Progress in Low Temperature Physics, Vol. 8

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PROGRESS IN LOW TEMPERATURE PHYSICS

VIII

This Page Intentionally Left Blank

CONTENTS OF VOLUMES I-VIIB VOLUME I

C.J. Gorter, The two fluid model for superconductors and helium I1 (16 pages) R.P. Feynman, Application of quantum mechanics to liquid helium (37 pages) J.R. Pellam, Rayleigh disks in liquid helium I1 (10 pages) A.C. Hollis-Hallet, Oscillating disks and rotating cylinders in liquid helium I1 (14 pages) E.F. Hammel, The low temperature properties of helium three (30 pages) J.J.M. Beenakker and K.W. Taconk, Liquid mixtures of helium three and four (30 pages) B. Serin, The magnetic threshold curve of superconductors (13 pages) C.F. Squire, The effect of pressure and of stress on superconductivity (8 pages) T.E. Faber and A.B. Pippard, Kinetics of the phase transition in superconductors (25 pages)

K. Mendelssohn, Heat conduction in superconductors (18 pages) J.G. Daunt, The electronic specific heats in metals (22 pages) A.H. Cooke, Paramagnetic crystals in use for low temperature research (21 paytu) N.J. Poulis and C.J. Gorter, Antiferromagnetic crystals (28 pages) D. de Klerk and M.J. Steenland. Adiabatic demagnetization (63 pages) L. Nkel, Theoretical remarks on ferromagnetism at low temperatures (8 pages)

L. Weil, Experimental research on ferromagnetism at very low temperatures (11 pages) A. Van Itterbeek, Velocity and attenuation of sound at low temperatures (26 pages)

J. de Boer, Transport properties of gaseous helium at low temperatures (26 pages)

VOLUME I1 J. de Boer,Quantum effects and exchange effects on the thermodynamic properties of liquid helium (58 pages) H.C. Kramers, Liquid helium below 1°K (24 pages) P. Winkel and D.H.N. Wansink, Transport phenomena of liquid helium I1 in slits and capillaries (22 pages)

K.R. Atkins, Helium films (33 pages) B.T. Matthias, Superconductivity in the periodic system (13 pages)

CONTENTS OF VOLUMES I-VIIB VOLUME I1 (continued) E.H. Sondheimer, Electron transport phenomena in metals (36 pages) V.A. Johnson and K. Lark-Horovitz. Semiconductors at low temperatures (39 pages) D. Shoenberg, The De Haas-van Alphen effect (40 pages) C.J. Gorter, Paramagnetic relaxation (26 pages) M.J. Steenland and H.A. Tolhoek, Orientation of atomic nuclei at low temperatures (46 pages) C. Domb and J.S. Dugdale, Solid helium (30 pages) F.H. Spedding, S. Legvold. A.H. Daane and L.D. Jennings. Some physical properties of the rare earth metals (27 pages) D. Bijl, The representation of specific heat and thermal expansion data of simple solids (36 pages) H. van Dijk and M. Durieux, The temperature scale in the liquid helium region (34 pages)

VOLUME 111 W.F. Vinen, Vortex lines in liquid helium I1 (57 pages) G. Carerj, Helium ions in liquid helium I1 (22 pages) M.J. Buckingham and W.M. Fairbank, The nature of the A-transition in liquid helium (33 pages) E.R. Grilly and E.F. Hammel. Liquid and solid 'He (40 pages) K.W. Taconis, 'He cryostats (17 pages) J. Bardeen and J.R. Schrieffer, Recent developments in superconductivity (1 18 pages) M.Ya. Azbel' and I.M. Lifshitz, Electron resonances in metals (45 pages) W.J. Huiskamp and H.A. Tolhoek, Orientation of atomic nuclei at low temperatures 11 (63 pages) N. Bloembergen, Solid state masers (34 pages)

J.J.M. Beenakker, The equation of state and the transport properties of the hydrogenic molecules (24 pages) Z. Dokoupil, Some solid-gas equilibria at low temperature (27 pages)

CONTENTS OF VOLUMES I-VIIB VOLUME 1v V.P. Peshkov, Critical velocities and vortices in superfluid helium (37 pages) K.W. Taconis and R. de Bruyn Ouboter, Equilibrium properties of liquid and solid mixtures of helium three and four (59 pages) D.H. Douglas Jr. and L.M. Falikov, The superconducting energy gap (97 pages) G.J. van den Berg, Anomalies in dilute metallic solutions of transition elements (71 pages)

Kei Yosida, Magnetic structures of heavy rare earth metals (31 pages) C. Domb and A.R. Miedema, Magnetic transitions (48pages) L. Ntel, R. Pauthenet and B. Dreyfus, The rare earth garnets (40 pages) A. Abragam and M. Borghini, Dynamic polarization of nuclear targets (66 pages) J.G. Collins and G.K. White, Thermal expansion of solids (30 pages) T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde, The 1962 'He scale of temperatures (35 pages)

VOLUME: V P.W. Anderson, The Josephson effect and quantum coherence measurements in superconductors and superfluids (43 pages) R. de Bruyn Ouboter. K.W. Tamnis and W.M. van Alphen, Dissipative and non-dissipative flow phenomena in superfluid helium (35 pages) E.L:Andronikashvili

and Yu.G. Mamaladze, Rotation of helium 11 (82 pages)

D. Gribier, B. Jacrot, L. Madhavrao and B. Farrioux, Study of the superconductive mixed state by neutrondifiaction (20 pages) V.F. Gantmakher, Radiofrequency she effects in metals (54 pages) R.W. Stark and L.M. Falicov. Magnetic breakdown in metals ( 5 2 pages) J.J. Beenakker and H.F.P. Knaap, Thermodynamic properties of fluid mixtures (36 pages)

CONTENTS OF VOLUMES I-VIIB VOLUME VI J.S. Langer and J.D. Reppy, Intrinsic critical velocities in superfluid helium (35 pages) K.R. Atkins and 1. Rudnick, Third sound (40 pages) J.C. Wheatley, Experimental properties of pure He3 and dilute solutions of He3 in superfluid He4 at very low temperatures. Application to dilution refrigeration (85 pages) R.1. Boughton, J.L. Olsen and C. Palmy, Pressure effects in superconductors (41 pages) J.K. Hulm, M. Ashkin. D,W. Deis and C.K. Jones, Superconductivity in semiconductors and semi-metals (38 pages) R. de Bruyn Ouboter and A.Th.A.M. de Waele. Superconducting point contacts weakly connecting two superconducton (48 pages) R.E. Glover, 111, Superconductivity above the transition temperature (42 pages) R.F. Wielinga, Critical behaviour in magnetic crystals (41 pages) G.R. Khutsishvili, Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities (30 pages) M. Durieux, The international practical temperature scale of 1968 (21 pages)

VOLUME VII A J.C. Wheatley, Further experimental properties of superfluid 3He (104 pages) W.F. Brinkman and M.C.Cross, Spin and orbital dynamics of superfluid 3He (86 pages)

P. Wolfle, Sound propagation and kinetic coefficients in superfluid 'He (92 pages) D.O. Edwards and W.F. Saam, The free surface of liquid helium (88 pages)

VOLUME VII B J.M. Kosterlitz and D.J. Thouless, Two-dimensional physics (64 pages) H.J. Fink, D.S. McLachlan and B. Rothberg Bibby. First and second order phase transitions of moderately small superconductors in a magnetic field (82 pages)

L.P. Gor'kov, Properties of the A- 15 compounds and onedimensionality (74 pages) G. Griiner and A. Zawadowski, Low temperature properties of Kondo alloys (58 pages) . J.

Rouquet, Application of low temperature nuclear orientation to metals with magnetic impurities (98 pages)

P R O G R E S S I N LOW TEMPERATURE PHYSICS EDITED BY

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

VOLUME.vrii

1982 NORTH-HOLLAND PUBI-ISHING COMPANY AMSTERDAM . NEWYORK * OXFORD

@ North-Holland Publishing Company - 1982

AII righu reserued. No part of this publication may be reproduced, stored in a rem’eual system, or nansmined, in any form or by any means, elecfronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner

ISBN: 0 444 86228 5

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK . OXFORD

-

SOLE DISTRIBUIORS FOR THE US A. AND CANADA.

ELSEVER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK.N.Y.10017

LIBRARY OF CONORES CATALOGING 5 5 - 14533

IN PUBLICATION

DATA

Filmset and printed in Northern Ireland at The Universities Press (Belfast) Ltd.

PREFACE

In this eighth volume of Progress in Low Temperature Physics I have tried again to pick out a few of the many topics which have been of great interest in low temperature physics since the previous volume was compiled. The subject of the first article - solitons -originated in 1834 when a solitary wave was observed as an isolated singularity moving with unchanging shape and velocity along a canal in Scotland. Like many other hydrodynamic phenomena observed or investigated theoretically in the nineteenth century, they enjoyed a long period of comparative rest, but since 1965 (when the word soliton was coined) they have appeared widely in all branches of physics. In Chapter 1, Professor Maki has concentrated on those aspects which are of particular interest in low temperature physics. Solid ’He has been the subject of extensive investigation since the late 1950s when predictions were made of a previously unexpected very large exchange interaction between the ’He spins. This continues to be interesting both experimentally and theoretically, and in the past few years an additional aspect has attracted attention - namely the solid-quantum liquid interface (both 3He and 4He). Striking experimental results include the interfacial energy of the solid 4He-substrate surface (solid 4He does not wet disordered substrates in preference to liquid), capillary waves, and the “roughening transition”. The last of these has been discussed theoretically in solid state physics since 1951 but only now observed experimentally, probably for the first time, in helium at around 1 K. Surprisingly, solid helium has only once before been reviewed in this series, as a small part of an article in 1961 on liquid and solid ’He. This situation has now been rectified by Professor Andreev in his article on Quantum Crystals. Another topic which appears widely in physics, astrophysics, and engineering is turbulence. The special interest in low temperature physics is that superfluid 4He provides an ideal quantum fluid of strictly zero viscosity to which many of the classical concepts are applicable. The role of turbulence in the dissipation processes in superfluid 4He has become gradually clearer since the work of Onsager and Feynman in 1949 and 1954 on quantised vortex lines (see the famous article by Feynman in Progress in Low Temperature Physics Vol. I, and the later development of the subject by Vinen in Vol. 111). It is a measure of the complexity of the phenomena that it is only in the past few years that a microscopic theory has emerged, and extensive detailed experimental work carried

X

PREFACE

out. Both theoretical and experimental aspects are discussed by Professor Tough in Chapter 3. Continual development of techniques to reach lower temperatures has in the past been found to play a vital part in making the most exciting discoveries. In the last chapter of this volume, Professor Andres and Professor Lounasmaa have reviewed current practices and some future possibilities in the production of temperatures of 1 mK or less. Until recently, the only such technique available for cooling other materials was the “brute force” demagnetisation of copper nuclei. Now, the use of hyperfine enhanced nuclear paramagnets, particularly praseodymium nickel five (PrNiJ is becoming more and more popular. It has certain considerable advantages over copper for some purposes, but not all. In addition to discussing the basis of the refrigeration process and the techniques, Andres and Lounasmaa compare the merits and disadvantages, and the appropriate use of the two methods. There seems little doubt that the use of enhanced nuclear refrigerants will make the 1 mK region much more easily accessible and attract a good deal more experimental work. This volume has taken rather longer to appear than I had originally planned. I think I can promise that the next one will come out after a much shorter interval, and will include some of the articles that I had hoped would be published in the present volume. I am grateful to many colleagues for discussion of what are the most significant current topics in low temperature physics, and to the authors for writing the articles. I am also grateful to Mr. Richard Newbury for assistance in indexing, and to the publishers for their help. Sussex, 1981

D. F. Brewer

CONTENTS VOLUME VIII

Preface

Ch. 1 . Solitons in low temperature physics, K . Maki

iX

1

1 . Introduction 2. Classical solitons 3. Solitons in one-dimensional systems 4. Classical statistical mechanics of the sine-Gordon system 5 . Quantum statistics of solitons 6. Correlation functions 7. Conclusion References

3 5 15 30 34 49 61 62

Ch. 2..Quantum crystals, A.F. Andreeu

67

1. Introduction 2. Quantum effects in crystals 3. Nuclear magnetism 4. Impurity quasi-particles - impuritons 5. Vacancies 6. Surface phenomena 7. Delocalization of dislocations References

69 69 72 80 100 112 127 129

Ch. 3. Superjluid turbulence, J.T. Tough

133

1 . Introduction 2. Theoretical background 3. Temperature and chemical potential difference data 4. Pressure difference data 5 . Second sound data 6. Ion current data 7. Fluctuation phenomena 8. The critical condition 9. Pure superflow, pure normal flow, and other velocity combinations References

135 143 155 165 171 180 189 200 207 216

xii

CONTENTS

Ch. 4. Recent progress in nuclear cooling, K. Andres and 0.V. Lounasmaa 1. Introduction 2. Brute force nuclear cooling 3. Hyperfine enhanced nuclear cooling 4. Two stage nuclear refrigerators 5. Comparison of brute force and hypefine enhanced nuclear refrigeration References

22 1 223

225 245 274 283 285

Author index

289

Subject index

297

CHAPTER 1

SOLITONS IN LOW TEMPERATURE PHYSICS* BY

KAZUMI MAKI Department of Physics, University of Southern California, Los Angeles, California 90007,USA

* Supported by National Science Foundation under Grant No. DMR76-21032. Progress in Lnw Temperature Physics. Volume VIII Edited by D.F. Brewer @ North-Holland Publishing Company, 1982

Contents 1. Introduction 2. Classical solitons 2.1. Mathematical solitons 2.2. Sine-Gordon solitons 2.3. Topological solitons 3. Solitons in one-dimensional systems 3.1. One-dimensional conductors 3.2. One-dimensional magnets 4. Classical statistical mechanics of the sine-Gordon system 5. Quantum statistics of solitons 5.1. Mass renormalization 5.2. Method of functional integral 5.3. Soliton energy and soliton density 5.4. Breather problems 5.5. The b4 system 6. Correlation functions 6.1. The sine-Gordon system 6.2. The b4 system 7. Conclusion

References

3 5 5

8 11 15

16 23

30 34 35

38 40 43 45 49 49 60

61 62

1. Introduction Since the notion of the “soliton” was introduced by Zabusky and Kruskal (1965) into physics, the word “soliton” has been used in a wider and wider context as time has passed. Therefore we shall start with a brief history of the “soliton”. Soliton physics was started by the analysis of the Korteweg de Vries (KdV) equation (Korteweg and de Vries, 1895), which describes the nonlinear water wave (the solitary wave) in a shallow water channel h t observed by Scott-Russell (1844). After the discovery of the remarkable stability of the nonlinear solutions of the KdV equation by Zabusky and Kruskal (1965), which led them to coin the name “soliton”, Gardner et al. (1967) showed that the initial value problem of the KdV equation can be solved completely by a few steps of linear operations (inverse scattering method). Then in rapid succession, it was shown by Ablowitz et al. (1973) that a similar method applies to other nonlinear equations: the cubic Schrodinger equation and sine-Gordon equation. It is now known that a large class of nonlinear equations are amenable to the inverse scattering method (Ablowitz et al., 1974). All of these nonlinear equations have localized solutions, which are remarkably stable. These are solitons in the most strict sense. We shall later refer to them as mathematical solitons, in order to distinguish them from other solitons, which do not necessarily require the underlying completely integrable differential equations. Parallel to these developments it was first realized by Finkelstein (1966) that some nonlinear solutions of quantum field theories can be classified in terms of homotopy classes: classification according to mapping between the real n-dimension physical space and the manifold formed by the field configuration of the ground states. Although Finkelstein called these solutions “kinks”, it is common practice among field theorists to call them “topological” solitons. This approach was generalized recently to quantum field theory (Patrascioiu, 1975; Belavin et al., 1975; Coleman, 1977) and to liquid crystals and superfluid 3He (Toulouse and Kleman, 1976; Mineyev and Volovik, 1978). Topological solitons are of particular interest in condensed matter physics. A condensed phase is characterized by order parameters. Furthermore, the ground state of the condensed phase is in general highly degenerate; there is a finite subspace of the order parameter space corresponding to the ground state. In this circumstance the topological solitons play an important role in the physical properties of the system. In particular, as

4

K. MAKI

first realized by Krumhansl and Schrieffer (1975), solitons play a crucial role in thermodynamics and dynamics of the quasi one-dimensional system. In the one-dimensional system, the distinction between the mathematical and topological solitons is almost superficial except that in the case of mathematical solitons the physical properties are determined in principle exactly. However, at low enough temperatures (Tee Es, where Es is the soliton energy, note we use the unit system A= kB = 1 hereafter) solitons behave as a new class of elementary particles. This approach has been extended to calculate the dynamical responses of the system (Kawasaki, 1976; Mikeska, 1978). For a recent review of this approach see Currie et al. (1980), for example. Some of the consequences of the classical statistical mechanics of one-dimensional systems have also been tested by molecular dynamic analysis of the system (Schneider and Stoll, 1975; 1978a; 1978b). We believe that quasi one-dimensional systems will provide a unique laboratory to test experimentally the physical properties of one-dimensional solitons. Indeed it is quite likely that the quantum field theory of nonlinear systems in 1+ 1 dimensions will be confronted with low temperature experiments on quasi onedimensional systems in the near future. The situation is not so fortunate for the condensate in higher space dimensions (i.e. two-dimensional and three-dimensional systems). No obvious physical system provides solitons with microscopic energy in more than two dimensions, although there is an example of pseudoparticles in a 2-D Heisenberg ferromagnet (Polyakov, 1976); the proper treatment of this object suffers uncontrollable infrared divergence (Jevicki, 1977) similar to that associated with instantons in SU(2) gauge field theory in 4-dimensions (Belavin et a]., 1975; t’Hooft, 1976). In parallel to the development of the statistical mechanics of solitons in the one-dimensional system just described, there have been remarkable advances in the quantum field theory of nonlinear systems in 1 + 1 dimensions. In the case of interacting bosons (Lieb and Liniger, 1963; Lieb, 1963) and some classes of Heisenberg antiferromagnet with spin 4 (Sutherland et al., 1967; Baxter, 1972) exact results for the energy spectra have been obtained by making use of Bethe’s ansatz (Bethe, 1931). More recently Dashen et al. (1974a, b, 1975) have developed the method of the functional integral and determined the energy spectrum of the sine-Gordon system, within the WKB approximation. Later the above WKB results for the sine-Gordon system were shown to be exact (Luther, 1976; Bergkoff and Thacker, 1979). Indeed the method of the functional integral can be easily extended to finite temperatures (Maki and

SOLITONS IN LOW TEMPERATURE PHYSICS

5

Takayama, 1979a, b; Takayama and Maki, 1979; 1980). Acutally, some ambiguities of the classical statistical mechanics of solitons are clarified in terms of the quantum statistics of solitons. Furthermore, it is shown that classical statistical mechanics applies only for the weak-coupling system. In section 2 we give some examples of classical solitons. Then we describe solitons of quasi one-dimensional systems in section 3. Section 4 is devoted to a brief description of the classical statistics on onedimensional nonlinear systems. The quantum statistics of solitons in one-dimensional systems are summarized in sections 5 and 6.

2.1. "Mathematical" solitons

We shall fust write down some nonlinear equations which appear most frequently in the literature (Scott e t al., 1973; Whitham, 1974): (a) Korteweg de Vries (KdV) equation,

41+a44,+4,=0.

(1)

(b) Cubic Schriidinger equation, (c) Sine-Gordon equation,

bI1-4xx+ m 2sin 4 =O.

(3)

As already mentioned the KdV equation describes an isolated water wave in a shallow channel. An interesting application of this equation to nonlinear third sound in a superfluid 4He film has been proposed by Huberman (1978). The cubic Schrodinger equation describes the spatial conformation of the electric field in nonlinear optics and possibly superfluid 4He in a linear capillary (Tsuzuki, 1971). Finally, the SG equation is ubiquitous in low temperature physics (Barone et al., 1971; Scott et al., 1973). Some examples of the SG systems and SG solitons will be given later in this section. All of the above equations possess, in addition to linear solutions, soliton solutions, which are summarized in table 1. Furthermore, these equations are amenable to the inverse scattering method (Ablowitz et al., 1974).

6

K. MAKI Table 1 Typical nonlinear equations and their soliton solutions

Equation

Dispersion of linear modes

Korteweg de Vries

o =k3

Cubic Schriidinger

o =k2

Sine-Gordon

o 2= k 2

Soliton 3u

-sech2[+1/2(x - or)]

e)”’ a

sech[a’I2(x - uI)]

+ m2

4 tan-’{exp[my(x - ut)D = (1 - u 2 ) - 1 / 2

The soliton solutions are characterized by the following. (1) They are localized solutions. (2) They behave like particles; in the absence of external perturbation they move with constant velocity without any change in their shape. For small perturbations they respond like Newtonian particles. (3) They are quite stable against large perturbations. For example, when two solitons collide with each other, they emerge after the collision with the same energies as before and without any change in their shape. Solitons suffer only a phase shift by colliding with each other. (4) There are exact N-soliton solutions. Readers will find detailed descriptions of the properties in a review paper by Scott et al. (1973). In order to illustrate the above properties, we shall consider the sine-Gordon system described by eq. (3). The SG system is remarkable for its additive topological charges; the SG soliton is a “mathematical” soliton and at the same time a “topological” soliton. This is intimately related to the fact that the gound state of the SG system is infinitely degenerate, which can be easily seen from a sine-Gordon Hamiltonian;

‘I

H = - d x { ~ ( x ) 2 + ( l i ~ / l i x ) 2 + 2 r n Z [ 1 d(x)n, -cos 2

(4)

where n(x)=a&/lat the conjugate field to 4 4 ~ )Since . -cos&(x) has minima at 4 = 0, * 2 r , * 4 ~ ,the ground state of eq. (4) is given by

4 = 0, *2T,*4T,f . . . .

(5)

A soliton is given by

4(x,

1 ) = 4 tan-’{exp[my(x - ur)B

(6)

SOLITONS IN LOW TEMPERATURE

PHysrcs

7

with y = (1 - u’)-”’ which is a moving domain wall with velocity u. Since the ground states at each side of the soliton are different, the soliton carries the topological charge 0:

while the antisoliton carries the topological charge 0 = -1. The total topological charge of the system QIoIal in the presence of N solitons and antisolitons is given by

which is conserved throughout the physical process. This is an example of the topological conservation (Coleman, 1977). In the SG system the solitonsoliton and the soliton-antisoliton scattering solutions are known (Seeger et al., 1953; Perring and Skyrme, 1962);

describes the scattering between two solitons with velocity u and -u, while

describes the scattering between soliton and antisoliton where y = (1 - u’)-1’2.

(11)

Furthermore, the bound state of the soliton-antisoliton pair is given by

where -j= (1 + u2)-’”. The above solution is called a “breather”, as it describes a localized oscillation of 4,. The initial value problem of the sine-Gordon equation is exactly solvable by means of the inverse scattering method (Ablowitz et al., 1973). Though we shall not go into the description of the inverse scattering method (ISM), ISM applied to the sine-Gordon system predicts that the final state consists of only solitons, antisolitons and breathers

K. MAKI

8

independent of the initial configuration of 4 at r = 0; the linear fluctuations disappear quite rapidly from 4. This seems to suggest that the basic elements of the SG system are solitons, antisolitons and breathers.

2.2. Sine-Gordon solitons We shall describe a few examples of the sine-Gordon solitons in this subsection.

2.2.1. Planar ferromagnet in a magnetic field

Perhaps the simplest example is a chain of planar spins coupled ferromagnetically and in the presence of a magnetic field in the plane of the planar spin. The magnetic field breaks the continuous symmetry of the spin direction. Let us consider a one-dimensional system described by the following Hamiltonian (Mikeska, 1978) n

n

n

where Sn is the spin vector on the site n. Here the chain is assumed in the z direction and the magnetic field is applied in the x direction. The above Hamiltonian with S = 1, J = 23.6 K and A = 5 K adequately describes CsNiF, in the paramagnetic phase (Steiner et al., 1976). Then in the temperature region T> 1 where (3 = (kB7T', n,(u) = e-@Es(k).

(55)

Villain (1975) has predicted that the longitudinal spin correlation function is dominated by the soliton. In particular, for 1q - ?r/al>> K, Villain (1975) obtained the dynamical structure factor

and K-' is the correlation length in the Ising limit (Ising, 1925) e-Ka= tanh(gJ/2).

(58)

The magnetic properties of quasi one-dimensional magnetic systems like CsCoC1, and K,Fe(CN), are well described by the Hamiltonian, eq. (52). Indeed a recent neutron scattering experiment on CsCoC1, (Hirakawa and Yoshizawa, 1979) is consistent with eq. (56). For momentum transfer q, very close to the reciprocal lattice vector (i.e., 19 - ?r/al > E : , and

for T C EZ,where EX = 8hm*g-’ is the classical soliton energy. Similarly the thermal average of c o s ( g 4 ) is calculated as 1 aa, 2 a4

(cos(g4))=--

for T > E:

=

4 141-8

-. . .,

lq13+& 1qIS

(107)


33

for T> 1) expansion of fo(z);

and yo= 1.78.. . . The first term in the exponent of eq. (132) reproduces the classical result, if g2 and m* in the classical theory are replaced by g’’ and m(0) respectively while the remaining terms give the quantum corrections. It may be useful to write eq. (129) as

where A = 4S(2AJ)”’ and 8(a-1)(1- D/2 IJJ)”’JS for the ferromagnetic and antiferromagnetic system respectively. In this form the second coefficient describes the genuine quantum correction.

5.2. Method of functional integral Substituting eq. (119) into eq. (117), we obtain;

which is now free of divergence. First of all, the ground state of the present system is degenerate and given by C#J = 2m/g and

n = 0, *l, *2, etc.

( 136)

Secondly, topological conservation requires that the topological charge

is a constant of motion. Therefore the total Hilbert space is decomposed into sectors characterized by integer 0.For example, the partition function 2 of the system is given by m

Z = ID(C#J)exp(-

Z,,

dr) = n=-m

and


39

where D,,(c#J) is the functional integral in the Nth sector. Here T is the imaginary time and @ = T I . In the low temperature region where the soliton energy Es is much larger than the temperature T, 2, is dominated by the soliton free state (i.e. purely bosonic excitations), although it includes the contribution from say a pair of solitons and antisolitons, while Zl is dominated by the single soliton term. Therefore in this temperature region we have Z'lZ, = Liis

(140)

where iis is the soliton density and L is the total length of the system. Within the same approximation Z is given by

1

z-z, (1+2Liis+-(2LrQ2+. .. 2!

)

The factor 2 in the exponent of eq. (141) arises from the fact that the contribution of the antisoliton t o the partition function is the same as that of the soliton. The thermodynamic potential of the system is then given by

R

= .n0-2p-'iis

(142)

where

Ro= -L-'@-' In Z,

(143)

and R, is the thermodynamic potential associated with the soliton free sector. The above analysis indicates that in the low temperature region (TccE,), Zo and 2,characterize completely the thermodynamics of the system. In the weak-coupling limit 2, can be calculated perturbatively (Maki and Takayama, 1979a) resu1,ting in

where o,=(Cgk2+ rn2)'I2

(145)

and g'' and fo have already been defined in eq. (130). The asymptotic

K. MAKI

40

behaviors of eq. (144) are given as

Ro = const. - @CJ'

rn

[Ko(Brn)+ 4 K o ( 2 ~ m ) ] 7r

+QC,'g'2(m/7r)2K~(gm)+Oe-38m

for T>m

i s= (EJ2mp)1‘2e-8Es

for T S m

= (1 y)@me-8Es-v

and = 2m ( p E , / 2 ~ ) ” ~ e - @ ~ s for

T>>m.

As already noted eq. (161) is in complete agreeement with the TMT result, eq. (115), if Ef in the classical theory is replaced by Es. Putting together eqs. (147) and (161), the thermodynamic potential of the sineGordon system for the intermediate temperature (m > E& the breathers cannot be considered elementary and eq. (144) will provide the appropriate expression for no. However, we don’t have yet a consistent theory which describes the two limits correctly. In the weak-coupling limit the breather mass spectrum at low temperatures can be obtained within perturbation theory. For T T >> m, where

M=M,,

s = 0 ’ )- Cik’,

q = TIE,

and

E,(k)=(M:+ Cik2)”’. In the low temperature region, the two magnon term consists of the second breather term with w = E,(k) and the two magnon continuum for 101 > [(2M)’+ C;k’]”’. At higher temperatures a central peak appears associated with thermal magnons for JwIC C,k (Maki, 1981) in addition to other contributions. In tables 8 and 9 we summarize the spin-spin correlation functions and thus determined for the planar ferromagnetic chain and the planar antiferromagnetic chain in magnetic fields described in subsection 3.2, respectively. In the tables, ns(w/k) is the density of the soliton with velocity u = w / k defined in eqs. (158) and (159). ns(w/k)= e-BEsy = (1 + y)Pme-BEsr

for T s m for T >> m

(252) (253)

K. MAKI

58

Table 8 Spin correlation functions in planar ferromagnet

Im x ( k . w )

with

In general these expressions describe satisfactorily both the neutron scattering data from CsNiF, in a magnetic field (Kjems and Steiner, 1978) and from TMMC in a magnetic field (Boucher et a]., 1980a, b) except that the theoretical expression for E t is slightly larger than the observed soliton energy EFP.For example, for CsNiF, in a magnetic field of 5 kOe, EB = 34 K, while EFp= 27 K (Kjems and Steiner, 1978). Similarly, in the case of TMMC in a field of 36.2 kOe, EZ = 12.3 K while E y p= 9.41 K (Boucher et al., 1980b). The discrepancies are always about 20%, which is very close to the mass renormalization for the magnon mass estimated in subsection 5.1. Since the same renormalization correction enters in the expression E, (see eq. (156)], the discrepancies may be interpreted in terms of the quantum fluctuations neglected in the classical theory. Table 9 Spin correlation functions in planar antiferromagnet

SOLITONS M LOW TEMPERATURE PHYSICS

59

However, if it is true, these renormalization corrections should also appear in the magnon mass in the system. Indeed the magnetic field dependence of the magnon masses as given in eqs. (127) and (128) may provide rather a unique check of the existence of mass renormalization. In the case of the antiferromagnetic chain, the transverse spin correlation function {S&>, has an enormous central peak associated with the soliton. This has been observed both by elastic and inelastic neutron scattering from TMMC [(CH3)4NMnC13]in a magnetic field by Boucher et al. (1980). In particular, eq. (238) describes quite well both the k and o dependence of the observed neutron scattering cross section (Boucher et al., 1980b). More recently, Boucher and Renard (1980) have measured the nuclear relaxation rate T;' of I5N atoms in TMMC in a magnetic field. Since T;' is proportional to the local spin correlation function, they found

c'= B,S2a-'u,2K/[(o/uo)2 +K 2 ] = B,S2(mgK)-'.

6.1.4. Breather contributions

We have seen that S:(k,w) is dominated by the first breather (i.e. the renormalization magnon), while the second breather appears in S .: In general it is possible to estimate the single breather contributions to the dynamical structure factor at least in the weak coupling limit. Most of the quasi one-dimensional magnetic chains in a magnetic field appear to be in this limit. In this limit, however, the intensities of the higher breather modes in the dynamical structure of factors are so small that, besides the well known magnon mode (i.e. the first breather mode), only the second breather mode is likely to be accessible experimentally in the near future. Nevertheless we shall give here for completeness the single breather terms in S:(k, o)and S L ( k , a)for rn 1, we have

h‘(2)=lil

(262)

and eq. (260) is approximated by

x&(z, r) = D ( z , i)+(40)2 e~p[-2ii~(z~+v~r~]”~]

(263)

where iis is the total soliton density and uo= (2/7$Es)’” is the thermal velocity of the soliton. The corresponding dynamical structure factor is


61

given by

where El(&)=(kZ+rn2)”* and K‘ = 2iis. The second term in eq. (264) gives rise to a large central peak in the quasi onedimensional Ising like system, with the width of the peak again controlled by the soliton density iis.

7. Condasion

We have described recent developments of soliton physics in low temperature condensed matter physics. The most important advance in our understanding of solitons has recently been made in quasi onedimensional systems, where a variety of theoretical works are available. Indeed, recent neutron scattering experiments have unveiled the existence of magnetic solitons in the quasi one-dimensional magnetic systems like CsCoCI,, CsNiF, and TMMC.For CsCoC1, a quantum description of the soliton is required, while for the latter two systems the classical theory appears to provide a semi-quantitative description of the dynamical structure factors, which are determined experimentally. It is proposed that quantum corrections to the classical theory may account for some discrepancies between the classical theory and experiment, although further work is required to clarify the present situation. The topological soliton in polyacetylene, furthermore, appears to correlate in a remarkable way magnetic and electronic properties of both undoped and doped polyacetylenes. For the time being quasi one-dimensional systems will continue to provide new types of solitons, which have not been covered in this article. Outside of the one-dimensional systems, however, the utility of the soliton concept appears to be quite limited. We don’t have any simple

62

K. MAKI

physical systems, which have point-like solitons in two and three dimensions. As for semi-macroscopic objects like vortex lines in superconductors and domain walls in superfluid 3He, classical theory suffices for describing their dynamics (no quantum effects). We have not described here the implication of solitons on the commensurate-incommensurate transition in two-dimensional systems, as there exists a recent review on this subject by Pokrovsky (1979).

Acknodedgments It is a pleasure to thank my friends and colleagues, Pradeep Kumar, Y.R. Lin-Liu, Hajime Takayama, and Steve Trullinger, who contributed to this work in a variety of ways. I have also benefited from discussion with Jean Paul Boucher, Duncan Haldane, Jim Loveluck, Mike Steiner, Tony Schneider, Erich Stoll and Jacques Villain on related subjects. I would like to thank the Institute hue-Langevin at Grenoble and the Laboratoire de Physique des Solides at Orsay for their hospitality, where most of the manuscript was written. 1 also gratefully acknowledge financial support from the Guggenheim Fellowship.

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Patrascioiu, A. (1975) Phys. Rev. D12, 523. Peming, J.K. and T.H.R. Skyrme (1962) Nucl. Phys. 31, 550. Pietronero. L., S. Strassler and G.A. Toombes (1975) Phys. Rev. B21, 5213. Poenaru, V. and G. Toulouse (1977) J. Phys. (Paris) 38. 887. Pokrovsky, V.L. (1979) Adv. Phys. 28, 595. Polyakov. A.M. (1976) Sov. Phys.-JETP 41. 988. Regnault, L.P., J.P. Boucher, J. Rossat-Mignod, J.P. Renard. J. Bouillot, and W.G. Stirling (1981) in Physics in One Dimension, eds. J. Bernasconi and T. Schneider (Springer, Berlin). Rice, M.J. (1979) Phys. Lett. 71A, 152. Rice, M.J.. A.R. Bishop, J.A. Krumhansl and S.E. Trullinger (1976) Phys. Rev. Lett. 36, 432. Scalapino, D.J., M. Sears and R.A. Ferrell (1972) Phys. Rev. B6. 3409. Schneider. T. and E. Stoll (1975) Phys. Rev. Lett. 35, 296. Schneider, T. and E. Stoll (1978a) Phys. Rev. B17, 1308. Schneider, T. and E. Stoll (1978b) in: Solitons and Condensed Matter Physics. eds., A.R. Bishop and T. Schneider (Springer, Berlin). Scott, A.C., F.Y.F. Chu and D.W. McLaughlin (1973) Roc. IEEE 61, 1443. Scott-Russell, J. (1844) Proc. Roy. Soc. Edinburgh 319. Seeger, A., H. Donth and A. Kochendorfer (1953) Z. Phys. 134, 173. Shankar, R. (1977) J. Phys. (Paris) 38, 1405. Shirakawa. H., T. It0 and S. Ikeda (1978) Die Macromol. Chem. 179, 1565. Snow, A., P. Brandt and D. Weber (1979) Polym. Lett. 17,263. Steenrod, N. E.. (1951) Theory of Fibre Bundles (Princeton University Press, Princeton). Steiner. M. and J.K. Kjems (1977) J. Phys. C10. 2665. Steiner, M.,J. Villain and C.G. Windsor (1976) Adv. Phys. 25, 87. Su. W.P., J.R. Schrieffer and A.J. Heeger (1979) Phys. Rev. Lett. 42, 1698. Su. W.P., J.R. Schrieffer and A.J. Heeger (1980) Phys. Rev. B22, 1101. Sutherland. B. (1978) Rocky Mountain J. Math. 8, 413. Sutherland, B., C.-N. Yang, and C.P. Yang (1967) Phys. Rev, Lett. 19, 588. Takayama, H. and K. Maki (1979) Phys. Rev. B20. 5009. Takayama, H. and K. Maki (1980) Phys. Rev. B21, 4558. Takayama, H., Y.R. Lin-Liu and K. Maki (1980) Phys. Rev. B21, 2388. t’Hooft, G. (1979a) Phys. Rev. Lett. 37, 8. t’Hooft, G. (1976b) Phys. Rev. D14,3432. Toulouse. G. and M. Kleman (1976) J. Phys. 37, L149. Trullinger, S.E. (1979) Sol. St. Commun. 29, 27. Tsuzuki. T. (1971) J. Low Temp. Phys. 4, 441. Volovik, G.E. and V.P. Mineev (1977) Sov. Phys.-JETP 46,401. Villain, J. (1975) Physica 79. B1. Walker, L.R., R.E. Dietz, K. Andres and S. Derek (1972) Sol. St. Commun. 11, 593. Weinberger, B.R., J. Kaufer. A. Pron, A.J. Heeger and A.G. MacDiarmid (1979) Phys. Rev. B20, 223. Whitham. G.B. (1974) Linear and Nonlinear Waves (John Wilcy and Sons, New York) ch. 17. Yang, C.N. (1968) Phys. Rev. 168, 1920. Yang, C.N. and C.P. Yang (1969) J. Math. Phys. 10, 1115. Yoshizawa, H. and K. Hirakawa (1979) J. Phys. 10, 1115. Zabusky, N.J. and M.D. KruskaJ (1965) Phys. Rev. Lett. 15, 240. Zeller, H. (1973) Adv. Sol. St. Phys. 13, 31.


CHAPTER 2

QUANTUM CRYSTALS BY A.F. ANDREEV Institute for Physical Problems, Moscow, USSR

Progress in LAW Temperature Physics, Volume VlII Edited by D. F. Brewer @ Nonh-Holland Publishing Company, 1982

Contents 1. Introduction 2. Quantum effects in crystals 3. Nuclear magnetism 3.1. Exchange interaction of nuclear spins 3.2. Spin diffusion and relaxation 3.3. Ordering of nuclear spins in 'He 4. Impurity quasi-particles- impuritons 4.1. Diffusion in a gas of impuntons 4.2. Diffusion of strongly interacting impuntons 4.3. Impuriton - phonon interaction 4.4. Two- and onedimensional impuritons 5 . Vacancies 5.1. Vacancies in 'He crystals 5.2. Zero-pint vacancies 5.3. Vacancies in 3He crystals 6. Surface phenomena 6.1. Equilibrium shape of crystal-liquid interface 6.2. Crystallization and melting 6.3. Crystallization waves 7. Delocalization of dislocations References

69

69 72 72 74 78 80 80 84 87 91 100 100 106 109 112 112 11R 122 127

129

1. Introduction Ordinary crystals are characterized by the purely vibrational motion of their particles about strictly defined positions of equilibrium -crystal lattice sites. Such a picture is quasi-classical because the particles turn out to be individualized by belonging to definite sites, while in a completely quantum description identical particles must be undistinguishable. The accuracy of the quasi-classical picture is, however, rather high. This is due to a relative weakness of quantum effects in most of the crystals, the criterion being the small ratio of the amplitude of the zero-point vibrations to the lattice parameter. There is a small group of the so-called quantum crystals (solid helium being the most clearly marked example) in which the amplitude of zero-point vibrations is abnormally high, so that the lattice parameter only exceeds this amplitude a few times. This fact directly results in quantitative anomalies of quantum crystals in which the energy of particle vibration is comparable to the total energy of the crystal even at zero temperature and the vibrations are highly anharmonic. The conventional approach to calculating the properties of quantum crystals, such as the energy of the ground state, compressibility and phonon spectrum, is, therefore, inapplicable. Self-consistent methods, accounting for the zeropoint vibrations (see the review of Guyer (1969)], have been developed to describe these properties, and the results obtained by these methods agree, at least qualitatively, with experiment. Of more interest, however, is a qualitatively new effect consisting in the change of the character of the particles’ motion. Since the amplitude of the zero-point vibrations in quantum crystals is high, there is an appreciable probability of quantum tunnelling of particles into the neighbouring lattice sites. The translational motion is, therefore, superimposed on the vibrational motion. This leads to a picture of quantum-mechanically undistinguishable, delocalized particles in the crystal, similar to that taking place in quantum liquids. This paper is a review of properties of quantum crystals, in which they manifest their unusual nature associated with quantum delocalization of particles. Theoretical and experimental results which have recently been obtained in the field undoubtedly show that here we have to deal with objects considerably different from ordinary crystals.

2. Quantum effects in crystals Let us consider, first of all, the quantitative characteristic of the role quantum effects play in crystals, and in what crystals maximal deviations

70

A.F. ANDREEV

from the ordinary quasi-classical theory are to be expected. As we have already mentioned, the relative magnitude of quantum effects is determined by the dimensionless parameter A - z / a 2 , where is the meansquare amplitude of the zero-point vibrations, a is the lattice parameter. The quantity A can easily be expressed in terms of the parameters of particles forming a crystal, if we estimate the amplitude of the zero-point vibrations by the formula Z-hlrnw where m is the particle mass, o - ( x / ~ ) ”is~the vibration frequency, x is the effective stiffness of the “spring” holding the particles in the equilibrium positions. The stiffness z is determined from the condition that the change in the potential energy z a 2 , when the particle is displaced from the equilibrium position at a distance of the order of the lattice parameter, is comparable to the characteristic energy of interaction U between these particles, i.e. Z U/a2. So we obtain A -(fi/a)(mU)-”*. In this form the parameter is known as the quantum parameter of de Boer (de Boer, 1948). It has the maximum value for the lightest and most weakly interacting particles. The largest values of A are reached for the following crystals: 3He (A = O S ) , 4He ( A = 0.4), hydrogen (A = 0.3), neon ( A = 0.1). In all other crystals consisting of particles of one sort the parameter A is vanishingly small. There are, however, important cases where the role of quantum effects is essential not for all particles forming a crystal, but only for some fraction of these particles. This refers, for example, to solutions of hydrogen in the lattices of some heavy metals (niobium, zirconium). Because of the small mass and comparatively weak interaction of hydrogen with the matrix atoms, its motion in the lattice is described by the parameter A, which now is not small, while the matrix atoms behave quite classically. Let us pay attention to the dependence of the parameter A on pressure. The energy of interaction U between adjacent particles in a crystal depends on their mutual distance. As the pressure grows, the lattice parameter a decreases and the interaction U increases, the quantum parameter increasing or decreasing depending on the behaviour of the product Ua*. Since the energy of interaction of neutral particles changes with the decreasing distance much faster than l/a2, the parameter A decreases as the pressure increases. So the maximal quantum effects in crystals are to be expected at minimal pressures. The applicability of the quasi-classical picture of a crystal is limited due to processes involving the quantum tunnelling of particles to adjacent lattice sites. We can make a rough qualitative estimate of the probability of these processes noting that the probability in question is the probability to find the particle at a distance u - a from the equilibrium position. At

2

QUANTUM CRYSTALS

71

low temperatures, when the zero-point vibrations play the major role, the probabilities of different values of the displacement are determined by the square of the modulus of the wave function of the oscillator ground state and obey the Gaussian law w(u) a exp(-u2/2z). The probability of tunnelling is, thus, proportional to the exponentially small quantity of the form exp(-A-'). We may say, therefore, that two different types of quantum effects exist in crystals. The effects, such as the zero-point vibrations, their contribution to the crystal energy, anharmonicity of vibrations, are proportional to some power of the de Boer parameter. These effects do not result in the delocalization of particles and do not destroy the quasi-classical picture of a crystal. Delocalization is an exponentially small quantum effect, so it may, in fact, be observed only in quantum crystals in which A 1. Since the tunnelling probability is exponentially small, its quantitative calculation is rather a difficult problem (Thouless,. 1965 ; Hetherington et al., 1967; Nosanow and Varma, 1969; McMahan, 1972; McMahan and Guyer, 1973; Avilov and Iordanskii, 1976; Delrieu and Roger, 1978) which have not yet been solved completely. In the simplest case the tunnelling process is the one in which two adjacent particles exchange their places (fig. la). In fairly closely packed crystal lattices, such as the hcp and bcc lattices of solid helium, this process is highly impeded by the absence of enough free volume. Therefore, tunnelling processes with many particles involved may be more probable (Thouless, 1965; Zane, 1972; McMahan and Guyer, 1973; Guyer, 1974; McMahan and Wilkins, 1975; Hetherington and Willard, 1975; Delrieu and Roger, 1978). The circular permutation of three (fig. lb) and four (fig. lc) particles requires, as can be seen from the figures, a smaller free volume. Of course, an increase in the number of particles participating in the process reduces its probability, but in this case such an increase does not, in fact, take place, because due to insufficient free volume the permutation of two particles (fig. la) is accompanied by a considerable displacement of the other particles. If a crystal consists of particles of one sort (e.g. 4He), the tunnelling processes considered cannot be observed directly, because they represent the permutation of identical particles. If the tunnelling particles differ from one another in some respect, delocalization of the particles results in qualitatively new observable phenomena. Depending on the character of this difference, there can be two types of phenomena associated with delocalization of particles in quantum crystals. In the first case, pure 3He being the example, the particles of the crystal are identical, but possess

-

72

A.F. ANDREEV

00 000

# o

O%%

000 00

00

(cl Fig. 1. Tunnelling permutations of two, three, and four particles in crystal.

non-zero nuclear spin and may differ in the spin projection. Delocalization of the particles manifests itself in this case as the appearance of the direct exchange interaction of the nuclear spins and nuclear magnetism of 'He crystals caused by this interaction. The phenomena of the second type are related to the behaviour of impurities and any point defects in the crystals. These phenomena are observed in pure form in 4He crystals, while in 3He the situation is more complicated because, as we shall see below, the phenomena of both the types are closely interrelated.

3. Nudear magnetism 3.1. Exchange interaction of nuclear spins The tunnelling permutations shown in figs. la,b,c represent the processes in which the coordinates of the particles are changed, but the spin

QUANTUM CRYSTALS

73

projections remain unchanged. Since in ’He-type crystals the particles are identical, the question, which particle is in a given lattice site, has no physical sense, and the tunnelling permutations can be represented as permutations of the spins corresponding to lattice sites, i.e. as an exchange interaction of spins. The Hamiltonian of this interaction for particles with spin f can be expressed in terms of the known Dirac operator

of the permutations of spins S, and Si referring to sites i and j , by formulas dependent on the character of the tunnelling processes. The processes of pair permutations of the nearest neighbours (fig.l a ) are described by the Hamiltonian

where summation is performed over all the various pairs (i, j ) corresponding to the nearest neighbouring sites. The operator Pijk of the circular permutation of spins of three sites i, j , k is reduced to successive permutations, first of spins jk and then ij, i.e.

Summation here is performed over all different crystallographically equivalent triples of lattice sites. The configuration of the triples has to be chosen from the condition of the “best permutability”, that is the magnitude of the corresponding exchange constant f3) should be as large as possible. The Hamiltonian of the four-particle exchange (fig. lc) can be written

74

A.F. ANDREEV

in a similar way

where summation is performed over quadruples of sites whose configuration corresponds to the maximal magnitude of f4'. There is a simple rule to determine the sign of the exchange constants J'", f3',f4)and, in generai, Jcn)(Thouless, 1965). Namely, Jcn) is negative for even n and positive for odd n. In fact, if J'"' is positive, the ferromagnetic ordering of spins corresponds to the minimal energy. But according to the Pauli principle, the system of fermions with parallel spins has a coordinate wave function which either does not change or, respectively, changes its sign with the even or odd permutation of the particle coordinates. Since the coordinate wave function corresponding to the minimal energy must be symmetrical (without nodes), and the circular permutation of an even or odd number of particles is the odd or even permutation, respectively, it is clear that J'") with odd n are positive (and with even n cannot be positive). Let us also note that J'"' are c-numbers only in the case where we neglect the coupling between the spin system and vibrational degrees of freedom of the lattice. Generally (see: Thouless, 1965; McMahan, 1972; McMahan and Guyer, 1973; Avilov and Iordanskii, 1976; Guyer, 1974) J(") are operators acting on the phonon variables.

3.2. Spin difision and relaxation First experimental evidence for the presence of the exchange interaction of nuclear spins in 3He crystals was obtained (see: Reich and Yu, 1963; Garwing and Landesman, 1964; Richards et al., 1965, Richardson et al., 1965) and the review article: Guyer et al., 1971) in the study of nuclear magnetic resonance. The relaxation times TI and T2of the longitudinal (with respect to the external magnetic field) and transverse, respectively, components of the nuclear magnetization, and also the spin diffusion coefficient 0,measured in NMR experiments are determined directly by

QUANTUM CRYSTALS

75

the character and magnitude of the spin-spin interaction. If the processes of particle delocalization are absent, the only type of this interaction is the dipoledipole interaction the characteristic energy of which is, by an order of magnitude, p2/a3-1O-'K, where p is the nuclear magnetic moment. This value is, in fact, the natural lower limit of the exchange interaction observed. If the constants f2), 5@', f4) appearing in eqns. (2)-(4) are less than lo-' K, the exchange interaction can be neglected. The dipoledipole interaction plays an important role in the presence of a stronger exchange interaction. The fact is that the Hamiltonian of any exchange interaction commutes with the total spin operator, i.e. any processes caused only by the exchange interaction cannot vary the total magnetization and, therefore, result in its relaxation. Spin diffusion is a process of transfer of magnetic moment in space without changing the total magnetization. Therefore, the spin diffusion coefficient is caused by a stronger exchange interaction. Let 1 / ~ ,be the frequency of spirt-flip processes at some lattice site due to exchange interaction with the neighbours. The time T~ is equal, by an order of magnitude, to h/J, where J is the order of the maximum of the constants in eqs. (2)-(4). Each of the processes consists in the transfer of quantum of the spin projection to a distance of the order of the lattice parameter, so the coefficient of spin diffusion due to the exchange interaction is

0,-a2/rc- a2J/fi.

(5)

Formula (5) is valid for fairly low temperatures. As the temperature increases, the thermally-activated mechanism of particle delocalization, characteristic of ordinary crystals, rather than the tunnelling exchange processes, should become more important (the nature of this mechanism will be considered in sections 5.1 and 5.3). The corresponding diffusion coefficient can also be written in the form D, a2/r,(T), where now the time T J T ) depends exponentially on the temperature T,(T)-exp(c/T) with some activation energy E. Fig. 2 presents experimental data of Reich and Yu (1963) on the temperature dependence of the spin diffusion coefficient in solid 'He for different values of the molar volume. At high temperatures the diffusion coefficient drops exponentially with decreasing temperature, i.e. in fact the thermally-activated diffusion takes place. In low-temperature region the diffusion coefficient does not depend on temperature for large molar volumes, and this constant value can be used to calculate the exchange interaction by a comparison with eq. (5). For 0,cmZ/s eq. ( 5 ) yields J lo-" K, which considerably exceeds the characteristic energy of

-

-

A.F. ANDREEV

76 20r

.

.

,

.

1

.

2248 2205 21.70 21 I0 20 I9

-

1

1

12

.

14

. .

F 1975 G 1947 H 1932

I6

*

1

l

1

18 2 0 2 2

T-'(K-l)

Fig. 2. Temperature dependence of the spin diffusion coefficient in solid 'He for various molar volumes (Reich and Yu, 1963).

dipoledipole interaction. It can be seen from fig. 2 that the lowtemperature limiting value of D,, and hence J, drops rapidly with decreasing molar volume, i.e. with increasing pressure, which is in complete agreement with what was said in section 2. So, the experimental data on spin diffusion show that the exchange interaction of nuclear spins K for takes place in 'He crystals, reaching the value of the order of maximal molar volumes. This conclusion is confirmed by experimental data on the relaxation times T, and T2. It was mentioned above, however, that the exchange interaction, as such, cannot result in the relaxation which, thus, takes place only due to dipole-dipole interaction, but nevertheless the presence of the exchange interaction can strongly affect the relaxation time. The physical reason for the effect of exchange interaction on spin relaxation is, in fact, the same as in the phenomenon of motional narrowing, known for ordinary crystals (see Abragam, 1961). We consider below a simpler case of this phenomenon associated with the transverse relaxation T2.Generally, the situation with the longitudinal time T, is more complicated (see the review by Guyer et al., 1971). The dipole-dipole interaction of a given spin with its neighbours can be considered as the action on that spin of the effective dipole field Hd equal, by an order of magnitude, to p / a 3 .The magnitude of the dipole field fluctuates by a value equal to the field itself, depending on some configuration or other of the spins in adjacent sites. In the field Hd the

QUANTUM CRYSTALS

71

Fig. 3. Temperature dependence of the relaxation time T2 in solid 'He for various molar volumes (Reich and Yu, 1980; Ganvin and Landesman, 1964; Guyer et al., 1971).

spin precesses with a frequency ~ d - p H d / h - ~ 2 / h a 3The . change in the configuration of the adjacent spins due to the exchange interaction o r thermally-activated flips of the spins occurs after the characteristic time T, introduced above. In the case of the exchange interaction 7,-h/J, while in the thermal-activation region T , < ~ / JThe . rotation angle of a spin for the lifetime 7, of a given configuration of the adjacent spins is determined by the formula 6q--ud7,Bp/Ju3, i.e. it is small because it does not exceed the ratio of the dipole-dipole interaction to the exchange interaction. The spin motion is, therefore, a random precession with a step Scp and the characteristic time 7,. The rotation angle for the time f >> T~ is given by the diffusion formula dt)

-[ ( ~ ( ~ ) ~ r / ~ , l ~ ' ~ .

The time of the transverse relaxation T2is determined from the condition

A.F. ANDREEV

78

cp(T2)- 1, i.e.

T i ' - 027,. Despite the dipole-dipc.: nature, the relaxation time T,, as well as the spin diffusion coefficient, is, therefore, inversely proportional to the exchange time T ~ . The experimental data on the temperature dependence of T2 for various molar volumes, shown in fig. 3, are in complete agreement with eqs. (5) and (6) and with the data of fig. 2 on the spin diffusion. Comparing figs. 2 and 3, we can see that experiments on measuring T2 permit the determination of the time 7c and the magnitude of the exchange interaction in a wider region of molar volumes. The maximal value of the exchange interaction K) and the frequency of tunnelling permutations of the particles are much lower than the Debye temperature of the crystal 8 3 2 0 K and the characteristic frequency Q/h of the particles' vibrational motion, respectively. The reason for this smallness is, in general, purely numerical, caused by fairly close packing of particles in helium crystals and does not contradict the general statement about their quantum nature - for other tunnelling processes, e.g. for tunnelling of vacancies (see section 5.1), this smallness is absent. As was shown in section 2, the exchange constants should be equal, in order of magnitude, to Q exp(-A-'), where A is the dimensionless quantum constant whose exact value depends on the particular type of the tunnelling process considered. The quantity A-' is known (Landau and Lifshitz, 1965) to be equal to (2/h)Im S, where S is the action integral taken over the classically inaccessible portion of the trajectory describing K corresponds to a fairly large the process in question. The value parameter A-' 12. For two or more exchange constants to coincide by an order of magnitude, the corresponding parameters A-' have to be equal, roughly speaking, within the accuracy of *l, which is less probable. It is natural to assume, therefore, that only one particular type of exchange plays a considerable role.

-

3.3. Ordering of nuclear spins in 3He Direct investigations of magnetic properties performed recently at ultralow temperatures yielded extensive information on the character of the exchange interaction in 3He crystals. The magnetic susceptibility measurements (Kirk et al., 1969; Bakalyar et al., 1977; Prewitt and Goodkind,

QUANTUM CRYSTALS

79

1977) on the melting curve in the temperature range 10-30mK exhibit the Curie-Weiss law with a negative (antiferromagnetic) characteristic temperature of 8,- (-2.9* 0.3) mK. If the exchange interaction were described by the simplest two-particle Hamiltonian, eq. (2), accounting for the interaction of the nearest neighbours, this would lead to the second-order phase transition into the antiferromagnetic state at 2 mK. The antiferromagnetic transition temperature is, in fact, equal to 1 mK (Halperin et al., 1974, 1978; Kummer et al., 1977). The investigation of antiferromagnetic resonance performed by Osheroff et al. (1980) has enabled the structure of the ordered state to be examined with rather a high accuracy. The antiferromagnetic structure is collinear. The nuclear spins located in the same (100)-type plane are aligned parallel, while in different (100) planes the alternation of the spins upup-downAown is observed (see fig. 4). Such a structure is unusual for bcc antiferrornagnets described by an ordinary Heisenberg exchange Hamiltonian. The characteristic feature of this structure is that it cannot arise from the paramagnetic state via the second-order phase transition. This follows directly from the analysis of possible changes in the symmetry under second-order phase transitions from the bcc-lattice state in orderingalloy-type systems performed by Lifshitz (see: Landau and Lifshitz, 1969). It can easily be seen that the same result is also valid for magnetic

Fig. 4. Magnetic structure of the antifemornagnetically ordered phase of solid 'He (Osheroff et al.. 1980).

80

A.F. ANDREEV

transitions. The experiment (Kumnier et al., 1977;Osheroff et al., 1980) confirms this result and clearly indicates the presence of the first-order transition. The elaborated theory of magnetic properties of solid ’He has been suggested by Roger et al. (1980).They proceed from the assumption of the multi-particle character of the exchange interaction. Since the observed properties of ’He cannot be described by only one type of the exchange, the above authors consider the two-parameter exchange Hamiltonian, containing simultaneously three-particle [see eq. (3)] and four-particle [see eq. (4)J exchange. Agreement with experiment is reached for f4’= -0.355 mK and f3’= 0.1mK. This Hamiltonian permits the explanation of the antiferromagnetic phase structure found by Osheroff et al., and consequently the fust-order phase transition, as well as the high-temperature behaviour of the magnetic susceptibility and magnetic heat capacity. In strong magnetic fields, H>0,4T, the theory (Roger et al., 1980)predicts the existence of a non-collinear ferrimagnetic phase of ’He separated from the paramagnetic state by the line of second-order phase transitions. The experimental observation of this phase would be a convincing argument in favour of the theory under discussion. The experimental evidence for the existence of the welldefined line of phase transitions in strong magnetic fields has not so far been available, though the data (Kummer et al., 1977; Godfrin et al., 1980) on the temperature dependence of the spin entropy reveal a noticeable “ferromagnetic trend”. It is also noteworthy, that the theory of Roger et al. is based on an a priori less probable (as was mentioned in section 3.2) possibility that two different types of exchange are characterized by the same (by an order of magnitude) exchange constants. Therefore, it is now not exceptional that the observed magnetic properties of solid 3He should be explained not only by the exchange interaction, but also by the interaction of spins with other degrees of freedom of a crystal (phonons, vacancies, etc.; see the review by Landesman (1978).

4.1. Diffusion in a gas of impuritons The direct way to detect the delocalization of particles, and not their spins, in a crystal is as follows. Let us consider a 4He crystal containing an

QUANTUM CRYSTALS

81

impurity atom of 'He. Owing to the above tunnelling processes this atom can migrate in the crystal changing places with the matrix atoms. Since the "He crystal is ideally periodical at absolute zero, the states of the impurity atom are specified in terms of the quasi-momentum p. The energy of the system E ( p ) is a periodic function of the quasi-momentum. The situation is completely analogous to the well known case of electrons in metals. The impurity atoms behave like quasi-particles - impuritons (Andreev and Lifshitz, 1969) or mass fluctuation waves (Guyer and Zane, 1970) -freely migrating through the crystal at a constant velocity. The most important characteristic of the impuritons is the width A of the energy band o r the frequency Alh of the tunnelling processes. Since these tunnelling processes are, in fact, the same processes which determined in section 3.2 the characteristic frequency 1 / ~ of , the spin flip, the band width can be estimated qualitatively by assuming A/h to be equal, in order of magnitude, to the spin flip frequency. So,we have A h/rc- lo-" K. We should, of course, bear in mind that this estimate is obtained without due account for the difference in the matrices of 3He and 4He and, in particular, for the fact that at low pressures 3He forms bcc crystals, while 'He - hcp crystals. The characteristic velocity of impuritons is u -aE/ap arc- lo-' cmls. It is noteworthy that the band width A and velocity u of impuritions are considerably lower than all other characteristic energies and velocities in the crystal. We shall see below that this fact makes the dynamics of impuritons rather distinctive. If the concentration of 'He impurities is fairly small, they constitute a rarefied gas of impuritons. Therefore, the simple arguments presented above permit an important conclusion on the character of the diffusion of impurities in quantum crystals (Andreev and Lifshitz, 1969). Namely, the so-called quantum diffusion characterized by the same specific features as the diffusion of particles in gases must take place. To calculate the diffusion coefficient 0,we may use the ordinary formula of the kinetic theory of gases D ul, where I is the free flight path of impuritons. At fairly low temperatures the phonons can be neglected, so the scattering of impuritons from one another plays the major role. The free flight path is I -(no)-' a3/ax,where n is the number of impurities per unit volume, x na3 is the concentration, o is the cross-section of the scattering of one impuriton from another. The diffusion coefficient

-

-

-

-

-

-

D Aa4/hm,

(7)

is, therefore, inversely proportional to the concentration and does not depend on temperature (Richards et al., 1972; Widom and Richards,

82

A.F. ANDREEV

1972; Grigor'ev et al., 1973a, 1973b, 1974; Landesman and Winter, 1974; Pushkarov, 1971, 1974; Yamashita, 1974; Kagan and Maksimov, 1974; Kagan and Klinger, 1974; Andreev and Meierovich, 1975; Huang et al., 1975). It is important to note that the scattering cross-section CT appearing in eq. (7) differs considerably from the square of the interatomic distance a2. This fact is related to the above specific features of the impuriton dynamics which are due to the smallness of the width of their energy band. In fact, let us consider the scattering of two impuritons. The total energy of the system is

E ~=zE ( p J + H p 2 ) + W r d ,

(8)

where p l , p 2 are the impuriton quasi-momenta, E ( p ) is the energy of an isolated impuriton as a function of its quasi-momentum, U(rl2} is the potential energy of the interaction, r12= rl - r 2 ; rl and r2 are the impuriton coordinates. As the colliding particles approach each other from infinity, the sum E ( p J + E(p,) changes, but it can differ from its value as t + --do by no more than 24, because A is the total width of the energy band. Since the total energy is conserved, the potential energy also cannot vary by more than 24. It is clear that the colliding impuritons cannot approach each other to within a distance of less than the interaction radius Ro determined from the relation IU(Ro)l-A. Since A is small compared with all other energies and U(w}= 0, the interaction radius Ro and scattering cross-section a - R g are large compared with a and a', respectively. At great distances the interaction of impurities, as well as any other point defects in a crystal, is the result of elastic interaction. An impurity creates in a crystal some deformation field, with which another impurity interacts. The theory of elasticity (Eshelby, 1956) gives the following expression for the interaction energy: WrlJ = Vo(n)(a/r12)',

(9)

where V&) is the characteristic energy of interaction dependent on the relative orientation of the impurities r = r I 2 / r l 2 .As can be seen from eq. (9), Vo is the energy of interaction of two impurities of 'He in the 4He lattice located at a distance r12- a. It must, therefore, coincide, in order of magnitude, with the temperature of separation of solid solutions of 'He4He, i.e. VO-1O-' K (Wilks, 1967) which agrees with the direct calculation (Guyer et al., 1971) of the interaction of 'He impurities based on formulas of the theory of elasticity.

QUANTUM CRYSTALS

83

10-10

104

10-~

10-2

x

Fig. 5. Concentration dependence of the diffusion coefficient for 3He impurities in 4He crystals with a molar volume of 21 cm3: 1, data of Richards et al. (1972); 2, data of Grigor'ev et al. (1973a).

Using eq. (9), we can find the interaction radius and the cross-section of the scattering of an impuriton from an impuriton

~ , , - - a ( ~ , / d ) " ~ , a-Ri-a2(Vo/d)2/3. The diffusion coefficient is (Pushkarov, 1971, 1974; Kagan and Klinger, 1974; Andreev and Meierovich, 1975)

Fig. 5 shows the experimental data of Richards et al. (1972) and Grigor'ev et al. (1973a) on the concentration dependence of the diffusion coefficient of 3He impurities in an hcp 4He crystal with a molar volume of 21 cm3. The data were obtained by the NMR technique, so, strictly speaking, the spin diffusion coefficient was measured. However, in the case of the small concentrations, considered here, the diffusion of spins occurs only due to the diffusion of impurity atoms themselves, so that both the diffusion coefficients coincide. The experimental points in fig. 5 fit well the solid straight line corresponding to the law Dx = 1.2 x lo-" c m 2 / s . Comparing this law with eq. (lo), we find the band width A-10-4K which agrees with the above estimate based on the NMR data for pure 'He. Moreover, the experiment confirms that the diffusion coefficient is independent of the temperature at T C 1.2 K.

a4

A.F. ANDREEV

Therefore, the experimental data correspond completely to eq. (10) based on the description of impurities as a gas of impuritons. As x + 0, the diffusion coefficient grows infinitely which is in reasonable agreement with the idea of the free motion of isolated impuritons.

4.2. Diffusion of strongly interacting impuritons The applicability of the above gas model is restricted by the requirement that the mean distance between the impurities must exceed considerably the interaction radius, i.e. a / ~ >>Rn ” ~ o r x5. Kramers (1965) observed the damping of second sound Helmholtz oscillations in pure normal flow and obtained results very similar to thermal counterflow in the same apparatus. A more extensive set of pure normal flow measurements have been recently reported by Ijsselstein et al. (1979) who also find results analogous to counterflow. Fig. 43 shows their data for the chemical potential at 1.7 K. Corresponding second sound data are shown in fig. 28. The progression from laminar, to states TI and state TI1 characteristic of thermal counterflow is evident here. Values of y 1 and y2 for the two turbulent states are also in fair agreement with thermal counterflow results.

9.3. Other V, and V, combinations

The extension of superfluid turbulence measurements to general (V,,V,) combinations has produced an almost bewildering profusion of data. It is

212

J.T. TOUGH I

I (

STATE T I I

P

15

o

l

0 0 0 0

A

n

E

STATE T I

0

0

\

0

al 10 C %

'D

00 0

v

?? 3. ala

0 5

0

ooo 0

0

I I

2

Fig. 43. The chemical potential difference A@ for pure normal fluid flow in a circular channel at 1.7 K (from ljsselstein et al., 1979; see also fig. 28). The results are characteristic of thermal counterflow.

possible to correlate these results with those from counterflow and pure superflow in at least a qualitative fashion by a crude application of some ideas previously introduced. Combining the expressions for the line density and the line drift velocity allows the identification of states TI, TZI, and T N to be extended into regions in the (V,, V,) plane. Suppose that the vortex line density can be written as

LA'2 = T(T)V,,

(78)

where r(n is a universal function of temperature and V, is the magnitude of the velocity between the normal fluid and the vortex lines. Using the observation of Ashton and Northby (1975) that the vortex line drift velocity in the superfluid rest frame can be written, eq. (60), as

SUPERFLUID TURBULENCE pKLhi2

213

then gives

where V is the relative velocity between the normal and superfluid. For pure superflow this result, eq. (79), agrees with the result for state TIV, eq. (77), with

For pure normal flow or thermal counterflow, eq. (79) fails to include the quantity V,, eq. (24), but is otherwise qualitatively correct for states TI and TII. The differences between yl, y2, and ys may be the result not only of the “equivalent” nature of the experimental vortex line density (see section 2.1) but of real differences in the drift parameter /3. Independent of these speculations, eq. (79), provides the means by which the states TI, TII, and TIV can be extended into regions in the (V,, V,) plane. The sketch in fig. 44 shows lines of constant L;”d constructed from eq. (79) in a portion of the (V,, V,) plane, at a particular temperature and for

Fig. 44. Lines of constant Li’2d in the (Vn, VJ plane constructed using eq. (79) with relative values of y , . y2, and y4 chosen roughly from experiment. Regions I, 11, and IV refer to the corresponding states of turbulence TI, TII, and TIV. The line density is zero in the shaded region where LA’2d < 2.5.

214

J.T. TOUGH

a channel of a particular size. The relative magnitudes of y,, y2,and y4 are chosen from experiment. The region of the plane where L;l2d

-d.

-0

250

2 0 -100

0

I

2

V, (cm/s) Fig. 45. The chemical potential difference A p measured as a function of superfluid velocity V, at various fixed values of the normal fluid velocity V,. The results are from a circular channel at 1.5 K (from Ijsselstein et al. 1979).

216

J.T. TOUGH

D

v o

0

VCap=3.23cm/s

A A Vcap=4.58cm/s Fig. 46. The chemical potential difference A p measured as a function of the normal fluid velocity V, at various fixed values of the mass flow velocity through the channel, Va,. The results are from a circular channel at 1.326 K (de Haas and Van Beelen. 1976).

configurations. The nature of these different states and their relation to each other will be the focus of experimental and theoretical work in superfluid turbulence in the future.

References Ahlers, G. (1974) Phys. Rev. Lett. 33, 1185. Ahlers, G. and R.P. Behringer (1978) Progress in Theoretical Physics, Suppl. 64. 186. Allen, J.F. and J. Reekie (1939) Proc. Cambr. Phil. Soc. 35, 114. Allen, J.F.. D.J. Griffiths and D. V. Osborne (1965) Roc. Roy Soc. 287, 328. Arms, R.J. and F.R. Hamma (1%5) Phys Fluids 8, 553.

SUPERFLUID TURBULENCE

217

Ashton, R.A. (1977) PhD Dissertation (University of Rhode Island) unpublished, Ashton. R.A. and J.A. Northby (1973) Phys. Rev. Lett. 30, 1119. Ashton. R.A. and J.A. Northby (1975) Phys. Rev. Lett. 35, 1714. Ashton, R.A. and J.T. Tough (1980) Bull. Am. Phys. Soc. 25, 533. Atkins, K.R. (1959) Phys. Rev. 116, 1339. Behringer, R.P. and G. Ahlers (1977) Phys. Lett. 62& 329. Bekarevich, I.L. and Khalatnikov (1961) Sov. Phys.-JETP 13, 643. Bhagat, S.M. and P.R. Critchlow (1%1) Cryogenics 2, 39. Brewer, D.F. and D.O. Edwards (1959) Proc. Roy. SOC. A251, 247. Brewer, D.F. and D.O. Edwards (1961a) Phil. Mag. 6, 775. Brewer, D.F. and D.O. Edwards (1961b) Phil. Mag. 6, 1173. Brewer, D.F. and D.O. Edwards (1962) Phil. Mag. 7, 721. Broadwell, J.E. and H.W. Liepmann (1969) Phys. Fluids 12, 1533. Campbell, L.J. (1970) J. Low Tem. Phys. 3, 175. Careri. G.,F. Scaramuui and J.O. Thompson (1959) Nuovo Cim. 13 186. Careri, G., F. ScaramuPj and W.D. McCormick (1960a) Proc. VII Int. Conf. Low Temp. Phys., Toronto, eds.. G.M. Graham and A.C. Hollis-Hallet (North Holland, Amsterdam) p. 21. Careri, G., U. Fasoli and F.S. Gaeta (1%0b) Nuovo Cim. 15. 774. Careri, G., F. Scaramuui and J.O. Thompson (1960~)Nuovo Cim. 18, 957. Careri, G.. W.D. McCormick and F. Scaramuzzi (1962) Phys. Lett. 1, 61. Careri, G., S. Cunsolo and M. Vicentini-Missoni. (1964) Phys. Rev. 136, 311. Chase, C.E. (1962) Phys. Rev. 127, 361. Chase, C.E.(1963) Phys. Rev. 131, 1898. Chase, C.E. (1965) Superffuid Helium, ed., J.F. Allen (Academic, London) p. 215. Chase, C.E. (1966) Proc. VII Int. Conf. on Low Temp. Physics, eds., G.M.Graham and A.C. Hollis Hallet (North Holland. Amsterdam) p. 438. Childers, R.K. and J.T. Tough (1973) Phys. Rev. Lett. 31, 911. Childers, R.K. and J.T. Tough (1974a) J. Low Temp. Phys. 15, 53. Childers, R.K. and J.T. Tough (1974b) J. Low Temp. Phys. 15, 63. Childers, R.K. and J.T. Tough (1975) Phys. Rev. Lett. 35, 527. Childers, R.K. and J.T. Tough (1976) Phys. Rev. 813, 1040. Cornelissen, P.L.J. and H.C. Kramers (1%5) Proc IXth Int. Conf. on Low Temp. Physics, ed., J.G. Daunt, D.O. Edwards, F.J. Milford and M. Yaqub (Plenum, New York) p. 316. Craig, P.P., W.E. Keller and E.F. Hammel (1%3) Ann. phys. 21, 72. de Haas, W., A. Hartoog, H. van Beelen. R. de Bruyn Ouboter and K.W. Taconis (1974) Physica 75, 311. de Haas, W. and H. van Beelen (1976) Physica 83B, 129. Dimotakis, P.E. (1974) Phys. Rev. A10, 1721. Dimotakis, P.E. and J.E. Broadwell (1973) Phys. Fluids 16, 1787. Dimotakis, P.E. and G.A. Laguna (1977) Phys. Rev. B15, 5240. Donnelly, R.J. (1964) Roc.Roy. Soc. 2814 130. Donnelly. R.J. and P.H. Roberts (1969) Roc. R. SOC. A312, 519. Douglas, R.L. (1964) Phys. Rev. Lett. 13,791. Fetter, A.L. (1963) Phys. Rev. Lett. 10, 507. Feynman, R.P. (1955) Progress in Low Temperature Physics, Vol. 1, ed., C.J. Gorter (North Holland, 1955) p. 17. Fineman. J.C. and C.E. Chase (1%3) Phys. Rev. 124, 1.

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Glaberson, W.I. and R.J. Donnelly (1%6) Phys. Rev. 141, 208. Goner, C.J. and J.H. Mellink (1949) Physica 15, 285. Gd?iths, D.J., D.V. Osborne and J.F. Allen (1965) Low Temperature Physics-LT-9 (Plenum. New York) p. 320. G f i t h s , D.J.. D.V. Osborne and J.F. Allen (1966) Superlluid Helium, ed., J.F. Allen (Academic, London) p. 25. Hall, H.E. (1960) Adv. Phys. 9, 89. Hall, H.E. and W.F. Vinen (1956a) Proc. Roy. Soc. A238, 204. Hall. H.E. and W.F. Vinen (1956b) Roc. Roy. Soc. lu38, 215. Hammel, E.F. and W.E. Keller (1961) Phys. Rev. 124. 1641. Hartoog, A. (1980) Physica B103,263. Henberger, J.D. and J.T. Tough (1981) Phys. Rev. BU, 413. Hoch, H., L. Busse, and F. Moss (1975) Phys. Rev. Lett. 34, 384. Ijsselstein, R.R., M.P. De Goeje and H.C. Kramers (1979) Physica 968, 312. Keesom, W.H.and G. Duyckaerts (1947) Physica 13, 153. Keller, W.E. and E.F. Hammel (1960) Ann. Phys. 10, 202. Keller, W.E. and E.F. Hammell (1966) Physics 2, 221. Khalatnikov, I.M. (1956) J E I T 30. 617. Khalatnikov, I.M. (1965) An Introduction to the Theory of Superfluidity (Benjamin, New York. 1965) Ch. 9. Kramers. H.C. (1%5) in: Supcrtluid Helium, 4.J.F. . Allen (Academic, London), p. 199. Kramers, H.C., T.M. Wirada and A. Broese van Groenou (1960) Proc. VII Int. Conf. Low Temp. Phys., Toronto, 1960, eds., G.M. Graham and A.C. Hollis-Hallet (North Holland, Amsterdam) p. 23. Kramers, H.C., T.M. Wiarda and G. van der Heijden (1973) Physica 69. 245. Ladner, D.R. and J.T. Tough (1978) Phys. Rev. B17, 1455. Ladner, D.R. and J.T. Tough (1979) Phys. Rev. B20,2690. Ladner, D.R., R.K. Childers and J.T. Tough (1976) Phys. Rev. B13, 2918. Ladner, D.R. (1980) private communication. Landau, L.D. and E.M. Lifshitz (1959) Fluid Mechanics (Pergamon. New York). Lucas. P. (1970) J. Phys. C2, 1180. Mantese, J., G. Bischoff and F. Moss (1977) Phys. Rev. Lett. 39, 565. Martin, K.P. and J.T. Tough (1980) Bull. Am. my;. Soc. 25, 533. Mehl. J.B. (1974) Phys. Rev. A10, 601. Mellink, J.H. (1947) Physica 13, 180. Meservey, R. (1962) Phys. Rev. 127, 995. Meyer, L. and F. Reif (1%1) Phys. Rev, 123, 727. Moss, F. (1977) private communication. Northby. J.A. (1978) Phys. Rev. BlS, 3214. Oberly. C.E. and J.T. Tough (1972) J. Low Temp. Phys. 7, 223. Ostenneier, R.M. (1980) private communication. Ostermeier, R.M.. M.W. Cromar, P. Kittle and R. J. Donnelly (1978a) J. Physique 39, C6-160. Ostermeir, R.M., M.W. Cromar, P. Kittel and R.J. Donnelly (1978b) Phys. Rev. Lett. 41, 1123. Ostermeier, R.M., M.W. Cromar, P.Kittel and R.J. Donnelly (1980) Phys. Lett. 77A,321. Peshkov, V.P. and V.J. Tkachenko (1962) Sov. Phys.-JETP 14, 1019. Pahkov, V.P. and V.B. Stryukov (1962) Sov. phy~.-JETp14, 1031.

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Piotrowski, C. and J.T. Tough (1978a) Phys. Rev. B17, 1474. Piotrowski, C. and J.T. Tough (1978b) Phys. Rev. B18, 6066. Peshkov, V.P. (1961) Sov. Phys.-JETP 13. 259. Pratt, W.P., Jr. and W. Zimmermann, Jr. (1969) Phys. Rev. 177, 412. Reif, F. and L. Meyer (1%0) Phys. Rev. 119, 1164. Ruelle, D., and F. Taken (1971) Commun. Math. Php. 20, 167. Rosenshein. J.S., J. Taube and J.A. Titus (1971) Phys. Rev. Lett. 26, 298. Schwarz, K.W. (1978) Phys. Rev. Bl8, 245. Schwarz, K.W. and C.W. Smith (1980) Bull. Am. Phys. Soc. 25, 533. Schlichdng, H. (1951) Boundary Layer Theory (Braun. Karlsruhle). Sitton. D.M.and F. Moss (1%9) Phys. Rev. Lett. 23, 1090. Sitton, D.M. and F. Moss (1972) Php. Rev. Lett. 29, 542. Slegtenhorst. R.P. and H. van Beelen (1977) Physica 90B, 245. Spangler, G.E. (1972) Phys. Rev. A5, 2587. Springett, B.E. (1%7) Phys. Rev. 155. 139. Staas, F.A., K.W.Taconis and W.M. Van Alphen (1961) Physica 27, 893. Swinney, H.L. and J.P. Gollub (1978) Phys. Today 31,41. Tanner, D.J. (1%6) Phys. Rev. 152. 121. Tough, J.T. (1980) Phys. Rev. Lett. 44. 540. Tough, J.T. and C.E. Oberly (1972) J. Low Temp. Phys. 6, 161. Van der Heijden, G.. W.J.P. de Voogt and H.C. Kramen (1972a) Physica 59, 473. Van der Heijden, G., J.J. Gieten and H.C. Kramen (1972b) Physica 61, 566. Van der Heijden, G.. A.F.M. van der Boog and H.C. Kramen (1974) Physica 77,487. van Alphen, W.M., G.J. Van Haasteren, R. de Bruyn Ouboter and K.W. Taconis (1966) Php. Lett. 20, 474. Vicentini-Missoni, M. and S. Cunsolo (1966) Phys. Rev. 144, 144. Vidal, F., M. LeRay. M. Francois and D. Lhuillier (1974) Low Temp. Physics-LT13. Helsinki, 1974 (Plenum, New York). Vinen, W.F. (1956) Conf. de Physique des Basses Temps, LT4, Paris, 1955 (Allier, Grenoble) p. 60. Roy. Soc. AUO, 114. Vinen, W.F. (1957a) ROC. Vinen, W.F. (1957b) Proc. Roy. Soc. AUO, 128. Vinen, W.F. (1957~)Roc. Roy. Soc. A242, 493. Vinen. W.F. (19578 Proc. Roy. Soc. A243, 400. Weaver, J.C. (1973) Phys. Lett. 43,397. Yarmchuck. E.J. and W.I.Glaberson (1979) J. Low Temp. Phys. 36,381.


CHAPTER 4

RECENT PROGRESS IN NUCLEAR COOLING BY KLAUS ANDRES Zenrralinstitut fur Tieftemperaturforschung der Bayerischen Akademie der Wissenschaften, Garching 8046, West Germany and

O.V. LOUNASMAA Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo 15, Finland

Progress in Low Temperature Physics, Volume V l l l

Edited by D.F. Brewer @ Nonh-Holland Publishing Company. 1982

Contab 1. Introduction 2. Brute force nuclear cooling 2.1. Basic equations 2.2. Cooling of conduction electrons 2.3. Cooling of 'He 2.4. Description of cryostats 2.4.1. The Onay machine 2.4.2. The Otaniemi double-bundle cryostat 2.4.3. The cryostat at Bell Laboratories 2.4.4. Other nuclear refrigerators for cooling 'He 3. Hyperfine enhanced nuclear cooling 3.1. Introduction 3.2. Hyperfine interactions in singlet ground state systems 3.2.1. Single ion properties 3.2.2. Hypefine interactions in singlet states 3.2.3. Singlet ground state systems without hyperfine interactions 3.2.4. Singlet ground state systems with hyperfine interactions 3.2.5. Exchange interactions with conduction electrons 3.2.6. Nuclear spin-lattice relaxation 3.3. High field behavior 3.3.1. High magnetic fields 3.3.2. Thermodynamics of hyperfine enhanced nuclear cooling 3.4. Experimental results 3.4.1. S w e y of properties of van Vleck paramagnetic materials 3.4.2. Reparation of praseodymium compounds 3.4.3. General cryogenic techniques 3.5. Description of cryostats 3.5.1. The Munich nuclear orientation cryostat 3.5.2. Cryostat for very low temperature magnetometry 3.5.3. Cryostat for cooling 3He with RNi, 4. Two stage nuclear refrigerators 4.1. The Tokyo cryostat 4.2. The Jiilich cryostat 4.3. The Otaniemi cascade refrigerator 5 . Comparison of brute force and hyperfine enhanced nuclear refrigeration Acknowledgments References

223 225 225 228 231 235 235 237 24 1 243 245 245 246 246 248 250 253 256 256 257 257 258 261 261 262 265 267 267 269 272 274 275 277 280 283 285 285

1. Introdaction Historically, the various pioneering experiments on adiabatic demagnetization have always been performed considerably later than when they were first proposed. This is true for the now classical adiabatic demagnetization of paramagnetic salts, which was suggested simultaneously by Debye (1926) and Giauque (1927) at a time when liquid helium was relatively scarce and when low temperature experimental techniques were not well developed. Soon after the first successful experiments by Giauque and MacDougall (1933) and by de Haas et al. (1933), C.J. Gorter proposed that it should, in principle, also be possible to use the nuclear magnetic moments in nonmagnetic metals for adiabatic demagnetization. Again, this proposal was premature for the then existing experimental techniques, and it was not until much later that Kurti e t al. (1956) performed the first nuclear adiabatic demagnetization of copper. This experiment was a cascade of two adiabatic demagnetizations. Chromium potassium alum was employed to pre-cool a bundle of fine copper wires to 12 mK in a field of 2 T. Under these starting conditions, the entropy of the copper nuclei, equal to R In 4 at high temperatures, was decreased only by 0.5%. Nevertheless, upon reducing the 2 T field to zero, nuclear spin temperatures of a few microkelven were observed for short times. For reviews we refer to Kurti (1982) and to Huiskamp and Lounasmaa (1 973). While this early experiment demonstrated the feasibility of nuclear magnetic cooling, it was clear that efficient refrigeration by nuclear adiabatic demagnetization requires much better starting conditions, i.e. larger initial magnetic fields and lower temperatures. Indeed, when high field superconducting.solenoidsand dilution refrigerators became available about a decade later, nuclear magnetic cooling soon developed into an important technique for refrigeration into the sub-millikelvin region. This method of cooling is also called the “brute force” technique, a somewhat unflattering expression which derives from the fact that the external magnetic field acts as such on the nuclear magnetic moments, without any enhancement caused by an internal polarization in the working substance. Because the nuclear magnetic moments are about 2000 times smaller than their electronic counterparts, 2000 times larger values of the ratio of the initial magnetic field to the initial temperature, &/Ti, are needed to reach the same percentage of entropy reduction AS in the nuclear spin system. With Bi/Ti= 600 T/K we obtain, for instance, ASiS = 5% in

224

K. ANDRES AND O.V. LOUNASMAA

copper; this small amount, however, is large enough for successful experiments. Initial temperatures below 20 mK and magnetic fields in the neighbourhood of 6 T are thus required for nuclear demagnetization. These starting conditions are nowadays readily available. The discovery of the superfluid phases in liquid ’He has made the development of nuclear refrigeration techniques both important and urgent. There is now a clearly defined need to reach temperatures below 1 mK. Many difficult but highly interesting other experiments have also been proposed at ultralow temperatures, notably by Leggett (1978). It is, therefore, not surprising that considerable progress has been made in nuclear refrigeration during the last decade. In nuclear demagnetization experiments it is quite a different matter to cool the nuclear spins only or the conduction electrons and the lattice as well. The situation becomes even more complicated if an external specimen, such as liquid ’He, must also be cooled. The temperature of the nuclear spin system can be reduced quite easily to 1 p K or even less, but in this case the conduction electrons may be several orders of magnitude wanner and the temperature of ’He still higher. These differences are caused by thermal barriers and by the omnipresent external heat leak. Meanwhile, around 1966, a different development started, making use of induced hypefine fields for nuclear refrigeration. This approach is possible for the so-called van Vleck paramagnetic materials, in which the electronic magnetic moments are quenched by crystalline electric fields but can be reinduced in part by external magnetic fields. This may lead to induced hyperfine fields which are much larger than the “brute force” applied field. As a result, the nuclear Zeeman splitting in van Vleck paramagnetic materials can be much enhanced, especially in rare earth compounds. Al’tshuler (1966) argued that the nuclear Zeeman splitting in thuliumethylsulphate, where the Tm3+ ion is in a nonmagnetic singlet ground state, should be nearly as large as the electronic Zeeman splitting. He was the first to suggest that this fact could be employed with advantage in magnetic cooling experiments. Independently, similar conclusions were reached from Knight shift measurements by Jones (1967) in van Vleck paramagnetic praseodymium and thulium pnictides; this led to the first successful hyperfine enhanced nuclear cooling experiments by Andres and Bucher (1968). In section 2 of this review we give a description of the brute force nuclear cooling method. We emphasize the experimentally relevant points, such as the difference between the temperature of the conduction electrons and the nuclear spins. We focus our attention especially on the

RECENT PROGRESS IN NUCLEAR COOLING

225

problem of cooling liquid 3He to the lowest temperatures and describe three cryostats in which 3He has been cooled to sub-millikelvin temperatures. In section 3 we discuss the method of hyperfine enhanced nuclear cooling. As no comprehensive review of this subject exists so far, we give a rather thorough description of the underlying physical principles. We also include a section on the preparation and properties of the various intermetallic van Vleck paramagnetic compounds which are of interest in this context. Again we discuss the performance of some cryostats employing hyperfine enhanced nuclear cooling. In section 4 we describe three two-stage nuclear cooling cryostats of which two use a hyperfine enhanced and a brute force cooling stage in cascade. In the third cryostat two brute force nuclear cooling stages operate in series; nuclear spin temperatures in the nanokelvin region have been generated by this machine. Finally, in section 5 we discuss the individual merits of both the brute force and hyperfine enhanced nuclear cooling methods. In this review we do not describe nuclear demagnetization experiments on CaF, and LiH, pre-cooled by dynamic polarization techniques. We refer to comprehensive papers by Goldman (1977), by Abragam and Goldman (1978) and by Roinel et al. (1978).

2. Brute force andear cooling 2.1. Basic equations

The basic relations describing brute force nuclear cooling are rather throroughly discussed, for instance, by Hudson (1972) and by Lounasmaa (1974). We refer to these textbooks for detailed information. In an external magnetic field B the 21+1 equidistant nuclear energy levels are given by

where pn= 5.05 x lo-'' A m2 is the nuclear magneton, g, is the nuclear g-factor (usually about 2), and rn runs from - I to + I with Z denoting the nuclear spin. The partition function of the system of nuclei is

226

K. AhDRES AND O.V. LOUNASMAA

where n is the number of moles of the sample, k is Boltmann’s constant, and No is Avogadro’s number; nNo is thus the number of magnetic nuclei in the specimen. The population of the mth energy level is given by

The entropy S = kd(T In Z)/aT, in the approximation q,, with equal but possibly opposite amplitudes because the wave functions have the same charge distribution and differ only in their phase. A natural consequence of this requirement is that the expectation value of ( J , ) in singlets is always zero. In cubic symmetry, the magnetic states with nonzero values of ( J , ) are always triplets or quartets, both singlets and doublets being nonmagnetic. For trigonal or tetragonal symmetry, there are only magnetic doublet and nonmagnetic singlet states. In the latter case the Zeeman splitting of the doublet states is often very anisotropic, since the crystal field tries to force the magnetic moment to lie along some symmetry axis, so that it cannot easily follow the magnetic field transverse to this axis. Similarly, the second order Zeeman shifts of the singlet states can be anisotropic. Formally this quadratic shift arises from the mixing of crystal field states by the perturbing magnetic field, and it is given by

Since the magnetic moment m, = -dE/dB, the van Vleck susceptibility xw of the singlet state n along the z-axis is

xw= m J H , = ~

P ~ ~ ~ P ~ B A , ;

(27)

is the permeability of free space (47rxlO-'Vs/Am). The magnitude of xw depends on the matrix elements of operator J between the state n and all other states, as well as on their energy separation from state n. For fields along the x- or y-axis, the corresponding sums A, and A, must be used. Often, the contribution from the first excited state dominates and we can write approximately p,o

is the dominant matrix element of J, between the where ai =(JlolJi ground and the first excited state and A = El - Eo.


248

3.2.2. Hyperfine interactions in singlet states The hyperfine term in the Hamiltonian can generally be expressed as Hhf = Al J,

(29)

where A is a tensor. In rare earth ions, the hyperfine energy A is, to sufficient accuracy, a scalar. The saturation hyperfine field is then given by

Hhf= -A(J)IPoRnPn,

(30)

where gn and kn are the nuclear g-factor and nuclear magneton, respectively. This is the magnetic field that the rare earth nucleus “sees” when the free ion is in its ground state in which ( J , ) = J . The field has a large positive contribution, of order 100 T/pB,from the orbital part of J and a small contribution, of order 1 T/pB, from the spin component of J, which is positive for the first half and negative for the second half of the rare earth series. In a singlet state ( J , ) = O and there is no hyperfine field at the nucleus according to eq. (30). In an applied field H, along the z-axis, on the other hand, a nonzero value of ( J , ) is produced by virtue of the van Vleck susceptibility. The resulting induced hyperfine field is given by -A (J) H f ----

POgn k n

-A h g nP n &@B

xwHa = h,XwHa*

(31)

where hf = - A / p o g n ~ , & p B and K = hrx,,. Parameter K is similar to the familiar Knight shift in metals, and I + K is called the hyperfine enhancement factor. In nonmagnetic metals, however, Knight shifts are typically of the order of 0.01, while K-values for rare earth ions in singlet ground states range typically from 5 to 100. It is this fact, which derives from the large values of A and xw in eq. (31), that explains the attractiveness of certain rare earth intermetallic compounds for nuclear cooling applications. Another way to determine the hyperfine enhancement of the nuclear Zeeman splitting in a singlet ground state is to calculate the change in the wavefunctions of the nuclear substates in the presence of hyperfine interaction. Denoting the electronic singlet state wavefunction with t+bo and the nuclear wavefunctions with \ZJ, the nuclear substates in the


249

singlet state are described by (33) in the absence of hyperfine interaction, i.e. when the electronic and nuclear coordinates are fully decoupled. The term A l * J introduces a coupling between the nuclear substates of different crystal field states, and leads to an admixture, proportional to I, of the 4f angular momentum to the ground state. The new wavefunction now has a nonzero expectation value of the 4f angular momentum given, to second order, by I$t)v

In)

($,I1 J I$,l> = -2A(A,I, +

+4 & ) ,

where the A i are defined by eq. (26). For cubic symmetry, A, A and we can write ($, I ) J

I$,

I) = -2AAI = J f .

(34) = Ay = A, =

(35)

The induced 4f momentum Jf is thus proportional to the nuclear spin I. According to eq. (27), we may express Jf also in the form (36)

J f = -AXwII/-4kk

It should be noted that in all rare earth ions, except gadolinium where the spin part of the hyperfine field dominates, the hyperfine energy A in the Hamiltonian A I.J is negative. In an applied field, the sum of the bare nuclear and the induced 4f angular momenta of the nuclear substates lead to a net Zeeman splitting

En = -gnknlnB(l - A X ~ / C C O ~ ~ C L ~ R J F B )

(37)

which, with -AXVv/FngnknRj/+j = hr~,y = K , leads to the same hyperfine enhancement factor 1 + K as described earlier in eq. (32). Even without any externally applied field, the hypefine interaction causes a shift in energy of the nuclear substates. This is due to coupling of the admixed 4f momenta with the nuclear spin. The shift can also be interpreted as a change in nuclear self-energy by the nuclear spin due to polarization of the 4f shell in the singlet state. In cubic symmetry, where the low field van Vleck susceptibility is isotropic, this polarization and hence also the self-energy are independent of the nuclear spin direction. The change in the self-energy when turning on the hyperfine interaction is then given by

E, =

(' AI - dJf(A)

=(-A2/2~Og~k~)12~n,,

where we have made use of eq. (36).

250


For noncubic crystal field symmetries, the van Vleck susceptibility is generally different in the three principal directions, and eq. (38) must be modified to read

This means that the nuclear self-energy is no longer independent of the orientation of I. The precessing spin in the different nuclear substates generates different 4f polarizations and self-energies, and the -latter depend now on For trigonal symmetry, for instance, we have xZ= XI[,xy= x = xl, and eq. (39) can be written as

c.

E, = ( - A 2 / 2 C L o g ~ C L ~ ) [ ( X , , - X l ) l f + X I I ( f +01.

(40)

This pseudoquadrupole splitting, which is proportional to the anisotropy of the van Vleck susceptibility, is superimposed on the ordinary nuclear quadruple splitting. E, has been known theoretically for some time (Zaripov, 1956; Mneeva, 1963) and it was first observed experimentally by EPR measurements in single crystals of Pr(S04)343H20 (Teplov, 1968).The pseudoquadrupole splitting is often found to be quite large, of the order of several millikelvins, in singlet ground state rare earth compounds with noncubic crystal field symmetries (Bleaney, 1973).

3.2.3. Singlet ground state systems without hyperfine interactions In any crystal containing a sizeable concentration of rare earth ions both dipolar and exchange interactions occur. Their magnitude, as mentioned above, is in general considerably smaller than the crystal field splitting and, consequently, the crystal field states are left intact, at least in the paramagnetic regime. For magnetic (i.e. degenerate) crystal field ground states, magnetic order will start below a certain temperature and some mixing between crystal field states will occur. For singlet states, no magnetic order can appear as long as the exchange interactions remain below a certain critical threshold. For larger values, the energy of the system can be lowered by spontaneously mixing ground and excited states below an ordering temperature. The critical exchange and critical temperature can be derived from the simple molecular field approximation, in which the exchange enhanced susceptibility is given by

x = xo/(l - Axo).

(41)

To simplify the arguments, we assume that the molecular fieId parame-


25 1

ter A is positive as for ferromagnetic interactions. It is related to Jo. the Fourier component of the exchange interaction Jii between ions i and j by

The condition for the beginning of spontaneous polarization at T = 0 is A,= 1/x0or, using eqs. (28) and (42)

Acxo= q, = 4Joa2/A= 1.

(43)

For values of A or q larger than A, or 1, respectively, the transition from the paramagnetic to this so-called induced moment state occurs at a finite temperature, namely when the crystal field susceptibility x(T ) has decreased to l / A . A plot of T versus l/xo thus also indicates the dependence of the transition temperature T, on the molecular field exchange parameter A, as shown in fig. 7 by the solid line. Owing to the weak temperature dependence of the van Vleck susceptibility at low temperatures, T, varies rapidly with h close to its critical value. Likewise, the magnitude of the induced moment at T = 0 is given in this approximation by the field dependence of the susceptibility xo(B = 0): the magnetization rn that develops in the ordered state is given by xo(m)= l/A and can be found as the crossing point of the crystal field magnetization curve with the line 1/A as shown in fig. 8. Due to the weak field dependence of xo, this moment increases rapidly with A near the critical value A,.

Fig. 7. A plot of T versus l/xo. which is identical to the plot of T, versus A. The dashed line shows schematically the latter curve when hyperfine interactions are included.

K. ANDRES A N D O.V. LOUNASMAA

252

8 Fig. 8. Plot of the van Vleck moment m J B ) versus the applied field. The induced moment at T = 0 is detcrmined by the crossing point of the l/A-line with m J B l

A peculiar property of an induced moment transition with near critical exchange interactions, A 2 l/xo, is that the downshift in the ground state energy below T,, which results from self-polarization or level mixing, as well as the thermal energy kT, are usually much smaller than the energy separation to the higher relevant crystal field states. Only a small amount of crystal field entropy is thus left at T,. In contrast to the normal ferroo r antiferromagnetic cases, induced moment transitions usually show only a small specific heat anomaly at T,. By going beyond the simple molecular field approximation and taking into account the crystal momentum ( k ) dependence of the exchange interactions ( J k ) , we can calculate a k -dependent susceptibility X k , which is enhanced by l/(l-xoAk)over the bare crystal field susceptibility x(,. The relation between J k and A k is Ak

(44)

= 2Jk//hg:@i.

Instead of using the enhancement by the molecular field parameter A,., we can describe x k more directly by defining a new crystal field excited state k whose energy gap A to the ground state is, for ferromagnetic interactions, reduced by the factor (1-xoAk). In the k-state the ions fluctuate from the ground state to excited crystal field states in a phasecorrelated way, the phase being given by k r. These fluctuations arise through the coupling of neighboring ground and excited state ions

-


253

through the exchange interaction. In the k-state, with energy Ak = fi%, crystal excitations thus move through the lattice with phase velocity w/k. The dispersion relation o ( k ) for ferromagnetic interactions is such that the k = 0 mode is lowest, and the transition to the induced moment state is characterized by a vanishing energy gap [d(O)=O].In the case of a helicoidal or antiferromagnetic induced moment state, it is a k# 0 mode whose energy gap goes to zero at T,. Theoretically, the collective crystal field excitations in singlet ground state systems were first treated by Trammel1 (1963) and subsequently by others (for a review, cf. Cooper, 1976). Experimentally, their existence was first demonstrated in the induced moment ferromagnet h3TIby Birgeneau et al. (1972) employing inelastic neutron scattering methods. Similar experiments under pressure in the van Vleck paramagnet PrSb by McWhan et al. (1979) have shown that at a pressure of 30kbar the energy of the x-point mode of the Brillouin zone goes to zero and the material enters an induced moment antiferromagnetic state. All neutron scattering experiments show that the lifetime of the collective crystal excitations is rather short, indicating that they are not true eigenstates, i.e. that their crystal momentum k is not really a good quantum number. This is presumably due to the neglect of the ion-lattice forces and interactions between the collective excitations themselves.

3.2.4. Singlet ground state systems with hyperfine interactions The effect of hyperfine interactions on a system of exchange coupled singlet ground state ions is again easy to see in the molecular field approximation, especially when assuming ferromagnetic interactions. The single ion susceptibility now consists of the sum of the temperature independent van Vleck susceptibility xo plus the enhanced nuclear susceptibility ,yne0,which follows Curie’s law [cf. eq. ( 5 ) ] xn.o= ~ o ( g i c ~ : / 3 k T ) I+( I1)(1+ K O ) ’ = A*/T

(45)

Here 1 + K O is the hyperfine enhancement factor in the absence of exchange interactions. The exchange enhanced total susceptibility is then given by (Andres et al., 1975a)

It is clear that even for undercritical exchange interactions

(Axo< 1)

K . ANDRES AND O.V. LOUNASMAA

254

there will be a low temperature at which x diverges and self-polarization begins. The nuclear susceptibility can thus always make the singlet ground state unstable against self-polarization. The transition temperature in this region is approximately given by

T, = AA*/(l

-

Axo)= A A * / ( l -

q),

(47)

where q = Axo is the critical exchange parameter discussed above. For q AI(Jf),

(48)

which again means that the nuclei remain disordered just below T, and only align at lower temperatures. This behavior, which was also predicted by Triplett and White (1973), is characteristic of near-critical electronnuclear magnetic ordering phenomena in singlet ground state systems. Interestingly enough, it has so far not been investigated experimentally in detail. For much smaller values of q, the transition temperature becomes linearly dependent on A or q. The induced moment below T, eventually approaches the value of the single ion hyperfine induced 4f moment JI [cf. eq. (36)]. In this case, the inequality in eq. (48) is reversed. The hyperfine coupling now dominates the exchange coupling, and the transition to the ordered state consists of a spontaneous alignment of hypefine induced 4f momenta, which are proportional to the nuclear spin I. The transition should then be similar to other magnetic transitions of exchange coupled localized momenta, and we would, for instance, expect a


255

large drop in the nuclear entropy at Tc and a corresponding anomaly in the specific heat. According to eq. (48) the borderline between cooperative nuclear and nuclear induced electronic transitions is given by

kTc=AI(Jf).

(49)

Using eqs. (36) and (28) this can also be expressed as

kTc= A2122a21A.

(50)

Nuclear ordering phenomena in van Vleck paramagnetic materials have been studied experimentally mostly in praseodymium intermetallic compounds. For P?+, we have A/k = 52 mK, I = 5, and a typical value of -2Aa2/A is 0.01; cooperative nuclear transitions can thus be expected to occur only below about 4mK. Indeed, such transitions have been seen calorimetrically in PrCu, at 2.3mK (Babcock et at., 1979) and in PrNi, at 0.40mK (Kubota et at., 1980). Experimentally it is difficult to determine the true sharpness of the transitions, owing to the long thermal relaxation times near and below T,. We should mention that Kubota et al.3 detailed analysis of the PrNi5 data does not reveal a net nuclear quadrupole splitting, a result which is consistent with NMR data on single crystals of PrNi, (Kaplan et al., 1980). It is not yet known whether this unexpected result is due to a cancellation of the bare and of the pseudoquadrupole splittings, or whether it is caused by the combined effects of the anisotropic crystal field and exchange interactions. Another theoretical approach, first used by Grover (1965), is to start from the collective crystal field states discussed in section 3.2.1 and treat the hypefine interaction as a second order perturbation. This leads, for each collective mode k, to additional interactions of order (A3/A)( l/r3) between any pair of nuclei separated by the distance r. It must be mentioned that even at T = 0 there is a zero point population of the collective modes. The sum over all values of k leads to an effective exchange interaction between nuclei, in a manner very similar to the Suhl-Nakamura (Suhl, 1959) interaction in an antiferromagnet; in this case the internuclear interaction is mediated by the zero point spin wave excitations. Using this scheme, Landesman (1971) calculated the nuclear ordering temperature in a van Vleck paramagnet in the limit of weak exchange interactions. For near critical exchange this method no longer works, because the energy gap in the collective crystal field excitation

256


spectrum becomes too small for the perturbation calculation to be applicable.

3.2.5. Exchange interactions with conduction electrons The exchange interaction with conduction electrons is mainly responsible for the short nuclear spin-lattice relaxation times that are observed in metallic singlet ground state systems. This process is discussed in more detail in the next section. It is also of interest how this interaction affects the single ion properties in a dilute metallic system. The 4f-5d exchange energy is often of the same order (-0.1 eV) as the crystal field splitting, but its effect on J (or rather on the projection of S on J) is much attenuated by the comparatively weak polarizability of the 5d electrons, which either form 5d virtual bound states or 5d bands. The shifts of the 4f crystal field levels, due to this exchange interaction Jf4, are proportional to the expectation value I(J,>I of each level and are thus zero for a singlet state. Again, however, there is a value of Jf4 above which it pays to spontaneously polarize even an isolated singlet state ion. This critical value is given approximately by

Usually, the smaller d-spin susceptibility xd dominates in eq. (51) and makes Jcri,.much larger than the critical exchange energy Jo in concentrated singlet systems [cf. eq. (43)].

3.2.6. Nuclear spin - lattice relaxation In rare earth ions the hyperfine coupling energy A determines a fastest time, of order 7 = A/h = 1ns, with which the nuclear spin can follow the fluctuations of the 4f electronic moment. In metallic compounds relaxation times of the local 4f momenta are typically in the region from microto nanoseconds at liquid helium temperatures, as has been demonstrated by various EPR measurements on Gd” diluted in such matrices. However, in a singlet ground state, where the electronic moment is quenched, the nuclear spin-lattice relaxation time T~ may be longer. In fact, T~ can be expected to depend strongly on temperature when, upon cooling, magnetic excited states are thermally depopulated.


257

Exchange interactions will generally contribute to a shortening of 7 , through fluctuations of thermally excited crystal field states. At low temperatures, where only the ground state is populated, T~ is given by the exchange coupling between the hyperfine induced 4f momenta [cf. eq. (36)] and conduction electrons. Consequently, 7 , should obey a Korringatype temperature dependence [cf. eq. (16)], where the constant K is several orders of magnitude smaller than it would be in the absence of hyperfine interactions. Explicit calculations for T~ in this case have been done by Tsarevski (1971), who obtains for PrBi K = 3 ps K. While nuclei of van Vleck paramagnetic ions in metallic hosts thus relax much faster than nuclei of ordinary metals, the same is not true in the case of nonmetallic materials. At low temperatures, where exchange fluctuations via excited magnetic crystal field states are no longer important, T , will be controlled mainly by the phonon modulation of the dipolar and exchange interactions between the hyperfine induced 4f momenta; the relaxation time can thus be expected to become much longer. Experimental information on 7 , and its temperature dependence in van Vleck paramagnetic compounds is still scarce. From NMR linewidth measurements on PrNi5 by Kaplan et al. (1980) one can estimate 7,’ 2 ps at 4.2K. Recent spin echo measurements by Satoh et al. (1981) in Pr,-*La,In3 between 1.2 and 4.2K yield a Korringa constant K = 270* 10 p s K for the praseodymium nuclei, independent of the lanthanum concentration.

3.3. High field behavior

3.3.1. High magnetic fields The hypefine enhancement of the nuclear susceptibility is reduced in high magnetic fields both because of the decrease in van Vleck susceptibility with increasing field and because of the paramagnetic saturation effect. If we denote the van Vleck magnetization in high fields by m,(B), the field dependence of the energy of the nuclear substates is given by

&(I?)

= -Bm,(B)

m,(B) + AZ, ~-

gJk

m,B,

where m, is the “bare” nuclear magnetization.

(52)

258


For the net magnetic moment of the substates we obtain m = - - =dE, m,(B)-AZ,

dB

dmw(B)ldB gJpB

+ mn

Apart from the van Vleck moment m,,(B), this is the same enhanced nuclear moment as in eq. (37), except that xl, is now the differential van Vleck susceptibility. Since xk always decreases in high magnetic fields, the hyperfine enhancement factor, or the magnitude of the hyperfine induced 4f momenta, also decrease with increasing fields. The paramagnetic saturation effect in the enhanced nuclear susceptibility sets in when the thermal energy is comparable to the enhanced nuclear Zeeman splitting, i.e. when

These effects were first corroborated experimentally by Genicon (1978) through magnetization measurements on PrCu, in high fields and at low temperatures.

3.3.2. Thermodynamics of hyperfine enhanced nuclear cooling In the paramagnetic regime the nuclear angular momentum remains a good quantum number and we can always write the free energy as a sum of the crystal field, the nuclear, and the usual lattice and electronic contributions. For nuclear cooling experiments, the temperature region of interest is below k T s g n p n B ( l +K ) , which is typically below 100mK for applied fields up to 10T. In the van Vleck paramagnetic regime at low temperatures the crystal field part of the free energy is independent of T, since excited crystal field states are no longer populated. -There is, however, still a strong field dependence of the free energy owing to the quadratic Zeeman shift on the singlet ground state. This effect is usually considerably larger than the free energy change due to the nuclear polarization or repopulation of the nuclear substates. It is of great practical importance that the quadratic free energy change occurs in a reversible way, ensuring that the van VIeck magnetization is a completely reversible function of the magnetic field. Even slight irrever-


259

sibilities can lead to an amount of heat production during demagnetization which is comparable to the nuclear heat of magnetization. Possible sources of such irreversibilities are magnetostrictive effects in strained polycrystalline samples or phase impurities. Since most singlet ground state materials are ordered compounds, the requirement for phase purity is especially important; neighboring phases are often magnetically ordered and exhibit irreversible magnetization curves. The thermodynamics of hyperfine enhanced nuclear cooling is the same as that described for the brute force method [eqs. (1-1411,except that the nuclear Curie constant A* [eq. ( 5 ) ] is now enhanced by (1+K)2.This factor, which is often as high as 200, naturally enhances the initial cooling entropy [cf. eq. (4)]a great deal, which is the main reason why nuclear cooling by means of van Vleck paramagnetic materials is so attractive. The requirements for high initial fields and low starting temperatures are thus relaxed very considerably; some numerical values are given in table 2. Corrections in calculating the cooling entropy can arise from a reduction of the hyperfine enhancement factor in high magnetic fields or from

Table 2 Properties of some van Vleck paramagnetic compounds. The effective residual field b and the cooling entropy density S/V are given for polycrystalline samples. T, is the nuclear ordering temperature XW

Crystal symmetry

(molar SI units)

mu,

Cubic Cubic Orthorhombic

PKUS

Hexagonal

PrPt,

Hexagonal

PrNi,

Hexagonal

cu

1.26 0.51 0.955 II= 13.2 1 = 1.88 11 = 0.88 I =2.70 (1 = 0.477 1 = 1.03

Cubic

Compound

R-b Pr Be 13

-

b

s/ v

T,

1+ K

(T)

(J/Km?

(mK)

20 8.7 15.3 198 29.1

0.053

0.11 0.033 0.096

1.o 0.06'

14.'

41.3

'"

16.4 1

0.136 0.45

o.21

2.4 40 (ferrom.)

0.11

0.155

-

0.065

0.102

0.00034 0.0022

"Calculated value. The calculated ordering temperature of PrBe,, given by Andres et al. (1978) was 8 times too high because the exchange integral from Bloch et al. (1976) was that between one praseodymium ion and its 8 nearest neighbors.

260


its anisotropy. In the latter case it is necessary to calculate the average of the square of K. For trigonal symmetry, this value is related to the average by a2+(4/3)a + 1

Wz)= (9/5)W2 4a2+4a

+

(55)

where Q = xl/xll is the ratio of the susceptibilities normal and parallel to the uigonal axis. The presence of a nuclear pseudoquadrupole splitting has an effect similar, but not exactly equal, to a residual field. The lowest temperature that can be reached after demagnetization to the field Bf is given approximately by

where the subscript i again refers to the initial conditions and b can be an effective residual splitting field or an exchange field. This relation is similar to eq. (9). For ferromagnetic interactions, it is possible to cool below the nuclear ordering temperature, as experiments on PrCu, (T, = 40 mK, Andres et al., 1975a) and on PrNiS (T,= 0.40 mK, Kubota et al., 1980) have shown. For antiferromagnetic interactions the situation is less clear. One important difference, compared with brute force nuclear cooling on copper, is the much shorter nuclear spin-lattice relaxation times that are encountered in hyperfine enhanced cooling materials, as pointed out above. Often T~ is so short (of order 10 ks at 1 K) that it cannot easily be observed experimentally. The electrons are thus always in local thermal equilibrium with the rare earth nuclei and the factor actually limiting the cooling power of the system is the electronic thermal conductivity. Since the materials in question are all ordered compounds, their thermal conductivity is, in principle, only a function of their purity. So far the highest thermal conductivities observed are 0.5 W/Km in R N i 5 (Folle et al., 1981); this value is about 1000 times worse than for copper. Therefore, in contrast to brute force nuclear cooling, where the lowest conduction electron temperature is given by the heat leak and the nuclear spin-lattice relaxation time Ed. eq. (19)], the minimum electronic temperature is now usually determined by the heat leak and by the thermal conductivity, at least for temperatures higher than the cooperative nuclear ordering temperature (cf. section 5 for a comparison of cooling powers in the sub-millikelvin range). Heat Bow to the cooling pill raises


26 1

its surface temperature and it is, therefore, advantageous to minimize Q per unit surface area by a suitable geometry of the cooling pill (cf. section 3.4.3). 3.4. Experimental results 3.4.1. Survey of properties of uan Vleck paramagnetic materials

Singlet ground states are most often found in praseodymium and thulium compounds, since for these ions the spin component S and its projection onto J are smallest among all non-Kramers rare earth ions. This leads to exchange energies which are often smaller than crystal field splitting energies so that singlet ground states can remain stable. If the crystal structure and the position of the neighboring ions as well as their charge are known, the crystal field splitting and the crystal field ground state can in principle be calculated, for example, by using the operator equivalent method of Stevens (1952). Often, however, especially in intermetallic compounds, such calculations do not predict the experimentally observed ground state. This is mainly due to two reasons: first, the neglect of the contribution due to the on-site 5d electron charge on the crystalline electric field and, second, the effective, usually negative charge on the ligand ion is often not well known. To a first approximation the effective ligand charge is related to the difference in the electronegativity of the rare earth and the ligand ions. This was shown for the first time in a systematic and comprehensive study by Bucher (1973). In practice, the first characterization of a van Vleck paramagnetic material is always obtained by means of a magnetic susceptibility measurement. The absence of any anomalies characteristic of magnetic ordering phenomena and the temperature independence of susceptibility at low temperatures ascertain a nonmagnetic singlet ground state. If the crystal field symmetry is known, the matrix element a of the angular momentum operator J between the ground and the first excited state is also given. By means of a specific heat measurement we can obtain an independent estimate of A from the low temperature end of the Schottky anomaly, according to

262


Here go and g, are the multiplicities of the ground and excited states, respectively, and A is their energy difference. Eqs. (28), (41), (43), and (57) then yield an estimate of the critical parameter q and indicate whether the singlet ground state is stable (lql< 1) or unstable ( q S1) against spontaneous self-polarization at still lower temperatures. Another way of determining A is inelastic neutron scattering, where the neutron magnetic dipole field induces transitions between the crystal field states. The energy difference between these states can then be obtained directly from an energy loss or an energy gain spectrum of the inelastically scattered neutrons. It is also possible to extract A from an analysis of the temperature dependence of any material property that reflects on which crystal field state the rare earth ion is in. Examples are the electrical resistivity, because the different crystal field states have different potential and spin scattering cross sections, and the elastic constants, because the individual crystal field states couple differently to lattice distortions. Luthi (1980) has shown that in many singlet ground state rare earth compounds the ultrasonic velocity shows pronounced anomalies at low temperatures from which, by theoretical analysis, the separation of the lowest crystal field states can be deduced. Table 2 summarizes data on various metallic van Vleck paramagnetic compounds: Cases with near critical o r over-critical exchange interactions, which show transitions to an induced moment state above the millikelvin range, have been omitted.

3.4.2. Preparation of praseodymium compounds Praseodymium metal often contains hydrogen and oxygen in the form of hydrides and oxides. When preparing intermetallic compounds by mixing and melting the constituents, the presence of these impurities leads to off-stoichiometry and results in phase impurities. This is particularly harmful if the impurity phase is magnetically ordered, because it then leads to an excess nuclear specific heat in zero field at low temperatures, as well as to irreversible heat production during magnetization and demagnetizat ion. In the first experiments on hyperfine enhanced nuclear cooling the praseodymium metal and the compounds made from it typically had residual resistivity ratios of 15 and 7, respectively. This led to poor thermal conductivities and thermal equilibrium times of order 1 h at 2 mK in the cooling pills. The best praseodymium metal which is currently


26 3

available from the Rare Earth Research Institute in Ames (Iowa, USA) has a residual electrical resistivity ratio of about 60, which significantly improves the thermal conductivities of compounds made from it (cf. table 2 and section 3.4.1). A good quality test of cooling compounds is a magnetization measurement at liquid helium temperatures. Starting in low fields, the magnetization should initially be a linear function of the field and should be reversible after applying large fields, i.e. remanence should be absent. Since molten praseodymium has a low vapor pressure, most intermetallic compounds can be prepared by various techniques, such as by melting in an arc furnace, in a tantalum tube in a vacuum furnace, o r in an induction furnace. Crucibles of tantalum or tungsten, if feasible from the metallurgical point of view, should be preferred over sintered ceramic crucibles, since the latter usually react with praseodymium. When preparing intermetallic compounds with heavy metals, such as F’rT13, it is good practice to stir the molten liquid in order to prevent enrichment of the heavy metal at the bottom of the container. In an arc or induction furnace such stirring is always present by eddy current forces. In resistively heated tantalum tubes stirring can be provided by positioning the tube horizontally and by making it slightly movable, as shown in fig. 9. By casting the melt into tantalum tubes one can make alloys in the form

VACUUM JAR

PINCHED To TUBE WITH MOLTEN SAMPLE

O-RINGS TO FOREPUMP CURRENT LEADS [COPPER 1 TO VAC ION PUMP

Fig. 9. Vacuum oven with a movable tantalum crucible.

264

K . ANDRES A N D O.V. LOUNASMAA

of long, narrow rods, which is usually the best geometry because of the higher surface to volume ratio as discussed earlier. It is also possible to obtain such a shape in an arc furnace which has long and narrow grooves in its copper hearth, especially if the surface tension of the melt is not too high. When making praseodymium rods this way, it is observed that higher purity material has a lower surface tension and is much easier to cast. When using inferior grade metal, a skin can be observed on the surface of the melt which increases the surface tension and makes casting of narrow rods difficult. An alternative way of casting rods in an arc furnace under such circumstances is shown in fig. 10 (Andres, 1978). A hi-arc furnace (from Centorr Associates Inc., Suncook, N.H. 03275, USA) was modified by building a movable rod into its copper hearth. By pulling the melt into the cold hearth it is possible, for instance, to make uniform rods of PrNi5, 5 mm in diameter and up to 5 cm long. In table 3 we list some relevant properties of various van Vleck paramagnetic praseodymium compounds that have been used for nuclear cooling. The various ways in which they can be prepared are also indicated.

O-RING SEALS BRASS HOUSING ELECTRODES

- OUARTZ WINDOW ARGON ARCS MELT WATER COOLING BRASS HOUSING COPPER HEARTH SUPPORT

COPPER PULL ROD

Fig. 10. A ui-arc furnace equipped with a copper pull rod for casting cylindrical samples.

RECENT PROGRESS IN NUCLEAR COOLlNG

265

Table 3 Some further properties of van Vleck paramagnetic compounds.The thermal conductivity u is given at 1 K

Compound

Method of preparation

Melting temp. (K)

pm,

Closed Ta crucible

1375

6.7

Ultrasonic soldering

Rather poor

Argon arc furnace

2175

0.7

Ultrasonic soldering

Good

PrBe13

Prcu,

Ta crucible, argon arc furnace

1235

0.17

Regular soldering

Good

Prcu,

Ta crucible, argon arc furnace

- 1100

Regular soldering

Good

PrPt,

Argon arc furnace

-2020

Regular soldering

PrNi,

Argon arc furnace

1638

u(W/Km)

0.5

Thermal con- Chemical tact to Cu stability

Good

Regular soldering

3.4.3. General cryogenic techniques Since the technique of hypefine enhanced nuclear cooling poses less stringent requirements on initial magnetic field Bi and the starting temperature Ti than the technique of brute force nuclear cooling, considerably more flexibility in the design of cooling stages is available. It is, in particular, often possible to modify existing refrigerators by adding a hyperfine enhanced nuclear cooling stage, which can be small in size. Conversely, larger cooling stages can be built either to effectively pre-cool brute force nuclear stages or to allow experimentation in the millikelvin range for long times. Special care must be taken to insure good thermal contact to the cooling compound. Best results are generally obtained by using soft solder joints with pure cadmium metal, which has a superconducting critical field of 3mT. Normally this field is small enough in order not to interfere with demagnetization. For hT13and PrBe,,, however, soldering must be done in an inert atmosphere using an ultrasonic soldering iron, as indicated in table 3. It is of advantage to assemble the cooling pill in the form of a bundle of

266


long, thin rods, with each of the rods soldered to copper wires. This geometry minimizes the heat flow per unit surface area and reduces the temperature difference between the surface and the center of the cooling rods. It also leads to a shorter thermal equilibrium time T, which is approximately given by

here C is the specific heat per unit volume, p is the thermal resistivity, and d is the longest dimension of the rod through which heat has to travel. As the nuclear specific heat is high, both before and after the demagnetization [d. eq. (6)],and the thermal conductivity l / p is often low, equilibrium times of several hours can be encountered around 1 mK. The shortest times so far, T = 5 min at 1 mK, were observed by Mueller et al. (1980) in PrNi, supplied by the Rare Earth Research Institute in Ames. Entropy increase caused by eddy current heating during demagnetization can usually be neglected compared with the large cooling entropy. Observed losses of polarization are often independent of the speed with which the field is swept, at least for rates not exceeding 400mT/min; losses are usually due to traces of phase impurities. The relatively rapid sweep speeds that are possible allow a simplified design of the switch isolating thermally the cooling pill: the switch, usually a ribbon of tin, can be located in the fringe field of the main magnet and it stays in the normal state during the first $ of the total sweep, without a significant increase of entropy during cooling. Adiabatic suspension of the cooling pill in order to obtain a low heat leak, the low temperature thermometry, and shielding against radiation and vibrations can be done in the same ways as for the classical demagnetization apparatus. Mechanically stiff supports are generally preferable. The use of plastic materials should be kept to a minimum owing to the difficulty of cooling them to low temperatures. If helium exchange gas is employed for precooling the apparatus to 4.2 K, the nuclear stage should be surrounded by at least two heat shields in order to minimize the condensation of gas residuals on the cold pill. It is important to shield both the pumping line vibrations and excessive acoustic noise from the cryostat. High frequency electromagnetic radiation, which usually affects only the resistance thermometers and rarely causes a direct heat leak into the nuclear stage, is best attenuated by low-pass filters in the electrical leads inside the cryostat.


267

3.5. Description of cryostats In what follows, we describe briefly the construction and performance of three hyperfine enhanced nuclear cooling cryostats in which PrCu, and PrNi, were used. The first two machines are examples of small cooling stages which were afterwards added to existing cryostats.

3.5.1. The Munich nuclear orientation cryosrat A classical demagnetization refrigerator, which employs 0.75 kg of chromium potassium alum and which was used by a nuclear physics group in Garching to cool routinely radioactive samples to 15 mK, was modified by adding a small PrCu, stage (Andres et al., 1975b). The arrangement is shown in figs. 11 and 12. Because the aim was to cool radioactive samples to temperatures as low as possible, the quality of thermal contact between the specimen and the nuclear cooling pill was of prime importance. The latter consists of three rods of PIC&, each 6 mm in diameter and 4 cm long and with a total weight of 19.5 g (0.045 mol); the rods were cast in tantalum tubes in a high vacuum furnace. Copper cold-fingers were soldered to the PrCu, rods (6.fig. 12) by means of indium metal using a regular flux. Temperatures were measured exclusively by employing y-ray anisotropy thermometers. Both the samples and the thermometers were in the form of thin metal foils and were attached to a cold-finger by means of Ga-In liquid eutectic alloy. During operation the first stage is demagnetized from 1.2 K and 3.6 T in about 30 min to zero field. The second stage, in a field of 2.4 T, then cools to 25mK in about five hours. After the second stage has been demagnetized to zero field in another 30 min, end temperatures around 2.5 mK are reached in the samples. A typical warm-up curve of the 6oCoNi thermometer is shown in fig. 13; it corresponds to an average heat input into the second stage of about 7 n W . The diagram in fig. 14 shows a rather large zero field entropy, which is mostly due to the nuclear pseudoquadrupole splitting (= 6 mK overall) resulting from the anisotropic van Vleck susceptibility in this material. The lowest temperature data in fig. 14 were actually obtained in a different cryostat with an AuIn, susceptibility thermometer and with a considerably smaller heat leak (Andres and Bucher, 1972). PrCu, is an ideal cooling material if large amounts of heat must be removed between 2 and 4 m K .

268


R-

CHARCOAL TRAP SHUT-OFF VALVE ALLAN BRADLEY RESISTOR

GUARD SALT SUPERCONDUCl'ING SOLENOID WORKING SALT

COIL FOILS GUARD SALT SUPERCONDUCTING SWITCH

r

m---

HELMHOLTZ SOLENOID

SAMPLE Ge(Li) DETECTOR

PrCu6 RODS PrcUrjSOLENOID

Fig. 11. Schematic view of the Munich cryostat with chrome-alum as the first and PrCu, as the second cooling stage (Atdres et at., 1975b).


269

SPEER CARBON RESISTOR SUPERCONDUCTING SWITCH SWITCH SOLENOID SPEER CARBON RESISTOR

HELMHOLTZ SOLENOID

COPPER STRIPS

Pr Cu6RODS COPPER STRIPS

5 cm Fig. 12. The Prcu, m l i n g stage of the Munich cryostat.

3.5.2. Cryostat for very low temperature magnetometry As another example (Andres et al., 1981) of the versatility of hyperfine enhanced nuclear cooling we show how the range of a magnetometer built into a dilution refrigerator has been extended from 15 to 1.5 mK by the addition of a small PrNiS stage (cf. fig. 15). A copper cold-finger was connected to the mixing chamber via a superconducting heat switch made


270

Fig. 13. A typical warm-up curvc for the '"CoNi nuclear orientation thermometer in the Munich cryostat.

of tin. The samples were attached to the cold-finger by means of Apiezon grease. The magnetization can be measured by a set of field and detection coils movable in the vertical direction. The PrNiS cooling stage (two bars of 35 g total weight) was soldered with cadmium to the lower end of the cold-finger. In order to keep the 1

1.u

I

I

I

*

-

IU

-

F

IUU

IUUU

TlmK)

Fig. 14. Entropy diagram of RCu,; the 2 T and 6 T curves are calculated. The zero field curves are from Andres and Bucher (1972. solid line) and from Babcock et al. (1979,

dashed line).

27 1


SC SHIELD SC PICKUP COILS

~

~

~

~

~

~

SC SOLENOID

Fig. 15. Schematic view of a magnetometer cryostat with a PrNi, cooling stage (Andres et al., 1981).

~

E

T

212

K . ANDRES A N D O.V. LOUNASMAA

indium shield around the pickup coils always in the superconducting state, it is important that the fringe field of the demagnetization solenoid is kept low. In the configuration shown in fig. 15, this means that the demagnetizing field must be less than 1.5 T. In spite of this low field and with a starting temperature of 17 mK, one reaches in the cold-finger end temperatures below 2 mK that can be maintained for about one hour. The first studies of the spin susceptibility of localized donor states in phosphorus doped silicon at very low temperatures have been carried out successfully with this apparatus.

3.5.3.Cryostat for cooling 3He with PrNis Fig. 16 shows a dilution refrigerator (model DRI 236, SHE Corporation, San Diego, California) with a built-in PrNi, cooling stage. The apparatus was used at the Bell Laboratories for the first specific heat measurements in the B-phase of superfluid 'He at low pressures (Andres and Darack, 1977). PrNi, has become the most widely used hyperfine enhanced cooling material, both because of its favorable physical properties and because of its good chemical stability and ease of handling (cf. tables 2 and 3). Although PrNi, has a hexagonal crystal structure, the anisotropy of the van Vleck susceptibility is not very large (xl=2x11) and leads to an estimated nuclear pseudoquadrupole splitting of only 1.3 mK. The observed total quadrupole splitting is, in fact, considerably smaller, for reasons which are not yet understood. Exchange interactions are small and produce nuclear ferromagnetic order only at 0.40 mK (Kubota et al., 1980). The material is thus useful for cooling into the sub-millikelvin range (Andres et al., 1974; Mueller et al., 1980). The mixing chamber of the dilution refrigerator contains a high surface area heat exchanger ( A = 10 m2) in the form of 100 silver wires coated with sintered silver powder. The cooling pills consist of seven rods, 6 mm in diameter and 5 cm long, and of total weight of 115 g (0.26 mol). The rods were cast in an argon arc furnace which was suitably equipped for extruding the molten material through the bottom of the copper hearth by means of a pull rod, as mentioned in section 3.4.2. The cooling pill was assembled by tightly packing the seven rods, by placing 1 rnm diameter copper wires into the open spaces between the rods, and by dipping the whole assembly in a bath of molten cadmium. This pill can typically be precooled overnight to about 17 mK in a field of 4T.After demagnetizing to 5 mT, end temperatures of 0.7 mK were


213

SC SOLENOID SC THERMAL SWITCH HEAT EXCHANGER GRAPHITE ROD MIXING CHAMBER SILVER SINTER GRAPHITE RODS

MAIN SC SOLENOID

PrNig RODS

Cu WIRES SUPPORT FRAME 1 K SHIELD VACUUM CAN

SC THERMAL SWITCH

SILVER CELL WITH Sll-VER SINTER

SILVER ROO Au In2-SAMPLE SOUID PICKUP COIL

SC SOLENOID

Fig. 16. A dilution refrigerator equipped with a RNi, amling stage for investigating supeduid 'He at Bell Laboratories (Andres and Darack, 1977).

214


T (mK)

Fig. 17. Entropy diagram of PrNi,; the 2 T and 6 T solid curves are calculated. The dash-dot line and the low field solid curves are from Folle et al. (1981); the dashed line is from Andres and Darack (1977).

observed with an AuIn2 susceptibility thermometer about two hours after the end of demagnetization. With a cell of 4 an3, 1.0 mK has been reached in 3He. For specific heat measurements, the liquid was thermally connected to the cooling pill through a small sintered silver heat exchanger of 0.6 m2 surface area, and to the AuIn, susceptibility thermometer via a similar heat exchanger of 6 m Z surface area. This construction resulted in a much longer thermal relaxation time between the 3He and the cooling pill (-lOh) than between the 3He and the thermometer (- 30 min at 1.2 mK) and made specific heat measurements possible without the use of a heat switch. The entropy diagram of PrNi, is given in fig. 17. Some of the data shown were obtained with the cooling pill described above. The material has a residual resistivity ratio of only 7 and hence has a rather low thermal conductivity which, at 1 mK, leads to thermal equilibrium times of about 2 h in the cooling pill. This made specific heat measurements at the lowest temperatures rather difficult. In fig. 17 we have also included more recent data on a better sample (Mueller et al., 1980; d. section 4.2), which had much shorter equilibrium times.

4. Two-stage nodear refrigerators We mentioned above that a hyperfine enhanced nuclear stage is very well suited for pre-cooling a brute force nuclear stage. To date this has been


275

experimentally verified by three groups (Hunik et al., 1978; Ono et al., 1980; Mueller et al., 1980). In what follows, we shall discuss the last two of these experiments in some detail. While both cryostats succeeded in reaching end temperatures in the microkelvin range, the first machine uses rather small quantities of cooling materials (14 g of PrCu, and 1.2 g of copper), while the second cryostat employs amounts larger by over two orders of magnitude (1.86 kg of PrNi, and 0.64 kg of copper). We then discuss an experiment on two stage brute force nuclear cooling which was carried out at the Helsinki University of Technology in 1979 and which for the first time generated nuclear spin temperatures in the nanokelvin range.

4.1. The Tokyo cryostat

The nuclear refrigerator of Ono et al. (1980) distinguishes itself by its simplicity (fig. 18). The first stage consists of six arc melted buttons of PrCQ which are soldered to copper wires. The actual stoichiometry used was PrCu,,*, with the hope that the excess copper would improve the thermal conductivity of the material without affecting its magnetic properties. The second stage was connected to the first by two 1.2mm diameter copper wires, 50cm long, and was simply made of another copper wire, 1.8mm in diameter. Demagnetization of the PrCu, from 20mK and 5.5T pre-cools the second stage to 3 m K in a field of 5.5T.Near the end of the first demagnetization, the second stage is automatically decoupled from the first by a lead heal switch located in the fringe field of the upper solenoid. The second stage is then demagnetized over a period of several hours to 28 mT and finally reaches a nuclear spin temperature of about 20 pK. This corresponds to an estimated final electron temperature of about 30 p K which can be maintained for a period of one to two hours. The copper nuclear spin temperature is measured indirectly via an attached 54MnAI nuclear orientation thermometer which is in a residual field of 28 mT at the end of the second demagnetization. The analysis of the data is based on the assumptions that the 54Mn nuclei are in good contact with the aluminium nuclei because of cross relaxation processes and that the aluminium nuclei are demagnetized to the same end temperature as the copper nuclei. There is a certain amount of doubt as to the correctness of these assumptions.

216


MlXlMG CHAMBER SiNTERED COPPER HEAT SWITCH MAGNET LEAD HEAT SWITCH CARBON RESISTOR THERMAL LINK THERMAL SHIELD HEAT SHIELD VACUUM CAN 1ST STAGE MAIN MAGNET 1ST NUCLEAR STAGE (PrCu6 or PrCutl 1 LEAD HEAT SWITCH THERMAL LINK

2ND NUCLEAR STAGE (Cu) 2N0 STAGE MAIN MAGNET NO THERMOMETER

Fig. 18. The cascade nuclear refrigerator at Tokyo (On0 et al., 1980).


211

4.2. The JiiIich cryostat

The two stage nuclear cooling cryostat of Mueller et al. (1980).shown in figs. 19 and 20, is considerably larger and more elaborate than the Tokyo machine. The first cooling stage consists of 60 arc-cast PrNi, rods weighing 1.86 kg (4.29 mol) and obtained again from the Rare Earth Research Institute at Ames. Six 1 mm diameter copper wires were soldered with cadmium to each rod. All rods were bundled as shown in figs. 19 and 20, the copper wires being arc welded into a copper cold plate located below the PrNi, stage. The cold plate is connected both to the mixing chamber of the dilution refrigerator, through a central thermal link (500 copper wires of 1 mm diameter) and a superconducting heat switch, and also to the copper cooling stage, via a heavy thermal link made of copper which is used also for mounting experiments. The copper cooling stage consists of 96 rods, each of 2 x 3 mm2 cross section and 25 cm long. Magnetic fields of 6 and 8 T are available for the first and second stage, respectively; the experimental space between the two stages is in a field compensated region at about S mT. The PrNi, cooling stage in this cryostat is the largest one built to date with the best quality material. Residual resistivity ratios between 20 and 30 were observed for the rods. This resulted both in an observed high degree of thermodynamic reversibility and in a better thermal conductivity at low temperatures. When operating the PrNi, stage alone and demagnetizing it from 6 T and 10 mK, an end temperature of 0.19 mK was reached (Folle et al., 1981), which is considerably below the nuclear ferromagnetic ordering temperature of 0.40 mK. The shorter thermal equilibrium times of this cooling pill, typically 5 min at 1 mK and 30 min at 0.55mK, allowed for the first time the calometric observation of spontaneous nuclear magnetic order in this material (Kubota et al., 1980). Largely due to the work of the Jiilich group, PrNi, is to date the best characterized hyperfine enhanced nuclear cooling compound. The refrigerating capacity of the PrNi, stage is very high; with a heat leak of 10 nW it would warm up from 0.25 mK to 1 m K in 17 days! When using both nuclear cooling stages, PrNi, is demagnetized first from 6 T and 25mK to 200mT, while a field of 8 T remains on the copper stage. After about lOh the two stages reach an equilibrium temperature of 5.5 mK, which means that 23% of the nuclear entropy has been removed from the copper stage. Demagnetization to 8 mT exponentially with a time constant of 2 h, then resulted in the lowest-ever

FIELD PROFILE OF MAGNETS

5 mT SPACE

VACUUM SPACE - LlQUlO HELIUM SPACE - MIXING CHAMBER OF DILUTION REFRIGERATOR

I

-A1

HEAT SWITCH 1

-MC

HEAT SHIELD

- 1 K HEAT SHIELD - VACUUM JACKET 6 T MAGNET-

- PrNi, DEMAGNETIZATION STAGE (ONLY 3 OF 60 RODS DRAWN)

- CENTRAL THERMAL LINK - THERMAL PATH TO PtNis

- Al

__I

HEAT SWITCH 2

5 mT SPACE

- EXPERIMENTAL SPACE - THERMAL PATH TO CU STAGE (ONLY 1 OF 3 LEGS DRAWN)

- CU DEMAGNETIZATION STAGE

Fig. 19. Drawing of the Jiilich two-stage nuclear refrigerator (Mueller et al.. 1980).


Fig. 20. View of the PrNi, cooling stage of the Jiilich refrigerator.

279

280


measured electron temperature, T, = 48 p K as recorded by a platinum NMR thermometer in the experimental chamber. The calculated nuclear spin temperature in the copper stage is 5 pK. With the observed total heat leak into the second stage, amounting to 1 nW, the calculated electron temperature in this stage is 9 pK. Assuming that the heat leak enters through the experimental chamber, the much higher electron temperature there, 48 pK, can be explained by the thermal resistances at contacts and in the copper cooling rods themselves. With a 1 nW heat leak, a conduction electron temperature below 60 p K can be maintained in the experimental chamber for several days. For further description of the Julich cryostat we refer to Pobell (1982).

4.3. The Otaniemi cascade refrigerator For studies of spontaneous nuclear ordering, i.e. nuclear ferromagnetism and nuclear antiferromagnetism, the starting entropy must, in general, be well below S,, the critical entropy at the transition to the ordered state. Only then, even after allowing for some losses during demagnetization, is there hope of reaching a temperature Tf below the transition point T,. We expect that S,=0.4 R ln(2Z-t 1). A high value of Bi/Ti, about lo" T/K, is thus needed for a simple metal like copper. With Bi = 7 T, Ti must be 0.7 mK or below. Re-cooling by nuclear refrigeration is necessary. In addition, for investigating the ordered state, the heat flow to the nuclei, coming from the conduction electrons at T, and proportional to T, - T. [cf.eq. (18)], must be as low as possible because in zero field the heat capacity of the copper nuclear stage [cf. eq. (6) with B replaced by b ] is very small. A cascade nuclear refrigerator for experiments of this type has been constructed at Otaniemi (Ehnholm et al., 1979 and 1980).The apparatus, which is schematically illustrated in fig. 21, consists of a powerful dilution refrigerator and two copper nuclear stages, all working in series. The mixing chamber reaches 6 mK without a heat load and 10 mK with the nuclear stages in high field. The first nuclear stage was made of 10 mol of copper wire, 0.5 mm in diameter and insulated with fiber-glass. The residual electrical resistance ratio is 700. Between the mixing chamber and the first nuclear stage there is a superconducting heat switch made of a piece of bulk tin. The second nuclear stage, which is also the specimen, was made of 2000 copper wires, 0.04mm in diameter and insulated by oxidation; the


28 1

. CONDENSER

.

snu

. LIQUID ‘He t

HEAT EXCHANGERS

MIXER HEAT SWITCH

FIRST NUCLEAR STAGE AND MAGNET

VACUUM JACKET HEAT SHIELD

SECOND NUCLEAR STAGE AND MAGNET 1-METAL SHIELD

SQUID

Fig. 21. Drawing of the Otaniemi cascade nuclear refrigerator (Ehnholm et al.. 1979 and 1980).


282

residual resistivity ratio is 200. The second stage is much smaller than the first, it weighs only 2 g (0.03 mol). There is no heat switch between the two nuclear stages; the wires were simply welded together. The magnetic fields for operating the nuclear stages are generated by two superconducting solenoids, producing 7.8 T and 7.3 T, respectively. The refrigeration procedure is as follows: after the heat switch has been turned on the first nuclear stage is magnetized to 8T. The dilution refrigerator then pre-cools the copper wire bundles to 10 mK overnight, after which the tin heat switch is turned off to isolate the nuclear stages. Next, the second stage is magnetized in 1h to 7.3T and, starting simultaneously, the first stage is demagnetized to 0.1 T in about 5 h, first rather rapidly and then more slowly towards the end of demagnetization. With a 1.3nW heat leak to the upper nuclear stage, the conduction electron temperature, measured from the lower end, is 0.25 mK. Because there is no heat switch between the nuclear stages, the conduction electron temperature in the specimen is fixed to 0.25 mK. The starting conditions for the second stage demagnetization are thus B i = 7.3 T and Ti= 0.25 mK, corresponding to an equilibrium nuclear spin polarization well over 99% in copper. Upon demagnetization to Bf = 0 in

1 -

0.8

-

5 0.6

-

0.4

-

7

sv,

0

0

1

I

I

]

Fig. 22. The entropy diagram of copper down to 50 nK in zero external tield (Ehnholm et a!., 1979).


283

10 min the nuclear spin system reaches 50 nK which is the lowest temperature ever produced. Conduction electrons remain at 0.25 mK. After demagnetization the nuclei in the specimen begin to warm up owing to heat leaking in via the spin-lattice relaxation process. In zero field the relaxation time T, = 20 min, so there is barely enough time for measurements. The nuclear spin temperatures were measured by applying the second law of thermodynamics, T = dQ/dS. It is interesting to note that a thermal switch between the two nuclear stages would make matters worse. In this case the external heat leak to the second stage would have to be absorbed by the relatively small number of nuclei demagnetized to Bf = 0. Their heat capacity is so small that the specimen would warm up in a few seconds with no time for experiments. For this reason the external heat leak must be conducted to the first stage where the large number of nuclei at B,= 0.1 T can easily absorb it. Experiments with this apparatus (Ehnholm et al., 1979 and Soini, 1982) have shown, so far, that the nuclear spin system of copper clearly tends to order antiferromagnetically in zero field but it also seems that the transition itself has not yet been found. This is probably due to large irreversibilities during demagnetization; the minimum entropy observed at 50 nK is 0.45R ln(21+ 1) (13. fig. 22) even though the starting entropy is close to zero.

5. Comprvisan of brute force and byperhe enb.nced nudear rehigemlion For quantitative comparisons between the relative merits of the brute force and hyperfine enhanced nuclear refrigeration techniques, we shall assume that copper and PrNi,, respectively, are employed as the working substances. Advantages of brute force nuclear cooling with copper are the ready availability of this material in high purity form with the resulting high thermal conductivity at low temperatures. If starting conditions BJT,= 600 T/K are available, the construction of a “nuclear bundle” of copper is simple because wires of 1 mm diameter can be used without the danger of excessive eddy current losses. A prerequisite for a single brute force nuclear stage is always a powerful dilution refrigerator which permits pre-cooling below 20 mK in a reasonable length of time. In practice it is difficult to reduce the nuclear entropy of copper by


284

more than 5% this way. However, even for cooling 5 cm3 of liquid ’He below 0.5 mK, 10 mol of copper is enough. If a large magnet is available, one can, of course, simply make a bigger nuclear bundle to increase the cooling capacity. The obvious advantage in hyperfine enhanced nuclear cooling is the strong polarizing field, which is usually at least 10 times larger than the applied field. Although the nuclear spin density in the applicable compounds is roughly 10times smaller than that in copper, the cooling entropy per unit volume is still roughly 10 times larger [cf. eq. (4)]. This makes it possible to build small and cheap nuclear cooling stages for reaching temperatures in the 1 mK range. F’rNi, stages can often easily be added to existing cryostats, an option which is usually not available for a brute force cooling stage. Disadvantages with PrNi,, on the other hand, are the not so ready availability of the materials, their poorer thermal conductivities, and their higher end temperatures. While the making of PrNi, rods is not difficult in general, a minimum of materials preparation facilities is required. PrNi, is available commercially from Dr. K.A. Gschneidner of the Rare Earth Research Institute (Ames, Iowa). With the best quality praseodymium metal, having a residual resistivity ratio of about 60, thermal conductivities of about OST, W/KZm can be obtained; this is still 1000times worse than that obtainable with copper. It is interesting to compare the cooling performance of copper with that of PrNi, in the sub-millikelvin regime. For copper the cooling power is governed by the spin-lattice relaxation time, whereas for PrNi, it is limited by the thermal conductivity of the material. Assuming a stage with 10 mol of copper and &/Ti = 600 T/K, eq. (18) can be rewritten to read

Q,,,

= 2.5T: W/K2,

(59)

where we have used T, = T,/2 for maximum cooling power [d.eq. (20)]. An equivalent PrNi, stage would contain 0.5 mol of the metal and consist typically of 25 rods, d = 5 mm diameter each and of a total surface area A = 200 cm’. With &/Ti= 600 T/K, PrNi, can be demagnetized to Tf= 0.2 mK and the nuclear stage can initially absorb heat through its surface at the rate

.

Q,

=

O.ST,A(T,- Tf) 2 4(T,Z- T,T,) W/K2 d/2

We have assumed that the thermal conductivity 0.5T, W/Kzm of PrNi, goes linearly to zero with temperature. Comparing eqs. (59) and (60) we


285

find that down to T, = 0.5 mK, PrNi, has a higher cooling power, while below this temperature the cooling power of the copper stage is higher. It thus seems that when large cooling powers are required at 0.5 mK and above, there is an advantage with PrNi,. Our comparison shows that it would be of great value to have a cooling material like PrNiS but with an ordering temperature about five times lower. This could significantly improve cascade nuclear cooling experiments, such as that one described in section 4.3, because the electron temperature of the second stage could be kept lower. At present, however, electron temperatures below 0.2 mK can only be generated by the brute force technique. An additional advantage of hyperfine enhanced nuclear cooling is that one can demagnetize to Bf = 0, since the effective internal fields are larger than 40mT. When using this option for doing experiments at low temperatures in near zero field, one must, however, keep in mind that the cooling pill retains a sizeable induction, typically of the order of 3 m T after demagnetization; this induction decays during warm-up. As additional examples we mention that a heat input of 6 = 3 n W warms 100g of PrNi, in zero field from 0.25 to 0.5 mK in 54 h (Folle et al., 1981). One cm3 of liquid 'He can be cooled from 25 mK to 0.3 mK with only 3 0 g of RNiS demagnetized from 6 T . With B i = 8 T and Ti = 12 mK for copper, and Bi = 6 T and Ti =25 mK for PrNi,, the cooling capacity of the latter per unit volume is twice the former. The starting temperature for copper must be reduced to 8 mK for the cooling capacities to be equal.

Acknowledgments We acknowledge with thanks information, comments, and criticism by 0. Avenel, H.M.Bozler, D.F. Brewer, G. Frossati, W.P. Halperin. M. Krusius, N. Kurti. D.D. Osheroff, R.E. Packard, F. Pobell, and R.C. Richardson.

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Kurti, N. (1982).Proc. LT-16. Vol. 3 (to be published in Physica B+C). Kurti. N., F.N. Robinson, F.E. Simon and D.A. Spohr (1956)Nature (London) 178,450. Landesman, A. (1971) J. Physique 32.67 1. Leggett, A.J. (1978)J. Physique 39, C6-1264. Leggett, A.J. and M. Vuorio (1970)J. Low Temp. Phys. 3, 359. Lounasmaa, O.V. (1974)Experimental Principles and Methods below 1 K (Academic. London). Lounasmaa, O.V. (1978)Physics at Ultralow Temperatures (Proc.Hakone Symposium, Phys. Soc. Japan) p. 246. Lounasmaa. O.V. (1979)J. Phys. El& 668. Luthi, B. (1980)in: Dynamical Properties of Solids, Vol. 3, eds., G.K. Horton and A.A. Maradudin, (North-Holland, Amsterdam). p. 243. Mast, D.B.. B.K. Sanna, J.R. Owers-Bradley, I.D. Calder, J.B. Ketterson and W.P. Halperin (1980).Phys. Rev. Lett. 45,266. McWhan. D.B., C. Vettier, R. Youngblood and G. Shirane (1979)Phys. Rev. BU), 4612. Mineeva, R.M. (1963)Sov. Phys-Sol. St. 5, 1020. Mueller. R.M., C.Buchal, H.R. Folle. M. Kubota and F. Pobell(1980)Cryogenics 20.395. Muething, K.A. (1979)Ph.D. Thesis, Ohio State University. Murao, T. (1971)J. Phys. Soc. Japan 31, 683. Murao, T. (1972)J. Phys. Soc. Japan 33, 33. Ono, K., S. Kobayasi. M. Shinohara, K. Asahi. H. Ishimoto, N. Nishida. M. Imaizumi, A. Nakaizumi, J. Ray. Y. Iseki. S. Takayanagi. K. Tenti and T. Sugawara (1980)J. Ldw Temp. Phys. 38,737. Osheroff, D.D. and W.O. Sprenger (1980)private communication. Osheroff. D.D. and N.N. Yu (1980),Private communication. Pobell, F. (1982).Roc. LT-16,Vol 3 (to be published in Physica B+C). Roinel, Y., V. Bouffard, G.L. Bacchella, M. Pinot, P. MCriel, P. Roubeau, 0. Avenel, M. Goldman and A. Abragam (1978)Phys. Rev. Lett. 41, 1572. Satoh, K., Y. Kitaoka. H. Yasuoka, S. Takayanagi and T.Sugawara (1981),J. Phys. Soc. Japan 50. 35 1. Soini, J.K. (1982)Ph.D. Thesis, Helsinki University of Technology. Sprenger. W.O.and M.A. Paalanen (1980).private communication. Stevens, K.W.H. (1952)Proc. Phys. Soc. (London) A65, 209. Suhl, H. (1959)J. Phys. Rad. 20, 333. Symko, O.G. (1969)J. Low Temp. Phys. 1. 451. Teplov, M.A. (1968)Sov. Phys.-JETP Lctt. 26, 872. Trammell, G.T. (1963)Phys. Rev. 131, 932. Triplett. B.B. and R.M. White (1973)Phys. Rev. B7, 4938. Tsarevskii, S.L. (1971)Sov. Phys.-Sol. St. 12. 1625. Veuro, M.C. (1978)Acta Polytech. Scand. Ph. 122, 1. Zaripov, M.M. (1956)IN. Acad. Nauk SSSR, Ser. Fiz. 22, 1220.


AUTHOR INDEX*

Ablowitz, M.J., 3, 7. 62 Balfour, L.S., 130 Abragam, A., 76, 97, 225, 129, 285, 287 Balibar, S., 126, 129 Ahrikosov, A.A., 13, 62 Baratoff, A., 10, 62 Adams, E.D., 64, 129. 130 Barclay, J.A.. 286 Ahlers. G.. 190, 191, 216, 217 Barone, A., 5. 10, 11, 62 Ahonen, A.I., 235, 238, 239. 285 Bar-Sagi, J.. 22. 62 Allen, A.R., 90, 129 Bartolac, T., 13, 62, 63, 286 Allen, J.F., 137, 165, 191, 192, 193, 194. Baxter, R.J., 4, 24, 62 195, 216. 218 Behringer, R.P., 190. 191. 216, 217 Als-Nielsen, J., 286 Bekarevich, I.L., 144, 217 Al'tshuler, S.A., 224, 245, 285 Belavin, A., 3. 4, 14, 62 Alvesalo, T.A., 240, 285 Berezinskii, V.L., 13, 62 Ambegaokar, V., 10, 62 Bergkoff, H., 4, 35, 62 Anderson, P.W., 14. 15. 62, 63 Berglund, P.M., 285 Andreev, A.F., 81, 83, 84, 86, 88. 91, 94, Bernasconi, J., 64, 65 99, 101, 104, 106, 108. 109, 110, 112, Bernier, M.E., 62 115, 121, 122, 127, 129 Bethe, H., 4, IS, 34, 63 Andres, K., 224, 245, 253, 259, 260, 264. Bhagat, S.M., 137. 217 267. 268. 269, 270. 271, 272, 273, 274, Bhatt, R.N., 286 65. 285, 286 Birgeneau, R.J., 253, 286 Andronikashvili, E.L., 129, 129 Bischoff, G., 218 Archie, C.N., 244, 130, 286 Bishop, A.R., 63, 64, 6.5 Arms, R.J., 149, 216 Bleaney, B., 250, 286 Asahi. K . , 287 Bloch, J.M., 259, 286 Ashton, R.A., 137. 139, 143, 145, 154, Bongen, E., 286 182, 185, 186, 187, 188, 208,210,212, Boucher, J.P.. 9.27.29, S5, 58.59, 63, 65 217 Bouffard, V., 287 Atkins, K.R., 180, 217 Boughton, R.I., 241, 286 Avenel, O., 13, 235, 236, 244, 62, 286, 287 Bouillot. J.. 63, 65 Avilov, V.V.. 74, 129 Bozler, H.M., 244, 62, 63, 285, 286 Brandt, P., 65 Babcock, J., 255. 270, 286 Brazovskii, S.A., 20, 63 Babkin, A.. 126, 130 Brewer, D.F., 137, 143, 158, 159, 160. 165, Bacchella, G.L., 287 166, 167, 174,202,204,206,244.217, Bacon. F., 228. 286 285. 286 Bak, P., 12, 62 Brewer, W.D..286 Bakalyar, D.M., 78, 129 Brinkman, W., 109, 129

* An italic number indicates the name appears in a reference list.

AUTHOR INDEX

290

Britton. C.V., 129 Broadwell, J.E., 143, 156, 159, 160, 217 Broese van Groenou, A., 218 Brubaker, N.R., 286 Buchal, C., 286, 287 Bucher, E.. 224, 245, 261. 270, 286 Busse. L., 218

Deville, G., 131 de Voogt, W.J.P., 219 Devreux, F., 64 de Vries. G.. 3. 64 Dietz, R.E., 65 Dimotakis, P.E., 143, 156. 157, 158, 159, 160, 192, 206, 217

Donnelly, R.J., 140, 180, 184. 200, 205. Calder, I.D., 287 Campbell, LJ., 144, 217 Careri, G.. 137, 139, 180. 182. 185, 202, 217

Castaing, B., 112, 126, 129 Chase, C.E.. 137, 156, 157, 159, 160, 200, 204, 205, 206, 207, 217

Chernov, A.A., 113, 115, 117, 129 Chester, G., 106. 108, 129 Childers. R.K., 137, 159, 160, 161, 165, 167,168,169,174,202,210.217,218

Chu. F.Y.F., 65 Clarke, T.C., 64 Cohen, M.J., 17, 63 Coleman, S., 3, 7. 35, 36. 63 Cooper, B.R., 253, 286 Cornelissen, P.L.J.. 202, 217 Craig, P.P., 144, 162, 163, 168, 203, 21 7 Critchlow. P.R., 137, 217 Cromar, M.W., 218 Cross. M.C., 15, 63. 131 Crowe, H.R., 63 Cunsolo, S., 137, 182, 191. 217. 219 Currie, J.F., 4. 15, 34, 42, 63 Dahm, A.J., 103, 131 Darack, S., 272, 273, 274. 286 Dashen, R.F., 4, 15. 35, 41, 43, 63 Daunt, J.G., 217 Davidow, D.. 286 de Boer, J., 70, 129 de Bruyn Ouboter, R., 217, 219 Debye. P.. 223, 286 de Gennes, P.G.. 12, 13, 21, 63 de Gceje, M.P., 218 de Haas, W.J., 137, 143, 166, 170, 202, 204,206,208,210,214,223,217.286

Delrieu, J.M., 71, 129. 131 Derek, S., 65 Derrick, G.H., 11, 63

217, 218

Donth. H., 65 Doring, W., 12. 63 Douglas, R.L.. 183, 217 Dufty, J.W., 64 Dundon, J.M., 235, 286 Duyckaerts. 165, 218 Dzyaloshinskii, I.E., 17, 20. 106, 108, 109, 63. 129

Edwards, D.O., 137, 143, 158, 159, 160, 165,166, 167,174,202,204,206, 130, 217 Ehnholm, GJ., 280, 281, 282, 283, 286 Eisenstein, J., 244, 286 Ekstrom, J.P., 286 Emery, V.J., 12, 17. 62, 64 Enz, U., 12, 13, 63 Esel’son. B.N.. 130, 131 Eshelby, J., 82, 129 Eska, G., 286 Esposito, F., 62

Fasoli, U., 217 Felner. I., 286 Ferrell. R.A., 110. 65. 129 Fetter, A.L., 205, 21 7 Feynman, R.P., 13, 145,201, 205,63, 217 Fick, E.. 246, 286 Fineman, J.C., 205, 217 Finkel’stein. A.M., 130 Finkelstein, D.. 3, 11, 119, 63 Fisher, D.S., 131 Folle, H.R., 260. 274, 286. 287 Fraass, B.A., 108. 111, 115, 130 Francois. M.. 219 Friedel, J., 13, 63 Frossati, G., 234. 244. 130, 285. 286 Fujita, T., 15, 63

AUTHOR INDEX Gachechiladze. LA., 129 Gaeta, F.S., 217 Garber,M., 130 Gardner. C.S., 3, 63 Garwin, R.L., 74, 130 Genicon, J.L., 258, 286 Giauque, W.F., 223, 286 Giezen. J.J., 219 Giffard, R., 131 Glaberson, W.I., 137, 141, 142, 143, 145, 162,163,168,172, 192,200,202,204. 205, 218, 219 Goalwin. P., 286 Gcdfrin, H., 80, 130 Gotdberg, H.A., 130, 131 Goldberg, I.B., 18, 63 ' Goldman, M.. 225, 285, 286, 287 Gollub, J.P.. 190, 191, 219 Goodkind, J.M., 78, 131. 286Gorter, C.J., 138, 144, 145, 165, 169, 223, 218 Gould, C.M., 13, 62, 63 Graham, G.R., 218 Grayevsky, A,. 287 Gredeskul, S.A., 110, 130 Greenberg, A., 98, 130 Greene. J.M., 63 Greene. R.L., 64 Grifliths, D.J., 192, 216, 218 Grigor'ev. V.N., 82, 83, 103, 130, 131 Grover, B.. 255. 286 Guinault, A.M., 234, 286 Gupta, N., 15, 31. 34, 63 Guyer, R.A., 69. 74, 76. 81. 82. 101, 130, 131 Haavasoja, T., 285 H a p , E., 286 Haikala, M.T.,285 Haldane, F.D.M., 18, 62, 63 Hall, H.E., 144, 145, 167, 171, 218 Halperin, W.P., 79, 130, 285, 287 Hamma. F.R., 149, 216 Hammel. E.F., 137, 143, 162, 168, 208. 217, 218 Harrison, J.P.. 286 Hartoog. A., 210. 217, 218 Hasslacher, B.. 63

29 1

Hatton, J., 131 Heald, S.M.,130 Hebral, B., 130 Heeger. A.J., 18, 63, 64,65 Heidenrich. R., 17, 63 Henberger, J.D., 155, 164, 176, 218 Heritier. H., 110. 130 Hetherington, J.H., 71, 101, 130, 131 Hiki, Y . , 129, 131 Hirakawa, K., 25, 63, 65 Hirth, J.P., 128, 130 Ho, T.L., 14, 64 Hoch, H., 140, 141, 192, 196, 197, 218 Hollis-Hallet, A.C..218 Hone, D., 64 Horowitz, B., 21, 23, 63, 64 Huang, W., 82, 84, 130 Hubennan, B.A., 5, 63 Hudson, R.P., 225, 286 Huiskamp, W.J., 223, 287 Hunik, R., 275, 287 Hunt, E., 131 Hutchins. J.D., 244, 287 Hwang, Y.C.,129 ljsselstein, R.R., 173, 177, 178, 179, 180, 208. 210, 212. 214, 218 Ikeda, S., 65 Imaizumi, M., 287 Iordanskii, S.V.,71, 74, 119, 129, 130 Iseki, Y.,287 Ishimoto. H., 287 king, E., 25, 63 Ito, T., 65 Jackiw. R., 63 Jackson, K.A., 113. 115, 130 Jacquinot, J.F., 286 Jauho. P., 228, 287 Jauslin, H.R., 65 Jevicki, A., 4, 14, 64 Johnson, J.D., 43, 64 Jones, E.D., 224, 245, 287 Joos. G., 246. 286 Jose, J.V.. 13, 64 Josephson, B.D., 9, 64 Kadanoff, L.P.. 64

292

AUTHOR INDEX

Kagan. Y., 82, 83, 88. 99, 119, 130, 131 Kaplan, N., 255, 257, 287 Kaufer, J.. 65 Kaup, D.J.. 62 Kawasaki, K., 4, 49. 64 Keesom, W.H., 165, 218 Keller, W.E., 137, 143, 162, 168, 208, 217, 218 Keshishev, K.O., 103, 108, 112, 122, 124, 126, 130 Ketterson, J.B., 287 Khaiatnikov, I.M., 144. 180, 217, 218 Kiely, J., 286 Kirk, W.P., 78, 130 Kirkpatrick. S.. 64 Kitaoka, Y . , 287 Kittel, P.. 218 Kjems. J.K.. 9, 27, 53. 58, 64, 65 Kleman, M.. 3, 11, 64, 65 Klinger. M.I., 82, 83, 88, 99, 130 Kobayasi, S.,287 Kochendorfer, A.. 65 Kondratenko, P.S., 129 Konter, J.A., 287 Kopnin. N.V., 15, 64 Korteweg. D.J., 3. 64 Kosterlitz, J.M., 13. 64 Krames, H.A., 286 Krames, H.C., 137, 159, 173. 176, 177, 178, 179,202, 208.210, 211, 217, 218, 219 Krinsky, S . , 64 Krivoglaz, M.A., 110, 130 Krumhansl, J.A., 4, 12, 15, 21, 49, 60, 63, 64. 65 Krusius, M., 244. 62, 285, 287 Kruskal, M.D., 3, 63. 65 Kubota, M., 255, 260. 272. 277. 287 Kumar, P.. 12, 62, 64 Kummer, R.B., 79, 80, 230 Kuper, C.G., 22, 110, 62, 130 Kurti, N.. 223, 285, 287 Ladner, D.R., 135, 137. 143, 155, 156. 160, 162, 163, 164, 168.202.204,206,207, 218 Laguna, G.A., 192. 217 Laloe, F., 112, 130

Landau, J., 126, 130 Landau, L.D., 78, 79, 105, 107, 113. 114, 118, 120, 143, 130, 218 Landesman, A,. 74. 80, 82, 84, 86. 255, 130, 131. 287 Larkin, A.I., 17, 63 Laroche. C., 129 Lederer. P., 110, 130 Lee, D.M., 13, 63 Leggett, A.J., 108, 224. 130, 287 LeRay, M., 219 Leung, K.M., 9, 29, 55. 64 Levchenkov, V.S., 129 Lhuillier, C., 112, 130 Lhuillier. D., 219 Lieb, E.H.. 4, 15, 17, 35, 64 Liepmann, H.W.,159, 217 Lifschitz, E.M., 78, 79, 105, 107, 120, 143, 129, 130, 218 Lifschitz, I.M., 81, 88, 91, 106, 108. 109, 110. 119, 130 Liniger. W.,4, 35, 64 Lin-Liu, Y.R.. 12, 62. 64 Lipson. S.G., 130 Loponen, M.T., 286 Lothe, J., 128, 130 Lounasmaa. O.V.. 223.225,230, 285,287 Loveluck, J., 62 Lucas, P., 144, 218 Luther, A., 4, 17, 43. 64 Liithi, B., 262. 287 Maattanen, L.M., 130 MacDiarmid, A.G., 65 M a c h u g a l l , D.P.. 223. 286 Magee, C.J., 62 Maita. J.P.. 286 Maki, K., 4, 5,9. 12, 13. 28. 29. 34, 35, 36. 39, 41, 43, 44, 45, 47, 49, 51, 53, 55, 56, 57, 59, 60, 64, 65 Maksimov. L.A., 82, 88. 99. 130 Manley, T., 286 Manninen. M., 285 M a n t a , J., 176, 180, 197, 198, 200. 218 Marchenko. V.I., 129 Martin, K.P., 142. 204, 218 Marty. D., 103. 131 Mast, D., 244. 287

AUTHOR INDEX Matisoo, J., 11, 64 Mattis, D.C., 15, 17, 64 McCormick. W.D.. 217 McCoy, B.M., 64 McGuire. J.B., 35, 44, 64 Mchughlin, D.W., 65 McMahan, A.K., 74, 131 McMillan, W.L., 12, 64 McWhan, D.B., 253, 287 Mehe, J.B., 171, 172, 176, 179, 180, 218 Meierovich, A.E., 82, 83, 94, 99, 101, 104, 129, 129, 131 Melik-Shakhnazarov, V.A., 129 Mellink, J.H., 135, 138. 144. 145, 165, 169, 218 Meriel, P., 287 Mermin, N.D., 11, 14. 64 Meservy, R., 206, 228 Meyer, H.,131 Meyer, L., 180, 218, 219 Mezhov-Deglin, L.P., 130 Michel. L.. 64 Mikeska. H.J., 4, 8, 9, 26, 27, 29, 49, 55, 56, 64 Mikheev. V.A., 90. 91, 130, 131 Mikhin, N.P., 131 Milford, F.J., 217 Mills, D.L., 64 Mineev, V.P.. 15. 101, 65, 131 Mineeva, R.M.,250. 287 Mineyev, V.P., 3, 11, 12, 14, 64 Miura, R.M.. 63 M a , F., 139, 140. 183, 184, 1%, 197, 199, 218, 219 Mueller, R.M., 266, 272, 274, 275, 277, 278, 130, 286, 287 Muething, 244, 287 Mullin, W.J., 96. 97, 130, 131 Murao, T., 254, 287 Nabarro, F.R.N., 13, 64 Nagaev, E.L.. 110, 231 Nagaoko, A., 109, 110, 131 Nakauumi, A., 287 Nakamura. T., 255 Naskidashvili, LA., 129 Nechtshein. M., 18, 64 Nelson, D.R., 64

293

Neveu, A., 63 Newell, A.C., 62 Newman, P.R., 63 Nishida, N., 287 Northby, J.A., 137, 139. 145. 154, 185, 186, 187, 188.195,198,199,200,212, 217, 218 Nosanow. L.H., 71. 230, 131 Nozikres, P., 112, 126, 129 Oberly, C.E., 200, 204, 206. 218, 219 Ohmi. T., 63 Ono, K., 275, 276, 287 Onsager, L., 13, 64 Osborne, D.V., 192, 216, 218 Osgood. E.B., 130 Osheroff, D.D., 13, 79, 80, 241, 242, 243, 244, 64, 131. 285, 287 Ostermeier. R.M., 141, 173, 175, 176, 179, 197, 198, 199, 202. 218 Owers-Bradley, J.R.,287 Packard, R.E., 285. 286 Paiaanen, M.A., 285 Patshin, A.Y.. 112, 115. 121, 122, 126, 129, 130 Patrascioiu. A., 3, 65 Paulson, D.N., 287 Pendrys, J.P., 286 Perring, J.K., 7, 65 Peshkov, V.P., 159, 204, 205, 206, 210, 218, 219 Petukhov, B.V., 119, 128, 231 Pickett, G.R., 234, 286 Pietronero, L., 17, 65 Pinot, M., 287 Piotrowskii, C., 146, 193, 194, 195, 219 Pirila, P.V., 228, 287 Pobell, F., 231, 281, 285, 286, 287 Poenaru, V.,65 Pokrovskii, V.L.,119, 128. 131 Polyakov, A.M., 4. 14. 62, 65 Pope, J., 131 R a t t , W.P., 183, 219 Prewitt, T.C., 78, 131 Pron, A., 65 Pushkarov. D.I., 82, 91, 101, 131

294

AUTHOR INDEX

Rasmussen, F.B., 130 Ray, J., 287 Reekie. J.. 137, 165, 216 Regnault, L.P., 9, 63, 65 Reich, H.A., 74, 75, 131 Reif, F.. 180, 218, 219 Renard, J.P., 59. 63, 6.5 Rice, M.J., 16, 17, 18, 109, 65, 129 Rice, T.M., 286 Richards, M.G., 74, 81, 83, 90.9598, 129, 131 Richardson, R.C., 74, 130, 131, 285 Riseborough, P.S., 64 Roberts, P.H., 140. 183, 184, 217 Robinson, F.N., 287 Roger, M., 71, 80, 129, 131 Roinel, Y., 225, 287 Rosenshein, J.S., 211, 219 Rossat-Mignod, J., 63, 65 Roubeau, P.. 235, 236. 244, 285, 287 Ruelle, D., 191. 219 Sacco, J.E., 96, 97, 131 Sai-Halasz, G.A., 103, 131 Sarma. B.K., 287 Sarwinski, R.J., 286 Satoh, K., 287 Scalapino, D.J., 15. 30, 34, 48, 65 Scarammi, F., 21 7 Schlichting, H., 172, 219 Schmidt, P.H., 286 Schneider, T., 4, 15, 49, 62, 64, 65 Schrieffer, J.R.,4, 12, 15, 18.49, 60,64, 65 Schroer, B., 63 Schwartz, A,, 62 Schwan. K.W., 135, 147, 149, 150, 151. 152, 153, 154, 157, 183, 188, 201,204,

219 Scott, A.C., 5, 6, 62, 65 Scott-Russell, J., 3, 65 Stars, M., 65 Seeger, A., 7, 65 Segur, H., 62 Seiler, R., 63 Shal'nikov, A.I., 103, 115, 130, 131 Shaltiel, D., 286 Shanker, R., 15, 65 Shirakawa, H., 18, 66

Shirane, G., 287 Shirley, D.A.. 286 Shul'man, Y.E., 130 Simmons, R.O., 130 Simon, F.E., 287 Sinohava. M., 287 Sitton. P.M., 139, 140. 183, 184, 196, 219 Skyrme, T.H.R., 7, 65 Slegtenhorst. R.P., 159. 208, 219 Slusarev, V.A., 130 Smith, C.W., 219 Smith, J.H., 183, 131 Smolic, E., 286 Snow, A., 18. 65 Soini, J.K.. 283, 286, 287 Spangler, G.E., 206, 21 9 Spohr, D.A., 287 Sprenger. W.O., 241, 242, 243. 244. 287 Springett, B.E., 183, 185, 219 Staas, F.A., 205, 211. 214, 219 Steenrod, N.E.. 11, 65 Steiner, M., 8, 9, 27, 53. 58, 62, 64, 65 Stevens, K.W.H., 246, 261. 287 Stirling, W.G.. 63, 65 Stolfe. D.L., 286 Stoll, E., 4, 15, 49, 62, 65 Strassler. S., 65 Street, G.B., 64 Struyokov, V.B., 210, 218 Strzhemechny, M.S., 130 Su, W.P., 18, 19, 65 Sugawara, T.. 287 Suhl, H., 255, 287 Sullivan, N., 101, 131 Sutherland, B., 4, 15, 24, 31. 34, 63, 65 Suzuki. H., 129, 131 Swift, J.W., 286 Swinney, H.L., 190. 191, 219 Symko. O.G., 288 Taconis. K.W., 211, 214, 217, 219 Takayama, H., 5,20,22, 34, 35.36.39,41. 45, 47, 49, 51. 53, 56, 60.62. 43. 4, 64, 65 Takayanagi, S., 287 Taken, F., 191. 219 Tanner, D.J., 183. 219 Taube, J.. 219

AUTHOR INDEX Templeton, J.E., 286 Teplov, M.A.. 250, 288 Terui. K., 287 Thacker, H.B., 4, 35, 62 Thomlinson, W.G., 130 Thompson. J.O., 217 't Hooft. G.. 4, 14. 65 Thouless, D.J., 13, 74, 64, 131 Thoulouze, D., 130 Titus, J.A.. 219 Tkachenko, V.J., 159, 204, 206, 218 Tofts, P.S.,131 Toombes, G.A.. 65 Tough, J.T., 135, 137, 142, 143, 146, 155, 156, 159, 160, 161, 162, 163, 164, 165, 167, 168, 169, 174, 176, 193,200,202, 204. 206, 208, 210, 217, 218, 219 Toulouse. G., 3, 11, 14, 62, 65 Trammel], G.T., 253, 288 Trickey. S.B., 64 Triplett, B.B., 254. 288 Trullinger, S.E., 41. 62, 63, 64, 65 Tsarevski, S.L., 257, 288 Tsuneto, T., 63 Tsmoka, F., 129, 131 Tsuzuki, T., 5 , 65 Tsymbalenko. V.L., 129. 131 Tynpkin, Y., 62 Uhlenbrock, D., 63 van Alphen, W.M., 205, 219 van Beelen, H., 137. 143, 159, 166. 170, 202,204,206,208,210,214,217. 219 van der Boog, A.F.M., 219 van der Heijden, G., 137. 143. 159. 165, 208, 210, 211, 214, 218, 219 van Haasteren, G.J., 219 Varma, C.M., 71, 256, 131, 287 Varoquaux, E.J., 62, 28s Vettier, C., 287 Veuro. M.C., 237, 244, 288 Vibet, C., 62 Vicentini-Miswni, M., 137, 182, 191, 217, 219 Vidal, F.. 176, 180, 219

295

Villain, J., 24, 25, 62, 65 Vinen, W.F., 135, 137. 144, 145, 147, 148, 152, 153, 154, 157, 158, 159. 160, 161, 163, 167, 171, 172, 173, 174, 175, 176, 177,178, 184. 195,201.204,206,218. 219 Volovik, G.E., 3, 11, 12, 15, 64, 65 Vuorio, M., 62, 287 Walker, L.R., 27, 65 Walstedt, R.E., 286 Weaver, J.C., 205, 219 Weber, D., 65 Weinberger, B.R., 18, 65 Weyhmann, W., 286 Wheatley, J.C., 287 White, R.M., 254. 288 Whitham, G.B., 5. 65 Wiarda, T.M., 218 Widom, A.. 81. 96, 97, 131 Wiersma. E.C., 286 Wilkins, J.W., 71, 131 Wilks, J., 82, 131 Willard. J.W., 71, 130 Williams. D.L., 287 Williams, F.I.B., 103, 131 Windsor, C.G., 65 Winter, J.M., 82, 84, 86, 130 Yamashita, Y., 82, 131 Yang, C.N.. 35, 44, 65 yang, C.P., 35, 65 Yaqub, M., 217 Yamchuck, E.J., 137. 141, 142, 143, 145. 162. 163,168. 172.192,200.202,204. 219 Yasouka, H., 287 Yoshizawa, H., 25, 63, 65 Youngblood, R., 287 Yu, W.N., 74, 75, 131, 287 Zabusky, N.J., 3, 65 Zane, L.I.. 71, 81, 130, 131 Zaripov. M.M.. 250, 288 Zeller, H., 17, 65 Zimmerman, W.. 183. 219


Anisotropy, 247. 250, 260, 272

Hyperfine interaction, 248. 253ff Hysteresis, 200

Brute force nuclear cooling. 223, 225ff Cooling power of refrigerators, 227, 229, 277, 284, 285 Counterflow radial, 183. 184 thermal, 135ff Critical heat flux, 148ff Critical velocity, 135ff Crystal-liquid interface, 1 12ff De Boer quantum parameter, 70, 71, 78 Defects point, 72, 88, 116, 127 linear, 11s. 116 Diffusion, 73ff thermally activated, 87ff vacancy induced, lOlff Dilution refrigerators, 272. 277 Dislocation, 13, 119, 127ff Eddy viscosity. 166ff Entrance length, 177, 179 Exchange enhanced susceptibility, 251). 253 Exchange interactions, 250. 253 fluctuations in superflow. 189ff

Impuritons, 8Off Induced moment states, 251 antiferromagnetic, 253 ferromagnetic, 253 Jons, 137ff, 180ff, 196ff and heat flush, 182 structure, 180, 183 Irreversibilities in nuclear m l i n g , 258, 259. 266, 283 Kapitza resistance, 231ff. 238, 243 Korringa relation, 11 Korteweg-de Vries (K de V) equation, 3, 5, 6 Lowest temperatures in nuclear cooling, 229, 234, 284, 285 Magnetic ordering. 251, 254 Magnetic soliton, 18, 24ff. 49, 61 Mass fluctuation waves, 81ff Mathematical soliton, 3, 4 NMR in solid 'He. 73ff Nuclear entropy, 267, 274, 282

t

Heat exchangers. 243. 274 Heat leak, 237, 238, 243, 277, 280, 285 Helmholtz oscillations, 176, 177 Hydrogen, dissolved in metals, 70. 91, 129 Hyperfine enhanced nuclear cooling, 224. 245ff cryostat, 267-274 Hyperfine enhanced Zeeman splitting, 249 Hyperfine field. 248

d4 system, 45ff. 60ff Polyacetylene, 18ff. 61 Praseodymium compounds, 245, 262, 272274 Pseudoquadrupole splitting, 250, 260 Quantum soliton, 34ff Quantum tunnelling, 69ff, 92ff. 127ff Quasi-one dimensional magnet, 23, 25, 50. 61

298

SUBJECT INDEX

Quasi-one dimensional systems, 4, 5 , 13ff. 61 Rare earth compounds, 262, 259, 265 Rayleigh-BCnard convection, 190ff Relaxation times in nuclear cooling, 228, 256 Roughening transition, 114ff Sample preparation in nuclear cooling, 262 Scattering cross-section, 82 Second sound, 1378. 171ff, 196ff dispersion in superfluid turbulence, 179, 180 velocity in superfluid turbulence, 179, 180 Sine-Gordon equation, 3ff. 23 Sine-Gordon system, 4ff. 30ff.43ff Single stage nuclear refrigerator, 235, 241 Singlet state, 247, 261 Spin diffusion in solid 3He, 738 Spin-lattice relaxation, 228, 256 Spin temperature, 228, 229, 282, 283

Thermal resistance, electronic, 231ff helium 11, 135 Kapitza. 231ff spin-lattice. 229 Thermal switch, 275. 277. 280. 282 Topological disorder, 16. 33, 56 Topological soliton, 3ff, 1 Iff, 61 Two stage nuclear refrigerator, 237ff. 274ff

van Vleck paramagnetism. 224, 247, 248. 253ff. 257. 261ff Vacancion, 101 Vacancy tunnelling, lOlff Vortex lines, 14, IS, 62. 139ff. 170ff. 182ff. 204ff self-induced motion, 149 Vortex ring, 150, 152. 201. 204

Zero-point vacancies, 106ff. 112 Zero-point vibrations, 69-71. 106

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