers of solid hydrogen). Nuclear polarisations of 30% or more in the lower bulb can be obtained with a laser power of about 100 mW. As shown in fig. 3.1, the measurement bulb was inside a small isolated inner cryostat of 5 cm inner diameter placed inside the main helium cryostat. In this way, the temperature of the bulb could conveniently be varied between 1.5 and 8 K. In some cases it was found useful or necessary to heat this cell to about 8 K for a few minutes in order to improve the quality of the cryogenic coating and allow polarisation transfer. The NMR system had two separate and orthogonal pairs of induction and detection coils. The latter coils had a diameter of 26 mm, and were wound with 40 turns each. The N M R frequency was about 12 kHz, which corresponded to a (vertical) magnetic field of about 0.4mT. The stability of this field was greatly improved (to about 10 I ' T) by using a vertical superconducting hollow cylinder to trap a fixed flux. This cylinder was a 1 mm thick layer of lead on the inside surface of a brass former of length 22 cm and a diameter of 9.2 cm. The shielding factor obtained in this way was about lo4 in good agreement with theory (Thomasson and Ginsberg (1976), Bardotti et al. (1964)). The field gradient was of the order of 2 x lo-' Tm-', depending on how the superconducting cylinder was cooled in the 4 x T vertical field produced by three horizontal coils of 1 m diameter. By adjusting the currents in three sets of compensation coils it was possible further to reduce gradients at the cell by a factor of about ten. The quantities measured in these experiments were the precession frequency and the transverse relaxation time T2 of a small transverse component of the nuclear polarisation at various positive and negative values of the longitudinal polarisation M. Reversals of M were produced by N M R n pulses, and 7r/20 pulses were used to sample M in an almost non-destructive way. The polarisations were deduced from the amplitudes of the responses to 7r/20 pulses by calibrating the detection system with the signal from a small circular coil. Accuracy was limited by the relatively high signal-to-noise and was in the region of 5-10% in M. For the particular cell geometry employed, the time constant for build-up of nuclear polarisation at 4.2 K in the lower cell was about 20 min. This time was effectively limited by a narrowed section of capillary in the connecting tube; a 5 cm length of 1 mm inner diameter included a 1 cm part whose inner diameter was only 0.3 mm just above the lower cell. Although the capillary increased somewhat the time required for M to rise to a steady limiting value, it also increased the confinement time in the lower cell and strongly reduced spurious frequency shifts due to spin diffusion within the connecting tube.

STRONGLY SPIN-POLARISED ’He GAS

91

After an acceptable value of M was obtained (20-30%), a series of NMR measurements was started. The procedure was to alternate TI10 pulses to measure T2 and the spin precession frequency, and T pulses to reverse M. Repeating this operation 5 or 10 times reduced M by a factor of about five, so that measurements of the spin frequency for a whole set of positive and negative values of M was obtained. T2was found to be practically independent of M. Typical values of T2 were in the range 0.5 to 5 s, and of A w / ~ T were in the range 1 to 10 mHz. The authors point out that great care had to be taken to avoid, or allow for, spurious frequency shifts arising in a variety of ways. Some types of shift were independent of M and these were eliminated by a suitable choice of experimental procedure. Other types, which were well understood, depended on M but fortunately not on 6Bo or T 2 .The procedure adopted at each temperature for dealing with these was to measure values of A 4 l M for various T2 (i.e., 6Bo) and then to extrapolate to zero T2 (i.e., infinite 6Bo). This is illustrated in fig. 3.2. The results for p are shown in fig. 3.3. The agreement with theory discussed above in section 1.9, and represented by the full curve in the diagram is very satisfactory. Later experiments were performed by Tastevin et al. (1985) using a so-called “direct” detection method, in which no static magnetic field gradient is necessary. The NMR coils were connected in opposition so that they created a pure rf field gradient with zero spatial average and excited (or detected) spin waves. Figure 3.4 shows the experimental arrangement in which two pairs of coils, each in opposition, are mounted around a

Fig. 3.2. Measurements reported by Nacher et al. (1984) at 2.5 K. The horizontal scale is the relaxation time T2of the transverse nuclear polarisation. 1/ Tz is proportional to the square of the field gradients over the sample. The vertical axis gives the phase shift A& (measured during a time T2)divided by the spin magnetisation M.

98

D.S. B E I T S ET AL.

Fig. 3.3. Measurements of the quality factor p of the spin waves as a function of temperature. The full line is theoretical (see section 1.9).The data are from Tastevin et al. (1985); see also Lhuillier and Leduc (1985).

spherical sample. The first pair was used for generating rf pulses creating an initial space-dependenttilt of the nuclear magnetisation, while the second picked up the induced signal. In ordinary NMR, the axes of the induction and pick-up coils are orthogonal to minimise their direct coupling; but in the arrangement shown in fig. 3.4 the cross-talk is minimised when the axes are at 45". The method offered some advantage in precision, particularly at temperatures above about 2 K. Below 2 K there were some unresolved discrepancies, as discussed by Tastevin et al. (1989, one possibility being the presence of a 'He monolayer. Spin waves have also been studied in similar work on spin-polarised hydrogen by Johnson et al. (1984) and on spin-polarised 3He dissolved in superfluid liquid 4He-II by Gully and Mullin (1984), and by Ishimoto et al. (1987).

3.2. EXPERIMENTS ON

THERMAL CONDUCTIVITY

This section is based primarily on the work described by Leduc et al. (1986), and Leduc et al. (1987). The theory of the thermal conductivity has been discussed in section 1.7. Figure 3.5 shows a general sketch of the experiment. The nuclear polarisation was obtained by optical pumping using the methods described above with a recently-developed LNA laser (Schearer et al. (1986)

STRONGLY SPIN-POLARISED 'He GAS

99

i ~ i Fig. 3.4. The induction coils (a) are connected in opposition and create an rf magnetic field with zero average value over the spherical sample, as shown by the magnetic field lines (dashed). The static field B, and the initial magnetisation are perpendicular to the plane of the figure. Just after a short resonant rf pulse, the direction of the transverse magnetisation varies over the sample as indicated by the large open arrows; it can be described by I = 1 diffusion modes of the transverse magnetisation. The detection coils (b), also in opposition and with their axes at 45" to the induction coils, receive a flux which is maximum at this initial time. For coils in opposition, the induction coupling in the absence of spins is minimised for a 45" angle, instead of 90" for coils in parallel. (From Tastevin et al. (1985).)

and Daniels et al. (1987)), capable of delivering 200-300 mW at the necessary wavelength of 1.083 p n . In the double-cell format, the two 'MPyrex glass cells were connected by a tube of 75 cm length and sealed after initial cleaning and filling with 'He (2.2Torr at room temperature) and H2 (0.62 Torr at room temperature). When the lower cell was cooled, the 'He pressure dropped to about 0.5 Torr, and the hydrogen formed a solid coating on the inner cold walls of the glass. The purpose of this procedure was to protect the polarised 'He against depolarisation as described in section 2.2. The same motive led to a cell design avoiding the use of any metal inside, and taking care that nearby wires were nonmagnetic and/or reasonably distant from the 'He. A cell optimised for the measurement of unpolarised 3He gas would have been quite different (see, for example, Betts and Marshall (1969)), and compromises had to be made. Figure 3.5 also shows an enlargement of the conductivity cell. It was a right circular cylinder whose end plates were 2 mm thick and 3 cm in diameter, separated by 1 cm, and with thick external silver layers intended to provide the best approximation to isothermal planes needed for sensible analysis of the data. There

I00

D.S. BETTS ET AL.

Fig. 3.5. Apparatus used by Leduc et al. (1987) to measure thermal conductivity. The 'He gas is contained in a double glass cell, in a homogeneous magnetic field B,. A weak gas discharge is maintained in the upper cell at ambient temperature where the atoms' nuclei are polarised by the method described in section 2.1. Spin diffusion downwards brings polarisation into the lower cell, and this is monitored by NMR. The lower cell is also shown on a larger scale, with carbon resistor thermometers R, and Rz.The upper and lower external surfaces are silvered and the lower is covered with a thin heating pad.

was a small orifice in the centre of the upper plate to allow passage of gases and polarisation. The glass side walls were about 0.5 mm thick; ideally they would have been much thinner than that in order to reduce the conduction of heat through the walls relative to that through the 'He to an insignificant magnitude, but in practice the primary constraint was mechanical. The temperature 0,of the upper plate was effectively thermally anchored to the pumped liquid 4He inner bath (see fig. 3.6) and could be varied at will

STRONGLY SPIN-POLARISED 'He GAS

101

Fig. 3.6. Sketched (not to scale) plan and side elevation of the conductivity cell showing the geometrical quantities S, S', I and I', together with the positions of the resistance thermometers measuring 8,and e,. Numerical values of these are used in eqs. (3.5) to (3.9).

in the range 1.3-4.2 K by adjusting the pumping speed. The temperature OPof the lower plate could be raised above 0, by means of a non-inductive winding of manganin resistive wire glued to the silvered surface with GE-7031 varnish, known for its favourable thermal conductivity (McTaggart and Slack (1969)). Heater powers were varied to achieve suitable temperature differences and were typically of the order of 1OOp.W for temperature differences in the region of 10-100 mK. 0, and 0,were measured using Allen-Bradley nominally 1 0 0 R resistors (originally investigated by Clement and Quinnell (1952)) which rise to about 1 kR, 2 kR and 20 kR at 4.2, 3 and 1.3 K respectively, offering suitable sensitivities in the range 0.2 R mK-' to 2 0 R mK-'. The pair of resistors were chosen to be wellmatched and were calibrated by reference to the vapour pressure of liquid 4He in the inner bath using the EFT-76 temperature scale (Durieux and Rusby (1983)). Resistances were measured with a bridge having sufficient sensitivity for use at the low powers necessary at low temperatures.

102

D.S. B E T S ET AL.

The upper plate of the conductivity cell was thermally anchored to the inner bath as described above. The rest of the cell was thermally insulated by the vacuum space which could be pumped to 10 6Torr, so that the heating power 0 supplied to the lower plate had to pass upwards through the glass and 'He to the inner bath. Raw data could be obtained in the form of 0, 8, and et at various T and M. A first approximation for the thermal conductivity K might be obtained then by using the basic relation

where S is the cross-sectional area of the contained 'He and 1 its length. This formula would be exact for an insulated solid cylinder with upper and lower surfaces taken as isothermal planes at 0, and OP, respectively. Problems of analysis always arise however with fluids because these have to be contained, and the containing walls themselves have the ability to conduct heat. In fact in the experiments being described here, a large fraction of the total heat conducted by the filled cell passed through the side walls. If those glass walls could be taken as having the perfect geometry of a cylindrical tube, and if in addition the two end plates could be taken as having infinite thermal conductivity, then a theorem due to Lazarus (1963) could be invoked to prove the exact relation with which to replace eq. (3.5). This would be (3.6)

where K , , ~is~ the thermal conductivity of the '"Pyrex walls, and S' the cross-sectional area of glass in the tube ( S as before is the area of the contained cylinder of 'He). Clearly if either K~~~ or S' were zero, then eq. (3.5) would be recovered. Otherwise K ~ ~has ~ toSbe' determined in a separate experiment in which the cell is evacuated. Unfortunately eq. (3.6) is exact only when the conditions mentioned above are exactly fulfilled. In the real cell the geometry is not perfect and the thermal conductivity of the end plates is finite. A further modification of eq. (3.6) can be made in a plausible though not exact way, by assuming not only that the upper and lower plane surfaces are isotherms (at 8,and @( respectively) but also that the inside upper and lower plane surfaces are isotherms (at, say, 0:and 0;respectively). It is then a straightforward exercise, bearing in mind Lazarus' theorem, to arrive at (3.7)

STRONGLY SPIN-POLARISED ’He GAS

103

where I’ is the combined thickness of the upper and lower end plates (S and 1 as before are the area and length of the contained cylinder of ’He, and S’ the cross-sectional area of glass in the tube). This modification would be strictly true for the geometry shown in fig. 3.6 and has the virtues that it reverts to the form of eq. (3.6) if l’=O with S’ finite and to the form of eq. (3.5) if I’ = 0 and S’ = 0. There remained a problem as to the correct ~ possibility ~ . would be to use published values procedure for finding K ~ One (e.g. Zeller and Pohl (1971)). Another would be to deduce it from measurements made on the evacuated cell, using eq. (3.7) with ~ 3 =“ 0;~ this however would be of dubious validity in the extreme case in which all the heat must flow through the tubular walls. Moreover, the geometry of the glass was unavoidably imperfect so that not only was K~~~ needed but also some way of choosing appropriately-averaged values of S’ and I’ (S and 1 may be taken as reasonably well known). The following procedure was adopted. Firstly the cell was evacuated so that the dominant thermal resistance was in the walls; the term containing I’ was therefore neglected in eq. (3.6), which then resolves to give the “measurable” result

Secondly the cell was filled with superfluid liquid 4He-II which effectively has almost infinite thermal conductivity relative to the glass so that the dominant thermal resistance was in the end plates; the term containing S‘ was therefore neglected in eq. (3.6), which then resolves to give the “measurable’’ result (3.9) These subsidiary “measurements” of S ‘ K p y , from (3.8) and K ~ I‘ from ~ ~ (3.9) / were then used in eq. (3.7) to obtain ~ 3 ” ~In. the case of unpolarised gas, the results could be compared with the accepted results of earlier authors (Betts and Marshall (1969) and Keller and Kemsk (1969)). The comparison reveals a small systematic discrepancy which is almost certainly due to the geometrical problems associated with the use of glass, and the resulting need to make approximations as described above. It was not a major concern in that the object of the experiment was to study diferences between ~ ( 0 ) and K ( M as ) discussed below. The magnetisation M of the gas was measured in the usual way with a set of crossed coils located in the evacuated space around the cell; free induction decay signals at 10 kHz could be detected after small tilting angle pulses in the induction coils. The separate calibration of the N M R signals

D.S. BETTS ET AL.

104

was obtained by comparison with optical measurements of circular polarisation of lines emitted by the discharge of the nuclear polarisation in the upper cell as described earlier (PavloviE and Laloe (1970)). The main measurements were concentrated on changes occurring in the thermal conductivity when M was deliberately destroyed. First the gas was polarised using the whole paraphernalia of laser and discharge. When an acceptable level of polarisation was achieved (typically 20-30%), the laser and discharge were switched off to avoid spurious heat inputs to the cell. Once this had been done, the time scale for action became restricted by the relaxation time (typically 30 min at 2.5 K, falling to 3 min at 1.8 K), and it was virtually impossible to take data below about 1.5 K. The temperature 9, was held as constant as possible manually (typically within 1 mK during the course of the measurement sequence) and Oe was continuously monitored under the influence of a steady heat flux (typically about 100 bW) and displayed on a chart recorder. The procedure was evolved so as to minimise dependence on measuring absolute values of conductivity. It involved destroying the polarisation M by a 7r/2 N M R pulse and observing the effect on as exemplified by fig. 3.7; SO( was typically 1 or 2 mK for 20-30% polarisation. The corresponding decrease SK in thermal conductivity was then finally obtained using eqs. (3.7) and (3.9).

3.5&0

1

depolarked

He 7112 NMR pulse

polarized ’He

3.535 t0

time

Fig. 3.7. A typical recorder trace of the resistance of RL (converted into temperature 6 , )as a function of time. The power 0 in this case was 107 p W and the temperature of the upper plate 8,was kept approximately constant at about 3.430 K. The initial nuclear polarisation, M = ? S % . was suddenly destroyed at time f,by an N M R pulse. The subsequent rise in temperature, about 2 m K in this case, provided raw data from which the conductivity change, K ( M )- ~ ( 0 )was . deduced. (From Leduc et al. (1987).)

STRONGLY SPIN-POLARISED 'He GAS

105

It was important to carry out three checks for consistency. Firstly, at a fixed temperature SK is predicted (see eq. (1.55)) to be closely proportional to M 2 for the conditions achievable in these experiments, and this was borne out in practice. Secondly, it was demonstrated that with a zero heating rate Q, SO, was also found to be zero, and this eliminates possible worries about spurious heat generated by the 1r/2 NMR pulse and/or the consequent depolarisation. Thirdly, there was the important and interesting matter of whether convection could safely be assumed to be absent, bearing in mind the fact that the fluid was being heated from below. Experimentally its absence was demonstrated by showing that the temperature difference across the cell was proportional to the heat current Q; non-proportionality would have been a clear signal that convection was present. We shall consider this matter in more detail in the following section, partly because of its intrinsic importance to the correct analysis of thermal conductivity and partly because a deliberate study of the onset of convection would be a feasible and interesting extension to the work described here. The experimental results obtained by Leduc et al. (1987) and in later work by the same group, yet to be published, are shown in fig. 3.8. Generally

Fig. 3.8. The thermal conductivity data of Leduc et al. (1987) in the convenient form ( K ( M ) M 2versus temperature, appropriate for low M. The circles and triangles represent data taken in November/Decembcr 1986 and June/July 1987, respectively, with the dotted area intended simply as a guide for the eye. For comparison, the predictions of Lhuillier (1983) are also shown with a dashed line, corresponding to the Lennard-Jones potential, and a dashed and dotted line to the Aziz potential. K(O))/

D.S. B E T S ET AL.

106

speaking, there is good qualitative agreement between theory and experiment. There are however significant differences, in particular the position of the maximum, which may point to the need to extend the theory to higher order. The most recent experimental work, not yet published, has confirmed that K ( M, T ) becomes independent of M at about 1.5 K as shown in fig. 3.8. It is clearly of interest to extend the measurements as far down in temperature as possible; the limitation is that the relaxation time becomes uncomfortably short below 1.5 K. 3.3. CONVECTION The thermal expansion coefficient a of 'He gas is positive, whether or not it is polarised, and may be assumed to be closely equal to the perfect gas value of 1/ T deg-'. Nevertheless heating from below, though it is a necessary condition for convective motion, is not a sufficient condition. For a given cell, filled with a given fluid under specified circumstances, the applied temperature difference has to exceed a certain well-defined minimum value before convection becomes the favoured mode. There is thus an onset condition, first studied by Benard and Rayleigh (see Saltzman (1962) for a collection of historic papers and references). There is a large literature on Rayleigh-Binard convection but suitable key references should include Saltzman (1962), Threlfall (1975), Ahlers and Behringer (1978), Busse (1978), Libchaber and Maurer (1978), Maurer and Libchaber (1979), and Drazin and Reid (1981). The experimental work of Threlfall (1975) on helium is particularly relevant to the present discussion. Data on the onset of convection are best shown in a plot of Nusselt number (Nu) versus Rayleigh number (Ra), these two dimensionless parameters being defined as follows. The Nusselt number is the ratio of the actual heat flow to that which would occur if conduction were the only mechanism. Thus, for a cylinder of cross section S,with axial heat flow, (3.10) where AT/Ax is the temperature gradient. The Rayleigh number is a more complicated combination of fluid properties including the isobaric thermal expansion coefficient (a),the density ( p ) , the isobaric specific heat (C,), the thermal conductivity ( K ) , and the viscosity (v),together with the external parameters g (the acceleration due to gravity), AT (the temperature difference between the upper and lower plates, and I (the distance between the plates). The dimensionless Rayleigh number is given by Ra =

'

gap C,,A 77' KT

(3.11)

STRONGLY SPIN-POLARISED 'He GAS

107

For sufficiently low values of AT, Nu = 1 independently of Ra, corresponding to the transmission of heat entirely by conduction. If AT is raised, there is observed to be a critical value of Ra, denoted by Ra,, beyond which Nu rises above unity, that is, the fluid begins to transmit heat by convection. The value of Ra, has been calculated for simple geometries and boundary conditions and is generally in the region of 1700. Threlfall (1975), in a systematic experimental study of the effect in gaseous helium in a cylindrical cell having a diameter of 5 cm and length 2 cm, found Ra, = 1630 as shown in fig. 3.9. In principle, therefore, it is possible to use eq. (3.10) with low AT so that Nu = 1 to find K, and then, by raising AT until the onset of convection is observed, to use eq. (3.11) with an appropriate value of Ra, to deduce 7.The idea is attractive because it allows a measurement of both K and 7 in a single experimental run. In practice it should also be possible in polarised 'He gas although the required numerical parameters are not ideally favourable. The main difficulty is that the optimum density, from the point of view of producing high polarisations, gives lowish values of

1630

Ra Fig. 3.9. Experimental data for gaseous 4He obtained by Threlfall (1975). The definitions of the Nusselt number Nu and the Rayleigh number Ra are given in eqs. (3.10) and (3.11), respectively. The cell geometry (diameter 5 cm; length 2 cm) was similar to, but not identical to, that of Leduc et al. (1987). The transition to convection is clearly seen, with a critical Rayleigh number of 1630.

D.S. BEITS ET AL.

1 OR

Ra for sensible values of AT. However, as an example of what might be possible, if the pressure and temperature of the unpolarised 3He in a cylindrical cell of 5cm diameter and 2cm length were 2Torr and 2 K respectively, then raising AT from 5 to 100 mK could be expected to sweep Ra/Ra, from about 0.19 through 1.0 (onset of convection) to about 3.8. Table 6 shows some other estimated possibilities at different temperatures, again for unpoiarised 'He at 2Torr in the same cell. The two columns correspond to the fairly arbitrary choices of AT = 5 mK (a suitable lower limit if measurement sensitivity is to be maintained) and A T = 50T mK ( T in K), a suitable upper limit if A T / T is not to amount to more than 5 % . Bouchaud and Lhuillier ( 1986) have given theoretical consideration to the Rayieigh-BCnard instability discussed above, and have also pointed out that there is a magnetic equivalent in which magnetisation is substituted for temperature and the Stern-Gerlach force for the gravity force. Such an equivalence arises from the formal similarity of the equations describing the two situations. Moreover the two effects can be superposed so that magnetic effects could for example be set to act either with or against the thermal effects. The equations governing this mode coupling were originally given by Lhuillier and Laloe (1982b). For 'He they take the forms J T = - K ( M , T ) V T + L l , ( M ,T ) 7 T M ,

(3.12)

+ LZl(M , T ) V In T,

(3.13)

JM

= -DO( T)VM

where JTand J M are the heat and polarisation current densities. We have already referred in some detail to the quantities K and Do, and Lhuillier (1983) gives plots of L,? and L,, which, like K and 7, show strong M dependence in the region of 2 K. These equations suggest some fascinating

Table 6

T

1.4

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3 .O

Ral Ra, for AT = 5 m K

0.75 0.45 0.29 0.19 0.13 0.10 0.070 0.053

0.040

Ra/ Ra, for AT = 5OTmK ( T i n K)

10.6 1.2 5.2 3.8 2.9 2.3 1.8 1.5 1.2

STRONGLY SPIN-POLARISED 3He GAS

109

experimental possibilities for investigating hydrodynamic instabilities in spin-polarised 'He gas, but these have yet to be attempted.

3.4.

L I Q U I D - G A S EQUILIBRIUM

It would be straying too far from the purpose of this chapter to discuss the properties of spin-polarised 'He in liquid form, interesting though these are. It is necessary however to remark that the production of polarised liquid is not altogether easy since the application even of very large magnetic fields only causes small polarisation. One method which has received attention makes use of effects at the solid-liquid transition, as originally proposed by Castaing and Nozikres (1979). An alternative, recently demonstrated by Tastevin et a]. (1988) and relevant to the topic of this chapter is based on the rapid liquefaction of spin-polarised 3He gas. The apparatus used is shown in fig. 3.10. The idea is first to polarise 'He gas using the methods described in section 2.1 and then to reduce the temperature quickly so that the initial gas pressure exceeds the vapour pressure and condensation occurs. Tastevin et al. (1988) have demonstrated that this method can lead to nuclear polarisations in excess of 40% in the condensed liquid at about 400 mK where the vapour pressure is very low. The relaxation time of the polarisation in the liquid is a few minutes. There are expected to be effects

- -

-

\\ II II

r r I I I I -

-

Fig. 3.10. The experimental arrangement used by Tastevin et al. (1988) for producing polarised liquid 'He. The container of the 'He sample is made of three different cells, connected by a long tube. The upper cell A is at ambient temperature, where the nuclear polarisation is produced by laser optical pumping. The intermediate cell B is at about 4 K and acts as a polarised gas reservoir. The lower cell C is immersed in the liquid 'He bath whose temperature can be varied by pumping; it is surrounded by the NMR pick-up coils.

110

D.S. BETTS ET AL.

02

04

06

08

1.0

M Fig. 3.1 1. The predicted phase equilibrium diagram for partially polarised ’He at T = 0.3 K. with a liquid density assumed to be independent of M,and with mechanical energies per atom assumed to be E = -2.5 K and E, = -2.2 K for unpolarised and fully polarised liquid respectively. With these parameters the liquid always has a smaller polarisation than the vapour with which it is in equilibrium but at T = 1.8 K, for example, this is reversed. Such diagrams depend very sensitively on 7, E and E , . (From Stringari et al. (1987).)

of polarisation on the phase diagram. These were first discussed by Lhuillier and Laloe (1979), and have recently been considered phenomenologically in some detail by Stringari et al. (1987). They have drawn some interesting conclusions about possible behaviours, although these depend strongly on the validity of the phenomenological model for free energy and on the currently unknown difference between the energy per atom in the unpolarised and fully polarised liquid states. As an example, fig. 3.1 1 shows the predicted interrelation between vapour pressure and polarisation at T = 0 . 3 K when particular assumptions are made: the liquid phase has a smaller polarisation than the vapour phase, a situation which can be reversed by raising the temperature. This example is relatively simple but more complicated “metamagnetic” effects might arise in which the liquid phase separates, with the two liquid phases having different polarisations. Such effects can also be represented in phase diagrams, of which Stringari et al. (1987) give a number of examples. Clearly, experimental data are needed.

STRONGLY SPIN-POLARISED ’He GAS

111

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Busse, F.H., 1978, Rep. Prog. Phys. 41, 1929. Byrne, J., and P.S. Farago, 1971, J. Phys. B 4, 954. Cameron, J.A., and G.M. Seidel, 1985, J. Chem. Phys. 83, 3621. Castaing, B., and P. Nozitres, 1979, J. Phys. (France) 40,257. Chapman, R., 1975. Phys. Rev. A 12, 2333. Chapman, S., and T.G.Cowling, 1970, The Mathematical Theory of Non-Uniform Gases, 3rd Ed. (Cambridge). Clement, J.R., and E.H.Quinnell, 1952, Rev. Sci. Instrum. 23, 213. Cohen-Tannoudji, C., and A. Kastler, 1966, in: Progress in Optics, VoI. V, ed. E. Wolf ( North-Holland, Amsterdam). Colegrove, F.D., L.D. Schearer and G.K.Walters, 1963, Phys. Rev. 132, 2561. Corruccini, L.R., D.D. Osheroff, D.M.Lee and R.C. Richardson, 1971, Phys. Rev. Lett. 27,650. Corruccini, L.R., D.D. Osheroff, D.M. Lee and R.C. Richardson, 1972, J. Low Temp. Phys. 8, 229.

Daniels, J.M., and R.S. Timsit, 1971. Can. J. Phys. 49, 525. Daniels, J.M., L.D. Shearer, M. Leduc and P.-J. Nacher, 1987, J. Opt. SOC.Am. B 4, 1133. De Boer, J., and E.G.D. Cohen, 1951, Physica 17, 993. De Boer, J., and A. Michels, 1939, Physica 8, 409. De Boer, J., J. van Kranendonk and K.Compaan, 1949, Phys. Rev. 76.998, 1728. De Boer, J., J. van Kranendonk and K. Compaan, 1950, Physica 16, 545. Drazin, P.G., and W.H. Reid, 1981, Hydrodynamic Stability (Cambridge). Dupont-Roc, J., M. Leduc and F. Laloe, 1971, Phys. Rev. Lett. 27, 467. Durieux, M., and R.L. Rusby. 1983, Metrologia 19, 67.

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Edwards, D.O.. and W.F. Saam, 1978, The free surface of liquid helium, in: Progress in Low Temperature Physics, Volume Vlla, ed. D.F. Brewer (North-Holland, Amsterdam), p. 283.

Emery, V.J., 1964, Phys. Rev. A 133, 661. Grad, H., 1958, Handb. Phys. 12, 205. Grimsrud. D.T., and J.H. Werntz Jr, 1967, Phys. Rev. 157, 181. Gully, W.J., and W.J. Mullin, 1984. Phys. Rev. Lett. 52. 1810. Halperin, W.P., F.B. Rasmussen, C.N. Archie and R.C. Richardson, 1978, J. Low Temp. Phys. 31, 617. Happer, W., 1972, Rev. Mod. Phys. 44, 169. Himbert, M., V. Lefevre-Seguin, P.-J. Nacher, J. Dupont-Roc, M. Leduc and F. Laloe, 1983, J. Phys. Lett. (France) 44, L523. Hirschfelder, J.O., C.F. Curtiss and R.B. Bird, 1954, Molecular Theory of Gases and Liquids (Wiley, New York) [reprinted 19641. Ishimoto. H., H. Fukuyama, N. Nishida, Y. Miura, Y. Takano, T. Fukuda, T. Tazaki and S. Ogawa, 1987, Phys. Rev. Lett. 59, 904. Johnson. B.R., J.S. Denker, N. Bigelow, L.P. Levy, J.H. Freed and D.M. Lee, 1984, Phys. Rev. Lett. 52, 1508. Keller, W.E., 1955, Phys. Rev. 98, 1571. Keller, W.E., 1957, Phys. Rev. IM,41. Keller, W.E., 1969, in: Helium-3 and Helium-4 (Plenum Press, New York) ch. 3, p. 61. Keller. W.E., and J.F. Kerrisk, 1969, Phys. Rev. 177. 341. Kilpatrick, J.E.. W.E. Keller, E.F. Hammel and N. Metropolis, 1954, Phys. Rev. 94, 1103. Laloe, F., ed., 1980, Proc. Conf. on Spin Polarised Quantum Systems, 21-26 April, 1980, Aussois, France, J. Phys. (France) Colloq. 0 ,suppl. 7, 1980. Laloe, F., M. Leduc, P.-J. Nacher, L.N. Novikov and G. Tastevin, 1985, Usp. Fiz. Nauk 147, 433 [Sov. Phys.-Usp. 28, 9411. Lazarus, R.B.. 1963, Rev. Sci. Instrum. 34, 1218. Leduc, M.. 1972, Thkse d'Etat (Ecole Normale Supirieure, University of Paris, France). Leduc, M.,G. Trinec and F. Laloe. 1980, J . Phys. (France) 41, C7-75. Leduc, M., P.-J. Nacher, S.B. Crampton and F. Laloe, 1983, in: Proc. AIP Conf. on Quantum Fluids and Solids,, 11-15 April. 1983, Sanibel Island, FL, U.S.A., eds E.D. Adams and G.G. lhas (AIP, New York) p. 179. Leduc, M., S.B. Crampton, P.-J. Nacher and F. Laloe. 1984, Nucl. Sci. Appl. 2, 1. Leduc, M., F. Laloe, P.-J. Nacher, G. Tastevin, J.M. Daniels and D.S. Betts, 1986, in: Proc. 14th Int. ConP. on Quantum Electronics, 9-13 June, 1986, San Francisco, U.S.A. (American Physical Society) p. 138. Leduc, M., P . J . Nacher, D.S. Betts. J.M. Daniels, G. Tastevin and F. Laloe, 1987, Europhys. Lett. 4. 59; see also Proc. 18th Int. Conf. on Low Temperature Physics, 20-26 August, 1987, Kyoto, Japan, Jpn. J. Appl. Phys. 26, suppl. 26-3, 213. Lefevre-Seguin, V., P.-J. Nacher and F. Laloe, 1982, J. Phys. (France) 43, 737. Lefkvre-Seguin, V., P.-J. Nacher, J. Brossel, W.N. Hardy and F. Laloe, 1985, J. Phys. (France) 46, 1145. Leggett, A.J.. 1970, J. Phys. C 3, 448. Leggett. A.J., and M.J. Rice, 1968, Phys. Rev. Lett. 20, 586. Lhuillier, C., 1983, Transport properties in a spin polarized gas, 111. J. Phys. (France) 44, 1. Lhuillier, C., and F. Laloe. 1979, J. Phys. (France) 40. 239. Lhuillier. C., and F. Laloe, 1980, in: Proc. of C N R S Colloq. on Spin Polarised Quantum Systems, 21-26 April. 1980, Aussois, France, ed. F. Laloe, J. Phys. {France) Colloq. C-7, suppl. 7, C7-51.

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Lhuillier, C., and F. Laloe, 1982a, Transport properties in a spin polarized gas, 1. J. Phys. (France) 43, 197. Lhuillier. C., and F. Laloe, 1982b, Transport properties in a spin polarized gas, 11. J. Phys. (France) 43, 225. Lhuillier, C., and M. Leduc, 1985, Ann. Phys. (France) 10, 859. Libchaber, A., and J. Maurer. 1978, J. Phys. Lett. 39, L369. Luszczynski, K., R.E. Norberg and J.E. Opfer, 1962, Phys. Rev. 128, 186. Mandl, F., 1957, Quantum Mechanics, 2nd Ed. (Butterworths, London) sect. 38. Massey, H.S.W., and R.A. Buckingham, 1938, Proc. R SOC.London Ser. A 168, 378. Massey, H.S.W., and R.A. Buckingham, 1939, Proc. R. SOC.London Ser. A 169, 205. Masuhara, N., D. Candela, D.O. Edwards, R.F. Hoyt, H.N. Scholz, D.S. Sherrill and R. Combescot, 1984, Phys. Rev. Lett. 53, 1168. Matacotta, F.C., G.T. McConville, P.P.M. Steur and M. Durieux, 1987, Metrologia 24, 61; briefer presentation, in: Roc. 18th Int. Conf. on Low Temperature Physics, 20-26 August, 1987, Kyoto, Japan, ed. Y. Nagaoka, Jpn. J. Appl. Phys. 26, suppl. 26-3, 1679. Maurer, J., and A. Libchaber, 1979, J. Phys. Lett. (France) 40, L419. McAdams, H.H., and G.K. Walters, 1967. Phys. Rev. Lett. 18, 436. McTaggart, J.H., and G.A. Slack, 1969. Cryogenics 9, 384. Meyerovich, A.E., 1978, Phys. Lett. A 69, 279. Meyerovich, A.E., 1983, J. Low Temp. Phys. 53, 487. Meyerovich, A.E., 1987, Spin-polarised 3He-4He solutions, in: Progress in Low Temperature Physics, Vol. XI, ed. D.F. Brewer (North-Holland, Amsterdam) p. 1. Meyerovich, A.E., 1989, Spin-polarised phases of 'He, in: Anomalous Phases of 'He, eds W.P. Halperin and L.P. Pitaevskii ( North-Holland, Amsterdam) to be published. Mollenauer, L.F., 1980, Opt. Lett. 5. 188. Monchick, L., E.A. Mason, R.J. Munn and F.J. Smith, 1965, Phys. Rev. 139, A1076. Nacher, P.-J., and M. Leduc, 1985, J. Phys. (France) 46,2057. Nacher, P.-J., M. Leduc, G. Trenec and F. Laloe, 1982, J. Phys. Lett. (France) 43, L525. Nacher, P.-J., G. Tastevin, M. Leduc, S.B. Crampton and F. Laloe, 1984, J. Phys. Lett. (France) 45, L441. Owers-Bradley, J., H. Chocholacs, R.M. Mueller, C. Buchal, M. Kubota and F. Pobell, 1983, Phys. Rev. Lett. 51, 2120. PavloviE, M., and F. Laloe, 1970. J. Phys. (France) 31, 173. Pinard, M., and F. Laloe, 1980, J. Phys. (France) 41, 799. Pinard, M., and J. van der Linde, 1974, Can. J. Phys. 52, 1615. Roberts, T.R., R.H. Sherman and S.G. Sydoriak, 1964, J. Res. Natl. Bur. Stand. Sect. A 68,567. Saltzman, B., ed., 1962, Selected Papers on the Theory of Thermal Convection (Dover, New York). Schearer, L.D., and G.K. Walters, 1965, Phys. Rev. A 139, 1398. Schearer, L.D., M. Leduc, D. Vivien, A.-M. Lejus and J. Thiry, 1986, IEEE J. Quantum Electron. QE-22, 713. Schiff, L.I., 1955, in: Quantum Mechanics (McGraw-Hill, New York) ch. 5. Silin, V.P., 1957, Zh. Eksp. & Teor. Fiz. 33, 495, 1227 [1958, Sov. Phys.-JETP6, 387, 9451. Silvera, I.F., and J.T.M. Walraven, 1986, Spin-polarised atomic hydrogen, in: Progress in Low Temperature Physics, Vol. X, ed D.F. Brewer (North-Holland, Amsterdam) ch. 3, p. 139. Slichter, C.P., 1980, The Principles of Magnetic Resonance, 2nd Ed. (Springer, Berlin). Stringari, S., M. Barranco, A. Polls, P.-J. Nacher and F. Laloe, 1987, J. Phys. (France) 48, 1337. Tastevin. G., P.-J. Nacher, M. Leduc and F. Laloe, 1985, J. Phys. Lett. (France) 46,L249. Tastevin, G., P.-J. Nacher, L. Wiesenfeld, M. Leduc and F. Lal0.5, 1988, J. Phys. (France) 49, 1

I14

D.S. BETTS ET AL.

Thomasson, J.W., and D.M. Ginsberg, 1976, Rev. Sci. Instrum. 47, 387. Threlfall, D.C., 1975, J . Fluid Mech. 67, 17. Timsit, R.S., and J.M. Daniels, 1971, Can. J . Phys. 49. 545. Timsit, R.S., J.M. Daniels and A.D. May, 1971, Can. J . Phys. 49, 560. Trenec. G., P.-J. Nacher and M. Leduc, 1982, Opt. Comrnun. 43, 37. Waldmann, L., 1958, Z. Naturforsch. a 13. 609. Zeller. R.C., and R.O. Pohl. 1971, Phys. Rev. B 4, 2029.

CHAPTER 3

KAPITZA THERMAL BOUNDARY RESISTANCE AND INTERACTIONS OF HELIUM QUASIPARTICLES WITH SURFACES BY

TSUNEYOSHI NAKAYAMA Department of Applied Physics, Hokkaido University, Sapporo 060, Japan

Progress in Low Temperature Physics, Volume XII Edited by D.E Brewer @ Elsevier Science hblishers B. V., 1989 115

Contents I . Introduction . . . . . . . ........................................... 2. Kapitza thermal bou nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition of the Kapitza resistance R, ................................. 2.2. General expression for the Kapitza resistance R, . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Anomalous behaviour of the observed Kapitza resistances . . . . . . . . . . . . . . . . . 2.3.1. The liquid 'He-sintered powder interface at mK temperatures . . . . . . . . 2.3.2. The liquid He-bulk solid interface above about 1 K . . . . . . . . . . . . . . . . . 3 . Fermi liquid theory of the Kapitza resistance ................................. 3.1. Heat transfer due to zero-sound excitations . . . 3.1.1. Emission of zero-sound from a small pa 3.1.2. Heat flux from a spherical particle at temperature 7 3.1.2.1. Energy current into liquid 'He ............................ 3.1.2.2. Kapitza conductance h, due to zero-sound excitations 1.2. Energy transfer due to inelastic scattering of single quasiparticles at th

....................... tween sintered powde ................... 4 . I . Heat exchanger using submicrometer metal particles ...................... 4.2. Soft phonon-modes in sintered powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer due to the effect of soft phonon-modes ...................... 5 . The magnetic channel of heat transfer between sintered powder and liquid 'He . . 5.1. Surface characteristics of submicron metal particles and surface magnetic impurities . . . . . . ............... ................ 5.2. Heat transfer due to magnetic coupling at mK temperatures . . . . . . . . . . . . . . . 6 . Thermal boundary resistance between liquid 'He-4He mixtures and sintered powder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Acoustic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Magnetic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Anomalous Kapitza resistance between liquid He and a bulk solid above about 1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Kapitza conductance h , and phonon transmission coefficient across the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. High-frequency phonon scattering at the liquid He-solid interface . . . . . . . . . . 7.2.1. Cause of diffuse scattering at the surface without liquid He . . . . . . . . . . 7.2.2. Specular versus diffuse scattering of bulk phonons . . . . . . . . . . . . . . . . . . 7.2.3. Diffuse signals in the time-of-flight reflection signals . . . . . . . . . . . . . . . . . 7.2.4. Reduction of the diffuse signal at the solid surface in contact with liquid He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . Summary . . . . . . . .................................... Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . ................................................

I16

117 118 118 120 124 124 126 128 128 128 130 130 132 135 140 140 141 144 149 149 152 158 159 162 165 165 168 171 175 176 179 186 187 189

1. Introduction

Heat exchange between dissimilar media is an important subject for experiments at low temperatures. When heat is transferred from a solid into liquid He or vice versa, a temperature discontinuity appears at the interface. Although this phenomenon was first discovered almost half a century ago, by Kapitza (1941) during his experimental investigation of the heat transport in superfluid 4He, it is only in recent years that low temperature physicists have studied this problem in detail, in connection with condensed matter physics. Indeed, various subfields of condensed matter physics such as surfaces, magnetism, disordered systems, etc., are involved in this. Initially, there was a suggestion (Gorter et al. 1951) that the phenomenon was related to the superfluid phase of liquid 4He (He 11). However, a similar temperature discontinuity also appears at the interface of normal liquid 'He and a solid (Lee and Fairbank 1959), and it has been established that the thermal boundary resistance is not peculiar to He 11. In fact, thermal boundary resistances have by now been reported for many interfaces between dissimilar media. These include the interfaces between nonmagnetic or magnetic solids and quantum media such as solid 'He or 4He, 3He-4He mixtures, superfluid 3He. In principle, a thermal boundary resistance or so-called Kapitza resistance occurs between any dissimilar media at low temperatures, even for the interface between two quantum media like the solid 4He-liquid 4He or liquid 4He-hydrogen gas interface (Huber and Mans 1982, Maris and Huber 1982, Peach et al. 1982, Castaing and Papoular 1983, Graf et al. 1984, 1985, Kagan et al. 1984, Goldman 1986). Khalatnikov (1952, 1965) presented a theory for the liquid 4He-solid interface based on the idea that heat is carried by phonons, but these phonons have a high probability of reflection at the interface due to the large acoustic mismatch between a solid and liquid 4He. This so-called acoustic mismatch (AM) theory predicts that the Kapitza resistance R K , whose definition will be given in section 2.1, is proportional to the inverse third power of the temperature ( T - 3 ) . However, the Kapitza resistances observed at the liquid 4He (or 'He)-solid interfaces are, in general, some orders of magnitudes smaller than the value of the AM theory in addition to disagreement with the predicted T - 3 temperature dependence. This anomalous behaviour, often called the anomalous Kapitza resistance, is well established experimentally in two cases: the liquid 'He-sintered powder interfaces at millikelvin (mK) temperatures and the liquid 4He (or 'He)-bulk 117

118

T. NAKAYAMA

solid interfaces above about 1 K. These have been long-standing unsolved problems in low temperature physics. To understand the underlying mechanisms for creating the anomalous heat conductance is of great importance for low temperature physicists. As far as the former problem at mK temperatures is concerned, the investigation seems to be entering a new stage in the past few years due to recent experimental and theoretical efforts, in comparison with the time when the reviews on this problem were written by Harrison (1979) and Nakayama ( 1984a). The present article treats mainly the investigations on the anomalous Kapitza thermal boundary resistances between liquid 'He and metal particles observed at mK temperatures. The problem of the Kapitza resistance above about 1 K will be surveyed in section 7. Section 2 begins with the definition of the Kapitza resistance R K . The AM theory is explained in a different manner from the conventional procedure based on phonon transmission and reflection at the solid-liquid He interface. This enables us to treat in a simple way the Kapitza resistance between a Jinite-size system such as small particles and liquid 'He, which is the main subject in section 4. In section 2, the experimental data for R K both at mK temperatures and above about 1 K are compared with the predictions of the A M theory. The Fermi liquid theory for the Kapitza resistance R K is explained in section 3, where the emphasis is laid on the theoretical treatment of the heat transfer mechanisms due to both zero-sound excitations and single quasiparticle excitations. Section 4 describes the effect of the elastic softening of sintered powders on the anomalous behaviour of the Kapitza resistance observed at the liquid 'He-sintered powder interfaces. Section 5 deals with another heat transfer mechanism originating from the magnetic coupling in connection with the surface characteristics of sintered metal particles. Section 6 discusses the problem of the Kapitza resistance between 'He-4He mixture and sintered powder. This is of great importance with respect to the possible discovery of the superfluid transition in the dissolved 'He in a dilute 'He-4He solution. The problem of the anomalous Kapitza resistance above about 1 K is described briefly in section 7 with emphasis on recent investigations using the new techniques of high-frequency phonon generation and detection.

2. Kapitza thermal boundary resistance 2.1. DEFINITION OF THE

KAPlTZA RESISTANCE R K

Consider the heat transfer from liquid He at temperature T, into a solid at T2 (fig. 2.1). The net heat flux across the interface Is given by the difference

KAPITZA THERMAL BOUNDARY RESISTANCE

Temp.

T2

119

T1

Fig. 2.1. Temperature discontinuity at the liquid He-solid interface. AO(T) is the net heat flux from liquid He at TI = Tz+AT into solid at T z .

of the heat flux per unit area AQ = OL( 7-11 - OS( T2)

(2.1)

I

os(

where Q,( Tl) is the heat flux from liquid He into the solid, and T2)the flux from the solid into the liquid, respectively. The net heat flux has to vanish when the temperatures are equal ( Tl = T2),namely QL( TI)= & ( T I ) . If the difference in temperature ( A T = Tl - T2)is small, eq. (2.1) becomes

where the heat conductance across the interface is defined as

The Kapitza resistance RK is defined as the inverse of the Kapitza conductance RK = l / h K . RK is usually expressed in units of m2K/W. In the above definition of RK, a temperature discontinuity ( A T ) is presumed at the interface. The validity of this assumption can be understood as follows: Provided that phonons are incident from liquid He into the solid, the critical angle of incidence, at which phonons transmit into the solid, becomes 0; = sin-'(cJoS) according to Snell's law, where cL and us are the velocities of phonons in liquid He and the solid. Hereafter, the subscripts 'S' and 'L' indicate the solid and the liquid, respectively. Since cL= 2.4 x lo4 cm/s for He I1 below 1 K and us is typically 5 x loscm/s, one 0: are totally reflected. The finds that 0',-3". Phonons incident at phonon transmission coefficient tLS from the liquid into the solid is determined from the ratio between the incident (A,) and reflected ( A , ) amplitude of a phonon as t,=

~-IA~/A,,~~.

(2.4)

T. N A K A Y A M A

120

The transmission coefficient tLs can be obtained easily in the case of normal incidence at O L = O , in terms of the acoustic impedance Z L = p L c Land Z s = psus, where p, and pLare the mass densities of the solid and the liquid, respectively. The result becomes

where the condition Z, > Z L is taken into account. The transmission coefficient tLs is equal to rsL in the case of the normal incidence, where tsL means the coefficient from the solid into the liquid. Since ps -- 10 g/cm3 and pL= 0.142 g/cm3, we have a very small transmission with tLs= 2 x Thus the large acoustic mismatch, in addition to the small critical angle (Of = 3”), severely limits the effective heat exchange between these two media. This is the reason why the temperature discontinuity appears at the interface. 2.2. GENERAL EXPRESSION

FOR T H E

KAPITZARESISTANCE RK

In deriving the expression for the Kapitza resistance RK in the acoustic mismatch (AM) theory (Khalatnikov 1965), a part of the lengthy calculation is taken up with estimating the angular integral with respect to the transmission coefficient fLS(0). Here we try to formulate R K from a different point of view (Nakayama and Nishiguchi 1981) from that of the AM theory (Khalatnikov 1965). The merit of this approach lies in being able to obtain the Kapitza resistance R K between finite-size system like a submicron metal particle and liquid He without much difficulty. The conventional derivation of RK will be dealt with in section 7, where the problem of the Kapitza resistance above about 1 K will be described on the basis of this formulation. Emission of sound waues from a small particle Consider the case in which a small particle with fixed center of mass is vibrating elastically in liquid He 11. This vibration causes a periodic compression and rarefaction of the density of liquid 4He near it. These propagate in the form of sound waves. The mean rate of emitted energy from the small particle is given in terms of the square of the fluid velocity u ( r ) (Landau and Lifshitz 1982), E = ipLcL Iu(r)I’ d S ,

(2.6)

where pL and cL are the mass density and the sound velocity of liquid 4He, respectively. The factor appears as a result of averaging eq. (2.6) over the period of the vibrations. The integral is performed over a closed surface surrounding the small particle. The energy carried away by sound (phonons)

KAPITZA THERMAL BOUNDARY RESISTANCE

121

is supplied by the kinetic energy of surface motion of the small particle. The velocity potential in liquid He is taken as the scalar function 4 ( r ) defined by u ( r ) =grad # ( r ) .

(2.7)

Sufficiently far from the spherical particle, having its center at the origin, the velocity potential may be written simply in the form

R being the radius of the particle. The functionf,(O, 4) is determined from the boundary conditions at the surface of the particle. By taking the gradient of eq. (2.8) in spherical coordinates, one obtains Dqb)=

exp(iqr - iqR) r2

(2.9)

The expression for the squared velocity at a distance r in terms of the surface velocity u ( R ) of the particle becomes (2.10)

Substituting eq. (2.10) into eq. (2.6) and taking the closed surface of the integration to be a sphere of radius r, the energy flux divided by 4rR2 is found to be (2.11)

where dl2 =sin 8 d e d4. At the surface of the small particle ( r = R), the fluid velocity u ( R , e , 4 ) must be equal to that of the surface motion of the small particle. Provided that the surface displacement of the small particle is defined as u ( R ) ,its time derivative is simply the velocity of the surface motion as uq(R)= Uq(R).

(2.12)

In the limit R + 00, eq. (2.1 1 ) reduces to the rate of emission of sound waves per unit area from a plane surface. As a result, the formula for the energy flux between a bulk solid and He I1 is obtained by replacing the radial component of the displacement u, by that of the z-component of Cartesian coordinates (see fig. 2.2). For deriving the formula of heat transfer, it is convenient to write down eq. (2.11) in terms of the quantized displacement (Nakayama and Nishiguchi 1981). If u,(r) represents the eigenmode

T. N A K A Y A M A

122

/Fig. 2.2. Definition of Cartesian coordinates in the limiting case of R --fa.

of elastic waves belonging to the eigenfrequency w J , the elastic waves can be quantized by replacing the amplitudes in u,(r) by the boson operators a, and a:, where J stands for a set of possible quantum numbers specifying the eigenmodes. Consider an isotropic solid with a plane surface in contact with He 11. Since He I1 has a small mass density ( pL= 0.142 g ~ r n - ~compared ) with that of a solid, the boundary conditions for the displacement field may be taken as those for a stress-free surface. Thus the displacement-field operator of a longitudinal phonon is written as u ( r )=

c eq(")Ps",

112

(a, + a:,) exp(iqll* x) cos q z z ,

(2.13)

Q

where e, is the polarization vector for a longitudinal phonon. Putting v i ( z = 0)= U i ( z = 0) into eq. (2.1 1) and taking the thermal average of the squared velocity at a finite temperature, the heat flux density & ( T ) from a bulk solid with a plane surface into He I1 is given by

O s ( ~=)I TP LTCrL [ e x p ( - P W

Ui(0)

U;(O)+I,

(2.14)

P

where H = I h o p f a , and P = 11kBT. The eigenfrequency wq in the solid equals to that of the longitudinal waves in He 11, i.e., wq = cLq, originating from the energy conservation law. Substituting eq. (2.13) into (2.14), and ) ~the Bose-Einstein distribution replacing the thermal average ( a ; ~ , by function n ( w , ) , eq. (2.14) is expressed as (2.15)

After the integration, we obtain Qs( T )=

r 2 p L c L k ;T4F 30ps v i h ' '

(2.16)

KAPITZA THERMAL BOUNDARY RESISTANCE

123

where us is the velocity of phonons in the solid and F takes the value of 2 in the present simplified case. For the eigenmodes in a half space, see the description given in section 7.2.1. Taking into account the contribution from transverse waves with polarization perpendicular to the surface, as well as surface waves localized in the vicinity of the solid surface, Khalatnikov (1965) performed a more rigorous calculation of &( T) and obtained n2pLcLk;T4F

QdT ) = 30psv:fi3

'

(2.17)

where vT is the velocity of transoerse phonons in the solid and F is a function of mass densities and phonon velocities of the solid and the liquid, respectively. For example, F takes the value of 1.57 m the case of the copper-liquid He11 interface. The numerical values of F for various materials are given in table 1 by Challis and Cheeke (1968). By differentiating eq. (2.17) with respect to T according to eq. (2.3), the explicit form of the Kapitza resistance R K is given by

RK

=

15fi3p,v: 2a2pLcLk;T3F '

(2.18)

The theory of the Kapitza resistance between normal liquid 'He and a solid was proposed by Bekarevich and Khalatnikov (1960), and Gavoret (1965), where sound energy from the solid is considered to be transferred at the interface to the collective zero-sound modes in normal liquid 'He. Taking into account the contribution from the transverse zero-sound (Fomin 1968), R K takes the form

RK=

15fi3p,v: 2n2pLk;T3(a,cL0+~ 2 c m ) F '

(2.19)

where cLoand cm are the velocities of longitudinal and transverse zero-sound in normal liquid 'He. The factors a , and a2take values of the order of unity and are expressed in terms of Landau parameters Fo and F,, whose expressions will be given in section 3.1.2. The form of eq. (2.19) implies that the physical origin of the heat transfer is identical with that of the solid-He I1 interface (eq. (2.18)). A detailed discussion of this point will be given in section 3.1.1. The Kapitza resistances R K obtained in eqs. (2.18) and (2.19) are expressed only by material constants, and are proportional to the inverse third power of temperature. In addition, it should be noted that, for bulk solids, the actual value of R K for liquid 'He becomes of the same magnitude as that of R K for He 11, since a, = 1.54, a2=0.64, cLo= 3 . 4 6 ~and ~ c w = uF, where uF= 1.8 (0atm)-3.9 (27 atm) x lo3 cm/s.

124

2.3. ANOMALOUS BEHAVIOUR

T. NAKAYAMA OF T H E OBSERVED KAPITZA RESISTANCES

2.3.1. The liquid 'He-sinrered powder interface at mK temperatures As described in the previous section 2.2, the Kapitza resistance increases with decreasing temperature. If the Kapitza resistance R K is proportional to T-' down to 1 mK as shown by eq. (2.19), the efficiency of heat exchange per unit area at 1 mK is lo9 times worse than that at 1 K. In order to overcome the difficulty of heat exchange at mK temperatures, it is necessary to make the contact area between a solid and liquid 'He large. For this reason, small metal particles with micron size are used in the heat exchangers of dilution refrigerators (Wheatley et al. 1968, Radebaugh et al. 1974, Frossati et al. 1977). which have come into wide use in low temperature laboratories. For example, powder (nominal particle size 700 A) compressed at room temperature at about 200 bar to a filling factor of 36% and then sintered at 180°C in vacuum for 30 min gives 1.3 m2/g (Rogacki et al. 1985). However, if the Kapitza resistance R K shows the T - j dependence as predicted by eq. (2.191, it is apparent that there still remains a difficulty of heat exchange at mK temperatures even using small metal particles of submicron size. Fortunately, this predicted T-' temperature law is not realized at mK temperatures as first pointed out by Avenel et al. (1973). They showed that the Kapitza resistance R K between various metals (Cu, Au, Pd) and liquid 'He observed at mK temperatures deviates considerably from the theoretical prediction of eq. (2.19) in both magnitude and ternperature dependence. This anomalous behaviour has been established by many research groups. Experimental work before 1979 is described in the excellent review by Harrison (1979), in which the data not only for various metal particles but also for magnetic insulators (CMN) are described in detail. Early data on R K did not indicate a magnetic field dependence up to several 10 mT (Andres and Sprenger 1975, Ahonen et al. 1976, Ahonen et al. 1978). However, Perry et al. (1982) reported the first instance of a magnetic field dependence (-0.8 T) of R K for a system of Pt powder packed in an equal volume of graphite powder, in contrast to the acoustic mechanism which predicts no field dependence. Subsequently, Osheroff and Richardson (1985) investigated in detail not only the magnetic field dependence but also the pressure dependence of R K between liquid 'He and compressed silver powder. They observed a strong increase in the thermal boundary resistance with the application of magnetic fields up to 9.4 T. The observed R K rises continuously with magnetic fields up to -0.2 T, but does not show a field dependence above 0.2T. This result implies that the data on R K above 0.2 T are attributed to the nonmagnetic channel. It should also be

KAPITZA THERMAL BOUNDARY RESISTANCE

125

noted that Osheroff and Richardson (1985) observed the Kapitza resistance RK to increase somewhat with applied pressure. These features will be discussed in detail in sections 4 and 5. The typical Kapitza resistances R, measured are shown in fig. 2.3, in which the data for silver particles are compared with the prediction for a bulk solid given by eq. (2.19). The common characteristic features of the data are the following: (i) Both the temperature dependence and the observed magnitude are in fair agreement with the theory (eq. (2.19)) above about 10 mK. (ii) As the temperature is decreased from 10 mK to 1 mK, the data deviate considerably from the theory for both the temperature dependence and the magnitude. Figure 2.4 shows the magnetic field dependence of RK observed by Osheroff and Richardson (1985). It should be emphasized that even the data at 0.385 T in fig. 2.4 deviate strongly from the theory for a bulk solid. It is evident that some unknown processes transferring heat effectively are present. This has been a long-standing unsolved problem in low temperature physics since its first observation by Avenel et al. (1973). The theoretical description on this subiect will be dealt with in detail in sections 4 and 5.

T(rnK1 Fig. 2.3. Thermal boundary resistance R K observed between silver particles and liquid 'He as a function of temperature. The symbols are taken from Andres and Sprenger (1975): (O), Ahonen et al. (1978): (0).The solid line represents the prediction from the AM theory for bulk silver (eq. (2.19)).

T. NAKAYAMA

I26

lo'

I

1

10

.

. . . 100

T(mI0

Fig. 2.4. Magnetic-field dependence of the Kapitza resistance R, between silver particles and liquid 'He at saturated vapour pressure. The filled circles (0)are the data of R, at zero magnetic field and the open circles (0) are R, at 0.385 T. The solid line denotes the theoretical prediction for the bulk silver-liquid 'He interface. These are taken from Osheroff and Richardson (1985).

2.3.2. The liquid He-bulk solid interface above about 1 K There is another interesting finding on the problem of the Kapitza resistance, which is quite different from the phenomenon described in the previous section 2.3.1. These are observed above about 1 K at the interfaces between liquid (or solid) He and a bulk solid. To show the deviation of the experimental data from the T-' law of eq. (2.18), the product R,T3 observed is plotted against temperature T in fig. 2.5, in which the results for solid or liquid 'He and 4He are taken from the work of the Illinois group (Anderson et al. 1964, 1966, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The discrepancy with the AM value (eq. (2.18)) is evident above 1 K. In addition, the observed values are much less than those of the AM theory even below 0.1 K. One should note that the magnitudes of the observed R K above about 1 K are essentially the same among liquid 4He (or 'He) and solid 4He (or 'He), while below 0.1 K the magnitudes of R K for these several forms of He are quite different (see fig. 2.5). These interesting observations are of 1960-1976 and this so-called Kapitza problem above about 1 K has attracted much interest among low temperature physicists for many years.

KAPITZA THERMAL BOUNDARY RESISTANCE

Lqud

-

'He

I27

-

10'

T(K)

Fig. 2.5. Thermal boundary resistance R, multiplied by T3between liquid 3He or 4He and copper under various pressures as a function of temperature. The symbols are for liquid 'He (O), solid 3He (0).liquid 4He (A), and solid 4He (A).Curves A, B and C are for liquid 'He, liquid 4He, and solid 'He, respectively. The data are taken from the work of the Illinois group (Anderson et al. 1964, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The arrows at the right side indicate the prediction from the acoustic mismatch theory (eq. (2.18)) for an ideal Cu surface.

Although the history of this problem is longer than that found at mK temperatures (Avenel et al. 1973), the underlying mechanisms are not yet fully understood. Many good reviews on this subject have been published (Pollack 1969, Challis 1974, Anderson 1976, Wyatt 1981, Zinov'eva 1985), which report the status of investigations when these articles were written. The problem is essentially how to understand the unexpectedly large transmission of high-frequency phonons ( 3100 GHz) across the interface between the He system and a bulk solid, where the He system is liquid or solid 'He and 4He. In ordinary experiments on heat transfer, information on the polarization, frequency and propagation direction of phonons are averaged out. As tunable sources of monochromatic phonons have become available, with which phonons with frequencies in the terahertz regime can be generated (see the book edited by Bron 1985), phonon spectroscopy with a high resolution has become another promising tool for the study of phonon scattering at the liquid He-solid interface. In section 7, recent efforts to understand the problem will be surveyed with emphasis on the application of high-frequency phonon generation and detection techniques to this problem.

T. NAKAYAMA

128

3. Fermi liquid theory of the Kapitza resistance 3.1. HEATTRANSFER

D U E TO Z E R O - S O U N D EXClTATlONS

3.1.1. Emission of zero-sound from a small particle into liquid -3He Liquid 'He below about 0.1 K behaves as a Fermi liquid. The theory of a Fermi liquid, developed by Landau (1957, 1958), postulates that the classification of energy levels remains unchanged when the interaction between 'He atoms is gradually switched on. It is well known, of course, that liquid 'He undergoes the phase transition into superfluid 'He at around 1 mK as first found by Osheroff et al. (1972). The thermodynamic quantities of liquid 'He can be calculated by taking account only of weakly excited energy levels lying close to the ground state. In liquid 'He these are single-particle excitations and zero-sound excitations. Bekarevich and Khalatnikov (1960) and Gavoret (1965) applied Landau theory to the problem of the Kapitza resistance. Toombs et al. (1980) have justified this phenomenological theory from the microscopic viewpoint by taking into account both zero-sound and single-quasiparticle excitations. This section deals with the heat transfer between a solid and liquid 'He due to zero-sound excitations in terms of the Landau Fermi liquid theory, using as illustration a small particle with radius R as mentioned in section 2.2. This treatment is not only useful to discuss the heat transferring mechanism between sintered powder and liquid 'He, which is the main subject matter of this article, but also to derive simply the Kapitza resistance R K at the liquid 'He-bulk solid interface as a limiting case of a large particle with radius R + co (fig. 2.2). The quasiparticle distribution function n ( p , x, 1 ) of liquid 'He in the nonequilibrium state is a function of time r and position x (we define x as the position vector from the center of a particle). Hereafter, the spin index u will be suppressed using the definition of the momentum p = ( p , (T). Consider the situation in which the distribution function differs by a very R being small amount 6 n ( p, x, t ) from the equilibrium one n o ( p ) at 1x1> IR(, the instantaneous position of the surface of the particle. In terms of 6 n ( p, x, t ) the kinetic equation is written as a w p , x, at

1)

+ 0;

a m p , X, ar

1)

= l[n(p')I,

(3.1)

where I [ n ( p ' ) ] is the collision integral and Sri(p, x, 1 ) is defined as an0(p) 6€p.

6ii( p , x, t ) = 6 n ( p, x, t ) --

a€P

(3.2)

KAPITZA THERMAL BOUNDARY RESISTANCE

129

Here 6ep is the variation of energy for a small deviation 6 n ( p ' ) . Defining up by the following equation: (3.3) 6ep can be expanded in terms of the Landau parameters keeping the first two terms of the expansion 6ep = 1f,' 6np' = P'

c ( f i + f ;P^

'

P^') vp'8( p'-pF)/uF

P'

-6,

(3.4) where p^ = p/IpI, and F i and Fi correspond to a spherical change of volume of the Fermi sphere and distortion of the sphere proportional to cos Op, respectively. The density of quasiparticle states at the Fermi surface N ( 0 ) is defined as N(O)=rn*pF/(.rr2h3).The parameters F i and Fi take the values of 10.07 and 6.04 at zero pressure. The moments vo and v 1are defined by the following integral with respect to solid angle Op = N(0)(fGvo+ffp^*v l ) = FivO+F;v1

(3.5)

The kinetic equation (3.1) is rewritten as

a VP -+up at

- grad( up+ h e p )= I [ v,]

At low temperatures, the mean free time of collisions 7 between quasiparticles is given by 7

-- lo-''/ T2s.

When the condition 1 holds, the collision integral can be omitted in eq. (3.6). This is satisfied in the temperature regime TC0.1 K, where there exists the zero-sound collective mode. It should be noted here that the mean free time is associated with inelastic scattering between quasiparticles and is not due to elastic scattering by the boundary. This becomes important when discussing zero-sound excitations in liquid 'He immersed in sintered powder. Consider a spherical elastic particle in liquid 'He whose surface is executing vibrations (Nishiguchi and Nakayama 1982). These vibrations act as a periodic perturbation to liquid 'He and cause a change in the distribution function as

T. N A K A Y A M A

I30

where R is the radius of the particle. f ( 0 , d ) is a dimensionless function depending on the polar( 0) and azimuthal(4) angle. Substituting eqs. (3.4) and (3.7) into eq. (3.6), one has the kinetic equation at a distance far from the origin ( r = 1x1Z= R )

+

+

( p - S ) up ( F~uO F3 vI * 6 )= 0,

-

(3.8)

where p = p^ 4 = cos 0,, s is the dimensionless velocity of a collective mode defined by s = w / upq, and vFis the Fermi velocity of the quasiparticle. The direction of q is taken to be parallel to that of x due to the spherical change of the distribution function. From eq. (3.8), up becomes

By taking the direction of ulr to be parallel to 9, the vectors written in terms of the row-vector representation

6 and v, are (3.10)

and v1 = ( h r .

Vlflr

hd).

(3.11)

The result is that the change of the distribution function due to the surface vibrations of a spherical particle gives rise to

3.1.2. Heat flux from a spherical particle at temperature T 3.1.2.1. Energy current into liquid ' H e The vibrating surface of a spherical particle produces excitations in liquid 'He as described in section 2.2, and the excitations carry away energy from the small particle. The energy current takes the following form according to the Landau theory

(3.13)

where we define d7=-

V

d'p .

KAPITZA THERMAL BOUNDARY RESISTANCE

Using the relations eP = uF(p -pF) +6ep, - S ( p - p , ) / v , , eq. (3.13) yields (vp6cp

J ( x ) = t]FN(O)

up = uFi,

and

+ 6e2) -.d o p pp4v A

131

ani/aep = (3.14)

Substituting eq. (3.4) into eq. (3.14) and performing the integration, the expression for J ( x ) in terms of moments vo and v1 is given by, J ( x ) = oFN(O)[(1 +$F;)F&vl+F;vl

*

(3.15)

[vz]],

where the second-order moment [ vzlVis defined by the relation (3.16)

For spherical zero-sound, it is sufficient to take into account the radial component of energy current J ( x ) , J,(x)=uFN(o)

[

F ; ( ~ + $ F ; ) ~ ~ ~ , ,1+ F~ ;l i -r. 8, d

1

.

i ~ 2 "

Using the number- and momentum-conservation law derived from the kinetic equation (3.8), one has Jr(x) = u F N ( O ) [ F ~ s L v ~ + F I S L v : , + F ? s T ( Y:g)]. (3.17) The fluid velocity u and the number density fluctuation 6 n are related to the moments vo and v, as follows: U=/

and 6n =

Up6?lp d T = N ( O ) O F ( 1 + f F s ) V l ,

I

6np d r = N ( 0 ) v o .

(3.18)

(3.19)

Using eqs. (3.18) and (3.19), the heat current J , ( x ) can be rewritten as

(3.20)

where the velocities of longitudinal and transverse zero-sound are given by cL= SLUF and cT=sTuF, respectively. By integrating eq. (3.20) with respect to a closed surface surrounding a small particle, the averaged energy flux E becomes E. = y /1 J r ( x ) r 2 d 0 4wR

T. NAKAYAMA

132

The energy currents J at the respective positions x and R are related to each other through the equation

Using this relation the averaged energy flux E becomes

E=-

'I

47r

J,(R)dO.

(3.21)

Because of the presence of viscosity in normal liquid 'He, the appropriate boundary condition is the nonslip condition, i.e. all components of fluid velocity near the surface equal those of the surface velocity of the spherical particle. It should be mentioned here that Ritchie et al. (1987) reported the measurements of both the real and the imaginary parts of the transverse acoustic impedance of an oscillating surface in contact with dilute solutions of 'He in 4He, as well as with pure 'He with and without 4He surface boundary layer. In particular, they emphasized the effects of 4He-rich surface layer on the boundary conditions. One can put the condition that the fluid near the moving surface oscillates in phase with the surface motion of the spherical particle. The condition is expressed as na,u

= U,

where a,. = (tir, tio, ti4) is the time derivative of the surface displacement of a spherical particle. u is the fluid velocity near the surface and n the number density of liquid 'He which is taken to be the global average value. From eqs. (3.18), (3.20) and (3.21) we obtain the energy flux E taking the time average, (3.22)

Here the prefactors a , and a2 are defined by (3.23)

where p L is the mass density of liquid 'He: p I . = nmHerand cL and cT are the velocities of longitudinal and transverse zero-sound, respectively. 3.1.2.2. Kapitza conductance hK due to zero-sound excitations The energy flux Q,( T) at a finite temperature T can be obtained by taking the thermal average of eq. (3.23) and summing over all the vibrational

KAPITZA THERMAL BOUNDARY RESISTANCE

133

modes of a spherical particle as

.

.

fJ=e.4

where the angular brackets denote the thermal average at T. The suffixes ‘s’ and ‘t’ denote the spheroidal and toroidal modes, respectively. J specifies the set of quantum number (I, m,0 ) .The nature of these modes and their eigenvalue equations are described in the Appendix. The thermally averaged squared-displacement can be written in terms of the Bose-Einstein distribution function nB(w,T ) as

Substitution of the expression for uJ,[and u ~into , ~eq. (3.24) leads us to the equation

1 2

J

x [a,CL(Ai)2G~~(R)+a2cT€(As) GsB9(R)+(A:)2G:(R)II.

(3.25)

The factors A: and G; are defined in the paper by Nishiguchi and Nakayama (1982). The prefactor (21+ 1) in eq. (3.25) denotes the degeneracy due to the sum over the quantum number m. Let us consider the heat flow from a small particle at temperature T + AT into liquid ’He at T. The Kapitza conductance hK is defined by

From eq. (3.25), the expression for the conductance becomes

I 2

J

1 2

J

+a,cT€(A,) Gso4(R)+(At 1 Gt ( R ) l l ,

(3.26)

where the numerical factors a, and a2 and the velocities of zero-sound cL and C, can be calculated from the Landau parameters Foand F;. Quoting the Landau parameters from Wheatley (1979, one can estimate a, = 1.54, a2= 0.64, cL= 3.456uF, and c,= vF, respectively.

T. NAKAYAMA

134

The solid curve in fig. 3.1 shows the calculated results (eq. (3.26)) of the Kapitza resistance R K between liquid 'He and one silver particle with R =0.5 p m in radius, where the eigenvalues up to 11 = 100 are taken into account, see Appendix. One sees from fig. 3.1 that the resistance exhibits a T - dependence above about 10 mK for R = 0.5 pm and the magnitude coincides with the bulk limit. The physical meaning of this result is evident because the dominant phonons contributing to the heat transfer in this temperature range have much shorter wavelengths than the size of a panicle. As a result, the shape of the particle is irrelevant and R K approaches the bulk limit. At low temperatures, the resistance R K increases exponentially with decreasing temperature. This exponential behaviour is caused by the size effect of small particles, i.e., considering the lowest eigenvalue from eq. (3.26) the asymptotic form of R K sufficiently below 10 mK becomes

where w o is the lowest angular frequency corresponding to the spheroidal mode with spherical symmetry. The Kapitza resistance observed for silver particles with R = 0.5 pm is plotted in fig. 3.1. Above roughly IOmK, the calculated resistance for

t Fig. 3.1. Calculated boundary resistance R , as a function of temperature between liquid 'He and a silver particle with 0.5 pm radius. The observed data from Andres and Sprenger ( 1975) are given for dirty silver particles with R = 0 . 4 p n in radius ( A ) and for clean silver particle with R = 0.55 pm (0).respectively. After Nishiguchi and Nakayama (1983).

KAPITZA THERMAL BOUNDARY RESISTANCE

135

R =0.5 pm agrees well with the observed data in both magnitude and temperature dependence. At temperatures less than 10 mK, there is a large discrepancy between experiment and the calculated value. This, of course, suggests the inadequacy of the isolated particle model to explain the experimental data. 3.2. ENERGYTRANSFER

DUE TO INELASTIC SCATTERING OF S I N G L E

QUASIPARTICLES AT THE INTERFACE

In addition to the excitations of collective zero-sound modes described in the previous section 3.1, single-particle excitations play a role in energy transfer across the solid-liquid 'He interface. In this section, the energytransfer mechanism due to inelastic scattering of single particles at the interface is described, following Toombs et al. (1980). They treated this problem from the microscopic viewpoint in terms of quantum mechanical perturbation theory, and justified the conclusions of the semiclassical treatment (excitation of zero-sound is the dominant heat-transfer channel) of Bekarevich and Khalatnikov (1960) afid Gavoret (1965). Consider the case in which a quasiparticle with momentum hk is approaching or leaving the periodically vibrating surface. This process gives rise to energy transfer across the interface due to the inelastic scattering of a quasiparticle. This situation is illustrated in fig. 3.2. As a first step for clarifying the physical basis of this process, the Kapitza conductance h , between a 'gas' of 'He atoms and a solid is treated in this section. This procedure will be generalized to the case of 'liquid' 'He taking into account the interaction between 'He atoms. If the solid surface is displaced by an amount u, along the z-axis, the energy change SE per unit volume of 'He gas is expressed by (3.27)

Fig. 3.2. Schematic diagram of inelastic scattering of a 'He quasiparticle from state k to k + p by a vibrating wall. q indicates the wave number of a phonon in a solid.

136

T. NAKAYAMA

where V = L3 is the volume of 'He gas. The stress tensor IZE is the energy density as understood by putting u, = L in eq. (3.27). The second quantized form of the stress tensor HS', can be obtained by combining the z-component of kinetic energy of an 'He atom with the following field operator (3.28)

Here the boundary condition at the interface is not considered, for simplicity. The explicit form of US', is written down by taking into account the conservation law of the particle number (Zubarev 1974), (3.29)

The interaction Hamiltonian is expressed in quantized form using eq. (3.27)

(3.30) where the displacement u, is given by the z-component of eq. (2.13). Using eqs. (3.28)-(3.30),the second quantized form of the interaction Hamiltonian becomes (3.31)

Here

where qll and pIIare the parallel components of the momenta at the interface. The Kronecker delta represents the momentum conservation law parallel to the interface as a consequence of the assumption of a plane interface. A is the contact area of the interface. The z-compoment of the polarization vector of the displacement expressed by eq. (2.13) is given by eqz. The k, p ) is represented by element Mo( (3.32)

The interaction Hamiltonian Hso yields the energy-transfer process in which the absorption or emission of a phonon with momentum hq in the solid is accompanied by the inelastic scattering of a 'He atom from state k to k + p (see fig. 3.2).

KAPITZA THERMAL BOUNDARY RESISTANCE

The heat flux density gas at T is written QSG=

1

137

oScfrom the solid at temperature T + AT into ’He

AoqW(k+k+p),

(3.33)

4p.q

where the transition probability W (k -* k + p ) is given by

-&+,(I

-h ) ( I + “q 116 ( E k + p - E k - hoq )

*

Here no( T + AT) is the Bose-Einstein distribution function and fk( T ) the Fermi distribution function at temperature T. Expanding &( T + AT) with respect to a small temperature difference AT, the net heat flux can be expressed by the relation = hKAT.

Using this, the Kapitza conductance hK is obtained as (3.34)

where

(3.35)

The above eq. (3.35) can be calculated by putting the appropriate condition q < k , because the phonon energy hoq is small enough compared with the Fermi energy EF at temperatures considered here. The result is o k4 RG( oq) = lF.

(3.36)

41r2

From eqs. (3.34) and (3.36), the Kapitza resistance and a solid takes the form

RK =

1 5psv2h’ lr2pLvFkiT’ .

RK

between 3He gas

(3.37)

The numerical estimate of eq. (3.37) becomes of comparable order with the conductance hK (or resistance RK) of eq. (2.19) which attributes it to excitations of the collective zero-sound modes. It should be emphasized, however, that this is not actually the case for ‘liquid’ ’He, in which the interaction between quasiparticles plays a key role. If this interaction is

T. NAKAYAMA

138

properly taken into account, i.e., the effect of a back-flow current of excited quasiparticles, the magnitude of the Kapitza conductance hK due to the inelastic scattering of single quasiparticles is severely reduced and becomes negligibly small compared with the conductance due to collective zero-sound excitations. The physical background of this conclusion is described in the following. Considering the mutual interaction of 'He atoms, the proper stress tensor of liquid 'He consists of the sum of eq. (3.29) and the following term, 1711(X) =;uG+(x)G+(x)G(x)G(x), (3.38) where a is the strength of the interaction potential defined by V ( x )= a 6 ( x ) . The sum of eqs. (3.29) and (3.38) becomes, in terms of the Hartree-Fock approximation,

(3.39)

where n, is the number density of 'He atoms in the liquid. The factor Fo (Landau parameter) is defined as Fo= i u N ( O ) , where N ( 0 ) is the quasiparticle density of states at the Fermi surface expressed by

where m* is the effective mass of a quasiparticle. From eqs. (3.31), (3.32) and (3.39), the unscreened matrix element taking into account the mutual interaction becomes h2 + fk:Fo] . (3.40) m The matrix element M(&,p ) refers to the scattering of quasiparticles with definite momenta. In our case, the vibrating solid surface produces a quasiparticle-quasihole pair which propagates into the system undergoing multiple scattering of other quasiparticle-quasihole pairs. This dynamical screening of the liquid-solid interaction is described in fig. 3.3, where the A4 ( k, p ) = -[( kz + :p, )'

--

&+ qk A

-

Fig. 3.3. Vertex correction incorporating the screening. The angular frequency w means that o f phonons in a solid, k and k + p indicates 'He quasiparticles approaching and leaving the \,lid surface, respectively. The definition of M ( k ) ,X o ( k ) , and A(&',k ) are given in the text. After Toombs et al. (1980).

KAPITZA THERMAL BOUNDARY RESISTANCE

139

black circle is the screened matrix element fi(k, p ) and the open circle the unscreened matrix element M(&,p ) of eq. (3.40). Introducing the scattering amplitude for a quasiparticle A(&,k', p ) , the screened matrix element can be expressed as &&, P)= M ( S P) -1M ( k ' , P)Xo(k',P M k , k',PI,

(3.41)

h'

where xo(k,p) is the particle-hole propagator of the noninteracting system expressed as

The scattering amplitude A(k,k') represents a process in which two quasiparticles with momenta hk and hk' exchange momentum hp and energy hw,. The scattering amplitude takes a simple form in the case of only one-nonzero Landau parameter Fo as follows (3.42)

where A = w/(uFp) and A+l g ( A ) = 1 - f A log-. A-1

After some manipulation one obtains the screened matrix element A?( k, p ) ,

where = i - i .The effect of mutual interaction is included in the screened matrix fi(&, p). Then the proper expression of R K due to single-quasiparticle excitations can be obtained by replacing Mi(&,p) for noninteracting 'He atoms in eq. (3.35) by A?(&, p ) . Figure 3.4 shows the calculated conductances where the two contributions due to single quasiparticles hip and collective zero-sound excitations h r are compared. The abscissa is the ratio between the velocities of zero-sound cLoand the Fenni velocity of quasiparticles uF as defined by s=cL0/vF.Because the actual value of s is about 3.5, we conclude that the contribution to the heat transfer from the inelastic scattering of a single quasiparticle is negligibly small. The physical origin of this result is due to the fact that the interactions between quasiparticles are very strong, and the back-flow current of excited quasiparticles modifies the effective heat transfer.

140

T. NAKAYAMA

S

Fig. 3.4. The conductance h',P from single particles. h z from zero sound. The solid curve represents the sum of these contributions: h: = h;'+ hF. The abscissa is the reduced zero-sound velocity s = cLo/ur. After Toombs et al. (1980).

4. Anomalous Kapitza resistance between sintered powder and liquid 'He at mK temperatures 4.1. HEATEXCHANGER

USING SUBMICROMETER METAL PARTICLES

The previous section 3 has dealt with the Fermi liquid theory of the Kapitza resistance R K between an isolated particle and liquid 'He. Noticeably the anomalous behaviours of the Kapitza resistance at mK temperatures are observed for sintered (or pressed) powders. Sintered powder constitutes an important part in the dilution refrigerators, which are now widely used in the laboratories. Thus the problem of the Kapitza resistance between sintered powder and liquid 'He is of interest not only in its own right but also from practical implications. The principle of the dilution refrigerator, which was originally suggested by London (1951) and London et al. (1962), is described briefly in the following. The development of the dilution refrigerator allows us to perform experiments at mK temperatures (see the reviews by Lounasmaa 1974, Betts 1976, Andres and Lounasmaa 1982, Richardson and Smith 1987). The first successful cryostats of this type were built by Neganov et al. (1966) and Hall et al. (1966).Cooling is produced in the mixing chamber by causing 'He atoms from the upper 'He-rich phase to move across the phase-separation boundary to the lower phase, which has 6.4% of 'He dissolved in superfluid He 11. This cooling process can be understood in analogy with evaporation (quantum mechanical): namely, the upper phase corresponds to liquid and the lower phase to vapor. The process can be made continuous by circulating 'He in the system with a pump at room temperature.

KAPITZA THERMAL BOUNDARY RESISTANCE

141

A heat exchanger is a vital part of the design of a dilution refrigerator, where the incoming liquid 3He is gradually cooled by the outgoing colder liquid. As described in previous sections, the Kapitza resistance is a severe problem in the design of a successful heat exchanger, making it important to maximize the heat-exchange area. Copper sinter was originally used in a heat exchanger by Wheatley et al. (1971) in their dilution refrigerator. Radebaugh et al. (1974) measured the Kapitza resistance of various metals, and introduced 'submicron silver powder', so-called Japanese ultrafine powder, to build more effective heat exchangers. Concerning the development of research on ultrafine particles in Japan, see a review by Hayashi (1987). The advantage of using submicron metal particles lies in making the contact surface area large. For example, the surface area of 10 g silver powders with size 700 A becomes almost 13 m2 (Rogacki et al. 1985). Frossati et al. (1977) constructed a dilution refrigerator using silver powder to reach 2 mK in the continuous mode. Submicron metal particles are used for thermal contact with the 'He system down to temperatures of 0.1 mK (Guenault et al. 1983, Owers-Bradley et al. 1983). However, it should be emphasized that if the T 3law of the Kapitza resistance of eq. (2.19) is valid at mK temperatures, the use of metal particles cannot be the final solution for designing an effective heat exchanger. The history of low temperature physics tells us that it is not necessarily legitimate to apply an argument based on the extrapolation of an experimental law measured at higher temperatures (Richardson 1981). The actual case is shown in fig. 2.3, where the temperature dependence of the Kapitza resistance R K around 10 mK changes dramatically from the predicted T-' law, i.e., at temperatures less than around 10 mK the resistance RK becomes roughly proportional to T ' .Thus it is evident that the use of submicron particles leads not only to a large contact area, but also to the appearance of an anomalous temperature dependence of the Kapitza resistance. This anomalous temperature dependence, roughly proportional to T ' ,is the key element for constructing a successful heat exchanger. 4.2. SOFTPHONON-MODES

I N SINTERED POWDERS

As shown in section 3.1, the internal vibrations of a particle of submicron size are irrelevant to effective heat transfer below about 10 mK. This is due

to the finite-size effect as understood from the following simplified arguments. The lowest frequency vo of 8 particle with diameter d is of the order of uS/2d, where us is the sound velocity in the particle determined from the elastic constants. For instance, the lowest vo becomes about 2 GHz for a particle of 1 pm in diameter. If heat exchange occurs from the excitation or absorption of zero-sound due to surface vibrations of a particle, the

142

T. N A K A Y A M A

resistance will increase exponentially at a temperature lower than that corresponding to vo. This argument has been given in section 3.1.2.2. At temperatures higher than that corresponding to the finite lowest eigenfrequency, the calculated resistance exhibits a T 3dependence and the magnitude is in agreement with the bulk limit. This is because the dominant phonons contributing to the heat transfer in this temperature regime have much shorter wavelengths than the size of the particle, i.e., the particle shape is irrelevant and the resistance approaches the bulk limit. The above argument was demonstrated quantitatively in the case of isolated particles for the normal liquid 3He as shown in fig. 3.1. However, when the particles are bridged in the sintering process, new low-energy vibrational modes associated with the characteristic property of sintered metal-powder should appear (Harrison and McColl 1977). It is expected that this might play a role in effective heat transfer. Sintered powders are characterized by particle size (or mass), bridges, pores, and an oxide layer (fig. 4.1). The various properties of sintered powder have been studied by some experimental groups (Iwama and Hayakawa 1981, Rogacki et al. 1985, Hayashi et al. 1986). Rogacki et al. (1985) made a systematic study of the influence of temperature, time, pressure and atmosphere during pretreatment and sintering on the surface area, packing factor, structure, hardness, and electrical conductivity of sinter produced from 900 8, Cu powder. Iwama and Hayakawa (1981) made a detailed study on the neck growth stage in the sintering of powders of Au, Ag, A1 and Cu. They observed that neck growth is severely affected in an atmosphere of 0, in all the materials. Hayashi et al. (1986) performed Mossbauer spectroscopy to investigate the dynamical properties of the sinter. Sintered particles of Au as well as Cu-Au and Ag-Au were employed in Mossbauer spectroscopy, Au being used in order to investigate the low frequency modes in sinters. The average particle diameter was 5008, for the Au and Cu-Au sinters, and 2500 A for the Ag-Au sinters. From the analysis of the width of Mossbauer spectra, they obtained a characteristic frequency (temperature) of the sinter of -25 mK. Frisken et al. (198 1 ) suggested a characteristic frequency corresponding to the Debye frequency of about 25 mK from the measurements of elastic constants. It should be noted that these two different experiments lead to the same magnitude of characteristic frequency. The low-energy vibrational modes are produced from the elastic deformation of the bridges connecting the particles. Kingery and Berg (1955) investigated the relation among the size of the bridges, annealing temperature and heating time in the sintering process. One can deduce the size of the bridge for the sintered particles from electron microscope photographs. Taking into account the sintering conditions used in the heat-transfer

KAPlTZA THERMAL BOUNDARY RESISTANCE

143

Fig. 4.1. T h e electron microscope picture of sintered Cu powder at ( A ) 80°C and ( B ) 130°C. Characteristic necks between grains grow after sintering at 130°C. By courtesy of Rogacki et al. (1985).

T. NAKAYAMA

144

experiments the radius R and length 1 are estimated to be of the order of 1/10 of the particle radius R. The necks could be modeled by rods with cross section ma2 and length I, and the maximum frequency corresponding to the Debye cutoff frequency can be estimated approximately from the elastic deformation of a rod. As suggested by Nishiguchi and Nakayama (1983), the maximum angular frequency can be expressed from the balance condition of forces as (4.1)

where M and E denote the mass of the particle and Young’s modulus of the rod, respectively. The frequency is estimated to be of order wE/ k B = 10 mK 20 mK for a silver particle 1 pm in diameter with Young’s modulus E = 8.27 x 10” dyne/cm2 and mass density ps = 10.49 g/cm. It should be emphasized that the value estimated above from the simple argument is of the same order as the frequency obtained by Frisken et al. (1981) and Hayashi et al. (1986). Thus the particles execute periodic vibrations around their equilibrium positions and these constitute a perturbation to the liquid ‘He in this temperature region. Although the detailed structure of the vibrational density of states (DOS) of the sinter is not an important factor in discussing the Kapitza resistance at mK temperatures, one should mention the characteristics of the DOS of sintered powders, which form a three-dimensional percolating network. It is anticipated (Deptuck et al. 1985, Maliepaard et al. 1985, Page and McCulloch 1986) that this system is fractal (in a statistical sense) at shortlength scales. The nature of vibrational modes in a percolating cluster has received considerable attention in recent years (see the review by Orbach 1986). From the scaling argument, Alexander and Orbach (1982) have conjectured that the DOS’s of percolating clusters in all Euclidean dimensions obey the universal law g ( w ) - w ” ’ in the regime above the characteristic frequency w,. Vibrational modes in this regime are called fracton. There exists also the phonon regime, where the DOS follows the conventional Debye law g ( w ) - w 2 below w,. The detailed description of the DOS for sintered powders will be given in section 6.1, where the Kapitza resistance between ’He-4He mixtures and sintered powder at subrnillikeluin temperatures is dealt with in connection with the characteristics of the DOS of sintered powders.

-

4.3. HEAT TRANSFER

DUE TO THE EFFECT OF SOFT PHONON-MODES

Let us consider the energy emission due to low-energy thermal vibrations of particles in sintered powder immersed in liquid ’He. The expression for

KAPITZA THERMAL BOUNDARY RESISTANCE

the energy flux eq. (3.22)

fi

I45

from one particle is written down in the same form as

[ u e ( R ) 2 + u , ( R ) 2 dR. ]

877

(4.2) Here the prefactors a, and a2 are defined as (4.3) and (4.4) Here pL is the mass density of liquid 'He defined by pL= nmHe, where mHe and n are the bare mass of a 'He atom and the number density of liquid 'He, respectively. cLand cT are the velocities of longitudinal and transverse zero-sound. When small particles in sinter vibrate without volume change, the surface displacement u in eq. (4.2) can be replaced by the particle displacement q from its equilibrium position using the identities u:( 0 ) = qz cos

e,

u;( 0 ) = qz sin

e,

u;( 0 ) = 0.

Here q is the small displacement of a particle from the equilibrium position (see fig. 4.2). The angular integral in eq. (4.2) can be performed in the following forms,

I I

lu:l2 do1477 = fq:,

Iuil' do1477 =$&,

@

Fig. 4.2. The definition of surface displacements. See the text for details.

T. N A K A Y A M A

I46

Substituting these relations into eq. (4.2), and when all three degrees of freedom of a particle are taken into account,

E =:,pL.(a,c,+2a2cT)141'. Taking into account the contribution from all particles, we obtain 1

E =PL(u,cL+ 2a,c,) c ( Q J * A , , 6s

(4.6)

I

where A, denotes t h e surface area of the ith particle with radius R,, and S is the total exposed area of the assembly of particles immersed in liquid 'He. The heat current OF(T )from sintered powder into liquid 'He is defined by the thermal average of eq. (4.6), OE(

T)

=z 1

~L(aicL+2a,c,) C ( 4 3 ~ 1 .

(4.7)

I

Recalling the fact that the Einstein model can describe well the phonon specific heat down to one-tenth of the Debye frequency (in our case this is wE), we can take a thermal average of (4,)2 of eq. (4.7) in terms of the Einstein model to describe the acoustic channel at rnK temperatures where we are interested in. The result is

where M, is the mass of the small particle on the ith site. The final result for OF(T) becomes

where use is made of

EL=-=---A 1 8

3 MIS Mo 4nR'ps'

Here M , is the average mass of particles in the sinter. The temperature derivative of eq. (4.9) yields the Kapitza conductance h: as (4.10)

where 8 = h w E / k Band R is the averaged particle radius. The total conductance h , between sintered powder and liquid 'He is given by the sum of two contributions hK = h E + h z ,

(4.1 1)

KAPITZA THERMAL B O U N D A R Y RESISTANCE

147

where h', denotes the Kapitza conductance due to the internal vibrations of particles given by eq. (3.26). Figure 4.3 shows the comparison between the experimental data and the calculated resistance R K = 1/ hK of sintered silver powder of radius R = 0.5 p m with characteristic temperature 0 = 15 mK. The total resistance exhibits a T-' variation above about 10 mK, and at temperatures less than 10 mK the calculated one shows a rather moderate temperature dependence. The essential point is that the particles in sinter are well described by the Einstein model, because we are seeing rather 'high' temperature properties of sintered powders even in the mK temperature region, originating from the relatively large mass of particles and weak couplings among particles. Let us consider the heat transfer mechanism due to single-quasiparticle excitations taking into account the effect of pores in the sinter. When powder particles in the sinter vibrate thermally at a finite T, quasiparticles are scattered inelastically by the vibrating wall and this scattering contributes to the energy exchange. Rutherford et al. (1984) described the pores in the sinter as a set of boxes of finite sizes, each of which contains 'He quasiparticles. Each box has a mass M and is free to vibrate as a three-dimensional oscillator, where energy is transferred to the 'He quasiparticles via the 10'

-

lo5

P

Y

z lo4 Y

10'

10' 1

10 T(mK)

100

Fig. 4.3. Contribution from the acoustic channel due to the coupling between zero sound and soft modes. The solid curve shows the calculated resistance with tl = IS mK. The dashed line represents the resistance from one small particle with R = 0.5 pm calculated in section 3.1.2. The observed data are from Andres and Sprenger (1975). After Nishiguchi and Nakayama ( 1983).

T. N A K A Y A M A

148

shaking of the box. They assumed that the frequency spectrum of vibrational modes of the sinter are distributed with a constant density of states over the frequency range 0 to wD, where w D corresponds to the Debye cutoff frequency of the sinter. The expression for h , obtained (Rutherford et al. 1984) is expressed as hK

=

5.6Am*’E:k;TD , Mn2h4

(4.12)

where D and A are the density of states of shaking box modes and the interface area of the box, respectively. They postulated the constant density of states of shaking box modes as (4.13) where N is the number of powder particles of diameter d in the sinter and uD is the bulk Debye velocity. The treatment is entirely analogous with that of Toombs et al. (1980) for ’He gas, in which the Debye density of states is used for describing their formula given by eq. (3.37), i.e., the result of eq. (4.12) can be rewritten by using the Debye density of states w 2V D ( w )= 27r2v;.

(4.14)

By replacing w with the dimensionless variable k g T / h w , one has

5.6Ak:kiT’I - 8h2(M / V ) n 4 v ; ’

h -

(4.15)

where the factor I means the following integral *

x’dx 7r4 -_ exp(x)-1 1 5 ’

(4.16)

Dividing by the interface area A and using the relation k: = 37r2n, the conductance h , is expressed by (4.17)

where a is a number of order unity. This is the same as eq. (3.37) in both magnitude and temperature dependence. Especially, it should be mentioned that eq. (3.37) was derived for the case of noninteracting ’He atoms (gas). Thus the physical ongin of the T-linear dependence of hK of eq. (4.12) comes from the assumption of a constant density of states for sintered powder. As stated in section 4.2, there are two frequency regimes characterizing the DOS of the sinter: the phonon regime ( w z ) and the fracton regime ( w ” ’ ) . For the sinters considered here the crossover frequency o, becomes

KAPITZA THERMAL BOUNDARY RESISTANCE

149

of the order of 10 mK as will be estimated in section 6.1, where the Kapitza resistance of 'He-4He mixtures will be discussed. The result of eq. (4.12) should be modified at mK temperatures. In addition, it should be noted that, if the multiple scattering of quasiparticle-hole excitations in liquid 'He is taken into account, the magnitude of the Kapitza conductance expressed by eq. (4.12) is much reduced as described in section 3.2, where the dominant heat transfer channel is shown to be through the zero-sound excitations rather than through single-particle excitations (see fig. 3.4).

5. The magnetic channel of heat transfer between sintered powder and liquid 'He 5.1. SURFACE CHARACTERISTICS

OF SUBMICRON METAL PARTICLES A N D

SURFACE MAGNETIC IMPURITIES

Metal particles used in experiments, especially Ag or Cu, would be covered with an oxide layer or other adsorbed gases such as O2and H 2 0with about 100 A thickness. There is some evidence, using various techniques, that oxygen with a paramagnetic moment is strongly adsorbed in a variety of forms: 0, ,0;.and 0-on sintered silver powder (Kummer 1959, Kobayashi et al. 1972, Ido and Hoshino 1974). Nishiguchi and Nakayama (1983) have pointed out that the oxygen impurities located at or near the metal surface play a crucial role in a magnetic coupling to the 'He nuclear spins. However, due at that time to a lack of experimental data on the dependence of R K on magnetic field up to -1 T, the estimate of the areal density of the adsorbed oxygens was not reasonable. In the case of the magnetic insulator (CMN in particular), Peshkov (1964) and Wheatley (1968) suggested the possibility of direct energy transfer from 'He quasiparticles to paramagnetic atoms in CMN or vice versa. In this connection, Potter (1976) and Nakayama (1984b) anticipated that adsorbed oxygens on the surface could form a magnetic adsorbed layer, and that magnetic coupling could occur between oxygens and 'He nuclear spins. There is no doubt that studies of the magnetic properties of 'He atoms adsorbed on metals are important in connection with the magnetic contribution to heat transfer across the sintered powder-liquid 'He boundary. Saito et al. (1985) reported the results of cw and pulsed NMR experiments for 3 He atoms adsorbed on small copper particles with an average diameter of 5 0 0 w . The linewidth and line shift were measured for both the 'He and 63 Cu nuclei. Figure 5.1 shows the first derivative of the absorption signal of 'He at 0.18 K and 9.4 MHz. The sharp lines are due to the liquid 3He marker. The

I so

T. NAKAYAMA

Fig. 5.1. First derivatives of N M R absorption for 'He adsorbed on Cu particles (Cu-A and Cu-B) and the marker as unshifted 'He. The inset shows an experimental cell with a capillary side cell for 'He as an unshifted mark. After Saito et al. (1985).

upper curve is for 2.7 layers of 'He adsorbed on sample Cu-A (oxygen content 0.25*0.05 at wt%) and the lower curve is for about 1.5 layers on Cu-B. The layers were thermally annealed in the region from 2.2 to 4.2 K for half a day. The scale for the external field sweep is 0.166 mT per division for the upper curve, and 0.503 mT for the lower. As the homogeneity of the in a sphere of 1 cm diameter, the shift electromagnet is better than 5 x ( d H / H ) is detectable within an experimental accuracy of 1 x In fig. 5.1, dH,/H (from the resonance field of liquid 'He) is evaluated to be -(3.2+ 1 . 0 ) ~ and dHB/H = -(1.2*0.3)x which are listed in table 5.1. The negarioe signs are of interest. Measurements of the Knight shift (Knight 1956) and linewidth for "Cu in samples Cu-A and Cu-B were done in parallel. The observed values are also listed in table 5.1, where the shift is defined as the same as that of the Knight shift. We see that the Knight shift for decreases with increasing oxygen content, while the absolute values of the line shift and linewidth for the 3He resonance grow larger with larger oxygen content in the Cu samples. The widths become broader with increasing oxygen content for the Cu samples. A small diamagnetic shift is observed for the adsorbed 'He. The experimental results

KAPITZA THERMAL BOUNDARY RESISTANCE

151

Table 5.1 Line shift and linewidth of NMR for "'Cu and 'He adsorbed on Cu samples, named Cu-A and CU-B,respectively. (*): As reference field, the resonance of liquid 'He was used. ~~

b3cu Knight shift

Sample, CU-A (0:0.2s wt%) Sample, Cu-B (0:0.45 wt%)

Adsorbed 'He

dH,-,(0.2 K ) (x T)

Shift (*)

( x 10-3)

2.26

6.2*0.1

-(0.032* 0.01)

2.10

6.7 *O.l

-( 0.120

(x

* 0.03)

dHp.,(0.2 K ) (X IO-~T) 0.45 (2.7 layers) 0.80 (1.5 layers)

concerning the linewidth and line shift of the adsorbed 3He are summarized as follows: (i) a larger width for larger substrate oxygen content, (ii) a larger shift for larger substrate oxygen content, (iii) a negative sign of the shift. Oxygens chemisorbed or adsorbed on the substrate play an important role in these features. For feature (i), it is possible to estimate the areal density no of chemisorbed oxygen from the linewidth data in a manner identical to the work of Nakayama (1984b). Assuming a dipole interaction between the 'He nuclear spin and the electron spin of the chemisorbed oxygen, the linewidth becomes of the order of d H = zpe/f3,where z is a number of order unity representing the effective coordination number and f is the mean distance from any localized spin to adsorbed 'He. The mean distance T is related to the areal density no of localized spins by I/f2 = no. Using the observed width d H = 0.08 mT from table 5.1, the areal density no becomes of the order of l O I 4 This agrees with the number density of surface oxygen calculated with the chemical analysis shown in table 5.1, 0.45 wt%, assuming that most of the oxygen is near the surface. For features (ii) and (iii), the observed shifts are small and their signs are inverted compared with the usual Knight shift. A reasonable hypothesis for explaining (ii) and (iii) is the existence of a small demagnetization field at the position of the adsorbed 3He due to the local magnetic moments of the chemisorbed oxygen. Perry et al. (1982) have reported that the thermal resistance between liquid 'He and Pt particles depends on the applied magnetic field at 0.8 T. The theoretical analysis using the Fermi contact coupling between conduction electrons and 'He nuclear spins was made by Perry et al. (1982) by changing the roles of spins in the formulation of the magnetic Kapitza resistance described by Leggett and Vuorio (1970), i.e., the localized electron

T. N A K A Y A M A

15'

spins and the nuclear spins of 'He quasiparticles are replaced by the localized nuclear spins of 3He adsorbed on Pt and the spins of conduction electrons in metals, respectively. The heat transfer between the adsorbed 3He and quasiparticles in the bulk 'He occurs through the exchange interaction. However, Nishiguchi and Nakayama (1983), Nakayama (1984a), and Hood et al. (1987) claimed theoretically that such a strong coupling can never be derived for actual systems. The theory proposed by Perry et al. (1982) requires a srrong magnetic coupling of the Fermi contact type between 'He nuclear spins and the conduction electron spins in Pt as large as lo2 to lo' times the value of the dipole coupling between the corresponding nuclear spins. The theory to explain the observation suggested, if such a strong coupling occurs, that the magnetic coupling could produce a Knight shift for the adsorbed 'He with a magnitude of d H / H 5 x 10 -'. Since the anomalous behaviour of Kapitza resistances below 20 mK has been established not only for Pt particles but also for other metal particles, such a large Knight shift could be observable as well as for 'He adsorbed on Cu or Ag particles, if the mechanism explaining the anomalous resistances below 20 mK is the same. Paying attention to this point, Saito et al. (1985) concluded that the Fermi-contact-type coupling suggested by Perry et al. ( 1982) is absent for the 'He-Cu system. Such a large Knight shift as suggested by Perry et al. (1982) was also not observed in the measurements on 'He adsorbed on small silver particles. This is consistent with the results of the Kapitza resistance between liquid 'He and Ag particles presented by Osheroff and Richardson (1985), who observed that the magnetic field and pressure dependence for the liquid "He-silver system was in conflict with the experimental results for the Pt-'He system (Perry et al. 1982). Finally, it should be mentioned that the theoretical work on the magnetic Kapitza resistance by Leggett and Vuorio (1970), which tried to explain the experiment for CMN (Abel et al. 1966), is the prototype for many subsequent works (Guyer 1973, Mills and Btal-Monod 1974a,b, Challis 1975, Nakajima 1978, Nakayama 1984b). Especially, Mills and Btal-Monod (1974b) investigated in detail the magnetic coupling between randomly distributed electron spins in a dilute magnetic alloy and 'He nuclear spins, in which they predicted that the Kapitza resistance R K should be proportional to T at very low temperatures.

-

'

5.2. HEAT TRANSFER

D U E TO M A G N E T I C C O U P L I N G A T

mK

TEMPERATURES

Osheroff and Richardson (1985) have found a novel magnetic field dependence of the Kapitza resistance between silver particles and liquid 'He in the temperature range 1-5 mK. The main features of their results are:

KAPITZA THERMAL BOUNDARY RESISTANCE

153

(i) the thermal boundary resistance RK observed increases with increasing pressure, (ii) the observed RK rises continuously with magnetic fields up to about 0.2 T, (iii) the observed RK does not show a magnetic field dependence above 0.2 T. It should be noted that all of these characteristic features are in contrast with the experimental results of the thermal resistances for the platinum particles-liquid 'He interface (Perry et al. 1982). The following is an analysis, according to Nakayama (1986b), of the origin of the magnetic field dependence of RK between sintered silver and liquid 'He observed by Osheroff and Richardson (1985). It is natural to consider, from the experimental evidence by Saito et al. (1985) and Osheroff and Richardson (1985) and the theoretical prediction (Nakayama 1984b), that the localized magnetic impurities (oxygens in a variety of forms: 02,O;, 0, 0-, etc.) play a crucial role for magnetic coupling with 'He nuclear spins. Since the thermal boundary resistance RK observed doubles from its zero-field value at about 200 G, this characteristic field should be a measure of the broadening of the energy splitting due to randomly distributed localized spins. The spacing between localized magnetic impurities estimated from the characteristic field is about P = 5 A (Osheroff and Richardson 1985). Above 2 kG, the Kapitza resistance RK observed does not vary with magnetic field, which implies that the magnetic coupling is suppressed due to the locking of localized spins and the thermal boundary resistance above 2 kG comes from nonmagnetic mechanisms. On the other hand, the RK observed at zero field is attributable to two conduction mechanisms (magnetic and nonmagnetic) since these two mechanisms provide independent paths for the heat transfer across the boundary. As a result, one can separate the thermal conductances (the inverse of RK) at zero field as follows: hK(total)= hK(mag)+ h,(nonmag),

(5.1)

where hK(total)denotes the conductance at zero field. Using this relation, the purely magnetic conductance hK(mag) is obtained by subtracting the observed values hK(nonmag) at 3.85 kG from hK(total) observed at zero field. Figure 5.2 shows the thermal boundary resistances: RK(nonmag), RK(mag) and RK(total), which are obtained from the data at saturated vapor pressure (SVP)observed by Osheroff and Richardson (1985). It should be noted, as seen from fig. 5.2, that the &(mag) values plotted by black triangles have a dip at about 2.5mK. We understand also that at temperatures above about 4 mK the nonmagnetic mechanism is important for effective heat transfer and below 4 mK the magnetic mechanism is dominant.

I54

T. NAKAYAMA

0 0 0

D Fig. 5.2. Thermal boundary resistance (R,) between silver particles and liquid 'He at saturated vapour pressure (Osheroff a n d Richardson 1985). The black circles ( 0 )are the data for are R,(nonmag) at 0.385 T. The R,(total) at zero magnetic field a n d the open circles (0) black triangles (A)denote the magnetic contribution of R,(mag). The solid curve is the theoretical result obtained by eq. (5.9). After Nakayama (1986b).

First, let us discuss the results of R,(mag) plotted by black triangles in fig. 5.2. Hereafter, the arguments are concentrated on the case of normal liquid 'He by treating the data at SVP above 1 mK, and the superfluid phase of liquid 'He is not considered here. Provided that the 'He quasiparticle with momentum hk approaches the interface and is scattered by flipping the 'He nuclear spin due to magnetic interaction with localized magnetic impurities near the surface (fig. 5.3), the Kapitza conductance h,(mag) = I/R,(mag) is expressed as

Fig. 5.3. Schematic illustration of inelastic scattering of 'He quasiparticles by the magnetic coupling with thy magnetic impurities.

KAPITZA THERMAL BOUNDARY RESISTANCE

I55

where f ( k ) is the Fermi distribution function for the 3He quasiparticles and W,, is the transition rate of a 3He atom from an occupied state k to an empty state k’. In eq. (5.2),the localized spins contributing to the transition are expressed by the two-level system with an energy splitting A with distribution n(A). The expression for n ( A ) is important in the present analysis and we shall discuss it later. Now the transition rate W f i .from the state k is written down as 21T W,. = - / ( M + A

1, k‘lH’lk, M ) 1 2 6 ( ~ k ,Q - + A),

(5.3)

H’ being the interaction Hamiltonian of the two-level system. The magnetic coupling between 3He nuclear spins and the electronic spins is taken to be dipolar type expressed as V=

cB I d x I d y d a B ( x - y ) s , ( x ) s B ( y ) ,

0.

a,~=x,y,z,

(5.4)

where duB denotes the dipole interacting between the nuclear magnetic moment pn of 3He atom and pe of localized electron spin, which is (5.5) daa (1 r 1) = [ pepn(r26aa - 3 rarB )I/ r 5 . In eq. (5.4), s(x) and S ( y ) are the spin densities for 3He nuclear spin and localized electron spin. By substituting the relation S p ( y )= 1,S,(y -I?”) and the second quantized form of s,(x), the dominant interaction Hamiltonian becomes,

where a: and ak are the creation and annihilation operators for 3He atoms with the momentum trk and spin c.The factor V is the volume of a half space occupied by liquid ’He. The symbol S: = S ; +is’, expresses the raising operator for the spin state characterizing a localized state at site R, in the effective mean near the surface. The factor J(k, k‘) is the Fourier transform of dipole interactions in a half space. It has been shown (Mills and BCalMonod 1974a) that the dipole interaction behaves like an effective contact interaction when the heat exchange is dominated by scattering with momentum transfer of the order of the ’He Fermi momentum p -pF. Therefore one can take J ( k , k‘) to be the contact type by setting the dipole interaction erg cm3. d ( x ) = J 6 ( x ) , where J takes the value of 0.99 x Substituting eq. (5.3)into eq. (5.2), one obtains the Kapitza conductance hK as (Nakayama 1984b, 1986b), n(A)A3dA hK = (5.7) exp(A/ k B T )- exp( -A/ k B T )’

T. NAKAYAMA

156

where use is made of

The exchange enhancement effect KeR of quasiparticles (Stoner enhancement) is included through eq. (5.7), which increases the conductance h K = RK'by one order of magnitude for pure 'He. Now let us describe the nature of the distribution function n ( A ) in eq. (5.7) according to Nakayama (1984b). The important point in the random spin system (dipole spin glass, for example) is the potential energy as a function of the simultaneously specified orientations of all of the spins. In this connection, the specific heat of the random spin system is well described by assuming the two-level system as shown by Villain (1979), which implies that the transition involves the simultaneous rearrangement of a small number of spins. The width of the energy distribution function n ( A ) of the above-mentioned two-level system can be estimated as 5 = z& i', where z is a number of order unity representing the effective coordination number and i is the mean distance from any localized spin to the nearest one. This estimate of is reasonable since the effective field acting on any given spin is dominated by the few spins which happen to be the closest (fig. 5.3). Following the discussion by Villain ( 1979), the non-vanishing distribution function at A = O is assumed. The energy distribution function n ( A ) in eq. (5.7) is taken as a Gaussian such as

a

n( A ) =

__ A J ~ no

exp( -A2/ A ' ) ,

(5.8)

where no is the areal density of the two-level system near the surface in a projected mean. Combining the density no and the width L,we can write the distribution function n ( A ) as a function of one variable from down A = zp:ny2. By replacing the variable A in the integral of eq. (5.7) with a dimensionless one x = A / k , T , one can find the characteristic feature of the temperature dependence of the resistance. Using a dimensionless variable x, eq. (5.7) becomes

The dimensionless width A / k , T in the distribution function in eq. (5.9) varies with the temperature, namely, the width shrinks with increasing temperature. Combining the temperature dependence of the integral with

KAPITZA THERMAL BOUNDARY RESISTANCE

157

T 2 in the prefactor in eq. (5.7), one can obtain numerically that RK has a minimum around T, = 6/2.5ks for the Gaussian distribution. By using the areal density of magnetic impurities estimated as no= 1/(5 A)’, we can evaluate the temperature T, as 2 3 mK where the resistance RK(mag)has a minimum. Here the relation 6 = zpczn;” is used. It should be emphasized that this T, is in good agreement with the temperature where the observed RK(mag)has a minimum as seen from fig. 5.2. At higher temperatures than T,, we have a simple expression for the resistance from eq. (5.9) as

-

(5.10) erg cm3, m* = By taking suitable values for the factors J = 0.99 x 3.01mHe, kF=7.86x lo7 cm-’, K:,= 12, and n0=4.6x 1014cm-2,we find RK(mag)Z0.47X 1O8T(m2K/W).

(5.11)

This agrees with the empirical relation above 2.5 mK ( R K= 0.45 x lo8 T(m2K/W)) plotted in fig. 5.2 by black triangles. At temperatures much smaller than T,, RK(mag) is very sensitive to the shape of the energy distribution function close to A =O. If a simple Gaussian distribution function is taken, one has RK(mag) T P 2 .The theoretical curve obtained from eq. (5.9) is shown by a solid line in fig. 5.2, where the values of parameters noted below eq. (5.10) are used. In view of our simplified model the agreement should be regarded as satisfactory. It is concluded here that the characteristic behaviour of the magnetic resistance R,(mag) observed is well described by the dipole coupling between randomly distributed magnetic impurities and ’He nuclear spins. As remarked earlier, the nonmagnetic contribution to the thermal conductances h,(nonmag) should be identified as the observed conductances above 2 kG because there is no sign of field dependence there. It should be emphasized that the observed data still deviate appreciably from the theory for bulk solids in both magnitude (about 10-lo2 times smaller) and temperature dependence (see the data at 3.85 kG and zero field are shown in fig. 2.4). This discrepancy could be explained from the coupling between the low-lying vibrational modes in the silver-particle system and the collective excitation in liquid 3He as described in section 4.3 (see fig. 4.3). The validity of the soft-mode picture for our particle system has been confirmed by recent investigations (Maliepaard et al. 1985, Deptuck et al. 1985, Page and McCulloch 1986, Lambert 1985, Burton and Lambert 1986). It should be noted that Osheroff and Richardson (1985) observed the pressure dependence of RK for high magnetic fields near 1 mK (see fig. 3 in their paper): they found that RK at 29 bar is somewhat larger than the

-

158

T. N A K A Y A M A

observed value at SVP. This experimental finding seems to reflect the existence of the superfluid phase of liquid 'He, since at such high magnetic fields the acoustic coupling is dominant and this coupling is weakened by the superfluidity (slip boundary condition). To summarize, the anomalous thermal resistance between metal particles and liquid 'He observed in the mK region are attributed to two compatible magnetic and nonmagnetic mechanisms, i.e., the origin of unexpectedly small thermal resistances is not unique but comes from both magnetic and nonmagnetic mechanisms in the mK region. In particular, it should be emphasized that the experiments by Osheroff and Richardson (1985) have revealed for the first time the magnetic and nonmagnetic contribution of the anomalous thermal conductance between small particles and liquid 'He at mK temperatures. Finally, we should give a comment on the Kapitza resistance between sintered powder and solid 'He. In this case both of the magnetic and the acoustic coupling are important as well as the case of liquid 'He. At high magnetic fields, the magnetic coupling is frozen and the acoustic coupling becomes dominant. Because the velocity of acoustic phonons in solid 'He (which transfer heat) is close to that of zero sound in liquid 'He, the same temperature dependence and magnitude of R K as that for liquid 'He are expected for solid 'He at high magnetic fields. In fact, these tendencies were observed by Mamiya et al. (1983) and Greywall and Busch (1987). For the magnetic channel, heat is transferred by spin diffusion in solid 'He (Guyer 1973, Morii et al. 1979).

6. Thermal boundary resistance between liquid ' H c + ~ H ~mixtures and sintered powder

The cooling of a dilute 'He-4He solution down to a few tens of FK has been of great interest associated with the possible discovery of the superfluid transition in the dissolved 'He. This cooling is, however, very difficult due to the problem of the Kapitza resistance between the refrigerant and the helium sample. To date, the lowest temperature of a mixture achieved using sintered Ag powder is close to 200gK (Chocholacs et al. 1984, Ishimoto et al. 1987). The Kapitza resistance R , between the mixture and Ag sinter at mK temperatures has been observed by many researchers (Frossati 1978, Osheroff and Cotruccini 1981, Ritchie et al. 1984, Chocholacs et al. 1984). The results indicate that the observed R K between dilute solutions of 'He in 4He and sintered Ag powder are proportional to T-I. The 3He-4He mixture represents a unique Fermi liquid, whose density, and hence degeneracy temperature, can be varied at will. Noticing this point, it has

KAPITZA THERMAL BOUNDARY RESISTANCE

159

been pointed out (Ritchie et al. 1984, Chocholacs et al. 1984) that the observed R K between dilute solutions of 'He in 4He and sintered Ag powder vary as T;' (see the data given in figs. 6.2 and 6.3 in section 6.2). It is natural to consider two independent channels of heat transfer for the system. These are due to acoustic and magnetic mechanisms, which will be discussed in the following subsections (Nakayama 1988). 6.1. Acousric

CHANNEL

In the case of normal liquid 3He, the collective mode (zero-sound) has played a role in opening up the acoustic channel, as shown in section 4.3. For 3He-4He mixtures, one should notice that the collective modes (phonons) in a 3He-4He solution do not couple with the vibrational modes of sintered powder, since at such low temperatures the metal particles of micrometer size in the sinter vibrate without volume change of individual particles (due to the size effect), i.e., the low-energy vibrational modes of the sinter do not excite phonons into the 'He-4He solution due to the slip condition at the boundary (in the case of liquid 'He, the nonslip condition leads to the excitation of zero-sound). The plausible acoustic channel is via the direct interaction between 3He quasiparticles and the low-energy vibrational modes in the sinter, which has been suggested by Rutherford et al. (1984).

-

Sintered powders with packing fraction of about 0.4 0.5 are normally used for facilitating the cooling of the dilute 3He-4He mixture into the mK and sub-mK temperature range, where the sinter forms a three-dimensional percolating network and takes a fractal structure at shorter length scale than some characteristic length 6 (in a statistical sense). It is suggested that the sintered powder is fractal at short-length scales (Deptuck et al. 1985, Maliepaard et al. 1985, Page and McCullough 1986). Alexander and Orbach (1982) have conjectured from the scaling argument that the density of states (DOS) of percolation clusters for all Euclidean dimensions exhibits universally the w dependence in the regime above the characteristic frequency w,. This regime is called thefracton regime (localized mode) or short-lengthscale regime. There is also the regime called the phonon regime or longlength-scale regime, where the DOS obeys the conventional law D ( w ) w 2 sufficiently below w, (fig. 6.1). This conjecture has been confirmed by computer simulation (Grest and Webman 1984, Yakubo and Nakayama 1987). Lambert (1985) has made a computer calculation of the DOS for sinters with varying percolation density p. The effective medium theory (Demda et al. 1984) predicts that the DOS is proportional to w 2 below some characteristic frequency w , and becomes constant above w , , which is

"'

-

T. NAKAYAMA

I60

* U C

OD

Fig. 6.1. The expected density of states (DOS) of percolating cluster. w, is the crossover frequency between the Debye density of states ( w ’ ) and the ‘fracton’ density of states ( w ” ~ ) .

expressed as w<wc,

D ( w ) = ( : ) * 3 w D9N - 2w,’.

D ( w )=

9N 3 w D - 2w,

;

W>Wcr

where the frequency oDis the cutoff frequency corresponding to the mode with wavelength equal to an inter-particle distance and N is the number of metal particles per unit volume, respectively. Putting w, = 0, eq. (6.2) yields the same density of states as eq. (4.13). From the above arguments, one may conclude that the characteristic frequency w , of the sintered powder is an important factor for predicting the power of the temperature dependence of the Kapitza resistance. It should be emphasized also that the percolation clusters become elastically soft as a function of the percolation density p. This is involved in the sound velocity vSinof the sintered particles, i.e., the magnitude of the sound velocity becomes as small as one-quarter or less compared with the corresponding bulk values (Robertson et al. 1983). One must note, from the experiments on ultrasonic propagation (Robertson et al. 1983, Maliepaard et al. 1985, Page and McCulloch 1986), that sound with frequencies of several 10 MHz can be propagated through the sinter. This implies, at least, that the characteristic frequency w , is larger than several 10 MHz, which corresponds to Rw,/k,> 1 mK. Let us estimate the characteristic frequency w , for a sintered powder with packing fraction f = 0.5, which is assumed to be composed of spheres (1000 A in diameter). This choice of the numerical values is quite reasonable because the sintered powders used in the experiments are made from silver particles with the nominal size 700 A. Provided that the system takes a

-

KAPITZA THERMAL BOUNDARY RESISTANCE

161

simple cubic composed of spheres (1000 A in diameter), this has the filling factor fo and packing fraction f is given by

f = nu?J= f o p ,

(6.3)

where u and n are the volume of one particle and the occupied number of particles in unit volume. The percolation density is defined by p. From eq. (6.3), p is estimated to be 0.96 for the sinter off = 0.5. Using this value of p and taking into account the dimensionality of the system, the average distance between vacant sites becomes -3a (this was estimated from the relation a ( l -p)-"'- 3a). As a result, the approximately estimated characteristic length 5, becomes of the order of 5000 A. This length is related to the characteristic frequency Y, by v,= u s / & . Since the sound velocity of the sinter with packing fraction f 0.5 is us = los cm/s (Robertson et al. 1983), we have Y,- 2 GHz, which corresponds to a few tens of mK in a temperature scale. We should note here that the value of T, = 15 mK recovers well the acoustic channel of the Kapitza resistance for pure 'He at mK temperatures as shown in fig. 4.3. Thus, at subrnillikeluin temperatures, the Debye-phonon picture eq. (6.1) is valid for sintered powders composed of the particle size of around 1000 A. The dissolved 3He atoms confined in pores diffuse into the adjacent pores through narrow connecting channels. In this process, the sizes of the pores and channels become an important factor for the diffusion of 'He atoms. The following is a discussion of the energy spectrum of 3He atoms in pores and the diffusion of 'He atoms through channels; special attention is paid to the size of pores and channels (Nakayama and Yakubo 1987). Pores in a sinter have irregular shapes and constitute a random network as seen from electron microscope pictures (fig. 4.1). Consider, first, the special case where 'He atoms are confined in small pores of l 0 0 A in diameter which are connected by cylindrical channels of 7 A in diameter and 10 A in length. The zero-point energy Eo of 'He quasiparticles in the channel becomes much higher than that in a pore, namely, E,= 1.41 K for d = 7 A. Since the Fermi energy EF of a 5 % 'He-4He mixture becomes 0.33 K, 'He atoms must tunnel through the potential barrier with a height of I K in order to diffuse into the adjacent pore. The tunneling probability 1/r is estimated to be lo5s-' in this case (Nakayama and Yakubo 1987). The condition for the localization of 3He quasiparticles w r > 1 in the pore holds for the temperature region around 1 mK, where w is the dominant angular frequency of thermal vibration of the sinter at temperature T. In the actual sintered powder used in the experiments, the average pore size D is of the order of 1000 or more, and the size of channels connecting pores is of the order of one-tenth of 0, i.e., 100 A. For this situation, 'He atoms can move freely through the narrow channels in contrast with the

-

-

-

T. N A K A Y A M A

162

above case. Thus the dissolved ‘He atoms in the sintered powder can be treated as free particles with a continuous spectrum. The formula for the Kapitza conductance hK obtained in eq. (3.34) can be applied to our system. Since we are interested in the millikelvin or submillikelvin temperature range, the phonon density of states of eq. (6.1), which is proportional to w ’ , should be taken into account. In this case, the conductance becomes, using the formula for the Fermi ‘gas’ derived in eq. (3.37),

h K=

cktktT’

(6.4)

15h2p,o:,,’

where c is a number of order unity and ps is the mass density of the sinter, respectively. The Kapitza resistance ( h K = 1/ R K ) is proportional to T-’ and TL’ ( - kE4) in contradiction to the experimental results. The magnitude of h , at 1 mK is also too small to explain the data. It should be noted here that us,, is the velocity of sound in the sinter, and eq. (6.4) is identical with eq. (4.17) when the mass density pL in eq. (4.17) is replaced by that of dissolved ’He atoms. 6.2. MAGNETICC H A N N E L

It has been reported that the Kapitza resistance R , of the interface between dilute ’He-4He mixtures and sintered powder varies as R K T in the millikelvin temperature region (Radebaugh et al. 1974, Frossati 1978, Osheroff and Corrucini 1981, Ritchie et al. 1984, Chocholacs et al. 1984). In addition to the above temperature dependence, Ritchie et al. (1984) and Chocholacs et al. (1984) have investigated the effect of varying the ‘He concentration in the solution, that is, the Fermi temperature TF, on R K . The results indicate that R K is proportional to T i ’ (fig. 6.2). These T-* and T i ’ dependences are not recovered from the acoustic channel mechanism in the submillikelvin temperature region as presented in section 6.1. In this section, let us consider another mechanism of heat transfer between ’He dissolved in He I1 and sintered powders. This is due to the magnetic coupling between ’He quasiparticles and magnetic impurities (i.e., 0 2 ,O ? , 0, 0-, etc.) in the vicinity of the surface of sintered powder (Nakayama 1984b), which has played an important role in the case of pure ‘He-sintered powder interfaces (see section 5 ) . One might suppose, at first sight, that magnetic coupling is irrelevant to the heat transfer due to the fact that ‘He atoms are preferentially adsorbed at the surface of the sinter, and interrupt the effective coupling. This is not always true for the following reasons. For example, even for the pure ‘He-sintered powder interfaces, the first few ’He adsorbed layer are localized at the interface and they are not important for direct coupling (dipole

-

KAPITZA THERMAL BOUNDARY RESISTANCE

I 10

20

I

I

50 100 200

500 1000 2000

163

I

TF(~K) Fig. 6.2. The Fermi temperature dependence of observed R, between a ' H e 4 H e mixture and silver particles. After Chocholacs et al. (1984). The data are taken from Ritchie et al. (1984): (A),Osheroff and Corruccini (1981): ( O ) ,Frossati (1978): ( O ) ,Chocholacs et al. (1984): ( A ) .

coupling) between 'He quasiparticles in bulk liquid 3He and localized magnetic impurities. That is, although adsorbed 'He atoms play a role as a second-order process in the perturbation theory, they do not contribute to the direct energy exchange (the first order) from the sinter into liquid 'He or vice versa. In addition, the observation of the TFdependence of R K for 3He-4He solutions is the clear evidence of the irrelevance of the first few adsorbed 4He layer to the heat transfer. This result is reasonable because the magnetic dipole coupling is proportional to the distance F 3and is not a short-range one. From the above argument one can apply the formula R K of eq. (5.9) to the case of the heat t.ransfer between dissolved 'He atoms and the sinter as well. The Kapitza resistance R K due to the magnetic coupling between dissolved 3He nuclear spins and magnetic impurities is expressed, using formula (5.9), as

In the above formula, the Stoner enhancement factor K c f l in eq. (5.9) is omitted because the exchange interaction between 3He quasiparticles in 4He is negligibly small. The energy distribution function n ( A ) in eq. (6.5) should take the same form used in section 5, which is expressed as

n0 n ( A) =7 exp( - A 2 / Z 2 ) , A&

164

T. N A K A Y A M A

where the width d of the distribution function n ( A ) can be estimated in a similar manner to that in eq. (5.8),

A= &if3,

(6.7)

where z is a number of order unity representing the effective coordination number and i is the mean distance from any localized spin to the nearest one. The width of the distribution function d / k , T in the integral of eq. (6.5)varies as the temperature. Combining n ( A ) with the factor x3/(ex- e-") &) in eq. (6.5),the integral can be easily obtained as r ~ ~ 7 ~ ~ / ( 4atdsufficiently low temperature T < d/k, where the width d takes a value of 5 mK as estimated in section 5. As a result, the Kapitza resistance'due to the magnetic coupling between dissolved 3He atoms and the sinter in the submillikelvin temperature region is expressed as

-

It should be emphasized that eq. (6.8) varies as T P 2and T,' (- k F 2 ) whose , dependences are in agreement with the experimental features reported by Ritchie et al. (1984) and Chocholacs et al. (1984). By taking suitable values for the factors for a 5% solution, J = 0.99 x erg cm3, m* = 2.46 m 3 , loa

5E

Y 106 l o 7 r

v h

cr'

Osherof f - Corruint 8 0 % Frossati et al 6 L */. Ritchie el d 0 3 o/. Rild7ie et d 1 2 .I.

0

A

%!I

'

'

"""' 1

'

'

"'.'.I

10

'

'

h , , d

100

T (mK) Fig. 6.3. Thermal boundary resistance R , between a 3He-4He mixture and Ag-sinters as a function of temperature (Frossati 1978, Osheroff and Corruccini 1981, Chocholacs et al. 1984, Ritchie et al. 1984). Straight lines are theoretically calculated by eq. (6.8) (Nakayama 1988).

KAPITZA THERMAL BOUNDARY RESISTANCE

165

EF= 4.57 x lo-’’ erg, and no = 4.0 x loi4cm-2, we find R K = 32 T-2 (m’ K/W) in the submillikelvin temperature region. This result is plotted by the straight line (below) as well as the experimental data in fig. 6.3. The upper line is for a 1.3% solution: RK=80 T-’(m2 K/W). It is remarkable that the observed T-’ and T;’ dependences of RK are recovered by the magnetic coupling model. If the magnetic coupling is a dominant channel between 3He-4He mixtures and sintered powders, the similar magnetic field dependence on RK with that for pure 3He will be observed. Although it is reported by Ritchie et al. (1984) that the observed R K for 3He-4He mixtures is insensitive to the magnetic field (-3 T) above 5 mK, it seems that this point requires further experimental investigation; especially the data at submillikelvin region are interesting.

7. Anomalous Kapitza resistance between liquid He and a bulk solid above about 1 K

7.1.

KAPITZA CONDUCTANCE

hK

A N D PHONON TRANSMISSION

COEFFICIENT ACROSS THE INTERFACE

The problem of the anomalous Kapitza resistance observed in the temperature range 1 K-2 K has a long history compared with that described in previous sections (Beenakker et al. 1952, White et al. 1953, Fairbank and Wilks 1955, Dransfeld and Wilks 1958, Challis et al. 1961, Kuang Wey-Yen 1962). Experiments on RK between liquid (or solid) helium and copper are described in section 2.3.2 and some typical results have been shown in fig. 2.5. The observed data deviate significantly from the acoustic mismatch prediction (RKT3= const.) above about 1 K. Thus, the anomalous Kapitza resistance observed above about 1 K presents an interesting problem as well as that in the mK temperature region. In section 2.2, the theory of Kapitza resistance has been dealt with from a different point of view from the conventional treatment by Khalatnikov (1952, 1965) and Little (1959), which incorporates the transmission coefficient t ( 0 ) of a phonon incident at angle 8 across the interface. The quantity t ( 0 ) is determined by applying the boundary condition at the interface, where the effects of the total or internal reflection of bulk (B) phonons as well as the surface waves propagating along the interface are involved. How the transmission coefficient t ( 0 ) is introduced in the AM theory is shown briefly in the following. When a phonon with energy h o is incident from liquid helium into a solid, an energy h o f L S ( 0is) transmitted to the solid. As a result, the heat

T. N A K A Y A M A

I66

flux

o,

from liquid He into a solid is defined by

where the angular integral ( 0 ) should be performed in the half space. The Bose distribution function for phonons is expressed as nB( f i w / k B T )in eq. ( 7 . 1 ) , and cL is the velocity of phonons in liquid He. Assuming the simple Debye density of states for phonons, which is valid at temperatures considered here, the integral of eq. (7.1) yields 0L.S

=

7r'pLkicLT' FLS, 30psh3V:

(7.2)

where the factor FLsis given bv

F,

=

ps 2Pl.

('> ' jo'rLS(

0 ) cos OL d(cos 0,).

(7.3)

CI.

If the transmission coefficient r ( 0 ) is not a function of frequency w , the heat flux will be proportional to p,and the product R K T 3 has no adjustable parameters, and hence is constant. From the data plotted in fig. 2.5, one sees that there are two distinct temperature regions in which the physical properties of R, are quite different, namely, T s 0 . 1 K and T a 1 K, i.e., there is a decrease by one order of magnitude in the quantity R,T3 as the temperature is increased from 0.1 K to 1 K. Characteristic features are as follows: ( i ) Above 1 K, RK is essentially the same for liquid 'He or 'He, solid 'He or 'He, while below 0.1 K the magnitude of RK for those several forms of He are quite different (see fig. 2.5). ( i i ) RK is very sensitive to surface treatment (Challis et al. 1961, Kuang Wey-Yen 1962, Johnson and Little 1963, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Opsal and Pollack 1974, Synder 1976, Rawling and van der Sluijs 1978, 1979). As seen from eqs. (7.1)-(7.3), the Kapitza conductance h,, defined by h K = aOLs/aT, involves information on the transmission coefficient f ( 0) through the factor FLs.Defining the average coefficient by the equation

oLs

f1.s

=2

1,)'

~ L S ( ~ Lcos )

0L 4 c o s 0 d ,

(7.4)

the conductance h , can be expressed approximately as

hK/ 7'

= aiLs,

where a takes a constant value determined by material constants.

(7.5)

KAPITZA THERMAL B O U N D A R Y RESISTANCE

167

In fig. 7.1, the Kapitza conductances h K divided by T 3 (eq. (7.5)) are plotted from the data of the Illinois group (Anderson et al. 1964, 1966, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The arrows indicate the theoretical values predicted by the AM theory for various phases and isotopes of helium. From the same figure, one can conclude that the transmission coefficient fLs depends strongly on temperature T above -0.1 K. This implies that the average transmission coefficient fLsdepends on the frequency w of incident phonons, in contrast with the AM theory. It is generally believed that, below 0.1 K, the modified AM theory incorporating phonon attenuation is .valid; in particular, the attenuation of Rayleigh waves plays a role in modifying the AM theory for ideal surfaces (Andreev 1962, Peterson and Anderson 1972, Haug and Weiss 1972). This point has been confirmed experimentally by Zinov'eva (1978,1980) and Zinov'eva and Sitnikova (1983). Since a in eq. (7.5) is proportional to pLcL, it is expected, if the acoustic mismatch picture is valid, that the data for h K / T 3depend on whether the He is solid or liquid. However, experimental data above 1 K show that the value of h K / T 3becomes almost the same for solid He and liquid He. These experimental facts suggest that some unknown mechanisms of effective heat transfer are present in this temperature range.

:?:.*:-i

Liqrid 4Hc--c

,

'O-", 01

1.0

T(K)

Fig. 7.1. The Kapitza conductance h , divided by T3as a function of temperature. The definition of symbols is the same as that in fig. 2.5. The arrows at the right side indicate the prediction from the acoustic mismatch theory.

T. N A K A Y A M A

168

-

At around 1 K 2 K, where the Kapitza resistance (or conductance) shows anomalous behaviour, excited phonons have a frequency range of 80- 160 GHz because their frequency is related to T through hw,,, = 3.8kBT. The average transmission coefficient above about 1 K, i.e., above 80 GHz, can be estimated from fig. 7.1 as

i= 0.1.

(7.6)

This value is two orders of magnitudes larger than the value from eq. (2.5) obtained by applying the acoustic mismatch boundary conditions. Thus, the Kapitza problem around 1 K-2 K is linked closely with the explanation of the anomalous features of high-frequency phonon transmission across the liquid He-solid interface.

7.2. HIGH-FREQUENCY PHONON HE-SOLIDI NT E R F AC E

SCAITERING

AT THE L I Q U I D

In the preceding section 7.1, it has been shown that the Kapitza conductance h , involves information about the transmission coefficient t ( 0 ) averaged over the polarization of phonons, propagation direction, and frequency w. In order to understand the origin of the anomalous Kapitza resistance, it is more desirable to observe directly the transmission (or reflection) coefficient t ( 8, w ) of high-frequency phonons with definite polarization, propagation direction, and frequency. In recent years, with the advent of high-frequency-phonon generation and detection techniques, it has become possible to study the scattering of phonons of known polarization, frequency and propagation direction at the interface. The generation of high-frequency phonons utilizes various techniques such as heat pulse, tunneling junction, and thermal conduction etc. (see the book edited by Bron 1985). Using these techniques, a great deal of effort has been devoted to the study of the phonon transmission mechanism at the liquid He-solid interface. Early attempts were made by Trumpp et al. (1972), Guo and Maris (1972,1974), Sherlock et al. (1972), Ishiguro and Fjeldly (1973), Swanenburg and Wolter (1973), and Kinder and Dietsche (1974). The heat pulse technique has the advantage that, through the time of flight analysis of a heat pulse, one can identify which modes and frequencies play a role for effective phonon transmission. Using the heat pulse techniques, Taborek and Goodstein (1979, 1980) succeeded in distinguishing clearly the phonon modes when reflecting at sapphire surfaces in contact with liquid 4He (fig. 7.2). Heat pulse experiments have confirmed that diffuse signals in phonons reflected from a surface are severely affected by placing liquid He (rather than vacuum) at the interface (Horstman and Wolter 1977, Weber et al. 1978, Folinsbee and Harrison

KAPITZA THERMAL BOUNDARY RESISTANCE

169

TimeIpsl

Fig. 7.2. Time-of-flight signals due to backscattering of phonons at a sapphire surface. The dotted line indicates the reduction of diffuse signals at the surface in contact with liquid 4He. The numbers 1.2 and 3 correspond to longitudinal (L),fast transverse (FT),and slow transverse (ST) phonon, respectively. After Taborek and Goodstein (1980).

1978, Taborek and Goodstein 1980, Marx and Eisenmenger 1981, 1982, Basso et al. 1984, Eisenmenger 1986). Important experimental facts obtained using these new techniques are summarized as follows: (i) Transmitted phonons into liquid He obey a cosine (Lambertian) law (Wyatt et al. 1974, Mills et al. 1975, Sherlock et al. 1975, Wyatt et al. 1976, Wyatt and Page 1978). See fig. 7.3. (ii) Surface irregularities are necessary for effective energy transfer (Weber et al. 1978, Basso et al. 1984, Mok et al. 1986). See fig. 7.4. (iii) Anomalous transmission into liquid He occurs above about 80 GHz (Sabisky and Anderson 1975) and there seems to exist an isotope effect between 3He and 4He (Koblinger et al. 1983, Heim et al. 1983). See fig. 7.5. (iv) Diffuse signals in time-of-flight reflection experiments: signals are severely affected by placing liquid He instead of vacuum at the interface. See fig. 7.2. As a first step, one must clarify the origin of diffuse signals observed in phonon reflection experiments at surfaces nor in contact with liquid He (Nakayama 1985, 1986a). Klitsner and Pohl (1986, 1987) have made some extensive experimental studies on the cause of diffuse scattering at Si-crystal surfaces by deposited thin films (without liquid He), through measurements of the thermal conductivity in the boundary scattering regime. Their results will be discussed at the end of this section along with the theoretical analysis of the cause of diffuse scattering.

T. NAKAYAMA

Fig. 7.3. The angular dependence of phonon emission from the cleaved surface of NaF to liquid ‘He at a heater temperature of 2.1 K. The measurements were made at 24 bar. After Wyatt and Page (1978).

Fig. 7.4. Phonon pulses reflected from LiF surfaces: dashed-dotted line, before cleaving in vacuum; solid line, freshly cleaved at I K; dotted line, with helium. After Weber et al. (1978).

KAPITZA THERMAL BOUNDARY RESISTANCE

171

Phonon Frequency [ G H t I 50

100

1

I

150

200

250

,

I

1

I

85 GHz

R

1 L

0 02

1

OC

T=lK I

I

06

I

I

I

08

1

10

I

12

Phonon Energy CmeV1 Fig. 7.5. Effective reflection coefficient R / R , as a function of phonon frequency. R is normalized to unity at 75 GHz ( R J . A strong breakdown of the phonon reflectivity at the ~ o l i d / ~ H e boundary occurs when the phonon frequency exceeds 85 GHz. The experimental situation is illustrated in the inset. After Koblinger et al. (1983).

7.2.1. Cause of di#iuse scattering at the surface without liquid He

The cause of diffuse scattering of the phonons actually lies in the surface irregularities which violate translational invariance parallel to the surface. Various surface irregularities can be considered: a rough surface, imperfections like dislocations in the vicinity of the surface, and surfaces covered by chemisorbed or physisorbed impurities. Hereafter, our arguments are concentrated on the case of rough surfaces, since the complicated surface state does not lend itself to simple understanding of the problem. Our task in physics seems to be to understand phenomena in as simple a way as possible. Provided that an isotropic elastic continuum occupies the half space z 2 0 with a stress-free boundary at z = 0, the displacement vector at a point x = (r, z ) and time t can be expanded in terms of eigenmodes (Ezawa 1971),

[ a J u J ( z exp(ik. ) r-iwt)+H.c.],

(7.7)

where ps is the mass density of the medium and J = (k,c, m ) labels a set of quantum numbers which specifies the eigenmodes of phonons, k is a two-dimensional wave vector, c the velocity of a waue front traversing the surface, and m specifies the mode. The sum over J in eq. (7.7) is defined as

(7.8)

T. N A K A Y A M A

172

where D, denotes the spectral range of the velocity c. R represents the Rayleigh mode (representative surface mode) whose amplitude decreases exponentially with the distance z from the surface. In eq. (7.7), u, and its Hermitian conjugate a: are the annihilation and creation operator of the ]-mode phonon, respectively. There arefive eigenmodes specified by 1 (Ezawa 1971). For the transverse (T) phonons, there exist two kinds of eigenmodes which have a velocity (of a wave front traversing the surface) spectrum c a cT, where C, is the velocity of T phonons. The first mode is the TH mode polarized parallel to of incidence and reflection is the surface [see fig. 7.6(a)]. The angle related to c by cot' 6 H = p 2 ( c )= (c/cT)' - 1 and the range of the velocity c is from cT (OH = ~ / 2 )to infinity ( & = 0). Note that c is related to the incident angle OH. The other T mode (referred to as the TV mode) consists of T phonons polarized in the sagital plane followed by evanescent pseudosurface-waves [see fig. 7.6(b)J. The velocity of a wave front c of this mode is confined in the finite range c T s c s cL, where cL is the velocity of longitudinal (L) phonons. Another mode consists of mixed longitudinal (L) and transverse (TV) waves with vertical polarization, which interact with each other through the surface. Let us consider a rough surface whose height from the plane ( z = 0) is given by a function f ( r ) , where r is the two-dimensional position vector. The bumps of roughness can be described as a mass density fluctuation. The spatial dependence of the mass density fluctuation p ( x ) is expressed by combining the roughness function f(I ) and the Heaviside step function as g(x) = goo(z +f(r ) ) , where go is the mean value of the mass density. For f ( r ) small compared with the wavelength of phonons, one can expand the step function as e(z+f(r)) = e(z)-f(r)Wd.

(7.9)

Thus, the random part of the mass density is separated as Ap( r ) = pof( r ) , where Ap( r ) has dimensions of g cm-'. The perturbed Hamiltonian due to

Surface

"

TH

I

\

TH

TV

I

!

TV

b

Fig. 7.6. ( a ) Transverse mode polarized parallel to the surface. This mode is denoted as the TH mode. (b) Transverse mode polarized in the vertical plane (TVmode). The longitudinal part i s localized in the surface.

KAPITZA THERMAL BOUNDARY RESISTANCE

173

the mass density fluctuation (Nakayama 1976, 1985, 1986a) is represented by

(7.10)

[a:(t)--J(f)l[a:'(f)--J'(f)l

where F ( k + k') is a two-dimensional Fourier transform of the roughness function f(r ) and k is the two-dimensional wave vector, respectively. The transition rate of the J-mode phonon into the J'-mode phonon is obtained as

X

1:

U j ( Z ) . U j f ( Z ) 6 ( Z ) d2

l2.

(7.11)

In eq. (7.11), the ensemble-averaged Fourier transform of the roughness function is defined by

( F ( k+ k')') = S

J

-

d r exp[i(k+ k') r](f(r)f(O)),

where S is the normalized area introduced by integrating over r. Following the formulation of scattering theory in quantum mechanics, the differential cross section of J-mode phonons is expressed in terms of (7.11) as, da(J+J')=

flw,y(J+ J ' ) 9

QJ

being the incident energy flux of J phonons with the velocity c,, given

QJ

- 12 P S / U J / 2 w 2 c J s -

(7.12)

Let us illustrate the case in which the transverse phonons polarized parallel to the surface (hereafter referred to as TH phonons) are incident at angle OH on a rough surface and are scattered into bulk and surface phonons (hereafter referred to as B phonons and R phonons, respectively). The displacement vector of TH phonons is written as (7.13)

T. N A K A Y A M A

174

where

UfH(Z)

=o.

Here wA is the angular frequency, k the two-dimensional wave vector parallel to the surface, and P’ = ( c/ c, ) ?- 1. The energy flux of TH phonons can be given from eq. (7.12) by

The cross section of T H phonons into TH phonons is obtained as (7.14) where 4 is the angle between the two-dimensional vector k and k‘, and the factor W is obtained by assuming the white noise for the correlation function as W=

(aG)’w4 87~’~:

’

where a is the characteristic length parameter of the surface roughness, and -

Ap is the average amplitude of surface density fluctuation. In a manner

identical to that used in deriving eq. (7.14), the cross sections of the other processes ( J + roughness -* J ‘ ) contributing to the time-of-flight spectra are obtained ( Nakayama 1986a). For the other important process converting TH phonons into R’ phonons (TH +roughness+ R’), one obtains d(r(TH + R’) =

TW

sinZ4ff d 4 2c,ci K

(7.15)

Here the numerical factors f,and K are written as fi

= 1 -2y77/(1+

v2L

K = ( Y - 7 I( Y - II + ~ Y T ~ ) / ~ Y T ’ , where y z = 1 - (cR/cT)’ and v 2= 1 - ( cR/cL)’. Although the cross section expressed by eq. (7.15) is not observable directly in the time-of-flight reflection experiment due to the position of the detector, R phonons converted at the rough surface should be rescattered into bulk (9) phonons because of the roughness, and constitute the diffuse signal in the time-of-flight experiments. The details will be given in section 7.2.3.

KAPITZA THERMAL B O U N D A R Y RESISTANCE

175

7.2.2. Speculur versus difluse scattering of bulk phonons

High-resolution time-of-flight phonon-reflection experiments have revealed that the reflected signals are composed of both specular and diffuse parts (Taborek and Goodstein 1980, Marx and Eisenmenger 1981, 1982, Burger et al. 1985). The scattering probability of a TH phonon incident at an arbitrary angle OH with angular frequency w into the diffuse part is expressed by hw T(TH + J’) hoT(TH + R) (7.16) tdiff =

x,,

+

1

QH

OH

where T(TH + J‘)is integrated over the scattered angle: T(TH + J’)=

II

y(TH -+ J ’ ) dx d4.

The first term of eq. (7.16) can be explicitly rewritten as

Here [JB] is the set of all J’s except R. The factor F, = I, + I , + I3 is the numerical constant which has the value 3.5, where the first term (II= 7r) corresponds to the process TH + TH, and 1, and I , correspond to the decay processes into TV phonons and L phonons. The transition rate of a TH phonon into a R phonon becomes

Hence, the first term of eq. (7.16) becomes t,(w,T H +

B) =

(a 6 ) 2 w 4 F 1

8 7 r 2 p ~ c ’~

(7.17)

and the second term is obtained as (7.18)

It is clear from eqs. (7.17) and (7.18) that the component of ‘the diffuse scattering increases with increasing frequency proportional to the fourth power of frequency. The component of the speculur reflection coefficient is obtained by extracting the part of the diffuse scattering from unity as

r,= l - ( t , + t z ) .

(7.19)

T. NAKAYAMA

I76

The sapphire surfaces (mildly anisotropic) used in experiments (Taborek and Goodstein 1980, Northrop and Wolfe 1984) have a roughness scale of the order of 6 = 100 A. This indicates that the surface can be treated with mean variation in depth and width of 6 = 100 with the areal density w = 0.56 -'. The characteristic length of roughness should correspond to the length scale a given in eqs. (7.17) and (7.18), and one can replace (ohp)' by w ( A M ) ? ,where the averaged mass of a bump is estimated as A4 = psa3= 3.99 x lo-'"g for a = 100 A. The resultant probability of diffuse scattering for TH phonons of frequency v in GHz yields for sapphire crystal, rd,ff = 1 0 - ' ~ ~ ~ ,

(7.20)

where the following values for sapphire are used: cL= 1 1 x lo5 cm/s and c T = 6 x l o 5 cm/s. Note that tdim is proportional to the fourth power of frequency. Equation (7.20) indicates that bulk phonons with frequency around 100GHz are scattered dominantly into a difuse part and, for sufficient low frequencies ( < 100 GHz), most of the incident phonons are specularly reflected. It should be emphasized that the probability of diffuse scattering rdirf cannot exceed unity. From this condition one can estimate the frequency regime where the present analysis is valid. In the case of a roughness parameter 6 = 100 A, one has the condition Y < 200 GHz. The value is reasonable because the corresponding wavelength of 200 GHz phonons becomes about 300 A. This implies that the present theoretical model is valid for phonons with larger wavelength than the roughness scale a. If the wavelength A of phonons becomes much shorter than the roughness scale a, the geometrical scattering by roughness becomes much more relevant. For A a one expects a crossover, which would correspond to the transition between specular and diffuse scattering.

-

7.2.3. Difuse signals in the time-ofiflight reflecrion signals

Let us consider the case in which the heater and bolometer are very small and close together. Figure 7.7 shows the geometry of our system with the definition of the thickness of crystal h and the polar coordinate r. Each element of the area d A = r d r d 4 on the top surface is irradiated by B phonons emitted from the heater of the Lambertian source and the element dA re-radiates the phonons. Since the heater and bolometer are assumed to be very small and close together, the bolometer detects only the phonons backscattered with the same angle as that of incident phonons, hence the element dA can be considered as a new source. Defining S ( r ) as the heat flux emitted by the heater and taking into account the time delay of arrival at the bolometer t = 2 d / c , , where d 2 = r 2 + h Z , the fraction of the reflected

KAPITZA THERMAL BOUNDARY RESISTANCE

177

rough surface

/

Source 0, Detector Fig. 7.7. Geometric arrangement of the system. The heater and bolometer are assumed to be close together.

intensity is expressed by dR, =

d a ( J + J ’ , c, 4) coS2 8 S ( t - 2 d / C ~ ~ ! ) ( r 2 + h2)2

(7.21)

+

Here ~ j j is, defined as ~ j j=, 2cJcJ‘/(cJ c J , ) .The cross section d a ( J + J ’ ) is obtained by eqs: (7.14)-(7.15). The diffuse signal R,(t) as a function of time is obtained by integrating eq. (7.21) over r (over the irradiated surface) and assuming the heat pulse to be described by a delta function

S ( t ) = S( t&. This is valid for a crystal of about 1 cm in thickness because heat pulses used in the experiments were of 10-100 ns duration. The relation between r in eq. (7.21) and x in eqs. (7.14)-(7.15) is obtained from

+

x 2= COt28 1 = (h/ T)’

+ 1,

x = C/ cJ.

Using this relation, the diffuse signal, which is a function of time, is represented by R, ( t ) = So

C J. J ’

jO2= 1r d4

dr

d o * ( J + J ’ ) COS28S ( ~ - ~ ~ / C J J , ) . (7.22) (r2+ h2)2

Here the definition of the cross section is d a * ( J + J’)= d a ( J + J‘)/dx d 4 . Component of Rphonons in difuse signals. B phonons scattered at the surface have a high probability of mode conversion into R phonons. To see the effect of mode-converted R phonons on the diffuse signals, we consider the transition rate of R phonons by roughness. The inverse of the lifetime of R phonons into B phonons can be obtained by using eq. (7.11) as well:

T(R+ B) =

2 Wf:o’F,

c:cRK

(7.23a) ’

T. N A K A Y A M A

I 7x

This transition rate is identical to that obtained by Maradudin and Mills (1976) except for the numerical factor. For the process ( R + roughness + R), the transition rate is obtained as (7.23b) From the ratio of eqs. (7.23a) and (7.23b), one can conclude that the transition rate of R phonons into R phonons is 3.21 times as large as that of R phonons into B phonons: f ' ( R + R) = 3.21/*(R+ B). The effective lifetime of R phonons taking into account eqs. (7.23a) and (7.23b) is 7R

=

s,

(7.24)

where v is in the GHz range. In deriving eq. (7.24) the values for sapphire are used. By taking v = 50-100 GHz, T~ becomes 10 ns. These results indicate that, when R phonons propagate along the surface, R phonons should be backscattered into B phonons with the lifetime 7R. Figure 7.8 shows the calculated shape of reflection signals. The thicknesses of crystals are taken to be h ;= 0.5 cm. For the calculation of the

-

0

1

2

3

Time(psec1 Fig. 7.8. Calculated results o f reflection signal taking into account all phonon modes ('urve A comes from mode conversion to B phonons. Curve B represents the component of the mode-convened R phonons. The thickness of the crystal is taken to be h = 0.5 cm. The peaks ( 1-61 correspond to the processes L -, L, L -,TV. L + TH, T V -,TV, TV + TH and TH + TH. respectively. After Nakayama ( 1986a).

KAPITZA THERMAL B O U N D A R Y RESISTANCE

179

time-of-flight reflection spectra, the parameters for sapphire are used by identifying the velocity of slow transverse phonons (ST) as that of TH phonons and the fast transverse phonons (FT)as TV phonons; cST= 6.0 x 10’ cm/s and cFT= 6.5 x los cm/s, and cL= 11.0 x lo5 cm/s. Curves 6, 5 and 4 correspond to the T phonons TH + TH, TH + TV, and TV-,TV. Curves 3, 2 and 1 are due to the L phonons L-, TH, L + TV and L-, L. Curve A comes from the mode conversion from B phonons to B phonons, and curve B shows the component of the mode-converted R phonons. The ratio of height between the curve A( B + B’) and B( B -,R-, B’) is defined by PJ-J*=

da(J +R+ J’) d a ( J + J‘) *

(7.25)

The ratio of height of curves 6A and 6B is PTH-TH = 0.2 and of 2A and 2B is ~ ~ ~ - ~ = (Nakayama 0 . 0 7 1986a). These are calculated from eq. (7.25). In fig. 7.8 curves 2 and 3 are rounded in comparison with the other processes. This indicates the absence of the forward scattering in the processes L + TH and L-,TV. This is attributed to the fact that the roughness has been simplified as the mass defects in the present calculation. 7.2.4. Reduction of the diffuse signal at the solid surface in contact with

liquid He It has become clear in the preceding section that the diffuse signals are composed of high-frequency phonons with the frequency above about 100 GHz. This is consistent with the experimental results of heat transfer, in which the anomalous heat conduction across the interface is observed above about 1 K (see fig. 7.1.). Burger et al. (1985) have observed only specular reflection and no He effect for low frequency phonons. For high frequency phonons a remarkable change of the diffusely scattered component was found when the Si surface was in contact with liquid He. Thus it is necessary to explain the reason why high-frequency phonons contribute to the effective heat transfer, by incorporating the surface irregularities. This subject has been studied theoretically in two different points of view: the modification of acoustic mismatch (AM) theory by incorporating surface irregularities (Little 1961, Adamenko and Fucks 1970, Haug and Weiss 1972, Peterson and Anderson 1972, Khalatnikov and Adamenko 1972, Sheard and Toombs 1974, Shiren 1981, Shen et al. 1981), and the quantum mechanical extension taking into account the interaction between phonons and the He system or adsorbed impurities (Toombs and Challis 1971, Anderson and Johnson 1972, Rice and Toombs 1972, Sheard et al. 1973, Cheeke and Ettinger 1976, 1979, Nakayama 1977, 1985, 1986a, Maris 1979, Kinder 1981, Kinder and Weiss 1986).

I80

T. NAKAYAMA

Let us consider the latter viewpoint, in which the quantum mechanical interaction between He system and phonons in a solid is taken into account. Apart from poorly defined surfaces such as metal surfaces (hard to handle theoretically), one can consider well-Characterized surfaces such as those of sapphire used in the phonon reflection experiments. It has been well accepted for these cases that the first few adsorbed layers of He are immobile at sufficiently low temperatures with a density similar to that in bulk solid He at a pressure of about 100atm (Brewer et at. 1965). At a location far from the range of the attractive substrate potential, the liquid He should maintain its bulk properties (superfluid). The He atoms between the first two adsorbed layers and bulk liquid are bound weakly to the substrate and their motion is quite restricted. In fact, how to describe this state is still open to question (e.g., Lauter et al. 1983). One possibility is to regard it as a dense fluid with no long-range order at temperatures around 1 K (Nakayama 1977, 1985) where most experiments of phonon reflection and transmission have been performed. An important task in understanding the phonon transmission quantummechanically is to determine the type and strength of coupling between phonons and the He system (Mans 1979, Nakayama 1985, 1986a, Haug et at. 1987). Under circumstances in which phonons are incident into the surface in contact with liquid He, the interaction has the effects of scattering and energy absorption. The two types of interaction considered for our system are the displacemenr-type coupling and the deformation coupling.

Displacement-rype coupling. The displacement-type coupling between adatom and substrate phonons has been studied for various problems since the work of Lennard-Jones and Strachen (1935). Here we consider a TH phonon incident at angle 0 to the surface (see fig. 7.6a). The interaction Hamiltonian can be represented by the quadratic form with respect to the relative displacement between the He atom and the substrate (7.26) where a is the coordinate of the He atom measured from the equilibrium position, and f is the coupling constant between the He atom and the substrate. The one-phonon (THmode) absorption probability is defined by

where is the incident energy flux defined by eq. (7.12) and hwd is the energy difference between the ground state and the first excited state of the He atom bound in the attractive potential from the substrate. Using first-order perturbation theory, the transition rate (s-') for normal incidence

KAPITZA THERMAL BOUNDARY RESISTANCE

of the TH mode phonon

r D

181

due to the coupling eq. (7.26) becomes, (7.27)

where N / A is the number of He atoms per unit area. In deriving eq. (7.27) the relation w i =f/mHe is used. As a result, one has the one-phonon absorption probability (7.28)

It should be noted that this absorption probability is expressed by the given physical parameters, and the sum rules presented by Mans (1979) can also be obtained from eq. (7.28). Using the sum rules, Maris (1979) has discussed in detail the phonon absorption by adsorbed He atoms. The two-level tunneling state (TLS) model was introduced for liquid 3He or 4He by Andreev (1978), Andreev and Kosevich (1978) in explaining the observed T-linear specific heats above the quantum degenerate temperature (no long-range order). The concept of the TLS’s for the He system is quite analogous in many respects to the TLS model in glasses originally proposed by Anderson et al. (1972) and Phillips (1972). The TLS’s are responsible for the universal low-temperature properties shared by all configurationally disordered systems. There are two essential differences in the TLS’s between liquid He and glasses. One of them is due to the high tunneling probability of the He atoms because of the large overlap of wave functions. The second one is that the density of states per unit energy n ( E ) of the TLS’s is larger than that of glasses by a factor of the order of lo4 as will be shown in the following. The TLS model has also been introduced independently by Nakayama (1977, 1985) for the He system close to the interface, where the positions of He atoms should be quite irregularly distributed (no long-range order). This is a particularly attractive possibility because it leads to lowenergy states, and it appears that a phonon should couple well to these states (Nakayama 1977, 1985, 1986a). The maximum energy difference Em can be estimated to be 1 0 0 K from the binding energy of the van der Waals potential to the substrate. The magnitude of the level density Y( E ) per one He atom becomes z / Em,where z is the number of neighbouring vacant positions. By taking z - 5 and Em 100 K, the density of states per atom can be estimated as v ( E ) = 3 x lOI4 erg-’. The number density of He atoms in the first few adsorbed layers is N 1015cm-’, so that the density of states per unit area no becomes

-

-

-

no= 3 x loz9erg-’ cm-’

.

T. N A K A Y A M A

182

This is quite large compared with the density of tunneling states i n glasses erg-' cm-'. per unit area: no= The absorption rate of a TH phonon by the TLS with a broad distribution of energy difference can be obtained by replacing the function S ( f i w k h w , ) N / A in eq. (7.28) by no. Taking account of the temperature dependence, one has (7.29) By using the explicit value of no= 3 x lo2' erg ' cm-*, the absorption probability I,, (corresponding to the transmission coefficient)becomes

I , ,= 6.9 x l 0 - ' v 2 , where v is expressed in GHz. If one considers a phonon of 100 GHz incident at the surface, the rate becomes ID= 6.9 x This is too small to transfer energy effectively in the frequency regime considered here, as concluded by Maris (1979). Dejonnorion coupling. The other important interaction arises from the coupling proportional to the strain called deformation coupling. We should bear in mind that the physical nature of He atoms close to the boundary includes the contribution from the He atoms and the substrate surrounding the He atoms. When a TH phonon is incident at the rough surface, the substrate surface atoms will be deformed by an incident phonon, i.e., the phonon works as a deformation coupling proportional to the strain enp.The He atoms close to the surface should change the states by rearranging the atomic configuration quantum-mechanically from this coupling. As a consequence, the He system close to the intertace has a new energy state E,-, which differs from the initial one E,. Because the spread of the wave packet of He atoms is small with respect to the spatial change of the strain, one can estimate the atomic energy difference as

(7.30)

The strength of deformation coupling constant g, = aE/ae,,p for the TLS can be estimated from eq. (7.30) by postulating the complete deformation of eOp= 1. Due to this deformation the change of the binding energy of the He atom should be of the order of the van der Waals potential so that the deformation coupling constant g becomes about 100 K. Thus, the interaction Hamiltonian between a phonon with the wave vector k and the He system

KAPITZA THERMAL BOUNDARY RESISTANCE

183

is expressed in the second quantized form, (7.31)

Hk = 8 k T k * x ,

where the operator a, is the Pauli matrix and 7, the strain of a TH photon. Using eq. (7.31) and assuming a wide distribution of the energy difference, the absorption probability of TH phonons becomes (Nakayama 1986a) fs = 27rg2nowtanh[ Psc:

21.

(7.32)

If one takes the values for the mass density of a solid and the velocity of a TH phonon for sapphire as ps = 3.99 g ~ m - ~c, ,= 6.0 x lo’ cm s-’, and g = 100 K, the absorption probability for frequency v in GHz is ts -- 2.56 x 10-6v.

(7.33)

For the incident phonon of v = 100 GHz, the absorption probability becomes t, -- 2.56 x This is too small to explain the experiments (-lo-’) as well as the case of the displacement coupling obtained in eq. (7.29). Apart from phonon absorption by the adsorbed He system, there is an interesting possibility that adsorbed air molecules constitute two-level tunneling states identical to those of glasses (Kinder 1981, Schubert et al. 1982, Basso et al. 1984, Kinder et al. 1985). The phonon absorption rate for this case is obtained as well by using eq. (7.32), which is the same as that obtained by Kinder (1981). From the density of states per unit volume obtained for fused silica, no= cm-3 erg-’, as a typical value, one can estimate the density of states per unit area as no= lo2’ cm-* erg-’. The deformation coupling constant is known to be of the order of g = 1 eV for glasses. Using these values, the scattering rate for frequency v in GHz yields t,=1.12x10-6v.

(7.34)

This is also too small to explain the phonon-reflection experiments as well as eq. (7.33). In eqs. (7.29), (7.33) and (7.34), it has been shown that the direct interaction process of phonon absorption (B phonons to He system) is negligible. These theoretical results are natural in some sense for the following reason. The cause of interaction (for example, TLS’s) postulated is present in the two-dimensional sheet. When bulk (B) phonons traverse the sheet, the number of TLS’s interacting with B phonons is so small that these states do not yield observable effects on the transmission or reflection coefficient of bulk phonons. This conclusion is consistent with the recent reports by Klitsner and Pohl (1986, 1987), who made a systematic study of phonon scattering at polished silicon surfaces by deposited thin films in the temperature range 0.05-2.0 K corresponding to dominant frequencies

T. N A K A Y A M A

184

from 5 to 180GHz. They prepared ex- or in siru-deposited thin films of metals, nonmetals, and condensed gas with average thickness from 2 8, to lo4 8, as well as a clean and a rough surface. Depositions and measurements are done in truly clean (ultra-vacuum) conditions using a remarkable method. In the course of these experiments, they found that diffuse scattering is associated with the islands of discontinuous thin film, or microscopic structural irregularities. There was no evidence of scattering by some kind of unknown interface state or individual atoms or molecules at the interface. Let us remind ourselves of the experimental evidence that diffusely scattered phonons'play a key role in the effective transfer of energy. It has been clarified that diffuse signals are due to two causes. One is the direct scattering of B phonons at irregular surfaces (B+ roughness -* B), and the other is due to the mode-converted R phonons ( B+ roughness .+ R -,B). Thus, another possibility for effective energy transfer occurs through the interaction between R phonons and the He system: The R phonons converted from B phonons at the rough surface interact with the He system, and the energy of R phonons is absorbed by the He system (see fig. 7.9). Note that R phonons propagate along the interface, and the scattering process is quite different from the case of B phonons mentioned earlier. I f the lifetime of mode-converted R phonons due to interaction with the He system ( T H ) is shorter than that of R phonons due to scattering by roughness ( T ~ ) , the phonon energy converted into R phonons should be transferred into the He system. As a consequence, the component of R phonons of diffuse signals (-20%) should vanish when liquid He is present. This process has been discussed in detail by Nakayama (1985, 1986a). in which both the displacement-type and the deformation coupling between R phonons and TLS's have been taken into account. The lifetime due to the displacernenttype interaction between R phonons and the He system is calculated in a straightforward way by replacing the surface displacement uTH(0) in eq. (7.26) by that of R phonons. Taking the numerical values for sapphire, we have the lifetime of R phonons with frequency i n GHz under the condition L

8

L

4

8

Fig 7.9. Two possible channels o f energy transfer. Process ( 1 ) represents the direct procesb, and ( 2 ) represents the R phonon mediated process of energy transfer.

KAPITZA THERMAL BOUNDARY RESISTANCE

A w ~> 2 k s r

185

*

7 H . d a 0.88 x i o - 2 V - 3 s. (7.35) The lifetime of R phonons due to the deformation interaction can be obtained by using eq. (7.31) and the wave function of R phonons. The numerical result for R phonons with frequency in GHz for sapphire gives

7H,s = 1.02 x 1 0 - ~y - 2 s. (7.36) It should be noted that the frequency dependence of eqs. (7.35) and (7.36) is different from that of bulk phonons. This is because of the fact that the energy density of R phonons is localized in the vicinity of the surface depending on its wavelength, i.e., the energy density is frequency dependent. By comparing the lifetimes of R phonons due to scattering by roughness and the defects (TLS), one can conclude that the mode-converted R phonons can be absorbed effectively by the He system for a frequency of about 100 GHz. If this is the case, the diffuse tail arising from the mode-converted R phonons vanishes when the surface is in contact with liquid He at around 100 GHz of phonon energy, i.e., the diffuse signal to the mode-converted R phonons (curve B in fig. 7.8) vanishes. This picture is also valid for the process postulating the presence of LTS in adsorbed air molecules. Finally, the possibility should be mentioned that adsorbed air molecules and He atoms combine together to form defect states (e.g., TLS’s), because it may be wrong to think of the exposed surface and He system as having a definite boundary. This means that adsorbed air molecules are not closely packed, and He atoms can penetrate between them. This possibility has been pointed out by Vuorio (1972), although the description is made in a quite different manner. In particular, if the wavelength of incident bulk (B) phonons is comparable with the thickness of these layers (matching layer), the interaction between B phonons and the He system must be affected, and this would constitute a highly absorbing layer for phonons. In any case, the modeconverted surface phonons play a key role in anomalous phonon transmission. It is worth mentioning the situation where a solid surface (characterized by roughness scale a ) in contact with liquid He receives phonons with much shorter wavelength than the roughness scale a (A < a ) . In this limiting case, the phonon transmission coefficient should equal, if only the acoustic mismatch boundary condition is taken, the value predicted by the acoustic mismatch theory for bulk solid (see the discussion in section 2.2). The calculation incorporating only surface roughness must recover this limiting conjecture. This section has reviewed the current understanding of the puzzling phenomenon of the Kapitza resistance above about 1 K, by enlightening the application of new techniques of generation and detection of highfrequency phonons. Although the problem has become clearer in recent

I86

T. NAKAYAMA

years due to the accumulation of experiments and theories, we have to admit that some detailed work has yet to be done to clarify the mechanism of high-frequency phonon transmission.

8. Summary The important interesting features of the Kapitza thermal boundary resistance have been reviewed. Section 2 was devoted to the introduction of the Kapitza thermal boundary resistance, and has cleared up the subject by comparing the existing data with the acoustic mismatch theory. In section 2.3, the anomalous Kapitza thermal boundary resistance observed at the liquid 'He-sintered powder interfaces at mK temperatures, reported for the first time by Avenel et al. (1973), has been discussed together with the anomalous Kapitza resistance observed at the liquid He-bulk solid interface above about 1 K. Section 3 has outlined the Fermi liquid theory for the Kapitza resistance, by illustrating energy transfer from a small particle into liquid 3He. In this section, the contribution from both the zero-sound excitation and the inelastic scattering of single quasiparticles have been explained theoretically. Sections 4 and 5 dealt with the main topics of this article: the Kapitza resistance between liquid 3He and sintered powder at mK temperatures. The acoustic channel of heat transfer has been described in section 4 by incorporating the characteristic features of elastic properties of sintered powders. In this section, it has been pointed out that the soft-phonon modes peculiar to sintered powder play an important part in the effective heat transfer. The contribution of the magnetic coupling to the effective heat transfer in the case of the liquid 3He-Ag powder interfaces was discussed in section 5 . Here, it was shown that magnetic coupling plays a relevant role in the anomalous heat transfer as well as the acoustic channel discussed in section 4. Especially, it is important to point out that magnetic impurities (i.e., 0 2 , 0;. 0, 0-, etc.) at the silver surface play a key role in the anomalous heat transfer. In section 6, the Kapitza resistance between liquid 'He-4He mixtures and sintered powder was discussed, illuminating the observations ( R K E ;' and T-') at millikelvin temperatures. Here, both the acoustic and the magnetic channels have been discussed in detail, and it has been suggested that the same magnetic coupling in the case of the pure 'He-Ag powder interface is important. In particular, it is remarkable that the observed Kapitza resistances for liquid 'He and 3He-4He mixtures are understood in a consistent way.

-

KAPITZA THERMAL BOUNDARY RESISTANCE

187

Section 7 reviewed the current aspects of investigations on the longstanding problem of the Kapitza resistance (probably from the time of the 1950's) about about 1 K, by paying attention to the application of new techniques of high-frequency phonon-generation and detection to this problem. There are two interesting problems on the Kapitza resistance, which have been omitted in this article. These are the Kapitza resistance between magnetic materials and liquid 3He, and the Kapitza resistance between two quantum media like the solid He-liquid He or liquid He-hydrogen gas interfaces. Experimental reports up to 1979 on the former problem are described in detail in the review by Harrison (1979). Recent experiments on this subject have been reported by Saito (1981) for CTS, Jutzler and Mota (1984), Batey and Main (1984) and Marek et al. (1986) for CMN, Fujii and Shigi (1987) for CPS, Jacquinot et al. (1986) for HoVO,, Mizutani et al. (1987) for TmVO,, and Ytterboe et al. (1987) for PrNiS. Although these topics are not yet fully understood, the rapid developments in the past several years encourage experiments and theories.

Appendix: Surface vibrations of a small particle

In order to calculate the energy flux given by eq. (3.22) from a small particle into liquid 3He, one must obtain the surface displacement u ( R ) from eigenmodes of phonons in a particle. The displacement vector u ( r , t ) is expressed through a scalar potential 4 , and vector potentials 42and 4', using the equation u( r, 1 ) = grad

4 , + curl 42+ curl 4'.

(A.1)

The first term represents the longitudinal mode with dilation, and the others are two transverse modes. Without loss of generality, the vector potential is taken as

4, = ( r , 0,O)4i,

i = 2,3.

(A.2)

The above definition of the vector potential gives us a good perspective for determining the eigenmodes in the spherical body. Consider a small particle in contact with liquid 'He. Since liquid 'He compared with that of the has a small mass density ( p L = 0.0815 g small particle, the appropriate boundary conditions determining the eigenmodes may be taken as those for a stress-free surface, which are expressed in the following forms u a P ( R )= O ;

a,p = r, 8, 4.

(A.3)

T. N A K A Y A M A

18R

This condition allows us two types of oscillations in a spherical particle: One is the spheroidal mode and the other is the toroidal mode. The spheroidal mode is expressed by the sum u J .= ~ grad

4 I + (YJ curl curl 43,

(A.4)

and the toroidal mode is UJ,, = curl 4 2 ,

(A.5)

where the lower suffices ‘s’ and ‘t’ mean the spheroidal and toroidal mode, respectively. The J acts for a set of quantum numbers specified by (1, rn, w), where 1 and rn give the order of spherical harmonics and w is the angular frequency. The factor aJ in eq. (A.4) indicates the ratio of the second to the first term. The potential functions are expanded by the associated Legendre polynomial P ;I(cos 0) and the spherical Bessel function j r ( x ) as

4,(r, r ) = C A:”’j,(k,r)P;I(cos0 ) cos rn4 1. m

x exp( -iwkr),

(rn s 1, j = 1 , 2 , 3 ) .

(A.6)

Here the suffix j means the longitudinal ( j = 1) and the transverse ( j = 2 , 3 ) modes, respectively. A;“‘ are the expansion coefficients. The wave number k, is defined by the relation @k = k,cj. From the boundary conditions expressed by eq. (A.3),the eigenvalue equations are obtained as

(A.7)

where $, = k, R and q = k,R ( i = 2 . 3 ) . The above equation is solved numeri2 k:/k: = cally with respect to the single variable using the relation v 2 / t = (2 P + A )IP. The displacement u obtained from the vector potential & has no radial component of oscillation as understood from the definition of eq. (A.2) expressed as ul(r,t)=curl$,=

(

0,-sl,:O:?~

-a? ):

KAPITZA THERMAL BOUNDARY RESISTANCE

I89

For instance, taking I = 1 and m = 0 in the potential eq. (Ah), we have

42= A:.oj,(k2r)P,(cos

0) e-'"'.

(A.9)

Substituting eq. (A.9) into eq. (A.8), the displacement u becomes u , ( l = 1, m =0) = (O,O, -A:"j,(k2r)

sin 0) e-i"'f.

(A.lO)

This yields the lowest torsional oscillation, which is called a roroidal mode. are called the The modes determined from the potentials 4, and spheroidal modes, which have the radial component of surface oscillation. The eigenvalue equation (A.7) has been solved numerically by Nakayama and Nishiguchi (1981), and Nishiguchi and Nakayama (1982) for silver and copper particle.

Acknowledgements I am grateful to my colleagues, K. Yakubo and N. Nishiguchi, for many valuable comments and discussions. I have benefitted greatly from conversations and correspondence with L.J. Challis, J.D.N. Cheeke, K. Dransfeld, W. Eisenmenger, G. Frossati, Y. Fujii, H. Fukuyama, J.P. Harrison, H. Kinder, M. Kubota, H. Ishimoto, H.J. Maris, F. Pobell, K. Rogacki, S. Saito, T. Shigi, F.W. Sheard, and A.F.G. Wyatt. I would like to thank K. Kimura and K. Yakubo for producing the typescript and figures. This research was granted in part by the Iwatani Naoji Foundation's Research Grant, the Suhara Memorial Foundation, and a Grant-In-Aid from the Ministry of Education, Science and Culture, Japan.

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Rutherford, A.R., J.P. Harrison and M.J. Scott, 1984, J. Low Temp. Phys. 55, 157. Sabisky, E.S., and C.H. Anderson, 1975, Solid State Commun. 17, 1095. Saito, S., 1981. Phys. Rev. 24, 459. Saito, S., T. Nakayama and H.Ebisawa, 1985, Phys. Rev. B 31, 7475. Schubert, H.. P. Leiderer and H. Kinder, 1982, Phys. Rev. B 26, 2317. Sheard, F.W., and G.A. Toombs, 1974, J. Phys. C 4.61. Sheard, F.W., R.M. Bowley and G.A. Toombs, 1973, Phys. Rev. A 8, 3135. Shen, T.J., D. Castiel and A.A. Maradudin, 1981, J. Phys. (France) 42, C6-819. Sherlock, R.A., A.F.G. Wyatt, N.G. Mills and N.A. Lockerbie, 1972, Phys. Rev. Lett. 29, 1299. Sherlock, R.A., N.G. Mills and A.F.G. Wyatt, 1975, J. Phys. C 8, 300. Shiren, N.S., 1981, Phys. Rev. Lett. 47, 1466. Swanenburg, T.J.B., and J. Wolter, 1973, Phys. Rev. Lett. 31, 693. Synder, N.S., 1976, J. Low Temp. Phys. 22, 257. Taborek, P., and D. Goodstein, 1979, J. Phys. C 12, 4737.

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Taborek, P., and D. Goodstein, 1980. Phys. Rev. B 22. 1550. Toombs, G.A., and L.J. Challis, 1971, J. Phys. C 4, 1085. Toombs. G.A.. F.W. Sheard a n d M.J. Rice, 1980, J. Low Temp. Phys. 39, 273. Trumpp, H.J., K. Lassmann a n d W. Eisenmenger, 1972, Phys. Lett. A 41,431. Villain, J., 1979, in: Ill-Condensed Matter, eds R. Balian, R. Maynard and G . Toulouse (North-Holland, Amsterdam) p. 522. Vuorio, M., 1972, J. Low Temp. Phys. 5, 1216. Weber. J.. W. Sandmann, W. Dietsche and H. Kinder, 1978, Phys. Rev. Lett. 40, 1469. Wheatley. J.C., 1968, Phys. Rev. 165, 304. Wheatley, J.C., 1975, Rev. Mod. Phys. 47, 415. Wheatley, J.C., O.E. Vilches a n d W.R. Abel. 1968, Physica 4, 1. Wheatley, J.C.. R.E. Rapp and R.T. Johnson, 1971, J. Low Temp. Phys. 4, 1. White, D.. O.D. Gonzales and H.L. Johnston, 1953, Phys. Rev. 89, 593. Wyatt, A.F.G., 1981, in: Nonequilibrium Superconductivity, Phonons, a n d Kapitza Boundaries (Plenum Press. New York) p. 31. Wyatt, A.F.G., and G.J. Page, 1978, J. Phys. C I t , 4927. Wyatt, A.F.G., N.A. Lockerbie a n d R.A. Sherlock, 1974, Phys. Rev. Lett. 33, 1425. Wyatt, A.F.G., G.J. Page and R.A. Sherlock, 1976, Phys. Rev. Lett. 36, 1184. Yakubo, K., and T. Nakayama. 1987a. in: Proc. 18th Int. Conf. on Low Temperature Physics, Jpn. J. Appl. Phys. 26, suppl. 26-3. 883. Yakubo, K., and T. Nakayama, 1987b, Phys. Rev. B 36, 8933. Ytterboe, S.N., P.D. Saundry, L.J. Friedman. C.N. Gould and H.M. Bozler, 1987, in: Proc. 18th Int. Conf. on Low Temperature Physics, Jpn. J. Appl. Phys. 26, suppl. 26-3, 379. Zinor'eva. K.N.. 1978, Pis'rna v Zh. Eksp. & Teor. Fiz. 28, 294 [Sov. Phys.-JETP Lett. 28,2691. Zinov'eva. K.N.. 1980, Zh. Eksp. & Teor. Fiz. 79, 1973 [Sov. Phys.-JETP 52. 9961. Zinov'eva. K.N.. 1985, in: Low Temperature Physics (Mir, Moscow) p. 78. Zinov'eva. K.N.. and V.I. Sitnikova. 1983, Zh. Eksp. & Teor. Fiz. 57, 576 [Sov. Phys.-JETP 57. 3321. Zubarev, D.N., 1974. Nonequilibrium Statistical Thermodynamics (Plenum Press, New York) p. 248.

CHAPTER 4

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS IN DRIVEN CHARGE DENSITY WAVE CONDENSATES BY

G. GRUNER Department of Physics and Solid State Science Center, University of Calgornia Los Angeles 90024, USA

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elseoier Science Publishers B. V., 1989 195

Contents

.................. I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Basic notions and observations in charge density wave dynamics . . . . . . . . . . . . . . . 2.1. The charge density wave ground state and model compounds 2.2. The dynamics of the collective mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Frequency and field dependent transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 3. Current oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 1 . General features . ............................................ 3.2. Current-frequency ............................................ 3.3. Size effects and fluctuation phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Broad band noise . . . . . . . . . . . . . . . . . . . . 4 . Models of charge density wave dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . I . The classical particle model . . . . . . . . . . . ........................ 4.2. Models including the internal degrees of 4.3. The tunneling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Interference phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Harmonic mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Subharmonic mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Nonsinusoidal and pulse drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Fluctuations and coherence enhancement . . . . . . . . 6 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of review papers . . . ... .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

196

197 201 20 1 205 211 217 217 220 223 226 227 227 232 237 239 239 246 255 259 262 265 266

1. Introduction

In highly anisotropic metals, electron-phonon interactions may lead to a new type of ground state at low temperatures with features both similar to and different from the well-known superconducting ground state. In contrast to superconductors, the condensate is formed of electron-hole pairs with a total momentum q = 2kF where kF is the Fermi wavevector, reflecting the 2 k , singularity of the Linhard response function in one dimension. The electron density has a spatial dependence, and in one dimension it is given by Ap = p o + p I cos(2kFx+4),

(1.1)

where po is the unperturbed electron density, p , and 4 are the amplitude and phase of the spatially oscillating charge density, called the charge density wave (CDW). The period of the charge density wave is determined by the Fermi wavevector and for a partially filled electron band, the CDW is incommensurate with the underlying lattice. Like the superconducting ground state, the charge density wave state also develops below a certain transition temperature (called the Peierls transition temperature Tp, Peierls 1955), and the mean field treatment of the thermodynamics of the phase transition is essentially the same as that of a superconductor. The central feature of both ground states is the development of a single particle gap A, which has a well-defined relation to the transition temperature, and of a collective mode which also determines the low frequency electrodynamic response. Charge density waves, such as given by eq. (1.1) have been observed in several inorganic and organic, linear chain compounds where the anisotropic crystal structure leads to highly anisotropic bands. The transition temperatures are, in general, somewhat below room temperature, and the appearance of single-particle gaps at *kF turns the metals into semiconductors, or semimetals with the gaps also observed by optical studies. The periodic modulation of the charge density leads also to periodic modulation of the atomic displacements, and therefore, the period can be determined by structural studies. The scattering intensity is proportional to A ( T), and available experiments are in fair agreement with a BCS-like temperature dependence of the order parameter (Fleming 1981). The collective mode can couple to the applied external electric field, and this for a dc electric field of sufficient magnitude, may lead to the translational notion of the condensate as envisioned by Frohlich (1954). For an 197

c. GRUNER

198

incommensurate C D W vanishingly small elctric fields would, in principle, be able to displace the collective mode. However, impurities, lattice imperfections, etc., pin the C D W to the underlying lattice, and this leads to a so-called threshold field E,, below which the condensate is pinned, and above which the current-carrying state, with nonlinear current-voltage characteristics is observed. For small impurity concentrations the threshold field is small, of the order of 10 mV/cm in various materials. The restoring force due to imperfections also shifts the oscillator strength associated with the ac response to finite frequencies, and the response can, in general, be adequately described by a simple expression, the same as that of a harmonic oscillator except at low frequencies, where additional contributions to a(@) occur, either due to the tunneling process or due to the internal degrees of freedom of the condensate. The pinning frequency oois much smaller than d / h , and the response is in general, overdamped, U ~ T >1. The effective mass m* is large, because in the dynamical response the kinetic energy of the electrons and ions has to be included, and values of m * / m , - lo3 with mb the band mass are typical. Because of the total momentum 2kF associated with the electron-hole condensate, the charge density wave ground state has a fundamental periodicity with the wavelength A = .rr/k,. The translational motion of the CDW, with a drift velocity Ud, leads to a characteristic frequency fo = v d / h ,which corresponds to the frequency related to the displacement of the condensate by one wavelength. The dc current is given by I = nevd with n the number of electrons in the collective mode. For a one-dimensional metal at T = 0, n = 2kF/ 71; and consequently the relation between current density per chain j andf;, is given by j/.f0 = 2.

These oscillations have first been observed by Fleming and Grimes (1979) and the linear relation, eq. (1.2) was later demonstrated by Monceau and coworkers (Monceau et al. 1980). The nonlinear and frequency dependent conductivity, together with the current oscillations, can be described by phenomenological equations of motion. Among the models, which assume that quantum effects can be neglected, the simplest, called the classical particle model (Griiner et al. 1981) is formally identical to that of the resistively shunted Josephson junction. Although the model does not account for all the details of the w and E dependent response it reproduces several of the main observations, including the current oscillations with the relation given by eq. (1.2). It is not surprising therefore, that a broad variety of observations can be made on materials with a C D W ground state, which are similar to those made on Josephson junctions. The combined application of ac and dc fields, for

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199

example, leads to various ac interference phenomena, some of which, in reference to the close relation to findings in Josephson junctions, are in general referred to as Shapiro steps. These have been investigated recently in detail, and observations on charge density waves have also been used to test various theories of nonlinear dynamical behavior, such as questions concerning transitions to chaos in deterministic systems. An alternative description of the nonlinear current-voltage characteristics is based on the concept of tunneling processes (Bardeen (1986, 1987) and references cited therein). The model gives a nonlinear current-voltage characteristic somewhat different from that of the classical particle model, and a periodic potential which is nonsinusoidal. The frequency dependence is accounted for by a formalism which has been applied earlier to superconductor-insulator-superconductor (SIS) junctions. With the form of the potential and of the I - V curve given, the main features of the current oscillations and of the various interference effects can be described without further assumptions. The experiments which study the dynamics of the collective mode can be classified as follows. 1. Frequency and electricfield dependent conductivity where the time average current ( j (1 ) ) is measured in the presence of dc or small amplitude ac drives. The dc conductivity is defined as a d , = (j(t ) ) / Edcwhere Edcis the applied electric field, and the small amplitude ac response is characterized by uaac = Re u ( w )+ i Im a ( w ) where Re a ( w ) and Im a ( w ) are the real and imaginary components of the conductivity. Here it is assumed, that linear response theory is appropriate, and the response to a sinusoidal ac drive E,, = E,, sin w t is also sinusoidal, with no higher harmonics. Because the dc response is nonlinear, it is obvious that this is not valid for arbitrary ac fields, and nonlinear ac effects may occur for finite fields. These experiments serve as tests of the various models which lead to predictions concerning the dc and ac response of charge density waves, and the ac response can also be used to evaluate the fundamental parameters of the problem, such as the effective mass m*,the pinning frequency wo and damping constant I / T. Frequency dependent conductivity studies have been performed over a rather broad frequency range, from audio to submillimeter wave frequencies, and by now the w dependent response of the pinned mode is fully characterized in many materials. Similarly, the dc conductivity has also been explored in detail, and compared with various theories. Experiments performed at finite fields and frequencies have also been conducted, and the nonlinear ac response has also been examined in detail in various materials. Such studies are supplemented by the measurements of rectification and harmonic mixing, both performed over a broad range of

200

G . GRUNER

frequencies and applied dc and ac fields. These studies will not be summarized here; it should be mentioned, however, that (perhaps not surprisingly) models which account for the dc and small amplitude ac response are successful in describing the response in finite fields and frequencies, and also in the presence of joint ac and dc excitations. 2 . Sepctral response in the current carrying stare. In the nonlinear conductivity region, the spectral features of the current include a large amplitude

broad band noise and current oscillations (often called narrow band noise N B N ) , with a fundamental given by eq. (1.2) and several harmonics f n = nso with slowly decaying intensity. The current oscillations have a finite spectral width, and also display temporal fluctuations which can be studied by examining the time dependence of the Fourier transformed current. The origin of the current oscillations has been studied by using imaginative lead configurations for current and voltage contacts and also by employing thermal gradients to break up the coherence which leads to current oscillations. The observation of current oscillations is suggestive for a significant coherence throughout the specimens, and it is expected that the range of current-current correlations (j(f , 0), j ( t, r ) )is comparable to the dimensions of the samples. The finite widths associated with the current oscillations (when viewed by employing a spectrum analyzer which detects the Fourier transformed current) and the broad band noise cannot be explained by external noise terms, and represent the dynamics of the internal degrees of freedom within the condensate. 3 . Interference experiments where both dc and a c field are applied, and where either the dc or the ac response is measured. Many of the observations are similar to those made on Josephson junctions, but the relevant frequencies for CDW dynamics are in the radio frequency, instead of in the microwave range. Features of mode locking between the intrinsic current oscillations and the externally applied ac field are closely related to the features of the current oscillations, but additional effects, such as synchronization and coherence enhancement by external drives can also be investigated by the joint application of dc and ac drives. Also, in contrast to microwave signals, waveforms different from sinusoidal can be applied at radio frequencies, and these can be used to examine phenomena such as transitions from the pinned to the current carrying state in detail.

This review focuses on the current oscillations and interference phenomena in an attempt to give an overview of the progress which has been made in the field, and to summarize the open questions, both from the experimental and from the theoretical point of view. The paper is

C U R R E N T OSCILLATIONS A N D I N T E R F E R E N C E EFFECTS

201

organized as follows. First, the basic notions and experiments on charge density wave dynamics are summarized, with emphasis on the frequency and field dependent response. This is followed by the summary of experiments on current oscillations, and by the discussion of attempts made to establish the origin of oscillating response in the nonlinear conductivity region. A short description of the various models is given, followed by a detailed discussion of the various interference effects which occur in the current-carrying CDW state driven by both dc and ac electric fields. The status of the field and a summary of open questions concludes this review. Several reviews on the various aspects of charge density waves have appeared recently and these are listed at the end of the paper.

2. Basic notions and observations in charge density wave dynamics 2.1. THECHARGE

DENSITY WAVE GROUND STATE AND

MODEL COMPOUNDS

The condensate which is the focus of this review has many features which are similar to superconductivity. Simple arguments which show the essential features of the ground state are as follows. Consider a one-dimensional ( 1 D ) metal. In the absence of an interaction with the lattice, the ground state is as shown in fig. l(a), where the electron states are filled up to the Fermi level and the underlying lattice is that of a periodic array of atoms with lattice constant a. As first pointed out by Peierls (1955) this state is not stable for a coupled electron-phonon system. In the presence of an interaction between the electron and the lattice, it is energetically more favorable to distort the lattice periodically with period A related to the Fermi wavevector kF, h = 7T/kF.

(2.1)

A lattice distortion with this period opens up a gap at the Fermi level, as shown in fig. l(b) where the situation appropriate for a half-filled band is drawn. As states only up to *kF are occupied, the opening of the gap leads to the lowering of the electronic energy. In contrast, the lattice distortion leads to an increase of the elastic energy, but in one dimension the total energy (electronic+lattice) is lower than that of the undistorted metal (this is the consequence of the divergent Linhard function at q = 2kF in 1 D). Consequently a distorted state is stable at T = 0 K. The gap opening also leads to the modification of the electron density, much in the same way as in the nearly free electron theory of metals. The electron density p = /+I2 will be a periodic function of the position x with the period given

G. GRUNER

202

-

0

0

0

0

0

0

0 'atoms

a

-1

mcta I

p(rl

00

00 >

20

00

OO\

atoms

insulator Fig. I . Peierls distortion in a one-dimensional metal with a half filled band. ( a ) Undistorted metal, ( b ) Peierls insulator.

by (2.1), and determined by the band filling. Thus, for an arbitrary band filling, the period of this modulated charge density (and the accompanying periodic lattice distortion, see fig. I(b)) will be incommensurate with the underlying lattice. At finite temperatures, normal electrons excited across the single-particle gap A screen the electron-phonon interaction. This in turn leads to a reduction of the gap as the temperature increases and eventually to a second-order phase transition, as in the case of a superconductor. The above features of this so-called Peierls transition and of the collective mode can be described by the mean field treatment of the one-dimensional electron-phonon Hamiltonian

H

=

trw:b:b,

c; V,, Re a,, measured at 3.2 MHz strongly increases, and the ac dielectric constant VT,rnV

VT,rnV

10

0

50

-

I

O:

200

400

Vn,rnV

Fig. 23. dc threshold voltage versus ac amplitude for several applied ac frequencies. The period of the oscillations with ac amplitude is roughly proportional to the ac frequency. T h e solid lines are curves calculated by using eq. (5.4). (After Latyshev et al. 1987.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

Nb9, T:42K

-

245

80,

oooo 0O0

dc boos volloge (mV1

Fig. 24. Real and imaginary part of the ac conductivity Re u ( w ) and dielectric constant c ( w ) and imaginary measured at w / 2 7 r = 3.2 MHz as a function of applied dc bias voltage. T h e threshold field is indicated by an arrow. (After Zettl and Griiner 1984.) 4

related to the ac conductivity by € ( l o )=

4 n Im a , , ( w ) 9

w

(5.7)

strongly decreases for increasing V,, . The dielectric constant approaches zero for large dc drive showing that there is no appreciable out-of-phase

G . GRUNER

246

component Re uac( o),on the other hand, approaches the high-frequency limit obtained from the frequency dependence of the small amplitude ac response (observed for the pinned charge density wave). In addition to the overall behaviors described above, figs. 24a and b also show that for NbSe3 both Reu,,(w) and E ( W ) have sharp anomalies for well-defined values of Vd, in the nonlinear conductivity region. Specifically, Re uacshows "steps" to higher conductivity values at V,, = 1.6, 2.3 and 3.3 mV. At these same values of vd,, shows well-defined inductive dips (Zettl and Griiner 1984, Fleming et al. 1985). The ac response has not been calculated using the Josephson equation, but arguments similar to those used in nonlinear circuit theory can be applied to account for the observations (Zettl and Griiner 1984). 5.2. SUBHARMONIC MODE

LOCKING

Interference phenomena have also been observed whenever pfex,= qf o for q integer, but not equal to one (Brown et al. 1985a). These are called the subharmonic Shapiro steps, and a rich array of such steps is displayed in fig. 25, again observed in NbSe3. A few of the p / q values are identified in

- 20

10

0

10

x)

sompk voltoqe V ImV) Fig. 25. Differential resistance of NbSe, with and without an applied rf voltage V r , . 7he

numbers indicate the various subharmonic steps. (After Brown et al. 1984.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

241

the figure; these were made by plotting I,,, versus we,[ and checking that the steps were p / q times that for the fundamental. In general, features with small p and q are more conspicuous and are both taller and wider. A complete mode locking would correspond to a plateau in d V / d I at the same level as the differential resistance below threshold. This can be understood as follows. The steps are regions of locking between the internal and applied frequencies when pfex,- q j n t is sufficiently small. If the locking within such regions is complete, the CDW velocity becomes fixed by fCxt and does not respond to changes in the applied dc voltage. Hence d V/d I rises to the linear resistance R o , which is due to the uncondensed electrons alone. This situation holds if the CDW velocity is coherent throughout the specimen. In reality the velocity coherence length must be finite, and Ant may vary spatially. If the variation of A, is larger than the width of the region over which locking can occur, locking will be incomplete, and d V/dI will rise to a level less than Ro. Observations under such circumstances are called interference “features”. Therefore, the height of d V/d I is expected to correlate with the degree of synchronization across the sample. The completeness of synchronization depends on various factors such as sample dimensions, ac amplitude and frequency, and in fig. 26, subharmonic peaks, several of them displaying complete mode locking, are shown (Hall and Zettl 1984). While complete mode locking is observed only for a few subharmonic interference peaks, in fig. 26, in subsequent experiments,

1

NbSe3

I

-16

I

I

0

15

bias current ( IJA 1 Fig. 26. Mode locked Shapiro steps in NbSe,. Over the mode locked region, d V / d I is independent of dc bias. The inset shows the subharmonic structure in detail, with corresponding p / s values. (After Hall and Zettl 1984.)

G. G R U N E R

248

performed in carefully treated NbSe, specimens up to 150 subharmonic steps between the threshold field and the 1 / 1 harmonic step have been observed (Thorne et al. 1988). The detailed dependence of the subharmonic steps on the ac amplitude and frequency was also investigated, and characteristic experimental results are displayed in fig. 29. Similarly to the Shapiro step corresponding to the fundamental (see fig. 22), subharmonic Shapiro steps also display a characteristic Bessel function behavior, and have in general an amplitude which is reduced when compared to the fundamental Shapiro step. Although it is anticipated that an analysis in terms of eq. (4.2) provides an adequate explanation for the experimental results displayed in fig. 29, such analysis has not been performed to date. The appearance of subharmonic mode locking has generated wide interest, mainly because within the framework of the classical particle model such as described by eq. (4.9) no subharmonics are expected for an overdamped response (Waldram and Wu 1982, Renne and Poulder 1974). Several, rather different explanations have been advanced to account for the subharmonic mode locking, emphasizing the importance of internal degrees of freedom, assuming a nonsinusoidal potential or making the ad hoc assumption that inertia effects are important in the current carrying state. In this latter case the time development of the phase 8 is described by a return map (Bak el al. 1984, Alstrom et a]. 1984) 8,-

I

= f ( o " ,0 ) .

(5.8)

where I / 0 is a "strobing rate" at which 8 is measured (R= fl/fdr,,,, is a convenient choice for ac driven systems). The reduction of the relevant coordinates provides the motivation for the study of simple maps and their relation to complicated dynamical systems. This idea has often been applied successfully for Rayleigh-Bernard instabilities in fluids. The subharmonic step widths, shown in fig. 25, can be analyzed analogously to that done for the circle map, where universal scaling properties have been conjectured for the quasiperiodic route to chaos (Bak 1983, Bak et al. 1984, Azbel and Bak 1984). The circle map is a one-dimensional map with the following rule for the phase

en+,= / ( 6 , ) = 6 , , + n - ( K / 2 n ) s i n ( 2 n B n ) ,

(5.9)

and i s considered a classic problem of competing periodicities, one coming from the phase space variable (0, is defined modulo 1) and the other from 0. For any value of K I the iterations of the map may converge to a limit cycle such that 8 , + , = p + O , for a nonzero interval R (indicating a p / 4 subharmonic). When K = 1 every rational R will result in this kind of trajectory, and the plateaus form a staircase structure such as shown in fig.

CURRENT OSCILLATIONS A N D INTERFERENCE EFFECTS

,

1 1 . .

m

249

I

s

I

NbSeg f = 25MHz

.#

*-

2'

- ....

.--

..-

0

1. Circle map k= 1

112

-

00

1.o

0.5

Fig. 27. Widths of interference peaks for a NbSe, sample and for the circle map, with p / 9 marked on the vertical axis. (After Brown e l al. 1986.)

27. The staircase is said to be complete if the number of steps in an interval

I of width greater than a discrimination level I-S(r) N(r)=----r

rf0,

D < 1,

r

obeys (5.10)

with S(r) the sum of the step widths that are greater than r. The exponent D defines the (fractal) dimension of the staircase. For K < 1 the staircase is not complete and D = 1. At criticality ( K = l), D = 0.87 was calculated (Bak et al. 1984) and a similar value was found by analog simulations (Yeh et al. 1984). while for K > 1 the staircase structure breaks down and the motion is chaotic. A staircase, derived from differential resistance curves similar to that presented in fig. 24, is also shown in fig. 27. For both the circle map and NbSe, system, the smallest steps have been left out of the figure, but the two staircase structures are qualitatively similar, and in both cases the steps are, in general, larger for smaller q values.

G . GRUNER

250

The experimental results displayed in fig. 24 have been used to evaluate the fractal dimension D (Brown et al. 1985a). The construction N ( r ) versus 1/ r is displayed in figure 28, leading to D = 0.91 0.03 in surprisingly good agreement with values calculated, and obtained from analog simulations. There are, however, several problems with such analysis. First, a complete staircase, D < 1 is expected only if the parameters of the system are measured at the phase boundary of the chaotic regime, i.e., for a well-defined V,, value. Experimental evaluations of D can also be limited by finite (instrumental or intrinsic) noise levels, smearing out the smaller steps. Also, computer studies of the sine circle map (Yeh et al. 1984), indicate that D-0.9 can be obtained over a finite range of the parameter K 1 if the smaller steps are not considered in the evaluation. This may occur for the CDW system as well. The good agreement between the fractal dimension, observed experimentally, and calculated on the basis of simple equations of motion, is most probably fortuitous, and cannot be regarded as evidence for a transition to chaos in this driven nonlinear system. An alternative, conceptually simple explanation for the subharmonic steps has been advanced by Bardeen (see Thorne et al. 1986a,b). The argument is based on the potential, given by eq. (4.14) and employs arguments used originally by Shapiro (1963) to account for the steps in the Josephson junctions irradiated by microwave fields. The theorem that no subharmonics are expected in the overdamped response holds only for a sinusoidal potential which does not have higher harmonic Fourier components. Any other potential gives both harmonic and subharmonic locking,

*

F"'"'

I

I

I

1 1 1 1 1 l

I

/

1

2

..

(Ilr)

Fig. 28. Experimental determination of the dimensionality D (see eq. (4))

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

25 1

and consequently the observations are not surprising, given V ( q ) as described by eq. (4.14). The approach which is used to calculate the magnitude of the interference regions is the following (Thorne et al. 1986, 1987a). The current carried by the collective mode is given by eq. (4.16), and the time dependence of the average phase is, in the presence of applied ac and dc fields d8/dt

=z - O d

-tA COS( W , , t t ) ,

(5.11)

where wd is the drift frequency wd = u,/ A and A is proportional to the amplitude of the applied ac current. The time dependence of the phase then is e(t)=w,t+(A/w)

sin(w,,,t)+Oo.

(5.12)

The pinning potential can be Fourier expanded, (5.13) where

(Y

= 1 for a periodicity given by the wavelength A. After some algebra m

m

V ( 8 ) = f a o +1

1

q = l p=-m

(

3

aqJ, qa-

~~~[(po-q~~)t+qaO (5.14) ~],

where Zp is the Bessel function of order p. The conditions of phase locking depend on the equations of motion for the collective mode, but in general, phase locking occurs if the time average pinning energy in the locked state is smaller than in the unlocked state. Without locking the time average pinning energy ( V ( 8 ) ) ,=fao,and for an additional polarization energy pwexr= (5.15) is obtained. The polarisation energy is less than zero for some range of O0, -8, < eo< 8,, and 0, determines the width of the step. V ( 0 ) as given by eq. (4.14) has been used to evaluate 0, for different p and q values. As expected, both the harmonics and the subharmonics display a characteristic Bessel function behavior in broad agreement with the experimental results. The period of the oscillations with ac amplitude A varies linearly with frequency and inversely with q also in agreement with the observations, as indicated by fig. 29. In general, the agreement between experiment and theory is satisfactory for a broad range of applied ac frequencies and

G . GRUNER

252

0.30

NbSe, Sample #1

T=12l K

5 MHz \

p/q: f

l/f

x

1/2

o

1/3

f

1/4

x 1/5

0.15

0.10

0.05

0.00

0

5

10

15

20

Fig. 29. Widths of selected steps in the dc I - V characteristic versus peak ac amplitude for applied ac frequencies of (a) 5 MHz and (b) 10 MHz. The period of the oscillations with ac amplitude vanes as I / q for the p / q step. The solid lines are guides to the eye. (After Thorne et al. 1987b.)

amplitudes (Thorne et al. 1987b). A similar agreement would, however, be obtained by any potential which has the same periodicity as that of eq. (4.14) and has smooth minima separated by cusps. A rather different concept has been developed by Tua and Ruvalds (1985) by Littlewood (1986), and by Matsukawa (1987) by extending the models which incorporate the internal degrees of freedom and which have been discussed in the previous section to account for the interference phenomena observed. Both conclude that internal degrees of freedom lead to subharmonic mode locking even under circumstances where single degree of freedom dynamics lead only to locking corresponding to harmonic frequencies. The classical dynamics of coupled domains has been used by Tua and Zawadowski (1984) to account for the current-voltage characteristics and for the finite size effects associated with the current oscillations. The extension of the model to combined dc and ac drives (Tua and Ruvalds 1985) leads to subharmonic steps, and to an apparent complete Devil's staircase. The overall amplitudes of the subharmonics agree well with the experiments, and the calculations also lead to a subharmonic structure which depends only slightly on the drive conditions, with no critical values dividing complete and incomplete staircase behavior, in striking agreement

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

253

with the experimental observations. The amplitude of the steps was found to decrease with the increasing number of domains, and the steps disappear in the thermodynamic limit. This feature of the results is in clear contrast with the hydrodynamic treatment of Sneddon et al. (1982) which predicts that (while the current oscillations are a finite size effect) interference peaks are recovered in the thermodynamic limit. The origin of this disagreement is not fully understood. The model where the internal degrees of freedom are incorporated following the Fukuyama-Lee-Rice model (Fukuyama and Lee 1978, Lee and Rice 1979) has also been used to study the response of charge density waves to a combination of dc and ac drives (Coppersmith and Littlewood 1986). As discussed earlier, second-order perturbation treatment in the hydrodynamic limit (Sneddon et al. 1982) leads to interference effects involving the fundamental and higher harmonics. Higher order perturbation theory leads to interference features (Coppersmith and Littlewood 1985a,b), i.e. sharp peaks in the d l / d V curves, but not full mode locking with plateaus as displayed in fig. 26. Complete mode locking is recovered by numerical simulations on finite size systems where the advance of the average phase ( 4 ) is monitored. The results of the calculations, which display advances A4/27r, which correspond to various mode lockings p / q , are shown in fig. 30. The detailed waveform near mode locking has also been calculated by Coppersmith and Littlewood. The overall tendency of the experimental results, which are suggestive so-called interference “features” (i.e. not complete mode locking) at high frequencies, and complete; mode locking with welldefined plateaus in d V/d l curves at low frequencies (Brown and Griiner, unpublished) is well reproduced by calculations which take the dynamics of internal degrees of freedom explicitly into account. Interference curves, calculated on the basis of single degree of freedom classical dynamics (called GZC) model, see eq. (4.2) and on the basis of the Fukuyama-LeeRice model are displayed in fig. 31, together with experimental curves, shown as the full line. The absence of well-defined “wings” in experiment and in numerical simulations based on the FLR model are taken as evidence of the dynamics of the internal mode. Both harmonic and subharmonic mode locking is recovered also by perturbational analysis of the FukuyamaLee-Rice model in the presence of combined ac and dc electric fields (Matsukawa 1987). The model leads to clear anomalies in the currentvoltage characteristics; whenever nq = mp, complete mode locking, such as shown for example in fig. 26, however, has not been recovered. The reason for this is not clear at present, it may be related to the deficiencies of the model itself (Bardeen 1988), or due to the breakdown of the perturbation theory. The conclusion has been sharply criticized recently (Thorne et al. 1987a,b,c), who argued that complete mode locking can also be obtained

G. G R U N E R

254

h

0

om m a 3

Om

5 -

- 2

O A 0

0 0 2

3

007

04 3

u =4 F = 16

OBBBOOI

A 55 IMPURITIES 0

25 IMPURITIES

0 10 IMPURITIES

n

0 0 71 1

O3 4

I

I

0.5

1

-

ton Fig. 30. Number of wavelengths moved per pulse (A4)/27r versus the pulse duration I,,, with a fixed Eon= 16 for two systems with U = 4, one with 55 degrees of freedom and the other with 10. Locking is demonstrated because (A4)/27r is always a rational fraction. Inset: Magnified portion of the plot, demonstrating that increasing the number of degrees of freedom causes the appearance of high order subharmonics. (After Coppersmith and Littlewood 1986.)

at high frequencies, in apparent disagreement with the classical descriptions of charge density wave dynamics. The disagreement between the various experimental results most probably reflects the difference in the degree of coherence in the materials which have been investigated. As a rule, in small and pure specimens, where the phase-phase coherence length may exceed the dimensions of the samples, a highly coherent response, with pronounced complete mode locking is observed even at high frequencies. In contrast, samples with less coherence

iJ F A

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

1. O r

Oa8-

E

H

U

5 U

0.6 -

I

I

\

0.4 -

I

'

.[ ..-..I

\ I

;

\

f

//

I

I

I

-........

/-

'\I

I

082 -

1

I

' \

I

I...* /

-

Experiment

-

' ........ --- FLR model

I OL

I

I

I

bc Y

r--

r--1

I

255

I

GZC model

~

4

l

l

l

I

6

l

l

l

voltage (mV)

l

l

2

4

6

voltage ( mV)

Fig. 31. Differential resistance d V / d l plotted versus dc voltage V, near the first harmonic feature Q. u wAc for NbSe, with ac frequency w,, = 25 MHz and ac voltage amplitude V, 50 and 75 mV. The sample's threshold voltage V,-2 mV. Also plotted are theoretical fits using the FLR deformable model (dotted line) and the GZC one degree of freedom result (dashed line). T h e tops of the peaks are not calculated for FLR because the perturbation theory breaks down when the change in d V/dl is large. (After Coppersmith and Littlewood 1986.)

-

-

display complete mode locking only at low frequencies. This difference, and the crossover between' the two behaviors have, however, not yet been investigated in detail. 5.3. NONSINUSOIDAL AND PULSE

DRIVES

In the majority of cases, interference effects have been investigated by applying a sinusoidal ac field E ( o )= Eo sin( weXtt) of various amplitude and frequency. Most of the relevant theories have also been worked out for this particular case, mainly because all experiments on Josephson junctions involving microwave fields have been conducted under such circumstances (Lindelof 1971). In the radiofrequency spectral range various

G . GRUNER

356

periodic waveforms can be applied, and this has been used (Brown et al. 1986a). to further investigate the various aspects of mode locking. For a periodic square wave drive which oscillates between two values, El and E 2 , the former for a time interval i,, the latter for i2, the fundamental period is i = i, + I,. t , , I , , and E l , E2 can independently be varied leading to a variety of conditions under which the dynamics of the collective mode can be investigated. The difference between a sinusoidal drive and pulse drive, the latter with I , = r2, is shown in fig. 32. The sharp spikes in the upper part of the figure correspond to the fundamental interference peaks, with the p / q = i subharmonics also evident on the figure. The amplitude of the interference peaks is approximately constant for the sinusoidal drive for small dc voltages, and starts to decrease when IEdc- Ea,I ET,applied for time r2 varied. Such experiments point to the importance of internal degrees of freedom in the dynamics of the collective mode. The measured step height is displayed in fig. 33 as a function of the time 1, spent below threshold. As expected, the step height approaches zero as i, (called i- in the figure) gets shorter and saturates the long time ( 1 , > CO) limit. For short times the step height 6 V can be described by the expression

hV-

vT(tI/TO)

(5.16)

where T~ is the characteristic relaxation time of the system. For an overdamped classical particle T~ = &T. In the long i2 limit simple arguments, (Brown et al. 1986a,b) based on the classical particle model lead to a step height

Computer simulations, with 1, = 0.25 ps, obtained by fitting the observed w dependent response to eq. (5.16), lead to the dotted line in fig. 33. The difference between the observed step height and that which follows from

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

257

1

-2vT

NbSe, T = 48K

loot 01

I

I

- 10

-5

5

0

10

somple voltoge (mV1 Fig. 32. Differential resistance curves for (a) sinusoidal ac drive and (b) square wave drive for the same amplitude. (After Brown et al. 1986a,b.)

the classical particle model is significant, and is most probably the consequence of the dynamics of the internal degrees of freedom (Brown et al. 1986a,b). A different type of interference effect is called, in general, “pulse duration memory”. The notion of pulse duration memory effect refers to the observation which for &,,Moo3 is displayed in fig. 34 (Fleming and Schneemeyer 1986). Square wave voltage pulses which drive the system from the pinned 0.4

,

I

1

1

r----------

->E A

I

‘

*., 5 .- 0.2 -

al

I

0 tl

I

.c

I

A

A

I

---- a- ,A- - = - r - -- - - A A

A A A A A

I

---- Tt=2.0w Simulation, ~ 0 . 2 5 ~ s A

8 A

‘ A

0 - I

I

Fig. 33. Step height amplitude versus waiting time 1. The dotted line is a fit to the simulation based on the classical particle model. (After Brown et al. 1986a,b.)

G . GRUNER

258

1

I

I

1 0

I

I

I

I

I

I

I

I

1

I

I

I

1

I

I

I 400

I

200

1

1

Fig. 34. (a) Current oscillations in response to a square-wave driving field of about lOE, (inset) in &,,MOO, at 45 K. We data was obtained in a current driven configuration and has been inverted; however the current oscillations are also clearly observed in a voltage-driven configuration. (After Fleming and Schneemeyer 1986.)

to the current carrying state lead to transient current oscillations with frequencies given by eq. (1.2). The amplitude of the oscillations gradually decreases with increasing time and for t + 00, the magnitude of the oscillations correspond to those which are measured under dc conditions by detecting the Fourier transformed current. This can be thought of as the consequence of gradual dephasing of the current oscillations which start with the same phase but which have slightly different frequencies depending on the local currents within the specimen. At the end of the pulse, there is a sharp upward cusp in the observed current, and this occurs for a broad

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

259

range of pulse durations t o . The cusp suggests that the system adjusts itself to the pulse duration in such a way that its velocity is always rising as the pulse ends. Apparently, the system “remembers” the length of preceding pulses, hence the name of “pulse duration memory effect”. The upward cusp, as displayed in fig. 34, is not observed for single pulses, although the current oscillations are evident. Clearly a single-particle description does not account for this finding as the current is determined by the derivative of the potential at the position of the particle at the end of each pulse, and this can be both negative and positive, depending on t o . Consequently, the current at the end of each pulse is expected to be a sensitive function of the pulse duration. It has been proposed (Coppersmith and Littlewood 1985a),that the pulse duration memory effect is the consequence of the internal degrees of freedom, where a large number of metastable states and a negative feedback mechanism play a crucial role. A simplified one-dimensional version of the equation of motion, eq. (4.18), has been examined numerically where a sequence of square wave pulses of length to is applied, with the time tombetween pulses long enough that the system relaxes to its appropriate metastable states between the applied pulses. The same model leads also to the subharmonic mode locking discussed earlier. The calculated response wave forms resemble closely those observed by experiment and this is taken as evidence for the importance of internal degrees of freedom in the dynamics. A further elaboration of the model suggests that when the system with infinite degrees of freedom is subjected to identical repeated pulses, it attempts to reach a state for which further pulses to not induce further changes. This fixed point is the least stable, and the pulse duration memory effect is the system’s signature of being on the verge of its region of stability.

5.4. FLUCTUATIONS AND

COHERENCE ENHANCEMENT

Fluctuation effects associated with the current oscillations, discussed in section 3, are suggestive of coupled domains of size L: which oscillate at approximately the same frequency fo =j / 2 e with random phase. The Gaussian distribution of the voltages of the narrow band noise displayed in fig. 14, is suggestive for independent phase dynamics for the various domains. The mode-locking phenomena discussed earlier can be understood on the basis of the dynamics of a single degree of freedom system; however, the volume dependence of the current oscillation amplitudes suggest that internal degrees of freedom, and finite size effects are important

G . GRUNER

260

(Mozurkewich and Griiner 1983). Also, interference experiments, performed for relatively small ac drives give mode locking, smaller than that expected for a single degree of freedom dynamics, and this has been interpreted as partial mode locking (Zettl and Griiner 1984). The reason for this is again most probably the absence of complete phase coherence throughout the specimens. It is then expected that phase synchronization by the applied ac field leads also to the reduction of incoherent phenomena, such as the broad band noise, or the narrow band noise fluctuations observed in the current carrying state. These have been investigated recently in detail (Bhattacharya et al. 1987, Sherwin and Zettl 1985). The broad band noise spectrum is significantly reduced during mode locking, with a total noise power reduced comparably to that observed in the pinned CDW state (Sherwin and Zettl 1985). This most probably is related to the freeze-out of the internal degrees of freedom, and may signal the enhancement of the dynamic coherence length which characterizes the current-current correlations. Such an effect has been recovered by a perturbational analysis of the FukuyamaLee-Rice model in the presence of combined d c and ac drives (Matsukawa 1987). The reduction is most probably the consequence of phase synchronization between the various domains. This reduces the dynamics to that of a single domain with no fluctuations and consequently no broad band noise. The noise reduction depends on the applied ac amplitude, as shown in fig. 35, where the total measured noise power is plotted as a function of ac current in this current-driven experimental arrangement. The gradual reduction of the noise power is suggestive for a gradual phase homogenization, and a complete phase homogenization is achieved for ac drive amplitudes

1.0

-?

--

0

0.8 -

-' 0.6 >= 0.4 -

*

i

NbSes T=48K

1

a,,/en=eN H Z

0 0

T

>*

0

0.2 -

I

j -

1

1

1

0

I

I l l l L

Fig. 35. Total broad band noise amplitude versus amplitude oran applied rrcurrent in NbSe,. Intense rf field suppresses the noise by homogenizing the CDW phase. (After Sherwin and Zettl 1985.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

261

which are typically about one order of magnitude larger than those which are required for complete mode locking. Fluctuations of the current oscillations are also influenced by the mode locking, and both the amplitude and frequency fluctuations are reduced. The system synchronized at the p / q = subharmonic steps displays reduced amplitude fluctuation reductions at short time scales, but may fluctuate between varous mode locked states (characterized by different oscillation amplitudes) over a large time interval. N o amplitude fluctuation reduction has been observed in a more detailed experiment (Bhattacharya et al. 1987), but the frequency distribution of the current oscillations was eliminated by mode locking. The phenomenon is shown in fig. 36. The upper part of the figure shows the fluctuations of the N B N in position and amplitude without the application of ac drive. When synchronization occurs, the N B N peak frequency does not fluctuate, but the amplitude fluctuation is unchanged. The absence of frequency distribution implies that at mode locking all

1.4

1.5 I .6 PEAK FREOUENCY (MHz)

1.7

PEAK AMPLITUDE (arb.units) Fig. 36. Hystogram of the temporal fluctuations in frequency (a) and amplitude (b) of a “bare” NBN ( V,, = 0) and a locked NBN p / q = 1/2 at o = 3 MHz). Results represent 500 scans of each case. In (b) the two hystograms are shifted arbitrarily for clarity. (After Bhattacharya et al. 1987.)

262

G. GRUNER

temporal fluctuations of the oscillation frequency are quenched, i.e., the velocity degrees of freedom are eliminated. Coherence enhancement and fluctuation suppression, such as observed in driven charge density wave systems, represent probably the most interesting consequences of nonlinear dynamics of many degree of freedom systems. Some aspects of this phenomenon, such as subharmonic mode locking, under circumstances where a single degree of freedom system would not lead to subharmonic locking, have been explored, and it remains to be seen whether the other aspects of the observations can be accounted for.

6. Conclusions

The current oscillation phenomena which occur in driven charge density wave systems in the nonlinear conductivity region and associated interference effects in the presence of dc and ac drives are clear manifestations of a new type of collective transport phenomenon, carried by an electron-hole condensate. The linear relation between the time average current and oscillation frequency reflect the fundamental 2kF periodicity associated with the electron-hole condensate, and the mere existence of current oscillations in macroscopic specimens is suggestive of macroscopic length scales involved. The collective mode is characterized by an amplitude and phase, and the length scales are related to the phase-phase correlation length, determined by the impurity concentration and the parameters of the collective mode, and the highly coherent reponse clearly demonstrates that they are comparable to the dimensions of the specimens investigated. In spite of a broad variety of experiments performed both in the time and in the frequency domain and conducted by employing under various combinations of ac and dc drive amplitudes, frequencies and waveforms, several unresolved questions remain. The linear relation between the time average currents and oscillation and oscillation frequency is by now well confirmed in all materials which display CDW transport phenomena; there is however, considerable uncertainty concerning the numerical factors involved. Experiments on TaS, suggest that simple arguments advanced in the Introduction are correct and the oscillation is related to the advancement of the phase by 27r, corresponding to the displacement of the collective mode by one wavelength. Careful experiments in other, structurally simple model compounds, such as (TaSe,)*I and &.,MOO, would be highly desirable. The relation, given by eq. (1.2) holds also only at T = 0, and experiments on NbSe, indicate that the ratio j / f o is proportional to the number of condensed electrons. In contrast, in TaS, the ratio was found to be only weakly temperature dependent (Brown and Griiner (1985). Further experi-

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

263

ments, in particular the transition temperature near Tp, are needed to clarify whether the proposed relation

is appropriate. Considerable controversy exists concerning the spectral width of the current oscillations, and the related broad band noise. Broadly speaking, larger spectral widths are accompanied by larger broad band noise amplitudes suggesting that the two types of incoherent effects are strongly related, and both care determined by the relative magnitudes of the phase-phase correlation length and dimensions of the specimens. One particular model (Bhattacharya et al. 1987) related the broad band noise amplitude to the spatial and temporal fluctuations of the CDW current, through the relation ( S V 2 ) = ( 6 I 2 ) R Nwhere R N is the resistance due to the normal electrons. Fluctuations in CDW velocity lead also to fluctuations in the oscillation frequency explaining the above overall correlation. Studies of the broad band noise amplitude performed in TaS, and NbSe, as a function of the dimensions of the specimens indicate that it is a bulk effect and that noise generation is due to randomly positioned impurities. In contrast, studies on rather pure NbSe, specimens indicate that macroscopic defects, such as grain boundaries, are the main source of broad band noise. Macroscopic defects, and also random impurities, have been suggested also as the source of the current oscillations. The most probable explanation for the variety of findings is that in general, both impurities and extended boundaries (such as grain boundaries, contacts, etc.) contribute the noise generation (both broad band and narrow band); but more detailed studies in alloys, or in irradiated specimens where the concentration of pinning centers is systematically varied, are called for to settle this issue. Fluctuation effects, and clear evidences that the current oscillations disappear in the thermodynamic, infinite volume limit demonstrate that description of CDW dynamics within the framework of single degree of freedom models is not appropriate. Various experiments however, suggest that the dynamics of the collective mode is characterized by macroscopic length scales. The static phase-phase correlation length (called the Fukuyama-Lee length) Lo is, for typical impurity concentrations and CDW parameters of the order of 10-100 pm, along the chain directions and one or two orders of magnitude less perpendicular to the chains. This leads to a typical “domain” size of 10-2-1 pm in broad agreement with estimates of the phase-phase correlation length on the basis of length dependent threshold field (Monceau et al. 1986, Gill 1982, Zettl and Griiner 1984, Borodin et al. 1986) current oscillations amplitudes (Mozurkewich and

264

G . GRUNER

Griiner 1983) and broad band noise (Bhattacharya et al. 1985, Richard et al. 1982) implies that the static and dynamic aspects of the problem are characterized by the same length scale (except perhaps near threshold, see Fisher ( 1985)). As Lo is expected to be inversely proportional to the impurity concentration, similar experiments on alloys would be of great importance. Interference effects confirm many of the conclusions which have been reached on the basis of the current oscillation studies alone, but also emphasize the formal similarity between the Josephson phenomena and nonlinear CDW transport. While for small amplitude ac drives fluctuation effects, volume dependences for interference effects and current oscillations go hand in hand, large amplitude ac drives lead also the phase homogenization, and to interference effects which survive the passage to the thermodynamic limit. Whether this corresponds to the conclusions which have been arrived at on the basis of the hydrodynamic treatment of the CDW dynamics (Sneddon et al. 1982) remains to be seen. Considerable theoretical activity, generated by the observation of subharmonic interference peaks, led to several fundamentally different proposals on the origin of subharmonic locking. Early analysis indicated that the concept of Devil’s staircase behavior, tied to inertial effects may be appropriate. By now this possibility is unlikely, with two remaining suggestions: nonsinusoidal pinning potential and internal degrees of freedom being able to explain a broad variety of interference features and mode locking. Most probably, both are important as there is no a priori reason why the pinning potential should be periodic, and also there is broad variety of independent experimental evidence for the importance of the dynamics of the internal modes of the condensate. The former effect may be more dominant for small specimens where pinning by boundaries such as the surface of the specimens and contacts are important, while the latter is more important for larger and impure materials where pinning is due to randomly distributed impurities. While the experiments led to several interesting theoretical questions, such as why internal degrees of freedom dynamics mimics many effects which are also the consequence of inertial dynamics, or the concept of minimally stable states - the hope that driven charge density waves can be appropriate model systems for general questions concerning nonlinear dynamics now appears to be remote. This is mainly because both randomly distributed pinning centers (which lead to a broadly defined localization problem) and extended pinning centers (leading to essentially a boundary problem) are in general equally important and cannot be easily separated. Several issues related to the dynamics of charge density waves have not been discussed in this rewiew. In particular, the charge density wave dynamics of specimens which display a so-called “switching” behavior (Zettl and Griiner 1982, Hall et al. 1986) has not been mentioned. The

CURRENT OSCILLATIONS A N D INTERFERENCE EFFECTS

265

phenomenon arises as the consequence of macroscopic defects which leads to dynamics similar to that observed in coupled Josephson junctions (Inoui and Doniach 1987). Several observations concerning transitions to chaos have also not been discussed and I refer to a recent review (Zettl and Griiner 1986) which covers this type of experiments. Mode locking phenomena also occur in the elastic properties (Bourne et al. 1986), a not too surprising observation given the fact that the CDW can be regarded as a coupled electron-phonon system, and the earlier observations on nonlinear elastic properties. Several models, with interesting dynamical properties have also not been discussed. Among them, the dynamics of Frenkel-Kontorova type models have been studied in detail (Sneddon 1984a, Coppersmith and Littlewood 1985a,b), and have been shown to display many of the features which are the consequence of models where random impurity pinning is explicitly taken into account, such models also lead to a broad variety of interference effects in the presence of combined dc and ac electric fields, discussed before.

Acknowledgements

I am grateful to John Bardeen, Sue Coppersmith, Peter Littlewood and Laszlo MihPly for many discussions, and to Stuart Brown for reading the manuscript. Support by the National Science Foundation Grant DMR 86-20340, is acknowledged.

List of review papers

Several review papers have appeared recently which cover the various aspects of charge density wave dynamics, see for example: Fleming, R.M., 1981, in: Springer-Verlag in Solid State Sciences, Vol. 23, Physics in One-Dimension, eds. J. Bernasconi and T. Schneider (Springer, New York). Griiner, G., 1983, Comments in Solid State Physics 10, 183. Griiner, G., 1983, Physica D 8, 1. Griiner, G., 1988, Rev. Mod. Phys. 60, 1129. Griiner, G., and A. Zettl, 1985, Phys. Rep. 119 117. Hutiray, Gy., and J. Solyom, eds., 1985, Lecture Notes in Physics, Vol. 217, Charge Density Waves in Solids (Springer, Berlin). Jerome, D., and L.G. Caron, eds., 1987 (Plenum Press, New York).

266

G . GRUNER

Monceau, P., 1985, in: Electronic Properties of Inorganic Quasi-OneDimensional Materials, ed. P. Monceau (Reidel, Dordrecht). Ong, N. P., 1982, Can. J. Phys. 60,757. Tanaka, S., and K. Hchinokura, eds., 1986, Physica B 143 (Yamada Conf. XV, Physics and Chemistry of Quasi-One-Dimensional Conductors). Zettl, A., and G. Griiner, 1986, Comments in Condensed Matter Physics 12, 265.

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CHAPTER 5

MULTI-SQUID DEVICES AND THEIR APPLICATIONS BY

Risto ILMONIEMI and Jukka KNUUTILA Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland

and

Tapani RYHANEN and Heikki SEPPA Electrical Engineering Laboratory, Technical Research Centre of Finland and Laboratory of Metrology, Helsinki University of Technology, 02150 Espoo, Finland

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elsevier Science Publishers B. V., 1989 27 1

Contents I . introduction . . . . . . . . . . . . . . . . . . . . . ................................... ..... 2 . SQUIDS . . . . . . . . . .......................... ............................. 2.1. Single-junction (rf) SQUlDs . . . . . . . . . . 2.1.1. General . . . ........................................... ......... 2.1.2. Rf SQUID in the hysteretic mode . . . . . . . . . . . . . . . . . 2.1.3. Discussion . . . . . . . . . 2.2. Double-junction (dc) SQUlDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Operation . . . . . . . . . . . . . . . ................................. 2.2.2. Problems with practical devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. The state of the art . . ................................... 2.3. Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications: biomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1 .1. Magnetically shielded rooms . . . . . . . 3.1.2. Gradiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Neuromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Origin of neuromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Spontaneous activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Evoked fields . . . . . . . . . . . . . . . . . . . . . . . ...................... 3.2.4. Clinical aspects of MEG . . . . . . . . . . . . . ...................... 3.3. Cardiac studies ........................................ 3.4. Other hiomagnet ........................................ 3.5. Multichannel neuromagnetorneters . . . . . . . . . . ..................... 3.5.1. Optimization of multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . 3.5.2. Existing multichannel systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Planar gradiometer arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. Use of multichannel magnetometers . . . . . . . . . . . . . . . 4 . Other multi-SQUID applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Geomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Physical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Accelerometers and displacement sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Monopole detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

213 213 214 214 216 219 280 280 284 286 289 292 293 293 295 296 291 298 299 300 302 303 304 305 310 319 323 326 326 328 329 329 332 333

1. Introduction

The Superconducting Quantum Interference Device (SQUID) offers unrivalled sensitivity for the measurement of low-frequency magnetic fields. Recent developments in the fabrication technology and electronics have made it possible to construct reliable low-noise SQUIDs. Consequently, magnetometers with many SQUIDs have become feasible in many applications and the number of SQUIDs is no longer limited by the difficulty of their use. Further reduction of SQUID noise is no more necessarily needed, since the noise limit in the state-of-the-art multi-SQUIDS seems to be determined by dewar materials, environmental low-frrequency noise, and other sources external to the SQUIDs. Additional recent interest in SQUID applications is caused by the possibility of fabricating them from the new high- T, materials. In this review', we start with an overview of the operation of rf and dc SQUIDs, stressing, in particular, the theoretical understanding of complete SQUID circuits. This is necessary for the design of practical devices. Main attention is focused to the operation of the dc SQUID and to recent progress in realizing practical dc-SQUID structures with flux-coupling circuits. A more thorough discussion of SQUID circuits is presented in a companion paper (Ryhanen et al. 1989). So far the largest field of application for SQUID arrays is biomagnetism, the study of magnetic fields originating in biological organisms. The subfield currently attracting most interest is neuromagnetism, where these techniques are applied to investigations of the central nervous system. A potentially important clinical use of SQUIDs in the future is in magnetocardiography (MCG), the recording of heart activity via magnetic measurements. We will give a brief overview of biomagnetic measurement techniques, illustrating them with several examples. Principles of SQUID magnetometers and their application to biomagnetism are discussed. Existing biomagnetic multiSQUID systems and some plans for future instruments are described. In addition, applications of SQUIDs in geomagnetism and in some physical experiments are briefly discussed.

2. SQUIDs

This chapter is a brief review of Superconducting Quantum Interference Devices (SQUIDs), which are formed by interrupting a superconducting 273

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274

ring by one or two Josephson junctions (Lounasmaa 1974, Tinkham 1975, Barone and Paterno 1982, Likharev 1986, van Duzer and Turner 1981). SQUI Ds have been studied intensively both theoretically and experimentally since the introduction of the double-junction interferometer (dc SQUID) by Jaklevic et al. (1964, 1965) and, in particular, since the invention of the rf SQUID by Silver and Zimmerman (1967). Both rf and dc SQUIDs are common in multichannel applications. In spite of the outstanding properties of SQUIDs as magnetic flux sensors, their impact outside research laboratories has remained modest. Imprudent operation of SQUIDs often leads to unexpected problems that discourage their use. Evidently, the complicated dynamics, caused by the strong nonlinearity and the lack of natural damping, is one of the main reasons for difficulties in practical applications. However, as this paper will show, these problems can be overcome. When discussing the suitability of a particular SQUID in a multichannel system, all the necessary circuits should be included in the analysis. In other words, the contributions to noise of preamplifiers, coupling circuits, postdetection filters, etc. should be as important objects of theoretical and experimental study as the SQUID itself. For example, theory predicts a flux noise of about Q,/& for a typical rf SQUID and about lo-"@,,/& for a dc SQUID; @,=2.07x10 "Wb is the magnetic flux quantum. However, in a practical measurement setup such figures are seldom reached. Excess noise is usually caused by the preamplifier in rf SQUIDs and by parasitic elements in the input circuits of dc SQUIDs. 2.1. SINGLE-JUNCTION (RF) SQUIDS 2.1.1. General

In an rf SQUID the low-frequency external flux in the SQUID ring is read out by superposing on it a high-frequency bias flux and monitoring the amplitude of the rf voltage by a preamplifier matched to the SQUID with a resonant tank circuit (fig. 1). The Josephson junction is typically described by the resistively-shunted-junction (RSJ) model, which consists of an ideal Josephson junction, a resistance R, and a capacitance C. The supercurrent passing through the ideal junction is related to the quantum phase difference across the junction cp by I, = I, sin cp where I , is the critical current of the junction. In a single-junction superconducting loop, cp = -27r( @+ n @ J / @,). If an external flux @a is applied to a SQUID loop with inductance L, the total flux Q = Qa+ LI, in the loop obeys the equation 2 ~ @Q,/

+ pL sin( 2 n Q l Q,,) = 2a@,/

Q, ,

(2.1)

k---Jq, MULTI-SQUID DEVICES

Tank circuit

275

SQUID

x

Josephson

junction 1,sinlp

Fig. 1 . A single-junction SQUID coupled to a resonant tank circuit. If BL= ZTLI,/#~>1 the SQUID is hysteretic, if BL< 1 it is nonhysteretic and called an inductive SQUID; I, is the critical current of the junction, Go is the magnetic flux quantum 2.07 x lo-'' Wb.

where p L = 2rLI,/ @, is the normalized inductance. The dynamics of the single-junction SQUID depends fundamentally on p L ; its influence on the @-@a characteristics is illustrated in fig. 2. When I , is not high enough to screen the ring, i.e., when p L < 1, @(@a) is single-valued; the SQUID is then nonhysteretic. Otherwise, .@( @a) is multivalued and the SQUID is hysteretic. The SQUID in the regime p L< 1 is called a nonhysteretic SQUID, an inductive SQUID, or simply an L-SQUID; its characteristics depend strongly on the parameters of the SQUID ring and the tank circuit. Consequently, the L-SQUID is rarely seen in practical applications. In contrast, the dynamics of the hysteretic SQUID is much less sensitive to parameter

Fig. 2. The total flux @ in the SQUID ring as a function of the applied flux @a for BL< 1 and for pL> 1. When BL> 1. the SQUID is hysteretic and transitions occur at @ = @== *((n + f ) @,, + LIJ.

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216

variations; therefore, it is preferred in most applications, including multichannel magnetometers.

2.1.2. Rf SQUID in the hysteretic mode Operation of the $SQUID in the dissipative regime A consequence of high B L is that the total flux through the SQUID loop becomes a multivalued function of the applied flux. In this mode, the flux may jump by about one flux quantum as depicted by arrows in fig. 2 . Sinusoidal flux excitation of sufficient amplitude causes the SQUID to traverse hysteresis loops; the flux transitions involve dissipation of energy that is proportional to the area of the hysteresis loop. Rf SQUIDS are thoroughly discussed in the literature (Zimmerman et al. 1970, Mercereau 1970, Nisenoff 1970, Giffard et al. 1972, Clarke 1973, Jackel and Buhrman 1975, chapter 3 of the companion paper: Ryhanen et al. 1989); only a brief review will be presented here. The peak voltage across the tank circuit depends on the amount of flux needed to excite flux jumps. The points for flux transitions can be approxiLI,) (see fig. 2 ) . Before a transition, depending mated by @,= * ( ( n on the branch where the system is, the amplitude of the tank circuit voltage approaches one of two critical values:

+a)@,,+

eT

where up is the pump frequency; the mutual inductance M between the SQUID ring and the tank circuit coil L, is proportional to the coupling Because of energy transferto the SQUID, the voltage constant k = M/&. in the tank circ?it periodically drops, rising up again until the next transition takes place at V,. For fixed rf excitation, a peak detector draws a triangular pattern with period @, as a function of dja as shown in fig. 3 . The energy AE absorbed in one complete cycle is approximately the area of the loop in the @- Q a plane divided by the loop inductance (Zimmerman et al. 1970, Jackel and Buhrman 1975); for high P L ,

c,,

ec2

as seen from fig. 2. Although and depend on the dc flux threading the SQUID loop (eqs. ( 2 . 2 ) and ( 2 . 3 ) ) , A E appears insensitive to the point of operation. Consequently, the effective impedance of the tank circuit must depend on the point of operation; the rf SQUID acts as a flux-dependent

MULTI-SQUID DEVICES

1

I

-2 a0

I

I

I

277

1

I

I

1

I

- a0

0

a0

200

Fig. 3. Ideal triangular flux-voltage characteristic of the hysteretic rf SQUID with periodicity of one flux quantum.

resistor. Suppose @a = @0/2: as long as the energy fed into the t?nk circuit during each rf cycle does not exceed A,: :he peak voltage V, remains unchanged, resulting in a plateau in the V,I, characteristics. An increase of power creates another plateau until an energy 2AE is fed into the system during every rf cycle. Correspondingly, at Qa = Go the first plateau ends when the energy 2AE is exceeded. Realistic staircase patterns, corresponding to the cases @ . = ( n + f ) @ , , and Q a = nQ0, are plotted in fig. 4. When k2QT> r / 2 , where QT is the quality factor of the tank circuit (Giffard et a]. 1972, Jackel and Buhrman 1975), the plateaus overlap and proper adjustment of the rf bias current implies a perfect triangular pattern as in fig. 3. Noise in the hysteretic rfSQUID Thermal noise causes flyctuations in the points of flux transitions, tilting the plateaus of the jdVT characteristics (see fig. 4) and increasing the equivalent flux noise. As shown by Kurkijarvi (1972), the uncertainty in the flux jumps decreases when the frequency of the sinusoidal excitation is increased. Kurkijarvi and Webb (1972) derived an expression for the equivalent flux noise and showed that the slope a of the plateaus in the staircase pattern is related to the intrinsic energy sensitivity E (Jackel and Buhrman 1975):

where

E

is associated with the equivalent flux noise @,, in the SQUID loop

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Fig. 4. The current-voltage characteristics of a hysteretic rf SQUID in the presence of thermal noise, for even and odd multiples of @,,0/2 of the externally applied flux 0".itf indicates the point of operation producing the triangular response illustrated in fig. 3. Q, is the quality factor of the tank circuit.

as

The intrinsic flux noise increases the noise temperature of the tank circuit. If noise from the preamplifier and the tank circuit is included as well, the experimentally determined value of a,a e x pcan , be used to estimate the equivalent input energy sensitivity (Jackel and Buhrman 1975): r

(2.7) where Ti @X/kgL is called the intrinsic tank circuit temperature; TT denotes the equivalent noise temperature of the tank circuit, and TAis the noise temperature of the preamplifier. Equation (2.7) is derived by assuming that the impedance of the loaded tank circuit equals the optimal impedance of the preamplifier (section 2.3) and that k2QT- 7r. Since the energy dissipation caused by the hysteretic loop is related to Ti simply by f k , T , a A E , the size of the loop should be as small as possible but sufficient to assure proper operation. TT is often determined by the preamplifier; thus T,, T,= TT and careful design of the preamplifier becomes imperative for a low-noise magnetometer. If wp = 27r x 20 MHz, L = 1 nH, and PL = 3, eqs. (2.5) and (2.7) predict that a 20.1 and T,= 300 K. Neglecting the tank circuit and the amplifier, an equivalent flux noise as low as (@:)"' = 1.8 x lo-' @,I& should be achievable, but without a cooled preamplifier this is impossible in practice. L-

L-

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A weak magnetic coupling results in a low flux-to-voltage conversion efficiency if it is not compensated for by a high Q-value of the tank circuit. On the other hand, tight coupling causes uncertainty in the flux transitions induced by tank circuit and preamplifier noise. Optimal choices of the mutual inductance and particularly of the product k2QT are discussed in detail by Jackel and Buhrman (1975). They argue that the best performance is obtained when k2QT exceeds unity. The same conclusion was drawn by Simmonds and Parker (1971) on the basis of computer simulations. Moreover, Ehnholm (1977) derived a small-signal model for an rf SQUID with complete input and output circuits and was able to show that the choice k2QT= 1 is a solid foundation for SQUID design.

Rf SQUIDS at high frequencies The sensitivity of rf SQUIDs can, in principle, be improved by increasing the pump frequency up,but the benefit is partly cancelled by the resulting higher preamplifier noise. When wp = w, = R / L,where w, is the characteristic frequency of the SQUID loop, the original absorption loop begins to deform, manifesting itself as a change in the tank circuit impedance. The flux sensitivity diminishes and the rf SQUID begins to resemble an LSQUID. Buhrman and Jackel (1977) concluded that proper adjustment of the SQUID parameters provides a low-noise rf SQUID even when w p > w , . High wp is tempting not only because it reduces the noise but also because it increases the signal bandwidth. It is, however, evident that a SQUID cannot reach the classical noise limit of the resistive loop, i.e., the thermal energy ( i k B T )divided by the noise bandwidth (aw,). SQUIDs operated at w,> w , have been studied both experimentally and theoretically by many authors (Kamper and Simmonds 1972, Kanter and Vernon 1977, Buhrman and Jackel 1977, Hollenhorst and Giffard 1979, Long et al. 1979, Seppa 1983, Vendik et al. 1983, Kuzmin et al. 1985, Likharev 1986). In principle, the high-frequency or microwave SQUID is suitable for multichannel applications since it can be made to a high-gain, low-noise magnetometer with a large signal bandwidth. The lack of reliable thin-film devices, the high cost of the electronics, and the existence of the dc SQUID, however, do not make it very tempting for applications where several channels are needed.

2. I .3. Discussion It is very important to keep in mind that a well-behaved rf SQUID can be constructed only by damping the junction properly. Incomplete damping may lead to multiple transitions and thus to excess noise. If the shunt

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resistance R is adjusted so that the Stewart-McCumber parameter pc= 2 a R ’ C f c / @” remains less than 0.7 (Ketoja et al. 1984a), the I V characteristics of the junction is nonhysteretic and considered well damped. The choice Pc< 1 ensures stable operation of the rf SQUID as discussed by Jackel and Buhrman (1975). Unfortunately, the extra resonances or the parasitic capacitance introduced by a tightly coupled signal coil may substantially decrease the effective damping; this will be discussed in more detail in connection with the dc SQUlD (section 2.2.2). The rf SQUID is based fundamentally on the hysteresis loop traversed as a result of the rf excitation. Energy dissipations reduce the dynamic Q-value of the tank circuit, broadening the signal bandwidth and also making the SQUlD characteristics predictable. The latter consequence is especially important in multichannel applications. With low bias frequencies, the flux-to-voltage conversion efficiency remains moderate and careful design of the preamplifier becomes one of the most important issues in the development of low-noise rf SQUID systems. The intrinsic flux noise can be reached by cooling the first amplifier stage; this is a suitable method in some applications, but hardly in multichannel systems because of increased helium boil-off. Recent developments show that dc SQUIDs are replacing hysteretic rf SQUIDs at least in multichannel magnetometer applications. It seems, however, that the discovery of high- T, materials makes the hysteretic rf SQUlD interesting again. 2.2. DOUBLE-JUNCTION(DC) SQUlDs 2.2.1. Operation

Much lower noise levels than with rf SQUlDs have been obtained with dc SQUlDs (Clarke 1966, Clarke and Fulton 1969, Clarke et al. 1976, Tesche and Clarke 1977, Ketchen 1981). An ideal dc SQUlD is a superconducting loop that has two identical Josephson junctions with critical current I, (fig. 5 ) . In principle, the dc SQUlD can be operated by measuring either the average voltage as a function of the external flux #a with constant bias current f, or by monitoring the current f as a function of @a with constant bias voltage The dc-SQUID loop is in the superconducting state for bias currents below a flux-dependent critical value; at higher currents a voltage over the SQUID appears. The properties of an autonomous dc SQUlD are normally described by two dimensionless parameters P L = 27rLfJ O0 and p, = 2 a R ’ C I J Go,representing the normalized loop inductance and the damping of the junctions, respectively (see also sections 2.1.1 and 2.1.3). The voltage

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28 1

Fig. 5. Equivalent circuit of the dc SQUID. The ideal Josephson junctions are characterized by their critical current I , . Each is shunted by a capacitor C, by a resistor R, and by a thermal noise generator I , . I is the bias current and L is the loop inductance of the SQUID. 8 is the phase difference of the macroscopic wave function of the superconductor over the junction.

and the circulating current oscillate at high frequencies, typically on the order of l-lOGHz, depending on @a and Z, and only the time average of the voltage is monitored. The Iv and O a v characteristics are obtained by integration over the period of one oscillation (Tesche and Clarke 1977, Imry and Marcus 1977, Ben-Jacob and Imry 1981, Ketoja et al. 1984b). In fig. 6 we show the Z v characteristics for p L = n- and pc = 0.3. Figure 7 shows the periodic behavior of the voltage 7 as a function of @ a . A more detailed description of the dynamics is presented in section 4.1.3 of the companion paper (Ryhanen et al. 1989). Higher values of p L and pc create more complex behaviour (Ben-Jacob and Imry 1981, Ben-Jacob et al. 1983, 1985, Ketoja et al. 1984b, 1987, Kurkijarvi 1985). When pc> 0.7, the Z v curves are divided into different voltage branches connected by hysteresis loops, leading to multiple solutions at the same set of parameters (Kulik 1967, Imry and Marcus 1977). Qualitaand Zv characteristics by rounding tively, the thermal noise affects the the point where the voltage state emerges. Because of thermal noise, the hysteresis also disappears, or the hysteresis loops are rounded. On the other hand, fluctuations between different states increase the excess noise in the system. If the SQUID loop inductance and the junction capacitance are negligible, i.e., p L< 7r and BE= 0, the total flux in the ring @ = @a, and the dc SQUID behaves like a single Josephson junction with a resistance R / 2 and = 21, cos( n-Qa/ Oo).Integrating V over the an effective critical current Ic,eff period T of the Josephson oscillation, the average voltage for Z Zc,cn is obtained (Tinkham 1975): V d t = Y [ l - ( ~ c o s ~2 ) 1/2 .

v

@=v

v=Lj:

]

7

In comparison with figs. 6 and 7, an increased inductance reduces the

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3

2

1/24 1

0

Fig. 6. Current-voltage characteristics of the d c SQUID with PI. = T and f3, =0.3. The solid line is obtained when the external flux CDo = @"/2, the dashed line when @, = @,,/4, and the dash-dotted line when CDa = 0. The straight diagonal line is the resistive curve of the SQUID, V = R1/2.

-

I

0.8-

, /

,--.

00--.

,/--'\\

\

\\

'

L.'

I-

\ '' . 4 '

\

-

\ 'i

-U

Fig. 7. Voltage as a function of external flux of a dc SQUID with pL = n and p, = 0.3. The dash-dotted line is obtained when the bias current i = 1 / 2 4 = 0.8, the solid line when i = 1.0, and the dashed line when i = 1.2.

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modulation depth of the effective critical current AZc.c,; for @ a = 00/2, a voltageless state of supercurrent exists. Because thermal noise rounding was neglected in eq. (2.8), the approximate characteristics do not yet indicate the point of operation maximizing the transfer function 3 P/3@,. Differentiating eq. (2.8), one obtains an estimate at the practical point of operation in the flux-locked-loop mode: TRZ,

-@,=@,,0/4,1=2Ic

d @ O *

In a similar way, the dynamic resistance is (2.10)

To release the assumption pL=O,we note on the basis of RSJ-model simulation of fig. 6 that AZc,effi/3L = IT) = 0.5 x AZc,err(jlL= 0) = I,. Thus a V/a@,a AZ,,, is reduced approximately by a factor of 2 for BL= IT. Using PL = 2rLZ,/ Q0 = IT, we obtain R

z@.=@00/4.1=21,

zL'

(2.1 1 )

The equivalent spectral density of the voltage noise power is

(;:I2

Sv = 4kBT - L 2 / (2 R ) + 4k0 TRdyn+ 4kBTARdyn,

(2.12)

where the first term is the contribution of the fluctuations of the circulating current generated by thermal noise of shunt resistors, the second term represents thermal noise across the SQUID ring, and TA is the noise temperature of the amplifier. Applying approximations (2.10) and (2.1 l ) , the energy resolution (2.6) becomes (2.13)

In comparison to the rf SQUID, eq. (2.7), we note that the energy resolution of a dc SQUID depends on the characteristic frequency o,= R/ L, which is normally much higher than the pump frequencies in rf SQUID magnetometers. Since, in addition, TA tends to increase with frequency, the dc SQUID appears superior. Neglecting amplifier noise and setting PL/ r = PC= 1, we find E = 1 2 k 0 T m . For practical reasons, Bc must be set below 1 (Knuutila et al. 1988); in a dc SQUID with C = 1 p F and L =0.2 nH the shunt resistance is about 5 R implying o,= 27r x 4 GHz. The noise temperature of a state-of-the-art

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0’04 0.02

t

1

I

I

HYSTERETIC JUNCTIONS-

0 0

0.5

1.5

1.0

2.0

2.5

3.0

Bc

-

Fig. 8. Dimensionless energy resolution E = J L / C O ; ~ Cas a function of / 3 c = 2 ~ R 2 C I c / @ , The solid line depicts the approximation (2.13), where the junction capacitance C and the loop inductance L are fixed, BL = n,and /3, is varied by changing R. The dashed vertical line refers to the critical value of p, for hysteresis. The squares are from hybrid computer simulations (de Waal et al. 1984). the circles from numerical simulations (Ryhanen et al. 1989).

amplifier can be as low as 2 K (section 2.3); its contribution is therefore negligible. According to eqs. (2.6) and (2.13), the flux noise in our example is ( CD;)~’*= 1.4 x O0/&. However, the presence of the coupling circuits deteriorates the performance (section 2.2.2). The sensitivity of the dc SQUID has been studied by computer simulations (Tesche and Clarke 1977, Bruines et al. 1982, de Waal et al. 1984). The optimized energy resolution was found to be nearly independent of p L and pc for.rr

t

$ W

40

-

n

FREQUENCY (Hz)

FREQUENCY ( H Z )

Fig. 27. (a) Subject noise measured with the Helsinki 7-channel gradiometer. The upper trace was recorded over the left temporal lobe with the subject having his eyes open; the bottom trace shows the corresponding noise spectrum without a subject. (b) Coherence for channels 36.5 mm (solid line) and 73 mm (dashed line) apart, recorded at the left postero-temporal lobe, with the subject having his eyes open. The dash-dotted bottom curve shows the coherence without a subject (Knuutila and Hamalainen 1988).

as the change in the volume of the five-dimensional hyperellipsoid describing the confidence limits of the maximum-likelihood estimate. In the study of Knuutila and Hamalainen (1988), an example is calculated; it was shown that this reduction is significant, demonstrating the advantages of simultaneously measured multiple recordings in obtaining more accurate estimates for the source parameters.

4. Other multi-SQUID applications 4.1.

G EO MA G N ETI S M

In addition to biomagnetism, multiple-SQUID systems are used in geomagnetism. The main advantages of SQUID magnetometers are low noise even with compact detection coils, a large dynamic range, a wide frequency band that starts from dc, and selectivity to specific field or field gradient components, allowing vector measurements. However, the need of liquid helium may cause problems at remote observation sites that are often necessary in geomagnetic studies; large dewars are needed to allow long maintenancefree operation. Small, closed-cycle cryocoolers would be most welcome; they must be battery operated and must not cause magnetic or mechanical disturbances in the measuring system, which has to operate in the open without shielding. The sensitivity of SQUIDS to rf interference and mechanical vibrations complicates their field use. Besides, the high cost of

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adequate multi-SQUID systems is perhaps the main factor limiting their use in geological research and surveying. The SQUID sensors for geophysical applications must have a very large dynamic range. The signals of interest are often only small variations superposed on large field fluctuations not relevant to the particular event under study. The slew rate must be high, even in the frequency range of several tens of kHz, to ensure operation without losing the flux lock. In addition, since the signals of interest extend to very low frequencies, low l/f noise and stable operation, free from flux jumps and drifts, are required. A compact integrated magnetometer with a high slew rate, .intended specifically to geomagnetic studies, has been introduced by Wellstood et al. (1984). In geomagnetic studies, usually all three components of the magnetic field are measured, and thus three SQUIDs are required. Measurements of the magnetic field gradient tensor may also be performed to detect possible anomalies; a gradiometer gives a strong signal when the sensor is moved over the boundary of the anomalous region. In magnetotelluric studies, the magnetic field and the electric field on the ground, caused by incident electromagnetic waves, are measured simultaneously. These waves, typically from 0.1 mHz to 100 Hz, are generated by ionospheric and thunderstorm activity. From the measured electric and magnetic field components as functions of frequency, the impedance tensor Z ( w ) can be determined. The analysis must usually be restricted to simple models such as plane waves and one- or two-dimensional structure of the ground. Because of noise, the estimates of the impendance tensor are, however, unreliable. To overcome this problem, a second measurement is carried out simultaneously at a site several kilometers away and transmitted to the first site by radiotelemetry. Since the noise sources at the two sites are uncorrelated, an unbiased estimate is obtained (Gamble et al. 1979). Correlation techniques can also be applied to improve the estimates, when an artificial variable-frequency electromagnetic source is used in magnetotellurics (Wilt et al. 1983). In addition to the distant reference signal, a phase reference signal is transmitted to the measurement station for lock-in detection. In this case, the reference measurement site senses mainly naturally-occurring geomagnetic fluctuations, which in this controlled-source paradigm constitute noise; this background noise scales roughly as l/f below about 1 Hz, easily outweighing the signal. Measuring the remanent magnetization or magnetic susceptibility of rock samples was perhaps the first application of SQUIDs to geological studies. In paleomagnetism one investigates the history of the earth’s magnetic field; another important area of interest is the identification of magnetic phases in samples. In both cases, a dewar with room temperature access to the

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measurement volume is needed; the devices have a coil to generate a magnetizing field and pickup coils to measure the remanent field components in the axial and transverse directions. Use of SQUID magnetometers in geophysics has been reviewed in detail by Clarke (1983). Several applications have been proposed and some feasibility studies made, although no full-scale tests have been carried out. Yet, for example, stress changes in the ground prior to earthquakes give rise to observable changes in the local magnetic field. The changes of the field are due to the piezomagnetic effect when the domains are reoriented under pressure. I n an experimental feasibility study, the gradients due to seismic waves were found to be typically on the order of 1 pT/m and thus easily detectable with a SQUID (Czipott and Podney 1987). Such a system requires an array of vector magnetometers with telemetry and data acquisition and processing capabilities; the high cost, however, limits the popularity of the method. The measurement of the magnetic field vector could also be applied to locate artificially induced fractures in the ground. These are made by injecting high-pressure propellant into a borehole; such fractures enhance oil or gas flow to a collecting borehole in weakly permeable rock regions. Similarly, in geothermal power plants water is pumped into the ground and high-pressure steam is collected via another borehole. If a magnetic tracer fluid is injected in the fractures, the resulting magnetic anomaly can be detected on the ground with SQUID magnetometers. In addition to measurement of the earth's magnetic field, SQUIDs are used in gravimeters to detect gravitational anomalies (see section 4.2.1). Geomagnetic magnetometers have been made commercially by BTi, CTF, and Cryogenic Consultants, Ltd.6

4.2. PHYSICAL EXPERIMENTS

SQUIDs have many applications in metrology and in various physical experiments where high sensitivity is required. Many of these rather nonstandard applications have been reviewed in detail by Cabrera (1978) and Fairbank (1982), and more recently by Odehnal (1985). Usually such measurement setups contain only a single SQUID; since we are discussing systems with multiple SQUIDs, we consider only a few specific examples where simultaneous recordings are important. In these experimental systems multiple SQUIDs are used mainly to measure different components of the magnetic field. Normally, no mapping

'Cryogenic Consultants, Ltd., Metrostore Building, 23 1 T h e Vale, London W3 7QS.

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of the field pattern is necessary; in some cases a simultaneous reference reading outside the experimental cell is needed. In many of the physics applications, SQUIDs are used to detect very small changes, usually requiring ultra-low-noise sensors. 4.2.I . Accelerometers and displacement sensors

SQUIDs can also detect very small displacements and accelerations. Devices of this type are based on a superconducting proof mass, suspended by springs or by magnetic levitation to allow free movement with respect to a superconducting coil. The inductance of the coil becomes very sensitive to its distance from the proof mass; the measurement coil is connected to a second, auxiliary superconducting coil, and a circulating current is persisted in the circuit. Changes in the inductance are then sensed by measuring the flux changes in the auxiliary coil (see fig. 28). In practical devices, the test mass is usually a superconducting diaphragm, and the coils are flat, as in the original work of Paik (1976). The design may then include several proof masses and coil sets connected to different SQUIDs. This enables the simultaneous measurement of sum and difference phases, different gravitational gradient tensor components, angular accelerations etc. The devices can be applied for detecting anomalies and for measuring the gradient tensor to improve inertial navigation (see, for example, Paik et al. 1978, Colquhoun et al. 1985); sensitivities better s-* HZ-’’~can be achieved. These instruments have been used for than sensing the movements of gravitational wave antennae (see, for example, Kos et al. 1977; McAshan et al. 1981) and for measuring transmitter movements in Mossbauer spectroscopy with subatomic precision of the m/& (Ikonen et al. 1983). Besides this inductance-moduorder of lated technique, piezoelectric and capacitive sensing with SQUIDs as pre-

-Proof mass

Detector coil

Auxiliary coil

Pickup

Input

coil

coil

I

Fig. 28. Schematic illustration of a superconducting accelerometer. A current I is persisted in the detector coil - auxiliary coil loop; changes of the detector coil inductance due to movement of the superconducting proof mass cause in the auxiliary coil a flux change, which is sensed by the SQUID.

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amplifiers have been used in gravitational wave antennae as well (Kos 1978, Cosmelli et al. 1987). 4.2.2. Monopole detectors

Superconducting detectors connected to SQUIDS have been applied in the search for magnetic monopoles predicted by grand unification theories. These particles are predicted to be very massive, too heavy to be produced at accelerators, and extremely penetrating. A magnetic monopole passing through a superconducting loop changes the flux threading the ring by 2Q0, providing a means of detection that is independent of the particle mass or velocity. These detectors should have as large an area as possible to enhance the probability for the passage of a monopole; then, however, the ambient field variations have to be carefully eliminated since an event in a detector, having an effective area of 1 m2 for example, would correspond to a 4fT field change. SQUID sensors with stable operation and freedom from flux trapping are thus required. So far, only one candidate event has been reported (Cabrera 1982). The apparatus consisted of a four-turn 5 cm diameter coil connected to an rf SQUID; it was surrounded by a 20cm diameter superconducting shield and a mu-metal cylinder, providing an ambient field as low as 5 pT and an 180 dB shielding factor for external magnetic fields. Ideally, a monopole traversing the four turns would give rise to a current of 8@,/L. in the loop for coupling to the SQUID, where L is the total inductance of the coil. However, a monopole traversing the superconducting shield will leave doubly quantized vortices at the intersections of its trajectory with the walls. This change in trapped flux reduces the observed signal; the reduction becomes a significant fraction of 2Q0 when the detector dimensions are close to those of the shield. In addition, the degree of degradation is dependent on the trajectory of the monopole, which creates a spread of magnitudes for possible signals instead of a single well-defined value. On the other hand, a small transient is found even from near-miss events. Later, the Stanford group reported a three-channel monopole detector having orthogonal coils with an area of 71 cm2 and a corresponding equivalent near-miss area of 476cm2 (Cabrera et al. 1983). Reliability in detection is based on a coincidence technique: a monopole passing through the detector should be seen in all loops. In a period of 150 days, however, no candidates for monopole events were found. The requirement of eliminating the effect of ambient magnetic field variations severely restricts the cross-sectional area of the sensor. However, the variations in the external field and the effects of the surrounding superconducting shield can be suppressed by twisting the coils to a

MULTI-SQUID DEVICES

0

1x1

33 1

2x2

h g . 29. Schematic illustration of the structure of two-dimensional planar gradiometers. The order of the gradiometer is given below the coils, and the clockwise and counterclockwise windings are indicated by pluses and minuses. The dashed lines show adjacent wires that can be left out.

gradiometric configuration. A monopole passing through only one of the loops still gives a full signal. Design of planar high-order gradiometers has been discussed by Tesche et al. (1983). Generally, an ( N + 1) x ( N+ 1) rectangular two-dimensional array is obtained by taking four N x N units with adjacent blocks oriented oppositely; a zeroth-order sensor is a simple loop (see fig. 29). An N x N array is insensitive to gradients of order P < 2N. In superconducting structures, adjacent loops having the same polarity may be joined. The construction may also be extended to polar and cylindrical geometries. Planar gradiometric sensors provide adequate rejection of external disturbance in ambient fields of the order of 1 FT (Tesche et al. 1983, Incandela et al. 1984). The latter device had two identical 60 cm diameter sensors. To be able to differentiate between real events and spurious signals even better, multiple planar detectors that cover the surface of a certain volume have been built. The six-channel device of IBM (Chi et al. 1984, Bermon et al. 1985) has independent fifth-order sensors on the sides of a 15 x 15 x 60 cm3 rectangular parallelepiped, providing a total area of 0.4 m2. A monopole passing through the volume causes signal in two and only two of the six coils; this kind of coincidence detection provides rejection also to events that might excite all of the detection coils in other geometries, as in the closely spaced set of parallel plates. Quite recently, a new dc SQUID detector of the same parallelepiped geometry of dimensions 26 x 26 x 380 cm3 and an effective area of 1 m2 (averaged over the solid angle 4 n ) has been reported (Bermon 1987); the Stanford group has constructed an octagonal-shaped detector with eight independent 17 x 521 cm2 gradiometer coils, having an effective area of 1.25m2, averaged over 4 n solid angle (Huber et al. 1987).

In addition, several anticoincidence detectors such as accelerometers, field sensors, cosmic ray detectors, power line analyzers, and rf monitors have been used. All the devices have been operated several hundreds of

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days; the null results imply an observed upper limit of 1.5 x lo-' m-* sr-2s-' (90% confidence level) for the magnetic monopole flux (Berrnon 1987). This is still three orders of magnitude higher than the theoretically predicted Parker bound (Turner et al. 1982). With the new superconductors operating at liquid nitrogen temperature it might be feasible to fabricate sensor arrays with a total area a few orders of magnitude larger than in the present devices; then the theoretical bound could be achieved during an operation of a couple of years.

5. Conclusions

Recent progress has made it posssible to build arrays of reliable, low-noise SQUIDs. First such systems, although with rather few channels, are already in use in neuromagnetism. It is expected that arrays with more than 20 channels will become available in the near future. In the state-of-the-art biomagnetic systems, SQUID noise is already negligible in comparison with noise from the dewar materials and from the subject itself. Therefore, there is no immediate need for further reduction of SQUID noise. However, reliability and insensitivity to rf interference will probably be improved as a result of further development. Also, the price of SQUIDs and the readout electronics is expected to drop significantly. The new high- T, superconductors offer some hope in eventually reducing the cost of operating SQUIDs, because liquid helium as a coolant could be replaced by liquid nitrogen. There are many problems on the way to practical high-T, SQUIDs, but one should keep in mind that up to 100 times inferior energy resolution than that in present state-of-the-art SQUIDs would be sufficient for neuromagnetometers. One reason for this is that the noise level in present SQUIDS is already significantly below that from other sources. In addition, the pickup coils in liquid nitrogen could be brought closer to the head, where the signals are stronger and the field profile carries more information about cerebral activity. Even if no high-7, SQUIDs will prove practical, detection coils made of these materials could be used to advantage. Simultaneously with improvements in instrumentation and as a result of it, the feasibility of using SQUIDs in fundamental brain research and in clinical applications has been demonstrated. Although the measurement of electric and magnetic fields are the only methods for non-invasive real-time monitoring of brain function, the eventual significance of neuromagnetism can only be speculated upon. MEG seems superior to EEG in that its interpretation is simpler and because the need of attaching electrodes to

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the scalp is avoided. However, because these two methods convey complementary information, they probably will often be used together. We live in a period of rapid change in the use of SQUIDS: the present 7-channel magnetometers will be replaced in a few years by systems with several dozens of SQUIDS. Faster data collection, provided by the possibility to record the whole field pattern at once, enables more complicated exoeriments and studies of patients, and improves the reliability of results. In addition to technical development of instruments, new ways of thinking are needed in the signal analysis and in the inverse problem studies to optimally utilize the measured signals. For the wide acceptance of the biomagnetic technique, it is crucial that clinically relevant results will be obtained.

Acknowledgements

We thank Olli V. Lounasmaa for suggestions and criticism; Toivo Katila, for sharing his views on cardiomagnetism; Riitta Hari, Matti Hamalainen, Matti Kajola, and Jyrki Makela for reviewing the manuscript. Financial support of the Academy of Finland and of Telecommunications Laboratories of Posts and Telecommunications of Finland is acknowledged.

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Stroink, G., B. Brown, B. Blackford and B.M. Horacek, 1981, in: Biomagnetism, eds S.N. Emt, H.D. Hahlbohm and H. Liibbig (De Gruyter, Berlin) p. 113. Sutherling, W., and D.S. Barth, 1987, in: Abstracts of the 6th Int. Conf. on Biomagnetism, Tokyo, August 27-30, 1987, p. 102. ter Brake, H.J.M., F.H. Fleuren, J.A. Ulfman and J. Flokstra. 1986, Cryogenics 26, 667. Tesche, C.D., 1982, J. Low Temp. Phys. 47, 385. Tesche, C.D., and J. Clarke, 1977, J. Low Temp. Phys. 29, 301. Tesche, C.D., C.C. Chi, C.C. Tsuei and P. Chaudhari, 1983, Appl. Phys. Lett. 43, 384. Tesche,C.D., K.H. Brown, A.C. Callegari, M.M.Chen, J.H.Greiner, H.C. Jones, M.B. Ketchen, K.K. Kim, A.W. Kleinsasser, H.A. Notarys, G. Proto, R.H. Wang and T. Yogi, 1985, IEEE Trans. Magn. MAG-21, 1032. Tinkham, M., 1975, Introduction to Superconductivity (McGraw-Hill, New York). Turner, M.S., E.N. Parker and T.J. Bogdan, 1982, Phys. Rev. D 26, 1296. van Duzer, T., and C.W. Turner, 1981, Principles of Superconductive Devices and Circuits (Elsevier, New York). van Nieuwenhuyzen, G.J., and V.J. de Waal, 1985, Appl. Phys. Lett. 46,439. Varpula, T., and T. Poutanen, 1984, J. Appl. Phys. 55, 4015. Vendik, O.G., Yu.N. Gorin and M.G. Stepanova, 1983, Pis'ma v Zh. Tekh. Fiz. (Sov. Tech. Phys.) 9, 1007. Vrba, J., A.A. Fife, M.B. Burbank, H. Weinberg and P. Brickett, 1982, Can. J. Phys. 60, 1060. Vvedensky, V.L., R.J. llmoniemi and M.J. Kajola, 1985a. Med. & Biol. Eng. & Comput. 23, Suppl., 1 I . Vvedensky. V.L., S.P. Naurzakov, V.I. Ozhogin and S.Yu. Shabanov, 1985b. in: Biomagnetism: Applications and Theory, eds H. Weinberg, G. Stroink and T. Katila (Pergamon Press, New York) p. 57. Weinberg, H., G. Stroink and T. Katila, eds, 1985, Biomagnetism: Applications and, Theory (Pergamon Press, New York). Wellstood, F., C. Heiden and J. Clarke, 1984, Rev. Sci. Instrum. 55, 952. Williamson, S.J., and L. Kaufman, 1987, in: Handbook of Electroencephalography and Clinical Neurophysiology, eds AS. Gevins and A. RCmond (Elsevier, Amsterdam) p. 405. Williamson, S.J.. M. Pelizzone, Y. Okada, L. Kaufman, D.B. Crum and J.R. Marsden, 1985. in: Biomagnetism: Applications and Theory, eds H. Weinberg, G. Stroink and T. Katila (Pergamon Press, New York) p. 46. Wilt, M., N.E. Goldstein, M. Stark, J.R. Haught and H.F. Morrison, 1983, Geophysin48,1090. Zimmerman, J.E., 1971, J. Appl. Phys. 42.4483. Zimmerman, J.E., 1977, J. Appl. Phys. 48,702. Zimmerman, J.E., P. Thiene and T. Harding, 1970, J. Appl. Phys. 41, 1572.

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AUTHOR INDEX Abel, W.R. 152, 189 Abel, W.R., see Wheatley, J.C. 124, 194 Abell, J.S., see Gough, C.E. 14, 43 Abragam, A. 83, 90, 94, 111 Achiezer, A.I. 26, 41 Adamenko, I.N. 179, 189 Adamenko, I.N., see Khalatnikov, I.M. 179, 191 Aeppli, G., see Harshman, D.R. 34, 43 Agranovich, V.M. 27, 28, 33, 41 Ahlers, G. 106, 111 Ahlfors, S., see Knuutila, J. 308, 313, 314, 318, 337 Ahlfors, S.,see Pelizzone, M. 302, 316, 338 Ahola, H. 290, 333 Ahonen,,A., see Knuutila. J. 285, 286, 308, 313, 314, 318, 337 Ahonen, A.I. 124, 125, 189 Ahonen, A.I., see Knuutila, J. 308, 316, 319, 320, 322, 325, 337 Aittoniemi, K., see Kalliomaki, K. 304, 336 Aittoniemi, K.,see Kalliomaki, P.-L. 304,336 Al-Shibani, K. 85, 111 Alexander, S . 144, 159, 189 Alexandrov, A S . 23, 41 Allen, P.B. 18, 41 Allender, D. 28, 37, 41 Alstrom, P. 248, 266 Ambegaokar, V. 232, 266 Ambegaokar, V., see Arani, M.R. 232, 266 Anayama. T.,see Ogushi, T. 8, 44 Ancill, R.J., see Ribary, U. 302, 338 Anderson, A.C. 126, 127, 166, 167, 179, 189 Anderson, A.C.. see Abel. W.R. 152, 189 Anderson, A.C., see Folinsbee, J.T. 126, 127, 166, 167, 190 Anderson, A.C.. see Peterson, R.E. 167, 179, 193 Anderson, A.C.. see Reynolds, C.L. 126,127, 167, 193 Anderson, C.H., see Sabisky, E.S. 169, 193

Anderson, P.W. 25,26,41, 181, 189,230, 266 Anderson, P.W., see Cohen, M.L. 17. 37. 42 Anderson, P.W., see Inkson, J.C. 37, 38, 43 Anderson, P.W., see Lee, P.A. 205, 206, 267 Andreev, A.F. 167, 181, 189, 190 Andres, K . 124, 134, 140, 147, 190 Ansaldo, E.J., see Harshman, D.R. 34, 43 Antervo, A., see Sams, M. 299, 300, 338 Aoyagi, M., see Nakanishi, M. 320, 321, 338 Arani, M.R. 232, 266 Archie, C.N., see Halperin, W.P. 54, I12 Ashburg, J.R., see Wu, M.K. 10, 44 Atsumi, K. 299, 333 Avenel, 0. 124, 125, 127, 186, 190 Azbel, M.Ya. 248, 266 Aziz, R.A. 49, 51, 52, 69, 75, 111 Bain, R.J.P., see Donaldson. G.B. 320, 334 Bak, Per 248, 249, 266 Bak, Per, see Azbel, M.Ya. 248, 266 Barbanera, S.. see Chapman, R.M. 299, 301, 334 Barbanera, S., see Modena, I. 301, 337 Barbi, R. 88, 111 Bardee, T.W.. see Faltens, T.A. 20. 42 Bardeen, J. 12, IS, 37, 38, 41, 42, 199, 219221, 237, 238, 253, 266 Bardeen. J., see Allender, D. 28, 37, 41 Bardeen, J., see Thorne, R.E. 219, 222. 226. 234, 237, 244. 250-253, 268 Bardotti, G. 96. 111 Barone, A. 274, 333 Barranco, M.,see Stringari, S. 110, 113 Barth, D.S.301. 333 Barth, D.S., see Sutherling, W.W. 313, 339 Barton, R., see Kos, J.F. 329, 337 Bashkin, E.P. 71, 111 Basso, H.C. 169, 183, 190 Batey, G.T.187, I90 Batlogg, B. 20, 21, 42 Batlogg, B., see Cava, R.J. 10. 19, 42

341

342

AUTHOR INDEX

Batlogg. B., see Harshman, D.R. 34, 43 Baym, G., see Hoddoson, L. 12, 43 Btal-Monod, M.T.. see Mills, D.L. 152, 155, 192 Beall, J.A., see Muhlfelder. B. 285, 286, 338 Beasley, M.R. 232, 266 Beatty, J., see Barth, D.S. 301, 333 Beauchene, P. 219, 266 Becker. W. 316, 333 Bednort, J.G. 9, 42 Beenakker, J.J.M. 165, 190 Beenakker, J.J.M., see Gorter, C.T. 117, 191 Behringer, R.P.. see Ahlers, G. 106, 111 Bekarevich, I.L. 123, 128. 135, 190 Ben-Jacob, E. 281, 333 Ben-Jacob. E., see Bardeen, J. 219-221, 266 Beno, M.A., see David, W.I.F. 32, 42 Bera, J., see Veuoli, G.C. 8, 44 Berg, M., see Kingery, W.P. 142. 191 Berglund, M.P., see Avenel, 0. 124, 125, 127, 186, 190 Berglund, P.M.. see Ahonen, A.I. 124, 189 Bergman, D.J., see Ben-Jacob, E. 281, 333 Berlincourt, T.G. 7, 42 Bermon, S. 331, 332, 333 Bermon, S.. see Chi. C.C. 331, 334 Bernstein, R.B. 55, 61, 111 Berthier. C., see Segransan, P. 221, 268 Bertotti, B., see Bardotti, G. 96. 111 Betts, D.S. 48, 99, 103, 111, 140, 190 Betts, D.S.,see Leduc, M. 98, 100, 104, 105, 107, 112 Beyermann, W. 215. 266 Bhattacharya, S. 225,226. 239, 260, 261, 263, 264. 266 Bierstedt, P.E., see Sleight, A.W. 8, 44 Bigelow. N., see Johnson, B.R. 98, 112 Bird, R.B.. see Hirschfelder, J.O. 55, 74, 77, 112 Black, W.C., see Abel, W.R. 152, 189 Blackford, B., see Stroink, G. 294. 339 Blatt, J., see Schafroth, M.R. 14, 44 Bleher, M. 235, 266 Bloom, S.B., see Brain, J.D. 304, 333 Bloomfield, P.E., see Lefkowitz, 1. 8, 9. 44 Bogdan, T.J., see Turner, M.S. 332. 339 Bohr, T., see Bak, Per 248, 249, 266 Bonn, D.A., see Kamaris, K. 32-34, 43 Borisov, N., see Neganov, B.S. 140, 193 Borodin, D.V. 211, 215, 263. 266

Bouchaud, J.P. 108, 111 Bouffard, V., see Jacquinot, J.F. 187, 191 Bourg, J., see Cabrera, B. 299, 330. 334 Bourne, L.C. 21, 42, 265, 266 Bourne, L.C., see Faltens, T.A. 20, 42 Boutaud, P., see Segransan, P. 221, 268 Bowley, R.M., see Graf, M.J. 117, 191 Bowley, R.M., see Sheard, F.W. 179, 193 Boyd, M.E. 74. 75, 111 Bozler, H.M., see Ytterboe, S.N. 187, 194 Brain, J.D. 304, 333 Brandt, N.B. 8, 42 Bray, J., see Allender, D. 28, 37, 41 Brewer, D.F. 180, 190 Brewer, D.F., see Ritchie, D.A. 132, 158,159, 162-165, 193 Brewer, J.H., see Harshman, D.R.34, 43 Breymayer, H.-J. 232, 266 Breymayer, H.-J., see Wonneberger, W. 232, 269 Brickett, P., see Ribary, U. 302, 338 Brickett. P.A., see Vrba, J. 305, 307, 339 Brittenham,G.M., see Farrell, D.E. 304,319, 335 Bron, W.E. 127, 168, 190 Brossel, J., see Barbe, R. 88, 111 Brossel, J., see Lefhre-Seguin, V. 85, 89.93, I12 Brown, B.. see Stroink, G. 294, 339 Brown, E. 8, 42 Brown, K.H.,see Tesche, C.D. 285,287,315, 339 Brown, S.E. 216,221,222.224.225,239,240, 242. 246, 249, 250, 256, 257, 262, 266 Brown, S.E., see Thorne, R.E. 222, 226, 250, 251, 268 Bruines, J.J.P. 284, 333 Brush, S.C. 55, 111 Buchal, C., see Owers-Bradley, J.R 71, 113, 141, 193 Buchal, Ch.. see Chocholacs, H. 58, 159, 162-164, 190 Buchanan, D.S. 312, 334 Buckingham, R.A. 61, I l l Buckingham, R.A., see Massey. H.S.W. 61, 113 Buhrman, R. 279. 334 Buhrman, R.A., see Jackel, L.D. 276-280.336 Bulaevskii, L.M. 40, 42 Bulaevskii, L.N. 24, 25, 30, 35, 42

AUTHOR INDEX Bulaevskii, L.N., see Buzdin, A.I. 19, 42 Burbank, M.B., see Vrba, J. 305, 307, 339 Burger, S. 175, 179, 190 Burger, S., see Mok. E. 169, 192 Burton, D. 157.. 194 Busch, P.A., see Greywall, D.S. 158, 191 Busse, F.H. 106, 111 Butler, S., see Schafroth, M.R. 14, 44 Buxton, J.L., see Fraser-Smith, A.C. 293, 335 Buzdin, A.I. 19, 42 Byrne, J. 63, 111 Cabrera, B. 299, 328, 330. 334 Cabrera, B., see Huber, M.E. 331, 336 Cais, M., see Gamble, F.R. 30, 36, 42 Callegari, A.C., see Tesche, C.D. 285, 287, 315, 339 Cameron, J.A. 75, 111 Campbell, M., see Incandela, J. 331, 336 Campitelli, F., see Chapman, R.M. 301. 334 Candela, D., see Masuhara, N. 71, 113 Capone, D.W., see David, W.I.F. 32, 42 Capone, D.W., see Kwok, W.K. 19, 43 Carelli, P. 285, 288, 320, 322, 334 Carelli, P., see Cosmelli, C. 330, 334 Carelli, P., see Modena, I. 301, 337 Carley, J.S., see Aziz, R.A. 51, 52, 69. 111 Castaing, B. 109, 111, 117, 190 Castaing. B., see Peach, L. 117, 193 Castellano, M.G., see Cosmelli, C. 330, 334 Castellano, M.G., see Paik, H.J. 287, 338 Castiel, D., see Shen, T.J. 179, 193 Cava, R.J. 10, 19, 40,42 Cava, R.J., see Batlogg. B. 20, 21, 42 Cava, R.J.. see Fleming, R.M. 219, 246, 267 Cava, R.J., see Harshman, D.R. 34, 43 Chaikin, P.M., see Griiner, G. 198, 227, 267 Challis, L.J. 123, 127, 152, 165, 166, 190 Challis, L.J., see Toombs, G.A. 179, 194 Chang, K.1.. see Bourne, L.C. 21, 42 Chapman, R. 90, 1I1 Chapman, R.M. 299, 301, 334 Chapman, S. 5 5 , 58. 111 Chaudhari, P., see Bermon, S. 331, 333 Chaudhari, P., see Chi, C.C. 331, 334 Chaudhari, P., see Tesche, C.D. 331, 339 Cheeke, J.D.N. 179, 190 Cheeke, J.D.N., see Challis, L.J. 123, 190 Chen, H.S.. see Batlogg, B. 21, 42

343

Chen, M.M., see Tesche, C.D. 285, 287, 315, 339 Chi, C.C. 331, 334 Chi, C.C., see Bermon, S. 331, 333 Chi, C.C., see Tesche, C.D. 331, 339 Chocholacs, H. 158, 159. 162-164, 190 Chocholacs, H., see Owen-Bradley, J.R. 71, 113, 141, 193 Choen, M.L., see Bourne, L.C. 21, 42 Christianson, R.V., see Bak, Per 248,249,266 Chu, C.W. 8, 42 Chu, C.W., see Huang, C.Y. 11, 43 Chu, C.W., see Wu, M.K. 10, 44 Clark, T.D. 242, 266 Clark, T.D., see Long, A. 279, 337 Clark, W.G., see Weger, M. 217, 222, 230, 231. 269 Clarke, G.R., see London, H. 140, 192 Clarke, J. 276, 280, 291, 328, 334 Clarke, J., see Gamble, T.D. 327, 335 Clarke, J., see Ketchen, M.B. 285, 287, 320, 336 Clarke, J., see Tesche, C.D. 280,281,284,339 Clarke, J., see Wellstood, F. 291, 327, 339 Clement, J.R. 101, 111 Cohen, D. 293, 298,304, 310, 322, 334 Cohen, D., see Kennedy, J.G. 310, 336 Cohen, E.G.D., see De Boer, J. 61, 111 Cohen, M.L. 17, 37, 42 Cohen, M.L., see Faltens, T.A. 20, 42 Cohen-Tannoudji, C. 80, 111 Colclogh, M.S., see Cough, C.E. 14, 43 Colegrove, F.D. 78, I I I Collins, R.T. 34, 42 Collins, T.C. 9, 42 Colquhoun, A.B. 329, 334 Combescot, R., see Masuhara, N. 71, 113 Compaan. K., see De Boer, J. 61, 111 Connolly, J.I., see Anderson, A.C. 126, 127, 167, 189 Cooper, L. 15, 42 Cooper, L.N.. see Bardeen, J. 12, 15, 42 Coppersmith, S.N. 234, 238. 253-255, 259, 265, 266, 267 Corruccini, L.R. 71, 111 Corruccini, L.R., see Osheroff, D.D. 158, 162-164, 193 Cosmelli, C. 330. 334 Cowling, T.G.. see Chapman, S. 55, 58, 111 Crabtree, G.W., see Kwok, W.K. 19, 43

344

AUTHOR INDEX

Crampton, S.B., see Leduc, M. 8 3 , 8 5 4 7 . 8 9 , 112 Crampton, S.B., see Nacher, P.-J. 72.86.9395,97, 113 Crocoll, E., see Drung, D. 320. 321, 335 Cromar, M.W., see Muhlfelder, B. 285, 286. 288. 338 Crommie, M.F., see Bourne, L.C. 21, 42 Cross, M., see Sneddon, L. 233,236,253.264, 268 Crum, D.B., see Williamson, S.J. 311, 318, 339 Cuffin. B.N. 302, 334 Cuffin, B.N., see Kennedy. J.G. 310, 336 Cunsolo, S.. see Kinder, H. 183, 191 Curtis, S. 299, 334 Curtiss, C.F., see Hirschfelder, J.O. 55, 74, 77. 112 Czipott, P.V. 328, 334 Daniels, J.M. 78, 81, 84, 85, 99. 111 Daniels, J.M., see Leduc. M. 98, 100, 104, 105, 107, 112 Daniels, J.M., see Timsit, R.S. 89,91,93, 114 Danilov, V.V. 291, 334 Danish. E.H..see Farrell, D.E. 304, 319, 335 David, W.I.F. 32, 42 Day, P.. see David, W.I.F. 32, 42 I>e Boer, J. 5 1 , 61. 1 1 1 De Ninno, A.. see Kinder, H. 183, 191 de Waal, V.J. 284, 285, 287, 320, 334 de Waal, V.J., see Bruines, J.J.P. 284, 333 d e Waal, V.J., see van Nieuwenhuyzen, G.J. 320, 339 DeConde, K., see Perry,T. 124,151-153, 193 Denker, J.S., see Johnson, B.R. 98, 112 Deptuck, D. 144. 157, 159. 190 Derka, I., see Maniewski, R. 294, 325. 337 Derrida, B. 159, 190 Diekmann, V., see Becker, W. 316, 333 Dietsche, W., see Basso, H.C. 169, 183. 190 Dietsche, W., see Kinder, H. 168, 191 Dietsche, W., see Weber, 1. 168-170. 194 Dinger, T.R. 17. 32, 42 Dinger, T.R., see Collins, R.T. 34. 42 Dinger. T.R., see Worthington, T.K. 32, 44 Disalvo, F.J., see Gamble, F.R. 30, 36, 42 Doettinger, S., see Mok, E. 169, 192 Dokoupil, Z., see Beenakker, J.J.M. 165, 190 Dolgov. O.V. 17-19, 29, 42

Donaldson, G.B. 320, 334 Donaldson, G.B., see Colquhoun, A.B. 329, 334 Donaldson, G.B., see Ketchen, M.B. 285, 287, 320, 336 Doniach, S., see Inoue, M. 265, 267 Doss, M.G., see Kamaras, K. 32-34, 43 Dransfeld, K. 165, 190 Dransfeld, K., see Challis, L.J. 165, 166, 190 Drazin, P.G. 106, 1 1 1 Dries, L.J., see Huang, C.Y. 11, 43 Drung, D. 320, 321, 335 Duggan, D.D., see Ong, N.P. 224, 268 Dumas, J., see Beauchene, P. 219, 266 Dupont-Roc, J. 81, 1 1 1 Dupont-Roc, J., see Himbert, M. 85, 86, 112 Duret, D. 305, 307, 335 Duret. D., see Nicolas, P. 294. 338 Duret, D., see Seppanen, M. 318, 338 Durieux, M. 101, 1 1 1 Durieux, M., see Matacotta, F.C. 75, 77, 113 Dynes, R.C., see Allen, P.B. 18, 41 Early, S., see Chu, C.W. 8, 42 Ebisawa, H., see Saito, S. 149, 150, 152, 153, 193 Eckert, J.C., see Ong, N.P. 217, 223, 268 Eckert, M., see Hoddoson, L. 12, 43 Edwards, D.O. 8 5 . I12 Edwards, D.O., see Masuhara, N. 71, 113 Efetov, K.B. 209, 267 Ehnholm, G.J. 279, 335 Ehnholm, G.J., see Ahola, H. 290, 333 Einstein, A. 12, 13, 42 Eisenmenger, W. 169, 190 Eisenmenger, W., see Burger, S. 175, 179,190 Eisenmenger, W., see Heim, U. 169, 191 Eisenmenger. W., see Koblinger, 0. 169,171, I92 Eisenmenger, W.. see Marx, D. 169, 175, 192 Eisenmenger, W., see Mok, E. 169, 192 Eisenmenger, W., see Trumpp, H.J. 168, 194 Eliashberg, G.M. 25, 40, 42 Emery, V.J. 26, 42, 70, 112 Emoto, H., see Furukawa, H. 316, 335 Endo, T., see Nakanishi, M. 320, 321, 338 Engel Jr, J., see Barth, D.S. 301, 333 Enpuku, K. 285-287, 335 ErnC, S.N. 294, 303, 320, 322, 325, 335 Estes. W.E., see Lefkowitz, 1. 8. 9, 44

AUTHOR INDEX Etienne, P.E., see Sinton, C.M. 302, 338 Ettinger, H., see Cheeke, J.D.N. 179, 190 Ezawa, H. 171, 172, 190 Fack, H. 232, 242, 267 Fairbank, H.A. 165, 190 Fairbank, H.A.. see Lee, D.M. 117, 192 Fairbank, W.M. 328, 335 Fairbank, W.M., see McAshan, M.S. 329,337 Faltens, T.A. 20, 42 Faltens, T.A., see Leary, K.J. 32. 44 Farago, P.S., see Byme, J. 63, 1 1 1 Farrell, D.E. 304, 305, 319, 335 Fenici, R.R. 303, 335 Fermon, C., see Jacquinot, J.F. 187. 191 Fife, A.A., see Vrba, J. 305, 307, 339 Fiory, A.T., see Gurvitch, M. 34, 43 Fisher, D. 234, 236, 264, 267 Fisher, D., see Sneddon. L. 233, 236, 253, 264, 268 Fjeldly, T.A., see Ishiguro. T. 168, 191 Fleming, R.M. 197, 198, 215-219, 237, 240, 246,257, 258, 267 Fleuren, F.H., see ter Brake, H.J.M. 324, 339 Flokstra, J., see ter Brake, H.J.M. 324, 339 Fogan, E.M., see Cough, C.E. 14, 43 Foglietti, V. 292, 335 Foglietti, V., see Carelli. P. 285, 288, 320, 334 Foglietti, V., see Cosmelli, C. ,330, 334 Folinsbee, J.T. 126, 127, 166-169, 190 Fomin, LA. 123, 190 Ford, P.J.. see Hall, H.E.140, 191 Forgacs, R.L. 289, 335 Frankel, R.B., see Huang, C.Y. 11, 43 Fraser-Smith, A.C. 293, 335 Freed, J.H., see Johnson, B.R. 98, I12 Freitas, P., see Collins, R.T. 34, 42 Friedman, L.J., see Ytterboe, S.N.187, 194 Friend, R.H. 30, 42 Frisch, H., see Incandela, J. 331, 336 Frisken, B. 142, 144, 190 Frohlich, H. 197, 267 Frolov, V.V., see Latyshev, Yu.1. 221, 267 Frossati, G. 124, 141, 158, 162-164, 190 Fuchs, I.M.,see Adamenko, I.N. 179, 189 Fujii. Y. 187, 191 Fukuda, T., see Ishimoto, H. 71,98, 112, 158, 191 Fukuyama, H. 209,232,253, 267

345

Fukuyama, H., see Ishimoto, H. 71, 98, 112, 158, 191 Fukuyama, H., see Mamiya. T. 158, 192 Fulton, T.A., see Clarke, J. 280, 334 Furayama, T., see Maeda, A. 216, 267 Furukawa, H. 316, 335 Furukawa, H.,see Shirae, K. 316, 318, 338 Gabovich, A.W. 8, 20, 42 Gallagher, W.J., see Collins, R.T. 34, 42 Gallagher, W.J., see Dinger, T.R. 17, 32, 42 Gallagher, W.J., see Foglietti, V. 292, 335 Gallagher, W.J.. see Worthington, T.K. 32.44 Gamble, F.R. 30, 36, 42 Gamble, T.D. 327, 335 Gammic, G . , see Lyding, J.W. 224, 225, 267 Gao, L., see Wu, M.K. 10, 44 Gardner, R., see Cabrera, B. 299, 330, 334 Gardner, R.D., see Huber, M.E. 331, 336 Gavoret, J. 123, 128, 135, 191 Geballe, T.H., see Chu, C.W. 8, 42 Geballe, T.H., see Gamble, F.R. 30, 36, 42 Geballe, T.H., see White, R.M. 16, 35, 44 Gehr, P., see Brain, J.D. 304, 333 Geilikman, B.J. 30, 42 Gershenson, E.M. 34, 42 Geselowitz, D.B., see Cuffin, B.N. 302, 334 Geselowitz, D.B.. see Grynszpan, F. 297.335 Geselowitz, D.B., see Miller 111, W.T. 302, 33 7 Gianolio, L., see Bardotti, G. 96, 111 Giffard, R.P. 276, 277, 288, 335 Giffard, R.P., see Hollenhorst, J.N. 279, 335 Gill, J.C. 211. 215, 263. 267 Gillson, J.L., see Sleight, A.W. 8, 44 Ginsberg, D.M., see Thomasson, J.W. 96, I14 Ginzburg, V.L. 5-7,9, 12-16, 18-20, 22, 2731, 35, 36, 38-40, 42, 43 Ginzburg, V.L., see Agranovich, V.M.27.28, 33, 41 Ginzburg, V.L., see Bulaevskii, L.M. 40, 42 Giri, M.R., see Morii, Y. 158, 192 Glukhov, N.A., see Kagan, Yu. 117, 191 Godfrin, H., see Frossati, G. 124, 141, 190 Godfrin, H., see Lauter, H.J. 180, 192 Goldman, V.V. 117, 191 Goldstein, N.E., see Wilt, M. 327, 339 Golovashkin, A.I. 9-11, 43 Gonzales, O.D., see White, D. 165, 194 Goodstein, D., see Kinder, H. 183, 191

346

AUTHOR INDEX

Goodstein. D., see Taborek. P. 168, 169, 175, 176, 193, 194 Gorbatsevich. A.A., see Ginzburg, V.L. 9, 43 Gorin, Yu.N.. see Vendik, O.G. 279, 339 Gor’kov, L.P. 35, 36, 43 Goner, C.T. 117, I91 Goubau, W.M.. see Clarke, J. 280, 291, 334 Goubau. W.M., see Gamble, T.D. 327. 335 Goubau, W.M., see Ketchen, M.B. 285, 287, 320, 336 Gough. C.E. 14, 43 Gould, C.N., see Ytterboe. S.N. 187. 194 Grace, J.D., see David, W.I.F. 32, 42 Grad, H. 5 5 . 112 Graf, M.J. 117. 191 Grant, P. 4, 10, 43 Greedan, J.E., see Kamaras, K. 32-34. 43 Gregg, J.F., see Jacquinot, J.F. 187, 191 Greiner, J.H., .see Tesche, C.D. 285,287.3 15, 339

Grest. G.S. 159, 191 Greywall, D.S. 158, 191 Grimes, C.C., see Fleming, R.M. 198, 215, 217, 218, 237. 267 Grimes, D. 304, 335 Grimsrud, D.T. 75, 112 Griiner, G . 198, 205, 215, 218, 227, 267 Griiner. G.. see Bardeen, J. 219-221, 266 Griiner, G., see Beyemann, W. 215, 266 Griiner, G., see Brown, S.E. 221, 222, 224, 225, 239, 240, 242, 246, 249. 250, 256, 257, 262, 266 Griiner, G., see Kim, Tae Wan 213, 267 Griiner, G . , see Mihaly, L. 216, 267 Griiner, G . , see Mozurkewich, G. 218, 222, 223. 227, 234, 260, 263, 264, 268 Griiner, G., see Reagor, D. 212, 213, 268 Griiner, C . , see Sridhar, S. 212. 213,235, 268 Gruner, G., see Thorne, R.E. 222, 226, 250, 25 1, 268 Gruner,G., see Weger, M. 217,222,230,231, ,769

Griiner, G., see Wu,Wei-Yu 215. 238, 269 Griiner, G., see Zettl, A. 215, 216, 224, 226, 239. 240, 243-246, 260, 263-265, 269 Grynszpan. F. 297, 335 Guenault, A.M. 141, 191 Guillon, F., see Frisken. B. 142, 144, 190 Guillon, F., see Robertson, R.J. 160. 161. 193 Gully, W.J. 71. 98, 1J2

Gully, W.J., see Osheroff, D.D. 128, 193 Gumnit, R.J., see Kennedy, J.G. 310, 336 Gunn, J.M.F., see David, W.I.F. 32, 42 Guo. C.J. 168, 191 Gurvitch, M. 34, 43 Gustafson, H.R., see Incandela, J. 331, 336 Guyer, R.A. 152, 158, 191 Gylling, R.G., see Avenel, 0. 124, 125, 127, 186, 190 Haario, H.. see Hamalainen, M.S. 308, 335 Hahlbohm, H.-D., see Erne, S.N. 294, 335 Haikala, M.T., see Ahonen. A.I. 124, 189 Hakuraku, Y., see Ogushi. T. 11, 44 Hall, H.E. 140, 191 Hall, R.P. 247, 264, 267 Hallstrom. J.. see Hari, R. 311, 335 Hallstrom, J., see Knuutila, J. 308, 313, 314, 318. 337 Halperin, B.I., see Ambegaokar, V. 232, 266 Halperin. B.I., see Anderson, P.W. 181, 189 Halperin, W.P. 54. 112 Ham, W.K., see Faltens, T.A. 20, 42 Ham, W.K., see Leary, K.J. 32, 44 Hamalainen, M., see Kaukoranta, E. 308,336 Hamalainen, M., see Pelizzone, M. 302,316, 338

Hamalainen, M., see Sams, M. 299, 300, 338 Hamalainen, M.S. 297, 308, 335 Hamalainen, M.S., see Ilmoniemi, R.J. 297, 336

Hamalainen, M.S., see Knuutila, J. 293, 308, 316, 319, 320, 322, 325, 326, 337 Hamilton, J.. see Buckingham, R.A. 61, 1 1 1 Hammel, E.F., see Kilpatrick, J.E. 61.74, 112 Happer, W. 80, 112 Harding. T., see Zimmerman, J.E. 276, 339 Hardy, W., see Harshman, D.R. 34, 43 Hardy, W.N.. see Lefevre-Seguin, V. X5, 89, 93, 112 Hari, R. 297, 299, 307, 311, 335 Hari, R., see Huttunen, J. 3 I I , 336 Hari, R., see Ilmoniemi, R.J. 296, 305. 306, 310, 318, 336 Hari, R., see Kaukoranta, E. 300, 307, 308, 311, 336 Hari, R., see Makela, J.P. 31 I , 337 Hari, R.. see Pelizzone, M. 302, 316, 338 Hari, R., see S a m , M. 299. 300, 338 Harris, J.W., see Farrell, D.E. 304, 319, 335

AUTHOR INDEX Harrison, J.P. 118, 124. 142, 187, 191 Harrison, J.P., see Deptuck, D. 144, 157,159, I90 Harrison, J.P., see Folinsbee, J.T. 168, 169, 190 Harrison, J.P., see Frisken, B. 142, 144. 190 Harrison, J.P., see Maliepaard, M.C. 144, 157, 159, 160, 192 Harrison, J.P., see Robertson, R.J. 160, 161, I93 Harrison, J.P., see Rutherford, A.R. 147,148, 159, 193 Harrison, W.T.A., see David. W.I.F. 32, 42 Harshman, D.R. 34, 43 Hatfield, W.E., see Leflowitz, I. 8, 9, 44 Haug, H. 167, 179, 191 Haug, R. 180, 191 Haught, J.R., see Wilt, M. 327, 339 Hayakawa. K., see Iwama, S. 142, 191 Hayashi, C. 141, 191 Hayashi, M. 142, 144, 191 He, Da-Run, see Yeh, W.J. 249, 250, 269 Hebral, B., see Frossati, G. 124, 141, 190 Hebral, B., see Peach, L. 117, 193 Heiden, C., see Wellstood, F. 291, 327, 339 Heim, U. 169, 191 Heim, U., see Koblinger, 0. 169, 171, 192 Heino, J.J., see Kelha, V.O. 293, 315, 336 Heinonen, P., see Malmivuo, J. 294, 337 Herr, S.L., see Kamaris, K. 32-34, 43 Henvig, R., see Drung, D. 320, 321, 335 Higgins, M.J., see Bhattacharya, S. 225, 239, 260, 261, 263, 266 Higo, S., see Ogushi, T. 11, 44 Himbert, M. 85, 86. 112 Hinks, D.G., see David, W.I.F. 32, 42 Hinks, D.G., see Kwok, W.K. 19, 43 Hirschfelder. J.O. 55, 74, 77, 112 Hoddoson, L. 12, 43 Hoen, S., see Faltens, T.A. 20, 42 Holiday, S., see Ribary, U. 302, 338 Hollenhorst, J.N. 279, 335 Holste, J.C., see Radebaugh, R. 124,141,162, I93 Holzman. G.N., see Gershenson, E.M. 34,42 Homan, C.G. 9, 43 Homan, C.G., see Brown, E. 8, 42 Honjo, Y., see Ogushi, T. 11, 44 Hood, K. 152, 191 Hor, P.H., see Huang, C.Y. 11, 43

347

Hor, P.H., see Wu, M.K. 10, 44 Horacek, B.M. 302, 335 Horacek, B.M., see MacAulay, C.E. 303,337 Horacek, M., see Stroink, G. 294, 339 Horstman, R.E. 168, 191 Hoshino, R., see Ido, M. 149, 191 Hostetler, W., see Ilmoniemi. R.J. 299, 312, 336 Hoyt, R.F., see Masuhara, N. 71, 113 Huang, C.Y. 11, 43 Huang, C.Y.. see Chu, C.W. 8, 42 Huang, S., see Chu. C.W. 8, 42 Huang, Z.J., see Wu, M.K. 10, 44 Hubacek, J.S., see Lyding, J.W. 224,225,267 Hubacek, J.S., see Thorne, R.E. 248, 268 Huber, M.E. 331, 336 Huber, T.E. 117, 191 Huber, T.E., see Mans, H.J. 117, 192 Huberrnan, B.A., see Beasley, M.R. 232, 266 Hundley, M.F., see Hall, R.P. 264, 267 Huttunen, J. 311, 336 Huttunen, J., see Peliuone, M. 302.3 16, 338

Ido, M. 149, 191 Ikonen, E. 329, 336 Ilmoniemi, R. 308, 311, 336 Ilmoniemi, R., see Ryhanen, T. 273,276,281, 284, 338 Ilmoniemi, R.J. 296, 297, 299, 305, 306, 310, 312, 318, 336 Ilmoniemi, R.J., see Chapman, R.M. 299, 301, 334 Ilmoniemi, R.J., see Hari, R. 297, 335 Ilmoniemi, R.J., see Kelhl, V.O. 293, 315, 336 Ilmoniemi, R.J., see Knuutila, J. 308, 316, 319, 320, 322, 325, 337 Ilmoniemi, R.J., see Sinton, C.M. 302, 338 Ilmoniemi, R.J., see Vvedensky, V.L. 299, 311,339 Imry,Y. 281, 336 Imry, Y., see Ben-Jacob, E. 281, 333 Incandela, J. 331, 336 Inkson, J.C. 37, 38, 43 Inoue, M. 265, 267 Irie, F., see Enpuku, K. 285-287,335 Ishiguro, T. 168, 191 Ishimoto, H. 71, 98, 112, 158, 191 Iwama, S. 142, 191

348

AUTHOR INDEX

Jackel. L., see Buhrman, R. 279, 334 Jackel, L.D. 276-280, 336 Jacquinot, J.F. 187, 191 Jaklevic, R.C. 274, 336 Janossy, A., see Beauchene, P. 219, 266 Jinossy, A.. see Brown, S.E. 224, 225, 266 Janossy. A.. .see Segransan, P. 221, 268 Janossy, A., see Wu, Wei-Yu 215, 238, 269 Jayaraman, A., see Batlogg, B. 20, 21, 42 Jaycox, J.M. 285, 320. 336 Jensen, M.H., see Alstrom, P. 248, 266 Jensen, M.H., see Bak, Per 248, 249, 266 Jerome, D. 36, 43 Jerome, D.. see Gor'kov, L.P. 35, 43 Jing, T.W. 224, 267 Johnson, B., see Ribary, U. 302, 338 Johnson, B.R. 98, 112 Johnson, R.C. 166, 191 Johnson, R.T., see Wheatley, J.C. 141, 194 Johnson, W., see Muhlfelder, B. 288, 338 Johnson, W.L., see Anderson, A.C. 126, 166, 167, 179, 189 Johnson, W.W., see Muhlfelder, B. 285, 286, 338 Johnston, H.L., see White, D. 165, 194 Jones, H.C., see Tesche, C.D. 285, 287, 315, 339 Jordan, R.G., see Gough, C.E. 14, 43 Jorgensen, J.D., see David, W.I.F. 32, 42 Jorgensen. J.D., see Kwok, W.K. 19, 43 Joutsiniemi, S.-L. 317, 336 Joutsiniemi, S.-L., see Hari, R. 307. 335 Jutzi, W., see Drung, D. 320. 321, 335 Jutzler, M. 187, 191 Kado, H., see Nakanishi, M. 320, 321. 338 Kagan, Yu. 117, 191 Kahan, B.G., see Shapligin, 1.S. 10, 44 Kaiser, M.B., see Zettl, A. 224, 269 Kajanto, M.J. 232, 267 Kajola, M., see Knuutila, J. 283, 286, 288, 291, 308, 313, 314, 318,320, 321, 337 Kajola, M.J., see Knuutila, J. 308. 316, 319, 320, 322, 325, 337 Kajola, M.J., see Vvedensky, V.L. 299, 311, 339 Kalem, C.B., see Ong, N.P. 2 17,223,224,268 Kalliomaki, K. 304, 336 Kalliomaki, K., see Kalliomaki, P.-L. 304, 336

Kalliomaki, P.-L. 304, 336 Kalliomaki. P.-L., see Kalliomaki, K. 304, 336 Kamaras, K. 32-34, 43 Kamper, R.A. 279, 336 Kanter, H. 279, 336 Kao, Y.H., see Yeh, W.J. 249. 250, 269 Kapitza, P.L. 117, 191 Karasik, B.S., see Gershenson, E.M. 34, 42 Karp, P. 309, 336 Karp, P., see Duret, D. 305, 307, 335 Karp, P., see Ikonen, E. 329. 336 Karp, P., see Seppanen, M. 318, 338 Kasai, N., see Nakanishi, M. 320, 321, 338 Kastler, A., see Cohen-Tannoudji, C. 80, 111 Katayama, M., see Furukawa, H. 316, 335 Katayama, M., see Shirae, K. 318, 338 Katayama, T., see Shirae, K. 318, 338 Katila, T. 303, 336 Katila, T., see Atsumi, K. 299, 333 Katila, T., see lkonen, E. 329, 336 Katila, T., see Maniewski, R. 294, 325, 337 Katila, T., see Montonen, J. 303, 338 Katila, T., see Seppanen, M. 318, 338 Katila, T., see Weinberg, H. 299, 339 Kaufman, L. 297.31 1, 336 Kaufman, L., see Curtis, S. 299, 334 Kaufman, L., see Pelizzone, M. 311, 338 Kaufman, L., see Romani, G.L. 305,307,324, 338 Kaufman, L., see Williamson, S.J. 297, 311, 318, 339 Kaufmann, R.,see Becker, W. 316, 333 Kaukoranta, E. 300, 307, 308, 311, 336 Kaukoranta, E., see Hari. R. 299, 335 Kaukoranta, E., see Sams, M. 299, 300, 338 Kawano, I . I . , see Ogushi, T. 1 1 , 44 Keene, M., see Gough, C.E. 14, 43 Keith, V., see Guenault, A.M. 141, 191 Kelha, V.O. 293, 315, 336 Keller, S.W., see Bourne, L.C. 21, 42 Keller, S.W., see Faltens, T.A. 20, 42 Keller, S.W., see Leary, K.J. 32, 44 Keller, W.E. 48, 61. 74, 75, 103, 112 Keller, W.E., see Kilpatrick, J.E. 61, 74, 112 Kennedy, C.J., see Guenault, A.M. 141, 191 Kennedy, J.G. 310, 336 Kennedy, J.S., see Ribary, U. 302, 338 Kemsk. J.F., see Keller, W.E. 103, 112 Ketchen, M.B. 280, 285, 287. 320, 336

AUTHOR INDEX

349

Ketchen, M.B., see Clarke, J. 280, 291, 334 Koch, R.H., see Foglietti, V. 292, 335 Kohjiro, S., see Enpuku, K. 287, 335 Ketchen, M.B., see Foglietti, V. 292, 335 Kojima, H.. see Morii, Y. 158, 192 Ketchen, M.B., see Jaycox. J.M. 285,320,336 Kopaev, Yu.V. 22, 43 Ketchen, M.B., see Tesche, C.D. 285, 287, Kopaev, Yu.V., see Ginzburg, V.L. 9, 43 315, 339 Kornhuber, H.H., see Becker, W. 316, 333 Ketoja, J.A. 280, 281, 336, 337 Khalatnikov,J.M. 117,120,123, 165,179,191 Kos, J.F. 329, 330, 337 Khalatnikov, I.M., see Bekarevich, I.L. 123, Kosaka, S., see Nakanishi, M. 320, 321, 338 Kose, V., see Fack, H. 232, 242, 267 128, 135, 190 Kosevich, Yu.A., see Andreev, A.F. 181, 190 Khomskii, D.I., see Bulaevskii, L.N. 24, 25, Kotani, M.. see Atsumi, K. 299, 333 35, 42 Khomskii, D.J., see Zvezdin, A.K. 24, 25, 44 Kourouklis. G., see Batlogg, B. 20, 42 Kourouklis, G.A., see Batlogg, B. 21, 42 Kilpatrick, J.E. 61, 74, 112 Kilpatrick, J.E., see Boyd, M.E. 74, 75. 111 Koyanagi, M., see Nakanishi, M. 320, 321, 338 Kim, K.K., see Tesche, C.D. 285, 287, 315, Kreitzman, S.R., see Hanhman, D.R. 34, 43 339 Krusius, M.,see Ahonen, A.J. 124, 189 Kim, Tae Wan 213, 267 Kuang, W.-Y. 165, 166, 192 Kinder, H. 168, 179. 183, 191 Kubo, R., see Teranishi, N. 234, 268 Kinder, H., see Basso, H.C. 169, 183, 190 Kubota, M., see Chocholacs, H. 158, 159, Kinder, H., see Schubert, H. 183, 193 162-164, 190 Kinder, H., see Weber, J. 168-170, 194 Kubota, M., seeOwen-Bradley, J.R. 71, 113, Kingery, W.P. 142, 191 141, 193 Kirschvink, J.L. 304, 337 Kubota, M., see Rogacki, K. 124. 141-143, Kirzhnits, D.A. 17, 37, 43 193 Kirzhnits, D.A., see Ginzburg, V.L. 12, 16, Kuchnir, M., see Incandela, J. 331, 336 18-20, 22, 27-30, 36, 38, 43 Kukharenko, Yu.A., see Bulacvskii, L.N. 30, Kishida, K., see Shirae, K. 316, 338 42 Klapwijk, T.M., see d e Waal, V.J. 285, 287. Kulik, 1.0. 281, 337 320, 334 Kleinsasser, A.W., see Foglietti, V. 292, 335 Kummer, J.T. 149, 192 Kleinsasser, A.W., see Tesche, C.D. 285,287, Kurkijarvi, J. 277, 281, 337 Kurkijarvi, J., see Ketoja, J.A. 280, 281, 336, 315, 339 337 Klemm, R.A. 233, 267 Klemm, R.A., see Bhattacharya, S. 225, 226, Kuvshinnikov, B.V., see Brandt, N.B. 8, 42 Kuzmin, L.S. 279, 337 239, 260, 261, 263, 264, 266 Kwok, W.K. 19, 43 Klemm, R.M., see Robbins, M. 233,235,268 Klitsner, T. 169, 183, 191 Knight, W.D. 150, 192 Kniittel, A., see Drung, D. 320, 321, 335 Laibowitz, R.B., see Collins, R.T. 34, 42 Knuutila, J. 283,285,286,288,291,293,308, Lake, G.M., see Hanhman. D.R. 34, 43 313, 314, 316, 318-322, 325. 326, 337 Laloe, F. 47, 48, 83, 86, 112 Knuutila, J., see Ilmoniemi, R. 308, 311, 336 Laloe, F., see B a h t , R. 88, I I I Knuutila, J., see Ilmoniemi, R;J. 297, 336 Laloe, F., see Dupont-Roc, J. 81, 111 Knuutila, J., see Ryhanen, T. 273, 276, 281, Laloe, F., see Himbert, M. 85, 86, 112 284. 338 Laloe, F., see Leduc, M. 81, 83, 85-87, 89, Kobayashi, S. 149, 192 98, 100, 104, 105, 107. I12 Koblinger, 0. 169, 171, 192 Laloe, F., see Lefkvre-Seguin. V. 85, 89, 91, Koblinger, 0.. see Heim, U. 169, 191 93.94, I12 Koch, H. 287, 337 Laloe, F., see Lhuillier, C. 48, 59. 60,62,63, Koch, R.H., see Collins, R.T. 34, 42 66, 67, 70-72, 74, 77, 108, 110, 112, 113

350

AUTHOR I N D E X

Laloe. F., see Nacher. P.-J. 72, 84-88,93-95, 97, 113 Laloe, F., see Pavloviac, M. 83, 104. 113 Laloe, F., see Pinard, M. 60, 113 Laloe. F., see Stringari, S. 110, 113 Laloe, F.,see Tastevin, G . 93.97-99, 109, 113 Lambe, J., see Jaklevic, R.C. 274, 336 Lambert. C.J. 157, 159, 192 Lambert, C.J., see Burton, D. 157.. 194 Landau, L.D. 13, 43, 120, 128, 192 Landau, L.D., .see Ginzburg, V.L. 14, 43 Larkin. A.I., see Efetov, K.B. 209, 267 Larsen, S.Y., see Boyd, M.E. 74. 75, 1 1 1 Lasarev, V.B., see Shapligin. I.S. 10, 44 Lassmann, K., see Burger, S. 175, 179, 190 Lassmann, K., ree Mok. E. 169, 192 Lassmann, K., see Trumpp. H.J. 168, 194 Latyshev, Yu.1. 221, 244, 267 Lauter, H.J. 180. 192 Lazarus, R.B. 102, 112 Leary, K.J. 32. 44 Leary, K.J., see Faltens, T.A. 20, 42 Leduc. M. 81,83,85-87.89,98,100,104,105, 107, 112 Leduc, M., see Betts, D.S. 48, 1 1 1 Leduc, M.. see Daniels, J.M. 78, 84. 85, 99, 111

Leduc, Leduc, Leduc, Leduc,

M., see Dupont-Roc, J. 81, 1 1 1 M.. see Himbert, M. 85, 86, 112 M., ree Laloe. F. 48. 83, 86, 112 M., see Lhuillier, C. 48.50.63-65.98,

113

Leduc. M., see Nacher, P.-J. 72.84-88.93-95, 97. 1 1 3 Leduc, M., see Schearer, L.D. 78, 98. 113 Leduc, M., see Tastevin, G . 93, 97-99, 109, I 1.3 Leduc, M., see Trenec, G . 8 1, 82. 114 Lee, D.M. 117, 192 Lee, D.M.. see C o m c c i n i , L.R. 71, 1 1 1 Lee, D.M.. see Johnson, B.R. 98, 112 Lee, D.M., see Osheroff, D.D. 128, 193 Lee, P.A. 205, 206. 223, 232, 253, 267 Lee, P.A.. see Fukuyama, H. 209, 253, 267 Lee, P.A., see Ong, N.P. 224, 268 Lefivre-Seguin. V. 8 5 , 89, 91, 93, 94, 112 LeTevre-Seguin. V., see Himbert. M. 85, 86, 112

Lefkowitz, 1. 8. 9, 44 Leggett, A.J. 71, 112, 151, 152, 192

Lehtinen, M.S., see Hamalainen, M.S. 308, 335 Leiderer, P.. see Basso, H.C. 169, 183, 190 Leiderer, P., see Schubert, H. 183, 193 Leinio, M., see Montonen, J. 303, 338 Leinonen, L., see Hari, R. 311, 335 Leinonen, L., see Huttunen, J. 311, 336 Lejus, A.-M., see Schearer, L.D. 78, 98, 113 Lekkala, J., see Malmivuo, J. 294, 337 Lekkala, J.O. 318, 337 Lennard, R., see Grimes, D. 304, 335 Lennard-Jones, T.E. 180, 192 Leoni, R., see Carelli. P. 320, 322, 334 Leoni, R., see Chapman, R.M. 301, 3.74 Leoni, R., see Modena, 1. 301, 337 Leoni, R., see Romani. G.L. 308, 312, 318, 338 Leory, K.L., see Bourne, L.C. 21, 42 Levinsen, M.T., see Alstrom, P. 248, 266 Levy, L.P., see Johnson, B.R. 98, 112 Lhuillier, C. 48, 50. 59-67, 69-72, 74, 77. 98, 105, 108, 110. 112, 113 Lhuillier, C., see Bouchaud, J.P. 108, 1 1 1 Libchaber, A. 106, 113 Libchaber, A., see Maurer, J. 106, 113 Liburg, M., see Neganov, B.S. 140, 193 Lifshitz, E.M., see Landau, L.D. 120, 192 Likharev, K.K. 274, 279, 288, 337 Likharev, K.K.. see Danilov, V.V. 291, 334 Likharev, K.K., see Kuzmin, L.S. 279. 337 Lindelof, P.E. 229, 242-244, 255, 267 Lindelof, P.E.. see Clark, T.D. 242, 266 Link, G.L. 225, 226, 267 Little, W.A. 28, 34, 36, 44, 165, 179, 192 Little, W.A., see Johnson, R.C. 166, 191 Littlewood, P.B. 235, 252, 267 Littlewood, P.B., see Coppersmith, S.N. 234, 238, 253-255, 259, 265, 266, 267 Liu, P.S. 40, 44 Llurba, R., see d e Waal, V.J. 284, 334 Lockerbie, N.A., .see Colquhoun, A.B. 329, 334 Lockerbie, N.A., see Sherlock, R.A. 168, 193 Lockerbie, N.A., see Wyatt, A.F.G. 169, 194 Lombroso, C.T.. see Kennedy, J.G. 310, 336 London, F. 13, 15, 44 London, H. 140, 192 Long, A. 279, 337 Lopez, L., see Fenici, R.R. 303, 335 Lounasmaa, O.V. 140, 192, 274, 337

AUTHOR INDEX Lounasmaa, O.V., see Ahonen, A.I. 124, 125, 189 Lounasmaa, O.V., see Andres, K. 140, I90 Lounasmaa, O.V., see Hari, R. 311, 335 Lounasmaa, O.V., see Kaukoranta, E. 300, 307, 311, 336 Lounasmaa, O.V., see Knuutila, J. 308, 313, 314, 318, 337 Loye, H.C., see Bourne, L.C. 21, 42 Loye, H.C., see Faltens, T.A. 20, 42 Ludwig, W. 319, 337 Ludwig, W.,see Becker, W. 316, 333 Lusznynski, K. 70, 71, 113 Lyding, J.W. 224, 225, 267 Lyding, J.W., see Thorne, R.E. 222,226,234, 237, 238, 244, 248, 250-253, 268 Lyding, J.W., see Tucker, J.R. 235, 269 Lynton, E.A., see Beenakker, J.J.M. 165, 190 Lyons, W.G., see Thorne, R.E. 222,226,234, 237, 238, 244, 248, 250-253, 268 Lyons, W.G., see Tucker, J.R. 235, 269 MacAulay, C.E. 303, 337 MacCrone, R.K., see Brown, E. 8, 42 MacCrone, R.K., see Homan, C.G. 9, 43 Madekivi, S., see Montonen, J. 303, 338 Maeda, A. 216, 226,267 Mager, A. 294, 337 Main, P.C., see Batey, G.T. 187, 190 Makela, J.P. 311, 337 Makela, J.P., see Han, R. 311, 335 Makela, J.P., see Pelizzone, M. 302, 316,338 Maki, K., see Ong. N.P. 217, 223, 268 Maki, M., see Mozurkewich,G. 218,223,227, 234, 268 Maki, M., see Reagor, D. 213, 268 Makijarvi, M., see Katila, T. 303, 336 Makijarvi, M., see Montonen, J. 303, 338 Maksimov, E.G., see Dolgov, O.V. 17-19.29, 42 Maksimov, E.G., see Mazin, 1.1. 20, 44 Maliepaard. M.C. 144, 157, 159, 160, 192 Malmivuo, J. 294, 337 Malmivuo, J.A.V., see Lekkala, J.O. 318,337 Mamiya, T. 158, 192 Mandl, F. 61, 113 Maniewski, R. 294, 325, 337 Maniewski, R., see Katila, T. 303, 336 Manning, J.S., see Lefkowitz, I. 8, 44 Mapoles, E.R., see Paik, H.J. 329, 338

351

Maradudin, A.A. 178, 192 Maradudin, A.A., see Shen, T.J. 179, 193 Marcus, J., see Beauchene, P. 219, 266 Marcus, J., see Segransan, P. 221, 268 Marcus, P.M., see Irnry, Y. 281, 336 Marek,D. 187, 192 Mans, H.J. 117, 179-182, 192 Maris, H.J.. see Graf, M.J. 117, 191 Maris, H.J., see Guo, C.J. 168, 191 Mans, H.J., see Huber, T.E. 117, 191 Marsden, J.R., see Williamson. S.J. 311, 318, 339 Marshall, R., see Betts, D.S. 99, 103, 111 Marx, D. 169, 175, 192 Mason, E.A., see Monchick, L. 61, 113 Masselli, M., see Fenici, R.R. 303, 335 Massey, H.S.W. 61, 113 Massey, H.S.W., see Buckingharn. R.A. 61, 111

Masuda, Y., see Mamiya, T. 158, 192 Masuhara, N. 71, 113 Matacotta, F.C. 75, 77, 113 Mathews, R.H., see Paik, H.J. 287, 338 Matkowsky, B.J., see Ben-Jacob, E. 281, 333 Matsukawa, H. 234, 236, 252, 253, 260, 267 Mattheiss. L.F. 20, 44 Matthias, B.T. 11, 38, 44 Maurer, J. 106, 113 Maurer, J., see Libchaber, A. 106, 113 May, A.D., see Timsit, R.S. 89, 91.93, 114 Mazin, 1.1. 20, 44 McAdams, H.H. 86, 113 McAshan, M.S. 329, 337 McColl, D.B., see Harrison, J.P. 142, 191 McConville, G.T., see Aziz, R.A. 51, 52, 69, 111

McConville, G.T., see Matacotta, F.C. 75,77, I13 McCourt, F.R.W., see Aziz, R.A. 49, 51, 52, 75, 111 McCullen, T., see Hood, K. 152, 191 McCulloch, R.D., see Page, J.H. 144, 157, 159, 160, 193 McCullough, J.R., see Sinton, C.M. 302, 338 McTaggart, J.H. 101, 113 Meerschaut, A., see Monceau, P. 204, 268 Mendoza, E., see London, H. 140, 192 Meng, R.L., see Huang, C.Y. 11, 43 Meng, R.L., see Wu, M.K. 10, 44 Mercereau, J.E. 276, 337

352

AUTHOR I N D E X

Mercereau. J.E., see Jaklevic, R.C. 274, 336 Metropolis, N., see Kilpatrick. J.E. 61, 74, I12 Meyerovich. A.E. 47, 53.60, 71, 113 Meyerovich, A.E., see Bashkin, E.P. 71. 111 Michaels, J.N., see Faltens, T.A. 20, 42 Michaels, J.N., see Leary, K.J. 32, 44 Michels, A., see De Boer, J. 51, 61, 111 Michelson, P.F.,see McAshan, M.S. 329,337 Migulin, V.V., see Kuzrnin, L.S. 279, 337 Mihaly. G. 215, 267 Mihaly, L. 216, 267, 268 Mihaly, L., see Beyermann, W. 215, 266 Mihaly, L., see Brown, S.E. 216, 224. 239. 249, 256. 257, 266 Mihaly, L.. see Mihaly, G. 215, 267 Miller, J.E., see Guenault, A.M. 141, 191 Miller, J.M., see Thorne, R.E. 222, 226,238, 250, 251, 268 Miller, J.M., see Tucker. J.R. 235, 269 Miller 111, W.T. 302, 337 Mills, D.L. 152, 155, 192 Mills, D.L., see Maradudin, A.A. 178, 192 Mills, N.G. 169, 192 Mills, N.G., see Sherlock, R.A. 168, 169, 193 Minakova.V.E.,see Latyshev.Yu.1. 221,244, 26 7 Miura, Y., see Ishirnoto. H. 71. 98, 112, 158, 191 Mizutani. N. 187, 192 Modena, I. 301, 337 Modena, I., see Chapman, R.M. 301, 334 Moilanen, M., see Kalliomaki. K. 304, 336 Moisseyev, D.P.. see Gabovich, A.W. 8, 20, 42 Mok, E. 169, 192 Molchanov, V.N.,see Sirnonov, B.I. 32, 44 Mollenauer. L.F. 81, 113 Monceau, P. 198,204,215,220,221,227,234, 239, 240, 263, 268 Monceau, P., see Richard, J. 226, 264, 268 Monchick, L. 61, 113 Montonen, J. 303, 338 Montonen, J., see Katila, T. 303. 336 Mooij, J.E., see Bruines, J.J.P. 284, 333 Morii. Y. 158, 192 M o m s , D.E., see Bourne, L.C. 21, 42 M o m s , D.E., see Faltens, T.A. 20, 42 Morrison, H.F., see Wilt, M. 327, 339 Mota, A.C., see Jutzler, M. 187, 191

Mota, A.C., see Marek, D. 187, 192 Moze, 0.. see David, W.I.F. 32, 42 Mozurkewich, G. 218,222,223,227,234,260, 263, 264, 268 Mozurkewich, G., see Brown, S.E. 225, 239. 246, 250, 266 Mozurkewich, G., see Link, G.L. 225. 226, 26 7 Mueller, R.M., see Chocholacs, H. 158, 159. 162-164, 190 Mueller, R.M., see Owers-Bradley, J.R. 71, lJ3, 141, 193 Mueller, R.M., see Rogacki, K. 124,141-143, 193 Muhlfelder, B. 285, 286, 288. 338 Muirhead, C.M.. see Gough, C.E. 14, 43 Mujsce, A.M., see Batlogg, B. 21, 42 Miiller, K.A.. see Bednorz, J.G. 9, 42 Mullin, W.J., see Gully, W.J. 71, 98, 112 Munn, R.J., see Monchick, L. 61, 113 Murphy, D.W., see Batlogg, 8. 21, 42 Muta, T., see Enpuku, K. 287, 335 Mutikainen, R., see Knuutila, J. 283, 286, 288, 291, 320, 321, 337 Nacher, P.-J. 72, 84-88, 93-95, 97, 113 Nacher, P.-J., see Daniels, J.M. 78, 84, 85, 99, I 1 1 Nacher. P.-J., see Himbert, M. 85, 86. 112 Nacher, P.-J., see Laloe, F. 48, 83, 86, 112 Nacher, P.-J., see Leduc, M. 83. 85-87, 89, 98, 100, 104, 105, 107, 112 Nacher, P.-J., see Lefkvre-Seguin, V. 85, 89, 91, 93, 94, I12 Nacher, P.J., see Stringari, S. 110, I13 Nacher, P.-J., see Tastevin, G. 93,97-99, 109, 113 Nacher, P.-J., see T r h e c , G. 81, 82, 114 Nad', F.Ya., see Borodin, D.V. 211,215,263, 266 Nain, V.P.S., see Aziz, R.A. 51, 52, 69, 1 1 1 Naito, M., see Maeda, A. 226, 267 Naito. N., see Maeda, A. 226, 267 Nakagawa, H.,see Nakanishi, M. 320. 321, 338 Nakajirna, S. 152. 192 Nakanishi, M. 320, 321, 338 Nakayama, T. 118, 120, 121, 149, 151-156, 159, 161, 162, 164, 169, 173, 174, 178-181, 183, 184. 189, 192

AUTHOR INDEX Nakayama, T., see Nishiguchi, N. 129, 133, 134, 144, 147, 149, 152, 189, 193 Nakayama, T., see Saito, S. 149, 150, 152, 153, 193 Nakayama, T., see Yakubo, K. 159, 194 Narici, L., see Emt, S.N. 325, 335 Narici, L., see Romani, G.L. 297, 312, 338 Naurzakov, S.P., see Vvedensky, V.L. 294, 339 Neganov, B.S. 140, 193 Nemoto, I., see Brain, J.D. 304, 333 Nenonen, J., see Katila, T. 303, 336 Nenonen, J., see Montonen, J. 303, 338 Neuhaus, M., see Drung, D. 320, 321, 335 Nicolas, P. 294, 338 Nisenoff, M. 276, 338 Nishida, N., see Ishimoto, H. 71, 98, 112, 158, 191 Nishiguchi, N. 129, 133, 134, 144, 147, 149, 152, 189, 193 Nishiguchi, S., see Nakayama, T. 120, 121, 189, 192 Noakes, D.R., see Harshman, D.R. 34, 43 Norberg, R.E., see Lusznynski, K. 70, 71, I13 Northrop, G.A. 176, 193 Notarys, H.A., see Tesche, C.D. 285, 287, 3 15, 339 Novikov, L.N., see Laloc, F. 48, 83, 86, 112 Nozibres, P., see Castaing, B. 109, 1 1 1 Numata, T., see Ogushi, T. 11, 44 Obara, K., see Ogushi, T. 8, 44 Obennayer, D.E., see Lauter. H.J. 180, 192 Ogawa, S., see Ishimoto, H. 71.98, 112, 158, I91 Ogg, R.A. 13, 44 Ogushi, T. 8, 11, 44 Ohtsuka, T., see Mizutani, N. 187, 192 Okada. Y., see Williamson, S.J. 311,318,339 Ong, N.P. 217, 223, 224, 268 Ong, N.P., see Jing, T.W. 224, 267 Ong, N.P., see Monceau, P. 204, 268 Ong, N.P., see Venna, G. 224, 269 Ono, R.H., see Muhlfelder. B. 285, 286, 338 Opfer, J.E., see Luszczynski, K. 70, 71, 113 Opsal, J.L. 166, 193 Orbach, R. 144, 193 Orbach, R., see Alexander, S. 144, 159, 189 Orbach, R., see Demda, B. 159, 190

353

O’Reilly, A.H., see Kamaris, K. 32-34, 43 Osheroff, D.D. 125, 126, 128, 152-154, 157, 158, 162-164, 193 Osheroff, D.D., see Corruccini, L.R. 71, 111 Osiecki, J.H., see Gamble, F.R. 30, 36, 42 Ostman, P., see Ahola, H. 290, 333 Ott. H.R. 26, 44 Owers-Bradley, J., see Chocholacs, H. 158, 159, 162-164, 190

Owers-Bradley, J.R. 71, 113, 141, 193 Ozhogin, V.I., see Vvedensky, V.L. 294, 339 Ozono, Y.,see Ogushi, T. 11, 44’ Paalanen, M., see Ahonen, A.I. 124, 189 Page, G.J., see Wyatt, A.F.G. 169, 170, 194 Page, J.H. 144, 157, 159, 160, 193 Page, J.H., see Frisken, B. 142, 144, 190 Page, J.H., see Maliepaard, M.C. 144, 157, 159, 160, 192 Paik, H.J. 287, 329, 338 Papoular, M., see Castaing, B. 117. 190 Papoulas, H., see Richard, J. 226, 264, 268 Panlla, P. 219, 268 Parker, E.N., see Turner, M.S. 332, 339 Parker, W.H., see Simmonds, M.B. 279, 338 Paterno, G., see Barone. A. 274, 333 Paterno, G., see Kinder, H. 183, 191 Paulson, D., see Buchanan, D.S. 312, 334 PavloviE, M. 83, 104, 113 Peach, L. 117, 193 Peierls, R.E. 197, 201, 268 Pelizzone, M. 302, 311. 316, 338 Pelizzone, M., see Hari, R. 311, 335 Pelizzone, M., see Williamson, S.J. 311, 318, 339 Peltonen, R.S., see Kelha. V.O. 293, 315,336 Penttinen.A.A., see Kelha,V.O. 293,315,336 Pergrum. C.M., see Donaldson, G.B. 320,334 Perry, T. 124, 151-153, 193 Peshkov, V.P. 149, 193 Peterson, R.E. 167, 179, 193 Pethick, C.J. 26, 44 Phillips, N.E., see Avenel, 0. 124, 125, 127, 186, 190 Phillips, W.A. 181, 193 Pickett, G.R., see Guenault, A.M. 141, 191 Pietronero, L. 234, 236, 268 Pinard, M. 60, 83, 113 Pines, D., see Pethick. C.J. 26, 44 Pisharody, R., see Gamble, F.R. 30, 36, 42

3 54

AUTHOR INDEX

Pizzella, V., see Erne, S.N. 325, 335 Plaskett. T.S., see Collins, R.T. 34, 42 Pobell. F., see Chocholacs, H. 158, 159, 162164, 190 Pobell, F.. see Owers-Bradley, J.R. 71, 113, 141. 193 Pobell. F., see Rogacki, K. 124. 141-143, 193 Podney, W.N., see Czipott, P.V. 328, 334 Pohl, R.O., .see Klitsner, T. 169, 183. 191 Pohl, R.O., see Zeller, R.C. 103, 114 Pollack, G.L. 127, 193 Pollack, G.L., see Opsal, J.L. 166, 193 Polls, A., ree Stringan, S. 110, 113 Polunin, E.A., see Kuzmin, L.S. 279, 337 Pomeranchuk, f.Ya., see Achiezer. A.I. 26,41 Porter, C.D., see Kamaris. K. 32-34, 43 Portis, A.M., see Monceau. P. 204, 268 Potter, W.H. 149, 193 Poulder, D., see Renne, M.J. 248, 268 Poutanen, T., see Varpula, T. 294, 339 Prance, R.J., see Long, A. 279, 337 Proto. G., see Tesche, C.D. 285,287.3 15.339 Pukki, J.M., see Kelha, V.O. 293, 315. 336 Quader 40. 44 Quinnell, E.H., see Clement, J.R. 101, 1 1 1 Radebaugh, R. 124. 141, 162, 193 Rae, A.I.M., see Gough. C.E. 14. 43 Raider. S.I., see Foglietti, V. 292. 335 Raivio, M., see Katila, 7. 303, 3-16 Raksheev, S.N., see Mazin, 1.1. 20, 44 Ramadan, B., see Kos, J.F. 329. 337 Ranninger, J., see Alexandrov, A.S. 23. 41 Rantala, B.. see Ahola, H. 290, 233 Rapp, R.E.. see Wheatley, J.C. 141, 194 Rasmussen, F.B., see Halperin, W.P. 54, 112 Rawling, K.C. 166. 193 Reagor, D. 212, 213. -768 Reagor, D., see Sridhar, S. 212,213, 235, 268 Reid, W.H., .see Drazin, P.G. 106, I 1 1 Reinikainen, K., .see Ilmoniemi, R.J. 296, 305, 306. 310, 318, 3.76 Reinikainen, K., see Sams, M. 299, 300, 338 Renard, M.. see Monceau, P. 198, 215, 220, 221, 227. 239, 240, 263, 268 Renard. M., see Richard, J. 226, 264. 268 Renne. M.J. 248, 268 Reynolds, C.L. 126, 127. 167, 193 Ribary, U. 302. 338

Ricci, G.B., see Chapman, R.M. 301, 334 Ricci, G.B., see Modena. 1. 301, 337 Rice, M.J. 179, 19.1 Rice, M.J., see Leggett. A.J. 71, 112 Rice, M.J., see Toombs, G.A. 128. 135, 138, 140, 148, 194 Rice, T.M., see Lee, P.A. 205, 206, 223, 232, 253, 267 Richard, J. 226, 264, 268 Richard, J., see Monceau, P. 198, 215, 220, 221, 221, 239, 240, 263, 268 Richards, M.G., see Long, A. 279, 337 Richardson, R.C. 140. 141, 193 Richardson, R.C., see Corruccini, L.R. 71, 111

Richardson, R.C., see Halperin, W.P. 54, 112 Richardson, R.C., see Osheroff, D.D. 125, 126, 128, 152-154. 157, 158, 193 Rietman, E.A., see Batlogg, B. 20, 21, 42 Rietman, E.A., see Cava, R.J. 10, 19, 42 Risken, H., see Breymayer, H.-J. 232, 266 Riski, K., see Ikonen, E. 329, 336 Ritala, R.K., see Ketoja, J.A. 280, 281, 336 Ritchie, D.A. 132, 158, 159, 162-165, 193 Robaszkiewicz, S., see Alexandrov, A.S. 23, 41 Robbins, M. 233, 235, 268 Robbins, M.O., see Bhattacharya. S. 226,264, 266 Roberts, T.R. 75, 113 Robertson, R.J. 160, 161, 193 Rogacki, K. 124, 141-143. 193 Roinel, Y., see Jacquinot, J.F. 187, 191 Romani. G.L. 297, 299, 305, 307. 308, 312, 318. 324. 338 Romani, G.L., see Chapman, R.M. 299. 301, 334 Romani, G.L., see E d , S.N. 320, 322, 325, 335 Romani, G.L.. see Modena, I . 301, 337 Rouxel, J . , see Monceau. P. 204, 268 Rupp, L.W., see Batlogg, B. 20, 21, 42 Rusakov, A.P., see Brandt, N.B. 8, 42 Rusakov, A.P., see Chu, C.W. 8, 42 Rusby, R.L., see Durieux, M. 101, I l l Rusinov, A.I.. see Kopaev, Yu.V. 22. 43 Rutherford, A.R. 147, 148, 159, 193 Ruvalds, J., see Tua, P.F. 234, 252, 268 Ryhanen, T. 273, 276, 281, 284, 338 Ryhanen, T., see Ketoja, J.A. 281, 327

AUTHOR INDEX Ryhanen, T., see Maniewski, R. 294,325,337 Ryhanen, T., see Seppa, H. 286,288,296,338 Saam, W.F., see Edwards, D.O. 85, 212 Sabetta, F., see Fenici, R.R. 303, 335 Sabisky, E.S. 169, 193 Saint-Lager, M.C., see Monceau, P. 215,221, 263, 268 Saito, S. 149, 150, 152, 153, 187, 293 Sakai, H., see Hayashi, M. 142, 144, 191 Salamon, see Quader 40, 44 Salmi, J., see Knuutila, J. 283, 286, 288, 291, 320, 321, 337 Salomaa, M.M., see Kajanto, M.J. 232, 267 Saltzman, B. 106, 113 Salustri, C., see Romani. G.L. 312, 318, 338 Salva, H., see Monceau, P. 221, 268 Sams, M. 299, 300, 338 Sandmann, W., see Weber, J. 168-170, 194 Sandstrom, R.L., see Collins, R.T. 34, 42 Sandstrom, R.L., see Dinger, T.R. 17,32, 42 Sandstrom, R.L., see Foglietti, V. 292, 335 Santikaya, Ya.S., see Latyshev, Yu.1. 221,267 Sarvas, J. 297, 308, 325, 338 Sarvas, J., see Hamalainen, M.S. 297, 335 Sarvas, J., see Hari, R. 307, 335 Sarvas, J., see Kaukoranta, E. 308, 336 Sasaki, W . , see Kobayashi, S. 149, 192 Sauls, J.A., see Perry, T. 124, 151-153, 193 Saunders, J., see Ritchie, D.A. 132, 158, 159, 162-165, 193 Saundry, P.D., see Ytterboe, S.N. 187, 194 Savitskaya, Ya.S., see Borodin, D.V. 21 1,215, 263, 266 Savrasov, S.Yu., see Mazin, 1.1. 20, 44 Sawada, Y., see Mamiya, T. 158, 192 Sawatzki, G., see Ludwig, W. 319, 337 Scaramuzzi, F., see Kinder, H. 183, 191 Schafroth, M.R. 14, 44 Schearer, L.D. 78, 90, 91, 98, 113 Schearer, L.D., see Colegrove, F.D. 78, 2 2 1 Scheer, H., see Erne, S.N. 294, 335 Schiff, L.I. 61, 223 Schlenker, C., see Beauchene, P. 219, 266 Schlesinger, Z., see Collins, R.T. 34, 42 Schneemeyer, L.F., see Fleming, R.M. 219, 240, 246, 257, 258, 267 Scholz, H.N., see Masuhara, N. 71, 113 Schomer, D.L., see Kennedy, J.G. 310, 336 Schrieffer, J.R. 25, 26, 44

355

Schrieffer, J.R., see Bardeen, J. 12, 15, 42 Schrieffer. J.R., see Klemm, R.A. 233, 267 Schrijner, P., see de Waal, V.J. 284, 334 Schubert, H. 183, 193 Schuller, I.K., see David, W.I.F. 32, 42 Schulz, H.J., see Jerome, D. 36, 43 Schumacher, G., see Frossati, G. 124, 141, 190 Schuss, 2.. see Ben-Jacob, E. 281, 333 Schweizer, R.J., see Heim, U. 169, 191 Scott, M.J., see Rutherford, A.R. 147, 148, 159, 193 Segransan, P. 221, 268 Segre, C.U., see David, W.I.F. 32, 42 Seidel, G.M., see Cameron, J.A. 75, 1 1 1 Semenov, A.D., see Gershenson, E.M. 34.42 Semenov, V.M., see Brandt, N.B. 8, 42 Senba, M., see Harshman, D.R. 34, 43 Seppa, H. 279, 286, 288, 296, 338 Seppa, H., see Ketoja, J.A. 281, 337 Seppa, H., see Knuutila, J. 283,286,288,291, 321, 337 Seppa, H., see Ryhanen, T. 273, 276, 281, 284, 338 Seppanen, M. 318, 338 Seppanen, M., see Maniewski, R. 294, 325, 33 7 Sera, M. 40, 44 Shabanov, S.Yu., see Vvedensky, V.L. 294, 339 Sham, L.J. 29, 37, 44 Shapiro, S. 239, 250, 268 Shapligin, I.S. 10, 44 Sheard, F.W. 179, 193 Sheard, F.W., see Toombs, G.A. 128, 135, 138, 140, 148, 194 Shearer, L.D., see Daniels, J.M. 78, 84, 85, 99, 1 1 1 Shen, T.J. 179, 193 Sherlock, R.A. 168, 169, 193 Sherlock, R.A., see Mills, N.G. 169, 192 Sherlock, R.A., see Wyatt, A.F.G. 169, 194 Sherman, R.H., see Roberts, T.R. 75, 113 Sherrill, D.S., see Masuhara, N. 71, 113 Sherwin, M. 239, 260, 268 Sherwin, M.S., see Bourne, L.C. 265, 266 Shigi, T., see Fujii, Y. 187, 191 Shinoki, F., see Nakanishi, M. 320, 321, 338 Shirae, K. 316, 318, 338 Shirae, K . , see Furukawa, H. 316, 335

3 56

AUTHOR INDEX

Shiren, N.S. 179, 193 Shlyapnikov, G.V., see Kagan, Yu. 117, 191 Shoji, A., see Nakanishi, M. 320, 321, 338 Short, K.T., see Batlogg. B. 20, 21, 42 Siegwanh, J.D., see Radebaugh, R. 124, 141, 162. 193 Sigmund, E.. see Haug, R. 180, 191 Silin, V.P. 60. 71, 113 Siltanen, P., see Katila, T. 303, 336 Siltanen, P., see Montonen. J . 303, 338 Silver, A.H. 274, 338 Silver, A.H., see Jaklevic, R.C. 274, 336 Silvera, I.F. 47, 113 Simmonds, M.B. 279, 338 Simmonds. M.B., see Kamper, R.A. 279, 336 Simonov. B.I. 32, 44 Simonov, N.A., see Kuzmin, L.S. 279, 337 Sinton, C.M. 302, 338 Sitnikova, V.I., see Zinov'eva, K.N. 167, 194 Slack, G.A., see McTaggan, J.H. 101, 113 Sleight, A.W. 8. 44 Slichter, C.P. 83. 94, 113 Smith, E N . , see Richardson, R.C. 140, 193 Smith, F.J., see Monchick, L. 61, 113 Sneddon, L. 233, 236, 253, 264, 265, 268 Sniguiriev, D.V.. see Danilov, V.V. 291, 334 Sobyanin, A.A., see Bulaevskii, L.M. 40, 42 Sobyanin, A.A., see Bulaevskii, L.N. 24, 25, 35, 42 Sobyanin, A.A., see Ginzburg, V.L. 35, 43 Soderholm, L., see David, W.I.F. 32, 42 Sokoloff, J.B. 234, 268 Soldatov, E.S., see Danilov, V.V. 291, 334 Somalwar, S., see Incandela, J. 331, 336 Soper, A.K., see David, W.I.F. 32, 42 Sprenger, W.O., see Andres, K. 124, 134,147. 190

Sridhar. S. 212. 213, 235, 268 Sridhar, S., see Reagor, D. 212, 213, 268 Stacy. A.. see Bourne, L.C. 21. 42 Stacy, A.M., see Faltens, T.A. 20, 42 Stacy. A.M., see Leary, K.J. 32, 44 Stager, C.V., see Kamaras, K. 32-34, 43 Stark, M., see Wilt, M. 327, 339 Stein, D.L., see Perry, T. 124. 151-153, 193 Stepanova, M.G.. see Vendik, O.G. 279, 339 Steur, P.P.M., see Matacotta, F.C. 75.77, I13 Stokes, J.P., see Bhattacharya, S. 225, 226, 239, 260. 261, 263. 264, 266

Slrachen, C., see Lennard-Jones, T.E. 180, I 92 Strassler, S., see Pietronero, L. 234, 236, 268 Stringari, S. 110, I13 Stroink, G . 294, 339 Stroink, G., see MacAulay, C.E. 303, 337 Stroink, G., see Weinberg, H . 299, 339 Stubbs, R.J., see Maliepaard, M.C. 144, 157, 159, 160, 192 Sueoka, K., see Enpuku, K. 285-287, 335 Sunshine, S., see Batlogg, B. 21, 42 Suresha, G.N., see Ogushi, T. 1 1 , 44 Sutherling. W., see Barth, D.S. 301, 333 Sutherling, W.W. 313, 339 Sutton, S., see Cough, C.E. 14, 43 Suzuki, H., see Mizutani, N. 187, 192 Swanenburg, T.J.B. 168, 193 Swithenby. S., see Grimes, D. 304, 33.5 Sydoriak, S.G., see Roberts. T.R. 75, 113 Symonds, A.J., see Brewer, D.F. 180, 190 Synder, N.S. 166, 193 Syskais, E.G., see Rogacki, K. 124, 141-143, I 93 Taber, M., see Cabrera, B. 299, 330, 334 Taber, M.A., see Huber, M.E. 331, 336 Taber, R.C.. see McAshan, M.S. 329, 337 Taborek, P. 168, 169, 175, 176, 193, 194 Taconis. K.W., see Beenakker, J.J.M. 165, I90 Taconis, K.W.. see Gorter, C.T. 117. 191 Takada, S.. see Nakanishi, M. 320, 321, 338 Takahashi, T. 40, 44 Takahashi, T., see Kobayashi, S. 149, 192 Takano, Y., see Ishimoto, H. 71,98, 112, 158, I91

Takayama, H., see Matsukawa, H. 234, 236, 26 7 Tamura, I., see Hayashi, M. 142, 144, 191 Tanaka, S.. see Maeda, A. 216, 226, 267 Tanner, D.B.. see Kamaras, K. 32-34, 43 Tastevin, G. 93, 97-99, 109, 113 Tastevin, G., see Laloe, F. 48, 83, 86, I12 Tastevin, G., see Leduc, M. 98, 100, 104, 105, 107, 112 Tastevin, G., see Nacher, P.-J. 72, 86, 93-95, 97, 113 Taylor, W.L., see Aziz, R.A. 51, 52, 69, 111 Tazaki, T., see Ishimoto, H. 71, 98, 112, 158, 191

AUTHOR INDEX ter Brake, H.J.M. 324, 339 Teranishi, N. 234, 268 Tesche, C., see Knuutila, J. 285, 286, 308, 313, 314, 318, 337 Tesche,C.D. 280,281,284,285,287,315,331, 339 Tesche, C.D., see Bermon, S. 331, 333 Tesche, C.D., see Chi, C.C. 331, 334 Tessema, G.X., see MihLly, L. 216, 268 Teszner, D., see Nicolas, P. 294, 338 Thkry, J., see Schearer, L.D. 78, 98, 113 Thiene, P., see Zimmerman, J.E. 276, 339 Thomas, N., see Cough, C.E. 14, 43 Thomasson, J.W. 96, 114 Thompson, K.,see Hall, H.E. 140, 191 Thomson, A.L., see Brewer, D.F. 180, 190 Thorne, R.E. 219,222,226,234,237,238,244, 248, 250-253, 268 Thorne, R.E., see Lyding, J.W. 224,225. 267 Thorne, R.E., see Tucker, J.R. 235, 269 Thoulouze, D., see Frossati, G. 124, 141, 190 Thoulouze, D., see Peach, L. 117, 193 Threlfall, D.C. 106, 107, 114 Tiky, C., see Lauter, H.J. 180, 192 Timsit, R.S. 81, 89, 91, 93. 114 Timsit, R.S., see Daniels, J.M. 81, 111 Timusk, T., see Kamaris, K. 32-34, 43 Ting, T.W., see Ong, N.P. 224, 268 Tinkham, M. 274, 281, 339 Toombs, G.A. 128, 135, 138, 140, 148, 179, 194 Toombs, G.A., see Rice, M.J. 179, 193 Toombs, G.A., see Sheard, F.W. 179, 193 Torng, C.J., see Wu, M.K. 10, 44 Tracht, A.E., see Farrell, D.E. 304, 319, 335 TrCnec, G. 81, 82, 114 TrCnec, G.. see Leduc, M. 81, 112 Trtnec, G., see Nacher, P.-J. 84, 85, 87, 88, 113

Tripp, J.H., see Farrell, D.E. 304, 319, 335 Trontelj, Z., see ErnC, S.N. 294, 335 Trumpp, H.J. 168, 194 Tsuei, C.C., see Bermon, S . 331, 333 Tsuei, C.C., see Chi, C.C. 331, 334 Tsuei, C.C., see Tesche, C.D. 331, 339 Tua, P.F. 234, 252, 268, 269 Tucker, J.R. 235, 238, 269 Tucker, J.R., see Thorne, R.E. 219, 222,226, 234, 237, 238, 244, 248, 250-253, 268 Tuomisto, T., see Nicolas, P. 294, 338

357

Tuomisto, T., see Seppanen, M. 318, 338 Tuomola, M., see Malmivuo, J. 294, 337 Turner, C.W., see van Duzer, T. 274, 339 Turner, M.S. 332, 339

Uchinokura, K., see Maeda, A. 216, 267 Ulfman, J.A., see ter Brake, H.J.M. 324, 339 Uspenskii, Yu.A., see Mazin, 1.1. 20, 44

Valberg, P.A., see Brain, J.D. 304, 333 van der Linde, J., see Pinard, M. 83, 113 van der Sluijs, J.C.A., see Rawling, K.C. 166, I93 van Dover, R.B., see Batlogg, B. 21, 42 van Dover, R.B., see Cava, R.J. 10, 19, 42 van Duzer, T. 274, 339 van Kranendonk, J., see De Boer, J. 61, 111 van Nieuwenhuyzen, G.J. 320. 339 van Soest, G., see Beenakker, J.J.M. 165, 190 Varma, C.M., see Anderson, P.W. 181, 189 Varpula, T. 294. 339 Varpula, T., see Seppanen, M. 318, 338 Veinstein, B.K., see Simonov, B.I. 32, 44 Vendik, O.G. 279,339 Veno, S., see Atsumi, K. 299, 333 Verma, G. 224, 269 Verma, G., see Ong, N.P. 217, 223, 268 Vernon Jr, F.L., see Kanter, H. 279, 336 Vetsleseter, A., see Avenel, 0.124, 125, 127, 186, 190 Veuro, M.C., see Ahonen, A.I. 124, 125, 189 Vezzoli, G.C. 8, 44 Vilches, A.E.. see Anderson, A.C. 126, 167, I89 Vilches, O.E., see Wheatley, J.C. 124, 194 Vilkman, V., see Knuutila, J. 308, 313, 314, 318, 337 Villain, J. 156, 194 Vivien, D., see Schearer, L.D. 78, 98, 113 Volkov, B.A., see Ginzburg, V.L. 9,43 Vollmer, H.D., see Breymayer, H.-J. 232,266 Vrba, J. 305, 307, 339 VujiwciC, G.M. 37, 44 Vuono, M. 185, 194 Vuono, M., see Avenel, 0. 124, 125,127,186, I90 Vuorio, M., see Leggett, A.J. 151, 152, 192 Vvedensky, V.L. 294, 299, 31 1, 339

358

AUTHOR INDEX

Waldmann, L. 60. 114 Waldram, J.R. 248, 269 Walraven, J.T.M., see Silvera, I.F. 47, 113 Walters, G.K., see Colegrove, F.D. 78, 1 1 1 Walters, G.K.. see McAdams, H.H. 86, 113 Walters, G.K.. see Schearer. L.D. 90.91. 113 Wang, K.Y., see Paik, H.J. 329, 338 Wang, R.H.. see Tesche, C.D. 285, 287. 315, 339 Wang, Y.Q., see Wu, M.K. 10, 44 Wang, Z.Z., see Monceau, P. 221. 268 Warnick, A., see Forgacs, R.L. 289, 335 Webb, R.A.. see Giffard, R.P. 276, 277, 288, 335 Webb, W.W., see Kurkijirvi, J. 277, 337 Weber, J. 168-170, 194 Weber, J.C., see Marek, D. 187, 192 Weber, W. 19, 44 Weber. W., see Batlogg, B. 20, 42 Webman, I . , see Grest, G.S. 159, 191 Weger. M. 217, 222. 230, 231, 269 Weinberg, H. 299, '339 Weinberg, H.. see Ribary, U. 302, 338 Weinberg, H., see Vrba, J. 305, 307, 339 Weiss. K., see Haug, H. 167, 179, 191 Weiss, K., see Haug, R. 180, 191 Weiss, K., see Kinder, H. 179, 191 Wellstood, F. 291, 327, 339 Welte, M., see Heim, U. 169, 1Y1 Welte, M., see Koblinger. 0. 169, 171, 192 Werntz Jr. J.H., see Grimsrud. D.T. 75, / I ? Westfall, R. 5 , 44 Wheatley. J.C. 124, 133, 141, 149, 194 Wheatley, J.C., see Abel, W.R. 152, 189 Wheatley. J.C., see Anderson, A.C. 126, 127, 167, 189 Wheatley, J.C., see Giffard, R.P. 276, 277, 288, 335 White, A,, see Batlogg, B. 21, 42 White, A.E.. see Batlogg, B. 20, 42 White, D. 165, 194 White, R.M. 16. 35. 44 Wiechert, H., see Lauter, H.J. 180, 192 Wiesenfeld, L., see Tastevin, G. 109, 113 Wilks, J., see Challis, L.J. 165, 166, 190 Wilks, J., see Dransfeld, K. 165, 190 Wilks. J., see Fairbank, H.A. 165, 190 Williamson, S.J. 297. 311, 318, 339 Williamson, S.J., see Atsumi, K. 299, 333 Williamson, S.J., see Buchanan, D.S. 312,334

Williamson, S.J., see Curtis, S. 299, 334 Williamson, S.J., see Ilmoniemi, R.J. 299, 312, 336 Williamson, S.J., see Kaufman, L. 297, 311, 336 Williamson, S.J., see Pelizzone, M. 311, 338 Williamson, S.J., see Romani, G.L. 299, 305, 307, 324. 338 Wilt, M. 327, 339 Wolfe, J.P., see Nonhrop, G.A. 176, 193 Wolter, J., see Horstman, R.E. 168, 191 Wolter. J.. see Swanenburg, T.J.B. 168, 193 Wong. C.C.K.. see Aziz, R.A. 49, 51, 52, 75, 111

Wong, Zheng-yu 17, 18, 44 Wonneberger. W. 232, 269 Wonneberger, W., see Breymayer, H.-J. 232, 266 Worthington, T.K. 32, 44 Wonhington, T.K., see Dinger, T.R. 17, 32, 42 Wu, Hang-sheng, see Wong, Zheng-yu 17, 18, 44 Wu, M.K. 10, 44 Wu, R.H., see Waldram, J.R. 248, 269 Wu, Wei-Yu 215, 238, 269 Wyatt. A.F.G. 127, 169, 170, 194 Wyatt, A.F.G., see Mills, N.G. 169, 192 Wyatt, A.F.G., see Sherlock, R.A. 168, 169, 1 93 Yakubo, K. 159, 194 Yakubo, K., see Nakayama, T. 161, 192 Yeh, W.J. 249, 250, 269 Yoffe, A.D., see Friend, R.H. 30, 42 Yogi, T., see Tesche, C.D. 285. 287, 315, 3.39 Yoshida, K.,see Enpuku, K. 285-287, 335 Ytterboe, S.N. 187, 194 Yu, Kin-Wah, see D e m d a , B. 159, 190 Zaitsev-Zotov, S.V., see Borodin, D.V. 21 1, 215, 263, 266 Zanzucchi, P., see Farrell, D.E. 305, 335 Zaremba, E., see Hood, K. 152, 191 Zawadowski, A., see Griiner, G . 198,227,267 Zawadowski, Z., see Tua, P.F. 234, 252, 269 Zawadzki, P., see Deptuck, D. 144, 157, 159, I90 Zeller, R.C. 103, 114

AUTHOR INDEX Zettl, A. 215,216,224,226,239,240,243-246, 260, 263-265, 269 Zettl, A., see Bardeen, J. 219-221. 266 Zettl, A., see Bourne, L.C. 21, 42, 265, 266 Zettl, A., see Faltens, T.A. 20, 42 Zettl, A., see Hall, R.P. 247, 264, 267 Zettl, A,, see Parilla, P. 219, 268 Zettl, A,, see Sherwin, M . 239, 260, 268 Zhang, K., see David, W.I.F. 32, 42

359

Zhang, K., see Kwok, W.K. 19, 43 Zhanov, Ya.A., see Latyshev, Yu.1. 244, 267 Zimmerman, J.E. 276, 288, 294, 339 Zimmerman, J.E., see Silver, A.H. 274, 338 Zinov’eva, K.N. 127, 167, 194 Zubarev, D.N. 136, 194 zur Loye, H.C., see Leary, K.J. 32, 44 Zvezdin, A.K. 24, 25, 44

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absorption processes 91-93 ac Josephson effect 239 accelerometers 329 acoustic mismatch theory 117, 120, 167 acoustic mode 205 active shielding 293 adsorption processes 91-93 alpha rhythm 298, 310, 311, 312, 317 amplifier noise 289 arrhythmias 303 asymmetric gradiometer 295 auditory evoked fields 299 auditory evoked response 316 auditory stimulus 299 autonomous dc SQUID 280, 286 Aziz 69, 75 Aziz interaction 51 Aziz potential 105

charge density wave (CDW) 197 charged Bose gas 14 circle map 248, 249 circular polarkation 83, 84 classical particle model 228 classical transport theory 46 clinical applications of MEG 300 coherence 200 coherence enhancement 200, 259 collision cross section 49, 55, 64 collision integrals 64,68, 73 colour-centre laser 82, 83, 85, 95 compensation coil 295, 306 computer simulations of dc SQUlDs 284 conductivity 70 confidence regions 307 convection 46, 106, 107 cooled preamplifier 280, 290 cordial resistance R 226 correlated noise 325 coupling 285, 288, 306 critical current 281 critical temperature in the BCS theory I5 cross section 72. 81 cryocooler 312 Cryogenic Consultants, Ltd 328 cryoSQUlDs 312 CTF 328 current carrying state 200 current-current correlations 200 current dipole 297-300, 307 current oscillations 198, 200, 281, 287 current-voltage characteristics 277,278,281 282 currents of neurons 297

balance 296, 306 balancing 296, 312 baseline 295, 305 beating resonance 287 P c 286 P L 286 P L 275 bias currents 280 binary collisions 54, 55, 63, 72, 81, 90 biomagnetic technologies 31 1 biomagnetism 292 bipolaron model of superconductivity 23 blue bronze b.,MoO, 204 Boltzmann equation 54, 55, 59. 60 broad band noise 200 BTi 328 bulk relaxation 85

damping 280, 288 damping constant 199 damping resistance 287 dc currents 304 dc SQUID 280, 291, 309

calibration 322 capacitive shunting 287 characteristic frequency 279, 283 characteristic lengths 46 361

362 de Broglie 52 de Rroglie wavelength 50 definition 48 demodulator circuit 291 density of states 144, 148 dewar noise 315 diazepam 302 dielectric 215 differential conductivity d l / d V 231 diffraction effect 53 diffuse signal 168, 176, 179 dilution refrigerator 124. 140 dipole 297, 299 dipole-dipole coupling 54 displacement sensors 329 Dornier Systems GmbH 319 double-D gradiometer 295. 310 double-loop dc SQUID 287 dye laser 81-83 dynamic impedance 290 dynamic resistance 283 dynamics of the CDW 205 eddy-current shield 294 EEG 296. 297, 299, 301 effective mass 199 electroencephalography (EEG) 296 electronic balancing 296, 3 1 I , 3 18 electronic circuit analogs 230 electronic polarisation 78 energy resolution 283-285. 287, 288, 306 epilepsy 301 equation of state 46 equilibrium polarisation 53 equivalent flux noise 277 evoked fields 299 excess noise 274, 279. 281, 284 exciton mechanism 21 exciton mechanism or HTS 26 external magnetic disturbances 292 l/f noise 289, 291, 292 Fabry-Perot etalons 82 Faraday cell 82 faVOUrdbk pressures 88 feedback loop 289 Fermi liquid theory 118, 128 ferromagnetic shield 293 field noise 306 first-order gradiometer 295, 310. 313

SUBJECT INDEX fluctuation suppression 262 flux feedback 324

flux-locked-loop 283 flux modulation 290. 292 flux-modulation 289 flux noise 274, 284 flux quantum 274 flux-to-voltage conversion efficiency 279,280 flux transformer 284, 306 flux transition 276 flux-voltage characteristics 277, 281 focal epilepsy 301 forward problem 298 fractional-turn SQUID 288 fracton 144, 148, 159 frequency dependent conductivity 198 gas of 'He 63 Gaussian noise 225 Gemini 312 geomagnetism 326 Gifford-McMahon cooler 312 gradiometer 295. 305. 306 gradiometer balance 296 gravitational wave antennae 330 grid spacing 308

harmonic mixing 199 harmonic mode locking 239 'He gas 59, 89.90, 94, 99, 100. 108, 109 'He-4He mixture 158. 162 hemochromatosis 304 high-frequency phonon 127, 168, 186 high-frequency rf SQUID 279 high-T, materials 288 high-temperature superconductivity (HTS)3 homogeneous head approximation 297 hydrogen isotopes 47 hyperfine structure 79-81, 84 hysteresis 215 hysteretic rf SQUlDs 280 hysteretic SQUID 275 IBM SQUlDs 315 identical spin rotation 72 identical spin rotation effect 63. 64, 65, 71 "identical spin rotation" effects 62 impedance matching 291 impurity pinning 21 1 incommensurate 198

SUBJECT INDEX inductance matching 306 inductive shunting 287 inductive SQUID 275 input current noise 291 integrated gradiometer 320 integrated magnetometer 321, 327 integrated sensors 319 integrity of sensory pathways 302 interaction potential 46, 49, 51, 56, 75 interatomic potential 52 interference experiments 200 interference “features” 247 interference filter 83 interhemispheric correlation 312 intermediate coupling transformer 288 intermediate transformer 322 intrinsic energy sensitivity 277 inverse problem 298 iron in the liver 304 isofield contour map 298 Josephson junction 232. 274 Joule-Thomson refrigerator 312 Kapitza conductance 118, 133, 137, 146 Kapitza resistance - above 1K 126, 165 -between ’He gas and a solid 137, 162 -between ‘ H e 4 H e mixture and sintered powder 158, 162, 164 -between liquid ’He and Ag particles 134, 152 - between sintered powder and liquid ’He 140 - magnetic field dependence 125, 153 kinetic equation 128 Knight shift 150, 152 krypton ion laser 81-83

A12 resonance 286 A/2 transmission line 286 Landau parameter 123, 129, 133, 139 late fields 303 lattice distortion 201 layered compounds 28 layered materials 39 Lazarus theorem 102 LC-resonances 287 Lennard-Jones interaction 51, 66 Lennard-Jones potential 105

363

linear chain compounds 197 liquid-gas equilibrium 46, 109 LNA laser 78, 98 local pairs 23 locating accuracy 322 lock-in detection 318 lock-in electronics 289 low-frequency noise 289 Lyot filter 82 magnetic coupling 149, 152, 155, 163 magnetic dipole coupling 90 magnetic disturbances 293 magnetic field isocontour maps 300 magnetic flux quantum 274 magnetic monopoles 330 magnetic shielding factor 293 magnetic susceptibility 327 magnetically shielded room 293, 310, 314 magnetocardiography 302 magnetoencephalography (MEG) 296 magnetometer 295 magnetopneumography 304 magnetotelluric studies 327 magnetotellurics 327 matching-transformer 288 Maxwell-Boltzmann distribution 55 MCG 302,309 MEG 299, 301, 313 metal-oxide ceramics 8 metastable atoms 78, 80 micropotentials 303 microwave SQUID 279 mode locking 200, 221,253 modulation depth 283 modulation frequency 289, 291, 292 monopole detectors 329 Mossbauer spectroscopy 329 multi-compartment model 302 multi-loop dc SQUIDS 288 multichannel neuromagnetometers 304 multiple solutions 281 multiplexing 318 mutual inductances 323 narrow band noise 217 Navier-Stokes equations 55. 56 NbSe, 204 negative differential resistances 215 “neon” superconductors 19

364

SUBJECT INDEX

neuromagnetism 296 NMR 78, 83, 95-98, 100, 103-105 noise temperature 278. 283, 292 nonlinear 198 nonlinear feedback circuits 232 “nonreproducible” superconductivity-type anomalies 9 normal liquid ’He 123, 132 normalized inductance 275 normalized loop inductance 280 nuclear magnetic moment of ‘He 54 nuclear polarisation 71,78,81,83,84,87,96, 98 Nusselt number 106, 107 off-diagonal gradiorneter 295, 309. 321 optical mode 205 optical polarisation 78, 87 optical polarisation of ’He nuclei 46 optical pumping 78, 79, 87, 88, 94, 98 optimization of magnetometers 305 onho-helium 79, 80 para-helium 78, 79 parasitic capacitance 280, 285, 287, 288, 321 parasitic resonances 288 Pauli principle 48, 50, 54, 69 Peierls transition temperature 197 peleomagnetism 327 percolation 160 phase-phase correlation length 21 I, 223 phase-sensitive detector 290 phase shift 49, 61, 74 phase-shift 69 phasons 206 phonon mechanism 34 PI controller 289, 290, 292 pickup coil 295, 306 pinning 215 pinning frequency 198 pinning potential 21 1 planar gradiometer 295 planar gradiometer arrays 320 planar gradiometers 331 polarisation 48 polarisation methods 46 position indication 322 preamplifier 278 preamplifier noise 278, 279, 283 primary current 297

probe-positioning indicator 312, 325 product k2QT 279 PSD 290 pulse drives 255 pulse duration memory effect 259 pump frequency 276, 279 quantum mechanical transport theory 46 quantum noise limit 288 quarter-wave plate 83 quasiparticle 128, 135, 147 Rayleigh-Binard convection 106 Rayleigh mode 172, 177. 184 Rayleigh number 106, 107 realistically shaped head model 297 rectification 199 relaxation oscillator 231 relaxation processes 46, 83, 89 relaxation time 8 5 , 88-90. 92-94, 96, 109 remanent 303 remanent magnetization 327 reproducible “neon” superconductivity 9 resistively-shunted-junction (RSJ) 274 resonance 280, 285, 286 resonant transformer 29 1 rffields 294 rf interference 294 rf shielding 294, 296 rf shunt 296 rf SQUID 274, 290 RSJ model 274 Schafroth model 24 Schafroth model (with localized pairs) 15 second-order gradiometer 296, 311, 312, 313 second virial coefficient 74-76, 77 shaking 293 Shapiro steps 241 shielding factor 293 shunt resistance 283 signal bandwidth 279 signal coil 285, 295, 306 signal-coil resonances 3 15 signal coupling 285. 288 single-particle excitation 128, 135 sintered powder 142, 160 slew rate 289-291, 309, 327 small particle 120, 124 spectral response 200

SUBJECT INDEX spherical head model 297 spheroidal mode 133, 188 spin diffusion 46. 62, 70, 71 spin magnetism 72 spin polarisation 94 spin-polarised 'He gas 47 spin-polarised 'He in liquid form I 0 9 spin rotation effects 46,93, 95 spin waves 46, 64,71, 93,95, 97, 98 spiral transmission line 286 staircase pattern 277, 290 staircase structure 248 Stewart-McCumber parameter 280 stray capacitance 285 subharmonic mode locking 246 subject noise 325 Superconducting Quantum Interference Devices (SQUIDS) 273 superdiamagnetism 9 surface 215 susceptometer 319 switching 215 symmetrical gradiometer 295 symmetry effect 53 synchronization 200, 247 tank circuit 274, 276, 290 temporal fluctuations 261 thalassemia 304 the tunneling model 237 thermal conductivity 46, 54, 56, 58, 62, 66, 69, 70, 98, 102, 103, 105

365

thermal noise 277, 281, 283, 286 thermal wavelength 52 threshold field ET 198 toroidal mode 133, 188 transfer function 283 transient oscillations 220 transmission coefficient 119, 165 transmission-line resonances 287, 288 transport properties 46 transport properties of dilute gases 54 transport properties of the gas 54 transport theory 56. 59 trazolam 302 triangular pattern 276 "true" or "genuine" superconductivity 10 tuned transformer 291 two-fluid description 215 two-level tunneling states 181. 183 Vacuumschmelze 294, 311, 312 vector magnetometers 316, 3 18 viscosity 46, 54, 56, 58. 62, 66, 67, 70 voltage plateaus 286, 288 voltage-to-flux transfer ratio 285 vplume currents 297 VT/,, characteristics 277 wall relaxation 85 Wol ff - Parkinson- White syndrome 303 zero sound 123, 128, 131, 132

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