PROGRESS IN LOW TEMPERATURE PHYSICS
VIII
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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
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ISBN: 0 444 86228 5
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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<(JA)I12,the spin becomes almost planar and can be parameterized by Sn = [ ( s 2 - 7r:)’12 cos 4,,, ( s 2 - 7r;)’”sin 4,,, 7rn)
(14)
which follows from the spin commutation relation. Here &,, is the angle Sn makes with the x direction. Substituting eq. (14) into eq. (13) and neglecting higher order terms in 7rn, we obtain
H=-JS2
CCOS(~, -&,+l)+A C T : - H ~ S ; . n
n
(16)
n
Furthermore, assuming that 4, varies slowly with n, we reduce eq. (16) to
SOLITONS IN LOW TEMPERATURE PHYSICS
9
a sine-Gordon Hamiltonian;
where a is the lattice constant. The soliton in the present system is a Bloch wall where the spin turns by 2m (i.e. 2m-twist). Since the magnetic soliton cames localized magnetization
M ( x )= (-2s sech2[my(z- ur)], 2 s sech[my(z
- ut)]tanh[my(z - u t ) ] ,
A-’myu sech[my(z - ut)l).
(18) The existence of solitons can be probed by neutron scattering experiments. Here we made use of a classical solution d(z, t) = 4 tan-’[exp my(z - u t ) ] , (19) where m = a-1(H/JS)1’2 y=
(1--
:;)-”’
and
co = u S ( ~ J A ) ” ~ . (20) Magnetic solitons in magnetic fields have been observed recently by inelastic neutron scattering on CsNiF, by Kjems and Steiner (1978). A very similar analysis of planar antiferromagnetic chain in a magnetic field (Mikeska, 1980; Maki, 1980; h u n g et al., 1980) also predicts magnetic solitons corresponding to the m-twist of spins. Unlike the case of the ferromagnetic chain, the magnetic solitons in a planar antiferromagnet give rise to a large central peak for the transverse spin correlation function. Very recently the large central peak was observed by elastic and inelastic neutron scattering from TMMC [(CH3)3NMnC13] in magnetic fields by Boucher et al. (1980) and Regnault et al. (1981). 2.2.2. Magnetic flux in a Josephson junction First let us consider two superconductors separated by a thin insulating barrier (e.g. a thin oxide film of thickness 10-20A). As shown by Josephson (1962, 1965) the pair of electrons (i.e. the Cooper pair) can tunnel from one superconductor to another leading to weak-coupling between two superconducting wave functions JI1 and 4f2 across the junction. In particular when d(=dl -d2)# 0, where dl and d2are phases
K. M A K I
10
of
and &, there flows a supercurrent J per unit area across the junction
J = Jo sin 4,
(21)
where Jo is a constant depending on the temperature (Ambegaokar and Baratoff, 1963) and the characteristics of the junction. Furthermore, if a voltage V is applied across the junction, the phase difference 4 increases as
b1= (2e/h)V.
(22)
There are the well-known dc and ac Josephson effects, respectively. Now let us consider a large area Josephson junction, which extends in the x-y plane. The large area Josephson junction is characterized in terms of a transverse (scalar) voltage V and a transverse current J (the surface current along the junction), which obey (Barone et al., 1971)
v v = u,, V -J=-CV,-Josin4, V 4 = -(2eL/h)J, = (2e/h)V,
where J and V are the two-dimensional vectors lying in the plane of the junction; the capacitance C and the reactance L of the junction are
C =K/d,
L = (2AL+d ) ,
(24)
where K is the dielectric constant, d is the gap between two superconductors, and hL is the London penetration depth. Eliminating V and 3 from eq. (23), we obtain
Lc4, - 4xx- 4yy= -(2elh)Wo sin 4
(25)
the two-dimensional sine-Gordon equation. The state of the junction is completely determined by 4. Of technical importance is the linear junction, which consists of two thin tapes of superconductors separated by a thin oxide layer. In this case eq. (25) reduces to the (one-dimensional) sine-Gordon equation. The soliton here describes moving flux lines along the junction. The flux line cames a single flux quantum
and @,,-2X lO-'G cm2.
SOLITONS IN LOW TEMPERATURE PHYSICS
11
The propagating flux quanta may be used as a signal unit in a future high speed computer (Matisoo, 1978). So far, in deriving eq. (26) the voltage loss across the junction is completely neglected. If the loss mechanism is included, a moving flux will decelerate or sometimes a flux may be annihilated by colliding with an antiflux (i.e. a magnetic flux with opposite sign). There are a few more examples of the sine-Gordon system in the literature (Barone e t al., 1971). However, the significance of these other models is not very clear at the present moment.
2.3. Topological solitons We shall describe here some examples of topological solitons. The basic tool here is the theory of homotopy; solitons and singularities are characterized by the element of the homotopy group of the order parameter space formed by the ground state of the system (Finkelstein, 1966; Toulouse and Kleman, 1976; Mineyev and Volovik, 1978; Mermin, 1979). In many cases the order parameter space corresponding to the ground state is equivalent to S, the hyper sphere in the n + 1 dimensional space. Some examples are shown in table 2. The distinction between soliton and singularity (Mineyev and Volovik, 1978) will be given later. Since the physical space surrounding a singularity is equivalent to S, (n = 2 for the point-like singularity, n = 1 for the linear singularity and n = 0 for the planar singularity, for example), the table of the homotopic group r,(S,,,) is extremely useful in determining the possible textures (solitons and singularities) of the system. This is given in table 3 where w,(S,,,)(Steenrod, 1951; Finkelstein, 1966) is the homotopic group of mapping from S, to S,. For example, the theorem due to Derrick (1964), that there is no point-like soliton in a real scalar field theory in more than one space dimension, is the consequence of the homotopy theory that there is no non-trivial mapping between S, (the n + 1 dimensional physical space) and a discrete set of points (the ground state of the real scalar field theory) for n L 1. Therefore in order to have a point-like soliton in the higher space dimension more than the one dimension, it is necessary that the structure of the order parameter space corresponding to the ground state of the system is adequately complex as seen from examples in table 2. In this sense superfluid 3He-A and 3He-B, which appear below 3 mK, provide us with systems with such complex order parameters.
K. MAKI
12
Table 2 Solitons and singularities in some condensed states Order parameter space Soliton Superfiuid “He and XY model Heisenberg ferromagnet Superfluid ’He-A (dipole-locked) Supertluid 3He-B (dipole-locked) Nematic liquid crystal
Singularity
S,
-
s2
Linear
Linear (vortex) Point-like
SO(3)
Linear
Linear
S,
Linear
Linear
P2 = S,IZ
Linear
Linear
Table 3 Homotopy group n,(S,,,) $ 1
2 3 4
1
2
3
4
z o o 0 o z o o o z z o 0 z: z: z
2.3.1. Planar solitons and domain walls The simplest topological solitons are domain walls -mbedd d in three (or two) dimensional space, which are characterized by a single space variable. The mathematical structure is the same as the one-dimensional soliton, and the stability of the soliton is guaranteed by the discrete degeneracy of the ground state. Domain walls in ferromagnets (Daring, 1948; Em, 1964) and ferroelectrics (Krumhansl and Schrieffer, 1975), Frederiks walls in nematic liquid crystal (de Gennes, 1974), Discommensurations in a charge density wave (CDW) condensate (McMillan, 1976; Bak and Emery, 1976), composite solitons in sugerfluid 3He-A and n-solitons in superfluid 3He-B (Maki and Kumar, 1977a, 1977b, 1978; Maki and Lin-Liu, 1978; h4ineyev and Volovik, 1978) are we11 known
SOLITONS IN LOW TEMPERATURE PHYSICS
13
examples. Composite solitons in 'He-A owe their existence to the nuclear dipole interaction energy which breaks the continuous symmetry associated with the spin-orbital orientation into the discrete symmetry. Similarly, the existence of n-solitons in 'He-B is due to the extremely weak symmetry breaking energy due to an external magnetic field. These solitons produce special resonance patterns in the nuclear magnetic resonance in superfluid 'He, which have been observed experimentally (Avenel et al., 1975; Gould and Lee, 1976; Osheroff. 1977; Gould et al., 1980; Bartolac et al., 1981). A brief review on solitons in superfluid 3He has been given recently (Maki, 1979). In addition there are many examples in the quasi one-dimensional system, which we shall describe in section 3. Most planar solitons are described by the sine-Gordon equation or its modified version.
2.3.2. Linear solitons and singularities The following are well known examples: edge or screw dislocations in a crystal (Friedel, 1964; Nabarro, 1967), vortex lines in superfluid 4He (Onsager, 1949; Feynman, 1955) and in superconductors (Abrikosov, 1957) and disclinations in nematic liquid crystals (de Gennes, 1974). The associated topological charges are the Burger vector, the circulation quantum h/m, the flux quantum dj0= h/2e, and the strength of the disclination respectively. Vortices in superfluid 4He and superconductors are classified by rrl(S1)= Z, (the mapping from S1 to S, where S1 is the circle); the internal space associated with the order parameter and the ring which encircles the linear singularity are both S1. All of them are linear singularities in the sense that the order parameter is strongly perturbed at the center of the singularity (e.g. the vortex core). Furthermore the interaction between the singularities has long range. In twodimensional systems, the linear singularities are point-like objects and control the topological order of the two-dimensional system (Berezinskii, 1970, 1972; Kosterlitz and Thouless, 1973, 1978; Jose et al., 1977). As to linear solitons some mathematical models have been considered by Enz (1977, 1978). In terms of homotopy the linear soliton is characterized by 7r2(S2)= Z (mapping from S2 to S2 where S2 is a sphere). In order to impose the condition of asymptotic uniformity the twodimensional space perpendicular to the linear soliton is complemented by a point at infinity, which turns the physical space into S2. On the other hand, the space of the order parameter is characterized by a unit vector
K . MAKI
14
n. which has arbitrary orientation. Then the space associated with n is also S2. Indeed, both in superfluid 3He-A and 3He-B we may find linear solitons (Mermin and Ho, 1976; Anderson and Toulouse, 1977; Mineyev and Volovik, 1978). In 'He-A the unit vector is f, the symmetry axis associated with the quasi-particle energy gap, while in 3He-B it is n, the axis of rotation around which the spin is rotated against the orbital part. The linear solitons in 'He-A are surrounded by superflow with circulation an integer multiple of the unit circulation h/2m where m is the mass of 'He atom. This circulation arises from the fact that the order parameter is not f but rather the triad (8,, i2,0. Actually the rotation of 8, and 8, generates the superflow. Because of the circulation the linear solitons in 3He-A are very similar to vortex lines in superfluid 4He. However, unlike the vortex lines in 4He, they are free of singularity; the amplitude of the order parameter is almost uniform all over the space. Therefore they are often called coreless vortices. Heisenberg ferromagnets in two-dimensional space provide another example. Again, the linear soliton is turned into a point-like object, the pseudo particle (Polyakov, 1976). In the continuum limit the Hamiltonian is given by
and n : + n f + n : = 1, where n indicates the spin direction. In the asymptotic uniform ground state with n = (0, 0, l), the pseudoparticle solution is given by
where t=x+iy and a and b are complex constants. Here n is parameterized as n = (sin 8 cos 4, sin 8 sin 4, cos 0).
(30) The pseudo-particle has energy E = 2 d S 2 which is independent of a and b. The classical statistical mechanics of the pseudo-particle appear to suffer uncontrollable infrared divergence (Jevicki, 1977) very similar to that encountered by the instanton (Belavin et al., 1975; t'Hooft, 1976a, b) in the SU(2) gauge theory in the four-dimensional Euclidean space.
SOLITONS IN LOW TEMPERATURE PHYSICS
15
2.2.3. Point- like solitons In order to have a point-like soliton in three-dimensional space, the order parameter space has to be S3, the solid sphere. This object is characterized by 7r3(S,)=Z, the mapping from S3 to S3,where the physical space complemented by the point at infinity becomes S,. The only example I know of in condensed matter physics is the point-like soliton in superfluid 'He A (Volovik and Mineev, 1977; Shankar, 1977; Fujita et al., 1978). where the triad (&, &, f, covers the solid sphere S3.This object is quite similar to the vortex ring in superfluid 4He in many aspects. The energy of the point-like soliton is proportional to R, its linear dimension, and its momentum is proportional to R2.This object may be pictured as a closed ring formed of a coreless vortex with double circulation unit. However, the point-like soliton appears to be dynamically unstable (Kopnin, 1978) due to the orbital viscosity in superfluid 'He-A (Cross and Anderson, 1975).
3. Worm m one-dimenaional system There are a number of advantages in studying one-dimensional systems. First of all, from the theoretical point of view the one-dimensional system is most easily analyzed. Exact results are known for many cases (Lieb and Mattis, 1966). For example, the classical statistical mechanics of onedimensional nonlinear systems are handled exactly in terms of the transfer integral technique (Scalapino et al., 1972; Gupta and Sutherland, 1976). In section 4, we shall summarize the results of the transfer integral technique for the sine-Gordon system. The ideal gas phenomenology pioneered by Krumhansl and Schreiffer (1975) and developed by Currie et al. (1980) more recently, where solitons are treated as a new class of elementary excitations of the system, provides a simple physical picture of the system. Furthermore at the classical level, the dynamical correlation functions can be studied by the molecular dynamic calculation (Schneider and Stoll, 1975, 1978a, b). However, in the one-dimensional system the quantum effects are in general not negligible and the above classical picture has to be supplemented by quantum statistical mechanics. Then for a limited class of systems, the Bethe ansatz (1931) allows the determination of the energy spectra of the system at absolute zero temperature. Some of these results have been reviewed by Lieb and Mattis (1966). More recently, the method of the path integral was developed by Dashen
16
K. h4AKI
et al. (1974a, b) to deal with the system with degenerate vacuum. The method is essentially a perturbation theory, but is extremely useful for the weak-coupling systems. Secondly, there are many condensed systems which can be considered as one-dimensional (i.e. the quasi one-dimensional systems). It is true that there is no stable one-dimensional system in nature. However, there are many quasi one-dimensional systems, where the interaction of electrons and ions or the interaction between spins in one direction (the chain direction) is overwhelmingly stronger than that in the other directions. Then under some circumstances these systems are treated as onedimensional. In general the one-dimensional approximation is very good as long as the temperature is much higher than the interaction energy between electrons and/or atoms on the different chains. Most of these systems undergo a second order transition at low temperatures, below which three-dimensional order develops. In the vicinity of this transition temperature, the one-dimensional approximation starts to break down. As is well known, a one-dimensional system cannot have an abrupt phase transition. However, in a system where the ground state is highly degenerate, the system exhibits a crossover from the symmetric phase (or the topologically disordered phase) at high temperatures to the topologically ordered phase at low temperatures. Indeed, in many cases the degree of the disorderdness is controlled by the solition density of the system. The crossover behaviour can be studied experimentally by measuring the thermodynamic quantities like specific heat, magnetic susceptibility and correlation lengths. Classical statistical mechanics will provide a semi-quantitative description of the crossover phenomenon. At much lower temperatures but still above the critical temperature, where three-dimensional order sets in classical theory is no longer adequate. In this temperature regime there is a fascinating possibility that the quantum field theoretical results in 1+ 1 dimensions may be confronted with experiments on the quasi one-dimensional systems. In the following we shall consider two distinct possibilities separately: one-dimensional conductors and one-dimensional magnets.
3. I . One-dimensional conductors The idea of solitons was fust introduced by Rice et al. (1976) in order to describe the low temperature conductivities of some organic conductors like K2Pt(CN),Bro,33H20(KCP) and tetrathiofulvalene-tetracyanoquindi-
SOLITONS IN LOW TEMPERATURE PHYSICS
17
methane ("F-TCNQ). The low temperature conductivities are described by an activated form. However, the observed activation energies of KCP and "F-TCNQ (Zeller, 1973; Pietronero et al., 1975; Cohen et al., 1977) appear to be much smaller than the quasi-particle energy gap in the CDW phase. Since in the CDW condensate, the 4 soliton (i.e. the soliton associated with the phase of the CDW order parameter) carries charge (Rice et al., 1976), it was proposed that the 4 soliton accounted for the anomalously small activation energy in organic conductors in the CDW phase. However, the soliton model appears to be unable to describe the non-ohmic conductivity in ?TF-TCNQ observed by Cohen et al. (1977). Furthermore, in the ordered phase the interaction between electrons on different chains may no longer be negligible. In this circumstance the picture of the one-dimensional soliton may no longer apply as well.
3. I . 1 . Solitons in electronic systems An interesting model has been proposed by Luther and Emery (1974) for one-dimensional interacting electrons, which is described by the following Hamiltonian:
H = UF 1 k(a,',ab -b,',&)+2L-' k.r
c
1Vpi(k)p,(-k) k
+ ss ' ~dxJrb(x)~~~~(xWl,.(x)~v,,~ss~ + V L L , ) ,
(31)
where aka and bks (uLand b&) are the annihilation (creation) operators of the right-going and the left-going electrons with momentum k f kF and spin S, & , ( x ) and +&(x) are the field operators for the right-going and the left-going electrons with spin S and
The above Hamiltonian is thought to be a generalization of the Tomonaga-Luttinger model, which is known to be exactly solvable (Mattis and Lieb, 1965; Dzyaloshinskii and Larkin, 1974). The new term [i.e. the last term in eq. (31)] describes the backscattering of the electron (2kF momentum transfer). Subsequently Heidenrich et al. (1975) have reported that the spin part of this model is mapped to the sine-Gordon
K. MAKI
18
Hamiltonian, where the soliton in SG theory is nothing but the quasiparticle in the original Hamiltonian with the energy gap 112 1 - v I14 A=8(%) 277 1 + v,, . (33)
(-)
However, Haldane (1979) has shown that the LE model is quite artificial as the last term in eq. (31) requires a non-Abelian gauge field; the last term may only exist in a system where the electron interaction is mediated by the gapless magnon. There is no serious application of this model to describe the electronic transport of quasi one-dimensional conductors.
3.1.2. Solitons in polyacetylene We shall conclude this subsection with the Su, Schrieffer, Heeger (SSH) model of solitons in polyacetylene. Recently, Rice (1979) and Su et al. (1979) have suggested that solitons in polyacetylene will play important roles in the magnetic susceptibility of undoped polyacetylene and the electronic properties of doped polyacetylene. In particular, Su et al. (1979, 1980) formulated a complete theory starting from the microscopic model, which includes the electron-phonon interaction. The conjugated organic polymer (CH),, polyacetylene exists either as trans(CH), or cis(CH),. although the trans(CH), has lower energy than cis(CH),. The existence of bond alternation along the polymer chain (i.e. alternating “single” and “double” bonds) is essential to the SSH model. In fig. 1, we depict the two degenerate dimerization patterns of trans(CH), as well as the dimerization pattern of cis(CH),. The existence of the doubly degenerate ground state of trans(CH), allows the topological soliton, which is a domain wall separating two distinct ground states. Su et al. (1979) have shown that the soliton can exist either as neutral with spin one-half or spinless with charge fe. The neutral solitons can account for the Curie electron spins per carbon atom in susceptibility corresponding to 3 x undoped trans-polyacetylene (Shirakawa et al., 1978; Snow et al., 1979; Goldberg et al., 1979). More recently proton NMR spin lattice relaxation experiments in undoped trans-polymer reveals that these spins are mobile and diffuse in the one-dimensional space (Nechtschein et al., 1980; Weinberger et al., 1979). This appears to further confirm the idea of the neutral magnetic soliton. On the other hand, as to the charged solitons in
SOLITONS IN LOW TEMPERATURE PHYSICS
19
a
b
C
Fig. 1. The dimerization patterns of polyaatylene; two degenerate trans(CH),, (a) and (b), and cis(CH,), (c). are shown.
doped polyacetylene the experimental situation does not appear to be so clear yet. The model Hamiltonian of Su et al. (1979) is given by ~ = - C ( t n + l , n C , + + I , s ~ ,+ h . c . ) + t ~ C ( y n + , - y n ) ’ + t ~ C N
a
Y:
(34)
n
with fn+l.n= tO-a(Yn+l-Yn)
(35)
where C&(&) creates (annihilates) a rr-electron with spin S on the nth CH group and yn is a configuration coordinate for the dispacement of the CH group. Here the transfer matrix fn+l,n is assumed to vary linearly with the distance between two CH groups, which may be justified since the difference in the distances between the double bond and the single bond is 0.08 A, while the average bond distance is 1.22 A. K and M are the
20
K . MAKI
elastic constant and the mass of the CH group respectively. In an undoped CH chain there is one w-electron per one CH group and the present model is identical to a one-dimensional charge density wave system with an exactly half-filled electron band. In particular, the ground state is doubly degenerate and can be given by
which correspond to two dimerization patterns shown in fig. 1. We shall consider in the following the continuum limit (Takayama et al., 1980) of the Hamiltonian eq. (34), which is given by
where w& = 4K/M A(z) = 4aY(z), g = 4 a ( ~ / M ) ” ~ , u F = 2r0a and
is the spinor representation of the electron field (Brazovskii and Dzyaloshinskii, 1976) and u,(z) and u , ( z ) describe the right-going and the left-going electrons, respectively. The continuum limit is applicable, when the site dependence of S(z)=(-l)”y, is much slower than a, the distance between adjacent CH groups or when Ao/(2to)<<1, where 24, is the dimerization energy gap. For polyacetylene 24, = 1.40 eV and W, = 41, = 10 eV, which implies Ao/(2to)-0.14. Therefore the continuum limit should be a good approximation. The Hamiltonian equivalent to eq. (36) was recently considered by Brazovskii (1978). Minimizing eq. (36) in $+(z) and A(z), we obtain
SOLITONS IN LOW TEMPERATURE PHYSICS
21
and
Eq. (38) is the same as the Bogoliubov-de Gennes (BdG) equation (de Gennes, 1966) for an inhomogeneous superconductor, while eq. (39) is slightly different from the corresponding equation in a superconductor due to the appearance of Re in the right hand side. This reflects the fact that the displacement field y(z) is a real field (Horovitz, 1980; Knunhansl et al., 1980). The sum in eq. (39) runs over up to the Fermi level, which is chosen to be E = 0, and u, and on are the normalized eigenfunctions of eq. (W,
Idz[u$(z)u..(r)+8:(z)a..(z)l=
an,+
(40)
For the time-independent A(z), the mean field energy of the system is given by
where again the s u m in the second term runs over up to the Fermi level. The ground state corresponds to the state with uniform dimerization [ A ( z )= d,J. Then eqs. (38) and (39) yield
where k is the wave vector of the quasi-particle measured from *kF, the Fermi momentum, and w = 21, is the band width. For the CH chain of polyacetylene we have A, = 0.7 eV and w = 5 eV (Su et al., 1979), leading to A = 0,25; the polyacetylene appears to be actually in the weak coupling limit. The dimerization energy per electron of the ground state is calculated as
where L is the total length of the CH chain.
22
K . MAKl
Putting A. = 0.7 eV and to = 2.5 eV in eq. (49, we obtain
A soliton in plyacetylene may be described by
(46) where 5 is the variational parameter. The above function (46) interposes between two ground state configurations A = *Ao. Making use of the method due to Bar-Sagi and Kuper (1972) to deal with the BdG equation, eq. (38) for A ( z ) given by eq. (46) with [ = to (=uf/Ao) is easily solved (Takayama et al., 1980). We find there is one bound state with A ( z )= A. tanh(z/t),
eB= 0
and
u ~ ( z=) iug(z) =&,”* wch(z/()
(47)
at the center of the energy gap. Furthermore, there are scattering states with €,
=f E k
u k ( z ) = (2L)-”2[[1 +(k,$o)2]-112(k(o+itanhz)*i]eikz 50
u k ( z )= -i(2L)-’I2[[1 +(k&JZ]-112(k50+i
(48)
where
& =[(Upk)2+A:]112.
(49)
Substituting eqs. (47) and (48) into eq. (39), it is easy to see that A(z) satisfies the self-consistent eq. (39). The soliton energy is then calculated from
Es=E&A(z)]=E,(AO)=-Ao.
2 7r
(50)
This soliton energy (-0.632Ao) is very close to that found by Su et al. (1979) in their numerical analysis. Furthermore the phase-shift analysis tells us that one state from the valence band is taken into the bound state due to the scattering. This implies that in the neutral system where no extra electron is added or taken away, the bound state has to be singly occupied, resulting in a spin 4 neutral soliton. Furthermore, making use of the eigenfunctions in eqs. (47) and (48), it can be shown that charge
SOLITONS IN LOW TEMPERATURE PHYSICS
23
neutrality is satisfied locally. On the other hand, when an extra electron is added to the system, this extra electron occupies the bound state, as it was only half filled in the neutral case. Since the bound state wave function does not contribute to the self-consistent eq. (39) (Horovitz, 1980), the soliton energy Es is still given by eq. (50). Since &
(5 1)
To summarize, within the present model, polyacetylene can have a topological soliton either as a neutral spin 4 soliton or as a charged spinless soliton. For the neutral soliton the special magnetization distribution is given again by eq. (511, where e has to be replaced by pB the Bohr magneton. As already mentioned, the topological soliton, which is mobile along the CH chain, appears to correlate nicely with magnetic and electric properties of undoped and doped polyacetylene. However, on the theoretical side only the single soliton has been treated until now. Further studies on the interaction between solitons and the possible metalinsulator transition, when the soliton density is increased, are certainly desirable.
3.2. One-dimensional magnets There are many magnetic insulators, where the interaction between spins in one direction (i.e. the chain direction) is particularly strong so that the systems can be considered quasi one-dimensional. For a general review of such magnetic systems see, for example, Steiner et al. (1976). In these systems when the temperature is not very low (i.e. T>>TN, where TN is the NCel temperature where three-dimensional order sets in), the systems are considered as one-dimensional. In many cases due to anisotropy in the exchange energy the system behaves successively like Heisenberg spin, planar spin and finally king spin as the temperature is decreased. In particular, the transition from planar spin to king spin is described in terms of the sine-Gordon equation for S 2 1 where S is the spin length.
24
K. MAKI
For the case S =4, there are many cases where the exact spin wave spectra are known (Baxter, 1972; Sutherland, 1978). However, we shall not go into these discussions here. We shall rather limit ourselves to three particular examples. 3.2.1. Zsing-like antiferromagnetic chain Perhaps the simplest magnetic system which contains solitonic excitation is the Ising-like antiferromagnetic chain described by the following Hamiltonian:
where unis the Pauli spin operator at the site n, and IcI<< 1 and J>O. The Nee1 ground state of this system is doubly degenerate. In the limit of small e, the approximate ground states are shown in fig. 2. Furthermore, as shown by Villain (1975), there is an elementary excitation '(magnetic soliton) approximately shown in fig. 2C. Then the second term in eq. (52) transfers the soliton (or domain wall) from one site to another. For small c, the soliton energy is obtained within the tight binding approximation as
Es(k) = J [ 1 + 2~ cos(2ak)], (53) where k is the wave vector associated with the moving soliton and a is
I f l t l t l r a
b
C
Fig. 2. The degenerate ground states of an king-like antiferrornagnet (a) and (b) and the soliton (c).
the lattice constant. For example, the soliton velocity u is given by u =aE,(k)/dk = - 4 ~ a J~in(2ak)
(54) and the soliton density with velocity u is given by the usual Bolmann function for #3J>> 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<
where K = 2a-'e-@'Zo(2(3J~) 4a uo = -Ii1(2BJ~)sinh(2(kJ)
4
(60) (61)
and Io(z) is the modified Bessel function. Here K-' is the correlation
K. MAKI
26
length, which incorporates the lowest order correction in E ; in the limit E tends to zero, K in eq. (60) reduces to J defined in eq. (58) in the low temperature limit. Furthermore, uo is the thermal velocity of the soliton. Eqs. (59)-(61) will be derived in section 6. The soliton gives rise to a large central peak near q = nla, of which the width is proportional to the soliton density iis where
Therefore, the detection of the temperature dependences of K and uo will provide definitive evidence of magnetic solitons in the Ising like antiferromagnetic chain.
3.2.2. Planar ferromagnetic chain in a magnetic field Following Mikeska (1978) let us consider the one-dimensional spin system described by
H
=- J C n
Sn* &,+I + A C(S:)’- H C SX,, n
(63)
n
where Sn is the spin vector on the site n. At low temperatures [T<< S(JA)*’2] the spin becomes almost planar (i.e. IS‘!<< S). Furthermore, the magnetic energy in the x direction breaks the planar symmetry. We have already seen in section 2.2.1 that in this temperature region and in the continuum limit, eq. (63) reduces to a sine-Gordon Hamiltonian (17). In particular the spin wave dispersion is given by 6Jk
= (2A)1’2[JS2(~k)Z + Hs]”2,
(64)
where a is the lattice constant and k is the wave vector. Furthermore, the soliton corresponding to the 2n-twist of spin is given by
d ( r ,t)[=tan-’(S:/SE)] = 4 tan-’[exp my(z - ut)], where m = a-’(H/JS)”’
(
:;)-”‘
y = 1--
and co = aS(2JA)”’
(65)
SOLITONS IN LOW TEMPERATURE PHYSICS
27
the soliton energy is given by
The magnetic soliton gives rise to a central peak (Mikeska, 1978) in the dynamical structure factor. This has been seen in the inelastic neutron scattering experiment on CsNiF3 in a magnetic field by Kjems and Steiner (1978). We shall describe later in sections 5 and 6 the thermodynamics as well as dynamics of the sine-Gordon system. For the purpose of formal analysis, it is more convenient to transform the sine-Gordon Hamiltonian into a standard form:
where m* is the bare magnon energy gap and gz is the dimensionless coupling constant. In the case of the Hamiltonian (17), we have m* = (2ASH)’”
(70)
g2 = S-’(2A/J)1’2.
(7 1)
and In the case of CsNiF3, we obtain g2=0.90. As we shall see later, g2 indicates the importance of the quantum corrections. In general the classical results like eqs. (68) and (70) are modified due to the quantum fluctuation in the case of the one-dimensional systems. This will be discussed in section 5 .
3.2.3. Planar antiferromagnetic chain in a magnetic field Another magnetic system of great current interest is an antiferromagnetic chain of spins described by n
n
n
where J <0. This Hamiltonian with S = 3, J = - 15 K, and D = 2.5 K describes quite well a quasi one-dimensional compound TMMC (Walker et al., 1972; Boucher et al., 1979). Furthermore, according to Walker et al. (1972), TMMC can be considered as a planar model below T = 20 K.
K. MAKl
28
We shall limit ourselves in this low temperature region. Again we can parameterize Sn as Sn= [ ( S 2 - T : ) ” ~cos 4,,, (s2-
sin &, T,]
(73)
with (nn,4,,,)= -iSnn,.
(74)
Substituting eq. (73) into eq. (72), we obtain
H=-JS2Ccos(4n-4n+,)-(J+D) C n
~ n ~ n + l
n
-JCT:-HsCcos4n, n
(75)
n
where higher order terms in T,, are neglected. The equilibrium spin configurations are given by from
sin &2n = -sin
cos &2n = cos h n + l =bh,
= (1
(774 ~ ~ 4 Z n = ~ 0 ~ 4 2 n + l = ! h r
sin d2,,= -sin
&2n+l
= -(I
-&h2)1’2, (77b)
where
h = H/(lJI S).
(78) The ground state is doubly degenerate. Unlike the case of a ferromagnet, there are nvo inequivalent sublattices and the passage to the continuum limit is more delicate. For this purpose, it is convenient to eliminate all spins at odd sites (Maki, 1980). This is formally achieved by functional integral over spins at odd sites. (79) Where H‘ is the decimated Hamiltonian, J is given by eq. (75) and r is the imaginary time. In the low temperature region (T<
2 + 1n
&2n
+cos 42n+2-h)
(80)
T?,+~:
i(2 IJI - ~ ) - ‘ ( & n
+ $2,
+2)*
(81)
SOLITONS IN LOW TEMPERATURE PHYSICS
29
The decimated Hamiltonian is obtained by substituting eqs. (80) and (81) into eq. (75):
H’ = - t IJI s2 C m ( 4 z n - 4 2 n + 2 ) + i(2 IJI - D)-’ C 4:n n
n
+! IJI s2h2C m2(4,,) + OW).
(82)
n
Finally in the continuum limit eq. (82) reduces to a sine-Gordon Hamiltonian;
In the antiferromagnetic chain the spin wave dispersion is given by && =s[2(2
1JI-D) IJ1]”2[(ak*)2+ah2]1/2
(84)
with
Furthermore, the soliton is given by 7l
dS(z, t ) = -+2 tan-’(exp[ 2
py( z
- ut)))
In the antiferromagnetic system the soliton corresponds to a 7r-twist of the spin. Furthermore, the spin correlation function ($(z, t)S,(O, 0)) reflects directly the topological order of the system (Maki, 1980; Mikeska, 1980; b u n g et al., 1980), the soliton gives rise to a large central peak in the dynamical structure factor. Such a central peak has been observed by recent inelastic neutron scattering experiments on TMMC in the presence of a magnetic fieId by Boucher et al. (1980). The dynamical structure factor of the antiferromagnetic chain in a magnetic
K. MAKI
30
field will be discussed in section 6. The Hamiltonian, eq. (83), can be again transformed into the standard form, eq. (69), with
[
m* = (1
-31 li2H
and 1I2
g'=? S (l-&)
In particular, for TMMC we obtain g'X3.19. The existence of magnetic solitons in the above three systems CsCoCI,, CsNiF3 and TMMC appears now firmly established from the experimental point of view. Furthermore, the qualitative behaviors of these solitions are quite consistent with the classical thermodynamics of solitons. However, at a quantitative level, there are a few discrepancies between the experiments and the classical theoretical model, of which clarification is certainly desirable. It is the author's belief that most of these discrepancies will be resolved, if the quantum mechanical mass renormalization is properly included in the theory (see section 5 ) .
4. Classical s t a t i s t i d mecbania of the sine-Gordon system As has been shown by Scalapino et al. (1972), the classical statistical mechanics of the one-dimensional system can be reduced to solving the eigenvalue equation. This is the transfer matrix technique (Th4T). Let us consider a onedimensional system described by the following Hamiltonian:
The thermodynamics of the system is determined by the partition function given by
where D(4) is the functional integral over 4 and H(4) is eq. (91) with 4, replaced by 47 with 7 the imaginary time. In the classical approximation [i.e. neglecting the non-commutativity of ~ ( = 4 , f i / Cand ~ ) 41 the functional integral is replaced by the ordinary integral over configuration
SOLITONS IN LOW TEMPERATURE PHYSICS
31
spkce. In order to carry out the configuration integral it is convenient to discretize the system. Then the configuration integral decomposes into the ordinary multiple integral. Then eq. (92) is approximately replaced by
J
n
n
where
The nn integrals in eq. (93) are easily done and we obtain:
z = Z,Z, Z,,
=(2h~/C~a/3)”~,
= (AolN,
(95)
where N = L/a, L is the length of the system and A. is the largest eigenvalue associated with the following integral equation
where
Furthermore, in the limit of small a (or large N ) , the integral equation (96) is reduced to a Schrodinger equation
Furthermore, in terms of A, the correlation length of the system is given by
6;’ = a-’ ln(Adh,).
( 100)
In the present limit, Z , and &, are independent of a, as should be the case. Furthermore, the a dependence of Z,, has no effect on thermodynamic quantities like entropy and specific heat. Therefore, the lattice constant a does not show up in the physical observables. As an example, here we shall consider the sine-Gordon system (Gupta and Sutherland, 1976), described by the standard Harniltonian. eq. (69).
K. MAKI
32
In this case, eq. (98) is reduced to Mathieu’s equation
or d2 I I ,+(a,- 2q cos 20)+” = 0,
do2 where o = j g 4
and Ef is the classical soliton energy. Then in terms of the lowest eigenvalue a, of Mathieu’s equation, the configuration part of the free energy is given by
Then making use of the asymptotic expression of a,, we obtain
for T > > 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)
SOLITONS IN LOW TEMPERATURE PHYSICS
33
for T
CJL=- 3g2 p-'(q2-gq4+$.$gq6) 8hC
for T Z E ~
(109)
for T<<EX. As can be easily seen from the above asymptotic expressions, the specific heat has a broad peak around T-iE,*. Furthermore, the correlation length associated with cosGg4) is given by
where b , is the lowest eigenvalue of the state with odd parity. The asymptotic expressions for .5/2 are given by 1
G/2
-g2p-' - 8hC,(1 - Is1+h2 +B Isl' -*q4
-&Iql')
for T > E t
The correlation length increases monotonically as the temperature decreases. However, the correlation function has an inflection point around T - f E t , which is characteristic of the crossover. Furthermore, in the low temperature region the correlation length is simply given in terms of the
K . MAKI
34
soliton density &;2
tis
as
= 4iis
(see section 6). From this iis is extracted as
Within the ideal gas phenomenology (Currie et al., 1980; Maki and Takayama, 1979a), the soliton contribution to the free energy is also expressed in terms of t i s as
F;'/L = -28- '15s
(116)
which agress with the last term of eq. (106).
5. Quantmn statistice of solitons
We have seen that the classical statistics of solitons in the onedimensional system is treated in terms of the transfer integral technique (Scalapino et al., 1972; Gupta and Sutherland, 1976). Classical statistics give qualitatively correct descriptions of the crossover from the high temperature disordered phase to the low temperature ordered phase. Furthermore, the heuristic ideal gas phenomenology (Currie et al., 1980). where the soliton is incorporated as a new class of elementary excitations reproduces the transfer integral technique results at low temperatures ( T c E:). In spite of this remarkable success, however, classical theory cannot handle the bound states of a soliton and antisoliton, for example. The above agreement between the transfer integral technique result and the ideal gas phenomenology is achieved by neglecting completely the contribution from the bound states. Furthermore, classical theory cannot tell us about the temperature dependence of the soliton mass or the magnon mass as these dependences arise from the mass renormalization. Quantum statistics will answer these questions in principle. In order to deal with the quantum field theory of nonlinear systems in 1 + 1 dimensions, two methods are available: (a) Bethe's ansatz (Bethe, 1931); which gives exact results but applies only for limited classes of
SOLITONS IN LOW TEMPERATURE PHYSICS
35
nonlinear systems (Lieb and Liniger, 1963; Lieb 1963; McGuire, 1964; Yang, 1968; Yang and Yang, 1969; Bergkoff and Thacker, 1979); (b) the method of the functional integral (Dashen et al., 1974a, b, 1975), which is applicable to any nonlinear systems, although the results are reliable in general only in the weak-coupling limit. Until recently the former method was used to solve the interacting boson gases (Lieb and Liniger, 1963; Lieb, 1963; Yang and Yang, 1969). Recently, the massive Thirring model, which is equivalent to the sine-Gordon system (Coleman, 1975), was analyzed by using Bethe’s ansatz (Bergkoff and Thacker, 1979). In principle, the quantum statistics of the sine-Gordon system will be formulated in this framework. The second approach is considered as the generalized quasi-classical approach. Indeed, the method of functional integration provides a natural framework for the quantum statistics of solitons (Maki and Takayama, 1979a, b; Takayama and Maki, 1979, 1980). We shall l i s t illustrate the method, taking the sine-Gordon system as example. 5.1. Mass renormalization
Let us consider a sine-Gordon system described by the following Hamiltonian ;
which is the same as eq. (69). Hereafter we shall take h and k, as unity for simplicity. For the quantum system in 1 + 1 dimensions, the only diverging diagrams are associated with a simple boson loop as shown in fig. 3. As shown by Coleman (1977), these divergences are eliminated by the mass renormalization. We shall write the last term in eq. (117) as (m*)’coskd) = m’N[cosg(d)l, where mz = (m*)’ exp( -4g’D)
Fig. 3. The divergent selfenergy terms in the sine-Gordon system in 1 + 1 dimensions.
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36
and
where w, is the Matsubara frequency, w(k) is the boson frequency with the wave vector k. Here we have made use of the relation (Coleman, 1977)
F(4)= N[exp(@ a2/W2)F(4)1 (121) Where N is the generalized “normal product” at arbitrary temperatures (Maki and Takayama, 1979a). In the continuum limit D diverges logarithmically, which is cut off at \ & \ = A . However, in the discrete systems described in subsection 3.2, eq. (1 19) gives the finite correction to the classical result. In order to calculate this finite correction, it is necessary to have the spin-wave spectrum for the discrete system. These are given for the ferromagnetic [eq. (63)] and the antiferromagnetic [eq. (72)] systems u(k) = {2[4JS2 sin2($ak)+ A - SH]r4JS2 sin2($ak)+ SHn”’
(122)
w(k) = S{2(2 ( J l - D) IJ( [4 sin2($ak*)+!la2 cos ak*n”’,
(123)
and respectively, while A in eq. (120) has to be replaced by lr/a and lr/2a respectively. Then Do ( D at T = 0 K) is calculated as
with q =[4JS/(4JS+H)]”2 for the ferromagnetic case and a similar expression for the antiferromagnetic case except that q is replaced by (1 - & I I ~ ) ” ~ . Here K ( z ) is the complete elliptic integral. In both cases q is very close to unity. Making use of the asymptotic expression of K ( z ) , we can approximate eq. (124) by 1 Do = -ln(64JS/H) for the ferromagnetic case, (125) 41r
1
= -ln(16(fi- l)JS/H)
21r
for the antiferromagnetic case. ( 126)
SOLITONS IN LOW TEMPERATURE PHYSICS
31
Finally the renormalized magnon mass at T = O K is given by m(0) = (2ASH)1’2(H/64JS)P”’6“
and
-$,)I
[
= (1
112
H(H/16(&- 1) IJI
~
)
p
’
~
~
~
for the ferromagnetic and the antiferromagnetic case respectively, where g2 has been given by eqs. (71) and (90). This is numerically the most important quantum correction. The magnon mass is reduced by the quantum fluctuation in the case of CsNiF, in a magnetic field of 5 kOe, as m(O)=0.906m*, while in the case of TMMC in a magnetic field of 32 k O e as m(0) = 0.60m*. As the same reduction factor appears in the soliton energy, it is quite likely that these corrections account for part of the discrepancies between the observed soliton energies and the theoretical values for CsNiF, and TMMC (see section 6). Furthermore, eq. (119) predicts the temperature dependence in m. To see the temperature dependence of m, it is more convenient to rewrite eq. (1 19) as m(T) = m(O)exp[-
k’2fo(Bm)l,
(129)
where g’2 = g 2 / (
1
--$)
and .
m
and KO is the modified Bessel function. As we shall see later, eq. (129) is valid for T K E , where Es is the soliton energy. It is useful to compare eq. (129) with the classical results for T S E : [i.e. eq. (lO8)J.Noting that
[m(T)/m(o)P = (cos(g4)) we obtain
(131)
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38
Here use is made of the high temperature (Brn >> 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
SOLITONS IN LOW TEMPERATURE PHYSICS
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
5.3. Soliton energy and soliton density Z1associated with the one soliton sector is evaluated as follows: we shall first consider the contribution of a soliton with velocity u, then we shall sum over the velocity of the soliton. The contribution of the soliton with velocity v is obtained by substituting for 4:
4 = 4&,
1)+
db, t ) ,
(148)
where &(z, t ) is a classical solution
~ J z ,t ) = 4g-' tan-'{exp[rny(z - ut)D and
and 4 is the fluctuation around the classical solution. Then the functional integral D,[&(z,t ) ] is replaced by the gaussian average over 6. In particular, 2,is given by
where P=(&/C0)u is the momentum of the soliton and ns(u) is the density of the soliton with velocity u. ns(u) is given by
ns(u) = expl-
Bns(u)l,
(152)
SOLITONS IN LOW TEMPERATURE PHYSICS
41
where
q f w ) = ln(2 sinh
5
0)
and D, ok and fo have been defined in eqs. (120), (145) and (133). Here A ( k ) is the phase shift of the linear fluctuation with wave vector k scattered by a moving soliton with velocity u. At T = 0 K, f2,(u; is nothing but the soliton energy with velocity 0:
which was first obtained by Dashen et al. (1975) within the framework of quantum field theory and m(0) is the renormalized mass [e.g., eq. (127)] at T = 0 K. The second term is additional renormalization. This term may be interpreted as the one arising from the renormalization of the coupling constant [eq. (130)l. Eq.(155) together with eq. (149) implies that at low temperatures (TS m) the soliton behaves like an elementary particle, which confirms a conjecture of Trullinger (1979). In the intermediate temperature region (rn << T<<Es) eq. (153) is written as (Maki and Takayama, 1979a) &(u) = E,y
- T In[( 1+ y)pm J -- g(3) @-l(pm)3. 3 (2n)3
(157)
The first term implies that the soliton energy is independent of temperature at least for T < c E , . This is not trivial as eq. (157) results from the cancellation of a variety of terms. The second term indicates that the soliton is no longer considered as elementary. The above expression is
K. MAKI
42
very similar to the one obtained within the classical ideal gas phenomenology (Currie et al., 1980) but differs in two respects: (a) Es in eq. (157) is Ez-T, where Ef is the unrenormalized classical soliton energy; and (b) y in the logarithm is set equal to unity. Because of this additional term linear in T in R,(u), there is a minor discrepancy (a factor of e) between the TMT result in section 4 and the ideal gas phenomenology. Quantum statistics removes this minor discrepancy completely. Furthermore, quantum statistics tells that if Ef in classical statistical mechanics is replaced by Es as given in eq. (157), classical statistics are valid at least in the weak-coupling limit and for T>>m. From eqs. (155) and (157). ns(u) and iis are calculated as ns(u)= e-8Esr
for T s m
+
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 << T<<Es) is given by
with yo = 1.78 . . . . Comparing this with the TMT result, eq. (106), we note that the first three terms correspond to the classical result, while the last two terms give the lowest order quantum corrections This will be seen more clearly from the specific heat, which is given by
C J L = -T aZ0/aTZ
SOLITONS IN LOW TEMPERATURE PHYSICS
43
where m, = m(0).The last three terms are the dominant quantum corrections in the intermediate temperature region. In general the thermodynamic quantities in this temperature region can be expanded in powers of g” and g’2T/m.Classical statistical mechanics takes care of all the terms in powers of g”T/m but neglects completely the higher order corrections in g’2. It is also possible to evaluate a variety of correlation functions in the present theoretical framework (Takayama and Maki, 1980; Maki and Takayama, 1980). This will be described in section 6. Before concluding our analysis of the sine-Gordon system, we shall consider breathers in the sine-Gordon system. 5.4. Breather problems As already seen in section 2, the sine-Gordon system allows a class of other nonlinear solitons called breathers in addition to solitons, and antisolitons. It is also noted that breathers may be considered as the bound state of a soliton and antisoliton. Until now we have neglected the breather degree of freedom in the functional integral and found perfect agreement with the TMT result in the classical limit, g‘’ + 0 and m/T+ 0. It is also easy to see that, if the breather contribution is added to the above result, it destroys completely the perfect agreement so far obtained (Maki and Takayama, 1979b). This is the gist of the breather problem. However, we cannot simply ignore the breather degree of freedom in the theory. For example Dashen et al. (1975) have shown that breathers have to be quantized at T = 0 K. Within the semiclassical approximation, they have shown that the breathers become a class of bosons with discrete mass spectrum:
M, = 2Es sin(ng”/ 16)
with n = 1,2,.
..,
<8~g’-~, (164)
where E , and g’’ have been defined in eqs. (156) and (130), respectively. Furthermore they have shown that there is no other bosonic excitation at low temperatures: the breathers exhaust the bosonic excitations of the system. Later the above mass spectrum was confirmed by Luther (1976), who exploited the relation between the bound-state spectrum of the XYZ chain (Johnson et al., 1973) and that of the sine-Gordon system. Indeed in the weak-coupling limit Dashen et al. (1975) have shown that the n = 1 breather (the first breather) is nothing but the renormalized linear mode,
44
K. MAKI
which is verified within perturbation theory. Furthermore, in the same limit, they have shown that the nth breather is the bound state of n first breathers. In fact, in the weak-coupling limit, eq. (164) is approximated as M 1= m(0)[1-&g‘2/16)2J (165)
The latter exhibits the bound state spectrum of the interacting bosons in 1+ 1 dimensions (McGuire, 1964; Yang, 1968), where the second term in eq. (166) is the binding energy. This suggests that there are two temperature regions where the breathers play different roles. In the low temperature region where the binding energy is much larger than the temperature (E$!>>T),all the breathers are elementary. In this low temperature region, Z,,the partition function of the soliton free sector, is completely given in terms of breathers dk no(=- 0 - 1 ~In ~2,) = (BC,) -ln(1 -e+E-(k)), (167)
cI “O
“-1
-.m
27r
where &(k)=(*+C;k2)”2 and Mngiven by eq. (164) and no is the largest integer less than 8.rrgP2. The expression should replace eq. (144). for example. On the other hand in the temperature region T >> 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<<&, eqs. (165) and (166) are generalized as (Maki and Takayama, 1980)
Unlike the soliton energy E,, the breather mass decreases rapidly as the M2 and M 3 temperature decreases. The temperature dependences of MI, are shown in fig. 4 as functions of TIE, for g’’< 1. For T = fE,these masses become less than one half of the corresponding masses at T = 0 K.
SOLITONS IN LOW TEMPERATURE PHYSICS
45
TI Fs Fig. 4. The temperature dependences of the lower breather masses & in the weakcoupling limit.
These breathers not only contribute to thermodynamic quantities like the specific heat at low temperatures, but also appear in a variety of correlation functions. The latter will be described in section 6. 5.5. The b4system A similar analysis is carried out for other systems (Takayama and Maki,
1979). We shall describe here the result for the 44 model with the Hamiltonian: H=$C,'!dr[4:+
C~4~-$(m*)24'+!A44]
(171)
where rn* and h are the bare mass of the 4 field and the coupling constant, respectively. As a classical field theory the vacuum of the present theory is doubly degenerate at 4 = *d$, where 4: is given by
4;
= m*A
-"'.
(172)
K. MAKI
46
However, quantum fluctuations give rise to a divergent contribution in the continuum model and mass renormalization is necessary. Substituting 4 = 40+dinto eq. (171) and eliminating the coefficient of 6, we obtain
where m2= (rn*)2 - 3hD = A4;
and N means the normal product. In the low temperature region (TccE,) where the soliton density is small, the thermodynamic potential is given by
R = 0, - @-lAs
(176)
where Ro is the thermodynamic potential of the soliton free sector and iis is the soliton density defined by
z,/z,= Liis
(177)
and 2,is the partition function of the one soliton sector. In contrast to the case of the sine-Gordon system there is no factor hvo in the coefficient of the second term of eq. (176). This is because in the 44 system the jump (cb0, -cbo) has to be always followed by the jump (-40,cbo) and vice versa. In this sense there is no antisoliton in the present system. The thermodynamic potential -L-’p-’ In Z,) is obtained in the weak-coupling limit (A/rn2cc 1) as
a,(=
where
SOLITONS IN LOW TEMPERATURE PHYSICS
47
and A is the cut off momentum, and fo has been defined in eq. (130). From eqs. (178) and (180), the asymptotic expression of f2, and m the renormalized mass are easily obtained. In the intermediate temperature (mo<
and
{
2 [2im
m=m0 1 - 7
+In
where y = 1.78. . . . The temperature dependence of Ro and m are very similar to those of the sine-Gordon system, especially in the weak-coupling limit. The classical kink (soliton) associated with eq. (171) is given by
Then substituting 4 = &+ 4 into eq. (171), the soliton related quantities are obtained as follows (Takayama and Maki, 1979):
Here A(&) is again the phase shift of the linear mode scattered by the soliton. The asymptotic behaviors of R,(u) are then given by
M u ) =Esy
for T a m
= E s y - Tln[(pm/31’2)(1+y)(2+y-’)]
(187) for T>>m, (188)
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48
Table 4 Renormalized masses (mo. m) of linear modes and soliton energy
where E s =2mi - - ( - - T3 ) m o1 3h 27r 4(3 )
Eq.(187) indicates that the soliton behaves like an elementary particle at T = 0 K, while eq. (188) implies that the soliton energy Es is independent of the temperature. The density of solitons with velocity u and the total soliton density are given then by ns(u ) = e-oEST
-- Bm (1 + y)(2+ y-l)e-BEsV
for T s m
(190)
for T >> m
(191)
and
respectively. Again the results for T > > m agree with the TMT result (Scalapino et al., 19721, when m* and Ei are replaced by m and Es. We have summarized the renormalized masses of the linear mode and Table 5 The density of soliton with velocity u; %(u)
SOLITONS IN LOW TEMPERATURE PHYSICS
49
soliton energy of the SG system and the d4system in table 4. The mass of the linear mode decreases linearly with the temperature as the temperature increases, while the soliton mass stays constant in both systems. Furthermore, ns(u), the density of solitons with velocity u for these systems, is summarized in table 5 . These soliton densities appear explicitly in the dynamical correlation function (section 6).
Dynamical correlation functions of the one-dimensional system are extremely valuable. On the one hand these functions contain detailed structures associated with the one-dimensional solitons. Furthermore, in the case of magnetic solitons these correlation functions are accessible by neutron scattering experiments. Some of these correlation functions have been analyzed in the framework of the ideal gas phenomenology (Krumhansl and Schrieffer, 1975; Kawasaki, 1976; Mikeska, 1978) and within the molecular dynamic calculation (Schneider and Stoll, 1975; Maki, 1981). In the following we shall describe the correlation functions in the framework of quantum statistics (Takayama and Maki, 1980). We shall limit ourselves to the correlation functions of the sineGordon system and the 44 system.
6.1. The sine-Gordon system
6.1.1. General formulation The ensemble average of an observable Q(4) is calculated as
is the functional integral in where 2 is the partition function and DN(4) the nth sector defmed in eqs. (138) and (139). In the temperature region T c E , , eq. (191) is further simplified as
where (Q)o is the thermal average of Q in the soliton free sector (i.e. the
K. MAKI
50
N = 0 sector), and
where &(z, 1 ) is the classical soliton solution with velocity u given in eq. (149) and ns(u) and Q,(u) have defined in eqs. (152) and (153), respectively. The factor two in the second term of eq. (195) arises from the fact that the antisoliton as well as the soliton contribute to (Q)l(u) and the momentum integral is over the soliton momentum P(=E,yu). Here it is assumed that
(Q>i(u) = (Q)-i(u).
(197)
Making use of eq. (195) we shall evaluate the correlation functions.
6.1.2. Correlation functions
The following correlation functions are of particular interest, since they are accessible by neutron scattering experiments in quasi one-dimensional magnetic systems (see subsection 3.2)
These correlation functions zre directly related to the spin-spin correlation functions of the quasi one-dimensional magnetic systems as we shall describe in table 6. To iliustrate the method, we shall calculate xcc. Replacing Q(z, t ) in eq. (196) by
Q = cos[g4(z, t)hsCgd(o, 011
(203)
SOLITONS IN LOW TEMPERATURE PHYSICS
51
Table 6 Spin correlations as correlation functions of sine-Gordon system
where
I
u2
X&(S)=~ dz’sin[g&(z+z’, r)Isin[g&(z’, O)] L -LIZ
f ( I ) = cosech’ 1 ( 1 coth I - l), g ( I ) = I cosech I(f - 2 coth i + 2 cosech’ I ) , and
f = my(z - at), Furthermore, in the weak-coupling limit the averages ( ): in eq. (204) are replaced by ( )o. Substituting these into eq. (195), we find
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52
where x;Jz. f ) and X : ~ ( Z , t ) are the corresponding correlation functions in the solition free sector; X 3 Z , 1)
= (cos[g4(z,
t)kos[g4(0,0)D”
(209)
These correlation functions are evaluated by simple perturbation calculations as
and x;%(z, t ) = e-g’D{sinh[g2D(z, t)l(N[cos g4(z, t)cos &(O, 0)D
+ cosh[g2D(z, r)l(nrlsin g 4 k Osin g4(O,O>D>,
(212)
where N is the normal operator introduced in subsection 5.1 and
and
Here D(k, w ) is the propagator of the linear modes at finite temperatures. The explicit form of D has already been giver. in eq. (120). It will be more useful to give here the Fourier transforms of x: and x: ;
SOLITONS IN LOW TEMPERATURE PHYSICS
where
2
no(k, 0 )= x [coth(!
0,) -coth(f
a,)]
+ ( R 1 + n 2 ) 2 - w 2[ ~ o t h ( f R , ) + c o t h ( ~ 0 ~ ) ] , "+"
(218)
and M , is the first breather mass (i.e. the renormalized magnon mass). Here we made use of the relation (119). Furthermore, the two magnon term in eq. (212) is calculated within the ladder approximation (Maki and Takayama, 1980). As is well known x: is dominated by the single is dominated by the two magnon term (i.e. the first breather) while magnon term. The intensity of the single magnon term in x:s decreases almost linearly as the temperature increases in the temperature region m << Tcc E,;
Actually the almost linear decrease of the magnon intensity in CsNiF, in a magnetic field has been observed experimentally by Steiner and Kjems (1977), which is consistent with the above expression. It is sometimes more convenient to rewrite eq. (208) as
where the first term is purely due to the bosonic excitations (i.e. the magnon term in the magnetic system), the second term is due to the soliton and the last term involves both bosons and solitons. In particular, we have
K. MAKI
54
Similar decompositions of other correlation functions are made:
x z ( z , t ) = 16mg-'
(234)
and
h ( z )= 2 coth f,
k ( z )= 5 cosech 2'.
(235) In eq. (232) we have summed over the higher order contribution in the soliton density ns(u), as the single term gives a divergent contribution for lzl>>m-'. For lzl>> 1, eq. (232) is further simplified as
and ~ ~ = ( 2 / 7 r @ Eis~ )the " ~ thermal velocity of the soliton. Eq. (237) implies that the correlation length of x+&, t ) is controlled by the soliton
SOLlTONS IN LOW TEMPERATURE PHYSICS
55
density iis. The planar antiferromagnetic chain considered in 3.2.3 is of particular interest because the transverse spin correlation is proportional to xi+. Therefore, x f c f ~is directly accessible to a neutron scattering experiment (Boucher et al., 1980a, b, Mikeska, 1980; Maki, 1980; k u n g et al., 1980).
6.1.3. Dynarnical structure factors Dynamical structure factors of the sine-Gordon system are obtained from eqs. (223)-(226). They are related to the Fourier transform of the correlation functions by the fluctuationdissipation relation;
S , (k,o)= 2( 1-e-*)-’8,Xm
(k,o).
(238)
In general we can write
S , (k,o)= S”,( k, o)+ S s ( k, w ) ,
(239)
where S k ( k , o)is the magnon term, while S G ( k , o)is the soiiton term. Furthermore, except for the case of S&(k, o),S%(k, w ) is proportional to the soliton density ns(u) with the soliton velocity u =o/k (Mikeska, 1978);
S s ( k , o)= 2 l(l-e-@”’)-*l-Es FA(k)fi(k)ns(o/k), 27r2k
(240)
where ns(u) has been defined in eqs. (158) and (159) and F,(k) and e ( k ) are the solitcn form factors associated with operators A ( 4 ) and B ( 4 ) . The soliton form factors are summarized in table 7. Table 7 Soliton contribution to operator A ( & )
K. MAKI
56
In table 7, we have not shown the soliton form factor associated with Scic, since it diverges in the infinite system. Indeed in this particular case the correlation length is controlled by the soliton density itself, as this correlation function is sensitive to the topological disorder. The soliton term of Sicf,(k, w) is then no longer given by eq. (239) but as (Takayama and Maki, 1980; Maki, 1980; Mikeska, 1980):
[
(Ey+
S&( k,w) = 2 I(1 - eCBW)-’(( m(T) 7 K ) k 2+ m 2m0 UO
K 2 ] 3 ’ 2 , (24 1)
where K = 4ris and uo is the thermal velocity of the soliton. In deriving the above expression, we made use of the approximate expression (237) for x:\,(z, I ) . In the temperature region T<<E,, the magnon contribution of S,(k, w) is dominated either by the single rnagnon term or the two magnon term (Maki, 1981) depending on the 4 parity; if A ( - & ) = - A ( 4 ) , the magnon term is described in terms of the single magnon term, while if A ( - d ) = A ( & ) , it is given by the two magnon term. Therefore we obtain;
where
where E , ( k )= (M:+ Cikz)1’2and
n(k,w) has been defined in eq. (217).
SOLITONS IN LOW TEMPERATURE PHYSICS
57
Im n(k,w ) has the following asymptotic expression;
g27r 1Im ~ ( k , o ) ~ = - { S [ o - E 2 ( k ) ] + S [ o + E z ( k ) ~ 64M
+
for T s m and -
64M
( F y} ’
(coth P M) 2{S[O - E,(k)]+ 6[0
+ E2(k)n
(6) ] 2
+ 8M3 ~~e(c,k-lwl)[l-
(249)
112
for Es >> 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 << T<<Es and in the weak-coupling limit (Maki, 1980)
60
K. MAKI
where
M
= MI,
and M,, has been defined in eq. (166).
6.2. The 44 system Let us consider another system described by the Hamiltonian, eq. (171j, which may be considered as a model for a one-dimensional ferroelectric system (Krumhansl and Schrieffer, 1975). Of particular interest is the correlation function x&(z, r) in this system defined by x @ ( z , ~= )
W ( O , 0)).
(259)
This correlation function is evaluated in a similar fashion as x f d C ( zr), in the sine-Gordon system and given by (Takayama and Maki, 1980);
and 4o and ns(v) has been given in eq. (174) and eqs. (190) and (191), respectively. Again in the limit 121>> 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
SOLITONS IN LOW TEMPERATURE PHYSICS
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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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
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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
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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).
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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
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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,
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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.
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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 x<
SU(r12)=U ( r I 2 + a ) - U(rI2)-a(av/ar,,)<
( 1 1)
Otherwise, the inhomogeneity of the potential energy will strongly influence the tunnelling probability and the energy spectrum E ( p ) will change considerably. Condition (11) is more than satisfied at rI2 Ro. In fact a4 a SU(R0)- vo--A lO-’A. R4, Ro
-
-
It is violated at distances rlzG R,, where R , - a ( V0/A)1’4.Thus, there is a range of distances R 1< r12< Ro where the interaction of impurities is
QUANTUM CRYSTALS
85
strong, but they can still be considered as interacting impuritons with the Hamiltonian, eq. (8). The region of concentrations in which the picture of interacting impuritons is valid is determined by the condition that the mean distance between the impurities should be less than Ro but greater than R , , i.e.
-<x<(2) vo A
3/4
or 1 0 - ~< x <
It is just the region of concentrations that should be considered in order to explain the experimental data presented in fig. 5. It is convenient to start the qualitative consideration of the motion of strongly interacting impuritons with the case of an isolated impuriton moving due to the action of a force F which is constant in space and in time. From the ordinary equation of motion p = F we determine the time dependence of the quasi-momentum p = po+ Ft, where po is the initial value of the quasi-momentum. If the force F is directed along one of the principal crystallographic axes, the energy E(p) and velocity u = aE/ap, which are periodic functions of the quasi-momentum with a period of hla, will also periodically change in time with a period of the order of h/aF. The mean velocity is, apparently, equal to zero. Therefore, when a constant force acts on an impurition it oscillates in space with a frequency of the order of aF/h and an amplitude of the order of u / ( a F / h ) - A / E Yet, there is an essential difference between the particular case considered and the general case when the force is directed arbitrarily. Since there is no periodicity in momentum space along an arbitrary irrational direction, all the three components of the impuriton velocity vary, in the general case, rather randomly. The trajectory of an impuriton is a random curve of the type shown in fig. 6. Since the potential energy -Fr cannot
Fig. 6. The trajectory of an impuriton due to a constant force.
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A.F. ANDREEV
change by more than A, the trajectory is entirely confined within a layer with a thickness of the order of L - A / F perpendicular to the force direction. The characteristic curvature radius of the trajectory coincides with L (by an order of magnitude). The impuriton moves along the trajectory with a velocity of u aA/fi. Hence, it performs the diffusion motion in the plane perpendicular to the force direction. The diffusion coefficient is, in order of magnitude, D - u L or
-
D
- ad 2/hF.
(12)
Such a motion, is, in principle, well known from the electron theory of metals. In the case of electrons, however, the diffusion motion discussed is, in fact, unobservable, because the distance L is always much greater than the free flight path due to a wide energy band of electrons. The above arguments enable the character of the motion of strongly interacting impuritons to be elucidated. The force F caused by their interaction is
Due to the action of this force the impuritons diffuse with a diffusion coefficient determined by eqs. (12) and (13) A2
D-- fia2Vo6 2 To calculate the diffusion coefficient of a 'He - 4He solution in the range of concentrations lO-'<x < it is sufficient to substitute the mean distance between the impurities a/x1I3 for t12in eq. (14). So, we have (Landesman and Winter, 1974; Andreev, 1975)
This formula agrees with the experimental data because it gives the same result as eq. (10) at the boundary of the applicability region, i.e. at x A/V,while the experimental accuracy is not high enough to distinguish between close laws x-' and x - ~ " in the interval lO-'<x< Therefore, in the region of higher concentrations the experimental data can be explained naturally by the specific diffusion of impuritons arising due to the action of the interaction forces.
-
QUANTUM CRYSTALS
87
4.3. Zmpuriton-phonon interaction We have considered so far rather low temperatures where the quantum diffusion was mainly determined by the interaction of impuritons with one another. When the temperature increases, the interaction of impuritons with phonons begins to play an important role. This reduces the free flight path of impuritons, so the diffusion coefficient has to decrease with the increasing temperature. This is the reason, however, why at fairly high temperatures the ordinary thermally activated diffusion, which grows exponentially with increasing temperature, becomes a more important diffusion mechanism than the quantum diffusion of impuritons. We have, thus, three characteristic temperature regions (fig. 7). In region I (low temperatures) the diffusion is limited by the impudon-impuriton scattering and does not depend on the temperature. In region 11 (intermediate temperatures) the diffusion is mainly limited by the impuritonphonon interaction, the diffusion coefficient decreasing with increasing temperature. Finally, in region 111 the thermally activated diffusion mechanism plays the major role. Since, apparently, the diffusion depends on the impurity concentration only in region I (small concentrations), the increase in the concentration leads to the narrowing of region I1 and eventually to its complete vanishing. The dashed curves in fig. 7 show how the diffusion temperature dependence varies at higher concentrations, x3 > x2 > x,. D
I ; I T-'
Fig. 7. Temperature dependence of the diffusion coefficient of impurities for various concentrations.
A.F. ANDREEV
88
Let us consider the phonon region 11. The time between collisions of an impurition with phonons is 7 - ( N p h m + ’ , where ]Vph-(T/$a)’ is the number of phonons per unit volume, c is the velocity of sound, a,, is the cross-section of the scattering of a phonon from an impuriton. At temperatures low as compared with the Debye temperature Q the longwavelength acoustical phonons play the major role. The cross-section of scattering of such phonons from point defects is known to be proportional to the fourth power of the phonon wave vector q T/&. Hence, a p h ~ ’ ( q a )a2(T/W4 ~ and the time between the collisions is 7 (a/c)(@/T)’-(h/@)(@/T)7. The expression for the diffusion coefficient D u27, contains, however, not 7, but the transport time 7, between the collisions which differs considerably from 7. The fact is that for all attainable temperatures the width of the impuriton energy band is smaller than the temperature. Under these conditions the impuriton quasimomenta are equal, by an order of magnitude, to h/a, while the phonon momenta are considerably smaller. It is known from the theory of low-temperature conductivity of pure metals that in this case the transport time differs from 7 by an additional factor (@/T)’.Thus, in the phonon region the diffusion coefficient
-
-
D
-
2 A 2 63 --ahO (-)T
is inversely proportional to the ninth power of the temperature (Andreev and Lifshitz, 1969). The region of applicability of eq. (16) is a point of interest. The free flight path of the impuritons is equal to 1 - u7 a(A/@)(WIY7.At temperatures T<
-
QUANTUM CRYSTALS
89
impuritons
where Z is the integral of collisions with phonons. The next key point is that the energy spectrum of impuritons is of the form E ( p ) = Eo + e(p), where Eo is a constant independent of the quasi-momentum, e(p) is a function of the quasi-momentum which is equal, in order of magnitude, to A and is, therefore, much smaller than the temperature. If T<
(18)
-n(q’)[l + n(q)lf(p’)}6(p+q-pJ-q‘)~[o(q)-o(q’)I,
where n(q) is the distribution function of the phonons, w(q) is their energy spectrum, W is the probability of the scattering of a phonon from an impuriton. Eq. (18) differs from the ordinary collision integral only in that we have neglected in the argument of the &function, expressing the conservation of energy, e(p) and e(p’) in comparison with the phonon energies because the latter coincide, in order of magnitude, with the temperature and therefore greatly exceed A. It can easily be seen that the Boltzmann equation in the form eqs. (17) and (18) holds if h / ~ >a)and A/T >>A (or I<< a) consists only in the fact that in the former case we could take into account e(p) in the argument of the 6-function in eq. (18), while in the latter case it would be an overestimate of the accuracy. In both the cases it is sufficient to use eqs. (17) and (18) to calculate the diffusion coefficient, which yields eq.
-
90
A.F. ANDREEV
0-1 0-2 4- 3
1.o
1.5
2.0
2.5
T-'(K-')
Fig. 8. Temperature dependence of the diffusion coefficient for 'He impurities in 4He crystals for various concentrations: 1. 6 x lo-'%; 2, 0.75% (Mikheev et al., 1977); 3, 5 x 10-*O/0 (Allen and Richards, 1977).
(16). The condition h / ~ < T < is, evidently, equivalent to the condition T < < O .Let us emphasize, however, that the above arguments do not prove the applicability of eq. (16) to the diffusion coefficient actually observed at temperatures up to the Debye temperature. In the derivation of this formula we assumed the thermally activated diffusion mechanism to be vanishingly small. In fact, the thermally activated mechanism becomes the major one at T<<9, as is the case with 'He - 4He solutions. Fig. 8 shows the experimental data on the temperature dependence of the diffusion coefficient of 'He impurities in hcp 4He crystals with a molar volume of about 21cm3 for different concentrations (Mikheev et al., 1977; Allen and Richards, 1977). Curve 1 corresponds to fairly small concentration x = 6 x lo-' (Mikheev et al., 1977). Here we can see the temperature regions I and 11. In the phonon region I1 the experimental points fit the power law T-" where n = 9* 1 (Mikheev et al., 1977), which is in complete agreement with eq. (16). Curve 2 corresponds to fairly high concentrations x = 7.5 x lo-' (Mikheevet al., 1977). In this case the curve exhibits the temperature regions I and 111; the intermediate phonon
QUANTUM CRYSTALS
91
region is absent, as it must be at higher concentrations. Curve 3 corresponds to an intermediate concentration value x = 5 x (Allen and Richards, 1977). Here all the three temperature regions can be clearly seen. When comparing quantitatively eq. (16) with the experimental data in region 11, we should bear in mind that the coefficient of T 9 depends very strongly on the Debye temperature. More exact calculations by Pushkarov (1974) show that the quantity equal to about ( 8 / 8 ) must appear in eq. (16) instead of 8. The comparison (Mikheev et al., 1977) of eq. (16) with the experimental data performed with due account made for this fact yields the width A of the impuriton energy band which does not contradict the above value. So, we may consider it to be a well established fact that impurities of 3He in 4He crystals behave like delocalized impuritons moving freely through the crystal. A rapid growth of the diffusion coefficient with decreasing temperature in the phonon region is the most clear manifestation of this fact. Let us note in this connection that if it is a question of the diffusion of impuritons in metals, e.g. the diffusion of hydrogen in niobium o r zirconium, a much slower law T-' (Andreev and Lifshitz, 1969) caused by the interaction of an impuriton with conducting electrons should be observed instead of the law T9. The time 7 between collisions of the impuriton with the electrons is 7 -(net)*)-', where n, N,(T/e,) is the number of electrons in the smearing region of the Fermi surface; N, is the total number of electrons per unit volume; eF, uF, are the energy and velocity on the Fermi surface; u a2 is the cross-section of scattering of the electrons on the impurity. The diffusion coefficient is, in order of magnitude,
-
-
D - tr2r--
fiAZ
PET'
where pF is the Fermi momentum of the electrons. 4.4. Two- and one-dimensional impuritons
In section 4.2 we have already been confronted with specific features of the impuriton dynamics caused by the smallness of the energy band width K as compared with the characteristic energy V,- lo-' K of A interaction of two impurities at the interatomic distance. If the distance
-
92
A.F. ANDREEV
-
between the impurities r12 is smaller than the interaction radius Ro a( VO/A)”’ but greater than R , - a ( V0/A)1’4, the impurities, as we have seen, perform a joint diffusion motion with the diffusion coefficient determined by eq. (14). Since at rI2
6 V b ) - vo(a/r12)3(p/r12)2.
(20)
-
If r12> R2, where R2- a( Vo/A)1’5,then for p a the value of S U ( p ) is smaller than the band width A. Under these conditions it is practically impossible for the impurities to migrate along n, but for the migration in
93
QUANTUM CRYSTALS
perpendicular directions the situation is completely analogous to that considered in section 4.2. The trajectories of the impurity atoms are random curves lying in planes perpendicular to the hexagonal axis. The curvature radius of the trajectory po is determined from the condition that the equality S U ( p ) - A should be performed for p-po i.e. poa(A/Vo)”2(r12/a)5’2. Hence, the impurity atoms located at a distance rt2 (satisfying the condition R , > rI2> R,) in the direction close to n perform a peculiar two-dimensional diffusion motion in the hexagonal plane of the crystal. The diffusion coefficient is, in order of magnitude,
D
- upo - (A3r~2/fi2Voa)1~2.
(21)
Let us emphasize that some accidental extremum of the function Vo(n) in the direction n is insufficient for such a motion. The crystal must be periodic in the plane perpendicular to n that holds only for isolated crystallographic symmetry axes. Let, finally, r,2
b
6
b
Fig. 9. Two-dimensional quasi-particle of two impurity atoms.
A.F. ANDREEV
94
,
. Fig. 10. One-dimensional quasi-particle of two impurity atoms.
properties of the crystal lattice. According to this symmetry the pairs of points AB, ABI and AB2 are crystallographically equivalent. The same is true for tunnelling of the first atom in plane PI from A into C, or C,. It is evident that due to the tunnel transitions of the type considered the system of two impurity atoms can move as a whole over the entire hexagonal plane, but not in the direction normal to it. Since these tunnel transitions occur with the exact conservation of the impurity energy, the motion is completely coherent. The pair of impurities behaves like a peculiar two-dimensional quasi-particle which moves as a free particle but only in the crystal hexagonal plane (Andreev, 1975). Since a given coordinate of one of the atoms implies six different positions of the other atom, the energy spectrum of the above two-dimensional quasi-particles contains six branches. Fig. 10 shows the configuration of two impurity atoms which can only move in one direction (Meierovich, 1975).Point A is the position of the first atom, point C is the projection of the second atom onto the same hexagonal plane. Here the first atom can tunnel from A into Al without changing energy. Next, the second atom can tunnel in its hexagonal plane from the point with projection C into the point with projection C1, and so on. The pair of impurity atoms behave in this case like a one-dimensional quasi-particle which moves freely along the only direction in the crystal parallel to AAI. The energy spectrum of these quasi-particles contains two branches. Fig. 11 is an example of a complex of three impurity atoms which constitute a one-dimensional quasi-particle (Meierovich, 1975). By displacing each time one of the atoms into a neighbouring lattice site, we can
QUANTUM CRYSTALS 0
0
95
0
0
Fig. 11. One-dimensional quasi-particle of three impurity atoms.
transform the initial configuration ABC through crystallographically equivalent configurations AB,C, AB,CI, AB2Cl into configuration AlB2Cl which differs from the initial one by a translation along the straight line AAI. Let us emphasize that in all the cases considered two- or onedimensional quasi-particles move through successive tunnel transitions of one of the impurities into the nearest lattice site. Therefore, the tunnelling frequencies and widths of the energy bands, and hence the velocity of motion, coincide, in order of magnitude, for all quasi-particles with the corresponding values for isolated impuritons. The possibility of the coherent motion of impurity atoms, which are the nearest neighbours in the crystal lattice, was noted by Richards et al. in connection with their experimental data (Richards et al., 1975) on the dependence of the spin-lattice relaxation time TI on the N M R frequency for 'He impurities in hcp 4He crystals. These data are presented in fig. 12. The spin-lattice relaxation depends strongly on the possibility of motion of neighbouring spins because the relaxation is the result of spin flips due to fluctuations of the pair dipole-dipole interaction which is caused by the relative motion of the spins. Since the dipole4ipole interaction reduces rapidly with increasing relative distance, the spin pairs occupying adjacent (neighbouring) lattice sites play the major role in the relaxation. The most interesting feature of the data presented in fig. 12 are resonance singularities at frequencies near 1.5 and 3.0MHz. This is a clear indication that pairs of neighbouring impurities move with characteristic frequencies
96
A.F. ANDREEV
10'
I
0
I
I
1.o
l
.
2.0 f(MHz)
,
,
.
l
,
3.0
,
,
,
4.0
Fig. 12. Spin-relaxation time for 3He impurities in 4He crystals with a molar volume of 3, 2 . 5 ~ 21 an3 at T=0.53 K as a function of frequency: 1, U.lo/~;2. 5 x (Richards et al., 1975).
of the order of 1 MHz, that is of the same order as the frequency A/h of tunnelling of isolated impurities. Mullin et al. (1975) and Sacco and Widom (1975, 1978) have calculated the contribution of 3He impurity pairs, which are nearest neighbours in a 4He lattice, to the spin-lattice relaxation rate F'.It was assumed in the calculation that the simplest two-particle process corresponding to fig. l a constitutes the main tunnelling mechanism (for impurities, which are nearest neighbours, the results depend, in general, on the character of the tunnelling process). In an hcp lattice there can be two types of nearest-neighbour pairs, and both of them must behave like two-dimensional quasi-particles. The first type is a particular case of fig. 9 corresponding to a situation where both impurity atoms lie in the same hexagonal plane (points B and C coincide). The second type is shown in fig. 13. In this case the impurity atoms are in neighbouring hexagonal planes. The points in fig. 13 correspond to lattice sites in one of the planes, the crosses to the projections of the lattice sites of the second plane onto the first plane. Point A is the position of the first atom, point C is the projection of the second atom. Here the first atom can tunnel without changing its energy into A, and AZ, and the second
QUANlVh4 CRYSTALS
97 0
+ 0
0
Fig. 13. Two-dimensional quasi-particle of two impurity atoms which are the nearest neighboun.
one into positions with projections C, and Cz.The results of the calculation (Mullin et al., 1975; S a m and Widom, 1975, 1978) confirm the qualitative picture of fig. 12, though they do not permit the unambiguous conclusion on the two-dimensional character of the impurity motion. Generally, there has so far been no experimental evidence for the two- or one-dimensional character of the motion of impurity pairs. It is of interest, in this respect, to consider the fine structure of the wings of the nuclear magnetic resonance line caused by impurity pairs (Andreev, 1976a). It is known (Abragam, 1961) that the dipole-dipole interaction slightly shifts the resonance frequencies o of a system of two spins with respect to the frequency yH ( y is the gyromagnetic ratio, H is the external magnetic field) of the resonance for an isolated spin w = ?H*-
3 hy2
4
-(1 - 3
e).
COS~
r:2
Here 8 is the angle between the magnetic field direction and the axis connecting the nuclei. Formula (22) shows that the frequency shift is a maximum for spin pairs with rI2 a. There can be, in general, three types of these pairs: immobile pairs in which one of the impurities cannot tunnel with energy conservation and to which eq. (22) is applicable literally; and pairs representing two- and one-dimensional quasi-particles. In the latter two cases the state of the system is determined by the quasi-momentum p (two- or one-dimensional) and the number v of the energy band. Such a state is a linear combination of all the possible localized states of the pair which are specified by the coordinate R of one of the atoms and the index i = 1,2, . . . , numbering the possible orientations of the axis joining the nuclei. The coefficients of this linear combination are of the form Ar(p)eic' and play the role of the wave function of the state (v, p) in the representation (i, R).The resonance frequencies are obtained from eq. (22) by averaging over the state (v, p). In the general
-
98
A.F. ANDREEV
case they depend on u and p and are equal to
3 hy2 o,(p)= yH*-(1 -3(cosz O)), 4 42 where
Here 8, is the angle between the magnetic ..A and the straight line joining the nuclei in the localized state i. The coefficients A;(p) satisfy the normalization condition i
The impurity pairs which are nearest neighburs in the lattice give the major contribution to the frequency shift. In hcp ‘He crystals they behave like two-dimensional quasi-particles and can be, as we have seen, of two types corresponding to figs. 9 and 13. In both the cases the index i takes three values, since both types of pairs have three different orientations (corresponding to configurations AC, AC, and AC, in fig. 9 and fig. 13). Let the magnetic field be directed along the hexagonal axis. Then cos 0, = O for all i for the first type pairs and cos Oi=(2/3)”2 for the second types of pairs. In this case the expression in parentheses in eq. (23) has the same magnitude but opposite sign for the pairs of the two types. The frequency spectrum is, thus, discrete and includes only two lines of equal intensity
The same result would be obtained for a particular field direction and in a rigid lattice, but in our case the quasi-particles move easily in the hexagonal plane and, hence, respond to the gradient of the magnetic field, and only to the gradient parallel to the hexagonal plane. It is essential that the frequency shift considerably exceeds the line width. The shift is of the order of h y 2 / a 3 lo4s.-’,while the times T2 observed (Richards et al., 1972; Greenberg et al., 1972) in weak 3He-4He solutions reach lo-‘ - 1s. If the magnetic field is directed at an angle to the hexagonal plane, the resonance frequencies depend on the quasi-momentum value and the spectrum becomes continuous. The shape of the spectrum, however, can
-
QUANTUM CRYSTALS
99
be calculated exactly, because the method, known in the theory of metals as the strong coupling approximation, can be used to calculate the coefficients A:(p). So a simple system of linear algebraic equations is obtained (Andreev, 1975) for determining the coefficients in the case of two-dimensional quasi-particles. For one-dimensional quasi-particles the spectrum is discrete whatever the magnetic field direction, because in this case there are only two orientations and the corresponding coefficients A: differ from each other only by the phase factor: JA;(p)l=lA;(p)l. Each of the onedimensional quasi-particles generates two lines w ~ . Y ~ H =
3hy2 * ~ [ ~ - ; ( c o s&+COS 4r12
e,)].
We should stress that the discrete structure of the lines can, of course, be observed only in single crystal samples.
Let us, finally, turn to the problem of calculating the lifetime of the above two- and one-dimensional quasi-particles. This time is associated with tunnelling processes, not accounted for so far, in which the interaction energy changes by an amount considerably exceeding the band width A. Thermally activated diffusion is the simplest example of such processes. The corresponding probability, however, drops exponentially with decreasing temperature. Therefore, at low temperatures the processes of quantum tunnelling, accompanied by a change in the phonon energy to compensate for a change in the interaction energy, must play the major role. Such processes can be of two types. The first one is the tunnelling of an impurity accompanied by the scattering of a phonon. The probability of this process wl,per unit time, can easily be estimated, using the results obtained by Kagan and Maksimov (1974) and Kagan and Klinger (1974)
The second process is the tunnelling of an impurity with the spontaneous emission of a phonon (Andreev and Meierovich, 1975). In this case the phonon energy is equal to the change of the interaction energy hw
-SU - Vo(u/r12)4.
The probability of spontaneous emission w2 is, as usual, proportional to the third power of the frequency of the emitted quantum and to the square of the overlapping integral of the wave functions of a tunnelling
100
A.F. ANDREEV
particle, i.e. to the square of the band width A
Unlike the first process which can proceed either with a decrease o r an increase in the interaction energy, the tunnelling with spontaneous emission can proceed only to states with lower impurity energy. It is impossible, therefore, for such configurations of impurity pairs which correspond to the minimum of the energy. For the most interesting case of quasi-particles with rI2 a (but not nearest neighbours which, probably, correspond to the absolute minimum of the interaction energy) the second process become the major one at T < V0(8/V0)4’7. In this low-temperature region the lifetime of most of the quasi-particles does not depend on the temperature.
-
5. Vacancies
5. I . Vacancies in 4He crystals As the temperature rises high enough, the processes responsible for the delocalization of atoms in a crystal become thermally activated. We have seen this from examples of the rate of spin relaxation G’ and spin diffusion Dsin ’He (the high-temperature portion of the curves in figs. 2 and 3), and also from the diffusion of 3He impurities in 4He crystals (region 111 in figs. 7 and 8). A possible mechanism in this case might be, in principle, ordinary classical processes of super-barrier transitions of atoms into adjacent lattice sites where two neighbouring atoms change their places or a circular permutation of three or four particles takes place (fig. 1). These processes differ from the above tunnelling processes in that they take place only if the energy transferred to atoms due to a thermal fluctuation is sufficient for a classical surmounting of the corresponding potential barrier. Like the tunnelling processes, super-barrier processes in closely packed lattices are extremely difficult due to the absence of enough free volume, while potential barriers which have to be overcome are rather high. They considerably exceed the energy e0 necessary to produce a vacancy in a crystal. That is why the basic mechanism of thermally activated translational motion of atoms in helium crystals is due to the existence of thermally activated vacancies, whose concentration is
QUANTUM CRYSTALS
101
proportional to exp(-E,/T), and tunnelling of atoms into vacant lattice sites. The tunnelling of an atom adjacent to a vacancy into a vacant lattice site is usually called the tunnelling of a vacancy into an adjacent lattice site. The thermally activated delocalization of particles observed at high temperatures is, thus, the result of the tunnelling motion of thermally activated vacancies in a crystal. The vacancy-induced motion of atoms in a crystal is of special interest. The fact is that vacancies in 4He crystals become, due to their periodicity, delocalized quasi-particles. The width of the energy band of these quasiparticles A,, determined by the vacancy tunnelling probability, considerably exceeds the band width of impuritons because in this case we deal with the tunnelling of a single particle, and under conditions when the process is in no way hindered by the absence of free volume. According to calculations of Hetherington (1968), who was the first to consider this problem, and of others (Pushkarov, 1971, 1974; Guyer et al., 1971; Guyer, 1972; Mineev, 1973; Sullivan et al., 1975) the value of d, has to be of the order of 1- 10K. Though at present we have no direct experimental evidence for the existence of vacancy quasi-particles, since vacancies are very difficult to observe, it is hardly a point of doubt because vacancies are much more mobile quasi-particles as compared with impuritons. Let us turn to the quantitative description of thermally activated diffusion of impurities (region I11 in fig. 7), considering it as vacancyinduced diffusion. The impurity migrates here via the following process (fig. 14). When moving through a crystal a vacancy may find itself in a lattice site adjacent to an impurity atom (see fig. 14a). Then the vacancy can tunnel to a site which is occupied either by a matrix atom (fig. 14b) or by an impurity (fig. 14c), and then the vacancy will travel far away from the impurity. In the case presented in fig. 14c the process is accompanied by the displacement of the impurity. Since the width A, of the vacancy band is much greater than that of the impuriton band, we can neglect the intrinsic tunnelling of the impurity and consider the process shown in fig. 14 as the quantum-mechanical scattering of a delocalized “vacancion” from a localized impurity (Andreev and Meierovich, 1975). The process of fig. 14b corresponds to elastic scattering because in this case the states of the target before and after the scattering coincide. The case of fig. 14c corresponds to inelastic scattering accompanied by the displacement of the impurity. The vacancy-induced diffusion of impurities is determined by the inelastic scattering probability. The diffusion coefficient is D a2v where v is the frequency of inelastic scattering events determined by the
-
A.F. M R E E V
102
0
0
0
0
0
0
0
(C)
Fig. 14. Interaction of a vacancy with an impurity atom.
-
relation v Nuuuin.Here uinis the cross-section of inelastic scattering of a vacancy from an impurity, Nu is the number of vacancies per unit volume, u is their velocity. At temperatures TBA,, we have Nu ~ - ~ e - ' d ~ , v aA,lh,
-
-
where co is the energy of the formation of a vacancy. So we obtain
If the thermally activated diffusion of impurities is caused by their interaction with vacancies, it follows then from eq. (26) that the activation energy does not depend on the sort of impurities in one and the same crystal. This result can be verified by a direct experiment, because not only isotopic admixtures but also ions can be introduced in a controllable manner and investigated in 4He crystals. The diffusion coefficient of ions can easily be calculated from the mobility measured in an external electric
QUANTUM CRYSTALS
103
10-8. D(cm2/s)
:\: k.0
10-10
0 0 . 0
0
’\
0x -- 231 1
0
-
0.6
0 0
1
0.8
2
1.o
L
’
1.2
’
’
1.4
T-’ (K-’ )
Fig. 15. Temperature dependence of the diffusion coefficient for impurities in ‘He crystals with a molar volume of 20.7cm3: 1, positive ions (Keshishev. 1977); 2. 0.75% 3He; 3, 2 x 17% 3He (Grigor’ev et al.. 1974).
field. These experiments were first performed by Shal’nikov (1965) and then by some other authors (Keshishev et al., 1970; Sai-Halasz and Dahm, 1972; Marty and Williams, 1973; Keshishev, 1977). Fig. 15 shows the experimental date of Keshishev (1977) on the temperature dependence of the diffusion coefficient of positive ions in solid 4He with a molar volume of 20.7cm3. The same figure presents the corresponding data of Grigor’ev et al. (1974) on the diffusion of isotopic impurities. In the temperature region 111 where the diffusion is thermally activated not only the activation energies, but also the absolute values of the diffusion coefficients coincide. The experimental data are described by the equation D = 6.6 X lo-’ exp(-9.5/T); the solid line drawn in fig. 15 corresponds to this equation. Comparison with eq. (26) yields the energy of formation of vacancies e 0 = 9 . 5 K and, if we assume A,- 1 K, the inelastic scattering cross-section uin-5x a’. Such a small crosssection can be explained as follows. The situation is similar to the case of the impuriton-impuriton scattering considered in section 4.1. The interaction of a vacancy with an impurity is of an analogous form U ( r ) - V ( U / ~ ) ~ , where V is some characteristic interaction energy. If V > A,, the vacancy
I04
A.F. ANDREEV
cannot penetrate into the region r A, the inelastic scattering cross-section is much smaller than a', while the total cross-section is much greater than a2. It is essential, however, that the inelastic process cross-section strongly depends, under these conditions, on details of the interaction of vacancies with 3He impurities and ions, so that coincidence between the inelastic cross-sections is rather strange. Only more thorough simultaneous experimental studies of the diffusion of 3He impurities and charge mobility might elucidate this problem. The above approach to the vacancy diffusion (or mobility) of ions based on the concept of inelastic scattering permits the establishing (at fairly low temperatures) of the dependence of the mobility on the temperature and electric field without making any suggestions concerning the specific structure of the ions (Andreev and Meierovich, 1975). Let a,(p) be the cross-section of the inelastic scattering of vacancy with a quasimomentum p , which is accompanied by the displacement of the ion by the vector u, connecting the initial lattice site with its n t h nearest neighbour. The energy of the ion and, hence, of the vacancy changes by eEq, where e is the ion charge, E is the applied electric field. The mean drift velocity of the ion is expressed in terms of the cross-sections as
Here u(p) is the vacancy velocity, n ( ~ is ) the equilibrium distribution . summation function of vacancies dependent on their energy E = ~ ( p )The in eq. (27) is performed over the nearest neighbours for which eEq, >O. Displacements of the ion by vectors (-un) are taken into account in eq. (27) as reverse processes. Expression (27) can be rewritten in the form
where the last integral is taken over the surface of constant energy. In the simplest case where all the vectors u, are crystallographically equivalent (in the bcc lattice, but not in the hcp lattice) this integral is independent of
QUANTUM CRYSTALS
105
n due to the lattice symmetry. Suppose that the temperature T is small as compared with the vacancy band width A,. In this case almost all the vacancies are located near the bottom of the band where their spectrum is quadratic and the velocity is small. According to a known result of quantum mechanics (Landau and Lifshitz, 1969, the cross-section of inelastic scattering of slow particles is inversely proportional to their velocity. Hence,
where eo is the energy corresponding to the bottom of the band, a is a constant. The distribution function can be considered to be the Boltzmann one, since e0 is of the order of 10 K and, therefore, is much greater than the temperature. We, finally, obtain
a (9 e-'JTzan[ 1 - exp( -eEo,/T)]. 167rfi' 312
u=
7r
n
In the region of weak fields eEa<
In the region of strong fields eEa >> T the situation is rather peculiar. For almost all directions of the field the drift velocity saturates and does not depend on IEI:
Since, however, the summation in eq. (30) is performed only over those n for which eEq, >0, the velocity u changes jump-wise any time when the vector E passes through a plane normal to one of a, under the variation of the field direction. The angular width of the transitional region is, in order of magnitude, T/eEa<<1. In this region the drift velocity changes rapidly both in magnitude and in direction. As an illustration, fig. 16 shows the dependence of the magnitude of the drift velocity and its
106
A.F. ANDREEV
Fig. 16. Dependence of the magnitude and direction of the ion drift velocity on the electric field orientation. 0 is the angle between u and a,, cp between E and a,.
direction on the vector E orientation for a plane square lattice with a period a. It is clear from eq. (30) that the maximal drift velocity u,, is in this case
It can be seen that the crystal lattice geometry is directly manifested in the angular dependence of the drift velocity.
5.2. Zero-point vacancies
We have already seen in the preceding section that the energy of the formation of a vacancy in solid helium and the width of the vacancy energy band are values of the same order. It is interesting to note in this respect that quantum crystals offer a theoretical possibility for the existence of the so-called zero-point vacancies which, like zero-point vibrations, exist in a crystal at absolute zero (Andreev and Lifshitz, 1969; Chester, 1970; Dzyaloshinskii et al., 1972). To illustrate this possibility, let us consider how the energy spectrum of a vacancy changes as the quantum tunnelling probability increases. In the
QUANTUM CRYSTALS
107
classical limit the vacancy is localized and has a certain energy Eo> 0. A small, but finite probability of the vacancy tunnelling into nearest lattice sites leads to the energy band of a finite width A,. Whenever A,<< Eo the energy spectrum can be calculated by using the strong coupling method which shows that the middle of the band coincides with the energy Eo of the localized vacancy. So it is clear that an increase in the tunnelling probability and a related increase in the band width lead to a decrease in the minimal vacancy energy eo corresponding to the bottom of the band. Therefore, in clearly marked quantum crystals, such as solid helium, a situation is possible when the minimal energy eo becomes negative. This means that the initial ground state of a crystal is, in fact, unstable with respect to the generation of vacancies. The correct ground state will correspond to the presence of a certain number of vacancies in the crystal. If the energy co is negative but small in the magnitude, then vacancies with quasi-momenta close to that corresponding to the bottom of the band have the negative energy. The energy E ( p ) of vacancies in this region, near the energy minimum, can be presented in the form
where M is some effective mass, p is the quasi-momentum reakoned from the value corresponding to the minimum. In a 4He-type Bose crystal which will be considered first vacancies are, obviously, bosons. To calculate their contribution (for small density) to the ground state energy of a crystal, we can, therefore, use the known formula (Landau and Lifshitz, 1969) for the ground state energy of a rarefied Bose-gas with the spectrum, eq. (31):
Here E, and N are the energy of the vacancies and their number per unit volume, respectively, f is the length of scattering of vacancies from one another. If f > O , i.e. for repulsive interaction of vacancies and small negative eo. eq. (32) has a minimum for small N which permits the calculation of the equilibrium density of zero-point vacancies No in a Bose crystal
N - - M leol O - 4.rrhZf *
(33)
108
A.F.
ANDREEV
The crystal containing zero-point vacancies must have many extraordinary properties: in particular superfluidity caused by the possibility of transfer of mass via superfluid motion of the “vacancion liquid” through the crystal (see: Andreev and Lifshitz, 1969; Chester, 1970; Dzyaloshinskii et al., 1972; Leggett, 1970). However, the experimental data available at present show that the possibility discussed can hardly take place in ‘He crystals. The experimental data on the vacancy-induced diffusion of ’He impurities and charge mobility presented in the preceding section show that the number of vacancies drops exponentially with temperature and this corresponds to the fact that their minimal energy is positive. This result, however, is obtained for the temperature dependence of the number of vacancies at a fixed pressure. Measurements of the temperature dependence of the charge mobility along the crystal melting curve show that the mobility is approximately constant [see Keshishev (1977)] down to the lowest investigated temperatures, of the order of 1 K. In ’He crystals quantum effects are more pronounced than in ‘He because the atomic mass of the former is smaller. But in this case too there are experimental data on the X-ray scattering (Fraass et al., 1977) indicating that the number of vacancies decreases more or less exponentially with temperature at 1 K. But this does not mean that ’He crystals contain no vacancies at absolute zero. The fact is that the behaviour of vacancies in ’He crystals has essential features associated with the nuclear spins of ’He atoms. If all the nuclear spins are aligned parallel, then the ’He crystal, as well as ‘He, is periodic and due to the tunnel effect the vacancies become delocalized quasi-particles with the band width of the order of 10K. The actual situation is, however, that down to lO-’K nuclear spins in ’He are disordered, so the crystal is not periodic. In such a crystal the vacancy is not a quasi-particle with a definite momentum. This case will be considered in the next section, and now we shall discuss the problem of zero-point vacancies for the simplest particular case, namely when all nuclear spins are aligned parallel. This state can be realized by applying a rather strong magnetic field. Since in the case considered the crystal is periodic, the energy spectrum of vacancies with a small (absolute value) negative energy eo corresponding to the bottom of the band can still be described by eq. (31). In a Fermi ’He crystal, however, the vacancies are fermions because one fermion - ’He atom - must be annihilated to create a vacancy. The ground state of fermions with the spectrum, eq. (31), corresponds, evidently, to the complete filling of all the states with a negative energy, i.e.
QUANTUM CRYSTALS
109
states with p < po, where po = (2M/eol)”2. The number of zero-point vacancies No per unit volume of the crystal is
N o = ( 2 ~ h )$7rp: - ~ = (2M )e0()3’2/67r2~3.
(34)
Their contribution to the total energy of the crystal is negative and is determined by
(2M ( E ~ ( ) ’ / * E , = [ ” ( e0+- p 2 ) - =d3p 2M ( 2 ~ f i ) ~ 307r2fi3M *
(35)
The zero-point vacancies in a Fermi quantum crystal can, thus, be described in terms of the Fermi-branch of excitations in the crystal energy spectrum which are located near some Fermi-surface. These excitations cause typically “Fermi-liquid” phenomena, such as a linear lowtemperature heat capacity, zero sound, etc. (Andreev and Lifshitz, 1969; Dzyaloshinskii et a]., 1972). The theory of these phenomena in quantum crystals can be developed [see Dzyaloshinskii et al. (1972)J without assuming the concentration of quasi-particles or their interaction to be small.
5.3. Vacancies in 3He crystals The behaviour of vacancies in 3He crystals depends significantly on the configuration of nuclear spins. At temperatures exceeding 1 mK and in not too strong magnetic fields the spins are distributed randomly so the vacancy quasi-momentum is not a good quantum number. In this case we can speak only about the density of energy levels of a vacancy v ( E ) = aN(E)/dE, where N ( E ) is the number of states with energy less than E. From the mathematical viewpoint the calculation of the function v ( E ) is equivalent to the so-called Hubbard model. The results obtained with the help of this model for the bcc-lattice by Nagaoka (1966) and Brinkman and Rice (1970) are shown qualitatively in fig. 17. The solid curve corresponds to the state with disordered spins, the dashed curve is the density of levels v ( E ) corresponding to the spectrum E ( p ) in the completely polarized (ferromagnetic) state. It is noteworthy that the energy of a vacancy in the disordered state always exceeds the energy e0, corresponding to the band bottom in the ferromagnetic lattice, by an amount of the order of the band width A,- 10 K. The situation is the same if we consider an antiferromagnetically ordered lattice instead of a disordered
110
CO
ENERGY
Fig. 17. Density of energy levels of a vacancy: 1, in disordered lattice of 'He; 2. In ferrornagneticallyordered lattice of 3He.
one. Generally, it was proved by Nagaoka (1966) that the bottom of the band co for the ferromagnetic bcc-lattice is the absolute minumum for the vacancy energy which is not attainable for any ordering of the spins different from the ferromagnetic one. Ferromagnetic ordering of nuclear spins in 3He crystals can, therefore, be accompanied by a considerable decrease in the energy of a vacancy. This situation is similar, in many rekpects, to the known problems on the behaviour of electrons in liquid helium (Ferrell, 1957; Kuper, 196 1) in disordered solids (Lifshitz and Gredeskul, 1970; Krivoglaz, 1974) or in magnetic semiconductors (Nagaev, 1968). It can easily be seen that a macroscopic region with completely polarized nuclear spins must arise round a vacancy in a 'He crystal (Andreev, 1976b; Heritier and Lederer, 1977). In fact, let the temperature T satisfy the condition J<< T C A, where J has the value of the order of the energy of exchange interaction of neighbouring spins, .I- K. In this case the major role is played by the vacancies near the bottom of the band where their energy spectrum is described by eq. (31). Suppose that inside the sphere of a radius R all the spins are completely polarized, while outside the sphere they are disordered. Then inside the sphere the Hamiltonian of a vacancy is E(p), and at the sphere boundary the wave function must vanish because, as was noted above, the vacancy in the disordered state has a great excess
QUANTUM CRYSTALS
111
energy. The ground state energy of the vacancy is in this case lr2 ti2 E=lEo+-2 MR2’
The radius R of the ferromagnetic sphere is determined from the condition that the free energy F = E - TS is a minimum. Here S is the change in the crystal entropy when the spins are ordered inside the sphere. It is clear that S = -N In 2 where N is the total number of 3He atoms within the sphere volume. The free energy is l r 2 ti2 41r F=€o+-+ nT- R 3 In 2, 2 M R ~ 3
(37)
where n is the number of ’He atoms per unit volume of the crystal. The condition that this expression is a minimum yields the equilibrium radius of the sphere
R = (lrh’l4MnT In 2)”’.
(38)
The number of particles N in the sphere volume is, in order of magnitude, (AJT)”’, i.e. it is large in comparison with unity. Each vacancy has a large magnetic moment A determined by A = N p = pn-
(39)
where p is the nuclear magnetic moment of 3He. Temperature dependence of the equilibrium number of vacancies N, in a crystal is determined by the expression exp(-FIT), where F is obtained by substituting eq. (38) into eq. (37). So we have
At high temperatures the second term contributes mainly to the expression in square brackets because this term is inversely proportional to a lower power of the temperature. This term is always negative, i.e. the number of vacancies drops rapidly with temperature independently of the sign of the first term, that is of the sign of the minimal energy of a vacancy in the ferromagnetically polarized state. So, the assumption that q, is negative will not contradict high-temperature data on the temperature dependence of the number of vacancies (Fraass et al., 1977). In this case the ferromagnetic state of a crystal is unstable (see the preceding
112
A.F.
ANDREEV
section) with respect to the formation of zero-point vacancies. This instability might, in fact, be observed in 3He crystals placed in a fairly strong magnetic field. The H - T diagram must, in this case, reveal a region corresponding to a crystal with zero-point vacancies (Andreev et al., 1977). At present, however, it is not clear whether this phenomenon actually exists or not. Here it is interesting to mention the remark made by Castaing and Nozieres (1979) in connection with the investigation of phase equilibrium crystal-liquid in 3He with polarized spins performed by those authors and also by Lhuillier and Laloe (1979). It is a matter of experiments carried out for times which are short in comparison with spin-lattice relaxation times both in solid and liquid phases. Both the times are quite macroscopic at low temperatures. If the polarization does not change in melting, the Pomeranchuk effect, i.e. an increase in the melting pressure of a crystal with the decreasing temperature, is absent and the region of solid state from the side of lower pressure expands. Since the width of the vacancy band, as well as probabilities of any tunnelling processes, increases with decreasing pressure (see section 2), this, according to what has been said in the preceding section, diminishes the minimal energy €0 and favours the appearance of the state with zero-point vacancies.
6. Surface phenomena 6.1. Equilibrium shape of the crystal-liquid interface Phenomena occurring at the phase boundary between solid and liquid helium appear to be the most clearly marked phenomena associated with the quantum nature of helium crystals. The neighbourhood of a quantum crystal with a superfluid liquid makes the processes of crystallization and melting very unusual (Andreev and Parshin, 1978; Keshishev et al., 1979; Castaing and Nozieres, 1980; Castaing et a]., 1980). These processes can be considered as supercrystallization or supermelting: because of the fairly low temperatures they proceed coherently and without considerable dissipation, i.e. similar to other quantum superphenomena. Even ordinary visual observation (Keshishev et al., 1979) shows that the boundary between solid and liquid 4He looks like a boundary between two nonmixed low-viscosity liquids continuously vibrating under the action of external perturbations.
QUANTUM CRYSTALS
113
Ordinary classical crystals always have a specific faceting at low temperatures, and crystallization and melting are rather slow thermally activated processes [see review papers: Chernov (1961) and Jackson (1975)J The physical reason for this fact is the difference in the symmetry of the arrangement of particles in a crystal and adjacent liquid or gas. A crystal is a periodic system, while a liquid or gas is a homogeneous one. It was shown by Landau (1965) that for this reason the surface energy at the crystal boundary is a fairly specific function of the direction of the normal to the surface. The derivative of this function is discontinuous everywhere. The faceting of a crystal, i.e. the fact that the function describing the crystal shape is not smooth, is a direct consequence of this property of the surface energy. Consider, following Landau (1965), the dependence of the surface energy of unit area a on the direction of the normal to the crystal surface for the case where the normal is close to one of the crystallographic faces of the crystal. Let a. be the energy per unit area of the original face. The surface which is slightly inclined to the face should be of the form shown in fig. 18. For small inclination angles cp the concentration of steps on the surface is small, so we can neglect their interaction and write the surface energy a of an inclined surface as the sum of a. and the energy of the steps which is equal to Bcpla, where a is the lattice parameter, p is the energy per unit length of an isolated step. If the surface is inclined to the opposite side, then cp is negative but the surface will contain steps with the “other sign” which have the same energy. For small (cp) the surface energy is, therefore,
P
a = ao+- Icp1.
a
So the derivative aalacp of the surface energy with respect to the angle cp has at cp = 0 a finite jump s ( d a / d c p ) equal to 2Pla. In the general case where the inclination of the surface is determined by two rather than one angle, the steps appearing on the surface are not rectilinear. The surface may contain kinks on the steps which are shown in fig. 19. It is easy to understand that the above statement on the discontinuity of \
- -I---
Ip
--rL ~
Fig. 18. Crystal surface with a small angle of inclination to the original face.
114
A.F. ANDREEV
Fig. 19. Step with kinks.
the angular derivatives of the surface energy is applicable to any face of a crystal, and not only to the (001) face shown in fig. 18. The shape and sense of “steps” and “kinks” can be different in either case; they are determined by the condition that the energy is a minimum. It is important that on surfaces close to some particular face of a given crystal the steps and kinks can have only one definite shape (Landau, 1965). The crystal faceting is a direct consequence of the discontinuity of the derivative aalacp. Proceeding from the minimum condition for the total surface energy, Landau (1965) showed that the length of flat sections with a given cp = cpo in the equilibrium shape of the crystal is proportional to the jump S(da/acp) at cp = cpo, i.e. proportional, as can be seen from eq. (41), to the step energy 0. At significantly higher temperatures the crystal faceting usually disappears and the surface energy becomes a smooth function of the direction. In this case we can speak about an atomically rough surface, as opposed to an atomically smooth surface of a faceted crystal. The phase transition from an atomically smooth to an atomically rough surface - the so-called roughening transition -occurs at quite a definite temperature which is different for various faces of a given crystal. As was said above, the step energy must vanish at the transition point. It is important to emphasize here that the definition of /3 as the step energy is not quite correct. At finite temperatures all the above relations must contain the free energy instead of energy, so 0 is, in fact, the free energy of unit length of the step and differs from the energy by the term - 7 3 , where S is the step entropy. At finite temperatures pairs of kinks of opposite sign appear on the step due to thermally activated processes. The concentration of these pairs grows with temperature, so their contribution to the entropy
QUANTUM CRYSTALS
115
also grows. It is an increase in the entropy S that makes the free energy 0 vanish at the roughening transition point. The transition entropy AS, i.e. the difference between the entropies of liquid (or gas) and crystal per particle, is a convenient qualitative criterion of the roughening transition in ordinary crystals [see: Chernov (1961) and Jackson (1975)l. The case AS B 1 corresponds to the smooth surface, while the case AS 6 1 corresponds to the rough one. For ordinary classical crystals the transition entropy increases infinitely with lowering temperature (due to an increase in the entropy per gas particle), so the crystal surface should be smooth at fairly low temperatures. Crystals of helium isotopes may be in equilibrium with liquid at arbitrarily low temperature. Since at absolute zero the entropies of both the phases vanish, the transition entropy is small in this case. The formal application of the classical criterion leads to the conclusion that the helium surface should be rough. Though the experimental data (Shal’nikov, 1965; Fraass et al., 1977) for temperatures above 1 K (the absence of crystal faceting, the absence of considerable supercooling even at high crystal growth rates) confirm this conclusion, the classical criterion is not applicable in this case. The rough, in the ordinary classical sense, surface cannot be in equilibrium at absolute zero because its entropy does not vanish. Experimental effects, as well as the fact of the existence of the crystal-liquid interface at absolute zero, can be explained by taking appropriate account of quantum effects. It turns out (Andreev and Parshin, 1978) that the phase boundary between the quantum crystal and quantum liquid may be in a particular state which is a quantum analogue of a rough surface. Since particles in a quantum crystal are delocalized, the microscopic structure of the interphase boundary cannot be represented so literally, as is done for classical crystals. However, such concepts as a step on the surface and kink on the step can also be introduced in the general quantum case. Here it is sufficient to use only the general properties of the boundary which follow from the symmetry of adjacent phases. For instance, a step on the surface corresponds to such a state of the boundary when its positions at infinity (to the right and to the left in fig. 19) are shifted by the elementary translation vector of the crystal, and the energy of the system has a minimal possible value. Since the crystal is periodic and the liquid is homogeneous, the shift by the translation vector transforms the boundary into an equivalent position, so the step is a linear defect on the surface and, in essence, only this property is important for what will follow. The crystal faceting, i.e. the discontinuity of the
116
A.F. ANDREEV
angular derivatives of the surface energy, is, obviously, related in the usual way to the step energy 0. Unlike classical crystals, however, in which P is always positive at absolute zero, so they are always faceted at low temperatures, in quantum crystals there are several possibilities. It will be shown below that apart from ordinary surfaces which are characterized by the positive energy of steps a quantum crystal can have surfaces which are characterized by exactly zero energy of steps. The state of a step on the surface is determined by the configuration of kinks on the step, the kinks being of two “signs”. Each kink can be considered as a point defect on the step. If the kink is shifted along the step by the translation vector, the energy of the system will not change because the transfer of matter from one phase to the other, which accompanies such a shift, does not contribute to the energy due to the equality of chemical potentials of phases being in equilibrium. Therefore, like other point defects (vacancies, impuritons) considered in preceding sections, an isolated kink in a quantum crystal behaves like a delocalized quasi-particle whose state is determined by quasi-momentum. Here we encounter the following point which is of major importance for further consideration. Let po be the value of the quasi-momentum corresponding to the bottom of the energy band. For an isolated kink at absolute zero this state is a stationary and ground state, and the kink velocity determined by the derivative of the energy with respect to the quasimomentum is equal to zero. Stationary states with p + p o and close in energy correspond to the kink velocity different from zero. So,an isolated kink on the step is an example of the system which has states arbitrarily close to the ground state, and these states are characterized by the continuous flow of matter from one phase to the other. If the width of the kink energy band A is rather large (roughly speaking, the following condition should hold: A / ~ > Ewhere 4 is the energy of a localized kink, d. similar arguments in section 5.2), then the energy near the bottom of the band will be negative. In this case kinks with p = po lower the total energy of the step. There will be spontaneous production of kinks of both signs, till their mutual interaction compensates for the negative intrinsic energy of the kinks. This will, obviously, take place at a kink concentration of the order of the atomic one. In this case the minimal step energy can be qualitatively expressed as
where
Po is the energy of a rectilinear, “bare” step without kinks.
QUANTUM CRYSTALS
p)
117
II
I
’
I
’
L.I,
c (b)
(a)
Fig. 20. Collisions between kinks.
The step whose ground state contains a large number of kinks is, like an isolated kink on the step, a system with stationary states close, in energy, to the ground state. These states correspond to a continuous motion of the step, i.e. they are characterized by the permanent flow of particles between the phases. This becomes clear if we remember what has been said above about the motion of isolated kinks and consider the character of possible processes of kink collisions. When kinks of either sign collide, they exchange their quasi-momenta (fig. 20a). In the case of kinks of one sign this is the only possible elastic process. If the kinks have opposite signs, another process with “umklapp” into the next row is possible, when each kink conserves its quasi-momentum (fig.20b). It is the latter process which causes the continuous motion of the step. In ordinary crystals the values of and a& are of the order of the difference in the energies of two adjacent phases, as calculated per one interatomic bond (Chernov, 1961). For the solid helium-liquid helium interface such an estimate yields E a@ 0.1 K. At the same time, for A we may expect values of the order of 1 K. It is quite probable, therefore, that the energy of the ground state of an isolated step is negative. In this case the atomically smooth surface of a crystal is unstable with respect to the production of steps. So. the equilibrium surface should be “quantumly rough”, i.e. represent a peculiar two-dimensional liquid consisting of delocalized steps of various configurations, including closed steps of a finite length. It is important that the number of steps of either sort in such a liquid is not fixed, but is determined from the minimum condition for
- -
118
A.F. ANDREEV
the energy. So in equilibrium the step energy p vanishes since it is the derivative of the total energy with respect to the number of steps. This means that the surface energy a is, in this case, a smooth function of the direction of the normal to the surface. Thus, it follows from the above considerations that the surface of a quantum crystal at absolute zero can be either quantumly rough or smooth. In the latter case it is characterized by the positive step energy p, but here two alternatives can exist. The step on the smooth surface can be, in turn, either rough or smooth, depending on whether the ground state of the step contains kinks. Naturally, one and the same crystal may have surfaces of various type, depending on their orientation, the surfaces with rather great indices always being rough. In fact, the results of Landau’s paper (Landau, 1965) show that the jumps of the angular derivative of the surface energy, and hence the energies of effective steps, for (1, N , , N2)faces, where N , , N 2 are large numbers, are caused by superstructures which appear in a rarefied system of steps and kinks. This, however, is impossible because of quantum delocalization of kinks and, hence, steps.
6.2. Crystallization and melting The character and rate of crystallization and melting depend significantly on the type of the surface. Let us consider first the case of a smooth surface. Since the step energy /3 is positive, the appearance of a new atomic row is associated with the overcoming of potential barriers. In fact, let, for certainty, the difference 6p = p, - p, of the chemical potentials of a solid p, and a liquid p I per unit mass be positive and small. Then, crystal growth is energetically more favourable. However, the formation of a nucleus of a finite size R for a new atomic row (see fig. 21) leads to a change in the system energy of
U ( R )= 21rRo - IrR2p,aSp
(42)
f
Fig. 21. Nucleus of a new atomic row.
QUANTUM CRYSTALS
119
where ps is the density of the solid. This value is positive for small R and becomes negative only for fairly large R satisfying the condition R > Ro, where The situation is quite similar to the problem of production of nuclei under the first-order phase transition from the metastable state considered by Lifshitz and Kagan (1972) and by Iordanskii and Finkel’stein (1972), and also to the problem of motion of dislocations considered by Petukhov and Pokrovskii (1973). At fairly low temperatures the growth of a crystal proceeds via the quantum sub-barrier formation of nuclei with R = Ro which then rapidly increase their size. Let us calculate the probability w of the formation of the nucleus of a new atomic layer, assuming the step on the surface in question to be quantumly rough. This case is the most favourable for crystal growth because, as we have seen in the preceding section, such a step implies the existence of stationary states corresponding to continuous motion of the step and lying arbitrarily close to the ground state. In other words, free motion of the step is possible in this case. In the case of a smooth step its motion is hindered, because this motion, in turn, is the result of the sub-barrier production of pairs of kinks of opposite sign. The energy U ( R )determined by eq. (42) plays the role of the potential energy of a nucleus. To calculate the kinetic energy K arising in the expansion of the nucleus at a velocity k, let us note the following. Since the densities of a solid ps and a liquid p, are different, the motion of the step at the velocity R should be accompanied by the flow of a mass ( p , - p , ) a R from the liquid to unit length of the step per unit time. The velocity of the liquid u ( r ) at distances r from the step satisfying the condition a << r<< R can easily be determined if we notice that at such distances the step can be considered rectilinear and the direction of the liquid velocity radial. From the condition of the incompressibility of liquid ~ d r )=~(ps-p,)aR , we find
The kinetic energy per unit length of the step is given by the integral
A.F. ANDREEV
120
which, as can be seen from eq. (U),has a logarithmic singularity and must be cut off on the lower limit at r a and on the upper limit at r R . Multiplying the above result by the step length ZTR, we find the total kinetic energy of the nucleus
-
(P* - m)’ PI
K=4a2-
a
2
In -. R a
-
(45)
Introducing the momentum
canonically conjugated to the coordinate R , we can write the expression for the total energy of a nucleus in the form (46)
where
M ( R ) = 8a2-( A -
PI)‘ PI
In R a
is the effective mass. The probability w of the formation of a nucleus is given by the known (Landau and Lifshitz, 1965) quasi-classical formula w -exp(
-f C’IPI dR),
(47)
where the integral is taken over the classical trajectory from R = 0 to R = ROat a fixed (equal to zero) energy. It can be seen from eq. (46) that in this case
IP(R)(= [ 2 M ( R ) U ( R ) ] ” 2 . Substitution into eq. (47) and calculation of the simple integral yield
The crystal growth rate is usually characterized by the growth coefficient X which specifies the linear relationship between the velocity of the crystal boundary V and the deviation Sp of the adjacent phases from
QUANTUM CRYSTALS
121
equilibrium:
v = xsp.
(49)
If the dependence V(Sp) is nonlinear, the growth coefficient must be defined as X = Iim 6~
av -. asp
(50)
Eq. (48) shows that for a smooth surface at zero temperature the probability of the formation of nuclei, and hence the crystal growth rate V, decrease as Sp + 0 approximately according to exp [-const./(Sp)*], i.e. much faster than Sp. So,the growth coefficient, as defined by eq. (50), for a smooth surface at absolute zero is equal to zero. Quite a distinct situation takes place for a quantumly rough surface whose ground state has a large number of steps (Andreev and Parshin, 1978). The growth of a crystal with such a surface proceeds both due to an increase or decrease in the area of surfaces bounded by closed steps and due to the formation of new atomic layers in the “collision” of two steps (this process is similar to the “umklapp” into a neighbouring row in the collision of two kinks, see fig. 20b). It is essential that these and reverse processes cause the existence of stationary states of the surface, however close to the ground one, in which continuous growth or melting of the crystal take place. The motion of a quantumly rough interphase boundary at zero temperature proceeds, therefore, without breakdown of the phase equilibrium. In other words, at zero temperature the growth coefficient becomes infinite. At finite temperatures the motion of a quantumly rough boundary is accompanied by dissipation caused by the interaction with thermal phonons (Andreev and Parshin, 1978). The energy dissipated per unit time at unit area of the surface is, in order of magnitude, Eph(V/c)V where c is the velocity of sound, Eph- nP/e3is the energy of the phonons, n is the number of atoms per unit volume, 8 is the Debye temperature. On the other hand, the same dissipated energy is S p h , where h-mnV is the mass transferred per unit time from one phase to the other, m is the atomic mass. So, we obtain V- rnc(03/P)Sp, i.e. the growth coefficient for a quantumly rough surface
X
- mc(e3/T4)
becomes infinite as T + 0 according to the law T-4.
( 5 1)
122
A.F. ANDREEV
It should be noted that regions of applicability of eq. (51) to 3He and 4He crystals differ considerably. In the case of 4He eq. (51) is applicable if phonons constitute the main type of thermal excitations in the liquid phase. This condition is known to be satisfied at temperatures below about 0 . 6 K . At higher temperatures the interaction with rotons, under the motion of the interphase boundary, plays the major roIe in the energy dissipation, so an exponential dependence of the form
x - exp(A,/T),
(5 2)
where A, is the roton gap should be observed instead of the power law. Eq. (51) can be applied to 3He crystals only at extremely low temperatures which are much lower than the 3He supertiuid transition point. At higher temperatures where liquid 3He behaves like a Fermi liquid the interaction with Fermi excitations is the main dissipation mechanism. The dissipation of energy is, in order of magnitude, p,nV V, where pF is the momentum at the Fermi surface; this corresponds to a growth coefficient independent of temperature
-
x -V i ' ,
-
(53)
where uF p,/m is the velocity of quasi-particles at the Fermi surface.
6.3. Crystallization,waues Weakly decaying melting and crystallization waves can propagate along a quantumly rough crystal-liquid interface (Andreev and Parshin, 1978; Keshishev et al., 1979). Outwardly these waves are similar to ordinary capillary waves at a liquid-vapour interface. There is, however, an essential difference consisting in the fact that capillary waves correspond to the motion of matter near the surface in the absence of evaporation and condensation processes. The waves under investigation are cornpletely caused by periodic processes of melting and crystallization. Let the initially flat crystal-liquid interface, z = 0, experience a displacement
t(x,,
t ) = S(t)exp(ik,x,)
in the direction normal to the interface. Here k,, x, are the twodimensional wave vector and coordinates x, y respectively, in the plane of the initial interface, p = 1,2.
QUANTUM CRYSTALS
123
Let us calculate the corresponding change AU in the potential energy
I
U = a(n)dS, where a ( n ) is the surface energy as a function of the direction n of the normal to the surface, dS is the surface element. For small [ we can assume n, = 1, n, = auax,,. dS = dx dy[l + $(a[/a~,)~], where a,,= adan,,
CY,”
a ( n ) = a(0)+ CY,
at -+ax,
1 a5 a5 a,” -2 ax, a&’
= a2alan,aq.
So, we obtain
where S is the total area of the surface. Since the densities ps and,& of the crystal and liquid differ from each other, the non-zero velocity of the interface 4 leads to the motion in the liquid (the crystal, naturally, remains at rest). It will be shown from the final result that the velocity of vibrations in question is much smaller than the velocity of sound. Under these conditions the liquid can be considered incompressible, i.e. the potential JI of the velocity u=VJI satisfies the Laplace equation AJI = 0. At z = 0 the mass has to be conserved P,(
awz)l,=o=
so we can express
-C(P,-~ in terms of
i:
The total kinetic energy of the surface is
Let us also calculate the energy E which is dissipated per unit time due to the finiteness of the growth coefficient. It was noted in the preceding section that the energy dissipated per.unit surface is 6pM, where fi is the mass flux across the phase boundary. To determine &, we may note that in the coordinate system where the surface is at rest the crystal moves at a
A.F. ANDREEV
124
velocity
-4,
so & = ps 4. We obtain, thus
where we took into account that Sp = t/%,according to eq. (49). Eqs. (54), ( 5 3 , and (56) coincide with similar equations for an oscillator with a fundamental frequency determined by w 2 ( k )=
~
'lk
( P s - PI)
'(ak'+a,,k,k,,)
(57)
and attenuation (imaginary part of the frequency) equal to
If we take into account the gravitational field g, an additional term p,gk/(p,-p,) appears in the right-hand side of the expression (57) for the crystallization wave spectrum. The dependence of the frequency on the wave vector is, therefore, similar to that for capillary-gravitational waves at the liquid-vapour interface, while the frequency dependence of the attenuation is quite distinct. The use of eqs. (51) and (57) for 4He in the low-temperature phonon region shows that in this case there is a wide range of wave vectors (T/O)'<
125
QUANTUM CRYSTALS
Fig. 22. Successive stages of the state of a surface after a strong perturbation.
(103~~-l) 20 -
10 -
5-
-
21
30
I
I
50
I
I
70
I
I
I
1
100
140 k(cm'')
Fig. 23. Dependence of the frequency of crystallization waves on the wave vector (Keshishev et al., 1979).
126
. A.F. ANDREEV
shown in fig. 23, agree well with the theoretical dependence u - - k 3 / * corresponding to the solid straight line in fig. 23. The experimental observation of crystallization waves proves that quantumly rough surfaces of 4He crystals do exist. This is also confirmed by visual observations of the state of the surface. In most cases the surface has a rounded shape corresponding to the equilibrium with due account made for gravitational and capillary effects. However, atomically smooth faces exist in addition to quantumly rough ones. According to observations by Keshishev, Parshin and Babkin this refers to the basal plane (0001). When the crystal orientation is such that the plane (0001) is almost horizontal, the interphase boundary contains an absolutely flat region of pure (0001) face. Such a situation is, probably, thermodynamical equilibrium, that is an indication of the positive energy of steps on the (0001) face. This fact is also confirmed by the data of Castaing et a]. (1980) on measuring the coefficient of sound propagation through the phase boundary. The fact is that at low temperatures propagation of sound is rather sensitive to the crystallization rate. If this rate is fairly high, then, as was shown by Castaing and Nozikres (1980), sound should be almost completely reflected from the phase boundary. In fact, in this case the phase equilibrium condition holds at the boundary that corresponds to a constant pressure as T + 0. The pressure must always be equal to the pressure of the liquid solidification, so the variable part of the pressure must vanish at the interface, i.e. the boundary condition for a sound wave at the interface coincides exactly with that for the case of reflection from the boundary with vacuum. It is clear that the reflection coefficient is equal to unity under these conditions. Castaing et al. actually observed very small transmission of sound, except the case where the phase boundary was very close to the (0001) direction. This result proves an abnormally small growth rate for the (0001) face that, according to section 6.2, corresponds completely to the statement that this face is atomically smooth. The experiments (Keshishev et al., 1979; Castaing et al., 1980) show, thus, that under equilibrium conditions and at low temperatures 4He crystals are partially faceted. Previously the faceting of helium crystals was observed (Landau et al., 1980) only under nonequilibrium conditions of rapid growth of crystals. As the temperature rises, the (0001) face becomes rounded. The corresponding temperature of the roughening transitions lies within the range 1.0-1.2 K, according various estimates (Keshishev et al., 1979; Landau et al., 1980; Balibar and Castaing, 1980).
QUANTUM CRYSTALS
127
7. Delocalization of dislocations
Another example of quasi-particles characteristic of quantum crystals can be obtained by considering a line dislocation lying in its slip plane at some angle to the crystallographic directions (Andreev, 1975). It is known that the dislocation line is not straight in this case. It consists of rectilinear segments located along the direction in which the energy of the dislocation line is minimal (this direction coincides with one of the principal crystallographic axes), and also of a certain number of kinks in the vicinity of which the dislocation passes from one valley to another (fig. 24). Each such kink can be considered as a point defect on the dislocation. Since the crystal is periodic along the principal crystallographic axis, this point defect in a quantum crystal becomes, due to tunnelling, a delocalized quasi-particle whose states are specified by the values of the one-dimensional quasi-momentum. Suppose that one kink exists on the dislocation with a definite quasimomentum. In such a state the kink is completely delocalized on the dislocation, it has an equal probability to be at all its points, and the dislocation itself turns out to be distributed uniformly between two neighbouring valleys. If a large number of kinks lie on the dislocation, then it is distributed over a large number of valleys. Therefore, the quantum-mechanical delocalization of kinks leads to the delocalization of the dislocation in the slip plane. If the density of kinks is rather small, they can be considered as a rarefied gas of quasi-particles. The necessary condition is that the mean distance n-' between the kinks (where n is the number of kinks per unit line of the dislocation) should be large as compared with the radius ro of the interaction between the kinks. The interaction radius is defined by arguments similar to those that were used in section 4.1 when considering the interaction of impuritons. Namely, at r = ro the potential energy V ( r ) of the interaction between two kinks must coincide, in order of magnitude, with the width Ak of the energy band of an isolated kink. In the case of kink the potential energy is inversely proportional to the first
128
A.F. ANDREEV
power of r (see: Hirth and Lothe, 1968) that is U(r)-S/r, where the constant S is, in order of magnitude, 6 pa4 and p is the shear modulus. So, we obtain ro-pa4/dk. The characteristic of a gas of kinks, which is observed experimentally, is their diffusion coefficient D. In the high-temperature region it is determined by collisions of kinks with phonons and with vibrations of the dislocation. The number of phonons and vibrations decreases with decreasing temperature and the diffusion coefficient reaches a limiting value determined by mutual collisions of kinks. When estimating this limiting value, we have to bear in mind the following. In the one-dimensional case the momenta acquired by quasi-particles due to a pair collision are unambiguously determined by the conservation laws. There are two unknown momenta and two relations for their determination (the conservation of energy and momentum). It is clear that the only solution is the trivial one corresponding to the collision in which the particles exchange their momenta. In this case the distribution function of the quasi-particle momenta does not change at all. In other words, the integral of pair collisions vanishes in the one-dimensional case, and all the kinetic phenomena are determined by triple collisions. The free flight path l3 corresponding to the triple collisions differs from the path 12- n-l due to pair collisions by a large factor ( n r J ' . Therefore, the diffusion coefficient of the kinks is, in order of magnitude,
-
D
- ul, --h
k
13--
A: hpa3n2'
So, instead of the ordinary law D-n in this case the diffusion coefficient is inversely proportional to the quasi-particle density squared. The motion of the kinks on a dislocation cannot provide for continuous motion of the dislocation itself due to the action of external forces. Such a motion must be accompanied by the production of pairs of oppositely directed kinks, and this production leads to the shift of a portion of the dislocation line to an adjacent valley. In ordinary crystals this process is a thermally activated super-barrier process, so the dislocation velocity due to the action of given force drops exponentially with temperature. It was shown by Petukhov and Pokrovskii (1973) that in quantum crystals the production of pairs of kinks is a quantum tunnelling process whose probability is independent of temperature. The measurement of internal friction is the most suitable method for the observation of the specific features in question concerning the dislocation behaviour. This method permits the investigation of the diffusion of
QUANTUM CRYSTALS
129
defects which cannot be observed by any other techniques. Additional possibilities arise for the case of quantum crystals (Meierovich, 1974). A specific mechanism of uniform relaxation of a non-equilibrium distribution of defects over quasi-momenta exists here along with the mechanism of relaxation of a non-uniform spatial distribution of defects characteristic of ordinary crystals. Though a large number of experiments has been performed up to now to measure the internal friction in quantum crystals of solid helium and in solutions of hydrogen in niobium (Suzuki, 1973, 1977; Andronikashvili et al., 1975, 1976; Tsymbalenko, 1977; Tsuruoka and Hiki, 1979), no clearly interpreted quantum features have been discovered yet.
References Abragam, A. (1961) The Principles of Nuclear Magnetism (Oxford UP, Oxford, England). Allen, A.R. and M.G. Richards (1977). Int. Quant. Cryst. Conf. Fort Collins, Colorado, p. C-83. Andreev, A.F. (1975) Sov. Phys. JETP 41, 1170. Andreev, A.F. (1976a) Sov. Phys.-Usp. 19, 137. Andreev, A.F. (1976b) Sov. Phys: JETP Lett. 24, 564. Andreev, A.F. and I.M. Lifshitz (1969) Sov. Phys.-JETP 29, 1107. Andreev, A.F. and A.E. Meierovich (1975) Sov. Phys.-JETP 40, 776. Andreev, A.F. and A.Y. Parshin (1978) Sov. Phys.-JETP 48, 763. Andreev, A.F. V.I. Marchenko and A.E. Meierovich (1977) Sov. Phys.-JETP Lett. -36. Andronikashvili, E.L., V.A. Melik-Shakhnazarov and LA. Naskidashvili (1975) Sov. J. Low Temp. Phys. 1, 290. Andronikashvili, E.L., LA. Gachechiladze and V.A. Melik-Shaknazarov (1976) J. Low Temp. Phys. 23, 5. Avilov. V.V. and S.V. Iordanskii (1976) Sov. Phys. JETP 42, 683. Balibar, S. and B. Castaing (1980) J. Physique Lett. 41, 329. Bakalyar, D.M., E.D. Adams, Y.C. Hwang and C.V. Britton (1977) Int. Quantum Cryst. Conf., Fort Collins, Coll. August 1977. Brinkman, W. and M. Rice (1970) Phys. Rev. B1, 1324. Castaing, B. and P. Nozitres (1979) J. Physique 40, 257. Castaing, B. and P. Nozieres (1980) J. Physique 41, 701. Castaing, B., S. Balibar, and C. Laroche (1980) J. Physique Lett. 41, 897. Chernov, A.A. (1961) Sov. Phys.-Usp. 4, 116. Chester, G. (1970) Phys. Rev. A2, 256. de Boer,J. (1948) Physica 14, 139. Delrieu, J.M. and M. Roger (1978) J. Physique Coll. 39, C6-123. Dzyaloshinskii, I.E., P.S. Kondratenko and V.S. Levchenkov (1972) Sov. Phys.-JETP 35, 823, 1213. Eshelhy, J. (1956) Sol. St. Phys. 3, 79. Ferrell, R.A. (1957) Phys. Rev. 108, 167.
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Fraass, B.A., S.M. Heald and R.O. Simmons (1977) Proc. Int. Quant. Cryst. Conf., Fort Collins, Colorado. Garwin, R.L. and A. Landesman (1964) Phys. Rev. 133, 1503. Godfrin. H. G. Frossati, A. Greenberg, B. Hebral and D. Thoulouze, (1980) Preprint. Greenberg, AS., W. G. Thomlinson and R.C. Richardson (1972) J. Low Temp. Phys. 8 , 3 . Grigor’ev, V.N., B.N. Esel’son, V.A. Mikheev and Y.E. Shul’man, (1973a) Sov. Phys.-JETP Lett. 17, 16. Grigor’ev, V.N., B.N. Esel’son, V.A. Mikheev, V.A. Slusarev, M.S. Stnhemechny and Y.E. Shul’man (1973b) J. Low Temp. Phys. 13, 65. Grigor’ev. V.N.. B.N. Esel’son and V.A. Mikheev (1974) Sov. Phys.-JETP 39, 153. Guyer, R.A. (1969) Sol. St. Phys. 23, 412. Guyer, R.A. (1972) J. Low Temp. Php. 8,427. Guyer, R.A. (1974) Phys. Rev. A10, 1785. Guyer, R.A. and L.I. Zane (197G) Phys. Rev. Lett. 24, 660. Guyer, R.A. R.C. Richardson and L.I. Zane (1971) Rev. Mod. Phys. 43, 532. Halperin, W.P., F. B. Rusmussen, C.N. Archie and R.C. Richardson (1974) Phys. Rev. Lett. 32, 927. Halperin, W.P., F.B. Rusmussen, C.N. Archie and R.C. Richardson (1978) J. Low Temp. Phys. 31, 617. Heritier, H. and P. Lederer (1977) J. Physique Lett. 38, L209. Hetherington, J.H. (1968) Phys. Rev. 176, 231. Hetherington, J.H. and J.W. Willard (1975) Phys. Rev. Lett. 35, 1442. Hetherington, J.H., W.J. Mullin and L.H. Nosanow (1967) Phys. Rev. 154, 175. Hirth, J.P. and J. Lothe (1968) Theory of Dislocations (McGraw-Hill, New York). Huang, W., H.A. Goldberg and R.A. Guyer (1975) Phys. Rev. Bll, 3374. Iodranskii, S.V. and A.M. Finkel’stein (1972) Sov. Phys.-JETP 35, 215. Jackson, K.A. (1975) in: Crystal Growth and Characterization (North Holland, Amsterdam). 21. Kagan, Y. and L.A. Maksimov (1974) Sov. Phys.-JETP 38, 307. Kagan, Y. and M.I. Klinger (1974) J. Phys. C7, 2791. Keshishev, K.O. (1977) Sov. Phys-JETP 45, 273. Keshishev, K.O., L.P. Mezhov-Deglin and A.I. Shal’nikov (1970) Sov. Phys.-JETP Lett. 12, 160. Keshishev, K.O., A.Y. Parshin and A. Babkin (1979) Sov. Phys.-JET? Lett. 30, 56. Kirk, W.P., E.B. Osgood and M. Garber (1969) Phys. Rev. Lett. 23, 883. Krivoglaz, M.A. (1974) Sov. Phys.-USP. 16, 856. Kummer, R.B., R.M. Mueller and E.D. Adams (1977) J. Low Temp. Phys. 27, 319. Kuper, C.G. (1961) Phys. Rev. 122, 1007. Landau, L.D. (1965) Collected Papers, Vol. I1 (Pergamon. Oxford). Landau, L.D. and E.M. Lifshitz (1965) Quantum Mechanics (Pergamon, Oxford). Landau: L.D. and E.M. Lifshitz (1969) Statistical Physics (Addison-Wesley, New York). Landau, J., S.G. Lipson, L.M. Maattanen, L.S. Balfour and D.O. Edwards (1980) Phys. Rev. Lett. 45, 31. Landesman, A. (1978) J. Physique, Coll. 39, C6-1305. Landesman, A. and J.M. Winter (1974) in: Proc. 13th Conf. on Low Temperature Physics, LT-13, Vol. 2 (Plenum, New York). Leggett, A.J. (1970) Phys. Rev. Lett. 25, 1543. Lhuillier, C. and F. Laloe (1979) J. Physique 40, 239. Lifshitz, I.M. and S.A. Gredeskul (1970) Sov. Phys.-JETP 30, 1197.
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Lifshitz, I.M. and Y. Kagan (1972) Sov. Phys.-JETP 35, 206. Marty, D. and F.I.B. Williams (1973) J. Physique 34, 989. McMahan. A.K. (1972) J. Low Temp. Phys. 8, 115. McMahan, A.K. and R.A. Guyer (1973) Phys, Rev. A7, 1105. McMahan, A.K. and J.W. Wilkins (1975) Phys. Rev. Lett. 35, 376. Meierovich, A.E. (1974) Sov. Phys.-JETP 40, 368. Meierovich, A.E. (1975) Sov. Phys.-JETP 42, 676. Meierovich, A.E. (1976) Sov. Phys.-JETP 44, 617. Mikheev, V.A., B.N. Esel’son, V.N. Grigor’ev and N.P. Mikhin, (lq77) Fiz. Nizk. Temp. 3, 385. Mineev, V.P. (1973) Sov. Phys.-JETP 36, 964. Mullin, W.J., R.A. Guyer and H.A. Goldberg (1975) Phys. Rev. Lett. 35, 1007. Nagaev. E.L. (1968) Sov. Phys.-JETP 27, 122. Nagaoka, A. (1966) Phys. Rev. 147, 392. Nosanow, L.H. and C.M. Varma (1969) Phys. Rev. 187, 660. Osheroff, D.D., M.C. Gross and D.S. Fisher (1980) Phys. Rev. Lett. 44, 792. Petukhov. B.V. and V.L. Pokrovskii (1973) Sov. Phys,-JETP 36, 336. Prewitt, T.C. and J.M. Goodkind (1977) Phys. Rev. Lett. 39, 1283. Pushkarov, D.I. (1971) Sov. Phys.-JETP 32, 954. Pushkarov, D.I. (1974) Sov. Phy.-JETP Lett. 17, 386. Reich, H.A. and W.N. Yu (1963) Phys. Rev. 129, 630. Richards, M.G.. J. Hatton and R. Giffard (1965) Phys. Rev. 139, A1991. Richards. M.G., J. Pope and A. Widom (1972) Phys. Rev. Lett. 29,708. Richards, M.G., J.H. Smith, P.S. Tofts and W.J. Mullin (1975). Phys. Rev. Lett. 34, 1545. Richardson, R.C., E. Hunt and H. Meyer (1965) Phys. Rev. 138, A1326. Roger, M., J.M. Delrieu and J.H. Hetherington (1980) preprint. Sacco, J.E. and A. Widom (1975) J. L o w Temp. Phys. 24, 241. Sacco, J.E. and A. Widom (1978) Phys. Rev. B17, 204. Sai-Halasz, G.A. and A.J. Dahm (1972) Phys. Rev. Lett. 28, 124. Shal’nikov, A.I. (1965) Sov. Phys.-JETP 20, 1161. Sullivan, N., G. Deville and A. Landesman (1975) Phys. Rev. B11, 1858. Suzuki, H. (1973) J. Phys. Soc. Japan 35, 1472. Suzuki, H. (1977) J. Phys. Soc. Japan 42, 1865. Thouless, D.J. (1965) Proc. Phys. Soc. 86, 893. Tsuruoka. F. and Y. Hiki (1979) Phys. Rev. B20, 2702. Tsymbalenko, V.L. (1977) Sov. Phys.-JETP 45, 989. Widom, A. and M.G. Richards (1972) Phys. Rev. A6, 1196. Wilks, J. (1967) Liquid and Solid Helium (Clarendon, Oxford,). Yamashita, Y. (1974) J. Phys. Soc. Japan 37, 1210. Zane, L.I. (1972) J. Low Temp. Phys. 9, 219.
This Page Intentionally Left Blank
CHAPTER 3
SUPERFLUID TURBULENCE* BY J.T. TOUGH Department of Physics, Ohio State University, Columbus, Ohio 43210, USA
* Supported
by U.S. National Science Foundation Grant #DMR 7925089.
Progress in Low Temperature Physics, Volume VZII Edited by D. F. Brewer 0 North-Holland Publishing Company, 1982
Contents 1. Introduction 2. Theoretical background 2.1. Macroscopic considerations 2.2. Microscopic considerations 3. Temperature and chemical potential difference data 3.1. Three turbulent states 3.2. The second superfluid turbulent state -TI1 3.3. The fmt superfluid turbulent state-TI 3.4. The third superfluid turbulent state - TI11 4. Pressure difference data 4.1. The Allen and Reekie rule 4.2. The fmt superfluid turbulent state-TI 4.3. The second supertluid turbulent state -TI1 4.4. The third superfluid turbulent state -TI11 4.5. The eddy viscosity and homogeneity 5. Second sound data 5.1. Propagation of second sound in superfluid turbulence 5.2. Transverse second sound attenuation data 5.3. Longitudinal second sound attenuation data 5.4. Second sound dispersion data 6. Ion current data 6.1. Early experiments 6.2. The ion-vortex line interaction 6.3. Recent experiments 7. Fluctuation phenomena 7.1. Classical and quantum mechanical turbulence 7.2. Mechanical probes 7.3. Ion and second sound probes 8. The critical condition 8.1. Primary and secondary transitions 8.2. The primary superfluid turbulent transitions 8.3. The secondary superfluid turbulent transition 9. Pure superflow, pure normal flow, and other velocity combinations 9.1. Pure superflow ( V , = 0): a fourth turbulent state 9.2. Pure normal flow (V,=O) 9.3. Other V , and V, combinations References
135 143 143 147 155 155 155
160 162 165 165 165 168 168 168 171 171 172 176 179 180 180 182 183 189 189 192 196 200 200 202 204 207 207 21 1 211 216
Although the most extraordinary property of liquid He I1 is superfluidity, it is rather the dissipative flow of this quantum fluid that has proven to be the more difficult problem to understand. Early work in this field sought to uncover a universal description of the various complex nonlinear phenomena that were observed in the laboratory. In this view, all forms of dissipation in He I1 would be regarded as a ‘breakdown of superfluidity” at some universal “critical velocity” associated in some way with quantized vorticity production. Such varied experimental observations as pure superfluid flow through microscopic slits and pinholes, thermal counterflow associated with heat conduction, the flow of the superfluid film, and the frictional forces on objects moving in the helium would all be expected to display similar dissipation and critical velocities. Although there certainly are some aspects of dissipation common to these various flow conditions, a universal explanation is simply unrealistic. The modern approach to He I1 dynamics has been quite non-universal and has sought to identify and model each particular form of dissipation. The present review is a critical analysis of the particular dissipative phenomenon known as superfluid turbulence. This type of dissipation is responsible for the large and nonlinear thermal resistance of liquid He I1 subjected to a large heat flux. Although a variety of experimental data have been variable for over three decades, it has only been recognized recently that various important aspects of supertluid turbulence are controlled by geometry (Ladner and Tough, 1978,1979). A principal goal of this review is to survey this vast accumulation of experimental data, to identify common characteristics, and to associate these features with the particular geometrical aspects of superfluid turbulence. Another recent advance has been the development of a microscopic theory of homogeneous superfluid turbulence (Schwarz, 1978). This theory contains all the qualitative features incorporated in the phenomenological models that have been used to interpret superfluid turbulence phenomena (Gorter and Mellink, 1949; Vinen 1957c), and in that sense justifies their use. The quantitative comparison of theory and experiment involves assumptions about the homogeneity and isotropy of the turbulence present in the particular experiments. Another goal of this review is to identify those features of the data analysis which involve these assumptions. It is important to initially establish a clear overview of the phenomena associated with the superfluid turbulent state, without extended discussion of specific details. The turbulent state is established by applying heat at a
J.T. TOUGH
136
probe ,-+T
+AT
heater Fig. 1. Schematic diagram of a thermal counterflow apparatus indicating the principal experimental probes of the superfluid turbulence in the Row channel.
rate Q to one end of a sample of He 11 contained in a flow channel (fig. 1).The channel has a length I, area A, small dimension d, and hydraulic diameter D (=4xArea/Perimeter). The helium supports the heat flux W ( = Q / A )by the well-known counterflow of the normal and superfluid components at average velocities V,, and V, respectively where V,, = WIpST,
(1)
Vs = Pn VnlPs,
(2)
and the average relative velocity of the two fluids is V = Vs+V,, = W/p,ST.
(3)
(The plus sign follows since V,, and V, are oppositely directed.) Here p,, and ps are the normal fluid, superfluid, and total fluid densities, T is the ambient temperature and S is the entropy density. The dynamical state of the He I1 is entirely determined by the four variables T, W, the small dimension of the channel d, and the channel shape. Information about the flow has been obtained from several types of measurement: the relative
SUPERFLUID TURBULENCE
137
increase of temperature, pressure or chemical potential (AT, AP, A p ) , the attenuation of second sound, and the influence of the flow on the transmission of positive and negative ions. Below a critical heat flux W,, the flow is always laminar and the only dissipation is provided by the normal fluid viscosity q. The pressure difference AP is observed to agree with the Poiseulle formula for the laminar flow of the normal fluid (Brewer and Edwards, 1959; Childers and Tough, 1975, 1976; van der Heijden et al., 1972a; Keller and Hammel, 1960),
APIA= GqlV,/dz, (4) where G is a factor which is known for each flow channel shape. The chemical potential difference is zero (Yarmchuck and Glaberson, 1979; de Haas and van Beelen, 1976), consistent with perfect dissipationless flow of the superfluid. The temperature difference is then determined by AP and A p and is given by AT, = APIpS = GqlV,lpSd2
(5)
again in excellent agreement with experiment (Brewer and Edwards, 1959; Keller and Hammel, 1960; Childers and Tough, 1976; Ladner and Tough, 1979; de Haas and van Beelen, 1976). The attenuation of second sound is found to be independent of W and determined by the normal fluid viscosity (Vinen, 1957a; Kramers et al., 1960). Both negative and positive ions are found to drift in the direction of the normal fluid flow (Ashton and Northby, 1975; Careri et al., 1959), and the speed of the positive ions is very nearly V, (Vicentini-Missoni and Cunsolo, 1966). The state of superfluid turbulence is only stable for a heat flux greater than a critical value W,, (Vinen, 1957d; Brewer and Edwards, 1961a; Childers and Tough, 1973, 1976; Chase, 1962). The pressure difference in the turbulent state is found to be only slightly larger than in the laminar state (Allen and Reekie, 1939; Bhagat and Critchlow, 1961; Brewer and Edwards, 1961b; Childers and Tough, 1975) as shown in fig. 2. The chemical potential (Yarmchuck and Glaberson, 1979; de Haas and van Beelen, 1976) and temperature difference (Vinen, 1957a; Childers and Tough, 1976; Ladner and Tough, 1979) are increased dramatically in the turbulent state, and are found to exceed the laminar value by an amount approximately proportional to as shown in figs. 3 and 4. The symmetry of the two fluid equations of motion indicates that if the increased dissipation can be described in terms of some new average force density, then this force must act with equal magnitude but opposite
J.T. TOUGH
138
T = 1.5 K I
Fig. 2. The pressure difference AP as a function of the heat current Q (Childers and Tough. 1975). The data in the turbulent region are only slightly greater than the extrapolation of the laminar data (broken line).
directions on the normal fluid and the superfluid (Gorter and Mellink, 1949). Such a mutual friction force also gives rise to an added second sound attenuation a' which increases approximately as (Vinen, 1957a), as shown in Fig. 5. The mutual friction force has a microscopic origin in the scattering of
SUPERFLUID TURBULENCE
139
7------1 6-1m-
El
5-
E
P 4P al
Y
-1m-
3-
-< D
2I I
1
I
50
I00
I50
w (mW/crn2)
Fig. 3. The cube-root of the chemical potential gradient (parallel to the flow) Vp,, as a function of the heat flux W (Yarmchuck and Glaberson, 1979).
normal fluid excitations from a random distribution of quantized vortex lines that define the microscopic structure of the superfluid turbulence. The most direct experimental evidence for this structure is found in ion experiments. Negative ion currents are found (Careri et a]., 1960a,c; Sitton and Moss, 1969, 1972; Ashton and Northby, 1973) to be strongly attenuated by the superfluid turbulence. Fig. 6 shows the fractional
Y
E
6
(p-wotts) Fig. 4. The temperature difference AT as a function of the heat current Tough, 1976).
0 (Childers and
J.T. TOUGH
140
E u
Y
w
(W ern-')
Fig. 5. The square-root of the excess second sound attenuation a' as a function of the heat flux W (Vinen, 1957a).
reduction of collected ion current as a function of the heat flux. The ion current data are in quantitative agreement with the calculations of Donnelly and Roberts (1969) for ion-vortex line trapping. In particular, the escape probability of negative ions trapped in superfluid turbulence is identical to that in rotating helium except for the spatial distribution of the vortex lines (Sitton and Moss, 1972). A recently discovered feature of the superfluid turbulent state (Hoch et al., 1975) is the presence of
I
I
I
I
50
100
150
200
i I
w (mwcm') Fig. 6. The fractional decrease in negative ion current Ajlj as a function of the heat flux W (Ashton and Northby, 1973).
SUPERFLUID TURBULENCE
141
fluctuations. The chemical potential (Yarmchuck and Glaberson, 1979), the second sound attenuation, and the negative ion current (Hoch et al., 1975; Ostermeier et al., 1980) all have been observed to fluctuate about a mean value. The mean-squared fluctuation amplitude increases approximately as W. The power spectral density has a roll-off frequency of between 0.1 and 1.0Hz and varies roughly as u - ~ . The properties of the superfluid turbulent state are determined by the temperature, the heat flux W,the channel size d, and the channel shape. The phenomena associated with the turbulence described qualitatively above have been observed over a broad range of heat flux, over the temperature range from 1K to TA and in flow channels of dimension extending from 1 cm down to 3 X cm. Experimental considerations tend to limit the ion and second sound experiments to larger channels. Temperature, pressure, and chemical potential measurements are generally limited to smaller ones. Modern experimental techniques have extended the temperature and chemical potential measurements
I
I
40
I
60 (pwotts)
50
b
I
70
Fig. 7. The temperature difference AT and the pressure difference AP as functions of the heat current 0 in a circular channel (Ladner et al., 1976). The data indicate the dramatic change in the turbulence at the second critical heat current. The broken lines are extrapolations of the laminar data. Note the suppressed zeros.
J.T. TOUGH
142 I
I
I
1
1
I
I
0
.30
20
15
10
V, (cm/sl
5
0
P
5
10
v
15
20
25
(crnls)
Fig. 8. Comparison of the temperature difference AT for a circular channel measured either in pure superflow (superfluid velocity V, plotted to the left), or in thermal counterflow (relative velocity V plotted to the right). There is no laminar AT in superflow, and only a single turbulent state. (Ashton and Tough, 1980).
(Yarmchuck and Glaberson, 1979; Martin and Tough, 1980) to channels as large as 0.25 cm. The analysis of these data indicates that there is no quantitative difference in the superfluid turbulent state over these two orders of magnitude range of channel size. Although the properties of the turbulence depend on the particular value of the heat flux W, that dependence may very well reflect a more complex variation with the individual velocities V,, V,, and V. Workers at the Kamerlingh Onnes Laboratory have pioneered experiments in which the normal and superfluid velocities can be independently varied rather than be fixed by the counterflow constraint, eq. (2). Observations of the temperature, pressure, and chemical potential difference as well as of second sound attenuation have been made in these complex flows. Of all the factors that influence the superfluid turbulent state, the shape of the flow channel is most profound and was the least expected. A
SUPERFLUID TURBULENCE
143
change of shape from circular to high aspect ratio rectangular cross section increases the magnitude of the mutual friction force by about a factor of four (Ladner and Tough, 1979), all other variables remaining unchanged. This is one reason for the apparent discrepancy of the mutual friction coefficients reported in the past. But channel shape has an even more important effect on the turbulence. In a circular channel, the superfluid turbulent state undergoes a dramatic modification at a welldefined second critical heat flux W,, (Brewer and Edwards 1961a, 1962; Ladner et al., 1976) as shown in fig. 7. The second turbulent state, well above Wc2, can also be described by a mutual friction force (Dimotakis and Broadwell, 1973). The past confusion of these two different states of superfluid turbulence has also resulted in rather discrepant values of the mutual friction coefficient. Only a single turbulent state is found in thermal counterflow in rectangular channels (Keller and Hammel, 1960; Ladner and Tough, 1978; Yarmchuck and Glaberson, 1979), or in pure superflow in circular channels (de Haas and van Beelen, 1976; van der Heijden et al., 1972b; Ashton and Tough 1980), as shown in fig. 8. A final goal of this review will be to catagorize the various distinct superfluid turbulent states (there are at least four) and compare their properties.
2. Theoretical background
2 , l . Macroscopic considerations In principle, the observed properties of superfluid turbulence reveal details of the microscopic structure of the turbulence, the distribution of quantized vortex lines. In practice, this connection is difficult to make and various phenomenological models have been introduced as expedients. It is necessary to make assumptions regarding the form of various microscopic quantities when averaged over macroscopic volumes. To some extent the data provide essential clues about the appropriate assumptions. Experiments in the laminar region are in excellent agreement with the two fluid equations (Landau and Lifshitz, 1959),
ps
at + p,(o,
V)o, - -& V P + p,SVT, P
(7)
144
J.T. TOUGH
where u, and us are the local time dependent normal and superfluid velocities. Assuming laminar time independent flow of the two fluids, so that the left hand side of eqs. (6) and (7)are zero, the temperature and pressure difference are given by eqs. (4) and (5) in excellent agreement with experiment as noted in section 1. In the presence of superfluid turbulence these equations alone can no longer define the dynamic state of the helium since they do not include the local interaction between the quantized vortex lines and the normal fluid (Hall and Vinen, 1956b; Hall, 1960; Campbell, 1970). The magnitude of this force on a single line, neglecting geometric factors arising from various vector cross products, is
where B is a temperature dependent coefficient which has been determined from second sound attenuation data in rotating helium (Lucas, 1970; Hall and Vinen, 1956a), K is the quantum of circulation, and uL is the local velocity of the vortex line. The overwhelming mathematical complexity that results from including normal fluid-vorticity coupling effects into eqs. (6) and (7) (Bekarevich and Khalatnikov, 1961) can be avoided by making certain approximations and assumptions (see for example Craig et al., 1963). Since the superfluid turbulence properties to be analysed are temporal and spatial averages, eqs. (6) and (7) can be re-written in the mutual fricrion approximation (Gorter and Mellink, 1949).
The brackets ( ) are to represent time averages of the axial component of the enclosed vector quantity, and the mutual friction force F,, is the time average of the axial component of f. Adding eqs. (9) and (10) gives
(VP) = qV2(un). (11) The experimental result that VP in superfluid turbulence is not substantially different from in the laminar region (the Allen and Reekie rule) suggests that the laminar solution to eq. (6) can be used for (u,) in eq. (9). This laminar mean flow assumption leads directly to a pressure difference
SUPERFLUID TURBULENCE
145
given by eq. (4), a temperature difference given by
AT = AT,+ lFsn/p,S,
(12)
and a chemical potential difference given by = lFsn/ps:
(13)
Analysis of the early temperature difference data by Gorter and Mellink (1949) indicated that F,, was proportional to third power of the relative velocity V and it has become traditional to write
although this will be shown to be only approximately true. The quantity A is called the Gorter-Mellink or the mutual friction coefficient. Using F,, as in eq. (14) leads directly (Vinen, 1957a) to an expression for the excess second sound attenuation by the superlluid turbulence, a’= AV2/2u2,
(15)
where u2 is the velocity of second sound. The real physics of the superfluid turbulence problem involves the microscopic distribution of vortex lines that determines the macroscopic mutual friction force. The connection between a distribution of vortex lines and Fsnwas first made by Vinen (1957~).Following a suggestion by Feynman (1955) that a state of superfluid turbulence might be a distribution of vortex lines, and the analysis of experiments on the attenuation of second sound in rotating He I1 (Hall and Vinen, 1956b), Vinen proposed that the superfluid turbulent state could be described by a homogeneous distribution of lines. If the steady state consists of an average length of line per unit volume Lo, and if the state is isotropic so that (03 = 0, then it follows directly from eq. (8) that Fsn
= (&spnK/2p)($U( V).
(16)
The factor 3 arises from an approximate averaging over directions to obtain the axial component of mutual friction, and V is the average relative velocity given by eq. (3). The result in eq. (16) provides the principal connection between experiments and theoretical determinations of the line density Lo, and its limitations must therefore be clearly understood. The ion experiments of Ashton and Northby (1975) have demonstrated
146
J.T. TOUGH
that the line distribution is not isotropic, and that the assumption (R)= 0 is incorrect. The experiments show however that (vL) is proportional to V, and that ( d / V is a function of temperature, b ( T ) . This temperature dependence is absorbed into that of Lo, if eq. (16) is used to analyse experimental data (Piotrowski and Tough, 1978a). The temperature dependence of Lo will therefore be universal only if b ( T ) is a universal function of temperature. If the distribution of vortex lines is homogeneous, then ( q )is solely determined by the microscopic line distribution and b(T) will be universal. If, however, the distribution of vortex lines is inhomogeneous, determined perhaps by the flow channel geometry, then b ( T ) will not be universal, and the temperature dependence of Lo obtained from the use of eq. (16) will vary with the flow channel geometry (see section 9.3). The question of homogeneity arises again when the factor V in eq. (16) is considered in detail. The assumption that has been made is
Note that the brackets denote the time average of the axial component, while V is an average over the channel cross section. The exact meaning of this assumption is not obvious, but the need to make it is perfectly clear. Since (v,,) is determined by the laminar mean flow assumption and is thus a function of position r in the flow channel, (vn-vs) will also be a function of r unless u,(r) is assumed to be exactly complementary to u,(r). If (u,,-u,) depends on r then, by eq. (16), F,, depends on r, and the equations of motion, eqs. (9) and (lo), must be extended to include transverse as well as axial components of, VT and VP. The recent experiments of Yarmchuck and Glaberson (1979) provide important information on this question. Using a very sensitive temperature gradiometer they were able to show that the transverse temperature gradient in the turbulent region was negligibly small. It must be concluded that the assumption in eq. (17) is empirically consistent, although without convincing justification. The implications of the simple equivalent assumption that vJr) and u,(r) are complementary will be considered in the discussion of the pressure difference data (section 4). Subject t o the substantial approximations and assumptions discussed above eqs. (12), (13), (15), and (16) provide the means by which experimental data can be reduced to give the vortex line density Lo. The analysis of ion data requires dynamical information and will be discussed in section 6.
SUPERFLUID TURBULENCE
147
2.2. Microscopic considerations The microscopic structure of the superfluid turbulent state is a description of the distribution of quantized vortex lines. For the experiments considered above it is only necessary to know the integral over the distribution giving the total length of line per unit volume Lo. If the distribution is spatially homogeneous then the density Lo cannot depend on channel size or shape, but only on T and W, [or the equivalent variable V, eq. (3)]. The function Lo(T,V) can be regarded as the dynamical equivalent of the equation of state for the superfluid turbulence. In the original model of Vinen (1957c), Lo(T, V) is determined directly but phenomenologically. In the theory of Schwarz (1978), the distribution function is obtained from a microscopic calculation and Lo is determined by integration over the distribution. The results of these two very different calculations are remarkably similar. The Vinen model considers a homogeneous distribution of vortex lines that at any instant of time has a density L and an average interline spacing I = 1/L”’. The density will increase due to line “stretching” or “production” processes occurring throughout the fluid. Modeling this process on the growth of a vortex ring in a counterflow, Vinen finds a line production rate (dLld r)p = ~ x I B (pnlp)L3’2V
(18)
where x, is a phenomenological parameter. Line is also being continuously destroyed throughout the turbulence. Vinen models this process after the mutual annihilation of two parallel and oppositely polarized vortex lines that are closer than the average separation I, and finds (dL/dt)d= X2KL2/2T,
(19)
where x2 is another phenomenological parameter. If the production and destruction processes are independent, then the net change in line density is given by the Vinen dynamical equation (dL/dt) = (dL/dt),- (dL/dt)d,
(20)
and the steady state line density Lo obtained by setting dL/dt equal to zero is Lo = y ’ V , where Y = TBPnX1IKPX2.
(21)
(22)
J.T.TOUGH
148
The ratio x,/x2 is a function of temperature that must be determined from experiment, but the velocity dependence of L,(T, V) is given directly by the model. A primary success of Vinen’s experimental work was to demonstrate that there was a critical heat flux W,below which the mutual friction force was zero. He showed (Vinen, 1957d) that a simple way to incorporate this effect into the line density model was to artificially exclude the production process from a layer of fluid within a distance of order I from the channel walls. A modified form of the steady state line density is obtained and is shown in dimensionless form as the solid line in fig. 9. The line density is unstable for V = V, where
L ‘I2( T, V,)d = 2,
(23)
and Lo=O for V< V,. Between 1.2VCand 4V, the line density is given within 2% by the approximate form
LA” = 1.037(V- VJ, 16
I
(24) I
I
YVd Fig. 9. The square-root of the vortex line density Lo as a function of the relative velocity V plotted in dimensionless form. The solid line is the Vinen result, and the dashed line is the approximate form, eq. (24). Note the distinction between the critical velocity V, and the intercept V,.
SUPERFLUID TURBULENCE
149
where
v, = 1.44fflyd and (Y is a number of order unity. Although a complete discussion of the critical velocity will be given in section 8, it is important at this point to note the difference between the quantities V, and V,. As seen in fig. 9, V, is the velocity at which Lo extrapolates to zero, and the critical velocity V, is the lowest velocity at which a line density exists. Combining eqs. (24) and (16) gives a mutual friction force of the form, eq. (14), with A = 1.06~By~/3p.
(26)
Schwarz (1978) has recently produced a remarkable theory in which the actual microscopic distribution function of the vortex lines is computed. He considers a homogeneous distribution of vortex lines in the presence of a uniform counterflow velocity V. Each unit length of line is subject to the local force f, eq. (8). Using the localized-induction approximation (Arms and Hamma, 1965) to simplify the equation of motion for a single vortex line of radius a0, it follows that the instantaneous motion of a point on a line subject to the force f is determined entirely by the local velocity ul. The microscopic structure of the vortex line distribution can then be written as a function A(q, t), where the total line density L(t) is the integral of A ( q , r) over q-space. Since the counterflow velocity V defines a preferred axis A is not isotropic. Using the result that Iu,( = B/R, where R is the local radius of curvature of the line and 0 = (~/4r)In(R/a,,) can be treated as a constant, an equivalent distribution function is A(R, 8, t ) where 8 is the angle between uI and V. The time evolution of a length AI of vortex line in the distribution is then rigorously determined by two types of processes which Schwarz calls “driving terms”, due to the force f, and “higher order terms”, due to the self-induced motion of the line. The driving terms alone lead directly to the equations
J.T. TOUGH
150
The physical meaning of these equations is readily seen when they are applied to a simple vortex ring of radius R moving in the forward (0 = 0) direction. If V > u1 = O/R, then the normal fluid is “pushing” the ring and it grows, eq. (27). If V < ul = @ / Rthe ring shrinks. The dynamic effect of eq. (28) is to rotate rings with O f 0 into the forward direction. Clearly if only “driving terms” are considered, any initial distribution of vortex line would grow without limit, with R + a ~ and 8 + 0. In terms of the distribution function A(R, 8, r), the effect of the driving terms can be written
The “higher order terms” that have not yet been considered are due to the self-induced motion of the lines, and will cause a kind of diffusion of A in ul-space. This will limit the runaway process generated by the “driving terms”. Unfortunately the “higher order terms” are too complex to be dealt with exactly. Instead Schwarz has given a dynamical argument suggesting that the effect of these terms is to produce a randomization in uI -space equivalent to line-line crossings over a characteristic distance 1 = 1/L1’2. Modeling this process leads to several very complex expressions for the relaxation effect on the distribution function, (dhldt),. The time evolution of A is now given by dh/dr = (dh/dt)d - (dA/dt),
(31)
an equation which contains no adjustable parameters. Beginning with some arbitrary initial distribution and a particular value V and T eq. (31) is numerically integrated forward in time. This is a monumental undertaking since (dA/dt) contains not only nonlinear terms, but GI and 1 which are averages over the distribution and must be included self-consistently. The “higher order terms” as modeled by Schwarz do in fact produce the desired relaxation and A approaches a well defined steady state Ao(R,8) independent of the initial conditions. Fig. 10a shows a steady state solution for a particular value of V and T. The distribution is strongly peaked in the forward direction, and about a radius of curvature R,,. Sampling distributions at other values of V and T Schwarz finds that the characteristic radius of curvature of each distribution Ro is proportional to the average interline spacing I, and
I/R, = c ( T ) .
(32)
This scaling of Ro with I is of course to be expected from self-consistency,
SUPERFLUID TURBULENCE 10
I
151
I
8 n -
0.909
E 6
0
0.545
m
0
Y
0. I 8 2
N
rY4
x
-0. I82
-0.545
2
0
3
2
I
R (10-4cm)
T = 2.OK ,V= 20cm Sec-'
03
L
j 1
0.0 I-0.0
0.5
1.0
1.5
2.0
r=R/Ro Fig. 10. The steady state vortex line distribution function A(R, 8 ) determined in the Schwarz theory. Fig. 10a shows that A is strongly peaked for a radius of curvature R,. and for motion in the forward (8 = 0) direction. Fig. 10b compares the distribution at extreme values of temperature and relative velocity and shows the approximate universality of the distribution when scaled with R , (Schwarz, 1978).
since I is the only characteristic length in the problem, and was specifically introduced to model the relaxation process (dA/dt),. It is as also found that when A is written in terms of the reduced variable r = R/R,, the reduced A is approximately a universal function of r and 6. This property is useful when computing the average of some property x ( R , 6 ) over the
152
J.T. TOUGH
distribution,
‘I
1x1 = -
L
x(R, 8)Ao(R, 8 ) R 2d R d R
since the integral over r and 8 will be approximately constant, independent of T and V. The approximate universality of the reduced distribution A,r213/c3L is shown in fig. lob. The microscopic picture of the superfluid turbulent state that is revealed by the Schwarz theory may be crudely pictured as a “gas of vortex rings” with radius Ro= I and moving in the forward direction. The distribution is maintained in a steady state by collisions that tend to randomize R and 8. As yet there have been no experimental tests of the microscopic aspects of the Schwarz theory, so macroscopic properties of the distribution must be considered. The total line density Lo(T,V) is obtained by a numerical integration of A. over R and 8, for various values of T and V. The results are shown in fig. 11 with straight lines drawn through the points corresponding to
L,,=a(T)(V-V$
(34)
v (cmsec-’1 Fig. 11. The steady state vortex line density Lo as a function of the relative velocity V as determined in the Schwarz theory (Schwarz, 1978). The result is the same as the approximate form of the Vinen result (dashed line, fig. 9).
SUPERFLUID TURBULENCE
153
Table 1 Quantities determined in the Schwarz theory of homogeneous turbulence
T(K)
a (s2/cm")
b
A (scrnlg)
y
1.2 1.4 1.6 1.8 2.0
7225 24 OOO 50 600 96 000 212000
0.24 0.33 0.39 0.44 0.5 1
20 46 69 86 188
70 120 166 219 304
(s/cm2)
where the function a ( T ) is tabulated in table 1. The result is remarkably similar to that obtained by Vinen, eq. (24) and fig. 9, by a very different method. The only difference between the Vinen and Schwarz result for the line density involves the quantity V, and this feature will be discussed below. The mutual friction force F,, can be obtained by averaging the axial component of the local force on a vortex line over the distribution A,. The result is shown in fig. 12 where straight lines correspond to the form of F, given in eq. (14). Values of A ( T ) obtained from the calculations are given in table 1. The drift velocity of the vortex lines can also be obtained by computing the average of ul over the distribution A,. The results gives U,=[UII=
b(T)V,
I
(35)
E
n
"0
5
v
I0
15
20
(cm sec-'1
Fig. 12. 'Ihe mutual friction force F, obtained in the Schwarz theory by averaging the force on a vortex line over the steady state vortex line distribution (Schwarz, 1978). The result is identical to the Gorter-Mellink form, eq. (14).
J.T. TOUGH
154
in agreement with the observations of Ashton and Northby (1975). Values of b ( T ) are given in table 1. The results in table 1 can be used to check the quantitative accuracy of eq. (16) for the case of homogeneous turbulence treated by the Schwarz theory. Since vL# 0, V in eq. (16) must be replaced by V- (uI1 = (1 - b )V, and eq. (16) is then equivalent to A = B ~ a ( 1 -b)3p. The results in table 1 indicate that this relation is approximately obeyed over the temperature range. The maximum discrepancy of about 20% at 2 K may be due to many causes, but indicates the degree to which eq. (16) is valid for homogeneous turbulence. As discussed above, the universal function b(T) must be absorbed into the temperature dependence of Lo. Using eq. (24) then yields
y(T)=[a(l-b)]”2/1.03
(36)
for homogeneous turbulence. Values of y are given in table 1. Finally it is a straightforward matter to show that when the fundamental dynamical equations (31) or (27) are averaged over ho the result is exactly the dynamical Vinen Equation (20). except of course the parameters x , and x2 are now known functions of temperature. The physical interpretation of the “destruction” term, eq. (19) is also rather different in the Schwarz theory. This term still measures the net rate at which line is removed from the turbulence, but the effect is more like line “shrinking” than “destruction” due to line-line collisions as suggested by Vinen. Collisions, in the Schwarz theory, simple serve to randomize and relax the distribution in ul-space. In the crude “ring gas” picture, there are large rings moving upstream getting larger [corresponding to (dWdr),] and small rings getting smaller [corresponding to (dWdr),] all according to eq. (27). Collisions between the rings simply keep the process from running away. The Schwarz theory must be regarded as a major contribution to the study of superfluid turbulence. Not only does it largely justify the various phenomenological models that have been used to analyse experimental data, but it provides a new and clear picture of the vortex line distribution and associated dynamics. Future experiments should be devised to test the various microscopic properties that are predicted by the theory. For the present it will only be necessary to determine how well the average properties of the superfluid turbulent state agree with experiment. The steady state line density is generally given by eq. (24) but there are important differences between the Vinen and Schwarz result aside from the obvious one that Schwarz has determined y ( T ) . In Vinen’s expression, V,, is a function of T and channel size d, eq. (25). It is not related to
SUPERFLUID TURBULENCE
155
the critical velocity V, in any fundamental way. It is perfectly reasonable to picture a line density that has V,=O, but a non-zero critical velocity (see fig. 9). For homogeneous turbulence, Vinen’s model would give V,=O and Vo=O. The result of the Schwarz theory, which only treats homogeneous turbulence, gives V, = constant, and no critical velocity. Clearly the quantity Vo is intimately related to the question of homogeneity. In the following survey of experimental results this connection will be considered in detail.
3. Temperature and chemical potential Mereace data
3.1. Three turbulent stares As noted in the introduction, the superfluid turbulent state in circular channels undergoes a dramatic mddification to a new state (fig. 7 and fig. 13) at a well defined heat flux Wc2. These two states will be referred to as the first and second turbulent states, or TI and TII. Henberger and Tough (1980) have observed these two states in square channels, but only a single state is found in large aspect ratio (10: 1) rectangular channels (Ladner and Tough, 1978) as shown in fig. 13. Until it is possible to further characterize the turbulence in these rectangular channels it will simply be referred to as the third turbulent state, or TIII. It will be necessary to examine the data from intermediate aspect ratio rectangular channels very carefully to determine which of the three superfluid turbulent states is present. The approach that will be followed in the analysis of the various experimental data thus reflects the philosophy outlined in the Introduction. Rather than assume a universal superfluid turbulent state, this review will seek to recognize and identify particular forms of turbulence. Using the theoretical ideas discussed in section 2 as a guide, the parameters of the three turbulent states will be obtained from the temperature and chemical potential difference data. These results will form the basis for the analysis of the other data in the following sections.
3.2. The second superjluid turbulent state - TII The results in fig. 7 and fig. 13 show the dramatic modification of the superfluid turbulent state that occurs at Wc2 in circular channels. For
J.T. TOUGH
156
RECTANGULAR
CIRCULAR
Y I
I
I
, E
Y
Fig. 13. Comparison of the temperature difference AT measured in circular and rectangular channels (Ladner and Tough, 1978). Two turbulent states (TI and TII) and two critical heat and Wc2) are present in circular channels (see also fig. 7), but only a single fluxes (W,, turbulent state (TIII) and a single critical heat flux ( Wc3) are present in the high aspect ratio rectangular channels.
W >> W,, the temperature difference can again be described by the mutual friction formalism, eqs. ( l l ) , (13), and (14) as shown in fig. 14. Some investigators (for example, Chase, 1962) have reported deviations from this W dependence, but in a detailed study of the large heat flux region Dimotakis and Broadwell (1973) have shown that this deviation is most likely an experimental artifact due to the strong temperature dependence of the various fluid parameters. These investigators employed a local probe of the temperature in the channel, and have given definitive proof that the local temperature gradient has the mutual friction form, eqs. (12) and (14), V T ( Z= ) W3A2~n/S(~sST)3,
(37)
SUPERFLUID TURBULENCE
157
where all the quantities are evaluated at the local temperature in the channel T ( r ) . Values of the Gorter-Mellink coefficient A,(T) determined from this work are shown in fig. 15. Other data that unquestionably belong to the same turbulence regime come from high heat flux measurements in circular channels. These results are listed in table 2 and are shown in fig. 15. These data span over an order of magnitude in channel size and are in good mutual agreement. The solid line in fig. 15 is drawn through the values of A calculated by Schwarz for homogeneous superfluid turbulence (table 1). The agreement m
10
I
I
I
I
I
0
1 c
6
Y
-
n -
1
0 0
0
-
0 OT=2.lIlK
-
T=1.312K0
6-
0
ID 4-
0
Y
0
oo
0
0
-
17 0
\
-
2-
(a> Oh5 0.1
0110 0.2
O(15 0.3
O.;O 0.4
0.;5 up& line 0.5 lower line
w (Wcrn-') n 1 c
E
0 \
Y Y
*I-
n
1
D
Y
0 0
Fig. 14. The temperature gradient in the second turbulent state (TII) is of the mutual friction form [eqs. (12) and (14)]. as shown by these data of Vinen (1957a). and Chase (1962). (figs. 14a and 14b respectively).
158
J.T. TOUGH
160 140
120
40
2o 0
I I I I I I I I I I I 1.2
I.4
I .6
I .8
2.0
T (K) Fig. 15. The Gorter-Mellink coefficient A, for the state TI1 from a variety of temperature differenceexperiments (table 2). The solid line is drawn through values of A calculated in the Schwarz theory (table 1).
of this collection of data with the homogeneous theory is excellent considering the lack of any adjustable parameters in the theory. Unfortunately it is not possible to confidently determine whether the parameter V, is constant as given by the homogeneous theory. The data of Dimotakis and Broadwell (1973) were obtained at very high heat flux and were insensitive to V,. Fig. 16 shows some data of Brewer and Edwards (1962) reduced by eq. (16) to give the line density Lo. Although these results are obtained from data just above W,, so that extrapolation errors are minimized, the curvature of the transition region and the slight nonlinearities in the results make an accurate determination of the intercept Vo very difficult. It seems clear however that V, is not zero. The temperature difference data of Vinen (1957a) were obtained in a rectangular channel of 2.7: 1 aspect ratio. Values of A determined from these data are shown in fig. 15, and are seen to be in good agreement with the other results. The determination of the critical heat flux by the “waiting time technique’? (section 5 ) also indicated a large “sub-critical turbulence” (state TI) for W< W,,as in the circular channels (see also
SUPERFLUID TURBULENCE
159
Table 2 Sources of the experimental data given in fig. 15,'and characteristic of state TI1 Symbol Reference ~
Channelsized (an)
Comments
0.318
Local gradient measurement
~~
Dimotakis and Broadwell (1973) Kramen (1965) van der Heijden et al. (1972). Peshkov and Tkachenko. (1962) Slegtenhont andvan Beelen (1979) Brewer and Edwards (1962)
0.029
0.0294 0.140 0.0216
V
Childen and Tough (1976)
0.0366 0.0108 0.0131
0
Broadwell and Liepman (1969) Vinen (1957a) Chasc (1962)
0.32 0.24 0.08
v 0
Channel 8 M a n long Values of A recalculated from thermal resistance data Averages over two different channels Cavitation method 2.7 : 1 rectangular channel Several channels in parallel
Chase, 1962). It must be concluded that the second turbulent state can be produced in rectangular channels of low aspect ratio as well as in circular channels. The channel length is apparently not critical for the existence of the second superfluid turbulent state. Brewer and Edwards (1962) reduced the length of one circular channel from 10 cm to 0.92 cm with only minor
"0
10
20
30
40
50
Fig. 16. The vortex line density Lo in state TI1 determined from measurements of the temperature difference in circular channels (Brewer and Edwards, 1962). The agreement with the theoretical result, eq. (24), is quite satisfactory although the intercept V,, is not well defined.
J.T. TOUGH
160
changes in the observed properties. Even more surprising are the observations of Dimotakis and Broadwell (1973) in a channel of ‘‘zero length”: - a circular orifice. They found a temperature gradient existed over an axial length of about one diameter, having all of the mutual friction characteristics of a long channel. Values of the Gorter-Mellink coefficient from the orifice, although not shown in fig. 15, are in surprisingly good agreement with the other results. These observations certainly preclude any possibility of extensive axial inhomogeneity.
3.3. The first superjfuid turbulent state - TI The existence of a turbulent state for W < W,, was recognized by many early workers (Vinen, 1957b, 1957d; Brewer and Edwards, 1961a, 1962; Chase, 1962), but the structure of this “sub-critical turbulence” has not been considered until rather recentiy (Childers and Tough, 1976; Ladner and Tough, 1978). The data for this state are limited to a small region at low heat flux (WCl< W < Wc2) where A T is small. Consequently, the determination of the parameters of this superfluid turbulent state is less precise than for the TI1 state. Rather than analyse the TI state in terms of the Gorter-Mellink coefficient A, it is useful to take a further step, eq.
-
h
I .5
I
.. . . ; .. .^
1.z
E
u
“0 1.0 -
T= 1.7
1.4
1.15.
v
N ‘ 0
0.5
Fig. 17. The vortex line density in the state TI showing the transition region to state TI1 (Ladner, 1980). In state TI. the line density is of the form, eq. (24), and V, is a function of temperature.
SUPERFLUID TURBULENCE
161
‘/ STATE
0@ @
I 0
1.0
1.2
1.4
1.6
1.8 2.0
T (K) Fig. 18. The line density mefficient y,(T) in state TI. The points are from temperature difference data in circular channels (Childers and Tough, 1976) reduced as in fig, 17 and fitted to eq. (24). The solid line is y(T) in the Schwarz theory (table 1).
(16) and introduce the vortex line density Lo. Considering the assumptions and approximations discussed in section 2, this Lo must be regarded as an “equivalent” vortex line density, since it is basically assumed in eq. (16) that the distribution is homogeneous and isotropic. The consistency of this assumption will be considered in detail below. Fig. 17 shows the equivalent vortex line density Lo deduced from the AT data of Childers and Tough (1976). The transition from the state TI to TII is clear in these results. The line density in state TI can be described by the functional form of eq. (24) and y1 and V, are then determined from a best-fit to the data. In fig. 18 are shown the values of y1 deduced from these data, along with the results from the theory of Schwarz (solid line), for homogeneous turbulence (table 1). The experimental results differ by about a factor of two from the theory, suggesting that the superfluid turbulence in state TI is not homogeneous and isotropic. The line density shown in fig. 17 differs from the Schwarz theory in another way. The intercept V,, though poorly defined, is clearly a function of both temperature and channel dimension, and roughly in agreement with the Vinen result, eq. (25) using a -- 1. Discussion of this result, closely connected to the critical velocity will be deferred to section 8. It is clear however that the homogeneous result given by the Schwarz theory is not consistent with these data.
J.T.TOUGH
162
3.4. The third supefluid turbulent s m e - TIZZ As shown in fig. 13, only a single state of turbulence is observed in large aspect ratio rectangular channels. This TI11 state is well defined by the experiments since it extends from the critical heat flux W,,to the largest heat flux studied. The “slits” used by Keller and Hammel (1960) can be considered to be long, extremely large aspect ratio rectangular channels with small dimension d on the order of a few microns. Although the temperature differences measured in these channels are very large, only the single state TI11 is observed. The experiments of Yarmchuck and Glaberson (1979) are also in high aspect ratio channels, but with dimension d of about a millimeter. Included in these results are local measurements of VT rather than the more usual temperature difference AT. Since these data are fully consistent with those for AT and Ap, it must be concluded that the gradient is uniform, and the turbulence is essentialty the same at all points along the channel. As in the case of the TI state, AT data can be reduced to give an “equivalent” line density Lo. Fig. 19 gives some results for a 0 . 0 0 3 2 ~ 0.032 cm channel at several temperatures. Again the functional form of LJT, V) given by eq. (24) appears to be correct. Values of the slope y 3 ( T )are given in fig. 20 along with the results from three other similar channels. Values of y J T ) determined from the small channels (Keller and Hammel, 1960; Craig et al., 1963) and the large channels of Yarmchuck
2 5 10 15 20
OO
V (cm/s) Fig. 19. The vortex line density Lo in state TI11 determined from measurements of the temperature difference in high aspect ratio rectangular channels (Ladner and Tough, 1979; Ladner, 1980).
SUPERF'LUID TURBULENCE
163
-
50t. 1 ' .0
1.2
1
1.4
1.6
1.8 2.0
Fig. 20. The line density coefficient -y3(T)in state TIII. The points are from data in various rectangular channels reduced as in fig. 19 and fitted to eq. (24). The open squares are from temperature difference data in small channels (Ladner and Tough, 1979). The large channel data of Yarmchuck and Glabenon (1979) are shown as solid squares (chemical potential difference data) and the open triangle (parallel temperature gradient data). 'Ihe temperature difference data of Craig et al. (1963)from ultra-small channels are shown as hexagons. The solid line is y ( T ) in the Schwarz theory (table 1).
and Glaberson (1979) are also shown in fig. 20. Considering these results span over two orders of magnitude in channel size, the mutual agreement must be considered excellent. The solid line in fig. 20 is drawn through the values of y calculated by Schwarz (table l), but the agreement with the data should be considered carefully before it is assumed that state TI11 is homogeneous turbulence. In an extensive series of measurements Ladner and Tough (1979) have shown that the intercept V, varies with both temperature (fig. 19) and channel size, whereas V, is constant for homogeneous turbulence in the Schwarz theory. The intercept V, can be determined much more reliably for these TI11 data than for the other turbulent states, and the results indicate that the Vinen expression (25) based explicitly on an assumption regarding inhomogeneity is approximately correct using a = 1. [Ladner (1980) has shown that a is probably a weak function of d. This result will be discussed more fully in section 8 along with critical velocity data.] Another reason to suspect the homogeneity of state TI11 comes from a
J.T. TOUGH
164
a 0 0
I a
N
rp 50
Fig. 21. Comparison of the line density coefficient in state TI (circles) and TI1 (squares) showing the scaling with the hydraulic diameter ratio Dld, eq. (38). Results from a square channel (Henberger and Tough, 1980) are shown as triangles and agree with the scaling.
comparison with state TI. Ladner and Tough (1979) have shown that states TI and TI11 may both be consistent with eqs. (24) and (25) but with a value of y that scales with the hydraulic diameter ratio of the flow channel Dld: y = yoD/d.
(38)
Fig. 21 shows that the function yo obtained from the circular and rectangular channel data is approximately universal. Using eqs. (38) and (25) also gives the intercept V,,correctly for both states. Results for a square channel (Henberger and Tough, 1981) which has the same hydraulic diameter ratio as the circular ones (Dld = 1) are also shown in fig. 21 and support this geometric scaling result. If the vortex line density in states T I and TI11 does scale with geometry as given by eq. (38) then it is certainly not homogeneous. If state T I were homogeneous, what then would be the nature of state TII? Yet, as fig. 20 shows, states TI1 and TI11 appear to have very nearly the same y ( T ) . If TI1 and TI11 are the same state, then the geometric scaling result shown in fig. 21 is fortuitous. If the scaling is correct, and TI and TI11 are simply
SUPERFLUID TURBULENCE
165
related by geometry, then the agreement of the TI11 results with the homogeneous theory (fig. 20) is fortuitous. Data from low aspect ratio channels could decide this issue. Taken as an empirical result, the scaling in eq. (38) combined with eqs. (24) and (25) yields a very useful dimensionless expression for the vortex line density in states TI and TIII:
LAi2d= 1.03yOVD - 1.48a.
(39)
4. Pressare difference data
4.1. The Allen and Reekie Rule
Certainly one of the most remarkable features of superfluid turbulence is the relatively negligible effect it has on the pressure difference A P (fig. 2). Early workers (Keesom and Duyckaerts, 1947; Mellink, 1947) verified the original observation of Allen and Reekie (1939) that A P remained proportional to W even when AT was increasing approximately as W3. Of course this observation led Gorter and Mellink (1949) to propose the mutual friction approximation, eqs. (9) and (10) and the laminar mean Row assumption leading to eqs. (4), (12), and (13). The recent experiments to be discussed below generally indicate that the Allen and Reekie rule, eq. (4), is not exactly obeyed, but that there is a small excess pressure difference that may be associated with the superfluid turbulence. One exception is the data of van der Heijden et al. (1972a) that give no indication of an excess pressure. These experiments may well be dominated by the anomalous effects associated with rough metal capillary walls (Childers and Tough, 1974a, b).
4.2. The first superjluid turbulent state - TI
The first systematic study of the excess pressure difference was done by Brewer and Edwards (1961b) using a Row channel of circular cross section (diameter 1.08 x 10-2cm). Simultaneous measurements of the temperature difference AT indicate that the A P data are predominantly
J.T. TOUGH
166
12
I
1.806K
t
4
TATETX,,
~
2
I
STATE T I I
/
2
,
O
n
“0
8
16
24
32 40 48
Fig. 22. The superfluid eddy viscosity 9.. determined from measurements of the pressure difference in states TI and ‘MI. The results are shown as a function of the dimensionless number LA‘2d at three different temperatures (Brewer and Edwards, l%lb).
from the second superfluid turbulent state ( W < W,,),although some results are clearly in state TI. Lacking any theory for the excess pressure Brewer and Edwards used the total measured pressure difference and eq. (4) to deduce an “effective viscosity” qea and then defined a “superfluid eddy viscosity” q, as rlr = “In-
“?en
(40)
where qnis the normal fluid viscosity. If q, is a property of the superfluid turbulence then it might be expected to depend on the line density Lo in some simple way. The line density is determined from the AT measurements using eqs. (12) and (16) as in section 3. Fig. 22 shows q s as a function of the dimensionless number LA’2d at three temperatures. A small pressure difference consistent with an eddy viscosity of a few micropoise has also been observed by de Haas and van Beelen (1976) in a 2.16 x cm diameter channel at 1.05 and 1.33 K. The results in fig. 22 show that in the first superfluid turbulent state (W,, < W < Wc2) the eddy viscosity is quite small, about 10% of qn, and the data tend to fall on a single line indicating a universal dependence on the line density Lo. The eddy viscosity in the TI state was studied in more
SUPERFLUID TURBULENCE
167
detail by Childers and Tough (1975,1976) using a capacitive pressure transducer in the heater reservoir which eliminated the vapor pressure correction, a major source of error in the previous experiments. Fig. 23 shows results for five separate data runs at 1.8K in a 1.31X cm diameter channel. The straight line through the points is the function r),
=P K ( A : L ~ ) ~ ’ ~
(41)
with A1 = 3 x lo-’ cm. The excess pressure at all other temperature studied (1.4K a T s 1.9K)was also found to agree with eq. (41)using the same value for A]. Experiments with a smaller (0.61X cm)channel were less conclusive, but strongly suggested that eq. (41)is again correct with the same value of A l . This result, eq. (41),is plotted in fig. 23 as a solid line and is in acceptable agreement with those data of Brewer and Edwards in the first turbulent state. If the concept of a superfluid eddy viscosity in the turbulent state is valid, as Vinen (1956)and Hall (1960) have suggested, then eq. (41)would appear to be a correct expression for the dependence of 9. on the line density Lo. The $-power dependence is striking, but not as enigmatic as the appearance of another length scale, A l . The temperature difference data from state TI suggested this turbulent state was not homogeneous. Perhaps A 1 is the spatial scale of the inhomogeneity. However, as will be discussed below (section 4 3 , if the turbulence is inhomogeneous then the concept of an eddy viscosity is less clearly defined.
Fig. 23. The superhid eddy viscosity q., determined from measurements of the pressure difference in state TI. The results are plotted in dimensionless form. The solid line has slope 3 (Childers and Tough, 1975).
J.T. TOUGH
168
4.3. The second supfluid turbulent state - TZZ
Ladner et al. (1976) have extended the measurements of Childers and Tough (1975) into the second superfluid turbulent region. The data are limited to the transition region only ( W e 1.5 Wc2) and are quite consistent with an eddy viscosity of the form, eq. (41). However the results in fig. 22 demonstrate that well past the transition region, the eddy viscosity in the second turbulent state is quite different than in the first state. The magnitude of qs is large, nearly half of q, at the highest line densities, making the concept of an eddy viscosity far less appealing. The dependence of qs on Lo in state TI1 appears to have the asymptotic form rls = f S ( T ) P 4 A , L Y 2 ) ,
(42)
where f,(T) is some weak dimensionless function of temperature and A2 is another length which must be of the order of A, if fs is of order one. The A T data (section 3.2) suggested that state TI1 may be homogeneous superfluid turbulence. If so, it is difficult to understand the importance of any length other than the interline spacing Lo1”. Perhaps the large excess pressure is in fact the result of a non-laminar flow of the normal fluid component. This possibility will be discussed below (section 4.5). 4.4. The third superfluid turbulent state - TZZZ
The only direct pressure difference measurements in high aspect ratio rectangular channels, state TIII, were made by Keller and Hammel (1960) and Craig et al. (1963). Unfortunately the sensitivity in these experiments was not sufficient to detect any small excess pressure, so comparison with states TI and TI1 is not possible. As noted in section 3.4, the A k data of Yarmchuck and Glaberson (1979) are quite consistent with the AT data, and thus provide indirect evidence that in the third turbulent state the excess pressure difference is small. It would be very interesting to know whether an eddy viscosity described by either eqs. (41) or (42) was present in the third state.
4.5. The eddy viscosity and homogeneity In discussing the question of homogeneity (section 2.1) it was noted that a variation of the apparent temperature dependence of the line density Lo
SUPERFLUID TURBULENCE
169
could signify superfluid turbulent states of different homogeneity. The line density in state I1 agrees very well with the Schwarz theory for homogeneous turbulence (fig. 15) while Lo in state I has a different magnitude and possibly a different temperature dependence as well (fig. 18). Can the pressure difference data reveal any further information about the homogeneity of these two states? Unfortunately the answer is probably no, although it is useful to consider the alternatives nevertheless, The procedure used to obtain qs from the pressure data rests on two assumptions (Gorter and Mellink, 1949; Childers and Tough, 1976). The first is that in the mutual friction approximation the superfluid equation of motion, eq. (lo), can be written as
0 = qsV2(us)%!-
P
( V P ) + p,S(VT) - Fsn.
The quantity (0,) is of course the time average of the axial component of the local superfluid velocity and is thus not subject to the irrotationality constraint. However, to be useful, eq. (44)must be supplemented by a second assumption giving the dependence of (0,) on position in the flow channel. To obtain eq. (40) it must also be assumed that V2(U,)
= V2(U,)
(45)
V
(46)
or that (0%)= (On> +
where V is the average relative velocity determined by the heat flux W, eq. (3). This complementary relation of (q)and (u,) is not only consistent with an eddy viscosity, but since the relative velocity ((u) = (us- u,)) is uniform, the line density Lo is homogeneous, eq. (17). Although the assumption, eq. (46), leads directly to a homogeneous line density and to an eddy viscosity, it is far from self-evident. Three simple possibilities for the average velocity fields (u,} and (us) are shown in fig. 24 along with the case of laminar flow. Case (a) is identical to laminar flow and results in an inhomogeneous line density and no eddy viscosity contribution to the pressure since (V2(us)= 0). Neither states I or state I1 have these characteristics. Case (b) is exactly the assumption, eq. (46), made by Gorter and Mellink (1949), and leads to a homogeneous line density and an eddy viscosity. Case (c) corresponds to a highly turbulent flow of the normal fluid and an approximately homogeneous
J.T. TOUGH
170
LAMINAR
(a 1
TURBULENT (b)
(C)
Averoqe velocity
I
relotive
non-uniform
uniform
opproximoteiy uniform
0
inhomogeneous
homogeneous
opproximoteiy homoqeneous
O
0
non-zero
0
7s
'In
non-uniform
line density
LO
1 lsV2 I effective viscosity 1
I n
In+
I
Fig. 24. A comparison of various quantities in laminar flow and in three possible turbulent flows.
line density. The superfluid eddy viscosity contribution to the pressure would be zero, as in case (a), but the "flattened" (u,) profile would be accompanied by an increased average normal fluid dissipation. In this case it is the normal fluid equation of motion, eq. (9), that would be modified in the mutual friction approximation by the addition of a force F,. This procedure has been enthusiastically supported by de Haas and van Beelen (1976) who reject the superfluid eddy viscosity concept. The experimental facts are ambiguous however. Only case (a) appears to be without empirical support. Between extremes of cases (b) and (c) are an infinite range of possibilities each with an effective viscosity and a line density profile. Clearly the temperature and pressure difference data, which are averages over the channel cross section cannot unambiguously determine which possibility occurs. These considerations do however make it obvious that superfluid turbulence may be universal in the sense that it can always be represented as a distribution of quantized vortex lines, but may nevertheless occur in various well defined states with different average macroscopic properties.
SUPERFLUID TURBULENCE
171
5. second s o d dab
5.7. Propagation of second sound in superfluid turbulence Vinen (1957a) was the fust to demonstrate that second sound suffers an excess attenuation a’in the presence of superfluid turbulence. He showed that a quantitative interpretation of the attenuation was possible in terms of a generalization of the phenomenological mutual friction force, eq. (14). If F,, is written F;n = A’pspn( V-
VO)’~,
(47)
where t) is the instantaneous relative velocity in the second sound field, then there is a linear attenuation a’= Alp( V- Vo)’/2uz.
(48)
Vinen found that the coefficient A’ obtained from measurements of a’ was identical with A obtained from temperature difference data (section 3). This agreement is probably fortuitous, as will be seen from the further discussion below. A detailed calculation of the attenuation based upon the theory of Hall and Vinen (1956b) and the model of superfluid turbulence as a tangled mass of vortex lines (section 2.2) of density Lo leads to a‘= BKL,/~u,.
(49)
This result is equivalent to eqs. (48) and (47) using eq. (16) and assuming A = A’. The interaction of second sound with quantized vortex lines has been critically re-examined by Mehl (1974) who showed that the inclusion of previously neglected phase effects leads to a second interaction parameter B,. Mehl finds that the attenuation a’is given by eq. (491, and the second sound (of frequency f = 4 2 ~ velocity ) changes by Au, where AuJuZ
= KB~LO/~U.
(50)
There are several difficulties that arise when eqs. (49) and (50) are used to determine Lo from attenuation or dispersion data. The quantities B and B2 are functions of both temperature and frequency. When considering temperature difference data (section 3) it is therefore necessary to use the dc (f = 0) values, while the second sound attenuation data must be analysed using the values of B and B, at the appropriate frequency. Vinen analysed his temperature data using values of B measured at about
172
J.T. TOUGH
1 kHz, and therefore the agreement of A and A' might be questioned. Recent dc measurements of B (Yarmchuck and Glaberson, 1979) show, however, that the difference between the values of B at 1 kHz and dc is only about 10%. Mehl (1974) has also pointed out that B and B2 in superfluid turbulence are actually different from the corresponding quantities in rotating helium, but again the difference is less than 10% (for T < 2 K). The biggest source of ambiguity in the second sound data will result from anisotropy or inhomogeneity of the turbulence. Different second sound modes sample effectively different portions of the counterflow channel. Further, experiments have been done with second sound propagating across the channel (perpendicular to the heat flow) and along the channel (parallel to the heat flow). Since all experiments are interpreted in terms of a homogeneous, isotropic line density, it can be anticipated that substantial ambiguities will appear in these data and this is indeed the case. Another source of difficulty is the actual size of the counterflow channel. Experimental considerations generally constrain the second sound experiments to channels with d > lo-' cm, whereas temperature, pressure, and chemical potential difference measurements are generally limited to d < lo-' cm.Since the distance along the channel that is required for the normal fluid to attain a parabolic Poiseulle profile (the laminar entrance length I,) is proportional to d Z (Schlichting, 1951) the homogeneity in large and small channels may be quite different, even if all other external parameters are the same.
5.2. Transverse second sound attenuation data The original experiments of Vinen (1957a) were done in low aspect ratio rectangular channels which acted as half wave resonators for second sound propagating perpendicular to the heat current and parallel to the large dimension of the rectangle. The second sound transmitter and receiver ran the entire length of the channel, and thus the attenuation is determined by the average of Lo over the channel length. The attenuation was of the form eq. (48) and the values of A' determined from the data are given in fig. 25 (see table 3). These results for A' are identical to the values of A determined from the temperature difference data in the same channel (fig. 15), and are representative of the second superfluid turbule n t state TII. The second sound attenuation was very much more sensitive to than was the temperature difference, and the sound data did
SUPERFLUID TURBULENCE
0
1.2
I .6
I .4
I .8
173
2 .o
T(K) Fig. 25. The Gorter-Mellink coefficient A' determined from second sound attenuation measurements. Results are given for the experiments listed in table 3. The solid lines show A ( T ) from the Schwarz theory (see fig. 15). and A ( T )for state TI.
Table 3 Sources of the experimental data for A' given in fig. 25
Symbol Referenn
Channel size (on)
Commenrs
V
Vinen (1957a)
0.24 x 0.65 x 10 lonz
0
Kramerr, et al. (1960)
0
Kramers (1965)
0.26diam. x 1.05 long 0.045 diam. x 0.2 long 0.102 diam. x ?? long
Transverse propagation. munterflow Longitudinal, counterflow
0
Ijsselstein et al (1979)
0.062 diam. X40 long
0
Os~ermeieret al. (1980)
1.Ox l.Ox401ong
Approx. 1 an channel between double resonator, longitudinal. pure normal flow ( V. = 0) Approx. 1 cm channel between double resonator. longitudinal, pure normal flow (V,= 0) Transverse. burst echos. counterflow
J.T. TOUGH
174
-
45/
OO
10
20
W, (mW/cm2)
Fig. 26. The “Vinen waiting time technique”. 7 is the time required for the amplitude of a standing second sound wave to decrease to one half of its initial value after the heat flux is increased from W, to W, >> Wc2. The channel is 0.4 x 0.783 x 10 cm long and the temperature is 1.4 K (Vinen, 1957d).
indicate a critical heat flux ( Wc2) which was not evident in the temperature difference data. Vinen (1957b, d) developed an even more sensitive technique which gives at least qualitative evidence for the existence of the first turbulent state for W < Wc2. A small heat flux W ,is established in the channel, and is then suddenly increased to a value W,>> Wc2. The delay time T required for the second sound amplitude to decrease to one half of its final value is measured as a function of the initial heat flux W,. Typical data are shown in fig. 26. The qualitative interpretation of these data is that the time required to simply increase Lo in the TI1 state is significantly less than the time required to generate the state from a vortex-free flow, or from state TI. Vinen attempted to make these results quantitative using a calculation based upon his phenomenological dynamic balance equation (20) and including a new term g ( u ) associated with the initiation of vorticity in undisturbed helium. Neither Brewer and Edwards (1962) or Childers and Tough (1978) have found any evidence for g ( u ) in the steady state, although the transient measurements of Vinen seem to require such a term. While the theoretical basis of Vinen’s calculation can be questioned, the results are internally consistent. The calculation relates the function 7(WJ to L,(V) with the results shown in fig. 27. The points shown as squares are directly from the attenuation data using eq. (49) and the circles are obtained from the delay time data (fig. 26). The latter results nicely overlap the more direct measurements and indicate the small line density present in the first turbulent state. The attenuation data
SUPERFLUID TURBULENCE
175
L .o
*O
0.5
I
1.5
Fig. 27. The circles give values of the line density Lo deduced by Vinen from the waiting time data in fig. 26. The squares gives Lo determined directly from the attenuation, eq. (49). States TI and TI1 are obvious (Vinen, 1957d).
of Vinen thus support the pattern of two superfluid turbulent states that has been developed from the temperature difference data in channels of various shape and over an order of magnitude smaller size. Ostermeier et al. (1978a, b) have recently developed a very sensitive technique for the determination of a‘ in a section of the counterflow channel that is only a few per Cent of the total length. A high frequency (f= 20 kHz) burst of second sound is generated in a transmitter on one wall, and a receiver on the opposite side of the channel detects the burst plus about eight subsequent echos. The echo signals are stored in a computer and analysed to obtain the attenuation a‘,and the line density Lo, eq. (49). It is found that Lo is quite inhomogeneous, being much larger at the entrance (the heated end of the channel). Further, the dependence of Lo on the relative velocity V is very different from eq. (24) in roughly the first half of the channel. At the channel entrance, Lo is almost independent of V. These results are not presently understood, but are almost certainly connected with the normal fluid entrance length mentioned in section 5.1. The channel in these experiments is square, 1cm X 1 cm, and even though it is 40 cm long, there is a substantial region at the entrance where the normal fluid velocity profile is almost “flat”, rather than parabolic. If the data from only the second half of the channel are considered, the normal fluid flow state should be fairly comparable to
176
J.T. TOUGH
that present in other experiments in smaller channels. The results for Lo deduced from the measurements at 20 cm and 40 cm are in fact virtually identical, and are in fair agreement with eq. (24). However, the values of A(T) obtained from these data (Ostermeier, 1980) and shown in fig. 25 suggest that perhaps the lint turbulent state TI is observed in these experiments. This state is to be expected for the square geometry (Henberger and Tough, 1981),but what is then anomalous is the conspicuous absence of a second critical heat current W,, and the state TII. Mantese et al. (1977) have observed the attenuation of second sound propagating transverse to the heat flux in a high (6:l) aspect ratio rectangular channel. The counterflow channel is rather complex, and is not long enough to eliminate the entrance length problem. The data are also limited to the single temperature 1.648 K. The dependence of the line density on relative velocity, Lo(V) is in very good agreement with eq. (24), but the amplitude y ( T ) is roughly a factor of two smaller than for state TI11 which should be present in this geometry. Since only about one thud of the second sound resonant cavity is filled with the turbulence in this particular experimental arrangement, this result is not surprising. Three resonators mounted across the channel give essentially the same line density, while three resonators along the channel length show an inhomogeneity comparable to that observed by Ostermeier et al. (1978a, b). This feature is again probably the result of entrance length effects. Vidal et al. (1974) have measured the attenuation and the change in velocity of second sound propagated perpendicular to the heat flux. The counterflow channel is rectangular, but the dimensions are not given. Results are obtained at 1.44 K and 1.52 K. The attenuation data appear to be anomalous (Mehl, 1974) but the velocity data are in excellent agreement with eq. (50) and again indicate a line density characteristic of the second turbulent state.
5.3. Longitudinal second sound attenuation data Kramers et al. (1960) pioneered the use of the Helmholtz resonator as a technique for investigating the attenuation of second sound propagated parallel to the heat flux. A conventional counterflow apparatus (fig. 1) is used, and a second heater in the reservoir provides an oscillating heat flux. The resonant frequency is determined by the geometry of the system. Vinen (1957b) actually observed Helmholtz oscillations in his
SUPERFLUID TURBULENCE
I77
transient measurements, but he cancelled out these longitudinal oscillations by the application of a second heat pulse. The first Leiden data (Kramers et al., 1960) were obtained with two different resonators each operating at about 30Hz. The counterflow channels were rather short, and probably suffered from entrance length effects. The attenuation was of the form, eq. (481, and the values of A determined from the attenuation are shown in fig. 25. As the Leiden group developed its capability to investigate velocity combinations other than thermal counterflow, the Helmholtz oscillators used for second sound measurements were changed to the “double” type (Kramers, 1965). These resonators were placed in series with the primary counterflow channel. It is not clear that the line density in the resonator is necessarily the same as in the flow channel. No results in pure counterflow are obtained in these experiments, but data for V,=O flows, which appear to be essentially the same as pure counterflow (section 9), are shown in fig. 25. Another series of experiments (Kramers et al., 1973) with a similar double Helmholtz resonator and flow channel gave second sound attenuation data that were nor in agreement with eq. (48) but instead indicated that the attenuation varied with the relative velocity as V”, with 1.66 < n < 2.44 over the temperature range 1.3< T<2.0 K. The recent experiments of Ijsselstein et al. (1979) again employ the double Helmholtz resonator to measure the attenuation and dispersion of second sound in the superfluid turbulence, and these experiments are supplemented by measurements of the temperature and chemical potential difference. The flow channel was circular (0.064 cm diameter) and the resonator had a frequency of about 100Hz. These attenuation data clearly reveal the two turbulent states T I and TI1 (fig. 28a) observed by Vinen in his experiments, and so prominent in the temperature difference data (section 3). Values of A determined from the attenuation data in the second turbulent state are shown in fig. 25, and are seen to be in fair agreement with the data of Vinen. The solid line in fig. 28a is calculated from eq. (48) using the values of A shown in fig. 25. Unfortunately, Ijsselstein et al. (1979) only give a very small amount of temperature difference data for V,= 0 flow that can be compared to the attenuation results. At T = 1.5 K the temperature difference data indicate a value of A that is about 20% larger than from the attenuation, and in better agreement with the results of Vinen. Possibly the line density in the double Helmholtz resonator is less than in the primary counterflow channel. The results for the Gorter-Mellink coefficient determined from second
J.T.TOUGH
178 4
-I
3
STATE T I
E
A@-
0 N
‘
0
-
2
Y
/ \
El
/
I
I
0
A
A
STATE
TI
A A
A A
A t A A
I
I
I
I
1
V
(b)
3
V
V V
8 N
3
V
2
VV
V
N ‘
4
VV
I V*
V 0%
0
I
”I
2
r
e
-
”--
3
I
4
V, (cm/s) Fig. 28. (a) The excess attenuation a’ of longitudinal second sound for pure normal fluid flow in a circular channel at 1.7 K (from Ijsselstein et al. 1979). The solid line is computed from eq. (48) using V, = O and A‘=61 cmlslg as given in fig. 25. The transition from states TI to TI1 is evident. (b) The dispersion of second sound determined simultaneously with the attenuation data in fig. 28(a) (from Ijsselstein et al. 1979).
sound attenuation and shown in fig. 25 are certainly less definitive than the results shown in fig. 15. Presumably all of these results are from the second turbulent state, although in fact only the experiments of Vinen (1957a,d), Kramers et al. (1965), and Ijsselstein et al. (1979) give any direct indication that the second turbulent state is present. The results from these three experiments are reasonably consistent, and give values of A’ which are comparable with those obtained from temperature difference experiments (fig. 15) at least at the lower temperatures. The systematic difference at higher temperatures could be attributed to the effects of inhomogeneity or anisotropy or to differences in the value of B
SUPERFLUID TURBULENCE
179
in rotating and turbulent helium. The attenuation data of Kramers et al. (1960) and Ostermeier et al. (1978a, b) were obtained in channels sufficiently short that entrance length effects could be expected. The small values of A' determined from these experiments are thus less anomalous, but still without explanation.
5.4. Second sound dispersion data There have been very few measurements of the dispersion or velocity change of second sound in superfluid turbulence. As noted in section 5.2,
t
I
- 0.04
0
I
0
1
I
I
I
2 3 4 V, (cm/s)
I
5
Fig. 29. The ratio of the dispersion (Au,/u,) to the attenuation (a')of second sound yields the ratio of interaction parameters B,/B. eqs. (49) and (50). The circles show results from the data of Ijsselstein et al. (1979) in pure normal fluid flow in a circular channel at 1.5 K. The dashed line gives the constant value for this ratio computed by Mehl(l974).
180
J.T. TOUGH
Mehl (1974) has shown that the data of Vidal et al. (1974) are in good agreement with his theoretical result, eq. (50). Ijsselstein et al. (1979) have also measured the dispersion of second sound in their very comprehensive study of different flow conditions. The dispersion data for the V,= 0 flow are shown in fig. 28b. It is clear that Au,/u, is virtually zero in the first turbulent state, but the dispersion increases rapidly with the relative velocity in the second state. According to eqs. (49) and (50),the ratio of the dispersion to the attenuation should be independent of velocity and proportional to BJB. Fig. 29 shows B2/B determined from the data of Ijsselstein et al. (1979) at T = 1.5 K. In the first state B J B is considerably smaller than given,by Mehl (1974) at this temperature, and in the second state, the ratio is not independent of relative velocity. This latter effect is many orders of magnitude larger than would be given by the “entrainment” calculated by Khalatnikov (1963, and appears to be without explanation. Although the second sound data clearly confirm the two superfluid turbulent states in low aspect ratio channels, the results have not provided any further information about either the homogeneity or isotropy of these states. Only one experiment (Mantese et al., 1977) has been performed in a channel which could support the third turbulent state, but the limited data and the geometry of the resonators make quantitative comparison with temperature difference data impossible. It would appear that further study of second sound propagation in superfluid turbulence could be quite valuable. 6. Ion m e n t data
6.1. Early experiments The use of ions as a probe of superfluid turbulence was discovered and developed by Careri and his co-workers at Padova (Careri et al., 1959, 1960a, 1960~).These early results were difficult to interpret, since the different macroscopic structures of the ions - the positive ion “snowball” (Atkins, 1959; Reif and Meyer, 1960; Meyer and Reif, 1961) and the negative ion “bubble” (Careri et al., 1960b) were unknown at the time. The experiments revealed several qualitative features however, which when interpreted in terms of our present knowledge of the ion-vortex line
SUPERFLUID TURBULENCE
1
181
Vn
T
S
i
I
c,+c,C\c,+c,
Fig. 30. Early ion experiments in superfluid turbulence. (a) Schematic diagram of the counterflow channel with radioactive ion source and four collecting electrodes C,-C,. (b) In laminar flow, both positive and negative ions are “flushed” in the direction of V,, decreasing the current to C, and increasing the current to C,. (c) In turbulent flow the positive ions continue to be “flushed” by the normal fluid, but the negative ions show a strong nonlinear behaviour. (d) The total negative ion current, obtained by summing all collectors, is decreased by the superfluid turbulence.
interaction constitute the strongest direct evidence that superfluid turbulence i s in fact a distribution of quantized vortex lines. These early experiments were performed in counterflow channels a section of which is shown schematically in fig. 30a. The channels were 6 cm long with low aspect ratio rectangular cross section - either 0.3 cm x 0.5 cm (#1) or 0.8 cm x 1.0 cm (#2). Four separate collecting electrodes were located on one channel wall, and a single electrode with an aparticle ionization source was on the opposite wall. Positive or negative ions produced by the source could be drawn across the channel at constant speed by the application of an electric field of the proper sign. The current at each collector could be measured independently, or the collectors could be connected together to measure the total ion current. Measurements in the laminar flow region ( W < Wc,) were made with both channels, but only the larger channel was sufficiently sensitive to give unambiguous results for the ion mobility. The data show that both positive and negative ions behave as simple impurities in the helium, and
182
J.T.TOUGH
are “flushed” in the direction of the heat flow at a speed very nearly equal to V,,, eq. (1).This “heat flush” produces a linear increase in the current collected at the downstream collector and a corresponding decrease in the current at the upstream collector as shown schematically in fig. 30b. The situation at high heat flux in the presence of superfluid turbulence is quite different as shown in fig. 30c. Positive ions continue to exhibit a simple heat flush, whereas the negative ions indicate a much stronger and nonlinear apparent drag in the direction of V,. The most dramatic observation of these early experiments was obtained by connecting all four collectors together. The total negative ion current was substantially reduced, fig. 30d. Apparently the superfluid turbulence traps some of the ions producing an attenuation of the current. The onset of trapping occurs at a fairly well defined m‘tical heat flux which is most probably W,, (Careri et al., 1960a). The qualitative results in figs. 30c and d can then be taken as representative of the second superfluid turbulent state. As noted, the positive ions continue to move with the normal fluid in this state. However, the ion drift velocity has been observed to be somewhat less than V, (Careri et al., 1960c, 1964) in some experiments and somewhat greater than V, in others (Vicentini-Missoni and Cunsolo, 1966; Ashton 1977). The size of the discrepancy is small, and may well be an experimental artifact.
6.2. The ion-vortex line interaction Clearly an understanding of the ion structure and of the ion-vortex line interaction is necessary before the ion data can be useful for understanding any properties of superfluid turbulence. Careri et al. ( 1 9 6 0 ~ ) suggested that the measurement of negative ion mobilities in rotating helium, where the density and spatial arrangement of the vortex lines was known, could be useful in revealing essential features of the interaction. Results of an experiment with a rotating space charge limited diode were reported by Careri et al. (1962). Negative ion currents perpendicular to the rotation axis were strongly attenuated by the vortex lines. Negative ions moving parallel to the lines, and positive ions moving in either direction, were unaffected by the rotation. The only possible explanation for the various observations was that the negative ions were trapped in a number of preferred positions in the rotating helium -on the vortex lines.
SUPERFLUTD TURBULENCE
183
Donnelly and Roberts (1969) have provided a successful theory of the interaction of ions and vortex lines. The trapping cross section u is calculated for both positive and negative ions as a function of temperature and electric field. Free-path effects limit the theory to temperatures greater than 1.3 K. Competing with the trapping process is the thermally activated escape of ions trapped on vortex lines. Donnelly and Roberts calculated the escape probability Pthand found that for negative ions it was negligible for temperatures below 1.6 K. Above 1.6 K, Pthincreased very rapidly with temperature, making the observation of negative ion capture essentially impossible for temperatures greater than 1.8 K (the “lifetime edge” phenomenon). For positive ions the thermal escape probability is very large for all temperatures above about 1.0 K. Ion experiments in rotating helium have shown that the capture cross section (Tanner, 1966; Springett, 1967) and the thermal escape probability (Douglas, 1964; Pratt and Zimmerman, Jr., 1969) are in excellent agreement with the Donnelly and Roberts theory. The qualitative results of the early ion experiments (fig. 30) are nicely explained by the Donnelly and Roberts theory and the model of superfluid turbulence as a distribution of quantized vortex lines. Positive ions basically do not interact with the turbulence, since the thermal escape probability is so high. The nonlinear negative ion current and the current attenuation are the result of space-charge limitation effects from charges trapped on the vortex lines between source and collector.
6.3. Recent experiments The early ion experiments were done without knowledge of the ion structure or the ion-vortex line interaction, and were thus rather qualitative although still very important. In constrast, the more recent experiments have used the properties of the interaction to advantage, and have extended our knowledge of the superfluid turbulent state. Pulsed ion techniques (Schwarz and Smith, 1980) even hold the promise of providing time and space resolved measurements of the line density. The first experiment to make use of the properties of the ion-vortex line interaction (Sitton and Moss, 1969) essentially c o n h e d and made quantitative the interpretation of the early experiments. An annular diode was used, with an incandescent tungsten filament on the axis serving as both a thermionic emission source and a heater. Radial counterflow has
J.T. TOUGH
184
not been previously studied, but the heat fluxes present in these experiments were so large, that there can be little doubt that superfluid turbulence of some form or other was present. Negative ions were drawn through vortex lines for a time sufficient to produce a steady state density of trapped charge, as indicated by a constant collector current. The voltage was then reversed and-following a burst of current due to untrapped ions-a slow decrease in the current of the form Ioexp(-Pr) was observed. This current was due to release of ions trapped on the vortex lines. The quantity P was found to be essentially identical to the thermally activated escape probability Pth calculated by Donnelly and Roberts (1969). These experiments provide the most direct quantitative evidence that the superfluid turbulence actually consists of a distribution of quantized vortex lines such as produced in rotating helium. The dynamic nature of the steady superfluid turbulent state (it is not clear which state, but the dynamic processes are probably similar in all three) was first demonstrated by another experiment of Sitton and Moss (1972). A short channel of circular cross section (2 cm diameter) contained a radioactive ion source and heater at one end and a collector grid downstream so that the channel could be operated as a space-charge limited diode. Assuming the ions trapped on the lines are essentially stationary, the diode current is proportional to the square of the source voltage and to the ratio R = pf/(pf+pt),where pf and pc are the free and trapped charge densities respectively. Sitton and Moss assume a dynamic balance between charge trapping and release: ( 5 1) pfvuL0= p t h & + ptid/LO, where u and Pth are the capture cross section and thermal escape probability of the Donnelly and Roberts theory, u is the free ion velocity, Lo is the length of vortex line per unit volume, and i d / & is the rate of ion release due to dynamic processes in the turbulence. From measurements of the diode current in the space-charge limited region, the ratio R can then be determined as a function of the heat flux, temperature, and pressure. The results depend critically on a model for the release rate id/Lo. Using Vinen’s result, eq. (19), gives
id/ Lo = X*KL0/27r.
(52)
At high temperatures, (above the “lifetime edge”) Plhbecomes very large and R + 1. There is essentially no trapped charge at these temperatures since the probability of thermal escape is so large. This result is in excellent agreement with the high temperature data of Sitton and Moss at
SUPERFLUID TURBULENCE
185
all heat fluxes. Further, the location of the “lifetime edge” is observed to vary with pressure in exactly the fashion observed by Springett (1967) in rotating helium. At low temperatures, Pth+ 0 and R + x 2 / ( x 2 + 2 ~ r n / K ) independent of Lo and of the heat flux. This saturation is also observed by Sitton and Moss, but only at the largest heat fluxes studied. In the saturation region the values of x2 obtained from the data are several orders of magnitude larger than expected. Much of this discrepancy is removed if the ion drift velocity rather than the thermal velocity is used for u in the data analysis, and in fact this is the proper choice. Uncertainties associated with the poorly understood trapping in zero field, the trapping of ions drifting other than normal to a vortex line, and with the extremely short channel length, make it unlikely that any quantitative results about the turbulent state can be obtained from these experiments, although the dynamic nature of the turbulence is clearly demonstrated. It should be also noted that in the Schwarz theory (section 2.2) there is no bulk line destruction process analogous to i d . The release rate i d / L O in eq. (52) could possibly represent release of ions at the channel walls. Ashton and Northby (1973) have repeated the original current attenuation experiments of Careri et al. (1959, 1960a, 1960c) using modern electronic techniques and guided by a much clearer picture of ion structure, the ion-vortex line interaction, and the nature of the superfluid turbulent state. Their apparatus was similar to that shown in fig. 30 except only a single large collecting electrode opposite to the radioactive source was used. With no heat flux, the negative ion current density was in. In contrast to the early experiments (fig. 30d), the attenuation Aj/jn was not observed to be a linear function of the heat flux. If it is assumed that the reduction of the ion current is due to a density N, of charge trapped on the vortex lines then it follows that
Aj = CN,,
(53)
where C is an experimental parameter determined by the geometry of the diode and the properties of the radioactive source. To relate N, to the heat flux a dynamic balance is again introduced similar to eq. (51):
( j o / e ) o ( L o -IN,) = N,Ld/Lo.
(54)
Here e is the ionic charge, I is a length of vortex line assumed to be obscured by previously trapped charge, and the temperature is assumed to be less than about 1.7 K so thermal escape is unimportant. Solving for
J.T. TOUGH
186
N, gives
Using eq. (53) along with eq. (21) for the line density Lo then gives
A plot of the attenuation data in the form l / A j against 1/ W2 = (p,ST/V)' should then reveal several important features. If Ld/Li is independent of W ,then the data will give a straight line with slope proportional to Uy2, and intercept proportional to id/Lg. Ashton and Northby's data are shown in fig. 31 for two temperatures and are indeed linear. The non-zero slopes indicate that If 0. Using values of A ( T )determined from the temperature difference data in the second turbulent state, section 3.2, and eq. (26), the slopes determine a value of the length I of approximately 3 x cm for all temperatures studied. This value of 1 is roughly the distance of closest approach of two thermal ions and is thus quite reasonable. From the intercepts in fig. 31 the value of i,/L; can be determined. As noted above this rate may represent ion release at the walls rather than the Vinen line destruction process, eq. (19). Using an approximate value for u (for ions moving normally to the vortex lines)
I
0
I
2 W-' ( I O - ' C ~ ~ / ~ W ' ) I
Fig. 31. The negative ion current attenuation as a function of heat flux W, plotted as suggested by eq. (56) (Ashton and Northby. 1973). The intercept gives the ion release rate, and the slope gives the length of line obscured by trapped charge.
SWERFLUD TURBULENCE
187
the intercepts in fig. 31 give values of x2 of about 0.15 and roughly independent of temperature. Ashton (1977) has measured the trapped charge density in a counterflow channel identical in size to that of Ashton and Northby (1973) but using a much more direct method. An ion source was developed which was recessed in the channel wall and could be turned on and off without affecting the electric field in the channel. The trapped charge density was measured by integrating the ion current after the source was switched off. Although the trapped charge measurement is more direct, and avoids the assumption leading to eq. (53), the interpretation still relies on the same assumptions as did eq. (55). Again using values of Lo and A ( T ) appropriate to the second turbulent state it is found that 1, the length of obscured line, is approximately twice the value found by Ashton and Northby (1973). Internal evidence in these data also indicate that I is constant and independent of Lo. The quantity x2, obtained from i.d/Li using the Vinen form, eq. (19), is only about 10% lower than found by Ashton and Northby. A far more sensitive test of this release rate is obtained from the charge decay time constant T . From eq. (54) it follows that 7-l
= Ld/LO
(57)
and if eq. (21) is used for Lo then a plot of T - ' against W2 should be linear. This is not found to be true for this data, and the rather significant discrepancies suggest that the release rate i d / & may well be associated with some non-homogeneous process. Ashton extended measurements of N,and T to higher temperatures where the thermal escape dominates the release rate. Although these results only cover a small temperature range, they are quite complementary to the lower temperature data. Using the ion source and counterflow channel described above but using a collector split symmetrically into upstream (toward the heater end of the channel) and downstream halves, Ashton and Northby (1975) were able to measure the average motion of the trapped charges-the vortex line drift velocity up When the collector currents are summed, the results for the trapped charge density N,and the decay time T described above are obtained. In the time constant measurement, a plug of vortex lines of length s (equal to the size of the source) is first produced, and then when the charging current is turned off, allowed to decay by the release process described by i d . If the charged plug is drifting downstream at uL, then since the fraction of the total current that reaches each collector will depend on the fraction of the remaining trapped charge in front of it, the downstream collector will receive an increasing fraction of the total
J.T.TOUGH
188
current. This effect is observed by running the split collectors in a difference mode. An entirely straightforward model gives (58)
= Const. + ( 2 u J s ) t
Gd-ju)/(jd+ju)
where jd and ju are the downstream and upstream current densities. This linear time dependence should persist for a time A t = (s/20L)after which the charged plug drifts beyond the upstream collector. The experimental data are in excellent agreement with this model and represent the first measurement of the vortex line drift velocity in superfluid turbulence. Considered as a function of the relative velocity V and temperature T, the experimental result for uL is uL = b(T) V,
(59)
in qualitative agreement with the Schwarz theory, eq. (35). However, the experimental values of b(T)shown in fig. 32 are significantly smaller than values calculated by Schwarz (table 1) and shown by the dashed line. Ashton and Northby give a dimensional argument that the average line drift velocity in the superfluid rest frame can only be determined by the and the circulation K and write average line spacing 1 =
0.4
>
0.3
‘I
w
1; 0.2 0. I
0
1.2
1.4
1.6
1.8
2.0
T (K) Fig. 32. 7he ratio of the average vortex Line drift velocity uL to the relative velocity V. Data are from Ashton and Northby (1975) and the solid line is from their dimensional argument. eq. (60).using 6 = 2.4. The dashed line is from the Schwarz theory (table 1).
SUPERFLUID TURBULENCE
189
where p is a constant, and uL is in the laboratory frame. The solid line in fig. 32 is obtained using = 2.4, with Lo computed from eq. (21) using values of y appropriate to the state TII. The disagreement with the Schwarz theory should not be taken too seriously, since the calculation of uL relies on the universality of the distribution, eq. (33), which is only approximate. The drift velocity measurements are certainly the most quantitative results to be obtained from ion data in superfluid turbulence. The importance of the more qualitative data should not be underestimated. In the light of our present knowledge of the ion structure and of the ion-vortex line interaction, the ion data provide the most direct evidence that superfluid turbulence is actually a distribution of quantized vortex lines. 7. Fluctuation phenomena
7.1. Classical and quantum mechanical turbulence The excess dissipation observed in AT (section 3) and second sound experiments (section 5 ) provides information about the macroscopic mutual friction force Fsn. This force is an average property of the superfluid turbulent state (section 2) which is described microscopically as a random distribution of quantized vortex lines. The ion experiments (section 6) strongly support this microscopic description. In what sense however is the superfluid “turbulent” in the classical use of the word? A turbulent flow is one in which the dynamical variables exhibit a chaotic or noisy time dependence. The transition to turbulence, as revealed by modern experimental techniques is incredibly rich in detail. If the “distance” away from the thermodynamic equilibrium state is given by some dimensionless variable X,then the flow at sufficiently small X represents a unique stable solution to the hydrodynamic equations. This solution becomes unstable for X greater than some critical value X,, and a new flow develops generally breaking the symmetry of the original flow. The average macroscopic dissipation changes abruptly at X, and then varies continuously with X. Although the average dissipation may show no sign of detailed structure in the flow for X>X,,the study of the fluctuations in the dissipation often reveals a succession of distinct flow states of increasingly complex character. At sufficiently large X a noisy o r chaotic time dependence appears, and the flow is turbulent.
J.T. TOUGH
190
A particular example of the transition to turbulence in a classical system is Rayleigh-Benard convective flow. In this system the fluid is contained between two horizontal parallel plates and is heated from below. The distance from equilibrium is described by the Rayleigh number R which is a dimensionless measure of the temperature difference across the cell. At sufficiently small R the stable solution to the hydrodynamic equations is one where there is no flow-a state of pure conduction. The instability of the purely conducting state and the onset of convective flow occurs when the Rayleigh number reaches the specific value R, = 1708. The effective thermal conductivity is given by the Nuselt number N which begins to increase at R, (fig. 33a) (Behringer and
.=1
I 5
4
2
3 2
I 0
I
I
0.10 0.05 FREQUENCY f (Hz)
1
0.15
I
‘ 0 20 40 60 80 I00 120 140
R/R,
Fig. 33. The transition to turbulence in classical Rayleigh-Benard convective flow. A fluid layer is contained between horizontal parallel plates and is heated from below. The Nuselt number N is the reduced effective thermal conductivity, and the Rayleigh number R measures the “distance” from thermodynamic equilibrium, and is proportional to the temperature difference between the plates. (a) N begins to increase from 1 as R exceeds the critical value R, and convection begins (Behringer and Ahlers, 1977). (b) N increases continuously with RIR, (Ahlen. 1974). (c) The temperature difference fluctuates about a mean value. The power spectral density of the fluctuations shows periodic flow with a single frequency for R / R , just greater than 1. (d) For larger RIR, a “quasi-periodic” flow with two frequencies and their harmonics is observed. (e) At sufficiently large RIR, broadband noise appears and the flow is “turbulent” (Swinney and Gollub, 1978).
SUPERFLUID TURBULENCE
191
Ahlers, 1977) and increases continuously with R (fig. 33b) (Ahlers, 1974). The temperature difference AT is not constant, but varies with time for R > R,. (The results in figs. 33a and 33b involve the average value of AT.) The time dependence of A T is conveniently described by & ( w ) , the power spectral density of the function AT(?)which reflects the time dependence of the convective velocity field. The results of a laserdoppler velocimetry study (Swinney and Gollub, 1978) of the time dependent flow velocity is shown in figs. 33c, d, e. For a small range of R at small RIR, the flow is strictly periodic and $ ( w ) consists of a single fundamental frequency and its harmonics (fig. 33c). For larger RIR, there is a range of R in which the flow is quasi-periodic with two fundamental frequencies in & ( w ) (fig. 33d). Finally, for larger RIR,, the power spectral density develops a broadband noise, and can be regarded as turbulent (fig. 33e). The transition to turbulence shown in figs. 33c, d, e is also observed in other flows, and is qualitatively in agreement with the theoretical considerations of Ruelle and Taken (1971). However, the pattern of instabilities leading to periodic, quasi-periodic, and chaotic time dependence is certainly not universally observed. Even in Rayleigh-BCnard convection, the transition to turbulence can be radically changed by relatively small changes in the system geometry (Ahlers and Behringer, 1978). In some cases, most notably the flow through a pipe, there is no sequence of well defined instabilities, and the turbulence begins immediately at X,. Superfluid turbulence, as described by the Schwarz theory (section 2.2) is obviously turbulent in the classical sense. Superimposed on the steady average superfluid velocity is a fluctuating component due to the quantized vortex lines maintained in a dynamic steady state of average density Lo. As in conventional fluid flow, the average dissipation increases smoothly above the critical heat flux (compare figs. 33a, b and fig. 13). The study of fluctuation phenomena in supefiuid turbulence is only now in its infancy, and there have as yet been no experiments which indicate how the transition to turbulence proceeds. The line density itself, regarded as a dynamical variable, probably fluctuates about a mean value Lo. Do line density fluctuations show a succession of periodic, quasiperiodic, and chaotic behaviour? The first observation of a fluctuating quantity in superfluid turbulence was made by Allen et al. (1965). These investigators measured the random deflection of a small quartz fibre suspended in the flow. The following year Vicentini-Missoni and Cunsolo (1966) reported fluctuations in a negative ion current in a turbulent counterflow, but the importance of these observations was not recognized until nine years later
192
J.T. TOUGH
when Hoch et al. (1975) made the first detailed study of these fluctuations. These investigators also introduced second sound attenuation as a probe of fluctuations. Velocity fluctuations in a counterflow jet have been observed by Dimotakis and Laguna (1977) using a phase shift technique. There appears to be no observation of fluctuations in the temperature difference although these would seem to be most easily detected and most readily interpreted. Very recently, Yarmchuck and Glaberson (1979) have reported substantial fluctuations in the chemical potential difference, although no systematic study of the phenomenon was undertaken.
7.2. Mechanical probes
In the pioneering experiments of Allen, Griffiths and Osborne (Allen et al., 1965; Grifiths et al., 1965, 1966) a quartz fibre with a small bob on the end formed a pendulum which hung in the center of the counterflow channel. The fibre was cm in diameter, and had a resonant frequency of about 5 H z . The channel had a low aspect ratio rectangular cross section (0.5 X 1.1 cm) and was only 5 cm long. Displacements of the fibre perpendicular to the heat flux were detected optically using two photomultipliers coupled to a high speed chart recorder. It was observed that when the heat flux was sufficiently large, the fibre was subject to substantial apparently random displacement fluctuations. Fig. 34a shows the rms displacement for two runs at 1.3 K (the dashed line in the figure and the arrows indicating W,,will be described below). Similar but less definitive results were obtained at other temperatures. A second type of measurement was designed to determine the circulation trapped on the fibre. Short (-5 ms) heat pulses were produced about every three seconds, and the transient response of the fibre was recorded. In the absence of a steady heat flux, the response was quite uniform in magnitude over a long time indicating a trapped circulation on the order of one quantum. As the heat flux was increased, the transient response decreased and was quite variable in size suggesting a trapped circulation that varied significantly over the three second repetition period. These observations were made quantitative by defining a persistence p of a series of transient responses as
SUPERFLUID TURBULENCE
193
h
E
0 0
I
0
Y
N
X
v
I
E
I
I
!
,“i I
,,
c
.Y,
I
e,
0.6
0
\
a“ 0.2
\
00
0
1
2
3
4
\
5
W (mW/cm2) Fig. 34. (a) The rms displacement (x2)”* of a quartz fiber suspended in a turbulent counterflow increases with heat flux W. The temperature is 1.3K and the channel is 0 . 5 1.1 ~ x 5 an long (Allen et al., 1965). The dashed lines are drawn to suggest a different response in states TI and TII. (b) The persistence, eq. (61), of the circulation trapped on the fibre. (Allen et al., 1965).The dashed lines again suggest a difference between states TI and TII.
where a, is the amplitude of the nth response. Clearly the persistence is a form of serial correlation coefficient and gives a measure of the constancy of the circulation trapped on the fibre. Results for p at 1.3 K are shown in fig. 34b (the lines and the arrows indicating critical heat fluxes are described below). At very large heat flux, the rms fluctuation level was so large that the pulse measurements were not able to resolve the trapped circulation. Another series of experiments employing mechanical probes, and in many ways equivalent to the work of Allen et al., was reported by Piotrowski and Tough (1978b). In these experiments a small “paddle”
J.T. TOUGH
194
2 .o
0 T=1.400K I.6 0 T=1.600K A
.-
u)
c
C
1.2
2)
2
z
.g
c
0.8
a
v
A
0.4
X
v
0 1
1
I
1
as a function of the relative velocity for several temperatures. The channel is 0 . 3 ~ 1 . 0 ~ 2.5 cm long. (Piotrowski and Tough, 1978). The solid line is calculated from the exponential power spectral density, eq. (63).
.
.
..
.. . . . . ” * ... .. was positionea in tne center 01 a counternow cnannei witn Its smaii dimension transverse to the flow. The “paddle” was supported on a superconducting niobium wire loop which served both as a spring and as an element in a flux transformer coupled to a SQUID. Four different channel/“paddle” combinations were investigated. In all cases the channel had a moderate aspect ratio rectangular cross section (ranging from 0.2 c m x 1cm to 0.3 c m x 1cm) but was only 2.5 cm long. The resonant frequency of the “padd1e”lspring combinations ranged from 28 Hz to 120 Hz. The sensitivity was such that paddle displacements of a few Angstroms could be resolved. The mean square displacement (x’) was determined by signal averaging of the squared SQUID output. Fig. 35 shows some data for (x’) as a function of the counterflow velocity V for two temperatures. Similar results were obtained in all channel/“paddle” combinations at all temperatures. The “paddle” response can be shown to be exactly that of a high Q oscillator driven by a random force F(r), with a power spectral density &(a). In particular the mean squared displacement is given by the value
SUPERFLUID TURBULENCE
195
of S, at the resonant frequency: (62)
(x2) = SF(00).
The strong dependence of (x’) on both T and V is consistent with a force spectrum of the form SF(w) = S0e--,
where the correlation time T = h(T)/v.
(63) T
is a function of T and V of the form (64)
The solid lines in fig. 35 were obtained from eqs. (62), (63), and (64) by fitting h(T) to the data. A correlation time can be obtained from the Vinen dynamical equation (20) (Piotrowski and Tough 1978b,Northby, 1978) which is of the form, eq. (64) with
h(T)= 4 T / K X 2 y 2 .
(65)
The values of h(T) determined from the experiments are consistent with eq. (65) using a value of xz of roughly 0.25. This is somewhat stronger support for the Vinen dynamical equation than the ion data results (section 6) since the term proportional to x2 need not represent annihilation of vortex lines but may simply represent the average decay rate as in the Schwarz theory. However, there are many models that can be invoked which give a characteristic time of the form, eq. (64). Moreover, the correlation time obtained from the Vinen equation also requires a Lorentzian rather than an exponential power spectral density (see section 7.3). Although the data are rather sparse, it is nevertheless tempting to consider a comparison of these two mechanical probe experiments. Each experiment employs a high 0 oscillator to detect a fluctuating force and thus measures SF(wo).The counterflow channels employed by Piotrowski and Tough were probably of sufficiently high aspect ratio to support the third superfluid turbulent state. Indeed, the data (fig. 35) indicate no change in the structure of (x2) as the relative velocity is increased. In contrast, the data of Allen et al. (fig. 34a) show a rather dramatic change at a heat flux of about 5mW/cm2. This flux is characteristic of W,, (section 9), marking the transition from state TI to TI1 in low aspect ratio channels. The dashed line in fig. 34a is drawn to suggest the presence of these two states in this experiment. Lines are also drawn on the persistence data in fig. 34b to suggest that the circulation about the fibre persists unchanged out to the critical heat flux Wcl, and then decreases in
196
J.T. TOUGH
the first turbulent state, vanishing at Wc2. Does this suggest that state TI is more ordered than TII? In particular, is state TI in any way analogous to the periodic or quasi-periodic flows observed in classical fluids preceeding the transition to chaotic turbulent flow? Detailed studies of the power spectrum in these states, using broadband rather than high Q detectors, will be required to answer these important questions.
7.3. Ion and second sound probes The negative ion current in a space-charge limited diode is strongly attenuated in the presence of superfluid turbulence (section 6). The attenuation results from the charge trapped on the quantized vortex lines. Fluctuations in the line density should then induce fluctuations in the diode current. Hoch et al. (1975) have observed these current fluctuations at 1.125 K using a diode similar to that of Sitton and Moss (1972). The fluctuations were observed to disappear for temperatures greater than the "lifetime edge," thus firmly establishing the vortex lines as the source of the fluctuations. The power spectral density was measured at several values of the heat flux, and the results are shown in fig. 36. At low frequencies S ( w ) varies as w - ' , and above a roll-off frequency wo = l h n , S ( w ) varies roughly as u - ~Although . these spectra appear to be completely unrelated to the mechanical probe results, the roll-off frequencies 200
'5 ; too t
.-
C
3
50
t
:'
8 0
8
A
q
W= 5rnw/cm2 A
N
3.
1
1
l
0.5 I
5
I0
I
A A
I 1
50
w (s-? Fig. 36. The power spectral density of the fluctuations in ion current traversing a turbulent counterflow at various values of the heat flux (Hoch et al., 1975). The arrows indicate roll-off frequencies calculated from eqs. (64) and (65).
SUPERFLUID TURBULENCE
197
are in fair agreement with eqs. (64) and (65). The arrows in fig. 36 indicate the roll-off frequencies calculated using x2= 3. These ion current data are much more directly related to fluctuations in the vortex line density, although the interpretation of the results may still be complicated by intrinsic fluctuations in the ion trapping and release process. The w - ’ response at low frequencies is certainly common to many systems but the high frequency w - 3 result is quite unusual. Hoch et al. also discovered that the amplitude of a standing second sound wave showed fluctuations about a mean value in the presence of superfluid turbulence. Since the attenuation is directly related to the vortex line density, eq. (49) it would appear that second sound is even more direct than ions as a probe of line density fluctuations. Unfortunately the initial experiments were only qualitative although there was a strong suggestion that the u-.’ spectrum was present at low frequencies. Mantese et al. (1977) have studied the fluctuations in second sound amplitude in short, high aspect ratio rectangular channels. The transducers were of the capacitance type which dissipate no heat, and had a high signal to noise ratio. Data were obtained only at 1.648 K, since at that temperature the second sound velocity is independent of temperature, and bath temperature fluctuations are not a problem. The average line density Lo (discussed in section 5 ) , the mean square line density fluctuation (SL2), and the power spectral density S L ( w ) were determined. In these experiments there was clear evidence of a critical heat flux (Wc= 3 mW/cm’) below which both Lo and (SL’) vanish. The results for (SL2) are shown in fig. 37a and indicate (SL’) increases roughly as W4.Some power spectral density data (Moss, 1977) are shown in fig. 38a, and these are similar to the ion results in fig. 36 suggesting 0 - l behaviour going over to u - 3at high frequencies. Somewhat similar second sound fluctuation results were obtained by Ostermeier et al. (1980) using the second sound burst technique and counterflow channel described in section 5 . Data for (SL’) and S,,(w) obtained at 1.25 K using the transducers near the channel center (at 25 cm) are shown in figs. 37b and 38b. There is no indication of a critical heat flux. It is possible that (SL’) is proportional to W4(dashed line) and dependence at high heat flux (solid line) in a way then goes over to a that is qualitatively similar to the results of Mantese et al., fig. 37a. The power spectral density results are again similar to Mantese et al. except at low frequencies where Ostermeier et al. find a plateau rather than a w - ’ dependence. In one sense these pioneering second sound fluctuation experiments are
J.T.TOUGH
198
- 10’
V
I
h
V
€
0
E
Y
A
“_I m v
10.0 r
0 aD
loo
0 X
Y
A N
-I bo
lo-’
1.0 =
v
+
t/ I L
I
1 0
W (mW/cm2)
100
0.I
10
I00
W (mW/cm2)
Fig. 37. Mean square vortex line density fluctuations determined from second sound attenuation measurements. (a) Results of Mantese et al. (1977) in a high aspect ratio rectangular channel at 1.648 K. Curves labelled 1 and 2 are from the analysis of Northby (1978). eqs. (66)and (67). (b) Results of Ostermeier et al. (1980) in a square channel at 1.25 K. The dashed line has slope 4, and the solid line slope 2.
disappointing: the rich structure found in the transition to turbulence in conventional fluids has not been observed. Does this mean superfluid turbulence is truly “turbulent” or chaotic from the onset? O r is there more detail yet to be discovered in future fluctuation experiments? The profound effects of geometry seen in the temperature difference data (section 3) are not evident in these fluctuation experiments. The fluctuations measured in the rectangular channel (Mantese et al., 1977) should be representative of the third superftuid turbulent state. The square channel measurements of Ostermeier et al. (1980) should show fluctuations of both the tint and second superfluid turbulent state. The data, like those for the line density Lo (section S), give no indication of a transition from TI to TII. Moreover, the form of the power spectral density is very similar to that for state TI11 (fig. 38). In another sense these experiments are truly exciting: a new probe of superfluid turbulence has been discovered which reveals dynarnical information about the turbulence.
SUPERFLUID TURBULENCE
199
Northby (1978) has attempted to analyse these data in terms of the Vinen dynamical equation (20). Although not entirely successful, the results are encouraging and are no less quantitative than the attempts to describe conventional turbulence from the Navier-Stokes equation. Northby (1978) assumes that a small random component of the counterflow velocity drives fluctuations in the line density as determined by the Vinen dynamical equation (20). For small fluctuations about Lo the equations can be linearized to give a Langevin equation and a Lorentzian power spectrum for the fluctuations. Except in the immediate neighborhood of the transition (W,) the result can be written
where the correlation time 7 is given in eqs. (64) and (65), and &(w) is the power spectral density of the random velocity fluctuations. A first approximation for the mean square fluctuation
J
( 6 ~=~ )s,(o) d o I
I
(a)
lo2
f 10' N
\
*I I
\ \
-E 10"
\ \
Y
3
Y
'nc lo-'
lo-: I
lo-'
lo"
w/27r (Hz)
lo-'
loo
10'
10'
w / 2 7 (Hz 1
Fig. 38. The power spectral density of the vortex line density fluctuations determined from second sound attenuation at various values of the heat flux. (a) Results of Moss (1977) in a high aspect ratio rectangular channel at 1.648 K. (b) Smoothed results of Ostermeier et al. (1980) in a square channel at 1.25 K.
200
J.T. TOUGH
may be obtained by replacing SJw) by its value at o = 0 and assuming it is independent of V. The results are given in fig. 37a as curve 1, and are seen to be in good agreement with the data at lower heat flux. The power spectral power spectral density, eq. (661, does not agree well with the data dependence at large frequency. Northby however (fig. 38), giving a has gone on to model the low frequency fluctuations in V, leading to the more realistic approximation
where the correlation time T,, = pnd2/q.rr2is about 56 s for the experiment of Mantese et al. The results, eqs. (66) and (67) then give a line density power spectrum going as w-4 at high frequencies. The results for SL2 are also somewhat improved at high heat flux as shown by curve 2 in fig. 37a. Northby has considered a further modification by including a feedback loop coupling fluctuations in L and V. Unfortunately the experimental data are inadequate to provide a test of the model details. 8. Tbe critical condition
8.1. Primary and secondary transitions Three distinct states of superfluid turbulence have been identified in the experiments discussed above. States TI and TI1 are found in low aspect ratio (predominantly circular) channels and state TI11 is found in high aspect ratio channels. The three states are all described by line density of the form given by eq. (24), but they differ in the magnitude and temperature dependence of the parameters y ( T ) and V,. There is a clearly defined critical heat flux below which the states TI, TII, and TI11 do not persist (fig. 13). There are two fundamentally different critical conditions: at W,, and W,, there is a “primary transition” from a turbulent to a laminar state. The “secondary transition” at W,, is from one state of superfluid turbulence to another. There is no hysteresis associated with the secondary transition, but a metastable laminar state can be observed for heat fluxes considerably greater than W,, or Wc3. The primary and secondary transitions differ also in response to rotation. Chase (1966) has shown that vortex lines introduced by rotation of the counterflow channel reduce Wcz, whereas Yarmchuck and Glaberson (1979) have
SUPERFLUID TURBULENCE
20 1
clearly demonstrated that W,, is increased. Some flow transitions in conventional fluids can be stabilized by modulation of the velocity at some appropriate frequency (see for example Donnelly, 1964). Oberly and Tough (1972) have shown that W,,can be increased over 50% by suitably modulating the heat flux. Modulation produces only a small and unsystematic effect on Wcl. Finally it should be noted that all critical heat fluxes are approximately inversely proportional to the channel dimension, indicating an extrinsic rather than intrinsic critical condition. The theoretical situation with respect to the critical condition is rather confusing, although not for lack of effort. There are over a score of expressions for the critical heat flux or critical velocity compared with only two theories for the steady superfluidturbulent state. In the theory of Schwarz (1978) the relative velocity V is assumed to be uniform and the distribution of vortex lines is assumed to be spatially homogeneous (section 2.2). This theory is thus not suited to considerations of the extrinsic critical condition in finite size channels. The line density obtained from the Schwarz theory (fig. 11) does contain an intrinsic critical velocity V,, although this may be an artifact of the numerical calculation. Indeed, there have been several experimental observations of superfluid turbulence at relative velocities much less than V,. Vinen (1957d) introduced a modification to his phenomenological theory (section 2.2) which leads to an extrinsic critical velocity V,. The idea was to modify the production term in eq. (18) in the dynamic balance equation (20) leading to the steady state line density of the form of eq. (24) and an explicit expression for V,. In the Schwarz theory, which reproduces the dynamic balance equation, the “production” and “destruction” terms are simply different aspects of the same dynamic process. It thus seems unlikely that the process of modifying only one term can be correct. However, the Vinen result, eq. (23), that there is a critical value of the line density at V, leads to a reasonably satisfactory description of the primary transitions at W,, and W,,(section 9.2) without having to adopt the Vinen model. It is not clear which, if any, of the many other critical velocity expressions proposed over the last twenty-five years are applicable to the critical condition for superfluid turbulence. Most of the proposals are either of the Feynman (1955) type where explicit expressions for vortex ring energy and momentum are employed, or of the critical “Reynolds Number” ty-pe (Staas et al., 1961) where various dimensionless numbers are proposed to be constant when evaluated at the critical condition. Comparison of the various critical velocity expressions with experimental data has been confounded by the existence of the three different critical
J.T. TOUGH
202
conditions in thermal counterflow alone. In some experiments, where both W,,and W,,are expected, only one critical heat flux is measured, and it is not clear which one (Cornelissen and Kramers, 1965; Careri et al., 1960a; Ostermeier et al., 1978a, b). Further uncertainty has resulted from the lack of unambiguous values for the Gorter-Mellink coefficient A(T) appropriate to the superfluid turbulent state of interest. Some of the proposed critical velocity expressions will be considered in the discussion of the secondary transition at W,,(section 9.3). 8.2. The primary superfluid turbulent transitions
Brewer and Edwards (1961a) have measured the critical heat flux Wcl in small glass channels of circular cross section. Their data are in excellent agreement with the results of Childers and Tough (1974) and de Haas and van Beelen (1976) in similar channels, and the dependence of temperature and channel size agrees well with the Vinen expression for V, (Childers and Tough, 1973). Ladner and Tough (1979) have measured the critical heat flux W,,in small glass channels of rectangular cross section (10d x d ) and have found the size and temperature dependence to be quite similar to that of Wcl. They pointed out that both of the primary transitions at W,,and W, could be regarded as taking place at a constant critical value of the line density:
(LA’2d),= 2.5.
(68)
The line density at velocities just above V, is small and difficult to determine with precision. The result, eq. (68) is obtained by fitting the line density at V> 1.2Vc to eqs. (24) and (25) and using the resulting y(7‘) and a to compute L;”d at the observed V,. Although this process is somewhat artificial, it allows a consistent estimate of (LAI2d),to be made in the critical region. Using the univerwithout detailed knowledge of sal expression for the line density in states TI and TII, eq. (39), along with eq. (68) leads to the estimate for the critical velocity
Vcd -[2.5
+ 1.48a(d)J(y0D/d)
(69)
where a ( d ) has been written to emphasize the weak d-dependence observed for the parameter a (Ladner, 1980). Data for W,,in large channels has been obtained by Yarmchuck and Glaberson (1979) and Tough (1980) has shown that the critical line density condition, eq. (6% remains valid. Results for W, in very small channels have been obtained
SUPERFLUID TURBULENCE
0.12
0
’
203
1
CHILDERS. 4 TOUGH 0.0131cm circ., W,,
1
YARMCHUCK c GLABERSON 0 . 0 9 2 ~l.13cm.,Wc3 o LADNER~TOUGH0.0032xOD32cm., W, LINES ARE :(L:2d&=2.5
A T
0. I 0
0.08 h
u)
\
N
E
u
D
0.06
>* 0.04
0.02
0
’ I.2
I
I
1.4
’ I.6
I
1.8
’
11
2.0
Fig. 39. The product of the Critical relative velocity and channel size (V,d)corresponding to the primary critical conditions at W,,and W,. Results are shown for three diverse experiments as a function of temperature. The lines are calculated from the critical line density condition, eq. (69).
by Craig et al. (1963). Since the temperature and relative velocity vary considerably over the length of these channels, the results are difficult to interpret, but eq. (68) appears to be only approximately correct. Various experimental data for the quantity V,d corresponding to the primary transition at W,,and W,, are given in fig. 39. The solid lines are computed from eq. (69) and correspond to a single critical value of the line density at the transition given in eq. (68). The agreement of these data with the critical line density expression is not as quantitative as would be hoped, but considering the range of experimental parameters,
204
J.T. TOUGH
the agreement is quite acceptable. Further support for the critical line density model has been given by Tough (1980) who has shown that the model can be extended naturally to provide a description of W,, in rotating counterflow channels in quantitative agreement with the data of Yarmchuck and Glaberson (1979). Although the critical line density condition, eq. (68), derives from the Vinen critical velocity expression, the Schwarz theory suggests the most physical interpretation of the result. Of course the superfluid turbulent states TI and TI11 are unlikely to be the homogeneous state described by Schwarz, even though the form of the line density is identical, eq. (24). The amplitude y ( T ) or A ( T ) , eq. (26), has a different temperature dependence than does the Schwarz result (fig. 22) and the geometrical scaling, eq. (38), indicates major inhomogeneity effects. Also the parameter a ( d ) must reflect the inhomogeneity associated with the normal fluid flow since a = 0 when V, = 0 (section 9). Nevertheless, the macroscopic properties of states TI and TI11 can be described by a steady line density Lo, and thus there is an intrinsic characteristic length 1 = L;’”. The geometrical significance of this length involves the particular distribution of vortex lines in the turbulence. The critical condition, eq. (68), indicated that the distribution cannot be maintained when I is approximately greater than d/2.5. If I represents some average radius of curvature or average vortex ring radius then this condition is intuitively very appealing.
8.3. The secondary superfluid turbulent transition The transition at Wc2 in circular and other low aspect ratio channels has been observed in many experiments beginning with those of Vinen (1957a, b, d). Chase (1962, 1963) has employed the Vinen waiting time technique (see section 5.2) to make a detailed study of the temperature, channel size, and geometry dependence of Wc2. Ladner et al. (1976) used a thermal relaxation time method to determine W,, in small circular channels, and Martin and Tough (1980) have extended this technique to channels an order of magnitude larger. This secondary transition can also be reliably observed in the temperature difference data of Brewer and Edwards (1961a, b, 1962) (see fig. 17), Oberly and Tough (1972), de Haas and van Beelen (1976) and Peshkov and Tkachenko (1962). The results from all these experiments are shown in fig. 40 (see table 4) as the value of V,d corresponding to Wc2. Other data that may very well correspond to W,, have been omitted from the figure since there is no
SUPERFLUID TURBULENCE
205
0.4
; 0.3 \
N
E V
v
U
”’
0.2
0.1
I
0
I .2
i
I
1.4
I
I
1.6
I
I
1.8
1
I 2.0
T (K) Fig. 40. The product of the critical relative velocity and the channel size (VJ) corresponding to the secondary critical condition at Wc2. Results are shown for a collection of experiments listed in table 4. The solid lines correspond to constant values of the “Reynolds Numbers” [eqs. (70)-(76)],normalized at 1.2 K.
compelling reason to believe W,, rather than W,, was measured. It is not likely that these data would greatly clarify the experimental situation anyway. Critical velocities of the Feyman (1955) type have been proposed by Peshkov (1961), Fetter (1963), Fineman and Chase (1963), Glaberson and Donnelly (1966), and Weaver (1973). A purely empirical result was also proposed by van Alphen et al. (1966). These critical velocities do not seem appropriate to the secondary transition at W,,since they are primary transitions: the onset of vortex line production in initially vortexfree flow. Staas et al. (1961) were the first to suggest a modified Reynolds Number as a scaling parameter for Wc2. They proposed that the quantity Ri=P V J h
9
(70)
J.T. TOUGH
206
Table 4 Sources of the experimental data for the secondary critical condition in fig. 40
Symbol
Reference
Channel size (cm)
Brewer and Edwards (1961a.b) Brewer and Edwards (1961a.b) Vinen (1957d) Vinen (3957d) Oberly and Tough (1972) de Haas and van Beelen (1976) Peshkov and Tkachenko (1962) Ladner et al. (1976) Chase (1962) Chase (1963) Chase (1963) Chase (1963)
0.0366 diam. 0.0108diam. 0.24 x 0.65 0.4 x 0.783 0.057 diam. 0.0216 diam. 0.274 dim. 0.0131 diam. 0.08 diam. 0.159diam. 0.262 diam. 0.051 ~ 0 . 1 0 3
when evaluated at Wc2, would give a constant result independent of temperature, channel size, and geometry. Similar “Reynolds Numbers” have been proposed by Dimotakis (1974), R2 = p,AVd
(71)
by Spangler (1972),
by Meseny (1962), RS = (p,p2/p,)”2Vd(A/~)”2
(74)
and by Tough and Oberly (1972),
Rd= pVd(A/r))1‘2.
(75)
All of these expressions attempt to characterize the state of the fluid preceding the transition at Wc2 (that is, the state TI) in terms of a modified density or modified viscosity. [Note that A-’, eq. (26), depends on geometry, eq. (38), and has units of viscosity.] The values of V,d corresponding to constant values of R , through R6 are shown as solid lines in fig. 40, normalized to the same value at 1.2 K. It is not clear that any of these results reflect even the temperature dependence of the data.
SUPERFLUD TURBULENCE
207
Another “Reynolds Number” was proposed by Chase (1965) in much the same spirit as those above
R , = p d ( -vnrl+ - rls , where q, is the eddy viscosity associated with the superfiuid turbulence in state TI (section 4.2). Because of the dependence of qs on V it is not possible to show this result in fig. 40, although the agreement with the data is no better than R6. Finally Ladner et al. (1976) proposed that R 1 may be the correct “Reynolds Number” but the critical value of R 1was a function of the line density at the transition. The data in fig. 40 do not appear to be any more unified when considered in this manner. The experimental results in fig. 40 indicate that Vcd depends weakly on d and also depends on geometry. This observation suggests that possibly W,, is associated with a constant value of LA’2d as were the primary transitions, eq. (69). The values of Vcd in fig. 40 do correspond to (L;”d), = 12, but the scatter is large and the results are thus inconclusive. It is possible of course that the secondary transition at W,, depends upon factors that have not been considered: channel wall roughness, entrance length effects, or even ambient vibration levels in the experiments. About all that can be determined at the present time is that the transition is strongly dependent on the normal fluid flow. No secondary transition is observed when V,, is zero (fig. 8, section 9). No secondary transition is observed in thermal counterflow in high aspect ratio channels where the normal fluid flow in state TI11 is one dimensional (fig. 13). The transition is only observed in state TI, where the normal fluid flow is two dimensional. The different inhomogeneity associated with the normal fluid flow fields somehow determines the properties of states T I and TIII. The secondary transition from T I to TI1 is then probably accompanied by a change in the normal fluid flow. Whether this is a “turbulent” transition as in classical pipe flow or simply the first (or second) of a series of increasingly complex secondary flows remains to be discovered.
9.
Pore emperflow, pure n o d eow, and otber velocity combinations
9.1. Pure superAow (V,, = 0): a fourth turbulent state Experiments in pure superflow, as well as other non-counterflow combinations of V, and V, have been pioneered by researchers at Leiden. The
J.T. TOUGH
208
flows are generated by the use of heaters in combination with super-leaks and sometimes supplemented with either a dual-plunger arrangement for direct mechanical drive (de Haas et a!., 1974; de Haas and van Beelen, 1976; Slegtenhorst and van Beelen, 1977) or with an evaporation “gaslink” (Kramers, 1965; Kramers et al., 1973; van der Heijden et a]., 1972a, b, 1974; Ijsselstein et al., 1979). Pure superflow is particularly interesting since there should be no inhomogeneity associated with the non-uniform flow of the normal fluid. Measurements of the temperature, pressure, and chemical potential difference, as well as of second sound attenuation have all been made in pure superflow in channels of various sizes but only in channels of circular cross section. [Keller and Hammel (1966) have studied isothermal superflow in rectangular channels, but since no mutual friction i s present, these results will not be considered.] The temperature difference and second sound attenuation data virtually all agree with a mutual friction force of the form, eq. (16). and with a turbulent vortex line density Lo given by eq. (25) (where of course V, is set equal to zero, and V = Vs).The line density Lo determined from temperature difference data of Ashton and Tough (1980) is given in fig. 41. Comparison of superflow and thermal counterflow data in circular channels (sections 3.2, 3.3. 5.2, and 5.3) reveals three major differences. The first difference is that in the superflow data V, = 0, as can be seen in fig. 41 (compare figs. 17 and 41). The meaning of Vo in counterflow 3
I
I
I
I
1
O0 O0
T zt.25K
00
a A
A 0
A
01 0
I
5
I
to
I
15
I
20
1
25
V, (crn/s) Fig. 41. The vortex line density in pure superfluid flow determined from the temperature difference data of Ashton and Tough (1980) at two temperatures in a circular channel. Only a single turbulent state (TIV) is observed with a line density of the form, eq (24) and with V, = 0 (compare fig. 17).
SUPERFLUID TURBULENCE
209
2 50
200
-
N
5
\
I50
ln
I00
50
01
1.0
I I. I
1
1
I
I
1.2
1.3
1.4
1.5
I
1.6
I
I
I
1
1.7
1.8
1.9
2.0
T (K1 Fig. 42. The line density coefficient y,(T) in state T N . The data are from the various sources listed in table 5. The line density coefficients for states TI and TI1 are shown for comparison. Only those data shown by solid symbols were obtained in channels also giving unambiguous counterhw results.
certainly is not clear, but the superflow results strongly suggest that V, is not an intrinsic property of the line distribution, but is rather associated with the non-uniform normal fluid flow and the average nature of the measured line density. The second difference in pure superflow data is that no transition is observed to a turbulent state of greater line density (compare figs. 17 and 41 for example). The single superfluid turbulent state observed in pure superflow could be closely related to either state TI or TII, but until this can be established without ambiguity this state will be denoted as TIV - the fourth distinct superfluid turbulent state. Since V,=O, the line density in state T N can be written LA’2 = y4( T )v,.
(77)
Data for yJT) obtained from a wide variety of experiments are given in fig. 42 (see table 5). The results for state TI (fig. 21) and state TI1 (table 1) are shown for comparison. The great degree of scatter in the experimental superflow data may be due to several causes including metal flow
210
J.T. TOUGH Table 5 Sources of the experimental data for the line density coefficient in state TrV shown in fig- 42 Symbol Reference
a X
+ 00 A
V 0
4
Ashton and Tough (1980) de Haas and van Beelen (1976) van der Heijden et al. (1972b) van der Heijden et al. (1972b) Kramen (1965) Kramen (1965) Ijsselstein et al. (1979). de Haas et al. (1974) de Haas et al. (1974) Peshkov and Stryukov (1962)
Channel size (cm) 0.0131 diam. 0.0216 diam. 0.0294 d i m . 0.01 diam. 0.029 d i m . 0.10 diam. 0.062 diam. 0.1 diam. 0.034 diam. 0.385 diam.
channels with rough surfaces (Childers and Tough, 1974a. b), stray heat leaks (Ashton and Tough, 1980) or incorrect determination of the actual superfluid velocity V,. Only those data in fig. 42 shown by solid symbols (de Haas and van Beelen, 1976; Ashton and Tough, 1980) were obtained in unobstructed glass channels in which the characteristic laminar-TI-TI1 structure in counterpow was also observed. For just these data it appears that y4 = 2y,, and the similar temperature dependence for states TI and TIV suggests a common origin. States TI and TI11 are similar in this respect, apparently scaling with the hydraulic diameter ratio [eq. (38), fig. 211. The scaling factor relating states TI and TIV may be a consequence of the vortex line drift velocity (section 9.3). (See discussion following eq. (16), and section 9.3). The third difference between thermal counterflow and superflow in circular channels is the incredibly small amount of critical velocity data in pure superflow. The reason for this is clear, however, if the critical condition for the vortex line distribution in state TIV is the same as for states TI and TI11 [eqs. (68) and (69)] since the corresponding critical superfluid velocities are very small and the critical condition is difficult to observe. For the data shown in fig. 8, the condition (L;”d), = 2.5 occurs at V,= 1.75 cm/s, at which point AT is unobservably small. Some experiments (Peshkov and Stryukov, 1962; Hartoog, 1980) have given strong evidence for a critical superfluid velocity V,. These results are all consistent with (L;”d), = 1.5. Considering the small amount of data, and the uncertainly in y4, this must be considered good agreement with the primary critical condition for states TI and TIII.
SUPERFLUID TURBULENCE
211
Another interesting feature of the pure superflow experiments is the observation of a pressure gradient along the flow channel such as observed in thermal counterflow (section 4). In most cases the pressure difference AP is too small to be measured with precision, but often it can be clearly demonstrated to be non-zero. Since F,, is an internal force and cannot produce a pressure gradient, these experiments imply that either there is an effective superfluid eddy viscosity q, and a superfluid body force ( A P a F, a V,qJ or an induced motion of the normal fluid with a zero mean value. In those experiments where A P has been determined with some precision, (van der Heijden et al., 1972b) the results are consistent with (F, a or with a superfluid eddy viscosity (q, This differs from the Li'3 dependence found in state TI [eq. (41), fig. 231 and the LA'2-dependence in state TI1 (fig. 22). The existence of an effective superfluid force has been determined quite unambiguously by Rosenshein et al. (1971), and is the key to understanding the unique flow f i s t studied by Staas and Taconis (1961). In this flow the chemical potential difference along the flow channel is zero: the force F,, is balanced by F,. In these experiments a pressure difference consistent with an L;I4 dependence of q, was also observed.
c"),
9.2. Pure normal flow (V,= 0) For temperatures less than 1.6 K, thermal counterflow may be expected to be essentially pure normal flow since Vn/Vs>5. 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 <2.5 is shown shaded, and represents the V,, V, combinations for which the primary critical condition is satisfied, and the line density cannot be maintained (section 8.2). The extensions of states TI and TIV into regions I and IV are indicated. The region corresponding to the state TI1 is presumed to occupy the area to the right of the line L;”d = 12. (See section 8.3). Naturally only qualitative information can be obtained from this diagram but it does provide a focus for the data in the (V,,, V,)plane. Several important paths through the plane should be noted. First is the path followed in a thermal counterflow experiment, and shown by the line labelled counterflow. Qualitatively there is a region of laminar flow (Lo= 0) followed by the first and second turbulent states. The results for a pure V, experiment would clearly be similar in this picture, since the path taken then is the line V,= 0. The path for a pure superflow experiment (section 9.1) is the line V, = 0 through the state TIV only. In the region below the line V, = V,the mutual friction F,,, and the superfluid force F, will be oppositely directed, and for a particular line density the net force on the superfluid (the chemical potential gradient) will be zero. The flow first studied by Staas and Taconis (1961) and later by van der Heijden et al. (1974) and de Haas and van Beelen (1976) occurs on this A p = 0 path, in state TI. It is not clear whether the A p = 0 line extends all the way to v, = 0. Other experiments have been constrained to different paths in the (V,,, V,)plane. In the work of Ijsselstein et al. (1979) V, could be varied while holding V, at various constant values. Data for the chemical potential difference are shown in fig. 45 and can be qualitatively understood in terms of the diagram in fig. 44. Of particular note is the abrupt drop in A p for the V,, = 4.0 and 4.75 cmls data as V, is increased. The diagram in fig. 44 suggests that this is the transition from state TI1 to state TI. Similar results are obtained by de Haas and van Beelen (1976), but V,) now following paths of constant mass flow velocity (V,,)in the (V,,, plane. These paths are lines parallel to the thermal counterflow line (V,, = 0), displaced by an amount f Vcap.Data for A p are shown in fig. 46 which again can be qualitatively understood in terms of the diagram in fig. 44. Particularly interesting are the data for V,, = +4.58 cmls which proceed through state TIV, cross the A p = 0 line, and then continue into state TI. The data for negative values of Vcsp suggest that the lines in fig. 44 can also be extended into the fourth quadrant of the (V,, V,)plane. Clearly the complexity of the various data in the (V,,, V,)plane renders
215
SUPERFLUID TURBULENCE
any complete explanation simply unrealistic at this time. The diagram in fig. 44 is probably useful for a coarse overview of the various flow states present, but may be based upon assumptions that will prove to be incorrect in flows other than thermal counterflow. Ijsselstein et al. (1979) have stressed the close connection between states TI1 and TIV. Is it only coincidence that yz = y4? De Haas and van Beelen (1976) have rejected the idea of a superfluid body force F,, but must accept instead a complex normal fluid flow induced by Fsn. There also has not been universal quantitative agreement between various experiments (see for example fig. 42). What is clearly revealed by the data however, is that several distinct states of superfluid turbulence can be produced in different experimental I
I
750 '
V,
0 '
0 A 0
0
0
0
(cm/s) 0 2 3 4 4.75
0
500
4
N
OD
E
V
V
0
0
\
c
0,
C
0
>
-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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J.T. TOUGH
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.
SLJPERFLUID TURBULENCE
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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.
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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,, << kT, then becomes
S = nR ln(2Z+ 1) - nA*B2/2T2,
(4)
where the nuclear Curie constant per mole is A* = NoI(Z+ l)&g;f/3k. In a constant magnetic field the nuclear heat capacity then, in the same approximation,
CB = nA*B2/T2.
(5)
[c,= T(ds/dT),] is (6)
It is important to note that C, is proportional to (B/T)*. The heat of magnetization that must be removed from the nuclear stage while it cools from a high temperature (approximated by T = 00) to Ti, in an external magnetic field Bi, is given by
AOM=
C, d T = -nA*Bf/Ti.
(7)
Eq.(2) shows that 2 and, therefore, also P(m), S, and C, are functions of BIT only for all values of B and T. During adiabatic demagnetization (AS=O) from Bi and T, to B,, P(m), S, and C, remain constant and, therefore, &/Ti = B,/T,; the final temperature is then given by Tf= Ti(B,/Bj).
(8)
If demagnetization is carried out to a low or zero field, B,in eq. (8) must be replaced by (B:+ b2)’l2 and we obtain
Tf = (T,/B,)(B:+ b2)”’,
(9)
where b is the effective internal field between nuclei arising from dipolar, quadrupolar, and/or exchange forces. Similarly, B 2 in eqs. (4) and (6) must be replaced by B 2 + b2 in low fields. For copper b = 0.34 mT. The ultimate low temperature which is reached when B, is reduced to zero is determined by spontaneous ordering of nuclear spins due to mutual forces, represented by b in eq. (9). In simple metals the ordering
RECENT PROGRESS IN NUCLEAR COOLING
227
temperature, 8 = p,b/k, is of the order of 100 nK. By comparison, spontaneous order occurs in CMN,the weakest paramagnetic salt, at 1.8 mK, so over four orders of magnitude in temperature can be gained by employing nuclear cooling. Furthermore, in paramagnetic salts the magnetic ions must be separated by neutral atoms, usually by crystalline water, to reduce their mutal interactions; molar volumes of salts are thus quite large, 300-400 cm3. Dilution is not necessary with nuclear spins: a high nuclear entropy per unit volume is thus achieved, the molar volume of copper being, for instance, 7.1 cm3. After demagnetization the nuclear spin system begins to warm up owing to the unavoidable external heat leak dQ/dr = Q. By observing that f = Q/C, we can calculate the time Al during which the nuclear spins warm from Tf to a higher temperature T. The result is
Ar = (nA*B:/Q)(T'- T1),
(10)
where eq. (6) has been employed. This relation shows that 1/T is a.linear function of time, provided that Q is constant. The total amount of heat, AQ, absorbed by the nuclear spin system during warm-up is A Q = [ C B d T = nh*B:(T;'-T-*).
(11)
This is very much less than the heat of magnetization AQ, [d.eq. (7)]. With an external heat leak the use of eqs. (8) and (9) for calculating Tf is not strictly valid because the demagnetization process is no longer adiabatic nor reversible. While the magnetic field is being reduced, energy will continuously flow from the outside to the conduction electron system, which passes it on to the nuclear spins. The adverse effects of Q on Tf can be reduced by demagnetizing rapidly but then the problem of eddy current heating, proportional to 6, might become serious in a metallic nuclear stage, even if it is made of thin wires. The effect of heat leaks during demagnetization can be calculated from the law of energy conservation, viz.
where Q = Q ( t ) includes all sources of heat, and At now is the length of demagnetization. A convenient measure of irreversibility is A(B/T) = (Bj/Tj)-(Bf/Tf), which in the ideal case is equal to zero. We find
A ( B / T )= r[Q(t)/nA*B(r)]dr.
(13)
228
K. A N D R E AND O.V. LOUNASMAA
For Q =constant and a(t)= constant eq. (13) reduces to
A ( B / T )= (Q/nA*b)ln(Bi/Bf).
(14)
2.2. Cooling of conduction electrons The speed with which nuclei reach local termal equilibrium among , spin-spin relaxation time (80 ms for themselves is characterized by T ~ the copper). The rate at which equilibrium is established between nuclear spins and conduction electrons is governed by T ~ the , spin-lattice relaxation time defined by d(T;l')/dt = - T ; ' ( T ; ~- T i ' ) ;
(15)
here T, is the nuclear spin temperature and it is assumed in this definition that the conduction electron temperature T, is kept constant. At low temperatures T ~ < < T which ~, is the necessary condition for the existence of a separate nuclear spin temperature. For most metals T~ is of the order of minutes at 1 mK, whereas for insulators the spin-lattice relaxation time is days or even weeks at this temperature. Metals must thus be used for brute force nuclear refrigeration; it would otherwise be difficult to remove the large heat of magnetization [cf. eq. (7)] during pre-cooling. The lattice heat capacity, proportional to T 3 , is totally negligible below 10 mK. We may, therefore, assume that conduction electrons and the lattice are in thermal equilibrium at all times. The small value of T~ in metals is due to conduction electrons which act as intermediaries between nuclear spins and the lattice. The energy transfer proceeds via the hyperfine interaction between the magnetic moments of conduction electrons and nuclei. Only electrons near the top of the Fermi distribution contribute. Their number is proportional to T,, is proportional to We.We thus have Korringa's relation that is, T ~ =TK . ~
(16)
The constant K is 1.1 s K for copper (0.4 s K in zero field), 0.086 s K for indium, and 0.030 s K for platinum. After combining eqs. (15) and (16) we find
The validity of Korringa's law at high polarizations has been discussed by Jauho and Pirila (1970) and by Bacon et al. (1972).
RECENT PROGRESS IN NUCLEAR COOLING
229
Owing to the external heat leak, which is first absorbed by the conduction electrons, T, will adjust itself to such a value that cooling due to the demagnetized nuclei just balances the external heat leak. If 0 is large, T, - T, will also be large and the conduction electron temperature will be relatively high. In order to determine T , -T , w e must calculate the rate Q,= (dO,,/dT,)(dT,,/dt) = CBTnat which the nuclear spins can absorb heat from the conduction electrons. By inserting C, and f,,from eqs. ( 6 ) and (17), respectively, we obtain
This equation gives the cooling power of nuclear spins. The relationship between T, and T,, is now obtained simply by equating Q,, with the external heat leak Q. We then find T,/T, = 1-+ K Q / n A * B : .
(19)
In small external fields, B: in eqs. (18) and (19) must again be replaced by B: + b2. We notice that T, >> T, if demagnetization is carried out to a low field. Adiabatic demagnetization of CMN can be extended to Bf=O because in this case the electronic dipolar field is sufficiently strong to provide a large heat sink in the salt. The optimal field &(opt) at which demagnetization must be stopped for reaching the lowest conduction electron temperature can be calculated from eqs. (8) and (19) by writing dT,/dBf = 0; this yields Bf(0pt)= [ ( K d / n A * ) ] ” 2 .
(20)
By inserting B,(opt) back into eq. (19) we find T,(min)=2Tn in the optimal case. The conduction electron and nuclear spin temperatures are thus, even in this case, quite different. The thermal resistance R,, between the nuclear spins and conduction electrons, proportional to 71 and thus inversely proportional to T,, can be calculated from eq. (18). We find
R,, = AT/Q = KTT/nA*B:T,,
(21)
where AT = T,- T,, and eq. (8) has been used again. For improving nuclear refrigerators it is, first of all, possible to change the “external” design parameters Bi, Ti, and n. With modern superconducting wires one can increase the initial magnetic field perhaps to 1 2 T
230
K . ANDRES A N D O.V. LOUNASMAA
but such magnets are expensive. A simple way to enhance the field by 20% is to operate the magnet at 2 K. This requires pumping of the main 4He bath, which should be carried out through a tube terminating near the top of the magnet. In this way only that part of liquid 4He which is in contact with the magnet is below the A-point; the surface of the bath remains near 4.2 K and at atmospheric pressure. This arrangement considerably reduces consumption of the bath liquid by evaporation. With present day dilution refrigerators (cf. Lounasmaa, 1979) T, can be decreased considerably below 10mK. However, in doing so two serious problems are encountered, both of which are connected with the relatively large heat of magnetization [cf. eq. (711 that must be removed before demagnetization. First, the refrigerating power of all dilution refrigerators decreases as TZand, second, the Kapitza thermal boundary resistance between liquid ’He and the mixing chamber walls increases as l/T3 when the temperature is lowered; we do not benefit here from the magnetic boundary coupling because all surfaces inside the mixing chamber are covered with a layer of nonmagnetic 4He. To enhance the entropy available for nuclear refrigeration it is best simply to increase n, that is, to make a larger nuclear stage. The heat of magnetization is, of course, proportional to n, but the dilution refrigerator can now operate at a higher temperature where its cooling power is adequate and where the Kapitza resistance is not prohibitive. The only serious drawback with a larger nuclear stage is that the size of the magnet increases as well. The performance of nuclear refrigerators can also be improved by changing the “internal” parameters A* and K . EQs (18)-(20) show that a large Curie constant and a small Korringa constant are advantageous. So far, however, copper has been used almost exclusively for brute force nuclear demagnetization, and it is easy to see why: this metal is cheap and readily available in suitable form. The thermal conductivity of copper is high and can be controlled over rather wide limits by an appropriate heat treatment of the wires. By varying the final field after demagnetization, a large range of temperatures becomes accessible very easily. The only other metal used so far for brute force nuclear cooling experiments is indium (Symko, 1969), which has a Korringa constant 10 times smaller than that of copper and a Curie constant twice larger. But these advantages by no means offset the practical difficulties involved in handling indium wires. Furthermore, because of the large quadrupolar field of indium, equivalent to b = 250 mT, the lowest attainable nuclear spin temperature for this metal is about 0.5 mK. With indium B, must
RECENT PROGRESS IN NUCLEAR COOLING
231
also be larger than the superconducting critical field, B, = 30 mT, because T , is very long in the superconducting state. Sometimes time-dependent heat leaks have been seen in nuclear refrigerators (Pobell, 1982). At least in some cases these appear to be due to proton relaxation in insulating materials employed in the construction of nuclear stages or experimental cells. A heat leak decaying with a time constant of several days is produced by this process. Internal generation of heat may also occur owing to the release of mechanical strains. The entropy of conduction electrons ( S , = 8.4 X lO-'nRT mol-' K-'for copper) can be ignored in comparison with the nuclear spin entropy; the heat load due to conduction electrons is thus so small that it has no effect in the derivation of eqs. (8) and (9). 2.3. Cooling of 'He
Cooling of liquid 3He below 1 mK has been, so far, the most important application of nuclear refrigeration. We shall, therefore, examine this special case in some detail. Fig. 1 is a much simplified schematic drawing of a nuclear refrigerator intended for research on liquid 3He. The whole assembly may be separated into three thermal reservoirs: the nuclear spin system, conduction electrons, and 3He. Because of heat leaks and thermal barriers, each subsystem will reach, in dynamic equilibrium, a different temperature: T,, T, and T3, respectively. The entropy of 3He (S3=2.11nRTmol-' K-' for liquid at zero pressure) can be an important fraction of the available nuclear spin entropy. S3 must then be taken into account in the derivation of eqs. (8) and (9); usually, however, T, is affected by less than 5 % . The experimental cell itself may also be a relatively large heat load if it is made of copper and exposed to a moderate magnetic field needed, for example, to carry out NMR measurements. A silver cell is preferable because the nuclear heat capacity of this metal is negligible. We have already discussed R,,, the thermal resistance between conduction electrons and nuclei [cf. eq. (21)]. Heat must also be transferred along the nuclear stage to cool the 'He cell. The relevant heat resistance Re, determined by the electronic thermal conductivity of the material of which the nuclear stage is constructed, is proportional to UTe. We have
R, = pl/.rrr2Te,
(22)
232
K . ANDRES AND O.V. LOUNASMAA
CONDUCTION ELECTRONS
NUCLEAR SPINS
RI
a ,rT,
T,
Fig. 1. Schematic drawing of a nuclear refrigerator for cooling liquid 3He.
where p, the thermal resistivity at 1 K, depends strongly on impurities: I is the length of the wire and r its radius. A good idea of the conductivity of a piece of metal intended for the nuclear stage may be obtained by measuring the electrical resistance at 300 K and at 4.2 I(;a ratio RRR between 400 and 1000, corresponding to p around 1000K2m/W according to the Wiedemann-Franz law, is often suitable when using copper. If the conductivity is very high excessive eddy current heating might occur during demagnetization. In actual experiments a temperature difference TJtopf - T,(bottom) will appear between the top and bottom parts of the nuclear stage. For a proper analysis of nuclear refrigerators intended for cooling 'He it is necessary to divide the heat leak Q into two parts: Qe, caused mainly by heat flow through mechanical supports, by vibrations, and by electrical disturbances, enters directly into the conduction electron system. Q3, mainly due to heat conduction along the liquid 3He column in the filling tube, electrical disturbances, and acoustical vibrations in the liquid, enters the 3He specimen and is then passed on to the conduction electron
RECENT PROGRESS IN NUCLEAR COOLING
233
system. The thermal bamer is the Kapitza boundary resistance between the cell walls and liquid 3He, defined by R K
= ATIQ3;
(23)
here AT is the temperature drop. T3 adjusts itself to such a value that heat flow through the boundary just balances Q3, that is
AT = T3 - T, = R K 43.
(24)
It should be noted that the heat conductivity of liquid 3He, typically 0.4/T3mW/m, approaches that of copper in the sub-millikelvin region so that the filling capillary tube should be thin (diameter <0.2mm) and long (about 100cm). Ultimately the total heat leak Q = Q e + Q s must be absorbed by the nuclear stage. Q. (24) shows that for reaching a low 3He temperature, T, must be low and the temperature drop T3- T, should be small. In order to satisfy the latter condition RK and Q3 should be reduced as much as possible. Decreasing the heat leak is difficult because frequently we do not know where Q3 comes from. Assuming that the heat transfer at the interface occurs via phonons only, we would expect a 1/T3 temperature dependence for RK. The boundary resistance below 1 mK would then be prohibitively high. Fortunately, the actual situation is much better. An additional thermal coupling mechanism, important below 5 m K , has been found between liquid 3He and several metals. It is assumed (Leggett and Vuorio, 1970) that this extra conductance of heat is due to a dipolar interaction between the nuclear spins of 3He and some localized electronic impurity moments on the surface of the metal. The boundary resistance then becomes proportional to 1 / T (cf. Lounasmaa, 1978), which is much better, and we find
R K= ~ K / A T ~ ,
(25)
where a Kis a constant which depends on the properties of the surface A between the metal and liquid 3He. A typical value for aK is 1000 m2 Kz/W. To make the boundary resistance sufficiently small it is necessary t o increase A greatly. For this reason copper or silver sinter, made of 1 p m diameter or smaller powder, is put into the experimental cell. The surface area can be increased to several mz per 1cm3 of the liquid; for details of suitable powders and of the appropriate heat treatment we refer to Frossati (1978). The sinter must be in good metallic contact with the nuclear stage.
234
K . ANDRES AND O.V. LOUNASMAA
All thermal resistances between 'He and the demagnetized nuclei (cf. fig. 1) thus have a 1/T temperature dependence, which simplifies the mathematical simulation of nuclear refrigerators intended for cooling 'He. Since the heat capacity of 3He in the Fermi liquid region is proportional to T, the thermal time constant RC, should be independent of temperature. In the superfluid phases C, decreases more rapidly with T so the situation, from this point of view, should be quite favorable for rapid cooling of ,He well below 1 mK. In order to reach the lowest possible ,He temperature demagnetization must usually be stopped at a field which is considerably higher than Bf(opt) given earlier by eq. (20). There is no point in reducing Bf further once there is no significant decrease of T,.In fact, the situation may again be optimized as soon as Qe, Q3, R,, Re, and RK are known. There are two basically different schemes for constructing an apparatus for nuclear refrigeration of liquid ,He. Either the coolant can be put, in powder form, directly into the 3He chamber, or it can be located outside the cell. The latter approach has been employed in most nuclear refrigerators which have been built so far. This arrangement gives a great deal of experimental freedom, especially when the properties of ,He are investigated as a function of the external magnetic field. Guenault and Pickett (1981) have refrigerated superfluid 'He well below 0.5 mK by mixing the coolant with the liquid. Mixing the coolant directly with ,He is a simpler arrangement, always employed when 'He is refrigerated by CMN.With this construction the cryostat becomes shorter and there are no problems with heat transfer along the nuclear stage. The drawbacks of this scheme are that the amount of refrigerant in relation to ,He is rather small and that metal powders cannot be used without insulation because of eddy current heating during demagnetization. Heavily oxidized copper powder would probably work. Other problems are that the heat of magnetization and the entire external heat leak will pass through the ,He-rnetal boundary; long pre-cooling times and a large temperature drop T3-Te are thus to be expected. We should emphasize the different roles played by Qe and d,:Qe, which is usually the larger of the two, mainly determines the time during which the apparatus stays cold whereas 6,determines, more than anything else, the temperature of ,He. Both these heat leaks are reduced if the dilution refrigerator is capable of cooling itself, after the nuclear stages have been thermally decoupled from it around 15mK, t o a lower temperature, possibly in the neighborhood of 4-5 mK.
RECENT PROGRESS IN NUCLEAR COOLING
235
2.4. Description of cryostats
In this section we shall briefly describe three successful nuclear refrigerators employed for cooling superfluid 'He below 1 mK. Some important properties of several more cryostats will be presented in tabular form. Details on ail these refrigerators can be found in the original publications cited herein. We shall not give a historical review except to mention that nuclear refrigeration was first successfully employed for cooling liquid 3He in 1973 by Dundon et al. (1973)and by Ahonen e t al.
(1974,1976). 2.4.1. The Orsay machine
Fig. 2 is a photograph of the cryostat built at Orsay near Paris (Avenel et al., 1976).The refrigerator became operational towards the end of 1976. The cryostat is supported by three rather massive legs which are isolated from the floor by means of automobile tyre inner tubes. The 4He dewar is 30 cm in diameter and about 2 m long. Pre-cooling is done by a dilution refrigerator which reaches 18mK in about 2 4 h with the 8 T superconducting magnet on. The nuclear stage was made of 45 mol of 0.5mm diameter copper wires with double fiber glass insulation and with an electrical resistivity ratio of 700. The wires are thus quite thick and their low temperature thermal conductivity is high. The nuclear stage was covered by silvered mylar and it extends about 40cm below the 'He cell. The construction of the nuclear stage, shown in fig. 3, is interesting: it was made of 19 wire bundles, each containing 271 copper wires closely packed and sintered together just below the 'He chamber. 61 of these wires per bundle enter the 3He cell with fine, 1 p m diameter copper powder sintered onto them in the form of a pill, 7 mm high and 9 mm in diameter. The total surface area of the 19 pills is calculated to be about 30 m2. Eight wires of each bundle are connected to the superconducting heat switch which provides the thermal path between the mixing chamber and the nuclear stage. The experimental cell is moulded out of Stycast and the volume for 3He is 9 cm'. The tube for the incoming 'He has a 0.2 mm inner diameter and it is 1.2m long. An epoxy tower, attached to the cell, contains three N M R coils for measurements on liquid 'He and one coil for N M R on
236
K. ANDRES AND O.V. LOUNASMAA
Fig. 2. Photograph of the Orsay machine (Avenel et al., 1976).
RECENT PROGRESS IN NUCLEAR COOLING
237
8 Cu WIRES TO HEAT SWITCH
Fig. 3. Constructionof the nuclear sub-bundles in the Orsay machine.
platinum. The tower is fixed to the main cell simply by a large thread made vacuum tight with a solution of soap and glycerol. The lowest 'He temperature reached in this apparatus is 0.31mK measured by means of a platinum pulsed N M R thermometer; the platinum powder is in thermal contact with liquid 'He only. The total measured heat resistance, R,,+ Re+RK, is rather low, 50/TK2/W. The total heat leak to the nuclear stage is 4nW, with de= 3 nW and Q3 = 1 nW. The time constant for equilibrium between the nuclear stage and liquid 'He is 30 min at 1mK, but becomes rapidly as long as several hours when the temperature is lowered further. There is some evidence for a nuclear heat capacity of the epoxy cell, caused by unbound protons or by impurities. The apparatus has kept a 'He sample near 0.5 mK for 8 h, below 1mK for 48 h, and below 3 mK for more than five days. 2.4.2. The Otaniemi double-bundle cryosrat
This refrigerator, which is described in detail by Veuro (1978), was commissioned in May 1977. In designing and building their apparatus the
238
K. ANDRES A N D O.V. LOUNASMAA
Otaniemi group could benefit from long experience with an older cryostat (Ahonen et al., 1976). Certain innovations were incorporated in the new refrigerator, the most important ones being a double-bundle nuclear stage and a silver 3He cell. The main aim was to reduce 4, by lowering the temperature of the surroundings. Fig. 4 is a drawing of the apparatus. Pre-cooling of the nuclear stage is done by means of a standard SHE dilution refrigerator. Three 10 mm thick and 5 cm long graphite rods, fixed to the mixing chamber, support a 18 cm long silver frame. This structure, in turn, supports the 29cm long nuclear stages and the 'He cell. Heat contact to the mixing chamber is via a tin heat switch. The 'He input tube of 0.1 mm inside diameter is coiled over a distance of about 1 m to make it sufficiently long so that 6,is reduced. The two nuclear stages were made of enamel insulated copper wires of 0.2mm diameter. The outer nuclear stage (15 mol) is annular in shape and it is attached, by means of a pressure contact, to the silver support frame. The inner stage (10 mol) is cylindrical. The purity of the wires is 99.999% and the residual resistance ratio RRR is 220. The nuclear stages are thermally connected to each other at their top ends by a lead heat switch, operated by the fringe field of the main magnet. All leads and tubes are carefully thermally anchored to the outer nuclear stage before they proceed further. The idea of employing two nuclear stages was borrowed from the old demagnetization cryostats which frequently used a main salt and a guard pill. In this way the heat leak is rather effectively prevented from reaching the inner parts of the apparatus. The cryostat works as follows. The dilution refrigerator first cools the nuclear stages, in an applied field Bi = 7 T, to Ti= 18 mK in 50 h. Demagnetization to Bf = 50 mT is then performed in about 3 h; when the field is 1.8T in the main solenoid, corresponding to a fringe field of 80 mT near the top of the copper bundles, the lead heat switch becomes superconducting whereby thermal contact between the two nuclear stages is broken. Until 0.5 mK the time constant of the apparatus is very short: the 3He specimen lags behind the nuclear stage by only a few minutes. Below OSmK, however, cooling proceeds slowly and the lowest liquid 3He temperature, 0.38 mK, is reached after 10 h. The apparatus stays below 0.5 mK for 30 h and below 1 mK for 52 h. Heat leaks and thermal resistances were studied in this apparatus rather carefully, with the following result: Qe = 0.9 nW, 6,= 0.2 nW, Re= 30/T K2/W, RK = 90/T Kz/W. Therefore, at T3= 0.40 mK, T3- T, = 45 p K [cf. eq. (24)]. There is a further temperature drop between the
Fig. 4. The Otaniemi double-bundle cryostat (Veuro, 1978).
K . ANDRES AND O.V. LOUNASMAA
240
EXCITATION A N D GRI\MENT COILS
ERtU!UETER COIL
QRAPHtTE SUPPORT
1 em
Fig. 5. The heat capacity cell employed recently in the Olankmi double-bundle refrigerator (Alvesalo et al., 1981).
sinter and the bottom of the nuclear stage; assuming that Qeenters at the top we find T,(top)-T,(bottom)= 100 p K . A calculation [cf. eq. (19), with K = 0.4 s K, n = 10 rnol, A* = 3.2X m4A K/V s moll shows that T,(bottom) - T, is small. We thus have T, = 0.25mK. Eq. (8), which does not take heat leaks into account, gives T,= 0.13rnK. It is thus clear that the main bottleneck in this cryostat is the electronic thermal conductivity of the silver and copper parts. A cell, constructed for measurements of heat capacity of normal and superfluid 3He in this apparatus (Alvesalo et al., 1981), is illustrated in fig. 5 . It was made of silver and connected to the inner nuclear stage via a
RECENT PROGRESS IN NUCLEAR COOLING
24 1
superconducting heat switch. The volume for 3He in the cell is 17 cm3 and the surface area of the silver sinter is lornZ. Two NMR thermometers, which employ the nuclear susceptibility of platinum and the electronic susceptibility of lanthanum diluted CMN ( = CLMN), were installed in the cell. 2.4.3. The cryostat at Bell Laboratories This apparatus, constructed by Osheroff and Sprenger (1980), is illustrated schematically in fig. 6 ; the cryostat has cooled liquid 3He to 0.22 mK. The commerical dilution refrigerator, manufactured by the SHE Corporation, has a cooling capacity of 45 p W at 100 mK. The minimum temperature reached in the mixing chamber is 10mK unloaded and 12 mK with both heat shields and all electrical leads attached. The mixer, equipped with a silver thermal exchanger of 1 2 m Z surface area, has an open bore on its axis; a tin heat switch is mounted inside. Contact to the nuclear stage is made through this switch and via a 3 mm diameter and 50 cm long silver rod. The copper bundle can be pre-cooled to 16 mK in about 60 h with the magnetic field on. The nuclear stage itself consists of 20000 insulated copper wires, 0.2 mm in diameter and 25 cm long; after annealing, the electrical resistivity ratio was about 400. The wires were Heliarc welded to a flange which fits into the lowest of three demountable joints working as a compressional collar. The joints are similar to those described by Boughton et al. (1967) except that the nylon ring was replaced by a BeCu ring and six pie-shaped tungsten spacers. The total electrical resistance of a demountable joint is about 15ndl. The nuclear bundle and the demagnetizing solenoid are positioned so that 15mol of copper is in a root mean square field of 6.7T. The two lowest demountable joints (d.fig. 6) are connected to each other by means of twelve thin silver plates, welded into appropriate flanges. The total electrical resistance between the 3He sample chamber and the nuclear bundle is 1 5 0 n 0 , about half of which is coming from welded joints. The long distance, about 40 cm,between the sample chamber and the nuclear stage allows the use of coaxial superconducting solenoids. At the site of the 3He specimen the NMR magnet will achieve a homogeneity of 1ppm over a 1cm3 sphere. The fields of the demagnetizing and NMR
242
K. ANDRES AND O.V. LOUNASMAA .DEMOUNTABLE JOINT -COLD FINGERS MIXING CHAMBER
PHENOL ISOLATION RODS
'NMR SOLENOID
3He SAMPLE CHAMBER DEMOUNTABLE JOINT GRAPHITE FLANGE HEAT SHIELS 3 m m SILVER ROD VACUUM JACKET -GRAPHITE FLANGE JOINT
. DEMOUNTABLE
-'He BATH
-DEMAGNETIZING SOLENOID -COPPER NUCLEAR STAGE
Fig. 6. The cryostat at Bell Laboratories (Osheroff and Sprenger, 1980).
solenoids interact slightly, with a maximum shift of 3 mT when the N M R magnet is in persistent mode, and a gradient of about 0.2mT/cm is present in the vertical direction at full demagnetization field. This gradient can be almost completely eliminated with a room temperature trim coil. In actual experiments, homogeneities of 2 ppm are typically achieved over a 1an3 sphere.
RECENT PROGRESS IN NUCLEAR COOLING
243
For their measurements on ’He Osheroff and his co-workers have employed five different sample chambers, each plugged into the cryostat by means of the middle demountable joint. Each 3He cell typically has a sintered silver heat exchanger of 100m2 surface area, with an effective particle size of 100nm and 40% packing factor, and constructed in a manner developed by Sprenger and Paalanen (1980). In addition, 1.3mm diameter silver rods are used for carrying heat through the sintered material and for thermal contact. The heat resistance between the sinter and the nuclear bundle is low, R,=7/TK2/W. The thermal boundary resistance is also small, typically RK= 7/T K’IW. The whole machine is vibration isolated from the surroundings by four bellows, 15cm long and 3.5cm in diameter, and pressurized on the outside with air. The cryostat itself has a very low rigidity; it will swing almost as a pendulum when the heat shields, attached to the 1 K plate and the mixing chamber, respectively, are demounted. The shields are spaced by a system of stretched and crossed silk threads at the bottom; in the bore of the demagnetizing solenoid the gap between them is only 1 mm. The low temperature portion of the cryostat is drawn rigidly to the mixing chamber heat shield using cotton threads and graphite spacers, in this way the cryostat is immune to low level vibrations but it will warm appreciably when liquid nitrogen is being transferred to the outer dewar. The stray heat input to the cryostat is low, Q,+ Q3 = 1 nW. The heat leak, however, is time dependent, varying as llr’; 1 0 n W is reached 7 days after demagnetization. The reason for the time dependency can possibly be traced to relaxation of mechanical strains in the two lowest demountable joints. The success of this refrigerator, cooling liquid 3He to 0.22mK, is evidently due to the low heat resistance between liquid ’He and the nuclear stage and to the small external heat leak. Incidentally, 0.22 mK is the low temperature record for cooling liquid 3He; it was obtained in one of the sample chambers used in this apparatus (Osheroff and Yu,1980).
2.4.4. Orher nuclear refrigerators for cooling ’He Many other nuclear refrigerators for studies of superfluid 3He have been constructed during recent years. Important properties of several of these machines are listed in table 1. For detailed information, we refer the reader to original publications cited therein.
Table 1 Examples of copper nuclear refrigerators for cooling liquid 3He Bell Lpbs ((Xheroff Sprengcr.
Berkeley (Eiscmtein Cornell et al., (Archie.
1980)
1979)
and
Nuclear stage n (moo Dim.of w i n s (mm)
Ti (mK)
B,(rms value, T) B, (nns value. mT) T,(minKmK) 6 (nW)
6,(nW RUT,(K’I\K) R T . (K2/W) Warm-up to 2 mK 01)
1978)
8 0.6
8
1
50 0.48 1.5
28 0.34 Cl
8 100 0.4 100 0.5 21 4 30
9 0.23 19
20
-
(Frossati, 1978)
21 0.5 19
15 0.2 16 6.7
0.22 0.4 1 7 7
GrcnoMe
2 50 0.4 150
7
ohiostatc (Muething
etal..
Los Angcles North(Bozlcr western etal.. (Mast ct al..
1978)
1978)
La Jolla (Knsius
17.5 0.25 14 5.5 1.5
14 0.20 16 8 50 0.35 1
-
1980)
19 0.34 20 6.4 50 0.39 1.5
-
40 30
-
>loo
100 17 cu 1000 35
Oraniemi Sussex
PI..
orsay (Avenel et al.,
(Vcuro.
(Hutchins.
1979)
1976)
1978)
1981)
et
40 0.18 27 7.7
0.4 5 0.3 60 5
45 0.5 12 8 30 0.31 3 1
I
>loo
25 0.2 18 5.3 30 0.38 0.9 0.2 90 30 100
15
0.5 20 8 80 0.7
} 3.5
’Heecll Amount of ’He (cm’) Type of sintcr Panick s k (nm) Surfam area (m’)
18
4
15
25
8.9
11
Ag
Ag
cu
As
cu
cu
100 100
10 40
30 107
40 60
flakes 215
flakes 45
16.7 Pd 2000 25
9
17
in
Cu’
43
cu
2000 30
100 10
loo0 8
RECENT PROGRESS IN NUCLEAR COOLING
245
3. Hyperline enhanced nudear cooling 3.1. Introduction The first experiments on hyperfine enhanced nuclear cooling followed the Knight shift measurements in thulium and praseodymium pnictides by Jones (1967). He showed that, owing to the combined effects of the hyperfine interaction and the high van Vleck polarizability of the praseodymium and thulium ions, the nuclear Zeeman splitting was much enhanced, typically by a factor from 10 to 100. The application of this effect for nuclear cooling purposes seemed very promising especially because the antimonides and bimuthides of praseodymium and thulium remain metallic down to T = 0 K, which thus ensures good nuclear spin-lattice contact even at low temperatures (cf. section 3.2.3). Indeed, large magnetocaloric effects were immediately obvious in the first experiments by Andres (1967), but thermodynamic irreversibilities due to phase impurities in the specimens prevented low end temperatures at first. Subsequent research showed that these harmful effects could be minimized in van Vleck paramagnetic intermetallic praseodymium compounds; the first successful nuclear cooling experiments on PrPt, and PrT13 were reported two years later by Andres and Bucher (1968). Soon after the 1967 experiments it was realized that the observed hyperfine enhancement effect was the same as the one predicted by Al’tshuler (1966) a year earlier to occur in various hydrated salts of the non-Kramers rare earth ions, such as thulium ethyl sulphate. Al’tshuler showed that the admixing ‘of the electronic momentum to the nuclear substates in the nonmagnetic singlet ground state by the hypefine coupling, which in principle is a second order effect, can be large in these salts due to the small crystal field splitting of the rare earth ion. He pointed out the potential usefulness of this effect for magnetic cooling experiments since the rare earth nuclei in these salts act like a paramagnetic system of small electronic moments with a low ordering temperature. Al’tshuler also argued that the spin-lattice relaxation time in these insulating salts could be shortened by the inclusion of a small concentration of paramagnetic Kramers ions. Because of the much higher thermal conductivity of metallic van Vleck paramagnetic materials, which is of prime importance for nuclear refrigeration purposes, research in this field has so far mainly concentrated on
246
K . ANDRES A N D O.V. LOUNASMAA
intermetallic compounds of praseodymium. In this review we first discuss, in some detail, the physics of hyperfine interactions in singlet ground state systems, both with and without exchange interactions between the rare earth ions. We then describe the high field behavior and the thermodynamics of hyperfine enhanced cooling. In the experimental sections we discuss materials preparation techniques and describe the performance of some hyperfine enhanced nuclear cooling cryostats.
3.2. Hyperfine interactions in singlet ground state systems 3.2.I . Single ion properties In atoms or ions with unfilled shells, the orbital and spin angular momenta of the electrons combine into multiplet states which are characterized by a net spin quantum number S, an orbital quantum number L, and a resulting total angular momentum quantum number J . In accordance with Hund’s rules, the multiplet of lowest energy is given by J = L f S, depending on whether the shell is less (-) or more (+) than half full. In a crystal the ion experiences interactions with its neighbors, which result in a partial or complete lifting of the ( 2 J + 1)-fold magnetic degeneracy of the ground state. We restrict our discussion to rare earth ions, in which the direct 4f wave function overlap is small and the largest perturbations affecting the 4f electrons are electrostatic fields, both from neighboring ionic charges and from the also disturbed 5d electron on the same ion, the latter being responsible also for most of the 4f-5d exchange interaction. These perturbing energies are of order 0.1 eV or less and are smaller than the intraionic spin-rbit interaction among the 4f electrons. Therefore, J remains a good quantum number, while J, does not. The crystal electric field then changes the degenerate free ion eigenfunctions (J, J , ) into new eigenfunctions, which can be written as linear combinations of (J, J , ) , and partly lifts their degeneracy. The possible residual degeneracies in the new crystal field states can be predicted by symmetry considerations alone (cf.,e.g. Fick and Joos, 1957). Actual calculations on the crystal field states and energies can be done using the elegant operator equivalent method developed by Stevens (1952), in which the crystal field Hamiltonian is rewritten in terms of so-called operator equivalents. These are combinations of the angular
RECENT PROGRESS IN NUCLEAR COOLING
241
momentum operators J, J,, J,, and J,, which have the property of remaining invariant under the symmetry operations of the crystalline field in quest ion. According to Kramer's theorem, singlet states are only possible for even values of J . For crystal fields of orthorhombic or lower symmetries, no degeneracies remain and all states are singlets. A general property of singlet states is that they must contain the free ion components (J, J , ) and ( J , J - , > 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.
K. ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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
K. ANDRES AND O.V. LOUNASMAA
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-
RECENT PROGRESS IN NUCLEAR COOLING
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
-
RECENT PROGRESS IN NUCLEAR COOLING
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 << 1, T, is a linear function of q and A, increasing faster towards q = 1 and finally joining the induced moment transition curve (6. the schematic dashed line in fig. 7). In the near critical regime (qS 1) the transition to the ordered state can be described as a nuclear-induced electronic transition. In this case, eq. (47) loses its validity and T, must be determined by finding solutions of the molecular field Hamiltonian with finite expectation values of the angular momentum ( J ) ; this amounts to solving a self-consistent equation. The first calculations of this kind were done by Murao (1971, 1972), who finds that for q 4 1 the induced moment below T, is small and rises to a considerably larger value only at much lower temperatures. We interpret the small induced moment below T, as a “van Vleck moment” (J), brought about mostly by exchange interactions, and its rise at lower temperatures as being due to the gradual alignment of the hyperfine induced moments Jf. For such nuclear-induced electronic transitions, the induced 4f moment Jf [eqs. (35) and (36)] satisfies the condition
kT, > 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
RECENT PROGRESS IN NUCLEAR COOLING
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
K. ANDRES AND O.V. LOUNASMAA
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.
RECENT PROGRESS IN NUCLEAR COOLING
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
K. ANDRES AND O.V. LOUNASMAA
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-
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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
K. ANDRES AND O.V. LOUNASMAA
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.
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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).
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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
K. ANDRES AND O.V. LOUNASMAA
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).
RECENT PROGRESS IN NUCLEAR COOLING
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).
RECENT PROGRESS IN NUCLEAR COOLING
Fig. 20. View of the PrNi, cooling stage of the Jiilich refrigerator.
279
280
K. ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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).
K. ANDRES AND O.V. LOUNASMAA
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).
RECENT PROGRESS IN NUCLEAR COOLING
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
K . ANDRES AND O.V. LOUNASMAA
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
RECENT PROGRESS IN NUCLEAR COOLING
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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(Roc. HakonC Symposium, Phys. Soc.Japan) p. 287. Hutchins, J. (198l), private communication. Jauho, P. and P.V. Pirila (1970)Phys. Rev. B1, 21. Jones. E.D. (1967)Phys. Rev. Lett. 19, 432. Kaplan, N., D.L. Williams, and A. Grayevsky (1980)Phys. Rev. B2l, 899. Krusius. M., D.N. Paulson and J.C. Wheatley (1978).Cryogenics 18, 649. Kubota. M.. H.R. Folle, C. Buchal, R.M. Mueller and F. Pobell (1980)Phys. Rev. Lett. 45, 1812.
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.
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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
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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
