PROGRESS IN LOW T E M P E R A T U R E PHYSICS V
CONTENTS O F VOLUMES I - I V
VOLUME I
c. J.
The two fluid model for superconductors and helium 11 (16 pages)
GORTER,
R. P. FEYNMAN,
I. R. PELLAM, A.
Application of quantum mechanics to liquid helium (37 pages)
Rayleigh disks in liquid helium I1 (10 pages)
c. HOLLIS HALLET, Oscillating disks and rotating cylinders in liquid helium 11 (14 pages)
E. F. HAMMEL,
The low temperature properties of helium three (30 pages) and K. w. TACONIS, Liquid mixtures of helium three and four (30 pages)
J. I. M. BEENAKKER
B. SERIN,
c. F.
The magnetic threshold curve of superconductors (13 pages)
SQUIRE,
The effect of pressure and of stress on superconductivity (8 pages)
T. E. FABER and A. B. PIPPARD,
Heat conduction in superconductors (1 8 pages)
K. MENDELSSOHN, J. G : DAUNT, A.
H. COOKE,
The electronic specific heats in metals (22 pages) Paramagnetic crystals in use for low temperature research (21 pages)
N. I. POULIS and
D. DE KLERK
Kinetics of the phase transition in superconductors (25 pages)
c. I. GORTER, Antiferromagnetic crystals (28 pages)
and M.
J. STEENLAND, Adiabatic
demagnetization (63 pages)
L. N&L,
Theoretical remarks on ferromagnetism at low temperatures (8 pages)
L. WEIL,
Experimental research on ferromagnetism at very low temperatures (11 pages)
A. VAN ITTERBEEK,
J. DE BOER,
Velocity and attenuation of sound at low temperatures (26 pages)
Transport properties of gaseous helium at low temperatures (26 pages)
VOLUME I1 Quantum effects and exchange effects on the thermodynamic properties of liquid helium (58 pages)
J. DE BOER,
H.
c. KRAMERS, Liquid helium below 1 OK (24 pages) and D. H. N. WANSINK, Transport phenomena of liquid helium 11 in slits and capillaries (22 pages)
P. WINKEL
K. R. ATKINS,
Helium films (33 pages)
B. T. MATTHIAS,
Superconductivity in the periodic system (13 pages)
CONTENTS OF VOLUMES I-IV
VOLUME 11 (continued)
Electron transport phenomena in metals (36 pages)
E. H. SONDHEUIER.
v.
A. JOHNSON
and K.
Semiconductors at low temperatures (39 pages)
De Haas-van Alphen effect (40 pages)
D. SHOENBERG, The
c. J.
LARK-HOROVITZ,
Paramagnetic relaxation (26 pages)
GORTER,
and (46 pages)
M. J. STEENLAND
c. DOMB and J. s. F. H. SPEDDING,
H. A. TOLHOEK,
DUGDALE,
Orientation of atomic nuclei at low temperatures
Solid helium (30 pages)
s. LEGVOLD, A.
H. DAANE
and L.
D. IENNINGS,
Some physical properties of
the fare earth metals (27 pages) The representation of specific heat and thermal expansion data of simple solids (36 pages)
D. BIJL,
and M.
H. VAN DIIK
DURIEUX,
The temperature scale in the liquid helium region (34 pages)
VOLUME I11 w. F.
VMEN,
G. CARERI,
Vortex lines in liquid helium I1 (57 pages)
Helium ions in liquid helium I1 (22 pages) and w.
M. I. BUCKINGHAM
M. PAIRBANK,
The nature of the L-transition in liquid helium
(33 pages) E. R. GRILLY K.
and E. F.
HAMMEL,
Liquid and solid 3He (40pages)
w. TACONIS, aHe cryostats (17 pages)
I. BARDEEN
and J.
M. YA. AZBEL’
R. SCHRIEFFER,
and I.
w. J. HUISKAMP and (63 pages)
M. LIFSHITZ,
Recent developments in superconductivity (118 pages) Electron resonances in metals (45 pages)
H. A. TOLHOEK,
N. BLOEMBERGEN, Solid
Orientation of atomic nuclei at low temperatures I1
state masers (34 pages)
The equation of state and the transport properties of the hydrogenic molecules (24 pages)
J. J. M. BEENAKKER,
z.DOKOUPIL, Some solid-gas equilibria at low temperatures (27 pages)
CONTENTS O F V O L U M E S I - I V
VOLUME IV
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. DOUGLASS JR.
and
0. J. VAN DEN BERG,
Anomalies in dilute metallic solutions of transition elements (71 pages)
KEI YOSIDA,
The superconducting energy gap (97 pages)
Magnetic structures of heavy rare earth metals (31 pages)
c. DOMB and
A. R. MIEDEMA,
L. N ~ E L ,R. PALITHENET A. ABRAGAM
L. M. FALIKOV,
and M.
J. G. COLLINS
and G.
Magnetic transitions (48 pages)
and B.
DREYFUS, The
BORGHINI,
Dynamic polarization of nuclear targets (66 pages)
K. WHITE,
T. R. ROBERTS, R. H. SHERMAN,
rare earth garnets (40 pages)
Thermal expansion of solids (30 pages)
s. G. SYDORIAK and F.
temperatures (35 pages)
G. BRICKWEDDE,
the 1962 SHe scale of
PROGRESS I N LOW
TEMPERATURE PHYSICS EDITED BY
C. J. G O R T E R Professor of Experimental Physics Director of the Kamerlingh Onnes Laboratory, Leiden
VOLUME V
1967 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
0 1967 N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y
- AMSTERDAM
No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher
Library of Congress Catalog Card Number: 55-14533
PUBLISHERS:
NORTH-HOLLAND P U B L I S H I N G CO. - A M S T E R D A M SOLE DISTRIBUTORS FOR U.S.A. AND CANADA: INTERSCIENCE PUBLISHERS, A DIVISION OF
J O H N WILEY & S O N S , I N C . - N E W Y O R K
P R I N T E D I N THE N E T H E R L A N D S
PREFACE
Looking back on five volumes of Progress in Low Temperature Physics, it seems worth comparing this latest volume with the earlier ones, and thus draw attention to the trends which emerge. It is immediately evident that chapters in the early books were shorter; then they averaged 20-25 pages, whereas the present average approaches twice this figure. To maintain a reasonable size, it has therefore been necessary to decrease the number of chapters per Volume. Thus, in the rapidly expanding field of Low Temperature Research, succeeding Volumes in this series cover an ever-decreasing fraction of the body of knowledge. Turning to the general features of the presentation, the more recent books contain relatively less text, an equivalent number of figures and tables, but more formulae and more references. This increased use of formulae is not due to an increasing representation of theoretical physicists among the authors, but rather to a tendency among the experimenters to use formulae more often in the presentation of data and in its interpretation. It is instructive to compare papers in the present Volume with those covering similar or related topics in the earlier books. Feynman’s paper in Vol. I on “Application of Quantum Mechanics to Liquid Helium” may be contrasted with Anderson’s chapter in the present Volume. Entitled “Quantum Coherence”, this stresses the relationship between superconductors and liquid helium and concentrates on important, novel aspects of “quantum fluids”. The more experimental papers in this Volume on Liquid Helium 11, presented by De Bruyn Ouboter, Taconis and Van Alphen, Andronikashvilli and Mameladze, might be compared respectively with those of Winkel and Wansink p o l . 11) and Pellam and Hollis-Hallett (Vol. I). The wealth of experimental data has increased greatly, but it is the improvement in relating theory to experiment which is quite striking; a result of developments which have taken place in the last few years, mainly in the U.S.A. and the Soviet Union. The short chapter by Cribier, Jacrot, Madhav Rao and Farnoux on the neutron diffraction analysis of niobium crystals in the superconducting
WI
PREFACE
mixed state has no direct ‘ancestor’ in this Progress Series, though suggestions of a magnetic microstructure date back to 1935. Only the paper of Faber and Pippard in Vol. I can be said to contain ideas which anticipate the rapid expansion in investigations and applications of the mixed state since 1960. Ganthmaker’s chapter and that of Azbel and Lifshtz in Vol. I11 are in several respects branches of the same young tree - electron resonance in metals. Though one might say that they scarcely belong to low temperature physics in a narrow sense of the term, the use of very low temperatures is just as essential for them as is the use of magnetic fields and the purity of the metals investigated. This applies also to the chapter of Stark and Falicov which has several links with Shoenberg’s paper on the De Haas-Van Alphen effect (Vol. 11). All of these papers include new and valuable information on Brillouin zones and Fermi surfaces. Finally, one may regard the chapter of Beenakker and Knaap as in some respects supplementary to another Leiden paper on solid-gas equilibria - that written by Dokoupil in Vol. 111. It covers, however, a much wider field of Fluid Mixtures and discusses also the links with recent theory. Several important fields in Low Temperature Physics are missing from the present Volume. Magnetism and temperatures below 0.3 OK are the most striking omissions. These gaps are closely connected with the reduction, mentioned above, in the number of chapters per volume, but are also due to some delay in the arrival of promised papers. I hope to redress the balance later, as in the case of the missing papers on Liquid Helium from Vol. IV. I want to express my thanks to the Leiden physicists and the foreign guests who by their valuable assistance made it possible to edit this Volume; particularly to Drs. K. W. Mess, who, among other things, prepared the subject index of this book, as well as that of Volume IV. C. J. GORTER
CONTENTS
Chapter
Page
I P. W. ANDERSON, THE JOSEPHSON EFFECT
AND QUANTUM COHERENCE
MEASUREMENTS IN SUPERCONDUCTORS AND SUPERFLUIDS
. . . . . . . . .
.
1
1. Historical introduction, 1. - 2. Elementary perturbation theory of the Josephson effect, 4. - 3. Coherence properties of coupled superconductors and superfluids, 11. - 4. Statics of finite tunnel junctions: magnetic interference experiments, 20. - 5. Systems other than tunnel junctions showing interference phenomena, 33. - 6. A.c. quantum interference effects, 36.
I1 R. DE BRUYN OUBOTER, K. W. TACONIS and W. M. VAN ALPHEN, DISSIPATIVE AND NON-DISSIPATIVE FLOW PHENOMENA IN SUPERFLUID HELIUM. Introduction, 44. - 1. Superfluidity, the equation of motion for the superfluid, 45. - 2. The critical superfluid transport in very narrow pores between 0.5 OK and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow, 54. - 3. Superfluid transport in the unsaturated helium film, 62. - 4. Dissipative normal fluid production by gravitational flow in wide channels with clamped normal component, 64. - 5. The dependence of the critical velocity of the superfluid on channel diameter and film thickness, 72.
44
111 E. L. ANDRONIKASHVILI and YU. G. MAMALADZE, ROTATION OF HELIUM11
. .
.
.. .
.
.
.......
. .
.
. .
. .
.
. . . . . .
Introduction, 79. - 1. Solid body rotation of helium 11, 80. - 1.1. Angular momentum and meniscus of rotation helium 11, 80. - 1.2. The thermomechanical effect in rotating helium 11, 83. - 1.3. The theory of the phenomena, 85. - 2. Dragging of the superfluid component into rotation, 91. - 2.1. Peculiarities of dragging of a quantum liquid into rotation, 91. - 2.2. Development of quantum turbulence on dragging helium I1 into rotation, 93. - 2.3. Relaxation time for the formation of vortex lines at small angular velocities of rotation, 96. - 2.4. Relaxation time for vortex line formation in rotating helium I1 on transition through the I-point. The mechanism of vortex line formation, 97. - 3. Observation of vortex lines and their distribution in uniformly rotating helium 11, 100. - 3.1. Experiments on establishment of vortex lines, 100. - 3.2. Direct observations of vortex lines in rotating helium 11,
79
X
CONTENTS
103. - 3.3. Irrotational region, 104. - 3.4. Distribution of vortex lines under a free surface, 109. - 3.5. On the normal component motion in a rotating cylindrical vessel, 110. - 3.6. The structure of the vortex line array, 113. - 4. Elastic properties of vortex lines. Oscillations of bodies of axial-symmetric shape in rotating helium 11, 114. - 4.1. The modulus of shear in rotating helium 11, 114. - 4.2. Anisotropy of elastieviscous properties of rotating heliumlI,ll5. -4.3. Hydrodynamics of rotating helium 11, 122. - 4.4. Hydrodynamics of small oscillations of bodies of axial symmetry in rotating helium 11, 125. - 4.5.
Sliding of vortex lines and collectivization of vortex oscillations, 134. - 5. The phase transition in rotating liquid helium in the presence of vortex lines, 137. - 5.1. The central macroscopic vortex, 137. - 5.2. Relaxation of vortex l i e s for the transition helium 11-helium I in the state of rotation, 139. - 5.3. The order of the phase transition in rotating liquid helium, 141. - 6. Decay of vortex lines and their stability, 144. - 6.1. Decay of vortex lines on stopping of rotation, 144. 6.2. Relaxation of vortex lines on change of temperature of rotating helium II, 146. - 7. Persistent currents of the superfluid component, 149. - 7.1. Discovery of persistent currents and the first observations, 149. - 7.2. Dependence of a persistent current on temperature. Superfluid gyroscopes, 153.
IV D. GRIBIER, B. JACROT, L. MADHAV RAO and B. FARNOUX, STUDY OF m s ~ ~ p ~ ~ c o m MMED u cSTATE ~ ~ vBY~NEUTRON-DIFFRACTION . . . . 161 1. Introduction, 161, - 2. Theory of neutron scattering by vortex lines, 164. - 3. Experimental conditions, 166. - 4. Experimental results with niobium, 171. - 5. Analysis of the results obtained with niobium, 171. - 5.1. Line shape, 171. - 5.2. Position of the peak, 174. - 5.3. Intensity of the peak, 175. - 5.4. Observation of only one Bragg peak, 177. - 6. Conclusions, 178.
.
.
V V. F. GANTMAKHER, RADIOFREQUENCY SIZE EFFECTSIN METALS . . . . . 18 1 1. Introduction, 181. - 2. Principles of the theory, 183. - 2.1. Anomalous skin-effect in zero magnetic field, 183. - 2.2. Anomalous skin-effect in a magnetic field, 185. - 2.3. Application of the ineffectiveness concept to the study of size effects, 193. - 3. Various types of radiofrequency size effects, 197. - 3.1. Methods of detection of size effects, 197. 3.2. Closed trajectories, 201. - 3.3. Helical trajectories, 207. - 3.4. Open trajectories, 216. - 3.5. Trajectories with breaks, 216. 3.6. Trajectories of ineffective electrons, 218. - 3.7. Conclusion, 219. - 4. Shape of line and various experimental factors, 220. - 5. Applications of radiofrequency size effects, 225. - 5.1. The shape of the Fermi surface, 225. - 5.2. Length of the electron free path, 229.
-
-
CONTENTS
XI
VI R. W. STARK and L. M. FALICOV, MAGNETIC BREAKDOWN IN METALS . . . 235 1. Introduction, 235. - 1.1. Pseudopotentialsand the nearly-free-electron model, 236. - 1.2. Dynamics of the electronic motion in a metal, 238. - 1.3. A diffraction approach to magnetic breakdown, 241. - 2. The theory of coupled orbits, 244. - 2.1. Amplitudes and phases at a MB junction, 244. - 2.2. Semi-classical transport properties, 246. - 2.3. Quantization of coupled orbits, 251. - 2.4. Theory of the De HaasVan Alphen effects in a system of coupled orbits, 256. - 2.5. Oscillatory effects in the transport phenomena, 261. - 3. Analysis of experimental results, 265. - 3.1. Hexagonal lattice of coupled semi-classical trajectories in magnesium and zinc, 265. - 3.2. Semi-classical galvanomagnetic properties of magnesium, 267. 3.3. Quantum mechanical galvanomagnetic properties of magnesium and zinc, 272. - 3.4. De Haas-Van Alphen effect in magnesium and zinc, 279. VII J. J. M. BEENAKKER and H. F. P. KNAAP, THERMODYNAMIC PROPERTIES OFFLUIDMIXTURES
.........................
287
1. Introduction, 287. - 2. Quantum liquids: zero point effects, 290. - 2.1. General remarks, 290. - 2.2. Apparatus, 292. - 2.3. Theory, 295. - 2.4. Phase separation for the systems Ha-Ne, HD-Ne and Da-Ne, 297. - 2.5. Ortho-para mixtures, 299. - 3. Classical liquid mixtures, 299. - 4. Gaseous mixtures, 301. - 4.1. Experiment, 301. - 4.2. Theory, 307. - 4.3. Comparison between experiment and theory, 314. - 4.4. Gas-gas phase separation, 317.
. . . . . . . . . . . . . . . . . . . . . . . . . AUTHORINDEX..
323
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .
330
This Page Intentionally Left Blank
CHAPTER I
TIHE JOSEPHSON EFFECT AND QUANTUM COHERENCE MEASUREMENTS IN SUPERCONDUCTORS AND SUPERFLULDS BY
P. W. ANDERSON BELLTELEPHONE LABORATORIES, MURRAYHILL,NEW JERSEY
CONTENTS: 1. Historical introduction, 1. - 2. Elementary perturbation theory of the Josephson effect, 4. - 3. Coherence properties of coupled superconductors and superfluids, 11. - 4. Statics of finite tunnel junctions: magnetic interference experiments, 20. - 5. Systems other than tunnel junctions showing interference phenomena, 33. 6. A. c. qua-ntum interference effects, 36.
1. Historical introduction The nomenclature, and to some extent the history, of the area of knowledge to be covered by this review has been confused by the very breadth of the achievement represented by the original letter by B. D. Josephson in which the first theoretical results were announced’. In this letter Josephson discovered not one but three (at least) distinguishable “effects”, to any one of which his contribution was sufficient that it could be - and has been named after him. As far as I can sort them out, these are: (1) The transmission of supercurrents through thin insulating or normal metallic barriers by means of quantum-mechanical tunneling. Without any intention to minimize the importance of Josephson’s other contributions, it seems to me that in order to reduce the amount of confusion in nomenclature, at least in this article, I should confine the term “Josephson effect” to this phenomenon, and let a “Josephson junction” be a device in which supercurrents flow by tunneling. In order to understand the implications of this discovery, Josephson had to investigate how such a junction would behave in the presence of applied electric and magnetic fields. In so doing, he realized two things : (2) That the supercurrents in different parts of the Josephson function References p . 42
I
2
P. W. ANDERSON
[CH.1, 8 1
could be forced to interfere destructively by application of an external magnetic field. This is an effect which I would like to call the “d.c. macroscopic quantum interference effect” or the “d.c. quantum interferenceeffect”, and it may be observed in many other situations than Josephson junctions. It is related to flux quantization and vorticity quantization in helium, and it may even have been observed unintentionally2 in some of the early experimental work on flux quantization, before Josephson’s letter. If it must have a name, it might appropriately be called the “Mercereau effect”, because of the many elegant applications of it which have been made3 by Mercereau and his colleagues. It is important to emphasize that, contrary to common opinion in the early days of the subject, this and the other interference effects have no necessary connection with the Josephson effect proper. (3) That the supercurrent in a Josephson junction under the right circumstances varies in time at a rate given by 2eV/h, and that this effect could be observed by irradiating the junction electromagnetically at this frequency or a subharmonic and measuring the d.c. current characteristic. Both of these have been called the a.c. Josephson effect, and the equation
hv
= 2eV
the Josephson frequency condition ;however, similar phenomena are of such wide occurrence that again it seems too confusing to call the whole complex of phenomena the a.c. Josephson effect. In this article we confine that name to the effect as observed in Josephson tunnel junctions, and use the wider term “a.c. macroscopic quantum interference” or “a.c. quantum interference” for the generalized phenomenon. A second circumstance which has complicated the literature and obscured the relationships among the various kinds of interference effect is that the second most important early paper in the field, again by Josephson, was never published; this is Josephson’s fellowship thesis4 in which he reexpressed his earlier results in more general and more physically satisfying terms, pointing out where the results were special to tunnel junctions and where they were a general property of coupled superconducting systems. It is in this paper that the generalized concept of quantum interference appears, as well as a number of special results and concepts of importance in the practical realization of these ideas. Many of these results were discovered independently and published by the present author 5, also unfortunately in a rather obscure place. Meanwhile the experimental demonstration of Josephson’s ideas proReferences p . 42
CH. 1, @ 11
JOSEPHSON EFFECT
3
ceeded fairly rapidly, after an initial lull of less than a year. The first published confirmation of the existence of tunneling supercurrents - the Josephson effect - and of the accompanying d.c. macroscopic interference effect was by Rowel1 and Andersons, though the effect had been observed but not recognized often previously. Rowel17 soon thereafter produced junctions which demonstrated the d.c. interference effect quantitatively. The a s . Josephson effect proper was demonstrated by S. Shapiro8 very soon thereafter (A. H. Dayem observed it independently but did not publish it), by the synchronization technique suggested by Josephson. Next occurred the elegant series of “two-junction” macroscopic interference experiments by Mercereau and colleagues 9, in analogy to the twoslit interference phenomenon in optics. Eventually, these experiments came to use simply superconducting contacts or bridges, verifying the theoretical indications that the tunneling phenomenon was inessential. In an attempt to demonstrate that experiments on thin film bridges by Parks10 were basically the d.c. interference effect, Anderson and Dayem11 demonstrated the a.c. effect on thin film bridges as well. No reasonable suggestion has yet been made for a physical realization of a true analogue to the ideal Josephson junction for superfluid helium. Nonetheless, the realization that Parks-Dayem bridges show the a.c. effect, and Mercereau contacts the d.c. interference effect, stimulated Richards and Anderson to attempt to demonstrate both using small orifices as the obvious helium analogue of the superconducting contact. So far, only the a.c. analogue experiment has given successful results12. The most recent chapter in the story has been the achievement of successful direct - i.e. incoherent, independent - detection of the radiated a.c. power from a Josephson junction. The Josephson-Anderson theories indicated that a stronger effect could be obtained with a synchronizing mechanism to monochromatize the frequency emitted, and this was realized in the “spontaneous step” phenomenon as observed by Fiskel3 in which internal electromagnetic resonances of the junctions become locked to the a.c. Josephson currents. These steps were used by Giaeverl4 in the first observation (by an indirect but perfectly genuine technique) and in two later more direct observations 15. If one may speculate on the “future history”, the direction of progress appears likely to be two-fold : first, more applications to specific devices, especially scientific instrumentation 16, and second, more variants of these phenomena are yet to be observed in liquid He.
References p . 42
4
P. W. ANDERSON
[CH.1,
52
2. Elementary perturbation theory of the Josephson effect proper It seems to lead the reader most gently into this novel conceptual structure to start with the rather straightforward theory of the tunnel junction as developed in Ref. 5. The basic calculation is extremely simple: one sets out merely to find the coupling energy between two superconductors which is caused by forming a tunnel junction between them. The tunnel junction is described by the “tunneling Hamiltonian” of Cohen, Falicov and Phillips 17
The states k and q are on the left and right sides of the tunneling barrier respectively; the phase relationship among the terms shown is required by time-reversal symmetry. We have not shown possible spin-flip terms which are, nonetheless, permitted by time-reversal invariance; they have no particularly striking effect, so far as is known. No completely satisfactory derivation of (2.1) from a rigorous point of view has yet appeared, to our knowledge. A number of correction terms are to be expected, representing tunneling with interactions (phonon, photon, or Coulomb) as well as multiparticle tunneling, but none of these appear at present to be very relevant to the Josephson effect, as opposed to ordinary incoherent tunneling, and their discussion does not belong in this review. Some derivations avoiding the tunneling Hamiltonian have been given 18119, especially the elegant and enlightening discussion by Josephson in terms of the temperature Green’s function theory of Gor’kov. The physical assumptions, though somewhat concealed by this technique, are essentially the same, as are the results: whether one wishes to introduce a tunneling Hamiltonian or a propagator (i.e. inverse Hamiltonian) which carries single particles across the barrier is simply a matter of taste. The former seems more suitable for the level of this article. and %, on left (1, k ) and right (2, q) We add to (2.1) the Hamiltonians sides respectively. For instance,
where Ek is the one-particle energy measured - for instance - from the bottom of the band, eV the mean electrostatic potential (including any long-range Coulomb effects of space charge) and Zin, the interparticle interactions reReferences p. 42
CH. 1,
8 21
JOSEPHSON EFFECT
5
sponsible, among other things, for superconductivity. Often, especially at finite temperatures, we may insert a fictitious chemical potential term p N to represent the effect of an attached reservoir of electrons, etc. Let us recall a few simple results from the B.C.S.theory20. Let
where in H:,ck is measured from the Fermi surface energy p. We assume for the purposes of simplicity - and powerful and sophisticated arguments are available to support that assumption in many cases - that 3‘:leads to a B.C.S. superconducting state of the conventional kind at or near absolute zero : (2.4) ‘y, ( A 1) = ( U k + ~ k c : c ~ kF)v a c (1) Y
n k
and similarly for (2)t. This ansatz and simple assumptions about Hint give us the usual results : k
E i = (ck - p)2 + 1A21. Here A is the energy gap, and Ek the quasi-particle energy. In one way (2.4)-(2.7) are simpler than the original B.C.S. - or for that matter the Gor’kov21 - theory, in that we choose the simple product eigenstate (2.4) rather than the state projected on a fixed number of pairs N,, which may be obtained by a transformation: 2n
e-iN1a!PI( A , e”) dq .
Y(N,) = 0
This is easily verified by substituting in (2.4) and using (2.6). The questions involved in why we use (2.4) rather than (2.8) are rather deep and basic to the whole subject, so aside from remarking that since we have tunneling it is not necessarily suitable to use (2.8) we will postpone them briefly. We follow the convention that - k ++ - k&,k tt k?. References p . 42
6
P. W. ANDERSON
[CH.1, 8
2
On the other hand, it is essential that we treat A (as we have) as complex *, if only because of (2.8): we must have a set of states complete enough to describe the fixed-iV, as well as fixed-A situation. The phase of A gives, in fact, the phase relationship between the component of (2.4) with N particles and those with N + 2, as we see from (2.5). The wave function (2.4) is essentially different from that of any normal substance in that it does contain a coherence between components with different total numbers of particles, and as will be described later it is this coherence which is vital to the whole concept of superfluidity and superconductivity. It is only the part of the Hamiltonian .Xy which is consistent with a timeindependent A , however. It was first noticed by Gor’kov21 that A has an intrinsic time-dependence which is different for samples at different electrostatic potentials. Josephson showed its relevance to tunneling, and since then several more complete discussions4~23,24 have been given. We follow Anderson, Werthamer and Luttinger23. They simply introduced the last term of (2.3) and noted that its only effect on a wave function of type (2.4) is to make A time-dependent according to the equation of motion
so that A
- lAl
e-
( 2 i / l ) (w
+ eY)t
(2.9)
Again, this merely signifies that while the last term of (2.3) has no effect on the components of Y of fixed N , , clearly the energies of states of different N differ by (2.10)
by definition, and therefore that a state with a definite phase relationship between components with different numbers of pairs must allow that phase
* It is necessary to distinguish carefully between this use of a “complex” energy gap and that of the Nambu-Eliashberg formalism used extensively in single-particle tunneling work (see, e.g., Schriefferaz).The “real” and “imaginary” components of A used here are, in that formalism, the “ri” and “r8” spinor components, dl and Aa, each of which may itself be complex when lifetime effects are included, the complex part representing decay in time. The true quasi-particle energy may be complex but is given by Ek2 = &k2 A l 2 As2. In most Josephson phenomena lifetime effects are not very important because of the low frequencies, and a proper calculation including them has not yet been attempted to my knowledge.
+
References p . 42
+
relationship to vary in time according to (2.9). Note that (2.6) ensures that the coefficient of &:k varies in time like A , which is of course what is necessary to given the proper time-dependence of A. Now let us do perturbation theory using the tunneling Hamiltonian (2.1) as the perturbation. Any reasonable barrier is tens of Angstr6ms thick, and or me-'' at best, so that the tunneling matrix elements are of order the only phenomenon of any interest must appear in the lowest non-vanishhaving no diagonal matrix elements. ing order, which is the second - ST Applying a typical term of (2.1) to the wavefunction (2.4), we get T~(C:C~
+
C ? ~ C - ~ )Y , ( A , ) yZ(dz)= T k q ( u k u q
+ ~,p~)(c:c?~) x (rest of Y ) . (2.11)
When there is no relative voltage between sides (1) and (2), the energy of this intermediate state is just the quasi-particle energy &+ Eq; it is obviously the state with an extra quasi-particle on each side. Thus the total coupling energy is just:
(2.12) The first two terms in the bracket are of no particular interest because they do not depend on the phase of A. By (2.6), however, the last one does, in fact its value is
k, 4
The relevant integral in (2.13) has been done elsewheres; the result is
(2.14)
K is a complete elliptic integral. The most trivial and yet most important consequence of (2.14) is just the idea that the energy does depend on the relative phase (rp1 - q Z )on the two Referencesp . 42
8
P. W. ANDERSON
[CH.
1, 5 2
sides, and in fact is a minimum when this relative phase is zero. This implies that if the coupling energy is strong enough these two approximately independent pieces of metal, connected only by very weak tunneling matrix elements, will find it energetically favorable to occupy coherent states like (2.4) with (ql-cp,)=O, rather than fixed-number states like (2.8). In the states (2.8), the phase-dependent coupling energy simply vanishes identically. Characteristically,if the tunneling current is appreciable this coupling energy is large relative to helium-temperature as well as zero-point fluctuations 5, and so, as we will review in the next section, the calculation using coherent states given here is correct. Before going on to that let us complete the perturbation theoretical discussion. The calculation of the energy perturbation in case there is a voltage difference across the junction is somewhat more complicated. The most complete calculation in the literature is that of Werthamer 25, following Riedel26. We will give here a simpler one related to Josephson’s original work. Of course, there is one major difference which is not discussed in any detail in any of these papers including Werthamer’s: that where the existence of the coupling energy provides a motivation for using the coherent states (2.4) in the zero voltage case, here the coupling energy is necessarily periodic in time and the assumption that the starting states should be the coherent ones (2.4) is not obviously valid. What should by rights be calculated is the current-current (or energy-energy) correlation function, which would, however, then depend on details of external circuitry, applied a.c. voltages, etc. The calculation we do should perhaps be thought of as follows: we calculate the energy - and, shortly, the current - using the coherent states (2.4) as a basis. Any state reasonably close to the equilibrium ground state can be made up as a wave packet of such coherent states (this situation is discussed in Refs. 24 and 5 and will be further elucidated in Section 3). In fact, it often turns out that even in the a.c. case the coherent states are approximately the correct ones. Even assuming the coherent states the calculation is not quite straightforward because the quasi-particle energy in the intermediate state depends on whether the quasi-particles were created by moving an electron from left to right or vice versa - i.e. whether a given quasi-particle, which is a coherent mixture of electron and hole, is in its “electron” or “hole” aspect. The first term of (2.11) creates a state of energy Ek+Eq+e(Vl- V,), the second Ek+Eq- e( V, - V,), relative to the starting state. The Hermitian conjugate terms have energy denominators which are the reverse, giving just the same References p . 42
CH. 1, 8
21
JOSEPHSON EFFECT
9
form of result but with a sum of energy denominators:
(2.12')
This expression increases as the voltage difference approaches the double energy gap, becoming singular there. Above that point, both the energy and the current contain complex parts, the interpretation of which has not been fully discussed in the literature. (But see Scalapino27.) Expressions for the current which probably suffice in all regions have been given by Werthamer26. The increase in current in the intermediate region has probably been observed by Grimes, Richards and Shapiro 28. So far we have merely done a second-order energy calculation. Of course, for finite voltage the quantity we calculate is time-dependent - it is simply the mean value of the tunneling Hamiltonian X T in the assumed states. In any real physical situation other contributions to the energy - external sources necessary to maintain the currents, etc. - will also be present. It will be shown in the next section by very general arguments that a phasedependent energy implies the existence of a supercurrent (2.15)
This relationship may be derived directly from second-order perturbation theory if we realize that the second-order energy is just the mean value of the tunneling Hamiltonian XTin the perturbed state:
(~E)~=((~Y,+~~),~,(~Y,+~'Y))~((~~Y,~,Y/~)+(~Y, As in (2.1 l), a typical term of XTis c:cq
+ c?qc-k.
(2.16)
A typical term of the tunneling current is given by using (as suggested by Cohen, Falicov and Phillips 17)
and the corresponding term of this to (2.16) is
References p . 42
10
P. W. ANDERSON
[CH.1,
2
Thus when calculating the current rather than the energy we must replace the coherence factor in (2.1 1) by
On the other hand, 6Y is unchanged, so that only one of the two coherence factors in (2.12) should have its sign changed. Thus we get for the current
and this is exactly the same as (2.12) and (2.13) except that the phasedependent factor Re (Ad,) = ldld2l C 4 P l - (P2) (2.17) This is precisely what we would get by applying (2.15). It is easily verified that the same expressions work in the frequency-dependent case of (2.12‘). The tunneling current, like the energy, is phase-dependent; this is in fact the point from which Josephson startedl. The interpretation can only be that the actual current which flows is determined by the external circuit, and that experimentally what we expect to see is a “tunneling supercurrent” phenomenon: we can draw any current consistent with (2.17) through a tunnel junction at zero voltage, or where there is a finite voltage there will be an a.c. tunneling supercurrent. Both of these predictions were verified shortly after Josephson’s paper came out. In order to understand these observations, it is first necessary to fully understand the meaning of the use of coherent states in our calculations, which is the purpose of the next section. The most interesting and most convincing demonstrations of these coherence effects all involve modulation of the supercurrents by electromagnetic or other forces, so that in the final sections, where the actual experimental work is explained, the emphasis will be on that aspect. The purpose of this section has been merely to make the References p . 42
CH.
1,o 31
JOSEPHSON EFFECT
11
most straightforward calculations on this simplest microscopic model which would give the reader a feeling for the source of these phase-dependent energies and currents with which the rest of the article will deal. 3. Coherence properties of coupled superconductors and superfluids
The first reasonably careful discussions of the basic theory of coherence in coupled superconductors were given independently by Josephson and Anderson5. Since then these have been expanded, the most complete, with a discussion of the similar case of superfluidity in He, being the recent review of Anderson 24. In the preceding section we showed that there exists a phase-dependent coupling energy between two superconductingsamples connected by a tunnel junction (assumed infinitely small spatially), when the states of the two superconductors are assumed to be coherent states, i.e. coherent superpositions of states of different numbers of pairs of particles: Y = C a NeiNqY,. N
(3.1)
We also showed that this phase-dependent energy implies the possibility of a supercurrent flowing between the two superconductors:
in this case. As was demonstrated in Refs. 5 and 25, this connection between phase and current is a consequence of the commutation relationships of the field operators : (Josephsonls gives another, thermodynamic, derivation)
“,$I
= -$9
(3.3)
which is equivalent for coherent states like (3.1) to the canonical commutation relationship between number and phase,
“,cpl=
-i,
(3.4)
which in turn implies the operator equivalences
As we said, the proviso that this holds only for states for which N is large and a fortiori for states which are superpositions of coherent states like (3.1)References p . 42
12
P. W. ANDERSON
[CH.
1, 5 3
must be made. Certain well-known difficulties occur for small N . The derivations of (3.4), (3.5), and (3.2) as given in Ref. 24 are almost trivial. First, we verify that (3.1) obviously has a definite fixed phase of the field operator ( IY, $ lU) = ei‘p u N -N‘l (IYN- 17 $ y N ) N
and we assume (as we are free to do) that the phases of a, and Y Nare chosen to make the former real as well as the matrix element. Second, it is clear that acting on (3.1) the operator -ia/acp has the same effect as N . The converse statement in (3.5) follows only if N may be taken as a quasicontinuous variable, which is the source of the small-N difficulties, of course. Finally, (3.2) is just the Hamiltonian equation of motion which follows from the fact that N and cp are canonical conjugate variables:
(3.6) or
( U = ( S ) ) . Of course, all these statements are equally valid for liquid He, where N is the number of helium atoms, $ their quantum field, as for superconductors where the reference is to number of electron pairs and to the pair field $$. The more explicit discussion given in Section 2 may remove any residual doubts in the reader that the pair field may be so treated. From this point of view, the Gor’kov-Josephson frequency equation (2.9) or dcp (3.7) dt aN is simply the other Hamilton’s equation of the pair implied by the canonical commutation relationship between number and phase. These various relationships hold just as well between small but macroscopic cells within a bulk superfluid as between fully macroscopic samples connected through a weak “Josephson” link. The corresponding coupling energy is the energy which maintains the internal coherence of the superfluid, the currents which flow according to (3.6) are the usual supercurrents nev, =ne(tt/m)Vcp, and (3.7) becomes the acceleration equation which controls the dynamics of the superfluid. Within the bulk of the superconductor, however, the various interference effects which Josephson suggested are not References p . 42
CH.
1, 8 31
JOSEPHSON EFFECT
13
easily observed because the coupling is too strong. The Meissner effect represents the successful attempt of the superconductor to exclude from its interior all phase differences which lead to supercurrents: Correspondingly, the zero-resistance phenomenon is the successful attempt to prevent temporal phase differences according to (3.7): dp/dt=O implies d U / a N is constant, i.e. the chemical potential p is constant. When, in type I1 superconductors, the magnetic field is increased to the point that phase differences and supercurrents do exist in the interior, the energies which control the resulting vortex structure are still so great that the structure becomes nearly microscopic in scale and has various types of rigidity which makes the direct observation of interference effects difficult. Similar considerations hold in liquid helium. Thus, while the coherence-interference phenomena in superfluids are perfectly general, the Josephson “weak link” - originally a tunnel junction as discussed in Section 2 - is essential to their direct observation, primarily because in such a weak link the response of the superfluid to phase perturbations is not controlling and we can read off the current corresponding to an “externally” applied phase difference. This discussion then brings us to the most fundamental question of principle with regard to these coherence effects. This is the following: we understand very well that in a strong-coupled situation such as the interior of a bulk superconductor the phasedependent coupling energy can maintain the phase coherence between different parts of the material. Only thermal agitation at a sufficiently high temperature, T,, can break this coherence. Whether we describe the coupled system mathematically, as a whole, by a totally coherent state, in which we assume (+) exists, or whether we use a fixed N or an incoherently mixed grand canonical ensemble is irrelevant, since the phase of the system as a whole is not observable. What processes, on the other hand, affect the phase coherence between two samples connected by a weak link in which we are modifying the phase relationship intentionally in order to observe interference? What maintains the coherence in opposition to the perturbations we may introduce? I believe the answer is threefold, depending on the type of interference experiment which is attempted. These are basically of three kinds: d.c., a.c. with synchronization, and a.c. without synchronization. The first two have been well understood in the past; the third is not complicated but no previous discussion has been given. In the first case, coherence in d.c. interference experiments is maintained by the phase-dependent coupling energy itself. As has been shown elsewhere24, a d.c. current source may be represented by a term in the MamilReferences p . 42
14
P. W. ANDERSON
[CH. 1,
83
tonian of the system depending linearly on the phase: &source
=
- 5 1 2 (cpl - cpd
and the Josephson current which flows is then obtained by minimizing the Since the phase sum of the phase-dependent internal energy and Nsource. relationships between different parts of a Josephson junction or between two junctions connected in parallel can be influenced by a magnetic field (by gauge invariance physical quantities can depend on the phase of the electron field only through q ( r ) - ( e / c ) r A .dl, where A is the vector potential) the phase-dependent energy can depend on the magnetic field, as first demonstrated by Rowel17 and exploited by Mercereau and his colleagues in many beautiful experiments. The question of what mechanisms may break up the coherence in the d.c. case was discussed in Ref. 5 as well as by Josephson29and Zimmermann and Silver30. The ultimate limitation by the zero-point fluctuations implicit in the fact that ( A N A q ) - l according to the commutation relation (3.4) is actually always completely negligible. The relevant frequency was first estimated in Ref. 5 but is most simply understood by pointing out, as Josephson 2o did, that the Josephson current is essentially inductive; i.e.
for small signals, where J = J1 sin cp cs: J,cp
This inductance per inverse unit area* matches the capacitance per unit area to give an LC resonance at a frequency
the “Josephson plasma frequency”, which is typically a few megacycles. The zero-point energy is then -Am, for a single small junction, which is much less than usable kT’s. That is, the electrostatic energy (involving C ) which tends to reduce the charge fluctuation A N is not strong enough to outweigh even a small coherent Josephson coupling which tends to make Acp small instead.
*
This inductance was actually measured directly by Silver et aL31.
References p . 42
CH. 1,
8 31
15
JOSEPHSON EFFECT
-
Much more relevant is the question of thermal fluctuations. Josephson currents of less than 0.1 PA correspond to coupling energies kT at helium temperatures and so are intrinsically unobservable at those temperatures. One or two orders of magnitude higher than that should still be quite hard to observe because of the rapid rate of phase slippage by thermal activation. That is (see Fig. 1) the combination of the current source energy and the CONSTANT CURRENT SOURCE J e
Fig. 1. Energy vs. phase for a Josephson junction with constant applied current. Thermal or zero-point fluctuation and/or magnetic fields can cause instability and “phase slippage”.
periodically phase-dependent energy can be represented as a “washboardlike” energy surface, with local minima but no absolute ones. Only if the local minima are very deep compared to kT will the supercurrent-carrying state be stable relative to one in which the phase slips and - by (2.9) - a voltage appears. References p. 42
16
[CH.1,s 3
P. W. ANDERSON
IC
SUPERCONDUCTING OR FILM
,/WIRE
JOSEPHSON JUNCTION
0
0
0
IC
0
AV (1) - AV (2) = f (H, I A p p
(a)
GENERAL CASE
SUPERCONDUCTOR
,
/I
'\'.
., - __-
(b)
/
/
SCHEMATIZED
Fig. 2. Schematic diagrams of the geometry of the Mercereau interferometer. (a) shows a superconducting loop containing two junctions to which may be applied fields or drift currents in order to change the relative phases &(l) and &(2) of the two junctions; (b) shows a possible path for the integration in the text, Section 4.
In the early experiments noise from the external circuit at room temperature was of importance. Josephson29 has given a thorough discussion of how to avoid that if desired. A final consideration which has been emphasized by Zimmerman and Silver30 is the question of the relative coherence between two Josephson or other junctions which are part of an interference experiment. CharacteristicReferences p. 42
CH.
31
17
JOSEPHSON EFFECT
ally, the Mercereau geometry (shown in Fig. 2) involves a pair of Josephson junctions connected in parallel in a superconducting loop around which one wishes to measure the phase difference caused by a current or vector potential. When the phase difference is 2nn, the junctions are in phase and twice the critical current of one can flow; when it is (2n+ 1) n , the currents are necessarily opposite, the energy coupling the phases of the two halves of the loop is reduced drastically, and a much smaller critical current is observed. The thermal fluctuation limit here, it is suggested, is the thermal fluctuation of the loop integral of phase. This may be simply computed from the point of vieuw of one of the junctions, which sees a superconducting loop of inductance L. The fluctuating voltage V obeys : dl V=Ldt ’ but also V controls the phase across the junction dq 2e dt
I/=--*
Thus we find
or 8e2L
( q 2 ) = fi2 ( 3 L I 2 ) = kT x
8e2L -* h2 ’
(3.9)
the fluctuation is proportional to the inductance of the loop. When ( q 2 ) 1, phase coherence will not easily be observed; this corresponds to an inductance of h2 x 2.5 x cgs = 2 cm N 2 x L--millihenry. 8e2kT N
Thus loops for use in Mercereau interferometers must either be wound noninductively (he has observed interference for non-inductive loop lengths of 1 m) or must be rather small. It is, however, not clear (Mercereau, private communication ; also Ref. 30) that this limitation cannot be evaded in practice, by sufficiently careful averaging procedures, by using the Josephson current in moderately stronglycoupled junctions to “tie down” the phase during most of the measurement process, or by appropriate filtering. N
References p . 42
18
P. W. ANDERSON
[CH.1,s 3
The second type of interference experiment which may be used to demonstrate coherence in superfluids is the “synchronized a.c.” experiment. This was suggested in the original letter of Josephson1 and first carried out by Shapiro * on Josephson junctions. Later the experiment was extended to superconducting contacts by Anderson and Dayemll, and it is so far the only successful interference experiment in superfluid He as carried out by Richards and Anderson 12. The principle of the experiment is to apply an alternating force and to observe simultaneously the junction’s d.c. I-V characteristic. At a voltage such that the Josephson frequency v, = 2eV/h
is in synchronism with the applied a.c. signal, power can be transmitted between the a.c. source and the Josephson system and therefore singularities in the I-V characteristic are to be expected. We will discuss the rather primitive state of the theory of this phenomenon elsewhere; our concern here is with the fact that in such an experiment, apparently, a definite phase relationship is maintained across the junction in spite of the presence of a finite d.c. voltage. The mechanism is, of course, one of synchronization; the coupling energy to the a.c. signal can be sufficiently great to maintain coherence against both thermal and zero-point fluctuations. Detailed study of even a simple case of the synchronization effect is an exercise in nonlinear, largesignal theory of noise and irreversible processes which has not as yet been carried out and is probably not at all simple; yet one can make some qualitative statements. One is that even a rather small synchronizing signal is adequate to outweigh thermal and zero-point fluctuations under the proper circumstances. For instance, if we think in terms of an externally applied a.c. supercurrent, which can be schematized as a term in the Hamiltonian like
the gain in energy when cp - rp changes by 2n in favorable phase witho, relative to unfavorable is of order hJ,le which is >> kT if J, is as much as a microamp6re. J, is of course limited to be less than the critical Josephson current. One can see no physical reason why the synchronization should not be essentially perfect, and more perfect the stronger the applied signal. All experimental evidence points in this direction. References p . 42
CH. 1, Ej 31
JOSEPHSON EFFECT
19
A special case is the “self-synchronized” Josephson system. In many cases, the Josephson junction itself may be able to sustain high Q electromagnetic resonant modes. In such a junction, the a x . Josephson current itself may be thought of as generating its own synchronizing signal, the relative phase being controlled by the resonant circuit. The result is the “step” phenomenon first reported by Fiskel3. It is notable that where with external synchronizing signals V is absolutely fixed by the applied frequency, in the “step” phenomenon the frequency and thus voltage may be “pulled” by varying the current. A fascinating variant of the self-synchronized technique has been given by Mercereau and Vant-Hull4I. Two junctions are connected in a loop with a long resonant strip line, and the harmonics ok the strip line frequency observed as voltage singularities across the pair of junctions. This technique seems to demonstrate coherence between junctions separated by over one meter of superconductors, and may be the most sensitive interference effect so far suggested. The final type of experiment is the “free-running” a.c. Josephson effect. It is hard to distinguish very strongly in principle between this and the “self-synchronized” case, since it certainly is true that there always will be a greater or lesser degree of feedback. In the case of point contacts or Dayem bridges, where the electromagnetic coupling to free space is relatively strong and the Q of any resonance reasonably low, we do not expect a very strong synchronizing effect and we can idealize the junction as simply running free at a mean voltage determined by the external circuitry. The measurements we can make all amount basically to measuring the correlation function of the Josephson current. If we measure the current a t any instant, we will get a finite value with some arbitrary phase, and in making such a measurement we automatically put the system in a wave packet of definite phase difference ‘pl - q 2 : since J = J , sin (ql -cpz), a measurement of J is equivalent to a measurement of ‘p1-’p2. Again, the dissipation of this wave packet is controlled by two kinds of perturbation: quantum-mechanical, and classical thermal fluctuations. The quantummechanical fluctuation is controlled by the electrostatic “kinetic energy”like term
which causes a wave-packet which starts out as a &function in phase to dissipate, just as the kinetic energy (A2/2m)/(i32/i3~2) causes a particle wave References p . 42
20
P. W. ANDERSON
[CH.
1, 5 4
packet to dissipate in x. The equation is
(3.10) where C is the capacitance. For a point-contact device such as is used by Dayem and Grimes32 C - lo2 cm and the time involved in complete dissipationis only -lo-' sec; they can never observe a spectrum narrower than 1 megacycle or so. For Josephson devices, on the other hand, C-104 and the spectra may in principle be very narrow, especiallyif the cavity resonances of the junction are employed to still further reduce the effective C (which is an equivalent result from a different point of view to the idea of selfsynchronization). If now a second measurement of current is made, the full amplitude of Josephson current will again be measured but its phase relationship to the previous measurement will depend on whether the measurement is made before or after the wave packet was dissipated. Thus the breadth of the spectrum depends on the mean time of dissipation of the wave packet. The thermal noise problem for this case is a fairly complicated non-linear one, but in practice the frequency spectrum will be controlled by the sum of the various noise voltages due to the resistance in the external circuit and the effective resistance of the junction device, and these noise voltages will have a role - probably the controlling one - in breaking up the coherence of the wave packet. Another way of doing the problem which would give similar answers would be simply to calculate the current correlation function ( J ( 0 ) J ( t ) ) , using if desired ordinary incoherent superpositions of states. This calculation, however, has to be carried to higher order than the type of calculation so far done. The major conclusion of this section is that it is correct in almost all cases to discuss the quantum coherence effects by means of coherent states with definite values of order parameters A or (I)) and definite phases. Almost all deviations from perfect coherence can be treated as thermal noise rather than as an intrinsic breakdown of the coherence assumption. Only in the last case of the free-running point contact device is there any possibility that this breakdown may be observed. 4. Statics of finite tunnel junctions: magnetic interference experiments
So far we have been discussing only an extremely small Josephson junction in which the phases q l and cppZon the two sides could be assumed to be References p . 42
CH. 1,f 41
21
JOSEPHSON EFFECT
single constants. Most of the interesting effects, starting from Rowell’s first observations 61 7, result when electromagnetic fields are allowed to enter the junctions and modulate the phase. The Josephson current at a particular point in a tunnel junction cannot depend only upon the phase difference of the order parameter on the two sides because that is not a gauge-invariant quantity. The current must not change if we add a gradient Vx to the vector potential A and at the same time add (2elhc)x to the phase. The most elegant way to write down the gauge-invariant Josephson current is in terms of the so-called “gaugeinvariant phase” w (called @ in Ref. 25), which is defined as the quantity which acts like a velocity potential for the supercurrent: (4.1) w is not single-valued, nor even the phase of a single-valued function; only its gradient is defined and unique. Nonetheless it is often a formally useful quantity. The Josephson current at a given spot x , y in a junction is given by
J ( x , Y ) = 51 sin [ w , - w2 (x, Y ) ]
(4.2)
9
where w1- w2 can only be defined in terms of a specific path, the shortest one from one superconductor to the other - and thus presumably the one followed by tunneling electrons WI
- ~2
=
s
s 2
2
Vw(x,y,z)dz = ~ p 1- ~2
1
- 2e PlC
A, dz .
(4.3)
1
The equations satisfied by w were written down in an elegant form by Werthamer25, making use of the fact that rp is the phase of a single-valued function so that V x V p is zero except possibly at vortex lines where the phase changes by 2n or 2nn on traversing the line: (VxV)w= - V x
2e +27cC6(r-rvorter)=- - H ( + 2 n Z 6 ) hC
(4.4)
and
a
2e
- ( V W )= - E at h
.
(4.5)
The second is just London’s acceleration equation for vs, leaving out as small thermal and Bernoulli terms. (As pointed out by London33, the ( u / c ) x H and References p . 42
22
[CH.1,
P. W. ANDERSON
84
SUPERCONDUCTOR 1
SUPERCONDUCTOR 2
motional terms cancel.) The presentation of Werthamer seems to be accurate for his purposes but does not make clear that w (his @) is not really the phase of any order parameter or Green’s functions (nor does the time equation have quite the same form as we have given). In component form, referring to a typical junction experimental geometry (Fig. 3) these two equations are
aZw a2w 2e ---- - - H , (+ &functions),
(4.4‘)
aZw --EE,. 2e --
(4.5‘)
azax
,ic
axaz
A
ataZ
Let us consider first a simple junction with field H, through the barrier region in the y direction. We want to compute the rate at which a field changes w1 - w 2 : 2
-ax (aw l
- w2)
a
= ax -
(J3) 1 2
2
dz
By London’s equation, (t&
a ax
-(wl References p . 42
= -(I&
=(2eH/rnc)A, so that 2e
- w 2 ) = -(2A Ac
+ d)ff,.
a. 1,8 41
23
JOSEPHSON EFFECT
Thus in the presence of a constant magnetic field in the plane of the junction, the maximum current it can sustain follows a single-slit diffraction pattern: +I
J,,, = J1
cos[32A -9
+ d)H,x
1
dx
for a junction of length I and width (in the y direction) W. This pattern was first observed in detail by Rowel17; it had been predicted by Josephson’. It is of course no coincidence that the dependence is on the number of flux quanta contained in the junction region, as we shall show. Since a flux quantum 2e/hC is -2 x lo-’ gauss-cm2, and 21even the earth’s field is adequate to perturb reasonable size tunnel junctions, which accounts for most of the early failures to notice the effect (as suggested by Anderson1.6). In large or high-current junctions the tunnel current is big enough to influence H,,: there is a kind of one-dimensional Meissner effect noted independently by Josephson4, Anderson6 and Ferrell and Prange34. If we insert the tunnel current J, = sin(w, - w 2 ) into Maxwell’s equation dHy 4nJ2 -=-ax c ’ we obtain
Linearizing the sin for small signals, this is d2A, --
ax2
1 --Az,
A;
(4.9)
in close analogy to London’s equation for the Meissner effect, with
(4.10) References p . 42
24
P. W. ANDERSON
[CH.1,s 4
Then (4.8) can be written wz)
aZ(W1-
1
sin(w, - w2). (4.8‘) 2; A junction larger than 2, excludes the field for small fields, and may break down entirely or exhibit more complicated effects for larger ones, rather than showing the full interference pattern of (4.8), a phenomenon also observed by Rowel17. It seems not to have been noted previously (except see De Gennes35 and Josephsonls, Appendix) that (4.8) has solutions corresponding to a onedimensional version of a single quantized vortex, as well as to a one-dimensional “Abrikosov array” of quantized vortices. That is, it exhibits in a particularly simple way the complete range of type I and type I1 superconducting behavior. It may be amusing and rather instructive in understanding the relationship between the interference phenomena and macroscopic superconductivity to describe these solutions briefly. Let w 1 - w2 =q. The first integral of (4.8‘) is
axz
AJ
=
dqo - = [2(1 dx
~~
+ 2 c - cos q)]*
(4.11)
which is a differential equation which may be solved in terms of an elliptic integral:
-- xo - [d(&p) (C + sin2 &p)-*.
(4.12)
J
We may distinguish two cases, which turn out to represent type I and type I1 behavior. TypeI: ascp+Oor2nn, dq/dxcCH-0. This is the only case in which, in a largejunction, the field can actually decay to a negligible value within the junction, i.e. J(a sin cp) and H+O together. By (4.1 1) this means C=O. The solution of (4.12) is then a degenerate limit of the elliptic function: x - xo (4.13) = log tan i c p . ~
AJ
(4.13) represents an isolated “linear vortex” at the point xo. The field ccdqldx and current ccdZq/dx2for such a solution are shown in Fig. 4. Note that the total flux J(dq/d.x)dx=2n; this corresponds to exactly one quantum of flux Q0 =hc/2e in the junction. References p . 42
CH. 1,s 41
JOSEPHSON EFFECT
25
Fig. 4. “Vortex” solution of Josephson equation for junction with applied fields. “H” and “J” are proportional to the field and current respectively: H is in units of Hci, J in units of UI.
To satisfy the boundary condition at the edges of the junction imposed by an external field, we may also use the solution (4.13) if H,,, is less than the maximum shown in Fig. 4, which is
(4.14)
This maximum field was noted by Ferrell and Prange34, who gave an equivalent solution to (4.13), as well as by Josephson 18. It represents one quantum of flux in a length nA,. Type 11: once H>H,, we cannot fit the boundary conditions with C=O. Set (4.15)
References p . 42
26
P. W. ANDERSON
[CH.
1, 8 4
and (4.12) is transformed into a standard Jacobian elliptic function : (4.16) or (4.17a)
T) kn, -
dq‘
2 x-xo dx =% dn(
=
2
(1
k2 sin2 $q’)*.
(4.17 b)
(4.17b) tells us that H,,
HC 1 =Happ= - = (1 k
+ C)*Hcl, (4.18)
Hmin= Hap,(1 - k2)* = Happ
The period of the vortex structure is given by the complete elliptic integral K, which is a logarithmically decreasing function as H increases above Hcl, approaching a constant as H+ co.
(x-xdh, Fig. 5. A “type 11” solution of Josephson equation. The parameters are Hex$ = Hms, 1.118 Hci, period = 2.257 15 = 1.44 H / @ o t . References p . 42
=
CH. 1,
41
JOSEPHSON EFFECT
27
As an illustration we have plotted out the structure in Fig. 5 for the case
k Z = l / ( l+C)=O.8; C=0.25. This corresponds to H,,,=1.118 !Icl.As Fig. 5 shows, H fluctuates by slightly more than SO%, but y ( x ) is surprisingly linear. An interference experiment at this field would lead to qualitatively similar results to k2+0, or +a, while for H/H,, greater than 1.5 or 2,H would be virtually constant and the screening effect unimportant. The biggest effect is on the period of the structure: K(0.8) is 1.44 times bigger than K(O), so the period is modified by 50%. The one dissimilarity with type I1 superconductivity we observe is that there is no natural Hcz. The supercurrents cannot be altogether stopped as long as the superconducting samples on the two sides retain their coherent fields, so the flux structure will last up to the critical field of the bulk samples. The net result is to point up the fact again that the Josephson effect is not a unique phenomenon unrelated to superfluidity as a whole but rather a kind of microcosmic manifestation of superconductivity in a particularly simple and understandable form. In this discussion it becomes clear that the Josephson interference pattern can be viewed as a question of fitting a vortex pattern into the appropriate boundary conditions, much as one might wish to discuss the thin film experiments of Tinkham36 and Parkslo. As in the optical case, while the simple “onsslit” interference pattern establishes the nature of the phenomenon quite adequately, much more beautiful and useful experiments can be done with more complicated systems, and in particular with the “two-junction” interferometer of Mercereaus. The principle here is again to study the critical current as a means of examining the relative phasing of the different parts of the Josephson current, but now two or more junctions are placed in parallel and their relative phasing controlled by electromagnetic fields in the loops connecting them. A rather general and simple way to understand such effects is to use Stokes’ theorem on the basic equation (4.4) for the gauge-invariant phase difference:
-
2e $Vw*dl= - - JH.dS( + 2nn). Ac
(4.19)
If we wish to compare the phase-differences w 1- w 2 at two junctions or two parts of a junction, AB and CD, in any interferometer circuit (see Fig. 2b on p. 16), we apply (4.19) to any circuit we may choose which includes the Referencesp . 42
28
P. W. ANDERSON
[CH.1, 8 4
(4.20) A
B
ABCD
For instance, in the single junction, 0, is everywhere parallel to the junction so that if we extend our circuits AC and BD perpendicular to the junction into the field and current-free interior of the superconductors, the line integrals on the right in (4.20) vanish and we recover the result we have already noted, that the phase difference depends only on the total flux in the junction between the points AB and CD, measured in flux quanta. Again,
Fig. 6. Mercereau interferometers (after Jaklevic et a1.9.s). (a) Cross-section of a Josephson junction pair vacuum-deposited on a quartz substrate (d). A thin oxide layer (c) separates thin (- loo0 A) tin films (a) and (b). The junctions (1) and (2) are connected in parallel by superconducting thin film links forming an enclosed area (A) between junctions. Current flow is measured between films (a) and (b). (b) The junctions (I) and (2) are connected in parallel by superconducting thin film links enclosing the solenoid (A) embedded in Formvar (e). References p . 42
CH.
1, B 41
29
JOSEPHSON EFFECT
in an interferometer circuit such as Mercereau's with a thick-film loop connecting two junctions, zlsl is very small and the total flux enclosed by the loop controls the phase difference. Fig. 6 shows the apparatus used by Jaklevic, Lambe, Mercereau and Silvers, 99 37 to demonstrate two-junction interference due to f H . d S in such a loop either from an external H or a small solenoid (the latter to demonstrate the reality of the AharanovBohm38 vector potential effect). The corresponding interference patterns are shown in Fig. 7. 4
I
>
d
H
-&
l
I
-500
l
l
-400
I
l
-300
l
l
-200
: l -100
I
I
0
I l 100
I
l
200
I
300
~ I
I I 400
I 500
I
I 800
MAGNETIC FIELD (MILLIGAUSS)
Fig. 7. Maximum supercurrent versus magnetic field for configuration similar to that of Fig. 6a with junctions of Sn-SnOa-Sn. For (a) the field periodicity is 39.5 mG, for (b) 16 mG. Approximate maximum currents are 1 rnA (a) and 0.5 mA (b). Configuration of Fig. 6b gives similar results but without the modulation envelope. (After Jaklevic et a1.s)
More interesting were two experiments to demonstrate the reality of the Su;dl terms in (4.20). In ones, one of the links AC was a very thin film of length I,, which was part of a loop exterior to the interferometer. The field of the exterior loop was well contained in it and contributed virtually no IH-dS to the interferometer loop. Then the phase difference is controlled by the surface current on the interferometer side :
s C
u;dl
=(use)-' lACj,= (n,e)- lAc($)(sinh
:)-',
(4.21)
A
where A is the penetration depth, t the thickness, and w the width of the thin film. Since n s N A - 2 , this gives a very strong dependence of the periodicity on A and thus on T which was observed. Fig. 8 shows the apparatus and results. References p . 42
-5
-10
-
0 5 10 DRIFT CURRENT
15
20
25
(ma)
7
Fig. 8 . (After Jaklevic et al.s) (a) Schematic of a junction pair (1) and (2) similar to Fig. 6, where the base film strip b carries a drift current which is returned beneath itself by a second base film b' designed to keep the field due to the drift current from the area enclosed by the junction loop. The insulating layers d are of Formvar. (b) Experimental trace of Zma, versus the drift current showing interference and diffraction effects. The zero offset is due to a static applied field. Maximum current is 1.5 mA. (c) Variation of observed driftcurrent period d h with temperature for two junction pairs of identical dimensions ( w = 0.5 mm and N.' = 8 mm). The curves are theoretical. The cross-section dimensions of the base film are 3 mm by 1100 *SO A.
CH. 1,O
41
31
JOSEPHSON EFFECT
Finally, a demonstration of the effect simply of mechanical motion was given, where us was caused by rotation of a circuit containing a junction pairso. It is interesting to go into the theory of this effect in a little more detail. If we rotate a bulk cylinder of superconductor, it is essential that over the sample as a whole us=ulaIIicc,because otherwise very large currents would flow. Since $Vq=O (or 2 n ~this ) means that
or
e
This is equivalent to the statement that the g-factor of the superconducting diamagnetism is exactly 2. The same B would appear in a rotating superconducting ring. Of course, under some circumstances B could be indeterminate to one or more additional flux quanta. In a ring containing Josephson junctions, finite currents can flow,and if in fact leA, we can make the opposite assumption that the magnetic field due to these currents is negligible, and that all the phase differencewill appear across the junctions. Then
s
Vq*dl- ( q l - q 2 )= 2nx
n
= 0,
k 1, ...
loop
where 'pi and q2 are the phase differences across the two Josephson junctions. This gives (m/A)2nR * ulaIIice 2nn = A q
+
so that the critical current of the rotating interferometer is a periodic function of Aq/2n = R 2 w m / k . (4.23) As Zimmerman and Mercereau30 point out, this experiment measures A/m and thus can be considered as a measurement of hjmc, the Compton wavelength; the value they quote is h/mc = (2.4 References p. 42
_+
0.1) x lo-'* cm.
32
[a. 1,g 4
P. W. ANDERSON
Josephson (private communication) has observed that there is an interesting “relativistic” effect which might be measured in this type of experiment. In both (4.22) and (4.23) the mass of the electron enters; the question is whether this mass is the rest mass or some kind of average inertial mass including kinetic energy corrections. Josephson argues that the correct mass is the rest mass corrected by the work function W,the energy difference between free space and the Fermi surface, i.e. that it is the total energy
2m0c2 - 2 w = 2m”c2 necessary to create a pair in the metal which determines the relevant mass m* which is to be used in (4.22) or (4.23). The argument is that the only gauge in which a Lorentz transformation gives simple results is that in which the potentials are zero outside the metal; then the frequency of J/ at rest (given by Aw =aE/aNpairs =2m*c2) is related to the wavelength in motion by a Lorentz transformation A la de Broglie. One nice way to see that not only the kinetic energy correction EF but also the potential energy V of the electron in the metal ( W =V-E,) must enter was also observed by Josephson. V may be thought of as the consequence of a surface electric field E in a layer d thick. The corresponding magnetic flux caused when this layer is rotated at a velocity u is @ = 2~
Rdv
-E C
RU
= 2n - V . c
The relative phase change caused by this is A-_q - _@J_ - R’w(2eV) 2n @o AC2 ’
the ratio of which to (4.23) is 2eV/mc2. As Josephson also points out, a rather larger effect can be induced by charging up the interferometer to high voltages, but this effect is rather trivially caused by the magnetic field of the moving charges. If it were wished to measure either of these effects, a basic limitation pointed out by Josephson is the difficulty of defining R of the interferometer in the rotating-interferometer experiment. Thus clearly the simple gyromagnetic experiment using an interferometer only as a fluxmeter and measuring the flux due to a rotating cylindrical sample will be much more accurate. The basic limitation there is thermal fluctuation of the flux, as discussed in the last section. The amount of flux is about 1 quantum/cps References p . 42
CH. 1,O 51
JOSEPHSON EFFECT
33
for a 1 cm3 sample, and the fluctuation in the same sample about 0.1 quanaccuracy we need very rapid and accurate rotation. tum; to get Perhaps more fundamentally interesting as an instrumental use of the interferometer is the measurement of gyromagnetic ratios in general magnetic materials. With conventional flux meters this is a very difficult experiment. Thermal noise is also a very severe limitation here, but it ought to be possible to get two or three-figure accuracy on ordinary paramagnetic materials. Of course, the interferometer has many other interesting device possibilities, especially as an extremely sensitive fluxmeter. One use which is not quite so obvious is as a sensitive ammeter. Because the input impedance is so low and its sensitivity to current so small, the energy sensitivity is many orders better than any other device, as demonstrated by CIarke 40.
5. Systems other than tunnel junctions showing interference phenomena Before going into the field of a.c. interference effects, a majority of which seem to have been observed in systems other than uniform tunnel junctions such as we have so far discussed, it may be worthwhile to have a brief section on these other types of “weak superconducting” or “weak superfluid” systems. In Josephson’s second paper 4, ZQ, the generalization from the pure Josephson effect to the general idea of “coupled superconductors” had already been made; and from the first it has been clear that it is not very easy to tell experimentally whether one is dealing with a tunneling supercurrent or a supercurrent flowing through a number of tiny metallic shorts. In fact many different systems have shown the extreme magnetic field sensitivity of critical current characteristic of the interference devices: tubular thin filmsz, flat long thin film bridges10, short thin film bridges in various geometries10.11, various kinds of point contacts32 and pressure contacts30, and even rather thick, wide films driven near critical current (Mercereau, private communication). Only one interference phenomenon has been observed in superfluid helium, the reason being that there are no good analogues to the Josephson effect except for the orifice geometry, and even that is either too strong a coupler or carries critical currents too small to measure conveniently12724. In the past there has been controversy in some instances about whether a given experiment on thin films in magnetic fields was better explained in terms of the static equilibrium of a system of vortices in the presence of the appropriate boundaries, or of a Josephson-like critical supercurrent resulting from interference between different parts of the supercurrent-carrying path 10. It must be clear from the theory we worked out in the last section, of the Referencesp . 42
34
P. W. ANDERSON
[CH.1,
55
Josephson junction in a magnetic field, that there is no distinction in principle between these two points of view. A Josephson junction exhibiting interference is a system of vortices in equilibrium under the appropriate boundary conditions. Conversely, the general theorem24 that the current is & = ( d U / d ( q , -qp,)>, the derivative of the energy with phase, indicates that a stable system of vortices - and therefore one which has a lower energy relative to other possible configurations with different phase difference ( q l - q Z )- can carry a larger supercurrent and thus will exhibit a lower effective resistance. As we deal with these systems we have also to lean more and more on the concept of “phase slippage”24*12. This concept proceeds from the Gor’kov-Josephson frequency condition (2.9) or, more generally, (3.7) :
That is, in any coupled superconducting system the appearance of a voltage (electron chemical potential ,u) difference can only mean that there is a corresponding rate of “phase slippage”. An applied current source represents a driving force causing the phase to slip, while the coupling energy tends to lock the phases together. In general, because of thermal and other fluctuations the phase will slip at a finite rate, which may be infinitesimal, as in a reasonably strong superconducting contact, or quite rapid, as in a thin film bridge very near T,. The more stable the structure of the vortex system at the contact and the less current is being forced through the slower will be the slippage, and thus the smaller the resistance shown - often, unmeasurably small. From this point of view all d.c. interference experiments are measurements of the rate of phase slippage vs. current and applied field. In the Rowell or Mercereau experiment, one has a sharp distinction between subcritical and supercritical currents, because the free energy barriers are rather high and the system is either stable or breaks down into very rapid flow above J,. In the Parks-Mochel10 experiments on thin film bridges, the barriers are relatively low (because near T,)and the dissipation high when phase slippage starts, so that one appears to be studying resistance in the transition region. In these experiments the resistance of a very narrow thin film bridge near T, is measured as a function of magnetic field, and found to exhibit quasiperiodic structure reasonably closely related to the fields at which an Abrikosov structure containing n whole vortices might fit in the film’s width (in a perpendicular field all thin films are type 11). The analogy with the References p . 42
CH. 1,
8 51
JOSEPHSON EFFECT
35
Rowel1 type of interference is clear: when a structure containing n vortices fits exactly into the film, the free energy is minimized, the structure is stable and phase slippage is hard. One might even suspect that the original Little-Parks experiment on thin film cylinders2 might well have a phase-slippage explanation. The usual explanation, that T,is modulated by the magnetic field, does not discuss the heart of the matter, which is by what mechanism the sample shows simultaneously resistive and superconducting properties. Phase slippage is occurring - then why not discuss directly how it is modulated by a magnetic field? Intermediate in character between point contacts and the Parks bridges is the Anderson-Dayernll thin film constriction. D.c. interference phenomena of great complication have been observed with these, but the main advantage, as with the point contacts, is the strong coupling to external electromagnetic fields which this geometry allows. The difficulty with this geometry for d.c. magnetic fields is that the entire film becomes type I1 in very small fields, and that therefore some resistance is developed almost everywhere. Only in nearly zero magnetic fields is the weak link clearly localized at the constriction. Virtually no theoretical work has been done on these thin film constriction devices. The critical currents have the right order of magnitude and temperature dependence to be explained by simple depairing, but a theory in terms of vortex motion across the constriction has not been ruled out. Particularly interesting is the strange and unexplained increase in critical current with a.c. power -the “Dayem effect”42. Now, we come to the various point contact devices. Many of the interferometer experiments of the Ford group31.39.44 were carried out actually not with tunnel junctions but with small Nb screws in pressure contact with Nb wires. Presumably, such a contact contains a thin oxide layer squashed between the metals (though a tiny true contact would probably serve as well) through which very local tunneling takes place. Point contact junctions have also been used by Grimes et al. in studies of the a.c. Josephson radiati0n29.9~. Clarke’s sensitive ammeter40 simply used blobs of solder on superconducting wires, making poor contacts which exhibited Josephson behavior. The final system possible is that in which the weak link is artificially created out of a strong one by driving high current, through it part of the time. This type of device shades imperceptibly into the measurement of flux quantization in truly macroscopic systems. While it can be quite useful as an interferometric device it does not really belong in the present article. References p . 42
36
[CH. 1,s
P. W.ANDERSON
6
6. Ax. quantum interference effects
In his original letter1 Josephson laid the groundwork for the theory and observation of the two closely-related alternating current interference effects: the appearance of a.c. supercurrents through weak links when a finite potential difference is applied across them, and the appearance of singularities in the d.c. I vs. Y characteristic at voltages such that: nhw = n’(pl - p 2 )
(6.1)
when external a.c. currents are applied at angular frequency w . Both of these phenomena are manifestations of the basic Josephson-Gor’kov (and one might as well add Einstein) relationship (3.7): dq dt
A-=p=-
aE dN
which simply states that the frequency of oscillation of the coherent matter field is given by the chemical potential (for electron pairs in the case of superconductivity). Since the supercurrent is a periodic function of q1- q2 (by the general principle of gauge invariance, all physical properties must be periodic functions of the phase with period 2n) this means that a.c. supercurrents must be associated with any voltage difference. In Section 3 we discussed rather fully why we assume that the state is a coherent one in these a.c. experiments; and the deeper background for the fundamental equations has been reviewed elsewhere24. Here we should like to first give an elementary discussion of the driven Josephson effect; write down the basic dynamical equations of the Josephson junction; and then to discuss briefly various experiments which have so far been done. By far the easiest, most general, and at the same time most accurate of the various interference experiments is the “driven” or “synchronized” a.c. effect. In the elementary derivation as given by Josephson1 one imagines a simple tunnel junction across which is maintained both a d.c. voltage Vo and an alternating voltage V, cos (w,t qo).The current (if w, and Yoare reasonably small, so that the Josephson-Riedel variation of 5, with o is not important) is given by
+
2e
= J~ sin -
(h
References p . 42
vat + 2e V, sin(o,t ~
hwa
+ cpo) .
)
CH. 1,
5 61
JOSEPHSON EFFECT
37
This is a simple frequency-modulated a.c. current which may be Fourier analyzed in terms of Bessel functions. In particular, whenever 2eV0 =nhw,, the current is perfectly periodic with frequency 0,/27c, so that all of the energy is in harmonics of 0,.One of these is the zeroth harmonic, or d.c., the amplitude of which is given by
when 2eV0 = nhw, . Jn is the nth order Bessel function. The important thing about (6.3) is that it is indeterminate: the mean current J, and thus the total power which is being fed from the as. source which maintains V , to the d.c. battery which maintains Vo
depends on the arbitrarily assigned phase angle cpo between the applied a.c. field V, and the spontaneous Josephson current at frequency 2eV0/h=coo. This means in turn that as we modify the characteristics of the external circuit which is supplying Vo so that the power drawn from it varies, the system can accommodate itself to such modifications by changing the phase cpo; thus for some large class of possible external circuits we expect to be able to observe different values of the current J f o r the same value of voltage V,: there will be a segment of the I-V characteristic where V is fixed at a finite value (1) and I varies, at many of the harmonics V0=nhwa. This phenomenon was indeed observed by Shapirog, who even verified the Bessel function dependence on V, in a rough way. For an example of the theory of this observation, let us imagine that the d.c. source is a constant voltage source V, applied through a resistance R which determines the load line:
As we vary V,, Vo will stay fixed and J will satisfy J=( V, - Vo)/Runtil J is greater than the appropriate critical current, which under ideal conditions will be given by
References p . 42
38
P. W. ANDERSON
[m. 1, Ei 6
To this simple discussion we need to make two addenda. First, suppose that the device involved is not a simple Josephson junction but, just for an example, a junction wide compared to Ap Then, the phase on the two sides can be varied only by passing “one-dimensional vortices” such as shown in Fig. 4 through the junction. The total current from such a vortex deep inside the junction is zero, but when it is near the surface a surface current will flow in the appropriate direction as discussed in our theory of the large junction. Thus the Josephson cursent as a function of phase or of time will be highly anharmonic. The equations of motion of such a system will be discussed briefly later; they are of such a complexity that no time-dependent solutions have been given, to my knowledge. In any case (6.2) will be replaced by
where f is some anharmonic but periodic function. Iff is anharmonic it is easy to see that the nth harmonic of 2eV0/h may beat with the mth harmonic of w, and give a d.c. current at “subharmonics” as well as “harmonics” of the fundamental voltage hwa/2e. This phenomenon was noted by Shapiro in some tunnel junctions, but is particularly striking in the measurements of Dayern11s42 on thin film bridges, as might well have been expected, since in the bridges the motions are almost certainly more vortex-like than sinusoidal. We show a typical I-V characteristic for an irradiated bridge in Fig. 9. Dayem makes the point (private communication) that it is probably because of the large harmonic content of his a.c. Josephson currents that radiation from his bridges was not directly observed. The second important point is that the basic nature of this type of effect is analogous to two rather similar nonlinear devices with whose operation we are fairly familiar, though in fact in each case the mathematical theory, beyond the simplest semi-intuitive considerations, is extremely complicated : the a.c. to d.c. power converter, and the synchronization circuit. We have already pointed out the a.c. to d.c. power conversion idea, and this analogy was extensively discussed in another review 24 : the “locomotive” analogy. The second analogy is to a synchronization circuit such as is contained in all cathode-ray oscilloscopes; we can think of the Josephson current as a freerunning oscillator, the frequency of which is read as the voltage Vo, and the a.c. signal applied as a “synch.” signal which triggers each cycle of the freerunning oscillator in phase synchronism with the external frequency, by means of the nonlinear coupling inherent in the Josephson effect. Both References p . 42
M. 1,9 61 2.
39
JOSEPHSON EFFECT
iiir I I
11
I
I
I
I
I
I
I
!
I !
!
I
1
SAMPLE NO.
9Ini
1
1.
I.
4
E
0
0
0
0.
2
’
’
2 2 3
4
1
1 5 4 3
.2L
r 3
2
3
Fig. 9. Experimental I-V curves for a Dayem bridge irradiated with microwaves at a frequency of 4.62 gigacycle. Relative microwave power in dB is the parameter. Voltage fluctuations in the steps are an experimentalartifact. (After Dayern and Wiegar~d~~.)
analogies indicate that the stronger the synchronizing signal, the more tightly controlled will be the Josephson frequency and therefore the voltage, and that for strong signals thermal noise will not be a factor. For this reason we can expect the relationship between V, and w, to be as exact as can be measured: this is, as originally emphasized by Josephson, by many orders the most accurate possible way to measure e/A. The synchronizationconcept (originally introduced in Ref. 5 ) indicateswhy the a.c. effect is so easy to see and so general. The relationship (6.1) between References p. 42
40
[CH.1,
P. W. ANDERSON
56
chemical potential and frequency is perfectly general, whether the phase changes with time by means of vortex motion, direct phase slippage as in a Josephson junction, flow of an Abrikosov structure, or whatever. Equally, we can expect that with strong enough applied synch. signals, almost any such motion can be forced to be regular and periodic. This is the reason we felt the best bet for an interference experiment in liquid He would be the driven ax. effect, and indeed the experiment did give positive results 12.This is the only interference experiment as yet successfully carried out in a precisely analogous way on both superconductors and helium 11. In view of the numerous experiments on the “synchronized” a.c. Josephson effect, there was no doubt that the a.c. Josephson currents existed, but the possibility of direct detection of radiation from them seems still to have stimulated very considerable effort and interest. The first successful attempt was by Giaeverl4 utilizing a second, different thin film device as his detector; thereafter two other groups reported completely external detection of the radiated powerl5. Intrinsic limitations on this power (coming essentially from the self-screening effect) are such that it can never be of much practical importance except possibly as a spectroscopic source in the infrared 29. All of these observations utilized the phenomenon of “Fiske steps”l3 which is undoubtedly a self-synchronization of the Josephson current to a.c. electromagnetic signals fed back from cavity modes of the tunnel junction structure. In discussing these it may be well to write down the equations of the dynamics of tunnel junctions. The dynamical term is easily inserted if we return to the derivation of equation (4.8) from the Maxwell’s equation for curl H . We must include the displacement current term in that equation if we are to allow time-dependence, so we start from
aH, 4x -=--J,+ ax c
1 ao,
--.
c
at
But now by (4.5’)
a% 2e -- -&,
ataZ
A
so that we get for the time-dependence of the phase shift a (wl
at References p. 42
- w 2 ) = -2ed E A
*
CH.
1,
5 61
JOSEPHSON EFFECT
41
Now we introduce the relationship (4.6)between phase shift and H,, into (6.5):
a2
kc
=
47T -
C
J1sin(w, - w2) +
he d2 -- (wl - w 2 ) 2edc at2
(e is the dielectric constant of the layer) which is the equation first written down by Josephson18 for cp =w1 - w 2:
where I , has been defined already (4.lo), and v, the effective electromagnetic wave velocity in the planar structure, is v=(
)e .
d e(2I d)
+
(6.7)
In general, this is 10-100 times slower than c, so that for usual junction sizes resonant modes of the purely electromagnetic 1.h.s. of (6.6)are to be expected in the microwave region. Josephson has given a very succinct discussion of two types of resonances which can be expected to follow from (6.6).Both may be understood best if we consider I , to be rather large. First is the Fiske step, which occurs at resonances of the 1.h.s. plus boundary conditions. We can think of the sin rp term as simply a weak driving term which generates a fairly large electromagnetic response, which in turn reacts back via the nonlinearity of sin cp to give a kind of self-synchronization at 2eV0 =hwres.Since the effective d.c. impedance is very low at a Fiske step, the breadth of the spectrum due to voltage fluctuations is relatively small and the radiation is easier to detect. A second type of resonance was discovered by Eck et al.43. In a magnetic field, as noted in Section 4, the junction takes on a periodic structure of wavelength about I QO/H,,(2A+ d ) . When this wavelength is equal to the electromagnetic wavelength A = 2 n v l o = hv/2eV
-
we can again expect a resonance, which turns out to be rather broad, and to show up only as a broad extra dissipation dependent on magnetic field. Dayem and GrimesS2 have observed radiation from point-contact tunnel junctions; here the resonant, low-impedance structure is avoided and relatively good coupling may be achieved to the radiation field. (That was one References p . 42
42
P. W. ANDERSON
[CH.
1
of the original purposes of the Anderson-Dayem structure l1, but the point contacts do not have the disadvantage of large anharmonicity.) Most interesting, however, is the inverse performance of these junctions as infrared detectors when biased near their critical currents so that the appropriate dynamical equations are near a point of instabilityzs. The appropriate equation for such a system (Werthamer and Simon, private communication) is azq drp + wo2 sin rp = J,, sin (oat+ q 0 )+ Jdc- R , (6.7) atz dt ~
where J,, is the applied signal of frequency o,,Jd, the bias current, and R drp/dt is the radiation and other resistance, When Jdo=o$, (6.7) may be expected to be extremely sensitive to the J,, term. It remains, in this as in most cases, my opinion that the great future of the a.c. Josephson effect lies not in the straightforward direction of using the radiated power but in their unique possibilities as nonlinear driven devices, as detectors, as phase-locked discriminator systems which convert an a.c. signal directly into a voltage, or in yet more interesting and complicated devices. Notes added in proof. The following are two developments of special interest occurring since the article was sent to press: 1 ) Kharana and Chandrasekhar45have repeated the measurements of Ref. 12 with improved equipment and results. 2) Parker, Taylor and Langenberg46 have determined 2e/h to a precision greater than previous measurements (2e/h=483.5913 f0.0030 megacycles/ microvolt), and great enough to be relevant to quantum electrodynamical effects. REFERENCES
5
8
B. D. Josephson, Phys. Letters 1,251 (1962). W. A. Little and R. Parks, Phys. Rev. Letters 9 , 9 (1962). R. C. Jaklevic, J. Lambe, J. E. Mercereau and A. H. Silver, Phys. Rev. 140, A1628 (1965). B. D. Josephson, Trinity College Fellowship Thesis (unpublished). P. W. Anderson, in: Lectures on the Many-Body Problem, Ravello 1963, Vol. 2, Ed. E. R. Caianello (Academic Press, 1964) p. 115. P. W. Anderson and J. M. Rowell, Phys. Rev. Letters 10, 230 (1963). J. M. Rowell, Phys. Rev. Letters 11,200 (1963). S. Shapiro, Phys. Rev. Letters 11, 80 (1963); S. Shapiro, A. R. Janus and S. Holly, Rev. Mod. Phys. 36, 223 (1964). R. C. Jaklevic, J. Lambe, 3. E. Mercereau and A. H. Silver, Phys. Rev. Letters 12, 159 (1964).
CH.
11
JOSEPHSON EFFECT
43
R. D. Parks, J. M. Mochel and L. V. Surgent, Phys. Rev. Letters 13, 331a (1964); R. D. Parks and J. M. Mochel, Rev. Mod. Phys. 36,284 (1964). P. W. Anderson and A. H. Dayem, Phys. Rev. Letters 13, 195 (1964). l2 P. L. Richards and P. W. Anderson, Phys. Rev. Letters 14, 540 (1965). l3 D. D. Coon and M. D. Fiske, Phys. Rev. 138, A744 (1965). l4 I. Giaever, Phys. Rev. Letters 14,904 (1965). l5 I. R. Yanson, V. M. Svistunov and I. M. Dmitrenko, Zh. Eksperim. i Teor. Fiz. 47, 2091 (1964) [English transl.: Soviet Phys.-JETP 20, 1404 (1965)l; D. M. Langenberg, D. J. Scalapino, B. N. Taylor and R. E. Eck, Phys. Rev. Letters 15, 294, 842 (1965); Proc. IEEE 54,560 (1966). IR J, E. Zimmerman and A. H. Silver, Phys. Rev. 141, 367 (1966); Solid State Commun. 4,133 (1966); and Refs. 3 and 30. l 7 M. H. Cohen, L. M. Falicov and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962). B. D. Josephson, Advan. Phys. 14,419 (1965). l8 P. G. DeGennes, Phys. Letters 5,22 (1963). 20 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108,1175 (1957). 21 L. P. Gor’kov, Zh. Eksperim. i Teor. Fiz. 36, 1918 (1959) [English transl.: Soviet Phys.-JETP 9, 1964 (1959)l. 22 J. R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964). P. W. Anderson, N. R. Werthamer and J. M. Luttinger, Phys. Rev. 138, A1157 (1965). a4 P. W. Anderson, Rev. Mod. Phys. 38,298 (1966). 26 N. R. Werthamer, Phys. Rev. 147,225 (1966). 26 E. Riedel, Z. Naturforsch. 19a, 1634 (1964). 27 D.J. Scalapino, Phys. Rev, Letters, to be published. 28 C. C. Grimes, P. L. Richards and S. Shapiro, Phys. Rev. Letters 17, 431 (1966). B. D. Josephson, Rev. Mod. Phys. 36, 216 (1964). 30 J. E. Zimmerman and A. H. Silver, Phys. Rev. 141, 367 (1966). 31 A. H. Silver, R. C. Jaklevic and J. Lambe, Phys. Rev. 141, 362 (1966). A. H. Dayem and C . C. Grimes, Appl. Phys. Letters 9, 47 (1966). 33 F. London, Superfluids, Vol. 1 (Wiley and Sons, New York, 1950) Section 8. 34 R. A. Ferrell and R. E, Prange, Phys. Rev. Letters 10,479 (1963). 35 P. G. DeGennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966) p. 240. 3~3M. Tinkham, Rev. Mod. Phys. 36,268 (1964). 37 R. C. Jaklevic, J. Lambe, A. H. Silver and J. E. Mercereau, Phys. Rev. Letters 12, 274 (1964). 8* Y. Aharanov and D. B o b , Phys. Rev. 115,485 (1959). 99 J. E. Zimmerman and J. E. Mercereau, Phys. Rev. Letters 14, 887 (1965). 4o J. Clarke, Phil. Mag. 13, 115 (1966). 41 J. E. Mercereau and L. L. Vant-Hull, Phys. Rev. Letters, to be published. A. H. Dayem and J. J. Wiegand, Phys. Rev., to be published. These results were repeated by A. F. G. Wyatt, V. M. Dmitriev, W. S. Moore and F. W. Shard [Phys. Rev. Letters 20, 1166 (1965)l. R. E. Eck, D. J. Scalapino and B. N. Taylor, Phys. Rev. Letters 13, 15 (1964). 44 J. E. Zirnmerman and A. H. Silver, Solid State Commun. 4, 133 (1966). QS B. M. Kharana and B. S. Chandrasekhar, Phys. Rev., to be published. 46 W. H. Parker, B. N. Taylor and D. N. Langenberg, Proc. Xth Conf. Low Temp. Phys., Moscow, 1966, to be published; Phys. Rev. Letters, to be published. la
C H A P T E R I1
DISSIPATIVE AND NON-DISSIPATIVE FLOW PHENOMENA IN SUPERFLUID HELIUM BY
R. DE BRUYN OUBOTER, K. W. TACONIS AND W. M. VAN ALPHEN KAMERLINGH ONNES LABORATORIUM, LEIDEN
CONTENTS: Introduction, 44. - 1 . Superfluidity,the equation of motion for the superfluid, 45. - 2. The critical superfluid transport in very narrow pores between 0.5 “K and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow, 54. - 3. Superfluid transport in the unsaturated helium film, 62. - 4. Dissipative normal fluid production by gravitational flow in wide channels with clamped normal component, 64. 5. The dependenceof the critical velocity of the superfluid on channel diameter and film thickness, 72.
-
Introduction Complementarily to earlier work reviewed in Progress in Low Temperature Physics on the properties of superfluidity (see also Andronikashvili and Mamaladze, and Anderson, in this volume) a research program of some interest has been accomplished in the Kamerlingh Onnes Laboratory. A number of flow experiments performed with He11 will be reported here. It concerns sub- and super-critical flow of the superfluid. The experimental work will be treated in four Sections (2-5) and is preceded by a short summary of the properties of a superfluid, including the equations of motion (Section I). The picture may facilitate the discussion of the presented material. The first series of experiments (Section 2) concern a new attack on the characteristics of the flow through pores. The gravitational flow is determined under the stringent condition that the temperatures at both sides of the “superleak” are known as precisely as necessary for an eventual fountain pressure correction. It was found that indeed real critical flow exists which is pressure and temperature independent, as it should be ; in contradiction References p . 76
44
CH.
2,p 11
FLOW PHENOMENA IN SUPERFLUID HELIUM
45
with many earlier experiments in which apparently small unobserved temperature differences gave rise to misleading results. The second study (Section 3) is directed to still smaller channel dimensions formed by the thickness of the helium film. It substantially increases the data on flow of the saturated and unsaturated films, especially at lower temperatures, down to 0.7 O K . In Section 4 a new technique is described which is introduced to detect succesfully the critical velocity in wide channels by means of measurements of the onset of energy dissipation in the superfluid flow using a calorimeter, whereas the dissipation above the critical velocity as a function of u, is found to obey a very simple formula for the observed friction. Finally in Section 5 a proposal is made for a general empirical relation between the critical velocity and channel dimension. It originates to a certain extent from the observation that earlier trials in this respect have been mislead by the fact that in wide channels often the turbulence of the normal fluid is taken as the appearance of a critical velocity of the superfluid. 1. Superfiuidity, the equation of motion for the superfluid
The equation of motion for the superfluid can, according to Landaul, be derived in the following way: in a simply connected amount of superfluid, the change in energy dU, caused by transport of a small quantity of superfluid dM, in or out of a considered part of the system, keeping the momentum p of the normal fluid constant with respect to the superfluid, is equal to (aU/aM), dM. Only superfluid is transported and the distribution of the normal fluid remains unchanged. Since the superfluid carries no entropy, one has (dUjdM),,, d M = & dM, in which p& is the chemical potential of He11 per unit of mass. The change of energy by a supeAuid transport of a unit of mass is equal to the chemical potential p. This quantity can be considered as the potential of the force acting on the superfluid; hence one gets the following equation of motion for the superfluid per unit of mass dvs -=dt
0 = - -Vp 1 VpM P
or
P S
= - -Vp
P
References p. 76
+ p,SVT,
+ SVT
R. DE BRUYN OUBOTER et
46
al.
[CH.
2, 5 1
whereas for duJdt =O (or V& =O), eq. (2) delivers the fountain relation of H. London: dp/dT=pS. However, superfluidity is restricted to certain possible types of flow by the restriction that the flow is irrotational (Landau', London2 and Feynman 3)
v A Us = 0.
(3)
This implies that the superfluid velocity us is the gradient of the classical velocity potential p; us=Vp and aus/at=(a/af)Vp=V dp/at. This gives us the following equation of motion (per unit of mass) for an ideal irrotational liquid
pM is the generalized chemical potential defined by pM = p k The equation of motion per 4He atom becomes
m -a U=S at
V(p0 + imo,2) = - VP,
+
41):.
(5)
in which rn is the mass of the 4He atom, p o =m& its chemical potential and p = m p M its generalized chemical potential (eqs. (4) and ( 5 ) are valid for a steady flow, when the normal fluid velocity u, =O), if du,/at =O the equation of motion gives us the theorem of Bernoulli VpM = v ( p i
+ 30:)
(6)
=0
(and if V A us was initially zero it will always remain zero). Instead of eq. (4) Landau considers also the more general case in which u,#O and pn#O ( p , ps= p ) and gets the equation
+
The general chemical potential p contains in this case also a kinetic energy term of the relative motion of the superfluid and normal excitations. For a steady flow
1
=o.
A unique feature of the superfluid is its single macroscopic quantum state for a11 condensed helium atoms. As a consequence of the Bose-Einstein condensation an enormous number of helium atoms are in exactly the same quantum state and have exactly the same wave function $,. According to References p . 76
CH.2,
5 11
FLOW PHENOMENA IN SUPERFLUID HELIUM
47
Feynman4 and Anderson5 t,bzt,bs can be interpreted as the density of the All condensed helium atoms move as a condensed helium atoms p s / p =l1,9~1~. unit rather than individually. Another way of reasoning is to interpret yS=,/(p,/p) e*@ as the internal order parameter of the condensed superfluid, which is a completely determined, single valued, complex quantity (Anderson5, Ginzburg and Landaus, Ginzburg and Pitaevskii?) with a phase factor ei@(Beliaevs) - irt/r *s = J(Ps/P) ei4= J(Ps/P) e (7) The frequency of the internal order parameter is given by v = - - =1 - 84 -
c1
h'
2.rcat
A new feature appears when a superfluid occupies a multiply-connected region. For instance, suppose that the superfluid moves in circles around a cylinder (in principle this quantum state can be reached by cooling down liquid helium in a rotating bucket, containing a cylinder in the middle, through its lambda-temperature). Such a flow has a circulation K =
fus*dsZ 0 S
around the cylinder, although V A us-0 everywhere in the superfluid. The phase 4 of the wave function t,bs = J ( p , / p ) eibchanges by an integral multiple of 2n in going around the cylinder
S
S
This implies that the circulation K around the cylinder is quantized, as first was predicted by Onsagero and Feynman3
s
S
(9) h cm2 sec-' . I C = ~ - = 0.997 x m In this case of cylindrical symmetry the superfluid velocity us is given by u , = ~ / 2 n r(r > radius of the cylinder). Vinen10 and later Whitmore and Zimmermanl1 have experimentally verified this relation for the quantized circulation around a small wire by measuring the Magnus force (or lift force) on the wire. Reppy and Depatielz References p . 76
48
R. DE BRWN OUBOTER
et al.
[CH.2, $ 1
and Mehl and Zimmerman 13 have observed long-living persistent currents of superfluid in powder-filled vessels and have observed that the total amount of angular momentum present in such a sphere can be changed reversibly solely by warming up or cooling down. The amount of angular momentum of the persistent current is proportional to the superfluid density ps ( T ) , which means that the circulation around each flow loop within the sphere remains constant at varying temperatures. Only the amplitude J [ p , ( T ) / p ] of the wave function $s changes by warming up or cooling down, and the condensation (or evaporation) of the helium atoms occurs in a state of definite phase, that of the superfluid condensate wave function. Up to now we have mainly considered regions that are multiply-connected in a straightforward macroscopic manner, for example superfluid flow around a solid cylinder, but as we will discuss now it is possible to have a region that is indeed multiply-connected by having a line through the fluid on which the amplitude of the wave function $s vanishes. From experiments one knows that it is possible to have, well enough above absolute zero and below lambda, normal excitations (phonons and rotons) as well as macroscopic excitations, of highly localized regions of vorticity, in the form of singularities in the velocity field. The “normal” core of such a vortex or vortexring (radius approximately a few A) is excludedfrom what we call the superfluid, in this way forming a multiply-connected region. Except for these “normal” cores the irrotational condition must hold throughout the superfluid. The pressure in the vortex near its centre can maintain a very small cylindrical hole in the superfluid against the action of the surface tension (Feynman3). Rayfield and Reif14 have shown that ions in superfluid helium at low temperatures can be accelerated to create freely moving charge carrying vortexrings, the ions being probably trapped in the core of the vortexring. They measured their energy E and velocity v and found a welldefined relation of v versus E (v E - ’) from which the circulation u and the core radius a can be determined by means of the relations
-
’
[
E = i p s ~R In
raR)-(:)] -
and v = L4nR [1.(s>-(a)],
(10)
in which R is the radius of the vortexring, u is found to be one quantum of circulation, u,, =hlm and the core radius a=1.3 A. These freely moving charge carrying vortexrings are only observed at sufficiently low temperatures (0.3 OK).At higher temperatures a pronounced energy loss is observed due to collisions with the normal excitations, especially the rotons, and the vortexrings are slowed down. References p . 76
CH. 2,
0 11
FLOW PHENOMENA IN SUPERFLUID HELIUM
49
Since V A v,=O in the superfluid, its velocity u, can be defined as the gradient of a scalar velocity potential in such a way that the potential may be identified with the phase 4 of the internal order parameter (or wave function) I),(Onsager and Penroseg, Feynman3) by means of the relation A us=
m
v4,
where (h/m) 4 can be identified as the classical velocity potential q =(h/rn) 4 per unit of mass. This can be verified since by substituting eq. (10) into eq. (9) one gets eq. (8). Eq. (11) is satisfied provided the superfluid velocity us does not change significantly over a distance less than some characteristic distance (“healing length” or “coherence length”) of the order of the atomic distance in the liquid. Sometimes one also uses the integral relation 2
(12)
The well-known De Broglie wavelength of the moving superfluid follows from eq. (1 2) as
A=-.
il
mu, Using eq. (1 1) one can transform eq. ( 5 ) in the following way
v
[ ::
h-+po++rnu, 2]
=v
[ :: ] = o . A-+p
Eq. (14) has the same form as the general Bernoulli equation (4) of classical hydrodynamics for an ideal irrotational liquidl5. Integrating eq. (14) gives that the term between the brackets is at any instant equal to the same constant throughout the fluid, which may be still a function of the time. However, this constant will appear to be time independent and equal to zero. Instead of starting with the classical equation of motion (l), one can better derive directly from the time dependent Schrodinger equation
(describing the single macroscopic quantum state, with the same wave function $s =,/(p,/p) eie for all condensed Bose particles and a potential References p . 76
R. DE BRLIYN OUBOTBR et al.
50
[CH.
2,8 1
energy p) the equation of motion5
a4
+ po + at
h-
34
=h at
+ p = 0.
Eq. (15) is valid if the superfluid velocity varies slowly with position and if the amplitude of the wave function I)s is practically constant. If one also considers a gradient in the internal order parameter, one has to add a term (Ginzburg and Pitaevskii ')
to the left-hand side of eq. (15), a more general situation. Such a term is of importance if we have a boundary between the superfluid and the wall or normal core of a vortex. Moreover, sometimes is added an additional term involving the total density, for example in the helium film near the wall. Following Andersonl6.5, one may take the line integral of this equation along an arbitrary path which is stationary through the superfluid from a point 1 to a point 2. Suppose that the end-points 1 and 2 lie in a region where the fluid is nearly at rest and undisturbed, then one may neglect the term +mu: and one gets the integrated form of this equation
which describes, even when dissipation is present, the connection between the change in the chemical potential and the average rate of production of vorticity. When two baths of superfluid helium are connected with each other by means of a superleak, in every path in the superleak which connects both baths and which remains superfluid, the phase of the order parameter, will slip relative to the other bath as long as there is a difference in the chemical potential Ap. On the average in 1 sec the phase will slip Av =A p / h times 2n. The phase slippage takes place by means of motion of vortexrings (or by vortices). Since I), is single-valued the phase 4 of a vortex will change by n . 2 on ~ going around its normal core. Experiments indicate that free vortexrings have n = 1 as quantum number of circulation. If Av = d p / h vortexrings (excited states) are produced during 1 sec the phase slippage can take place in a proper way (Andemonla). The vortexrings produced in this way stream into one of these two baths and are slowed down by collisions with rotons (most likely radiating second sound). The vanishing vortexrings become References p . 76
CH. 2,g 11
FLOW PHENOMENA IN SUPERFLUlD HELIUM
51
probably eventually rotons by themselves’@‘. In this way the difference in the chemical potential energy is converted into heat by producing normal fluid. According to Andersonle*5 dissipative effects in superfluids follow from this concept of phase slippage and the gradient in the generalized chemical potential p=po++mu: gives the average rate of vortex production. Normal fluid counter flow and the dissipation phenomena have probably no influence on the phase slippage concept. Richards and Anderson16 have observed this rate of vortex production (phase slippage) at a small orifice by synchronizing the vortex motion to an ultrasonic frequency (the so-called a.c. Josephson effect) in which an a.c. flow is superimposed on the d.c. flow. Furthermore one can formulate a law of induction of circulation (vorticity) by integrating the equation of motion (1) along a closed integration line in the superfluid
S
S
when superfluid flows through a We now determine the critical velocity small orifice, with radius r, from one helium bath to another when there is a difference in its chemical potential A p between both baths, by supposing that A v = A p / h vortexrings (eq. (16)) are produced in 1 sec. The amount of energy which flows per unit of time from one bath to the other is equal to ijss
2
n, A y = E Av or &, =
E n,nr2h ’
n, is the number of condensed helium atoms per unit volume (ps=nsm). Suppose E is the energy necessary to form one vortexring with radius R =r, which is given by eq. (10)’ substituting for E eq. (lo), for Av eq. (16), and for K = K ~=h/m in eq. (17), gives
This expression gives an explanation for the critical transfer rate, and is nearly equal to the familiar Feynman’s expression17
We would like to remark that eq. (18) is independent of LIP (see Section 3). The values predicted by these equations are, however, not in agreement with References p . 76
52
R. DE BRWN OUBOTER et
[CH.2, 5 1
al.
the experimentally observed critical velocities. We come back to this point later in Section 5, in which is discussed the dependence of the critical velocity of the superfluid on the channel diameter and film thickness. The boundary condition for the superfluid gives a special problem (Ginzburg and Pitaevskii 7). When a superfluid is moving along a solid wall its tangential velocity us cannot gradually decrease to zero since V A v, =O. Just the opposite case we have in a viscous normal fluid flow, where v, =O at the wall. In the bulk superfluid the internal order parameter p , / p = l$J2 in general is constant and unequal to zero. However, generally on a solid wall and at the free liquid surface and on the axis of a vortex line the amplitude of the order parameter p , / p =1t,!~,1~ =O is zero. Summarizing the boundary condition one gets: in the bulk
at the wall PAP = l$,I2 = 0
A m
us= - V 4 # 0
superfluid t
=
I$,(T)I’
z0
fi
v A V, = 0 +
normal fluid
PnlP = 1
P,(T)ip
us = - v4 # 0 m pn/p
+ p,/p = 1 ,
p,/p # 0
(normal component) u, = 0
+
Poiseuille’s law
--f
v, # 0
For a normal fluid (and also for the normal component of HeII) the equations are much more complicated, because the effects of viscosity ( q ) have to be included and the liquid is no longer rotation-free V A u = 0. The flow of an ordinary incompressible viscous fluid is governed by the Navier-Stokes equation of motion
When all acceleration terms are zero (du/dt=O) (or nearly equal to zero) one gets Poiseuille’s equation for laminar flow vp = qv2v,
(21)
for which the exact solution is known. At high enough velocities in the completely turbulent region (du/dt#O) no exact solutions are known. In retrospect of a later discussion of effects of turbulence (Section 5) which are References p. 76
CH. 2,
5 11
FLOW PHENOMENA IN SUPERFLUID HELIUM
53
observed in He11 by Staas, Taconis and Van Alphenl*, one can define two dimensionless Reynolds numbers in the way introduced by these authors for a type of flow with a circular cross-section 2rpB Pr3 Re,=and Re,= t ~ p , 1 41
in which ij is the mean velocity over a cross-section and r is the radius of the tube. The relation between Re, and Re, for a turbulent flow can be obtained from the empirical equation by Blasius and is then represented by Re, = 4.94 x loL3
(23)
In laminar flow through a channel with circular cross-section the law of Poiseuille becomes rz (or: ij = - -Vp).
Re, = Re,
8rl
(24)
The transition from laminar to turbulent flow has been studied by Reynolds. The value of the critical velocity for onset of turbulence corresponds usually to a Reynolds number Re, =2rpij/q between 1200 and 2300. For the normal and superfluid component together one has an equation of motion of the form of eq. (20) du, dun + Pn- = - v p dt dt
Ps-
+ qnv2un,
in which q,, is the viscosity of the normal component. Combining eq. (25) with the equation of motion for the superfluid (eq. (2)) one obtains 'the Navier-Stokes equation of motion for the normal component dun dt
P -=--
Pn P
Vp
- p,SVT + ~ , , V 2 ~ , ,
(eq. (14) remains unchanged). Furthermore, sometimes one introduces the idea of mutual friction. The first description of the interaction between the normal and superfluid component was made by Gorter and Mellinkl9Y2Oin order to describe some aspects of thermal conduction in moderately narrow channels. They proposed a mutual friction force of the following form F sn = ~ ~ n ~ s I unIz u s (8s
References p . 76
-uJ*
R. DE BRWN OUBOTER et al.
54
[a. 2,§ 2
Adding this extra friction force to the hydrodynamical equations, they obtain
du, p,dt
+ pndun = - Vp + qnV2un dt ~
(eq. (25)).
However, in classical hydrodynamics the phenomenon of turbulence, for instance; does not require a revision of the fundamental hydrodynamical equations, but rather a consideration of special types of solution of the ordinary Navier-Stokes equation. The phenomenon of mutual friction is present in HeII, but this does not imply the necessity of adding those mutual friction terms in the equation of motion (eq. (25)). One may expect to find mutual friction between the normal and the superfluid when vortices are present in the superfluid, since the excitations constituting the normal fluid will be scattered by a vortex. With respect to this picture of mutual friction Hall and Vinen20 have investigated the attenuation of second sound in rotating helium. These experiments are fully discussed by Vinen20 in Progress of Low Temperature Physics, Volume 111, Chapter 1. 2. The critical superfluid transport in very narrow pores between 0.5 OK and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow An advantage in studying the characteristics of superfluid flow in narrow pores (“superleaks”) is the possibility to obtain a pure superfluid flow as the normal component is quite immobile due to its shear viscosity qn. Moreover very high superfluid velocities (us
CH.2,g 21
FLOW PHENOMENA IN SUPERFLUID HELIUM
55
Fig. 1. The apparatus in which is measured the critical superfluid transport in different powders. Left: for measuring at ordinary *He bath temperatures above 1 OK.Right: for measuring below 1 OK by means of a SHe cryostat. (V) reservoir, (M) gauge glass, (Tv) thermometer. temperature reservoir, (Tb)thermometer temperature bath, (D) displacer (B) bellows, (S) superleak and (Ts) thermometer temperature superleak.
References p. 76
56
R. DE BRUYN OUBOTER et al.
[CH.
2, 5 2
sizes of 0.7 and 0.5 pm and two bits of vycor glass, each with a mean pore size in the neighbourhood of 60 A. The apparatus in which these experiments are carried out is shown in Fig. 1. The apparatus on the left-hand side is used at 4He temperatures, and consists of a superleak ( S ) , a bellows (B), and a reservoir (V). The superleak ( S ) is pressed in a chamber (0.7 mm in diameter and l cm long), drilled in a brass block, with a powder-filled entrance on each side, in order to restrict the critical effects to the narrow part of the superleak. The copper bellows (B) connects this superleak with the reservoir (V). Due to the large surface of the bellows the temperature change is kept low when the superfluid flows through the superleak out of or into the bath. In spite of this, the Kapitza resistance at the surface of the bellows is still large enough to produce a temperature difference with a magnitude of 10-50K
cm
2
t
.t
0
_ _ _ _ _ _ ~ ~
40
P°K
the K a p i t z a temperature
difference ovar tho bellows
1
Fig. 2. Above: the level differenceZ between reservoirand bath as a function of the time t . Below: the Kapitza temperature difference AT over the bellows as a function of the time 1.
References p . 76
CH.2, 5 21
57
FLOW PHENOMENA IN SUPERFLUW HELIUM
at a flow rate of 1 cm3/min. By means of measuring this temperature difference the effective pressure head can be calculated by correcting the gravitational pressure head for the fountain pressure. This is demonstrated in Fig. 2, where the levels of bath and reservoir are plotted as a function of time. The Kapitza temperature difference A T over the bellows is measured, and also plotted in the figure. The corrected bath level for this temperature difference is given by the dashed line in the figure. It is remarkable that already the flow rate is independent of the pressure head. The second apparatus shown on the right-hand side of Fig. 1 is used to determine the flow capacities of jeweller's rouge superleaks down to 0.5 OK. The copper block in which the superleak is placed contains also a 3He cryostat and is surrounded by a vacuum jacket. By means of raising or lowering the displacer (D) a level difference between the reservoir and the bath can be created. An important result obtained in all the experiments, is that the flow capacities of jeweller's rouge and vycor glass are independent of the effective pressure head.
2000
I500
1000
500
F
t 0
05
T
-10
2 0 OK
15
Fig. 3. The critical transfer rate as a function of temperature for different powders. (0) A60, = 1.2 mm; (0)TH Delft, = 0.7 mm; vycor 11, = 4 rnm; (0) TH Delft, q5 = 1.2 mm; (A) vycor I, 4 = 4 mm.
+
References p . 76
(m)
+
58
R. DE BRUYN OUBOTER et
ul.
[CH.2, 8 2
In Fig.3 the critical flow rates for vycor glass and two different powders, labelled “Am” and “TH Delft” are shown as a function of temperature. Powder “A60”, being of larger grain size ( ~ 0 . pm) 7 shows a negligible change in lambda-temperature, and a constant critical flow rate at temperatures lower than 1.5 “K. The two powders “TH Delft” ( ~ 0 . 5 pm) show lambda-shifts of about 0.1 OK and 0.2 “K, and also the vycor glass gives shifts. The critical flow rate is still increasing down to the lowest temperatures at which the measurements have been performed. The reason that the two curves given by the open squares and the filled circles do not coincide
rot
0.0
47.d
t 0
Fig. 4. The streamlines (solid) and the lines of equal velocity potential or phase (dashed) for an irrotational fluid in a Venturi tube. References p . 76
CH. 2,9 2)
FLOW PHENOMENA IN SUPERFLUID HELIUM
59
can be understood from the fact that both superleaks differ in diameter. Using the same press, the smallest superleak has sustained a higher pressure, and is therefore more closely packed. This results in an increase in lambdashift and a decrease of the critical flow rate due to a smaller void space between the grains. From the lambda-shift of the vycor glass the mean pore diameter can be estimated22 at 60 A. Using for the void space a value of 25% of its total volume as given by the manufacture^-23, the actual superfluid velocity in the pores can be calculated to be of the order of 20 cm/sec at 1.4 OK,which is in agreement with the values given by Atkins17. The mean open cross-section of the jeweller’s rouge is determined from the observed level oscillations, produced by the kinetic energy in the superleak in a flow experiment when the bath and reservoir level equalize. With the help of these values superfluid velocities of about 30 cm/sec are found. So that reasonable agreement between the two estimations is obtained. The results can be found in Fig. 18 of Section 5, where they are compared with other data on critical velocities and fit satisfactorily. Due to the possibilities to obtain high superfluid velocities in these powders, one can try to verify whether a Pitot or Venturi tube gives the correct Bernoulli pressure difference. Meservey24 has shown from Bernoulli’s equation (6) that a Venturi tube or a Pitot tube will have practically no level difference for an irrotational superfluid. Whereas the Navier-Stokes equation (20) for a viscous flow, when properly corrected for the drop in pressure along a tube by friction, gives a level difference g A z = + A u 2 . The fringing field in a standpipe for an irrotational liquid was calculated by Meservey and in Fig. 4 are shown the streamlines (solid) and the lines of equal velocity potential or phase 4 (dashed). Everywhere in the liquid jSu, ds =O. If the flow field everywhere is irrotational there will be a finite
Fig. 5. Left: the Venturi tube for a viscous fluid (q # 0) V tube for an irrotational superfluid V A References p . 76
A
on # 0. Right: the Venturi
Us = 0.
60
R. DE BRUYN OUBOTER et
al.
[CH.2, 5 2
velocity at the surface of the liquid in the standpipe. However, for even a moderate height of the liquid in a standpipe, the velocity falls off exponentially with height and so in the usual case the term +ul is practically zero in each pipe. This implies that the level difference approaches zero as indicated in Fig. 5. A Venturi tube with three standpipes (A, B, C), one at each end and one at the constriction (see Fig. 6), in connection with a reservoir, is completely filled with compressed jeweller’s rouge. The parts of the standpipes are partially filled with the compressed powder. A superfluid flow through the constriction is effectuated by raising or lowering the apparatus relatively to the bath level. The results of these experiments are shown in Fig. 6 for three typical situations. First a sudden level difference is applied by quickly raising the apparatus over a few cm, after which the position of the levels in the three standpipes are plotted as a function of time. The level A is always at the same position as the level in the reservoir, the levels B and C are always equal with each other and with the bath level, apparently the critical constriction is located between A and B. Although the velocity in the constriction near standpipe B is about 30 cm/sec and tp,u,2~0.45cm helium, no level difference between B and C is detected. Secondly, instead of raising, the apparatus is suddenly lowered over a few cm. Again the levels B and C are equal with each other and the bath level, and A is equal to the level in the reservoir, showing that the critical constriction still is located between A and B. In both cases the same critical flow rates are found. Apparently irregularities in the density of the powder are such that between A and B the critical velocity is reached first, and therefore the potential energy, originating from the pressure head is dissipated between A and B (see Section 4). At the bottom of the figureis shown what happens when the apparatus is continuously raised in such a way that the flow rate is just below the critical one (~ 29c m / s e c ) .Now all levels A, B and C are equal in spite of the high velocity in the constriction ( x 2 9 cm/sec). This experiment indicates or even confirms that the flow through the superleak is irrotational (see eq. (3)). The remaining possibilities to measure the Bernoulli pressure is to measure the pressure with a membrane gauge at the entrance of the standpipe or using a Rayleigh disk (Pellam25126).Pure superfluid flow (u,
CH. 2, $21
61
FLOW PHENOMENA IN SUPERFLUID HELIUM
A BC
A,
cm
P-3.7
. .
mm * v=v,
mi n
---_--___
4
apparatus not moving mi n
t
v
A , --37
.t
------
-
. .
mm , v =v,
rnin
apparatus not moving min
t
apparatus moves in such a way t h a t : A,,-35
t
. .
mm, v
min
Fig. 6. The superfluid flow through a powder-filled Venturi tube. A, B and C are manometers in connection with both wide sides and the constriction. Above: emptying the reservoir with critical flow rate, the levels B and C are equal with each other and the bath. In the middle: filling the reservoir with critical flow rate, the levels B and C are again equal with each other and the bath. Below: the apparatus is raised in such a way that the critical flow rate is just not exceeded; all levels are equal.
References p . 76
R. DE BRWN OUBOTER et
62
[CH.2, 6 3
al.
torque, t =+a3p,o,Z, exerted on a Rayleigh disk with a radius a in a superfluid “wind” with velocity n, is determined by measuring the deflection angle c1, and values for the superfluid fraction p , / p are obtained in agreement with the values (1 -p,/p) of Andronikashvili27. The critical velocities are obtained by observing the moment when a resulting force appears. The values obtained in this way will be discussed in Section 5, Fig. 18, on the critical velocities. 3. Superfluid transport in the unsaturated helium film
A special type of “channel size” is formed by the thickness of the helium films. The flow of the film can be divided in two regions: that of films under saturated and unsaturated vapour pressure. Here we concentrate our attention mainly on the unsaturated films. Recently Fokkens, Taconis and De Bruyn Ouboter 28 investigated the behaviour of superfluidity in unsaturated films, by measuring the heat conduction of the film, adsorbed on the inner wall of a glass tube (see Fig. 7b), as used by earlier investigators in the higher temperature range29-81. One side of this tube is kept at a constant temperature by means of a ’He cryostat, at the other side a thermometer and a
OK
0.90
0.85
Tui
0.80
d
lh
Fig. 7. (a) The upper thermometer temperature T, as a function of the heat current 0,at a constant lower temperature of 0.8 OK,for different values of the insaturated vapour pressure p/po. (b) The apparatus for measuring the superfluid transfer rate in the unupper thermometer temperature, (H) heater, (a) glass tube in saturated helium film;(T,,) which the transport of the film is measured, (b) filling tube, (c) 3He cryostat, (d) 3He pump tube. References p . 76
CH. 2, § 31
63
FLOW PHENOMENA IN SUPERFLLJIDHELIUM
heater are mounted. As long as the pressure is high enough the film is mobile and the behaviour of the temperature at the upper side T,,as a function of the heat input Q is shown in Fig. 7a, for different pressures PIP,, in whichp, is the saturated vapour pressure. This graph is one out of a series for various temperatures. Here 0.8 "K is chosen. The linear part must be ascribed to the Kapitza resistance, as inside the apparatus the temperature can be expected to be constant because of the superfluidity of the film. The linear rise originates from the heat resistance between the wall and the film. It indicates that this "Kapitza" resistance is not located in the bulk liquid, but even exists between the wall and a very thin helium film of a few layers thick. A critical transport rate 0, is reached at the point where the departure from the linearity starts, indicated by arrows. Subsequently in Fig. 8a are shown
1o-'w
50
QC
1
A
LonggMeycr
o
Brewer & Mcndclrrohn
0 Fokkcnr *.a
I
0 O-
PIP0
0.5 (a)
0.5
T
1.5
OK
2.5
(b)
Fig. 8. (a) The critical heat current & as a function of p/po at various temperatures. (b) The unsaturated vapour pressure ( p / p ~below ) ~ which the film is no longer superfluid, as a function of the onset temperature T.
these critical heat currents Q, as a function of the unsaturated vapour pressure p/po for several temperatures, and from extrapolation to 6, =O one finally obtains for all temperatures the critical values of the unsaturated vapour pressure (p/p,),, below which the film is no longer superfluid (see Fig. 8b). The so-called onset of superfluidity is indicated by (p/p0),. The experimental data for the critical heat current Qc are discussed with respect to the temperature and the total entropy of the film (Sf). Values of the critical velocity us,, can be derived if only the thickness is known. The thickReferences p . 76
64
R. DE BRWN OUBOTER et
al.
[CH.2, 5 4
ness is estimated from the values mentioned by Brewer, Symond and Thomson32, used in their measurements of the specific heat of the adsorbed layers. It is remarked that as soon as the critical flow rate is exceeded the superfluid flow can be connected to the difference in the chemical potential Tu
LIP= / S f d T . To
Plotting the logarithm of the difference in the chemical potential (dp) as a function of the logarithm of the transfer rate (p,v,d) for different film thicknesses, one obtains a set of parallel straight lines. Experimentally it is found that the relation A p =Bv: fits the experimental data, with a constant power n = 11. Hence in comparison to wide channels a much higher friction is observed (see also Section 4). Expressing the transfer rate in reduced units of the critical transfer rate, one obtains one straight line when the logarithm of the difference in the chemical potential A p is plotted as a function of the logarithm of this reduced transfer rate. The values obtained for the critical velocity are plotted in the last graph in Section 5 and compared with the data for wider channel dimensions.
4. Dissipative normal fluid production by gravitational flow in wide channels with clamped normal component In Section 1, page 50, is discussed the formation of vorticity in the presence of a gradient in the chemical potential p and the resulting dissipative heat production. Vermeer, Van Alphen, Olijhoek, Taconis and De Bruyn Ouboter33 have investigated this dissipative heat production (production of normal excitations) by means of gravitational superfluid flow through an adiabatic system. Their apparatus, drawn in Fig. 9, consists mainly of a copper chamber (C) mounted in vacuum and carrying a thermometer (T) and a heater (H). This chamber is connected at one side through a superleak (S) with the helium bath, and at the other side again through a superleak ( S ) to a glass reservoir surrounded by the bath, in which the height of the liquid level can be changed. A sudden rise or lowering of a displacer (D) produces a level difference 2. The corresponding potential energy AE,,,, =frrR2pgZZ effects a difference in the chemical potential Ap=mgZ somewhere in the chamber. The resulting energy dissipation in this chamber can be detected by following the temperature rise. In Fig. 10 two runs are shown. The temperature of the copper block is plotted as a function of time. From the two humps, the first one is due to a sudden electrical heat input of 2000 erg, References p . 76
CH.2, 8 4)
65
FLOW PHENOMENA IN SuPERnUID HELIUM
AE
KR2P9
AD-
-
- -
- -
-
____-- - ...............................
5 -$
-
- .............................
B
Fig. 9. Schematic diagram of the apparatus. (D) displacer, (B) bellows, (S) superleaks, (C) chamber, (T) thermometer and (H) heater.
the second one is effected by a sudden level change causing a potential energy difference of 2360 erg. The temperature returns each time to the same bath temperature by a carefully adjusted heat leak due to some exchange gas. At chamber diameters of a few mm nearly all the available potential energy developed in the chamber is dissipated as heat during the flow. However, for diameters of about 15 pm only about 60% of the applied potential energy is dissipated into heat, and 40% is transferred into kinetic energy (Bernoulli). Investigating the effect in more detail these measurements are extended to a continuous procedure by a steady superfluid flow, which is effected from the reservoir to the bath or reversely by slowly raising or lowering the displacer (D) in the reservoir in such a way that the level difference between reservoir and bath remains constant and therefore a constant difference in the chemical potential A p is realized (see Fig. 1 la). The energy dissipation is detected by the rate of the temperature rise of the chamber during such a stationary flow. In this case no exchange gas is used and the copper chamber is situated References p . 76
66
R. DE BRUYN
1388 OK
r
t!
2000 erg in 5 0 sec
[CH.2, 8 4
omom et ul.
'7
1 387
I. 386 6 min t,2
1.388 OK
\
1.387
Tt
1.386
lowering
I
-t
1 1
L 6 min
Fig. 10. Chamber temperature as a function of time, showing at 11 an electrical heat input of ZOO0 erg, at t r the effect of a sudden level difference with a potential energy equal to 2360 erg.
in high vacuum. The method of measurement is demonstrated in Fig. 1lb. At tl the displacer (D) starts to move, the pressure head builds up, together with an increase of the temperature. At t2 the pressure difference has reached a stable value and remains constant till t3 when the displacer is stopped. During this period the temperature increase is linear in time. The rate of energy dissipation (l?) is calibrated by producing a known amount of heat per second by means of heater H in the calorimeter. At r4 the difference in pressure has vanished, and the calorimeter gradually cools down due to its heat leak. In Fig. 12 the results for various flow velocities and temperatures are shown for a chamber of 0.44cm diameter and with a length of 1 cm. At velocities lower than the critical one, no energy dissipation is observed contrary to the situation at higher velocities where the energy dissipation appears. It is a function of the superfluid velocity only and it is temperature independent. A dissipative pressure can be defined as the energy dissipation per unit superfluid transport, 8/0u,, the pressure responsible for the dissipation. This dissipative pressure varies squarely with the superfluid velocity References p. 76
CH. 2, g 41
I moll
I l l
67
FLOW PHENOMENA IN SUPERFLUID HELIUM
t,
I
I
I
t2
0 0
0 6
0.4
.".I 0
2
3
4 min
(b)
Fig. 11. (a) The apparatus for the continuous measuring procedure; (D) displacer, (G) gearbox, (M) motor, (B) bellows, (S) superleak, (P) plunger, (C)chamber, (T) thermometer and (H) heater. (b) The rise of the temperature, AT, of the chamber as a function of time.
(as is shown in Fig. 13), yielding the experimental relation
B being temperature independent. 0 is the area of the chamber or slit. One of the other channels which is examined is a slit with a slit width of 15 pm made by filling the former chamber with a plunger. For this slit the energy dissipation as a function of superfluid velocity is shown in Fig. 14. Again for velocities lower than the critical one there is no energy dissipation and the earlier mentioned quadratic behaviour of dissipative pressure on the superfluid velocity is found. Due to the much higher velocity range the nondissipative Bernoulli pressure can no longer be neglected, and consumes nearly 40% of the observed pressure head, but still the sum of dissipative References p . 76
R. DE BRUYN OUBOTER et al.
68
[CH.
2, 8 4
TABLE 1 d (cm)
0.44 0.13 0.015 0.0015
B (g/cm3)
us,c(cm/sec)
2 0.5 0.3 0.2
0.95 1.7
3.0 6.0
pressure and Bernoulli pressure is just equal to the observed pressure A
as is demonstrated in Figs. 15 and 16. The results of four examined channels which differ in diameter can be tabulated as is shown in Table 1. The critical velocities are temperature independent and obey the experimental relation v ~ =, d-'~
(c.g.s. units)
as will be discussed in Section 5. The rate of energy dissipation is smaller I
I
0
20
0
erglr
0 0
0 0
A 10 I
chamber
E0 l
0
I3
8
d = 0.44cn
Fig. 12. The energy dissipation I? as a function of the superfluid velocity us at different temperatures. Chamber diameter d = 0.44 cm, chamber length = 1 cm. References p . 76
CH.
2,O 41
69
FLOW PHENOMENA IN SUPERFLUID HELIUM I'
I
,1.225
dyne
cm'
OK
A - 1.320'K
4!f
0 = 1.709K' = 1.830'K
%Im
10
I
db A A A 0
0.2 0.2
A
"S
1.O
cm sac
5
Fig. 13. The dissipation pressure L?/Ochvsas a function of the superfluid velocity us at different temperatures. Chamber diameter d = 0.44 cm, chamber length = 1 cm.
than the loss of potential energy per second (see Fig. 15), the difference being just equal to the energy involved in the Bernoulli pressure which will be most likely dissipated outside the calorimeter (see Fig. 16). This is demonstrated in an experiment, in which the calorimetric circuit is formed by a chamber and a slit in series (see Fig. 17). They differ in diameter (0.4 cm and 0.02 cm), but their dimensions are chosen in such a way that both parts have almost the same critical flow capacity. Now the energy dissipation is no longer independent of the direction of superfluid flow through the system, as is shown in Fig. 17. When the helium flows first through the slit and then through the chamber, the kinetic energy, obtained from the Bernoulli pressure in the slit, attributes to the energy dissipation
References p . 76
R. DE BRWN OUBOTBR et
70 I
I
[CH.2, 8 4
al. 1
I
sI it t c
0 0
vs
+
to
20
cmlr
30
Fig. 14. The energy dissipation g a s a function of the superfluid velocity us at T = 1.26 OK. cm2, slit length = 1 cm. Slit width d = 15 pm, surface area Osl = 1.9 x
in the chamber. However, in the reversed direction of flow, this pressure can only dissipate the involved energy at the other side of the closely connected superleak outside the calorimeter and therefore less energy
is detected in the calorimeter. Hence the differencein the dissipative pressure is equal to + p , ~ : (see ~ Fig. 17). It appears that the coefficient B is not very sensitive for the length of the channel. When the chamber is completely filled with compressed jeweller's rouge, no energy dissipation is observed (see Section 2). The quadratic dependence of dissipative pressure on the superfluid velocity is apparently only valid for channels with a diameter larger than 10 pm. In general the dissipative pressure can be written as ps LIP= Apdis-v:, n being larger than 2 for channels with a diameter smaller than 10 pm. Still the critical velocity obeys the experimental relation (see Section 5) us,c= d-' (c.g.s. units). References p . 76
CH.2,f 41
71
FLOW PHENOMBNA IN SUPERFLUID HELIUM
dyne -
cm'
60
-
,
l
l
r
(
l
l
l
l
f
l
'
d,15pm
0
1 = 1.26OK 00
40
-
0
-
l
-
-
-
0
20
'
0
0
-
Fig. 16. The applied pressure head dPobs as a function of the calculated pressure head Ape which is equal to the sum of dissipative pressure and the Bernoulli pressure. The straight line is drawn at 45".
References p . 76
72
R. DE BRUYN OUBOTER et
al.
[CH.
2, $ 5
60
dyne an’
40
20
E OSI “s
0 c
superfluid velocity in t h e s l i t
Fig. 17. The dissipative pressure for a chamber and a slit in series as a function of the superfluid velocity in the slit at 1.25 O K , when the displacer is raised or lowered, the difference being,nearly equal to +pSu2.Chamber diameter = 0.4 cm, chamber length = 0.5 cm, slit width = 0.02 cm and slit length = 0.5 cm.
This work is in reasonable agreement with the carefully performed experiments on isothermal flow through narrow slits (0.3 ,um
5. The dependence of the critical velocity of the superfluid on channel diameter and film thickness The dependence of the critical velocity v ~ of, the ~ superfluid on channel versus don a logarithmic scale and often diameter dis usually plotted as vSsdd divided into two region~~~134, as is shown in Fig. 18. In region “S” the References p . 76
CH. 2,851 I
1
73
FLOW PHENOMENA IN SUPERFLUID HELIUM I
T= 1.4
lo-'
I
I
I
I
I
I
I
OK
d
t cm
Fig. 18. The dependence of the product of the superfluid critical velocity and channel ~ the d ) channel diameter (4. (@) Fokkens, film flow; (A)Pellam, "superdiameter ( ~ ~ ,on fluid wind tunnel"; (0) Chase, heat conduction, T + Ta, un -+ 0 );.( Chase, heat conduction at 1.4 "K; (A) Van Alphen, adiabatic flow rate; (0) Van Alphen, energy dissipation technique; (r)Kramers, second sound attenuation in pure superfluid flow; (0)Van Alphen, critical flow through jeweller's rouge; Keller and Hammel, isothermal flow; (----) heat conduction, normal turbulence; (-.-.-.-) us,c = (2h/md) [In (4d/a) - 31; ( 0 ) data from preceding reviews of Atkins, and Hammel and Keller.
(v)
diameters of the channel or the film are smaller then lOpm, the critical velocity is found to vary nearly with the fourth root of the channel diameter, and is temperature independent. In region "L" the diameters are larger than 10 pm, the product of critical velocity and diameter is nearly constant or slowly increasing and is temperature dependent. In this latter region the critical velocity is often considered as the velocity necessary to create vortexrings of the Feynman type (see Fig. 18 and also Section 1, eqs. (17)-(19)). Van Alphen, Van Haasteren, De Bruyn Ouboter and Taconis35 have noticed that region "L" (the horizontal part of the figure) does not represent critical velocities of the superfluid, but is only erroneously interpreted as such, whereas in reality it is related to the onset of classical turbulence in the References p . 76
74
R. DE BRUYN OUBOTER
et ul.
[CH.2, 8 5
normal fluid (see Section 1, eq. (23)). In experiments in which care is taken that the normal fluid stays at rest (see Section 4), the us,c is found to be much larger. Plotting these large values of us,c in the figure, then they form a continuation of the first general, temperature independent, relation between and d. They elucidate this by means of the following remarks on region “L”: All experiments performed in this region “L” are carried out in measuring equipment in which the normal fluid flow cannot be neglected. This is the case in heat conduction and oscillating disks experiments, in isothermal flow, or in oscillations in a U-tube, contrary to the behaviour of the normal fluid in the first region “S”, in which, due to the much smaller dimensions, the normal component is quite immobile. For this second region “L”, for instance in the case of heat conduction, one can calculate the onset of classical turbulence by means of the critical Reynolds number (see Section 1, eq. (22)), containing the normal fluid velocity u,, the total density p (Staas, Taconis and Van Alphenla), the normal viscosity q n and the diameter d
!,I
Revm= dpun ,
1200 < ReonO < 2300.
Using the heat conduction relation that psus=pnunone gets for the misleading “critical” velocity of the superfluid Pn 9 a
%, c = Re,=, - Pn d P ’
In the figure the dashed lines enclose the region of the onset of normal turbulence between the Reynolds numbers 1200 and 2300 at a temperature of 1.4 “K. As soon as the normal component is sufficiently clamped, which is usually achieved by application of superleaks in the flow path, the observed v ~ is, ~ much larger, is temperature independent and obeys the same diameter dependence as in region “S”. This is the case in two experiments by Van Alphen et al. (see Section 4). One in which the onset of superfluid friction is detected calorimetrically by measuring the energy dissipation of the flow (see Section 4), and the other by measuring the critical flow rate in a thermally isolated ~apillary4~, in which the normal flow is suppressed by means of a superleak. The same is the case in Pellam’sa5*26superfluid wind tunnels in which the torque on a Rayleigh disk is studied (see Section 2). Also from the work of Chases6 one is able to derive large critical velocities References p . 76
CH. 2,
8 51
FLOW PHENOMENA IN SUPERFLUID HELIUM
75
of the right order of magnitude in the temperature region very close to the lambda-point, with the help of a veocity-temperature diagram, where the normal fluid velocity is negligible. This feature is already indicated by Chaseso. As the diagram is given for a temperature of 1.4 OK, it is assumed here that v ~is ,independent ~ of temperature, which is strongly justified from the available data. It is exphasized that all superfluid critical velocities, obtained in experiments in which the normal component is clamped, obey the same temperature independent expression v ~= ,Cd-', ~ 6
with C% 1 cm' sec-', d ranging from a thickness in the unsaturated film of a few A up to a few cm as is present in a superfluid wind tunnel. As is discussed in the first section (eqs. (17)-(19)), the breakdown of pure superfluid potential flow is due to the creation of vortices, but realizing the disagreement between experiment and theory (Section 1, eqs. (17)-(19)), it is not clear how the vorticity is created. In the theory, discussed so far (see Section l), it is supposed that the superfluid flows along an ideal surface with a finite velocity, which is constant over the cross-section of the tube. However, if the surface is rough, the superfluid is forced to flow around the roughness with a high velocity and has an extra kinetic energy, proportional to the square of its velocity at the boundary (Andersons). The creation of vortices in superfluid helium remains up to now an unsolved problem. A possible mechanism for the generation of vortices and the dependence of the critical velocity on channel diameter is given by Craig37. Assuming that the vorticity necessarily is created at the wall, the available kinetic energy of the fluid, within a nucleation time z, must be large enough to create a vortex, which can flow freely into the liquid. Instead of eq. (17) of Section 1, Peshkov38 and Craig37 postulate an equation of the following form +PSd, c
us, C O T
=E 9
(27)
in which 0 is the area of the channel, and E the energy of a vortexring. For this energy E they use the expression for the energy of a vortex close to the wall
(y:
~ = + p-
( ):
RIn 1 + -
mip
(y ( -
R l n I+- 2n:v,n)7
(**)
obtained by Fineman and Chases@,by taking into account the image forces. R is the radius of the vortexring, nearly coinciding with the radius of the References p . 76
76
R. DE BRUYN OUBOTER et
al.
[CH.2
channel and r is the distance between the vortexcore and the wall. Again a is the core radius; r is determined by the velocity at which the vortexring is no longer bound to the wall %c
h < 4xmr ~.
Combining eq. (27) with eq. (28) yields
The argument of the logarithmic term is not very sensitive for the critical velocity and just easily satisfies the experimental data as the empirical relation v,,c = Cd-'.
References L. D. Landau, J. Phys. USSR 5,71(1941) [Men of Physics: L. D. Landau, Vol. 1 , Low Temperature and Solid State Physics, Ed. D. Ter Haar (Pergamon Press, London, 1965) p. 541; L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1959) Ch. 16. 2 F. London, Superfluids, Vol. 2 (John Wiley and Sons, New York, 1954). 3 R.P. Feynman, PIogress in Low Temperature Physics, Vol, 1 . Ed. C. J. Gorter (NorthHolland Publ. Co., Amsterdam, 1955) Ch. 2. 4 See, e.g., R. P. Feynman R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 3, Quantum Mechanics, Ch. 21. 5 P. W. Anderson, Proc. Ravello Spring School: Lectures on the Many Body Problem, Vol. 2, Ed. E. R. Caianiello (Academic Press, New York, 1964) pp. 113-135; P. W. Anderson, Rev. Mod. Phys. 38,298 (1966); P. W. Anderson, Quantum Fluids, Ed. D. F.Brewer (North-Holland Publ. Co., Amsterdam, 1966) p. 164. 6 V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 12,1064 (1950); V. L. Ginzburg and L. D. Landau, Physik. Abhandl. Soviet Union 1 (7) (1958) Tieftemperatur Physik [English trans].: Men of Physics, L. D. Landau, Vol. 1, Ed. D. Ter Haar (Pergamon Press, London, 1965) p. 1381. V. L. Ginzburg and L. P. Pitaevskii, Zh. Eksperim. i Teor. Fiz. 34,1240 (1958) English trans].: Soviet Phys.-JETP 7,858 (1958)l; V. L. Gimburg, Zh. Eksperim. i Teor. Fiz. 29,244 (1955) [English trans].: Soviet Phys.JETP 2, 170 (1956)l; Physica 24, S136 (1958); L. P. Pitaevskii, Zh. Eksperim. i. Teor. Fiz. 35,408 (1958); 40,646 (1961) [English transl.: Soviet Phys.-JETP 8, 282 (1959); 13, 451 (1961)l. 1
.
CH.
21
FLOW PHENOMENA IN SUPERFLUID HELIUM
77
S. T. Beliaev, Zh. Eksperim. i Teor. Fiz. 34,417 (1958) [English transl.: Soviet.-JETP 7, 289 (1958)l. L. Onsager, Nuovo Cimento Suppl. 6, 249 (1949); 0. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956); 0. Penrose, Phil. Mag. 42, 1373 (1951). l o W. F. Vinen, Proc. Roy. SOC.(London) A 260,218 (1961); W. F. Vinen, Progress in Low Temperature Physics, Vol. 3, Ed. C. J. Gorter (NorthHolland Publ. Co., Amsterdam, 1961) Ch. 1. l1 S. G . Whitmore and W. Zimmerman, Jr., Phys. Rev. Letters 15, 389 (1965). l2 J. D. Reppy and D. Depatie, Phys. Rev. Letters 12, 187 (1964); J. D. Reppy, Phys. Rev. Letters 14, 733 (1965); J. R. Clow and J. D. Reppy, Phys. Rev. Letters 16, 887, 1030 (1966). l3 J. B. Mehl and W. Zimmerman, Jr., Phys. Rev. Letters 14, 815 (1965); J. B. Mehl and W. Zimmerman, Jr., Bull. Am. Phys. SOC.11, 479 (1966). l 4 G. W. Rayfield and F. Reif, Phys. Rev. Letters 11, 305 (1963); G. W. Rayfield and F. Reif, Phys. Rev. 136, All94 (1964); F. Reif, Sci. Am. 211, 116 (December) (1964). 15 L. Prandtl and 0. G. Tietjens, Fundamentals of Hydrodynamics,Ch. 10, eqs. (1) and (2) ; L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1959) Ch. 10, eqs. (9) and (3). l8 P. L. Richards and P. W. Anderson, Phys. Rev. Letters 14, 540 (1965); P. W. Anderson and A. H. Dayem, Phys. Rev. Letters 13, 195 (1964); R. L. Richards, Quantum Fluids, Ed. D. F. Brewer (North-Holland Publ. Co., Amsterdam, 1966) p. 180. "%ee discussion between F. Reif and D. Pines in Quantum Fluids, Ed. D. F. Brewer (North-Holland Publ. Co., Amsterdam, 1966) p. 254. 1 7 K. R. Atkins, Liquid Helium (Cambridge Univ. Press, London, 1959) p. 116, eq. (4.41) and p. 201, eq. (6.58). 1 8 F. A. Staas, K. W. Taconis and W. M. Van Alphen, Physica 27, 893 (1961); H. Schlichting,Boundary Layer Theory (Pergamon Press, London, 1955) p. 401 ; L. Prandtl and 0. G. Tietjens, Applied Hydrodynamics, p. 31. I @ C. J. Gorter and J. H. Mellink, Physica 14, 285 (1948). 20 H. E. Hall and W. F. Vinen, Proc. Roy. SOC.(London) A 238,204, 215 (1956); W. F. Vinen, Progress in Low Temperature Physics, Vol. 3, Ed. J. C. Gorter (NorthHolland Publ. Co., Amsterdam, 1961) Ch. 1; E. M. Lifshitz and L. P. Pitaevskii, Zh. Eksperim. i. Teor. Fiz. 33,535 (1957) [English transl.: Soviet Phys.-JETP 6, 418 (1958)l. 2 1 W. M. Van Alphen, J. F. Olijhoek and R. De Bruyn Ouboter, Proc. Conf. Liquid Helium, St. Andrews, August 1965, to be published. 22 D. F. Brewer, D. C. Champeney and K. Mendelssohn, Cryogenics 1, 1 (1960). 23 M. E. Nordberg, J. Am. Ceram. SOC.27,299 (1944). 24 R. Meservey, Phys. Fluids 8, 1209 (1965). 25 P. P. Craig and J. R. Pellam, Phys. Rev. 108, 1109 (1957). 28 T. R. Koehler and J. R. Pellam, Phys. Rev. 125, 791 (1962). 27 E. L. Andronikashvili, Zh. Eksperim. i Teor. Fiz. 18, 424 (1948); E. L. Andronikashvili and Yu. G. Mamaladze, Rotation of HeII, see this book, Ch. 111. 28 K. Fokkens, K. W. Taconis and R. De Bruyn Ouboter, Physica 32,2129 (1966). 8
R. DE BRUYN OUBOTER et
78 aD 30
31 32
a3
al.
[CH.2
R. Bowers, D. F. Brewer and K. Mendelssohn, Phil. Mag. 42, 1445 (1951). E. Long and L. Meyer, Phys. Rev. 98, 1616 (1955). D. F. Brewer and K. Mendelssohn, Proc. Roy. SOC. (London) A 260, 1 (1961). D. F. Brewer, A. J. Symonds and A. L. Thomson, Phys. Rev. Letters 15, 182 (1965); A. J. Symonds, D. F. Brewer and A. L. Thomson, Quantum Fluids, Ed. D. F. Brewer (North-Holland Publ. Co., Amsterdam, 1966) p. 267. W. Vermeer, W. M. Van Alphen, J. F. Olijhoek, K. W. Taconis and R. De Bruyn Ouboter, Phys. Letters 18,265 (1965); W. M. Van Alphen, J. F. Olijhoek, R. De Bruyn Ouboter and K. W. Taconis, Physica 32,1901 (1966).
34
35
37
39
40
W. E. Keller and E. F. Hammel, Physics 2,221 (1966); E. F. Hammel and W. E. Keller, Cryogenics 5,245 (1965). W. M. Van Alphen, G. J. Van Haasteren, R. De Bruyn Ouboter and K. W. Taconis, Phys. Letters 20, 474 (1966). C. E. Chase, Phys. Rev. 127, 361 (1962); C. E. Chase, Proc. VIIIth Intern. Conf. Low Temperature Physics, London, 1962, Ed. R. 0. Davies (Butterworths, London, 1963) Ch. 3, p. 100. P. P. Craig, Phys. Letters 21, 403 (1966). V. P. Peshkov, Soviet Phys.-JETP 13,259 (1961). J. C. Fineman and C. E. Chase, Phys. Rev. 129, 1 (1963); A. L. Fetter, Phys. Rev. Letters 10, 507 (1963). W. M. Van Alphen, W. Vermeer, K. W. Taconis, and R. De Bruyn Ouboter, Proc. IXth Intern. Conf. Low Temperature Physics, Columbus, Ohio, 1964, Eds. J. G. Daunt et al. (Plenum Press, New York, 1966) p. 323.
C H A P T E R I11
ROTATION OF HELIUM 11 BY
E. L. ANDRONIKASHVILI and YU. G. MAMALADZE INSTITUTE OF PHYSICS, ACADEMY OF SCIENCES OF THE GEORGIAN SSR,
TBILISSI, USSR CONTENTS: Introduction, 79. - 1. Solid body rotation of helium 11, 80. - 2. Dragging of the superfluid component into rotation, 91. - 3. Observation of vortex lines and their distribution in uniformly rotating helium 11, 100.- 4. Elastic properties of vortex lines. Oscillations of bodies of axial-symmetric shape in rotating helium 11, 114. 5. The phase transition in rotating liquid helium in the presence of vortex lines, 137. 6. Decay of vortex lines and their stability, 144. - 7. Persistent currents of the superfluid component, 149.
-
Introduction Many aspects of the rotating helium I1 problem were already considered in several reviews published in previous volumes of Progress in Low Temperature Physics (in particular by Feynman in the second chapter of the first volume and by Vinen in the first chapter of the third volume). Naturally we shall not repeat the material of those reviews and only short references to their contents will be given, where it will be required, to make the topic under discussion complete. During the past few years the study of helium I1 rotation has been continuing to progress rather rapidly. The general interest in this problem is clear. Rotation is related with the most profound problems of the theory of superfluidity. The question concerns the very nature of a superfluid liquid, i.e. the medium, the properties of which are determined apparently, as it was emphasized already by F. London, by its complete inability to perform nonpotential motions, rather than by the absence of viscosity. At present we can speak of a rather distinct separation of the problem into two different kinds of rotation: 1) motion determined by the presence of Onsager-Feynman’s vortex lines, which carry single quanta of circulation (ro=2nh/m),and 2) potential rotation in the doubly connected region withReferences p . 156
79
80
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.
3, 8 1
out vortex lines. Accordingly our review is divided into two (highly unequal) parts. The first part (Sections 1 to 6) is devoted to rotation of helium I1 in rotating vessels and the second part to persistent currents of the superfluid component in a vessel at rest. All the facts, known at present, concerning both cases of rotation just mentioned, confirm completely the conception of Landau-Onsager-Feynman, according to which the curl of the superfluid component velocity is equal to zero everywhere curl us = 0 (A) excluding some isolated singular lines (vortex lines) on which it is equal to infinity. This equality is closely associated with the assumption of the existence of some wave function, the gradient of the phase of which determines u,(u,=AVq~/m). The analogy between superfluid helium and the electron liquid of superconducting metals becomes closer and closer. This is especially so in the case of superconductors of the second kind, where penetration of a magnetic field is accompanied by formation of quantized lines of the magnetic flux. The mixed state of such superconductors is similar to the state of rotating helium I1 and the conception of Landau-Onsager-Feynman, described in sufficient detail in Feynman’s paperl7, has very much in common with the theory of
Ginzburg-Landau-Abrikosov-Gorkov. Even this very superficial survey of the questions associated with the rotating helium I1 problem shows that investigations performed in this direction are not limited to the study of a particular (though a very curious) case of motion of only one liquid possessing very peculiar properties. The meaning of such investigations is much broader. They concern directly the foundations of the quantum theory of the condensed state and they are even more important because the creation of a microscopic theory of helium I1 is not yet finished. Though these aspects have been greatly developed recently, in this review theoretical problems will be elucidated to the extent required for the explanation of the experimental facts. The aim of this review is to describe the main experimental facts, not considered earlier in the Progress in Low Temperature Physics, from the single point of view of the LandauOnsager-Feynman conception.
1. Solid body rotation of helium I1 1 . 1 . ANGULAR MOMENTUM AND
MENISCUS OF ROTATING HELIUM
11
As early as 1941 Landau1 suggested that the temperature dependence of the References p . 156
CH. 3,
8 11
81
ROTATION OF HELIUM I1
angular momentum of helium I1 rotating together with the cylindrical vessel with angular velocity oo be measured. According to his assumption the superfluid component of helium I1 with effective density p s should be motionless due to the absence of frictional forces and the condition (A), while the normal component with effective density pn could be in a state of rotation. Therefore the angular momentum (reduced to unit height of the vessel) should be equal to L = 4p,nR4~o (1.1.1) instead of its classical value L,, corresponding to solid body rotation of the liquid (u=wor): L, = $ p n R 4 0 , . (1.1.2) In formulae (1.1.1) and (1.1.2) R is the radius of the cylinder and p is the total density of helium I1 ( p = p s + p n ) . For the same reasons it should be expected that the meniscus of rotating helium I1 should be different from that of rotating helium I. Indeed, in helium I1 the centrifugal force should act only on the rotating normal component, while the force of gravity acts on both the components. On equilibrium of hydrostatic and centrifugal pressures one should arrive a t an expression for the meniscus depth of helium I1 h = -P- n- W;R’ P 29 ’
( 1.1.3)
instead of the well-known formula describing the meniscus of the liquid rotating as a whole w;RZ h=-. (1.1.4) 29 Here g is the acceleration of gravity. The temperature dependence of the angular momentum of rotating helium I1 (Lccp,) and the temperature dependence of its meniscus depth (hocp,) should be observed till critical velocities of the order of 60 m/sec are reached when, according to the theory, an interaction between the wall of the vessel and the superfluid component takes place and as a result, new thermal excitations should be generated. It seemed that nothing could prevent the success of the experiment suggested by Landau, once the independence of the two kinds of motion (superfluid and normal) had been confirmed by Andronikashvili’s experiments 3, per2g
References p . 1%
82
E. L. ANDRONKASHVILI AND YU. G. MAMALADZE
[CH.
3, 4 1
formed with an oscillating pile of disks. Peshkov’s experimentsd, proving the existence of second sound predicted by Landau l, in which ps and pn oscillate in antiphase, had also indicated the possibility of effecting Landau’s suggestion. That is why it is not surprising that physicists were greatly puzzled, when they learned that attempts to find temperature dependence of the meniscus depth, made by Andronikashvili5 in 1948, by Osbornee in 1950 and by Andronikashvili and Kaverkin7 in 1955, failed. All the experiments were made under similar conditions: a transparent cylindrical vessel with a radius about 1.5 cm, partially iXed with liquid helium, was rotated with an angular velocity from 3 to 100 sec-l. The maximum linear velocity at the periphery of the vessel reached 70 cm/sec. Under such conditions the meniscus of helium I1 was not different from that of helium I and was determined by formula (1.1.4) with great accuracy. Not to return to the question on the shape of the meniscus of helium 11, we shall mention that later measurements 89 9 of the meniscus curvature were made at lower velocities of rotation 1.58 sec-’ and 0.29 sec-’. In both cases, in spite of the great sensitivity of the experiments (a light beam was reflected from the meniscus), they did not manage to distinguish the shape of the equilibrium meniscus from a paraboloid. If one can speak of a difference in the shapes of helium I1 and helium I meniscuses, it happens only at high velocities of rotation (about 30 sec-I), and the parabolic meniscus has the slight conic pit7 shown in Fig. 1. We should pay attention to this conic pit as it will be mentioned later. Landau’s assumption was not confirmed in the direct experiments which were made by Hall10 either. Hall has observed complete participation of the superfluid component in the rotation of the vessel. In his experiment, described in a previous volume11, a pile of closely spaced disks, completely binding the normal component, was placed into a cylindrical vessel. The angular momentum of helium I1 was measured by determining the torque needed for acceleration and deceleration of the liquid. Later such investigations were extended by Hall12 and also by Reppy, Depatie and Lanels and Reppy and Lane14, who observed rotation of a cylindrical vessel, filled with helium 11, round a vertical axis. The vessel, hanging from a magnetic suspension, was given (for a fraction of a second) a rotational impulse and then the time dependence of its angular velocity was watched, while it’s angular momentum was transferred to helium 11. It was found that the equilibrium angular momentum of the liquid in these experiments was always a classical one; in other words, the liquid rotated with the Referencesp. 156
CI-Z. 3,8
11
ROTATION OF HELnmi I1
83
vessel as a whole. In the experiments described the minimum velocity oo was equal to 0.05 sec-' 1.2. THE THERMOMECHANICAL EFFECT IN ROTATING HELIUM 11 The first studies of rotating helium 11, in which the experimenter's expectations ended in a striking failure, made physicists wonder whether the breakdown of superfluidity happened during twisting of the vessel. Perhaps then such conditions are created in which thermal excitations are generated in immense numbers and the superfluid component disappears in the rotating vessel. Andronikashvili and Kaverkin7 performed an experiment to elucidate this fact. They observed the fountain effect in rotating helium 11. A capillary packed with rouge was fixed in a vertical position in the centre of the rotating vessel and was illuminated with a light beam. It was found that the fountain effectnot only takes place in rotating helium 11, but also that it is indistinguishable, within experimental error, from the same effect in helium I1 at rest. This experiment clearly showed invariability of p J p , at all the velocities of rotation used.
Fig. 1. The meniscus of rotating helium I1 at velocities of the order of tens of radians per second.
Referencesp . 156
84
E. L. ANDRONIKASHVILI AND YU. G . MAMALADZE
[CH.3,
51
Much more detailed experiments on the fountain effect in rotating helium I1 were made later by Zamtaradze and Tsakadzel5 with the aim of establishing the nature of some peculiarities of helium I1 behaviour near the I-point. These studies were made both in the centre of the rotating vessel and at its periphery. In the latter case their device (Fig. 2) was a cylindrical vessel with a coaxial tube of somewhat smaller diameter, fixed in such a way that its lower edge did not reach the vessel bottom. The lower portion of the annulus so formed was filled with rouge and the difference of helium I1 levels between the vessel and this annulus, arising as a result of rouge heating, was measured. The experiments performed have shown that the height which the helium rises above the liquid surface in the vessel is precisely the same both in a state of rotation or at rest. Hall and Vinen’s experiment, where the velocity of second sound was measured at different angular velocities of rotation and the invariability of this magnitude was established, also shows within the errors of the experiment (of the order of 1 %) the independence of the density ps on the velocity of rotation11.
Fig. 2. The device for the measurement of the thermomechanical effect in rotating helium 11.
References p . 156
CH. 3 , 8 11
ROTATION OF HELIUM I1
85
1.3. THETHEORY OF THE PHENOMENA Thus the breakdown of superfluidity is not observed in rotating helium I1 in the sense of Landau's critical velocities and the discrepancy between Landau's initial assumption and the results of the performed study seemed to cast doubt on some conclusions of the theory. At present the well-known hypothesis of Onsagerl6 on the existence of vortex lines with quantized circulation in the superfluid component, which was later developed by Feynman 17, has helped superfluid hydrodynamics out of difficulties existing for several years. Not repeating the conclusions and main equations, describing the properties and number of quantized vortex lines in rotating helium 11, we shall only indicate that the first vortex line on the vessel axis appears at the critical velocity of rotation determined by the formula of Arkhipov-Vinen l *19~ wcl = --
a_ _ -
m ( R 2 - a:)
R In -. a,
(1.3.1)
This velocity has nothing in common with the critical velocity of the breakdown of superfluidity according to Landau. In formula (1.3.1), m is the mass of a helium atom, R is the radius of the vessel, a, is the radius of the vortex core (ao-4 x lo-* cm). It is easy to calculate that for R- 1 cm we have w , , ~ I o sec-'. -~ Let us note that at a velocity slightly exceeding wcl, the number of vortex lines per unit of the vessel cross-section cannot yet be determined by Feynman's formula17 (1.3.2) Indeed, according to the definition, wcl is the velocity at which one vortex line should appear in the vessel. However, substituting L O , = W , ~ into formula (1.3.2) we should obtain, for the total number of vortex lines in a vessel whose radius is not extremely small ( R S a,), the magnitude nR2N~ln(R/ao), which is appreciably larger than unity. Kiknadze, Mamaladze and Cheishvili have calculated the critical velocity mkl, starting from which formula (1.3.2) becomes valid. We shall reproduce their derivation as it will allow us to explain, by the way, the reason why the angular momentum of rotating helium I1 has the classical value of magnitude L,, determined by formula (1.1.2). The existence of vortex lines in the quantity required by formula (1.3.2) is thermodynamically favourable if their contribution to the free energy References p. 156
86
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
E-Lw, is negative. Hence 4
[CH.3,
81
1 = EVILV 9
where E, and L, are the contributions made by vortex lines to the energy and the angular momentum of the liquid respectively. E,
= E,,
+ E',
where ESb=&np,u~R4 is the energy of the solid body rotation of the superfluid component, and E', which is given by
A b E' = KR'NE = nR2wops- In -, (1.3.3) m a0 is the excess energy of vortex singularities reduced to unit height of the vessel. Here h' b E = np, -1n m 2 a, is the energy of unit length of a vortex line17, b is its effective radius [nb2=N-* and b=(h/rnwo)*]*. Each vortex line in a cylindrical vessel at the distance r from its axis makes the following contribution to the value of Lv** A (1.3.4) L , = np, (R' - r'). Therefore R
[
ma0
L, = L , -2nr dr = ;fpsnR4w,. 7th
(1.3.5)
0
As the angular momentum of the normal component rotating as a whole is determined by a similar formula (1,l. 1) with the substitution of p, instead of ps, we have in complete agreement with the experiment L = L,
+ L, = L,.
* The usual estimation of the effective radius of the region, in which the superfluid component rotates round a given vortex line, can be made more accurately by substituting bl = ba/r instead of b in the expression for e (bl is the radius of the region in which the contribution of the nearest vortex line to us exceeds that made by other vortex lines equal to wor). Then In(b/ao) in formula (1.3.3) is replaced by In(b/ao) - In(R/z/Zb). At small velocities of rotation b/ao@R/z/& and the second logarithm can be neglected. However, when wo 10 sec-1, the magnitude ln(R/z//eb), though it does not change the order of the magnitude of E' so far, ceases to be negligibly small. ** In the derivation of formula (1.3.4) one takes into account the fact that a reflected vortex line with a circulation - 2nh/rn at a distance R2/r from the axis of the cylinder corresponds to the considered one. The formula is written for point vortex lines and that is why it is valid in the approximation R a S ao2.
-
References p. 156
CH. 3,
8 11
87
ROTATION OF HELIUM ll
Let us return to the calculation of the magnitude coil. Dividing the expression for E, by L,, we obtain I
0,l
4A In b mR2 a,
= __
-
4w,,
.
(1.3.6)
The last approximateequation is valid when we have b/a, % R/b ;this inequality always holds under the conditions for formation of the first vortex lines. We have already noticed that the velocity coC1 is very small for vessels of normal size. Therefore the velocity ohlis small for them as well. The formation of the first vortex line and an increase of vortex line number to values corresponding to formula (1.3.2) takes place in a very narrow interval of low velocities of rotation. That is why, almost at any real velocity of rotation, helium is far into the supercritical regime and contains a lot of vortex lines. There are exceptions only when devices with extremely small radii are used. It is easy to obtain formula (1.1.4) describing the shape of the meniscus of rotating helium I1 as well, if one proceeds from the conception of OnsagerFeynman of quantized vortex lines.
I I
I
I I
I
I I
I1 I I
I I I I
Fig. 3a. The distribution of velocities of the superfluid component in rotating helium 11. The dashed line, passing through the origin of the coordinates, corresponds to the law u. ---coo x r. The solid lines show schematically the real distribution of velocities. The figure is plotted in arbitrary units. In fact the radii of the vortex cores are usually much smaller than the vortex separations, the velocities in the immediate vicinity of these cores are much larger than oor, but almost everywhere vS - &O x r is closer to zero than it is shown in the figure.
References p . 156
88
E. L. ANDRONXASHVILI AND YU. G. MAMALADZE
[CH.3,
81
To explain the parabolic shape of the meniscus one is reminded that, for the density of vortex lines determined by Feynman’s formula, the velocity of the superfluid component is almost everywhere determined by the expression us w
zr2Nr0 = wor = v,. 2nr ~
(1.3.7)
In other words, helium I1 rotates on the average like a solid body and the difference between the velocity of the superfluid component and the mean velocity exists essentially only in the immediate vicinity of vortex lines (Fig. 3a). Then the shape of the meniscus, as was already mentioned, is determined by the equilibrium of the centrifugal and hydrostatic pressures and the equation of the free surface has the form 2 2
z = wor 129
(1.3.8)
(the z-axis is directed upwards). Deviations from this equation are essential only in microregions in the vicinity of the vortex line intersections with the free surface. But such microregions are quite undetectable visually. However, this explanation has a self-contradiction. It is implied that the superfhid component performs a stationary potential motion *. But for such a motion Bernoulli’s law should be valid, the use of which would clearly lead to the wrong conclusion that the meniscus shape should be convex (corresponding to the equation z = - a i r ’/2g). Mamaladze and Cheishvili have done a more detailed treatment and showed that this contradiction does not really exist. The problem is that the distribution of velocities in helium I1 placed into a uniformly rotating vessel is, if one can say so, “tied” to vortex lines, moving together with the vessel and the normal component. The motion is stationary in the rotating frame of reference in which it is not potential. And in the laboratory system of reference, in which the motion is potential**, it is not stationary. This is easy
* One should not be surprised that on calculating us as the velocity of the potential rotation, we have obtained the expression with the curl different from zero: curl wo x r = 2w0. It is due to the fact that the difference us - <06), which is small itself, has a curl equal to - 200, which is not small in value. ** The complex velocity potential of the superfluid component in the rotating cylinder is determined by the equation h Z -Z k w=-Ch-n’ mi n
where Z k are the coordinates of vortex lines and Z‘k are the coordinates of their reflections = R 2 / [ Z k l r arg Z‘k = arg zt), Z k and Z’k are time variables.
([Z’kl
References p . 156
CH. 3,
11
ROTATION OF HELIUM I1
89
to see if, for instance, we imagine the rotation of the diagram of motion, represented in Fig. 3b, round an arbitrary point and if we reproduce mentally the sequence of events at some fixed point of space. Under such conditions, the hydrodynamical equation of motion has a Cauchy integral instead of a Bernoulli integral, In particular, to Landau's
Fig. 3b. The motion of the superfluid component in rotating system of reference. The picture is taken from the paper of Kleiner et al.131 and describes the distribution of the density of superconducting electrons and magnetic field in a superconductor of the second kind when H 5 HCz.The possibility of a direct use of this figure in the case of helium I1 rotating with the velocity 00 5 u e 2 = 4.3 x 1010(T~- T) sec-1 is proved in Ref. 57 (Section 3.6). The lines of constant density of the superfluid component are shown with the corresponding values of density on them (in units of the maximum density). These lines are at the same time the lines of relative velocity us - wo x r. The centres of the concentric circumferences, where ps = 0 are the vortex points which form a lattice with triangular symmetry. In the centres of the triangles formed by the three nearest vortex lines the maxima of the wave function are located, round which the reverse rotation of the superfluid liquid (Section 3.6) takes place. Nore added in proof. The value of wc2 - the maximum angular velocity compatible with superfluidity - must be replaced by wCz= 1.06 x
4
lo1' (TA- T)a sec-l (see note added at the end of this chapter).
References p . 156
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E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH. 3, 4 1
equation corresponds such a form of Cauchy integral 2 vs
A ac ~+-~1 -
m at
+ m 2
-
Pn -(us - u,)
2p
2
+ f ( t ) = 0,
(1.3.9)
where hq/m is the potential of velocity of the superfluid component (p is the phase, u,=AVq/m), p is the chemical potential (in the considered case 8p =mgdz), f ( t ) is an arbitrary function of time. Denoting by the symbol 6 the difference in values of some magnitude at two points (at rest), we obtain
[; - ;; 2
gdz = 6
-
-(us
-]'JU
h acp -m 6 (%)
( 1.3.10)
Using this formula, one can get the correct equation of the free surface *. Here the contribution of the term with dp/ar is very essential, taking into account the fact that the motion is non-stationary. It is just this term which brings the magnitude w:r '/g into the expression for z, added to the negative expression -w;r2/2g, which, as it was recently noticed, follows from Bernoulli's law [that is, from formula (1.3.9), where ap/ar is put equal to zero andf(t) is constant]. Omitting the details of the derivation of the free surface equation, we shall dwell on the calculation of the magnitude aq/ar. The phase q is a single-valued function of the coordinates if the surface of the cylinder and all the vortex points (the picture of motion in the plane perpendicular to the z-axis is considered) are connected by a line of the cut. This imaginary cut is made from an arbitrary point of the boundary through all the vortex points in turn to the central one and represents something like a spiral. As one moves along the cut in some direction along one of its edges and in the opposite direction along the other edge, the phase changes in magnitude by 211. times the number of the passed vortices. That is why the difference of phases between two edges at the distance r from the axis of the vessel is 2n (1.3.11) 'pz - 401 = - m o ~ - n r 2 . 2 n= - - mmor2. A During the period of rotation different points of the liquid located on the circumference with the radius r pass through a fixed point of space one after
* Such a derivation of the equation of rotating helium II meniscus makes its parabolic shape one of the phenomena in which a change of the liquid level is observed due to creation or motion of vortex lines (Allen1~9,Anderson and Richards130). References p . 156
CH. 3 . 5 21
91
ROTATION OF HELIUM I1
another beginning from one of the edges of the cut till the point on the other edge. If the circumference just mentioned does not pass too close to the nearest vortex lines then the change of the phase, caused by the passage through different points of the liquid, takes place rather smoothly, and at the fixed point of space (1.3.12)
This expression, providing, as was already mentioned, the correct equation of the free surface*, is obtained due to the fact that the distribution of phases over an arbitrary circumference is determined by all the vortex lines which it contains. It is one of the manifestations of coherence of a superfluid liquid state which is caused by the potential character of its motion. 2. Dragging of the superfluid component into rotation 2.1. PECULIARITIES OF DRAGGING
OF A QUANTUM LIQUID INTO ROTATION
The first observations of helium I1 being dragged into motion were made by Andronikashvili (published in a paper7). It was established that when the motor which brings the vessel containing helium into uniform rotation is suddenly started, the periphery of helium I1 is dragged at first into motion with the vessel while the central part of the meniscus remains flat (see Fig. 4). In the light of the considerations stated at the end of Section 1.3 this means that vortex lines are generated at the periphery of the vessel and do not immediately penetrate the region adjoining the axis of rotation. Gradually the flat area of the surface becomes narrower and at last the meniscus forms a paraboloid, the shape of which at small and moderate angular velocities does not differ from the parabolic shape of the meniscus of rotating classical liquids. And only at angular velocities of the order of 30 sec-' a conic pit mentioned in Section 1.1 is formed on the lower part of the paraboloid. The velocity of propagation of the vortex front under the conditions of this experiment is about lo-' cm/sec. The process of helium I1 being dragged into rotation was studied by Hall in the experiment already mentioned with a pile of disksla; by Eselson, Lazarev, Sinelnikov and Shvetsal (an impulse of rotation is suddenly given
* There are, on the free surface, sharp but extremely narrow craters in the immediate vicinity of vortex lines. They are considered in most detail by Hallz0,who took the surface tension into account. References p . 156
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E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.3,
32
to a pile of disks hanging from a magnetic suspension); by Pellam22123who estimated the velocity of propagation of the vortex front by observing the delay of the deflection experienced by the Rayleig disk placed at different distances from the periphery of the vessel ; by Craig24 who used a torsional balance with the disks suspended perpendicularly to the liquid flow at the ends of a beam; and by others (see also Section 2.3). The general conclusions made on the basis of these experiments are the following: a) an increase of the time for dragging helium I1 into rotation on a decrease of the angular velocity of the vessel rotation; b) existence of a finite velocity of the propagation of the front of rotation (in contrast to the front of deceleration which propagates almost instantaneously); c) slow progress of the initial stage of the process of bringing a superfluid into rotation, but rapid completion of
Fig. 4. The gradual growth of the rneniscus of twisting helium 11.
rotation (in contrast to deceleration which begins rapidIy, but completes extremely slowly). In many studies a difference was noticed in the kinetics of dragging classical and quantum liquids into rotation. We will dwell in more detail on the experiments made by Reppy, Depatie and Lanel3 and Reppy and Lane25. In their experiments a hollow cylindrical References p . 156
CH. 3,5 21
93
ROTATION OF HELIUM 11
vessel, containing liquid helium and hanging from a magnetic suspension, received a rotational impulse lasting 0.5 sec. Here the vessel acquired some initial velocity. Due to the redistribution of the angular momentum between the vessel and the liquid, deceleration of the suspended system could be observed. The character of this deceleration was determined by the laws for dragging into rotation, valid either in a classical liquid or in a quantum one. These laws, as it is seen in Fig. 5 , were quite different.
.
D
0 0
0
I
0
4
8
12
16
2.63'K o
1.36-K
20 100 x s e c
Fig. 5. Twisting of classical and quantum liquids in the experiments of Reppy et aI.l3. The velocity of the vessel decreases because of the transfer of its angular momentum to the liquid.
While there is general agreement, the quantitative comparison of the results of the experiments on dragging helium I1 into rotation is often impossible because of the fact that the conditions under which the experiments were made and the techniques of measurements were quite different. 2.2. DEVELOPMENT OF QUANTUM TURBULENCE ON
DRAGGING HELIUM
11 INTO
ROTATION
The existence of the difference between the processes of dragging into rotation of classical and quantum liquids is shown especially clearly in the simple, but rigorously quantitative experiments of Tsakadze 26, and Tsakadze and Cheremisina27.They have measured the time dependence ofthe meniscus depth Az by recording it with a cinema camera, switched on by means of an electronic circuit every 3 sec. As it is seen from Fig. 6, the character of the meniscus growth is quite different for helium I and helium 11. For helium I, References p . 156
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E. L. ANDRONMASHVIJJ A N D YU. G. MAMALADZE
I
[CH.3, 5
2
log A z
Fig. 6. Kinetics of the process of meniscus deepening of classical (curve a) and quantum (curve b) liquids according to the data of Tsakadze and CheremisinaZ7.
20 30
I
0
5
10
15
20
25
30
35
40
l z w .
S-TfR
Fig. 7. The time of dragging helium I into rotation (curve a) and that of helium I1 (curve b) according to the data of Tsakadze and Cheremisina2’. Different notations of the experimental points on the curve a correspond to different values of 00 and R,showing that the time of dragging helium I into motion does not depend on each of these magnitudes, but depends only on their ratio. In the case of helium II the similar dependence is not universal. References p . I56
CH. 3,
p 21
95
ROTATION OF HELIUM I1
for instance, the dependence of the magnitude log LIZ on time is a straight line with a break at some instant (curve a). The length of the first straight portion decreases linearly with an increase of the angular velocity. At the A-point the character of the meniscus growth changes sharply and already at T=2.17 OK (only 0.002 OK lower than the I-point) the dependence log d z = f ( t ) is a smooth curve (curve b). It is interesting to note that the time of twisting of helium I is, in all the studied cases, shorter than that of helium I1 (Fig. 7). That means undoubtedly that formation of vortex lines in the superfluid component decelerates the process of turbulent twisting of the normal component (the Reynolds number under the conditions of these experiments is in the range Re- 5 x 105-2 x lo'). The quantum turbulence suppresses the classical one. In the experiment of Tsakadze and Cheremisina the dependence of time for bringing helium I1 and helium I into rotation on the angular velocity and on the radius of the vessel was studied. The results are shown in Figs. 7 and 8.
120
t
t(sec)
110
100
90 60
:I
70
40
1.3
1.4
1.5
1.6
1.7
.
1.8
19
2.0
2.1
2.2
TPK)
Fig. 8. The temperature dependence of the time of dragging helium I1 into rotation according to the data of Tsakadze and CheremisinaZ7.Different notations of the experimental points correspond to different values of wo and R,the products of which are written above the curves. The time of dragging helium I1 into rotation does not depend on each of these magnitudes, but on their products, i.e. on the linear velocity.
Explaining the results of their investigations the authors proceeded from an assumption that the number of quantized vortex lines n, generated at the unit length of the periphery per unit time and moving to the axis of rotation, is proportional to ua. Here u is the time-averaged relative velocity of the normal and superfluid components, which changes from w,R at the beginning of the process to zero at its end, and u is some exponent. So ncc(w,R)". References p . I56
96
[CH.3,
E. L. A N D R O M K A S H U I AND YU. G. MAMALADZE
52
Then, the process of the establishment of the stationary regime on formation of an equilibrium number of vortex lines NR equal to rnw,R2 / A [according to eq. (1. 3. 2.)], would require a time t , determined by the relationship NR (2.2.1) t oc - cc (w,R)'-*. Rn Thus for any law of the form ncc ua, a universal dependence of the time of twisting t on the product of wo and R, i.e. on the linear velocity, arises in complete agreement with the experiment (Fig. 8). A numerical estimation shows that a is of the order of a- 1.3. TIME FOR THE FORMATION OF VORTEX LINES AT SMALL ANGU2.3. RELAXATION LAR VELOCITIES OF ROTATION
It was mentioned in Section 2.1 that the initial stage of the process of dragging helium I1 into rotation proceeds rather slowly, In the studies, mentioned in the same section, made by Reppy and Lane25, it is shown that, when the angular velocity of rotation is decreased, the initial portion of the corresponding curve becomes more and more like a horizontal plateau ( ~ ~ 5 0 . 2 sec-l, Figs. 9 and 10). At angular velocities w,-0.1 sec-' the authors have observed a long delay at the beginning of helium I1 twisting, which could take some thousand seconds (their data are taken at T r 1.2 OK for which p , / p x 3 % ) . At lower angular velocities (w0x0.065sec-') they did not manage to bring helium I1 into rotation at all during the experiment (- lo4 sec). However, as Reppy and Lane14 have shown, such delays do not occur when there is an artificial irregularity on the inner surface of the cylindrical vessel.
0.06
OC
1 1 1 1 1 1 1 1 1 1 1
0
2
4
6
0
10
k sec
Fig. 9. Dragging of helium I1 into rotation at small velocities of rotation of the vessel (compare Fig. 5) according to the data of Reppy and Lane25.
References p . 156
CH. 3,5 21
ROTATION OF HELIUM I1
91
Fig. 10. The same thing as in Fig. 9. The curve A is obtained on twisting of the vessel containing recently rotated liquid. The curve B is obtained under the conditions when the liquid, before the vessel was twisted, was at rest for a long time in the state of helium I. Helium 11 prepared in such a way was not dragged into rotation after the vessel obtained four impulses in succession. Dragging has taken place only after the fifth impulse, after which the velocity of rotation reached about 0.2 sec-l.
2.4. RELAXATION TIME
FOR VORTEX LINE FORMATION IN ROTATING HELIUM
ON TRANSITION THROUGH THE
A-POINT.
I1
THE MECHANISM OF VORTEX LINE
FORMATION
The study of formation of Onsager-Feynman’s vortex lines on cooling of uniformly rotating helium I lower than the A-point was made in detail by Andronikashvili, Bablidze and Tsakadze 26-30. To this end a radial mode resonator was used, in which the waves of second sound, generated on the surface of the inside cylinder, were reflected from the inner surface of the outside cylinder and received by a detector wound on the inside one. The resonator was tuned to some temperature; then the liquid helium was heated to T=2.21 OK at which temperature it rotated for about 30 min and then, without stopping rotation, it was cooled to the temperature at which second sound was tuned to resonance. By means of this device attenuation of second sound waves was observed. It was established that on transition through the A-point, vortex lines in rotating helium I1 are formed after considerable delay. Depending on the rate of cooling the relaxation phenomena accompany the process of rotation till 2.12 OK. Lower than this temperature one could References p . 156
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[CH.3 , g
2
not observe non-equilibrium phenomena at any velocity of rotation. The relaxation time was found to be dependent on the angular velocity wo and on the value of undercooling (TA- T). The following empirical formula was established (see Fig. 11): _ _,JA-0,1 _ a
t = ~ ~ e ,
(2.4.1)
where a= 1.18 sec-', wC1is the critical angular velocity of rotation in the vessel with given characteristic dimensions (ac1 emo), to is a parameter depending on the temperature, which changes from 1100 sec at T= 2.165 OK to 500 sec at T=2.125OK. On cooling rotating helium I1 lower than the L-point, there can arise two situations: either vortex lines are formed at once or there can exist another
1200
1000
aoo
600
400
200
Fig. 1 1 . The time of formation of an equilibrium array of vortex lines after helium I is cooled till the temperature shown at each of the curves, according to the data of Andronikashvili et a1.2gsS0. Referencesp. 156
a.3 , s 21
ROTATION OF HELNM I1
99
type of motion, which can be called an intermediate one and which, with the lowering of temperature or in the course of time, is transferred into a stable array of vortex cores, aligned parallel to the axis of rotation. A slow decrease of the second sound amplitude in the course of time, observed in the experiments with the radial mode resonator, cannot resolve this question in favour of either assumption. A second axial mode resonator had to be constructed to answer this question. In this resonator a generator of second sound was fixed to the top of the device and the receiver to its bottom. From Hall and Vinen's work 31 (see also Ref. 11) it was known that propagation of second sound along vortex lines does not lead to additional attenuation of its amplitude. Therefore, if the vortex lines oriented along the axis of rotation would be the only reason for additional scattering of second sound (in the radial mode resonator) then, in their presence, rotating helium I1 should seem transparent for thermal waves propagating in the axial mode resonator. The experiments performed by Andronikashvili, Bablidze and Tsakadze have shown that, in the immediate vicinity of the &point, rotating helium I1 is opalescent to second sound propagating along the axis of rotation. After some time or on moving away from the bpoint it becomes more and more transparent. And finally (after about 200 sec for a velocity of rotation oo= 1.76 sec-' and T=2.168 "K, when T- 300 sec) the amplitude of second sound reaches its resonance value. The use of resonators with different geometry shows that the change of the inner surface of a rotating vessel does not influence the formation of the array of vortex lines. Thus an intermediate type of motion can exist, which makes rotating helium I1 isotropic with respect to the propagation of second sound. Apparently, at this stage vortex nuclei are formed in rotating helium I1 and vortex lines are gradually formed from them. It is interesting to note that, according to eq. (2.4. I), the smaller is the angular velocity, i.e. the fewer are the vortex lines, the longer is the time during which vortex nuclei can form a vortex line. This is clear, since the mean diffusion path till a vortex line is encountered is longer for each nucleus when there are few vortex lines. It is also an interesting fact that the time required to form a vortex line on transition through the I-point is comparable in order of magnitude with the time of vortex formation and decay taking place on sudden starting or stopping of rotation of the cylindrical vessel; for instance, at oo= 1.76 sec- ' References p . 156
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100
-
ANDRONLKASHVILI AND YU. G. MAMALADZE
[CH.3,
53
and the finite temperature T=2.168 “K, the time intervals are equal to 300 sec and 70 sec respectively. This makes us think that vortex lines in this case as well are formed from vortex nuclei (or decay on them). But the turbulent regime of the normal component in this case contributes to the shortening (3 - 4 times) of the relaxation time. In other words, the classical turbulence promotes the development of quantum turbulence. The conception of formation of vortex nuclei finds its theoretical treatment in Iordanski’s work32. It is shown there that when there is sufficiently high relative velocity of motion of the normal and superfluid components, vortex formation takes place in the latter. This formation passes through an initial stage of nucleus creation (the author considers the process of creation of ring vortices). The intensity of this process is calculated.
-
3. Observation of vortex lines and their distribution in uniformly rotating helium I1 3. I , EXPERIMENTS ON ESTABLISHMENT
OF VORTEX LINES
As it was already mentioned in Sections 1.1 and 1.3 the existence of vortex lines causes a peculiar distribution of velocities in helium I1 which, for a sufficient number of vortex lines, imitates the solid body rotation of the liquid. Such a rotation is detected in “integral” experiments (Section 1. I), in which the liquid meniscus was observed and the angular momentum was measured. But there are many experiments in which the local vortex singularities are rather clearly pronounced. We shall not describe here in detail the well-known experiments of Hall and Vit1en3~911on second sound wave scattering on vortex lines in rotating helium 11. As it is known these experiments have led to a convincing confirmation of vortex line existence, to the evidence that their circulation is equal to the magnitude 2nAlm and also to the determination of the force of mutual friction between the normal component and superfluid vortex lines. Later, the elementary circulation was determined in the direct experiments of Vinen l91339 l1. One of the methods of vortex line study, which became recently rather widespread, is the study of motion of charged particles through rotating helium 11. This method is based on intensive absorption of negative (but not positive) ions by vortex lines. The cause of the “mutual attraction”, existing between vortex lines and negative ions, is apparently the similarity of their physical nature. The former are node lines of the wave function of the liquid. References p . 156
CH. 3 , g 31
ROTATION OF HELIUM I1
101
The latter, as it is assumed, are hollow spheres with the dimension of the order of ten Angstroms, involving electrons. The capture of negative ions by vortex lines of rotating helium I1 was found by Careri, McCormick and Scaramuzzi34. They have observed an appreciable decrease of the current intensity of negative ions, formed by a radioactive source210P0,while moving along the radius. This decrease of the current intensity is proportional to the angular velocity wo. The motion of ions along the axis of rotation, on the contrary, does not depend on the angular velocity. The obtained data give information regarding the formation of a volume charge in the gap between the electrodes, caused by the capture of negative ions by vortex lines (Fig. 12). n
Fig. 12. The dependence of the number of negativeions (in millions per cubic centimetre), trapped by vortex lines, on the applied voltage according to the data of Careri et
In the experiments of Douglass35 the mean time of negative ion capture by vortex lines was measured. It obeys the following empirical relation z
cc exp ( E , / k T ) .
(3.1.1)
This allows one to estimate the depth of the potential well Eo, which traps ions. According to Douglass E0=0.012 eV. Tanner, Springget and Donnelly36~37 have made a number of measurements on the dependence of the capture cross-section of negative ions by References p . 156
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[CH.
3, 8 3
vortex lines on temperature and applied voltage and the obtained data were compared with the theoretical calculations of Donnelly38. Let us note that, according to Donnelly, the difference of the cross-sections of capture of a negative ion and a positive one by a vortex line is not very large. But the differenceof the probabilities of the trap escape is sufficiently large. Because of that, a decrease of the current intensity of positive ions is impossible to observe and it can be discovered only at T < 1 O K . The existence of vortex lines, arranged in a definite geometrical order, was shown in Chase’s experiment39. Chase rotated a horizontally placed tube with a diameter of 2.62 mm round a vertical axis; the tube was packed with 70 wires (diameter 0.25 mm). These wires formed a great number of narrow parallel channels of irregular form in the cross-section. Heat was passed along these channels, filled with helium 11. The critical value of the heat flux w, should, according to Vinen11, depend on the presence of vortex lines. Measuring w, as a function of the square root of the angular velocity of rotation, Chase has observed that this magnitude changes by steps (Fig. 13) and the first one of them is characterized by an appreciable hysteresis, which is absent at higher angular velocities. The data were obtained under the conditions of accelerated rotation (open symbols) and decelerated rotation (solid symbols). There are no vortex lines on the first step. Their sudden appearance corresponds to formation of a stable configuration of vortex lines; the distance between them (nh/rno,)% is 0.34 mm. This value is com0.05
b
I .O
15
2.0
5
mi (rad/sec)*
Fig. 13. The dependence of the critical flux of heat on the velocity of rotation in the experiment of Chases9. References p . 156
CH. 3, § 31
ROTATION OF HELIUM ll
103
patible with the size of the wires. The vortex line spacings, equal to 0.17 mm and 0.1 1 mm, correspond to the transitions to following steps, i.e., the vortex line spacings for two different steps are to each other as I:+:+. Such a situation could arise only in the case when there would be a discrete succession of stable configurations of vortex arrays, distributed in one, two, three etc. rows between the wires. Unfortunately Chase has not obtained the same clear results for the channels of other shape and larger size40. It is possible that the formed vortex lines are “blown out” in these channels by the thermal flux, or that the conditions of their attachment differ sharply from those existing in the experiments with thin wires of the first capillary used by Chase.
3.2. DIRECT OBSERVATIONS OF VORTEX LINES IN ROTATING HELIUM I1 Quite recently Steyert, Taylor and Kitchens41 managed to observe directly vortex lines in a rotating cylinder where tiny snow flakes composed of a hydrogen-deuterium mixture were condensed in helium 11. Watching the snow flake motion through a microscope, they recorded their observations by means of a cinema camera. In such a way they observed the motion of snow flakes round vortices and thus established the existence of vortex lines and measured their circulations. In contrast to the results of Hall and VinenS1911, they have found that there are not only single circulations (in units of 2nA/m), but also multiple and non-integral ones as well (Fig. 14). The largest of the observed circulations is equal to 10.5 (2nh/m). The dimensions of the loops along which snow flakes were moving exclude the possibility of flowing round several vortex lines. However, from the experiments of others42, it is known that under some conditions the snow flakes move with a velocity different from that of the superhid. Therefore this 10
a ROTATING CYLINDER
a MOVING WIRES 5
0
Fig. 14. A hystogram of Steyert et al.*l describing the distribution of the number of the observed vortex lines over circulations.
References p . 156
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[CH.
3,
53
method should be recognized as the wrong one for the numerical calculation of the circulation, though the existence of vortex lines is shown by it in the most direct way. As far as Fig. 14 is concerned we think that it sooner describes a random distribution of the number of snow flakes over velocities than the distribution of the number of vortex lines over different circulations. 3.3. IRROTATIONAL REGIONS
The uniform distribution of vortex lines in a volume of rotating helium I1 experiences an appreciable distortion near the walls of the vessel. This was already noticed by Hall44, who, on minimizing the free energy of the liquid, has shown that some region free of vortex lines should exist near the side of the cylinder. The motion of the superfluid component is determined here by the relation u, = r / 2 n r ywhere Tis the sum of circulations of all the vortex lines existing inside the vessel. In this case both the curl of the local velocity and the curl of the averaged velocity are equal to zero. This was the reason why such regions are usually called irrotational ones. According to calculations of Khalatnikov45 the radius of the irrotational region Ri, outside of which the velocity is determined by the formula uszcooR2/r,is determined by the equation (3.3.1)
where b = (h/mwo)f. Bendt and Oliphant46 have shown that, for rotation of helium I1 in the annulus formed by two coaxial cylinders which rotate with a common velocity, the irrotational region should be created round the inside cylinder as well. Proceeding from the considerations of minimum free energy E - Loo, they have calculated the velocity dependence of the radius of the irrotational region and the value of the circulation in it. It was found that the velocity circulation round the inside cylinder can reach hundreds of thousands of the quantum value 2nAlm. The appearance of an irrotational region round the inside cylinder was also considered in the work of Kemoklidze and Khalatnikov47. Not considering the details of the procedure of minimization realized by Bendt and Oliphant, and Kemoklidze and Khalatnikov, we shall only note that the existence of an irrotational region and the great circulation in it can be explained in the following simple way. First of all vortex lines cannot approach very close to the surface of the cylinder because, interacting with References p . 156
CH.
3 , 5 31
ROTATION OF HELIUM 11
105
their reflections, they would acquire too high a velocity*. But this contradicts the requirement for the equilibrium motion of vortex lines together with the vessel and the normal component. Further, the velocity of the superfluid component should be close to cuor outside of the irrotational region. But this velocity is determined by the circulation along the circumference with the radius r ; this circulation is, in turn, a sum of circulations over the circuits involved. Therefore we should have 2nr Hence the expression of the circulation in the irrotational region is obtained in terms of the radius of this region ri (Mamaladze):
r x 2nw,r:
= 7crzNTo x n r T N I , .
(3.3.2)
In other words, the circulation in an irrotational region compensates for the lack of vortex lines with elementary circulation I', arranged with the density N [see eq. (1.3.2)] **, for the points located far from this region. If the existence of the inner irrotational region is thermodynamically favourable, then it is natural to pose the question as to the possibility of its appearance when there is no inner cylinder, instead of which the liquid could form a hollow cylindrical core. Kemoklidze and MamaIadze48 have shown that it is impossible because of two reasons :the radius of such a core, corresponding to the minimum of the free energy, should have a size smaller than the interatomic distance while the circulation should not be quantized and would be equal approximately to l,5ro. However, the existence of an irrotational region was found to be thermodynamically favourable, when there 'was the usual Feynman vortex line of unit circulation on the axis of the rotating vessel. The radius of this region was six times as large as the radius of a region which corresponded to one vortex line at their normal density( ri z 6b). An irrotational region surrounding the inside cylinder was discovered experimentally by Tsakadze4Q.He has used a cylindrical vessel with a rod placed along its axis (Fig. 15). Another vessel was put on the rod in such a way that
* This idea used by Hall44 to explain the existence of the outer irrotational region, can be applied with the same success to the vicinity of the inside cylinder. ** The minimization of the free energy leads to the following expressionfor r(Ref. 47): ri2 - r12 r = 2 7 ~ In 0~ (riZ/r12)' ~
When ri - rl Q r1, where rl is the radius of the inside cylinder, this formula coincides with the estimation (3.3.2). References p . 156
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[CH.
3, 0 3
there was a narrow gap between the bottoms of the inside and outside vessels. The superfluid liquid from the inside vessel (not glued to the rod) had to leak through this gap due to the difference of the levels. Being in the gap between the bottoms, it had to flow in the horizontal direction. If there were vortex lines in the gap between the bottoms, then the velocity of the flow out had to depend, due to the mutual friction (see Ref. 11 and Sections 4.2,4.3 and 4.4 of this paper), on their density, i.e. on the velocity of rotation. If, with a
n
Fig. 15. Tsakadze’s device4O used to find an irrotational region surrounding the insidi cylinder. References p . 156
CH. 3,
# 31
ROTATION OF HELIUM ll
107
decrease of the velocity of rotation accompanied by an increase of the radius of the irrotational regjon46, vortex lines would leave the gap between the bottoms, the velocity of the outflow would become constant. Tsakadze sucked in the liquid into the inside vessel by means of the thermomechanical effect and then measured the time of its outflow as a function of the velocity of rotation (Fig. 16a). The break on the observed curve means that, when
1t
t ” (sec)
Fig. 16. (a) The dependence of the time of outflow of helium I1 from the inside vessel (Fig, 15) on the velocity of rotation; tn is the time of the outflow in non-rotating device. T = 1.75 O K . (b) The same dependence as in (a), when T = 1.46 OK. Rdfierences p . I56
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[CH.3,
83
1.5 sec-I, the radius of the irrotational region ri becomes equal to the outer radius of the inside vessel, which is 0.8 cm. The radius of the rod is rl =0.5 cm. Thus ri - rl ~ 0 . cm. 3 On decrease of temperature, the same value of ri -rl corresponds to the velocity w o w 10 sec-l (Fig. 16b). Unfortunately these data agree unsatisfactorily with the calculations of Kemoklidze and Khalatnikov (and even worse with those of Bendt and Oliphant), apparently because of the difference of geometrical conditions under which the experiments were performed and the calculations were made. In the device of Tsakadze the liquid flows from a narrow gap between the bottoms into a volume confined by the free surface from the top, which is much higher than the bottom of the inside vessel. Meanwhile the calculations of Bendt and Oliphant, as well as those of Kemoklidze and Khalatnikov, were made for the case of liquid flow between the parallel planes. A similar method was used by Tsakadze to find an irrotational region when there was no inside cylinder present. With this aim in mind he has used a W,,NN
Fig. 17. Tsakadze's device4e used to find an irrotational region surrounding the axis of rotation. References p. 156
CH. 3 , o 31
ROTATION OF HELIUM I1
I09
variation of the device just described, in which there was no axial rod, and the liquid leaked from a capillary, placed on the axis of rotation, flowing through the gap between its lower edge (the capillary had a flat section) and the bottom of the vessel (Fig. 17). The results are shown in Fig. 18. The radius of an irrotational region was found to be equal to ri w0.1 cm when wox 1 sec- '. The calculations of Kemoklidze and Mamaladze give the value ri ~ 0 . 0 cm 8 for these conditions.
I
t-tn
(set)
Fig. 18. The dependence of the time of outflow of helium I1 from the capillary (Fig. 17) on the velocity of rotation.
3.4. DISTRIBUTION OF VORTEX LINES UNDER A FREE SURFACE The existence of a free surface of the liquid can be also a special reason for non-uniform distribution of vortex lines. A vortex line in equilibrium should be perpendicular to the free surface, not to be drifted by the tangential component of the vortex tension. But in this case it will not be parallel to the axis of rotation, as the surface has a non-flat meniscus near the point where vortex lines have to bend. In this connection Kemoklidze and Mamaladze50 have considered the solution of the hydrodynamical equations for rotating helium I1under the corresponding boundary conditions for solid surfaces and for a free surface. They did not manage to solve the problem completely because of great difficulties in the calculations, but they have shown that the solution ( u s ) = w or and ]curl (u,)l= w, =2w, (the conReferences p . 156
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[CH.3,
83
stancy of the density of vortex lines, Section 4.4) is incompatible with the parabolic meniscus. Small deviations from this solution show a tendency to increase (curl u s ) (an increase of the density of vortex lines) on approaching the axis of rotation, and to make the meniscus deeper near this axis, in the immediate vicinity of which the deviations cease to be small. Apparently the conic pit of the meniscus observed by Andronikashvili' is a consequence of these circumstances. Let us note that considerations on the shape of the meniscus, mentioned in Section 1.3, also confirm the necessity for the meniscus to become deeper in the region where the vortex line density increases. As to the immediate vicinity of the axis of rotation, we should rather assume that an increase of the vortex line density is replaced there by an irrotational region and a single vortex line is situated on this very axis. The results of recently mentioned work by Kemoklidze and Mamaladze48 can be used as a confirmation of such an assumption. Though they concerned the case of the liquid, bounded with flat surfaces, they should be valid deep under the liquid meniscus as well. 3.5.
O N THE NORMAL COMPONENT MOTION IN A ROTATING CYLINDRICAL VESSEL
The natural assumption of all the authors, describing the process of helium I1 rotation, was that the motion of the normal component in a uniformly rotating vessel proceeds according to the laws of the classical hydrodynamics. This conception was in doubt because of an experiment performed by Pellam22*23 in which the Rayleigh disk, being at rest in the laboratory system of coordinates and located at an angle of 45" with respect to current lines, responded to helium I rotation by deflecting through a definite angle. On transition through the A-point the deflection became zero, and when the temperature was further decreased, the deflection of the disk was proportional to ps (Fig. 19). This experiment seems to mean that the whole normal component stops at the I-point. Due to this stopping and under the conditions when ps is negligibly small, no forces act on the Rayleigh disk. For such an interpretation of Pellam's experiment the normal component should be considered to remain at rest in the whole range of the temperatures under investigation corresponding to helium 11. Naturally such a model of rotating helium I1 was quite unacceptable for everybody including Pellam himself. Therefore his experiment stimulated numerous studies of the problem thus created. In particular, just in this connection, Reppy, Depatie and Lane13 made measurements of the angular momentum of helium 11, confirming the data of Hal110112 on complete dragging of helium I1 into rotation. References p . 156
CH. 3,
$31
ROTATION OF H E L N M I1
111
Craig was justified in remarking that "integral" experiments of this type do not explain the mysterious result obtained by Pellam, but only confirm a well-established fact that helium I1 rotates as a solid body. With the aim of studying "local" velocity distribution in the liquid, Craig24 has used a torsional balance with two disks perpendicular to the current lines of rotating helium 11. The force due to the pressure on these disks was found to be practically independent of temperature in the range 1.3 "K to the A-point. Thus, having confirmed once more the joint rotation of both of the components, Craig nevertheless could not indicate the reason for the Rayleigh disk behaviour
TEMPERATURE
( O K )
Fig. 19. The temperature dependence of the Rayleigh disk deflection in Pellam's experiment22sz8(compare Fig. 21).
Craig53 later suggested a hypothesis that, on flowing round the Rayleigh disk at large angles of attack (45"), the potential flow is replaced by a Helmholtz flow. Using a suspension of a solidified hydrogen-deuterium mixture (cf. Section 3.2) Kitchens, Steyert, Taylor and Craig54 have confirmed this hypothesis. They have really observed tearing off jets and formation of eddies behind a plate at rest, immersed into rotating helium 11. These phenomena developed more and more intensely at an increase of the attack angle from 27" to 45" and 90". Referring to the data of Craig and Pellam43, as well as of Kohler and Pellam51, Kitchens et al. state that under such conditions (a velocity of the order of several centimetres per second) the superfluid component flows potentially. Hence the phenomena observed by them are connected with the motion of the normal component. This speaks once more in favour of the point of view that the normal component of rotating helium References p . 156
112
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.
3, 8 3
is not at rest. However, the independence of the effect on temperature made the authors refrain from looking for an explanation of Pellam’s experiment. Varying the experimental procedure of his experiment with the Rayleigh disk, Pellam55 has again obtained a curve similar to that shown in Fig. 19 and stated that the rotating normal component does not interact with the disk in an incomprehensible way. The experiment of Tsakadze and Shanshiashvili56 has shown that some unexplained mechanism which reduces the action of the normal component on the Rayleigh disk to zero is connected with the illumination of the disk. The disk was used as a mirror, to detect its own deflection in all the experiments of PelIam22.233 55. Tsakadze and Shanshiashvili reproduced Pellam’s experiments in two variants: 1) when a light beam fell directly on the Rayleigh disk as in Pellam’s experiments and 2) when a light beam fell on a mirror, fixed to the same rod as the Rayleigh disk, but above the helium level (Fig. 20). They have shown
Fig. 20. The device of Tsakadze and Shanshia~hvili~~. (1) the Rayleigh disk; (2) rod; (3) elastic suspension; (4) second mirror. References p. 156
CH. 3 , s 31
113
ROTATION OF HELIUM 11
that the dependence of the disk deflection on the temperature is not observed only in the second case (Fig. 21). Thus Pellam’s strange results are connected entirely with the heat produced by the light beam*. 3.6. THESTRUCTURE OF THE VORTEX LINE ARRAY Considering an infinite volume of helium 11, the normal component of which rotates according to the law u, = oar, Kiknadze, Mamaladze and Cheishvili57 have shown that vortex lines of the superfluid component can create “solid”
1
0.5
0
‘K)
Fig. 21. The deflection 9 of the Rayleigh disk on illumination of the lower mirror (open circles) and upper one (solid circles) (Fig. 20). 91 is the mean deflection in helium I (compare Fig. 19: 9 / 9 1= penr/p.*).
configurations with two-dimensional lattices, rotating together with the normal component without any change of the mutual arrangement of vortex lines. They have found that the phenomenological theory of superfluidity58159, applied to the study of rectangular vortex lattices, is completely equivalent to the theory of two-dimensional lattices of fluxoids due to Abrikosoveo. Therefore the conclusion on comparable energy advantages of the lattice with the triangular symmetry (Fig. 3b) is directly applied to vortex lines in helium 11. Besides they managed to clear up some details of velocity distribution in helium I1 rotated by vortex lines. In particular, it was found,
* Note added in proof, Recently, Pellam132 has shown that his results are not changed if the time of illumination of the Rayleigh disk is much less than the period of the oscillations of the disk. The disagreementbetween the experimentsremains unexplained and the reason of the absence of interaction between rotating normal liquid and Pellam’s device is not yet clear. References p , 156
114
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.3,
34
that between vortices near the points where the maximum of the wave function takes place (Fig. 3b), the velocity of the superfluid component is distributed according to the relation (3.6.1)
vs=wo x rm,
where r,,, is the radius vector of the wave function maximum. Thus the vicinities of the maxima are similar to the hypothetical domains of Pellam61 in which the linear velocity is constant (but not the angular one) that provides the equation curl us =O. In the rotating system of coordinates the superfluid liquid rotates with the velocity -w, round the point r, in a direction opposite to that of its rotation round vortex points us - oox r = - wo x (r
- r,).
(3.6.2)
Tkachenko62 has come to the same conclusions on the “solidity” of twodimensional lattices of vortex lines and on the minimum free energy of the triangular lattice in the case of a classical perfect non-compressible liquid. Later he63 has given evidence for the stability of the triangular lattice, small perturbations in which are drifted to the infinity, and the sum of the squares of the perturbations increases extremely slowly (in contrast with the case of unstable lattices for which the sum of the perturbation squares increases rapidly). In contrast with Pinkus and Shapiro64, Tkachenko has shown that the triangular vortex lattice has a definite spectrum of the normal oscillations, and a transverse wave can travel along it. Thus doubts often expressed on the existence of a regular geometrical structure of vortex lines in rotating helium I1 can be considered essentially shaken. The search for a triangular lattice in a sufficiently rapidly rotating vessel, under the conditions for which irrotational regions occupy a small part of its volume, is a real experimental problem. 4. Elastic properties of vortex lines. Oscillations of bodies of axial-symmetric shape in rotating helium I1 4.1. THEMODULUS OF SHEAR IN ROTATING HELIUM I1 Vortex lines have a certain tension which gives rise to their elasticity. This tension coincides numerically with the energy of unit length of a vortex line and can be expressed17 by the following formula (cf. Section 1.3) h2
b
(4.1.1) m 2 a, Due to elasticity of vortex lines one can assume that rotating helium I1 E = Z P ~-ln-.
References p . 156
a. 3, Q 41
115
ROTATION OF HELIUM I1
should have a modulus of shear with respect to a deformation transverse to the axis of rotation. In other words, it should resist the torque round this axis. This paradoxical fact was really established by Andronikashvili and Tsakadze65166, who have chosen different indicators of elastic properties of rotating helium 11; in one case it was a pile of disks and in the other a light disk, moving with the same angular velocity a,,as the vessel, and simultaneously performing axial oscillations with some frequency Q. As we see in Fig. 22, the frequency of oscillation of a light disk grows linearly with an increase of the angular velocity of rotation, i.e. with the number of vortex lines. And only when 2w0/Qm0.2 does the curve exhibit a tendency to saturation. The essence of this phenomenon will be explained later.
-
x 10Ssec-1
Fig. 22. The dependence of the frequency of the disk oscillations on the velocity of rotation (a0 is the frequency of oscillations in vacuum). Curve a represents experimental data of Andronikashvili and Tsakadzees;curve b gives the results of calculationsaccording to formula (4.4.12).
The experiment of Andronikashvili and Tsakadze has established quite clearly the existence of the modulus of shear in helium 11, which is seen in an elastic counteraction to the torsional deformation round the axis of rotation. At the same time as these authors65, Hall67 has found the change of the oscillation period of a pile of disks with variable disk spacing44.His experiments are described in detail by Vinen in the third volume of Progress in Low Temperature Physicsll. 4.2. ANISOTROPY OF
ELASTIC-VISCOUS PROPERTIES OF ROTATING HELIUM
11
The next step in the study of vortex lattice properties was a series of experiments on the determination of damping of axial symmetrical bodies which References p . 156
116
E. L. ANDRONIKASHVILI A N D YU. G . MAMALADZE
[CH.3 , s 4
perform simultaneously rotating and oscillating motions in helium I1 (undertaken by Andronikashvili, Tsakadze and their collaborators). Most of these experiments are described in a review6*. First of all damping of rough disk oscillations was studied6g*70;a disk with grains of sand on its surface (the size of grains was about 50 pm) oscillated round its axis. The main part of the device (Fig. 23), used in these experiments and in the experiments with a pile of disks (Section 4.1), was a vessel consisting of rigidly connected wide and narrow cylinders (a tube). The lower part of the vessel (i.e. the wide cylinder and the lower part of the tube) was immersed into a Dewar vessel. The upper part of the tube was above the Dewar cap. An oscillating disk or a vessel with a pile of disks was placed into a wider cylinder filled with helium 11. They were attached to a glass rod which was
y1-
photomultiplier
other bodies suspended
Fig. 23. The device to study axial oscillations of a disk, a pile of disks or a cylinder in rotating helium 11.
References p . I56
CH. 3,s 41
117
ROTATION O F HELIUM I1
suspended by an elastic fibre to the top of the vessel (the whole fibre was at room temperature). The described system rotated without any mechanical connection with the outer moving parts of the device. It was brought into rotation by electromagnets which were connected by means of belts with the motor and also with special devices, which provided uniformity of rotation (an additional constant loading and an oil damper). Damping of the oscillations was measured by means of a chronometric electron circuit which allowed to determine the decrease of the velocity of the oscillating suspended system by observing the increase of the time of flight of a light spot between two photomultipliers71. In Fig. 24 this damping is shown as a function of the ratio 2oO/l2.
110 100
90 80
70 60
50 40
30 20 10
0
20
40
60
80
100
120
140
2W.xrd n
Fig. 24. The velocity dependence of oscillation damping of a rough disk in rotating helium 11. 6, 6, and 60are the logarithmic decrements of damping in rotating helium 11, in non-rotating helium I1 and in the vacuum respectively. The lower curve is obtained at a frequency of oscillation 52 = 0.581 sec-l, the upper one at 0 = 0.361 sec-1. T = 1.78 OK (Andronikashvili and Tsakadzeenfi 'O).
The comparison of these data with those, obtained for classical liquids [water (Fig. 25), helium I] shows that the character of the velocity dependence of the damping S for quantum and classical liquids is quite different under the conditions of disk oscillations. While 6 is practically independent of oo at 200/f'250.5 and a deep minumum at 2w0/12= 1 is characteristic of classical References p . 156
118
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[a398 .4
(6-6.)x 10'
0
0.5
10
15
20
2.5
3.0
2 a. n
Fig. 25. The velocity dependence of oscillation damping of the disk in rotating water (Mesoed and Tsakadze73).
50
45 40 35 30 25
20 15 10
5
0 0 5 10 15 20 25 30 35 40 45 50 5 5 60 65 70 75 2w./n10a
Fig. 26. The velocity dependence of oscillation damping of a smooth disk in rotating helium (Andronikashvili et aI.'O). The lower curve is obtained at 52 = 0.551 sec-l, the upper one at Sa = 0.368 sec-' (compare Fig. 24). T = 1.78 "K. References p . 156
a. 3,9 41
119
ROTATION OF HELIUM ll
liquids*, there is a maximum in helium I1 at 2w0/Q!250.2. Having passed through the maximum, the curve, while not reaching the level characteristic of helium I1 at rest, has a plateau and when 2w0/f2 % 1 begins to increas sharply. The damping of a smooth disk behaves rather dflerently (the lower curve in Fig. 26). At the initial porfion of the curve the damping increases appreciably slower: having passed the maximum it drops to the value 8, corresponding to non-rotating liquid. On decreasing the frequency of oscillation the smooth disk begins to behave as a rough one (compare the upper curves of Figs. 26 and 24). It becomes clear from this experiment that the conditions for vortex line Y 1.221
t'
1.20. 1.18
'
1.16
.
0 2 0 4 06 08 1 0 12
wo x 1Osec-1
Fig. 27. The velocity dependence of damping of cylinder axial oscillations in rotating helium II. The points are the results of measurements made by Tsakadze and Chkheid~e'~. The straight lines are plotted according to formula (4.4.3)when T = 1.48 OK (the lower line) and T = 1.75 "K(the upper line). The index y denotes the magnitude 62-61
z
Symbols are explained in association with formula (4.4.3).
* Such a character of the dependence of 6 on 200/Swas predicted earlier by Mamaladze and Matinyan'z and confirmed with great accuracy by measurements of Mesoed and Tsakadze73, made with water (and helium I). The normal component of helium I1 should behave in the same manner if it had not interacted with the vortex lines of the superfluid component. References p . 156
120
E. L. ANDRONIKASHVILI A N D YU. G . M A W A D Z E
[CH.3,
54
attachment to the disk surface are very important and that the extent of vortex attachment depends on the frequency of disk oscillation. But independent of the conditions of attachment, of the temperature and of other conditions, the maximum in the curve S = f ( o o ) is always found at 2o&! 5~0.2. This value coincides with that attained when the dependence of the frequency of the light disk oscillations on velocity of rotation (Fig. 22) shows a tendency to saturation. As Tsakadze and Chkheidze74 have shown, quite a different dependence of damping on the angular velocity is observed in the case when a cylindrical surface performs axial oscillations superimposed on rotation. The device is in principle the same as in the previous case: a hollow cylinder, turned upside down, was suspended instead of a disk (Fig. 23). The results of this experiment are shown in Fig. 27. A linear increase of electromagnet
-,
Ll,,TkU
,!
to the vacuum pump
cojl rotating the cylinder.
Fig. 28. A device used to study vertical oscillations of a cylinder. (In fact the magnetic arrow is perpendicular to the rectangular coil rotating the cylinder and the cylinder i: nearer to the disk than it is shown.) References p . 156
CH. 3,
41
ROTATION OF HELIUM I1
121
damping is observed on an increase of the velocity of rotation. A classical liquid, under such conditions, behaves as a non-rotating one. The difference of the behaviour of helium I1 and that of a classical liquid is quite comprehensible in this case: the layer of the normal component, adjacent to the wall of the cylinder at the distance of the order of the penetration depth of a viscous wave, drags all the vortex lines which pierce this layer into motion due to mutual friction. This is why the damping of a cylinder, oscillating round its axis, is different from that in a liquid at rest and increases proportionally to the angular velocity, i.e. proportionally to the density of vortex lines. At last in avery fine experiment, performed by T~akadze7~-77, it was shown that the viscous properties of classical and quantum liquids behave in the same way when up and down oscillations of a cylindrical surface are superimposed on its rotation. That corresponds to the motion of a solid surface along vortex lines. Tsakadze’s device, shown in Fig. 28, was similar to that shown in Fig. 23, with the only difference that the elastic fibre was replaced by a solid rod and was led out into a hermetically closed chamber with an analytical balance. The connection between the rod and the balance beam was realized by the magnetic attraction of two steel balls, which did not prevent the rotation of the rod to the lower end of which the oscillating cylinder was attached. The rotation of the cylinder and the rod was effected by means of a circuit with a current fixed on a rotating vessel and a magnetic arrow attached to the bottom of the cylinder. As a result of this experiment the independence of damping on the velocity of rotation both in classical liquids and in helium I1 was established. In this case the layer of the normal component, oscillating together with the inside cylinder, slides along vortex lines without dragging them into its motion. In the last experiment of this series, a disk placed below the overturned cylinder performed forced axial oscillations and generated elastic waves running along vortex lines (Tsakadze76978). This disk is shown below the oscillating cylinder in Fig. 28. The transverse waves change the character of the interaction of rotating helium I1 with the vertically oscillating cylinder. In this case, a linear dependence of damping of the cylinder on the velocity of rotation (i.e. on the number of vortex lines) is observed. As is shown in Fig. 29, the velocity of rotation, at which 200/Q1wO.2 (Q1 is the frequency of the disk oscillations), is connected in this case as well with the existence of a certain peculiarity: the break on the curves obtained by Tsakadze. A short summary of the data described in this Section is given in Fig. 30. References p . 156
122
E. L. ANDRONIKASHVLI A N D YU. 0. MAMALADZE
[CH.3,
34
6X1OZ.
60 59
5.6
5.7
5.6 5.5 5.4
5.3 5.2 5 .l
5.0
,
4.2 4.1 4.0
3.9 38
3.7 36
3.f 3.4
w o x 103 sw-1 Fig. 29. The velocity dependence of vertical oscillations of a cylinder when an axially oscillating disk is placed below it. The arrows show the values of the velocity of rotation, calculated according to formula (4.5.3): (30 = f91/(2rnv8/72ch l), 91is the frequency of the disk oscillations (TsakadzeT6’7%).
+
4.3. HYDRODYNAMICS OF ROTATING HELIUM 11 On the derivation of equations of hydrodynamics of rotating helium 11, a complicated picture of motion, caused by the existence of vortex lines (Figs. 3a and 3b), is averaged. All the physical magnitudes are averaged over a volume which contains a sufficient number of vortex lines to smooth the sharp “splashes” in the distribution of these magnitudes. Then the necessity of taking into account numerous boundary conditions for each vortex line is avoided. However, as a result of such an averaging, the surface forces which References p . 156
CH. 3,O 41
123
ROTATION OF HELIUM I1
'k&k W.
W.
I 2u..rl
I
_.
u.
I
I
W.=q2w.=n
W.
Fig. 30. The summary of the data on the velocity dependence of damping for different orientations of the oscillating surfaces and the direction of oscillation with respect to the axis of rotation (the first column) in a classical liquid (the second column) and in helium II (the third column).
act on each vortex line acquire the aspect of volume ones and should be included into the hydrodynamkal equations. Therefore the equations of the averaged hydrodynamics of rotating helium I1 are more complicated than those of the two-fluid hydrodynamics of Landau. Such hydrodynamics of rotating helium I1 are formulated in the papers of Hall67144, Mamaladze and Matinyan 79, Bekarevich and Khalatnikov80 (see the reviews l1, 45 as well). If to limit ourselves to the consideration of the phenomena, taking place at a constant temperature and at small values of the relative velocity Iu, -u,l (in comparison with the velocity of second sound), the averaged hydrodynamics of rotating helium I1 are determined by the following set of equations. References p . 156
I24
[CH.3,
E. L. ANDRONIKASHWLI AND YU. G. MAMALADZE
84
The equation of motion of the superfluid component [the explanation of the symbols is given after formula (4.3.7)]: 8%
at
+ (u,~V)u,+ v
0 1 , x~ curl - = -- V ( P U
P
+ qsw)+ sVT + F,, .
(4.3.1)
The equation of motion of the normal component: aun Ps F,, . + (un*V)un - V, VU, = --1 V ( P + ?,a)- - sVT - P S
P
at
Pn
(4.3.2)
Pn
The equation of discontinuity for both of the components, considered as incompressible and mutually non-converting liquids : div us = div u, = 0.
(4.3.3)
The equations of transport of vortex lines, establishing the relationship between us and the velocity of the vortex line motion uL*:
(4.3.4) The boundary conditions for un and us:
u,
- u,
= 0,
(us - u,)N = 0
(on the solid surface).
(4.3.5)
(on the solid surface).
(4.3.6)
The boundary condition for uL:
(uL
- u,)~ = aN x N
0
0
0
w
x - - a'N x
-
The boundary condition on the free surface: O'ikNk
=0
(on the free surface).
(4.3.7)
k
In formulae (4.3.1)-(4.3.7), w denotes the curl of the superfluid component velocity (the curl of the mean velocity is different from zero, compare Fig.
* With the purpose of establishing such a relationship, the condition of the equilibrium of the Magnus force, mutual friction and straightening effort, acting on the curved vortex lines may be used as well w x (us - OL)
References p. 156
+
V ~ Wx
W
curl - - Fsn = 0 . W
(4.3.4a)
CH.
3 , § 41
ROTATION OF HELIUM I1
125
3a), v, = q,/p, = &/PSI'(later it will be shown that the constants v, and qs in many respects exhibit themselves as kinematic and dynamic viscosities of the superfluid component, though they are determined by the vortex line tension and, strictly speaking, characterize their elasticity); s is the entropy of unit mass; Fs, is the force of mutual friction acting on unit mass of the superfluid component : w
u,-uv,-v,curlL
w
(4.3.8)
B, B' and B, are the coefficients of mutual friction; v, is the kinematic viscosity of the normal component; P is the pressure; u, is the velocity of a solid surface; the index t in formula (4.3.6) denotes the tangent component of a vector; N is unit vector normal to the solid surface; a and a' are the coefficients of the vortex line sliding along the solid surface (a = a' =0 corresponds to complete attachment, a=a'= 00 corresponds to complete sliding); oikis the tensor of the momentum flux:
q, =pnvn is the dynamic viscosity of the normal component. 4.4. HYDRODYNAMICS OF SMALL OSCILLATIONS OF BODIES OF AXIAL SYMMETRY IN ROTATING HELIUM 11
The equations of the previous section have a stationary solution corresponding to the uniform rotation of the vessel * : u, = u, = oox r ,
w =20,
(4.4.1)
(we mention once more that we speak of averaged magnitudes, though the symbol of averaging is omitted for simplicity). In the experiments, which were described in Sections 4.1 and 4.2, small harmonic oscillations are superimposed on the uniform rotation (4.4.1). The
* Euler's equation for a classical ideal liquid does not define unambiguouslythe behaviour of the liquid in a rotating cylinder. h our case the normal component motion determines that of the superfluid one by means of the term Fenin eq. (4.3.1). Under the stationary conditions Fsn should be equal to zero and curl w/w = 0. References p. 156
126
E. L. ANDRONIKASHVILI A N D YU. G. MAMALADZE
[CH.
3,g 4
equations of rotating helium I1 hydrodynamicscan be linearized with respect to relatively small perturbations of the stationary regime. This allows one to solve rather simply hydrodynamical problems on oscillations of axial symmetrical bodies, superimposed on their rotation, performed together with the liquid. To this purpose, after the determination of the velocity distribution corresponding to given boundary conditions, the tensor of momentum flux is calculated (4.3.9). This determines the momentum of force (or a force) with which the liquid acts on the oscillatingbody, and hence determines the changes of the frequency and damping of body oscillations caused by its interaction with the liquid. The simplest problem is the problem on oscillations, performed by the up-down movement of a cylindrical surface. The normal component is here dragged into a cylindrical transverse viscous wave with vertical direction of the oscillations. It either interacts or does not interact with the superfluid component depending on the difference of the coefficient of mutual friction B, from zero in formula (4.3.8), as the terms with the coefficients B and B’ describe only the mutual friction in directions perpendicular to a vortex line. Therefore the change of damping of a vertically oscillating cylinder is described by the following formula*’ QO
~--aoCcl+
Q
wo Ps ---&, 2P
(4.4.2)
where 6 is the logarithmic decrement of damping, a0 and Do are the vacuum values of damping and frequency respectively. For that reason, therefore, the method of vertical oscillation of a cylinder was suggested as a convenient one to determine the coefficient B,. Tsakadze’s experiment 75-77 (Section 4.2) has shown that B,, determined in such a way, is equal to zero within the errors of the experiment which are equal to L-0.025(at T = 1.86 OK,.when B- 1). The equality B, =O corresponds to the idea that the mutual friction force is caused by scattering of thermal excitations, representing the flow of the normal component, from vortex lines. The same result was obtained in the experiments of Hall and Vinen with second ~ 0 u n d 3 ~ . Besides, ~ ~ * . the
* In the experiments of Snyder82 and Bruce*Son second sound, attenuation values of Bc different from zero were observed. These values changed depending on the conditions of the experiment. Snyder found that B, changes between zero (30% of all the data) and 0.25, while Bruce found values between 0.01 and 0.025. Perhaps non-zero values of B, exceeding the experimental errors appear as a result of the existence in the device of vortex lines which are not parallel to the axis of rotation. References p . 156
CH. 3,
8 41
ROTATION OF HELIUM I1
127
equality B, =O brings formulae (4.3.4) and (4.3.4a) into agreement with eq. (4.3.1). However, B, can be different from zero due to zero, thermal or forced oscillations of vortex lines. This was confirmed by Tsakadze's experiment~~6-78 (Section 4.2). In the case of axial oscillations of the cylinder, the normal component is dragged into a cylindrical transverse viscous wave. In this case the direction of oscillation is perpendicular to the axis of rotation. The mutual friction drags into this wave the vortex lines which, practically without distortion, oscillate in the same direction (within the penetration depth of a viscous wave). Then the mutual friction gives a contribution to the damping of the cylinder oscillations.The contribution is described by an approximate formu~
4
:
where 6, and 6, are the logarithmic decrements of damping on the immersion of a part of the cylinder surface with the length l2 or ll into the liquid; Z is the moment of inertia of the oscillating system. The agreement of formula (4.4.3) with the data of Tsakadze and Chkheidze74 is shown in Fig. 27. Let us note that the experimental values of the coefficient B (Refs. 31,11 and 74) and the results of the theoretical calculations of this coefficient31.11~85~86 agree well with each other. Let us also note that under the conditions of weak mutual friction, the wavelength 9 and the penetration depth 1 of the cylindrical waves, excited by oscillating cylinders, are described to a good approximation by formulae (4.4.4)
In the opposite case, when there is a very strong mutual friction, the situation is changed. For instance, in the case of axial oscillations, if (o,/sZ)(ps/p)B91, formula (4.4.3) is replaced bya formula obtained by Mamaladze : 6, -6, 2 n 2 ~ 32v,p -t ____(4.4.5) ' 1,
- 11 --(3-) I
i.e. it seems that we deal with a non-rotating liquid with the viscosity, q,, but with the density p (and not p,,). The wavelength and the penetration References p . 156
128
E. L. ANDRONIKASHVILI A N D YU. G. hfAhfALADZE
[CH.3,
84
depth are given by 2?=2d=($
(4.4.6)
The most complicated case for hydrodynamical consideration is the case of axial oscillations of the disk (Section 4.2). The oscillations of a flat solid surface perpendicular to the axis of rotation generate a transverse wave travelling along the axis of rotation in helium 11. This wave consists of four circularly polarized waves. Two of them propagate, in the main, in the normal component if the mutual friction is weak. They will be called n(+) and n(-) waves. They are viscous waves whose circular polarizations have opposite directions. Naturally, the laws of dispersion are different for them as the direction of polarization can coincide with that of rotation or be opposite to it. The two other waves are caused by oscillations of the vortex line ends, attached (more or less strongly) to the disk surface. For weak mutual friction such oscillations propagate, in the main, in the superfluid component. They will be called s(+) and s(-) waves, They are elastic waves with mutually opposite circulations. The wavelengths and the penetration depths of n(*) and s(*) waves are determined by complex wave numbers* (4.4.7) (4.4.8)
These formulae are written in the linear approximation over the coefficients of mutual friction (complete formulae are given in paper87 and review6S). Besides in the formulae for kh-) and k! - ) we assume that ll2 - 2w,,l4 Q6/271. Formulae (4.4.7) and (4.4.8) represent a good illustration of the statement made earlier that v, definitely plays the role of the kinematic viscosity of the superfluid component. However, it should be emphasized that, in spite of this formal resemblance of the formulae for k‘,*’ and ki*’, viscous waves n(*) and elastic waves d*) behave quite differently. The relationship, usual for viscous waves in a classical liquid72, remains valid for n(*) waves: (4.4.9)
+
* The complex wave number k = CT is determines the wavelength 3’ = 2741~~1 and the penetration depth A = 1/7. The expression for the square of the wave number unambiguously determined k for a definite choice of the sign of 5. We always imply that 7 > 0. References p. 156
CH. 3,9 41
129
ROTATION OF HELNM I1
According to this the wave practically attenuates at the first half wavelength. The wave s ( + ) attenuates very strongly as well. Moreover, due to the large wavelength, oscillations at all the points take place practically with a common phase and we have here, essentially, even less right to call it a “wave” than in the case of the n(*) wave. The wave s(-) has a similar character if 213, >8. Only when 20,<8, does the wave s(-) behave as a typical elastic wave. The wavelength and the penetration depth are determined in this case (when 20,
(4.4.11)
where only the first terms of the expansion over B and B‘ are taken into account. Let us emphasize that the magnitude v, enters the formula for the penetration depth A:’) as well as the corresponding formula for A:-) for the condition 2 0 , > 52 in the same manner as vn in eq. (4.4.9); it really plays the role of the kinematic viscosity in both of them. This magnitude does not depend on the temperature and has the order of magnitude 11968 v, 10- cm2 secIn the range of temperatures used for the experiments described in Sections 4.1 and 4.2, it is close to the order of magnitude of v,, the square root of which determines ,I$ Therefore *). all the waves, excluding the wave s(-) for 20, c 52, have approximately the same penetration depths and attenuate in a thin layer adjacent to the disk surface. Only when 20, < 52 does a deeply penetrating wave s(+ ) appear; its penetration depth is larger than the wavelength. This wave can exhibit itself in resonance phenomena, connected with the possibility of arranging a certain number of half-waves (or quarter-waves) between two oscillating surfaces (or between an oscillating disk and the free or non-oscillating surface). Such resonance phenomena were observed in experiments of Hal167944, and Andronikashvili and Tsakadze8*. They are included in reviews11.68 and we shall not describe their details. We shall only note that, in accordance with the character of n(*) and s(*) waves just described, the resonance phenomena are observed only when 213,<8. Therefore we do not agree with Lin’s remark89 on the classicalnature of the differencebetween the phenomena taking place at 213,<52 and 2w,>sZ. An example of such a classical difference is shown, for instance, in Fig. 25. However, the resonance phenomena which N
References p . 156
’.
I30
E. L. ANDRONIKASWUI AND YU. G. MAMALADZE
[CH.3,
4
Lin refers to are impossible in classical liquids whether 2w0Q * They are due entirely to the elastic vortex wave s ( - ) ~ which is the only wave required for observation possessing (when 2w, 4 2 ) the property A:-) > 9:-), of resonance phenomena. The numerical values of v, obtained in the resonance experiments are: vS=(8.5+1.5)x cm2 sec-l (Hall67), v,=8 x cm2 sec-’ (Andronikashvili and Tsakadzess), vs=(9.7+0.3) x cm2 sec-’ (Hall44). According to the new preliminary data of Nadirashvili and Tsakadze92: vn=(8.83f0.25)cm2 sec-’. Just the deeply penetrating wave s(-) causes the peculiar interaction of an oscillating disk with the rotating liquid. It carries away most of the energy, lost by the disk; therefore, the damping of the disk in rotating helium I1 becomes similar to damping of charge oscillations (or those of a dipole) the energy of which is carried away by an electromagneticfield (radiation damping). The mechanism of vortex damping of the disk oscillations is caused by the fact that the wave s(-) gives a definite orientation to the vortex distorted by it in such a way that the tension of the vortex line slows down the motion of the disk93 (Fig. 31). The formulae describjng the change of the character of the disk oscillations are rather cumbersome*7~6*4 96. We shall give them in a very simplified form, neglecting the mutual friction and vortex sliding**: n2
{
- ng = lTR4* - (+qnpn)f[(a+ 20,)* ~
21
+ (qsp,)+ 61 200 (52 + 2w$
*
+ (52 - 200)*] + ,
when 20, < s2 ,
(4.4.12)
when 20, > s2,
(4.4.13)
Deep penetrating waves appear in a classical liquid only in an exceptional case 152 - 2wol g DS/2n. They were predicted by Mamaladze90 and were really observed by Mesoed and Tsakadzeel. In the experiments67,8*. 44 the condition of existence of such waves was not fulfilled. ** Under the conditions of the experiments with an oscillating d i ~ k ~ ~ described ~70, in Section 4.2, the penetration depth As(-) when 2w0 < D is rather large (- 1 cm), but still it is less than the distance from the disk to the non-rotating parts of the device. The formulae given here concern just this case of oscillations in an “infinite” liquid. References p . 156
CH.
3, 8 41
131
ROTATION OF HELIUM I1
+ (q,p,>+ 2WO (52 - 2 0 , ) ~ 52 --
,
when 2 0 , < !2,
(4.4.14)
when 2w0 > Q .
(4.4.15)
The magnitude qs= psvshere clearly plays the role of the dynamic viscosity of the superfluid component. It is clearly seen that (since sliding is neglected) the wave s(+) and, when 20,>52 the wave s(-) as well, give a contribution only to the change of the frequency of oscillations, while the wave d-), when 2w0c52, contributes to the change of damping. If one substitutes formally qs =0 (or if one considers the case when ps=0) formulae (4.4.12)-(4.4.15) are reduced to the formulae for disk oscillations in a rotating classical liquid (without the corrections for eddies, calculated in Ref. 72). The vortex damping term in formula (4.4.14) contains the product wo J(0-204). In this product 0,describes an increase of the number of vortex lines with an increase of the velocity of rotation. J(52 - 2 0 , ) decreases thus describing the drop of the decelerating action of the vortex line tension in connection with an increase of the s(-) wavelength, due to which the vortex line slope with respect to the disk surface becomes smaller. These circumstances give a qualitative explanation of the main peculiarity of the graphs in Figs. 24 and 26, i.e. of the maxima which distinguish these curves from those in Fig. 25. However, the product uoJ(Q-2~0,) has a maximum when 2u0/12= 5, and the quantitative explanation of the experimental maxima in the approximation to which formulae (4.4.12)-(4.4.15) are written is impossible (see Section 4.5). The mutual friction between the components of helium 11, as well as vortex line sliding, which are neglected in these formulae, play very different roles in the described phenomena. The mutual friction, in principle, is very essential. Only the mutual friction dissipates the energy taken from the disk by the wave s(-), providing that the assumption of an infinite liquid is valid. But the quantitative contribution of the mutual friction to the change of the frequency and oscillation damping under the real conditions of the experirnents65.679 691 709 88 is comparatively negligible (for instance, it could have provided not more than 10 % of the damping of the oscillations of the rough disk as shown by the maxima of the curves in Fig. 26). Exceptions are only References p . 156
132
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.3,
84
Y
The influence of the tension of a distortedvortex line on the disk oscillations. the tension, Et is its projection on the surface of the disk, f (+) andfc-) are the components of the vector et caused by the participation of a vortex line in s(+) and s(-) waves. Rotating vectors in the table denote f (+) and f I-). The vertical vector directed along the axis a, denotes the direction of the disk motion. When the initial point of this vector is placed on the circumference or in its centre, it corresponds either to the motion of the vortex line to the position of the equilibrium or to a deflection from this position respectively. The table does not take into account sliding and mutual friction. Then the s(+), and when 2w0 > i.2 the s(-) wave as well, slow down the deflection of the disk from the position of the equilibrium and accelerate its return to this position. This increases the frequency of oscillations. When 2w0< f2, the s(-1 wave continuously decelerates the motion of the disk and increases the damping of its oscillations. Fig. 31.
E = ( W / W ) E is
References p . I56
CH.
3, 41
ROTATION OF HELIUM I1
133
for a not sufficiently studied region 2wOz:52which we omit in this review, and the region T-TA. In the latter case the smallness of p s according to formula (4.1.I ) weakens the immediate interaction of the disk with the vortex lines as realized by their tension. On the other hand when T-T,, the coefficients of the mutual friction do not tend to zero; on the contrary, large values of these are possibleg4. Therefore the role of the mutual friction increases in this region. Probably this explains why the difference between the disk damping in rotating helium I1 and in helium I1 at rest which changes with temperature as p, when T S 2 OK ceases to decrease according to this law on further rise of temperature. It tends to a value different from zero when T= TA(Andronikashvili, Mesoed and Tsakadze95; Fig. 32).
Fig. 32. The dependence of the damping of disk oscillations (at the maximum of Fig. 24) on temperature. The dotted line corresponds to the dependence of the type (6 cc cc ps. The solid line is plotted according to the experimental data of Andronikashvili et aL95 (circles). Lower than the I-point the circles show the stationary values of damping. Above the A-point they correspond to the damping observed as a result of the transition helium I1 - helium I in the state of rotation. The stationary damping above the bpoint is shown by a circle with a cross (see Section 5.2).
In the other cases (excluding T- TAand 20, %Q), the mutual friction gives only relatively small corrections, and a more detailed analysis of the phenomena considered is possible only when sliding of vortex lines is taken into account. This will be done in the next section. Not to return to the problem of mutual friction, let us note in conclusion that formula (4.4.10)could have been used for the experimental determination References p . 156
134
E. L. ANDRONIKASHVILI A N D
YU.
G. MAMALADZE
[CH.
$5 4
of the magnitude B‘. However, so far the accuracy of the resonance experiments of L?’!-) measurement was not sufficient for this purpose (see, for instance, Ref. 44). The theoretical calculations of B’ were made by Lifshitz and Pitaevski85and made more accurate by Iordanski86. The data of Snyder’s experiment 82 (second sound in a cubic resonator) * apparently agree in the main with the calculations of Iordanski, indicating the absence of an increase of B’ in the region of low temperatures as predicted by Lifshitz and Pitaevski. The prediction by Pitaevski94 of the growth of B’ when T-, TAis confirmed by Snyder’s measurements. Apparently, outside of the region T- T,, B’ is tens to hundreds of times as small as the coefficient B, which has a magnitude of the order of unity (see Ref. 11). 4.5. SLIDINGOF VORTEX LINES AND COLLECTIVIZATIONOF VORTEX OSCILLATIONS
The strong influence of conditions of vortex line attachment to the disk surface on the velocity dependence of the character of its oscillations was mentioned in Section 4.2 while describing the experimental data (cf. Figs. 24 and 26). The cause of such an influence is quantitatively clear from the considerations, developed in the preceding section, on the mechanism of the disk interaction with vortex lines. A more detailed analysis shows that this interAction depends very strongly on the numerical values of the coefficients of sliding u and a’ encountered in the boundary conditions (4.3.6)**. Even small values of a increase damping of oscillations considerably 96, especially when 201, %zD. The physical cause of this phenomenon is a shift of the phase of the wave s(+) arising as a result of sliding. Because of this shift the wave s(+) contributes together with the wave s(-) to damping [that changes formulae (4.4.14) and (4.4.15)]. Large values of the coefficients a and a’ on the contrary lead to a decrease of the contribution made by vortex lines to the damping of a disk. Such a decrease takes place, due to a weakening of vortex line distortion by waves running along them. The contribution made by vortex lines to the frequency of oscillations is suppressed by sliding96 both at small and large a and a’.
* Unfortunately we have no more available data on the final results of this experiment, which are mentioned as preliminary ones in Ref. 82. * * The coefficient a was measured in the experiments with an oscillating pile of disks, performed by Hall67944 (see also Ref. ll), and in a more direct experiment of Gamtsemlidze et d.97.In order of magnitude these data agree with each other and with the estimation of Bekarevich and Khalatnikovao: a tiB/m8, where 6 is the size of protuberances of the surface. However, the experimental values of a depend on the character of the surface motion as well (for instance, on the frequency of oscillations). The coefficient a’ under the real conditions of experiments should be negligibly small: a‘/a B‘/B (Ref. 80).
-
-
References p. 156
CH.
3,g 41
ROTATION OF HELNhf I1
135
Just this decrease of damping, caused by an increase of sliding, is the reason for the maximum shift on the curves, shown in Figs. 24 and 26 at 2w0/G?s50.2with respect to the maximum of the product wo ,/(2w0 -a) at 20,/i2 -0.7. An increase of sliding at 2 w 0 / 0 x 0 . 2 is not connected with the nature of sliding and is caused by the properties of vortex lines. It is due to the phenomenon of collectivization of vortex oscillations, the essence of which is the following (Mamaladze, see e.g. Refs. 69 and 68). The superfluid component, surrounding a vortex line, takes part in its oscillations within the effective radius 98*94: b' = l/a, (4.5.1) where Q is a (real) wave number of the wave running along a vortex line*. In the case of the disk oscillations in rotating helium I1 the magnitude 27r/9:-) should be substituted for a in formula (4.5.1), because the wave s(-) is the only one the size of which exceeds the thickness of the layer adjacent to the disk, where the complete viscous dragging of the normal component into the motion of the disk takes place. Then (2w0
At small velocities of rotation this magnitude is smaller than the vortex separation which is equal to (nA/rnwo)*.Under such conditions oscillations of separate vortex lines are independent of each other. However, on an increase of wo a situation arises where the effective radii of oscillations overlap and oscillations are collectivized. This takes place at a velocity of rotation which will be denoted as Go, where 2G0 -
n
1 (2mv,/nh)
+ 1 25 0 . 2 .
(4.5.3)
In the experiments, described in Sections 4. I and 4.2, the collectivization of vortex lines has exhibited itself: i) in the stopping of the linear growth of the square of the frequency of the disk oscillations (Fig. 22), ii) in the appearance of maxima in the plots of the disk oscillation damping as a function of the velocity (Figs. 24 and 26), iii) in breaks of the plots of the velocity
* It should be emphasized that the effective radius of oscillation is different from the effective radius b = 1/ (hlrnwo), determining the region of liquid rotation round a vortex line in formula (4.1.1). References p . 156
136
E. L. ANDRONIKASHVJLI AND YU. G . MAMALADZE
[CH.
3, 5 4
dependencies of damping of a vertically oscillating cylinder, below which the disk generated transverse waves (Fig. 29). And later more than once we shall have to notice different peculiarities of the behaviour of rotating helium I1 at wo=cij0*. Now it should be explained why the phenomenon of collectivization of vortex oscillations is accompanied by an increase of vortex lines sliding along the disk surface. The problem is that, when wo
u0 x I O ~ S ~ C - I
Fig. 33. The results of the numerical calculations according to the formulae, completely taking into account both mutual friction and ~ l i d i n g ~(A7 *=~6~ - 60).The upper curve is plotted at a = 0, the lower one at a = 00, the middle curve corresponds to the lower experimental graph in Fig. 24 and follows from the theoretical formula for a variable coefficient a (see Fig. 34).
* Let us note that the consideration given just now of the phenomenon of collectivization of vortex oscillations is out of the domain of hydrodynamics averaged over multivortex volumes, and the velocity CEIO or the distance b' are not characteristic parameters of this theory. Gopal99 has obtained an expression for the wave number of the wave s(-) without averaging over volumes containing many vortex lines. If we use our symbols, Gopal's formula coincides with the expression for kd-) (4.4.8)when w o < 60and begins to deviate from it when 00 60.
-
References p . 156
CH.
3 , 8 51
ROTATION OF HELIUM I1
137
vortex lines (including vortex lines from the region beyond the periphery of the disk) begin to inffuence each other. It is just this which leads to an effective increase of sliding. These considerations are confirmed by computer calculations of the disk damping according to the formulae given in Refs. 87 and 68. These formulae take into account the mutual friction and the coefficient of sliding a. (In principle, a' could also have been taken into account if one substitutes a by a+ia' in the mentioned formuiae. But as a' is always much less than a, one practically uses only the coefficient a.) In Fig. 33 the calculated curves are shown for the velocity dependence of the disk damping for different assumptions of the value of a. And in Fig. 34 an empirical velocity dependence of the coefficient a is shown, providing the agreement between the theory and the experiments. The minumum of a is situated at the same spot as the maximum of 6, i.e. at oo=Go. aQ-* x 10acm se& 56 52 48
. ' '
44. 40 ' 36 32 r 28
-
,/
!.{, 8 . 12 4 .
.-'
5. The phase transition in rotating liquid helium in the presence of vortex lines 5.1. THECENTRAL MACROSCOPIC VORTEX
Studying rotation of liquid helium, Andronikashvili as early as 1948 has found that, under certain conditions, a central macroscopic vortex is observed in rapidly cooled rotating liquid helium. This vortex pierces the liquid to the References p . 156
138
E. L. ANDRONIKASHVlLI AND YU. G. MAMALADZE
[a. 3, § 5
very bottom and forms something like a hollow axis of rotation in rotating helium (see, e.g., Ref. 7). Andronikashvili's observations were confirmed later by Donnelly, Chester, Walmsley and Laneloo, who observed formation of a central macroscopic vortex both in helium 11and in helium I. The conditions for such a vortex formation were studied in detail by Tsakadzelo1T102 who established by means of experiments using water as a model that it is a purely classical formation, having nothing to do with quantum properties of the liquid. A certain ratio between the rate of vapour formation in the liquid volume and the velocity of its rotation is required for its formation. Thus such a vortex cannot be formed in helium 11. It is always formed in helium I, in which it has a shape of a string with a diameter
Fig. 35. Photograph of the central macroscopic vortex in helium II (Tsakadzelo2)and the calculated shape of the meniscus at the same velocity of the vessei rotation and circulation of the vortex line r = loacmasec-l. References p . 156
CH. 3,5 51
ROTATION OF HELIUM II
139
of the order of 2 mm. This “string” does not remain stationary but twists and oscillates. Nevertheless it can be observed in helium I1 if one rapidly cooles rotating helium I with the central vortex till the temperature is lower than the bpoint. In helium I1 it alongates immediately and stops to oscillate and the meniscus acquires the shape shown in Fig. 35. After some time, nevertheless, the central vortex tears away from the bottom of the vessel and begins to shorten, forming gradually a conic pit on the lower part of the parabolic meniscus. One can manage to preserve this vortex line provided the temperature was not lower than 2.09 OK. It is a characteristic fact that the conic pit, typical of a quantum liquid (Fig. l), is conserved for a long time in helium I as well. It can be observed there up to T= 3 “K on rapid heating from Tc T,. The described phenomena can be interpreted as supercooling of classical kinds of motion on transition of a liquid into a quantum state, for which these kinds of motion are non-equilibrium ones. One can observe an opposite phenomenon, i.e. a superheating of quantum kinds of motion, non-equilibriurn ones for the classical liquid (transition of helium I1 into helium I in the state of rotation). 5.2. RELAXATION OF VORTEX LINES FOR THE TRANSITION HELIUM 11-HELIUM 1 IN THE STATE OF ROTATION
Experiments of Andronikashvili, Mesoed and Tsakadze95 have shown that a “vortex” damping of a disk, rotating together with helium and performing, in addition, axial oscillations (Sections 4.2 and 4.4), remains constant by virtue of its character above the I-point as well. But to observe such a phenomenon the transition helium 11-helium I should be effected in the state of rotation. However, all the attempts to find vortex damping in liquid helium, brought into rotation above the I-point and then not cooled till T < T,,failed (Fig. 32). The character of the relaxation processes, connected with the decay of vortex lines at T> T,, is seen in Fig. 36, corresponding to the velocity coo=do and obtained on the basis of the data of Andronikashvili, Gujabidze and Tsakadze103-106. These experiments seem to show that vortex damping remains quite unchangeable in helium I till a vortex line suddenly tears off from the disk surface. It leads to a decrease of damping to values smaller, not only than those in helium 11, but even than those in helium I. Gradually damping begins to increase again and reaches the value characteristic of helium I. a
References p . 156
140
[CH.3,
E. L. ANDRONIKASHVILI AND W. G. MAMALADZE
85
The character of the relaxation processes can be understood if one recollects that damping of a disk oscillating in helium I1 at 20,
s - 6, oc
,i”z”
+ Js2
0 J -(
20,
+ Jq,p, 7Ja - 2w,, ~
- 20,)
(5.2.1) and, in helium I, it should be
6 - 6,
K
J9p 2
(J-0
+
(5.2.2)
The second term of eq. (5.2.1) describes vortex damping; the first term of the same equation describes damping connected with the normal component. Eq. (5.2.2) describes damping in a classical viscous liquid. After the vortex line tears off, the term with J?,ps disappears rapidly, while &?,p, should increase to the value But that takes place rather slowly, as the part of the liquid “entering the composition of vortex lines” is gradually dragged into the regime of motion, characteristic of a classical liquid. The correctness of such a treatment is confirmed by the fact that on rotation of helium TI with the velocities w, c 0,and w, >6,the length of the first portion of the curve in Fig. 36 is considerably shorter, the same as in the case of a smooth disk with weakly attached vortex lines. The length of the first portion of the curve of the time dependence of
4%.
t
t0
-
Fig. 36. The relaxation of vortex lines on the transition helium I1 helium I in the state of rotation. to is the instant of transition, 61 is the equilibrium damping in helium I (Andronikashvili et al.loploE). References p. 156
CH.
3,5 51
ROTATION OF HELIUM I1
141
damping is decreased with an increase of temperature of helium I. One did not manage to observe a metastable array of vortex lines above T = 2.27 OK.
5.3. THEORDER OF THE PHASE TRANSITION IN
ROTATING LIQUID HELIUM
Appreciable supercooling and superheating of metastable kinds of motion made experimenters look for the cause of these phenomena either in a shift of the A-point or in an appearance of the phase transition of the first order in rotating helium, instead of transition of the second order, characteristic of liquid helium at rest.
Fig. 37. One of the picnometers used by Andronikashvili and Tsakadze108p10B.111 to measure the density of rotating helium.
The use of the thermomechanical effect by Zamtaradze and Tsakadzel5 has not led to determination of any shift of the A-point. Searches for a shift of the I-point by a jump of heat loss, undertaken by Bablidze, Tsakadze and Chanishvili l07, have shown that the A-point remains unshifted to an accuracy of + 5 x 10-4 OK. Theoretical studies of Kiknadze, Mamaladze and Cheisvili57 have established later that the expected shift of the A-point, at the maximum availaReferences p. 156
142
E. L. ANDRONIKASHVILI A N D YU. G . MAMALADZE
[CH. 3,
05
ble velocities reached in the experiment at present (of the order of lo2 sec- I), should not exceed the magnitude of the order of lo-' OK*. Quite clear results, showing that the phase transition of the first order takes place in rotating helium 11, were obtained by Andronikashvili and Tsakadze108+109. They have used the method of a rotating picnometer, sensitive enough to detect the slightest changes of helium density, occurring at a change of temperature of angular velocity. On rotation of a picnometer, shown in Fig. 37 with the velocity oo= 30 sec-', the value of pwo-p remains equal to zero in the range 2.18 OK to 2.172 OK.However, on further cooling a jump shown in Fig. 38** appears.
4 1
I
2l
I
I I
2
I
1 '
0 -
1
I .
L I
I
,
Fig. 38. The jump of the density at the point of the phase transition of liquid helium. pa, is the density at wo = 30 sec-1, p is the density at wo = 0.
The comparison with Kerr's curve l10, obtained for the temperature dependence of density (Fig. 39), allows one to make the conclusion that the density of liquid helium changes at the I-point by a jump of 0.02%. Despite the opinion of many physicists who expected that the existence of vortex lines should make helium I1 less dense, rotation, on the contrary, increases its density111#10*9109(Fig. 40). As it is seen in this figure, the observed increase of density is larger, the lower is the temperature. One did not manage to observe similar effects in helium I. The natural attempt to explain the observed facts by centrifugal pres-
* Note added in proof. Now this estimation must be taken as 10-7 O K (see note added at the end of this chapter). ** Each point in this figure, as well as in Fig. 39, is a result of measurements made at a constant temperature. References p . 156
CH. 3,
8 51
143
ROTATION OF HELIUM I1
sure fails here: the value of the effect, which was found to be equal to (dp/p),, 4x requires that the coefficient of compressibility ( I / p ) (ap/aP), be two orders of magnitude larger than its real value (the centrifugal pressure at the periphery is of the order of 2 x atm and the N
coefficient of compressibility is of the order of lo-* atm-'). In addition, the
0.1466
1
'
0.1463 2.13
2.14
2.15
2.16
2.17
2.18
2.19T('K)
Fig. 39. The density of rotating helium (circles) and helium at rest (points) according to the data of Andronikashvili and Tsakadzelov. The solid line shows Kerr's datallO.
I*'-
40
. P
x105 1.42'K
32
1.?4' K
24
2.05.K
16
e -c--
0
100
200
300
400
500
600
700
000
(sec900 w'. w'.(sec-?
Fig. 40. The dependence of the density increase of liquid helium on the velocity and temperature.
References p . 156
144
E. L. ANDRONIKASHVILI AND YU. G. MAMALADZE
[CH.3,@ 5
compression of helium I1 by centrifugal pressure should increase and not decrease with the rise of temperature, as actually observed. As the density is the first derivative of the thermodynamical potential, its discontinuity allows one to make an unambiguous conclusion that liquid helium experiencesa phase transition of the first order on heating in the state of rotation. This result, in principle, agrees with the theoretical consideration of the phase diagrams of rotating helium I1 on the plane w,-T (Kiknadze, Mamaladze). However, the unexpectedly large value of the experimentally observed jump is not theoretically explained so far. 6. Decay of vortex lines and their stability 6.1. DECAY OF VORTEX LINES ON STOPPING OF ROTATION It was already mentioned in Section 2.1 that there is a difference between kinetics of vortex line creation on twisting of the vessel and their disappearance on stopping of the rotation. This difference was noticed in many studies in which the process of deceleration was observed. The first information on the kinetics of helium I1 deceleration is to be found in the paper of Andronikashvili and Kaverkin'. In a suddenly stopped cylindrical vessel the meniscus begins to become flatter at the walls. The conical crater shown in Fig. 1 is the last to disappear. For an initial velocity of rotation equal to 30 sec-l, the time of the meniscus disappearance is four times as short as that of its growth. A similar ratio for the duration times of the processes for bringing helium I1 into rotation and deceleration was observed by Craig as well in the paper already cited 24 on the action of helium I1 on the disks of the torsional balance. Deceleration proceeds 3 - 4 times as fast as twisting, though the angular velocities in these experiments and in the experiments of Andronaskshvili and Kaverkin were quite different. Pellam 221 23 has also noticed that deceleration proceeds muoh sooner than twisting and the onset of the process of deceleration takes place without any delay after the stopping of the vessel. However, a fast deceleration is characteristic only of the initial stage of the process, which, when almost complete, proceeds more and more slowly over rather a long time, just as it has been already noticed by Hall12. The very long completion stage of the process of deceleration is characteristic not only of these cases when, in the stopped vessel, a persistent current remains (we shall speak about that in Section 7.1). The point is that, as it has been References p . 156
CH.3,
8 61
ROTATION OF HELIUM I1
145
discovered on more careful consideration, the decay of vortex lines proceeds according to an exponential law. Studying the current of negative ions in a first rotating and then stopped vessel, filled with helium 11, Careri, McCormick and Scaramuz~i34,11~ have measured the period of half-decay of vortex lines which, depending on temperature, changes from z=510 sec at T=0.87 OK to ~ = sec 3 at T=2.12 OK. This dependence corresponds to the law zcc l/Ne, where No is the density of the thermal excitations of helium I1 at a given temperature. The existence in the stopped vessel of an array of vortex lines parallel to the axis of rotation certainly does not correspond to the equilibrium state of helium 11. Acting on each other, vortex lines tend to continue the uniform rotation. However, the normal component, being decelerated by viscous forces as well as by protuberances on the bottom and the top of the vessel at rest, prevents such a motion of vortex lines. Therefore, it would seem that the law T C C ~ / is N ,naturally explained by the role of the mutual friction in the process of the vortex decay. Nevertheless, because of reasons which are at present not quite clear, this law is not confirmed in the experiments made with the use of another technique. For instance, in Bablidze's experiment113 in which one could estimate the number of vortex lines from the degree of attenuation of second sound caused in a radial mode resonat0r~~y~9, the dependence of the period of half-decay on temperature was not observed at all (Bablidze has obtained T x 30 sec). And in the experiment of Gamtsemlidze et al.114 the dependence of z on temperature was even reversed with respect to the data of Ref. 112: the period of half-decay jncreases with an approach to the &point. Gamtsemlidze et al. have used the method of axial disk oscillation damping. In their earlier study115 which used the same method, some interesting peculiarities of the process of vortex decay were observed, which probably will elucidate the causes for different behaviour of z on using different techniques for its determination. In Ref. 115 it is shown that an extra damping of the disk oscillations, caused by the rotation of the liquid, changes after the stopping of the vessel according to the exponential law o = n , e x p ( - In T l2) , (6.1.1) where the period of half-decay z changes suddenly from z1= 70 sec to z2 = 55 sec (Fig. 41) at the moment of time t,, which satisfies the relationship: o , e x p ( - T In 2 t l ) = c i j o .
References p . 156
(6.1.2)
146
E. L. ANDRONIKASHVILI A N D YU. G. MAMALADZE
Fig. 41. The relaxation of vortex lines after stopping the rotated vessel according to the data of Gamtsemlidze et aL1I5. The time measured from the instant of the vessel stopping is plotted along the abscissa. a is the difference of the dampings a = 6t - 6,. The values of the velocities till stopping, for the different curves, are (A) w o = (0.10 & 0.02) sec-', (B) 00 = 0.24 sec-1, (C) w o = 0.48 sw-l. Under the conditions of this experiment, wo = 0.11 sec-1.
Here, coo is the angular velocity of rotation before the stopping of the vessel, t , is the time measured from the moment of stopping and Go is the angular velocity of collectivization of vortex oscillations, determined in Section 4.5. Thus, if the collectivizationof vortex oscillations ceases as a result of reduction of the number of vortex lines caused by their decay, then the period of the half-decay shortens. This shows that oscillating vortex lines are more stable in comparison with non-oscillating ones (the experiments described in Section 5.2, in which the maximum of the relaxation time in helium I was established105.106 at 0 ~ = 6lead ~ ,to the same conclusion, see also Section 6.2 and Fig. 43).
6.2.
RELAXATIONOF VORTEX LINES ON CHANGE OF TEMPERATUREOF ROTATING HELIUM
11
On a change of temperature of uniformly rotating helium I1 the number of vortex lines remains constant [see formula (1.3.2) for the density of vortex lines]. But the amount of the superfiuid component taking part in vortex line motion changes. References p . 156
a. 3, I 61
ROTATlON OP HELIUM ll
147
Fig. 42. The relaxation of vortex lines at helium I1 cooling in the state of rotation and at rest. (The position of the minimum with respect to the point t z depends on the duration of the process of cooling.)
Fig. 43. The dependence of the time duration of the process shown in Fig. 42 on the velocity of rotation.
The study of relaxation processes on the change of temperature was undertaken by Gudzabidze and Tsakadze 116. Here, they have found that the normal component on a lowering of temperature, being converted into the superfluid one, is dragged into the motion of already existing vortex lines over a very long time, in some cases reaching 40 minutes. This process of vortex line enrichment by the superfluid component was studied by observing the time dependence of an extra damping of the disk, oscillating in helium I1 and performing a uniform rotation with it. It is seen in Fig. 42 that, on the curve of the change of damping with the time, there are two portions. One of them is connected directly with the process of temperature lowering, in particular with the change of the viscosity of the normal component. This period lasts References p . 154
148
E. L. ANDRONIKASHVILI A N D YU. 0. MAMALADZE
[CH.3,
86
about 5 minutes. The other portion is connected only with the process of vortex enrichment by the superfluid component and its duration time reaches 30-35 minutes. Fig. 43 shows that the time of vortex line enrichment increases with an increase of the number of vortex lines, reaches the maximum at 2w0/f2x0.2, i.e. again when the collectivization of vortex oscillations takes place, and after that begins to decrease. Starting from velocities wo-0.25f2, the time of vortex line enrichment does not already depend on the angular velocity, but remains’strictly constant. Thus a vortex line is the most stable with respect to dragging new amounts of the superfluid component under conditions of the strongest fastening (Section 4.5,Fig. 34). The phenomenon of losing the superfluid component from a vortex line on a rise of temperature was studied by the same experimenters. The process of exclusion of the normal component from a vortex line, occurring on heating rotating helium 11, proceeds more smoothly than on cooling (Fig. 44). The dependence of time for the losing of the superfluid component on
I i
tI
I
I
Tlme
I
I I 11
I t2
t3
Time
Fig. 44. The relaxation of vortex lines on heating helium I1 (compare Fig. 42).
the angular velocity has the same character and even the same time characteristics as in the case of vortex line enrichment. The corresponding curve literally reproduces Fig. 43. And again the vortex lines under the conditions of least sliding are the most stable in the sense of their decay. The special role of the point wo=Go in these phenomena is explained by the fact that the amplitude of the oscillation is the largest at the optimum fastening of a vortex line. A liquid which “flows into” such a vortex line or “leaves” it should most strongly change the References p. 156
CH. 3 , s
71
ROTATION OF HELIUM ll
149
character of its motion; this explains the maximum of the relaxation time. The problem of vortex line decay on heating of rotating helium I1 was considered by Andreevll'. He has shown that, on heating, two vortex lines are formed instead of one: a superfluid vortex line and a normal one with coinciding cores. The normal vortex line gradually spreads due to viscous forces. From the analysis of dimensions, Andreev has estimated that the duration of this process is of the order of ten minutes. 7. Persistent currents of the superfluid component 7.1. DISCQVERY OF PERSISTENT CURRENTS AND
THE FIRST OBSERVATIONS
The first attempt to find a persistent annular current of the superfluid component was made by Andronikashvilills. The main part of the device used by him to this purpose was a pile of parallel light disks hanging from an elastic fibre, with a small disk separation (0.2 mm) and embodied by a thin aluminium jacket. The same pile showed itself to be a very good indicator of the normal component motion while the superfluid component was at rest2Jl119. It seemed (and later this was confirmed) that it will be a useful instrument to observe the opposite state of affairs, i.e. when the superfluid component rotates and the normal one is at rest. The following experiment was made with this purpose in mind. The pile rotated uniformly at a temperaturealittle lower thanT,. Duringtherotationcooling took place till T= 1.5 OK and after that the system was smoothly decelerated. It was assumed that the superfluid component, formed from the rotating normal one in the process of its cooling, continued to rotate, while the normal component was decelerated rapidly on stopping of the closely spaced solid surfaces. The final stage of the experiment was to heat the liquid till 1.65 OK.The electromagnetic brake of the pile* by that time was switched off. It was assumed that the rotating superfluidcomponent, being converted into a normal state, would be decelerated and should turn the pile by some measurable angle * *. But this assumption was not justified. At present the failure of this experiment is quite clear. The velocity wo-3 sec-', with which the pile of disks was rotated, corresponds to approximately 6000 vortex lines per square centimetre. A rather long time is required for their decay (Section 6.1). It is clear that the superfluid com-
* In this experiment deceleration and uniform heating of the device were effected by an electromagnetic field (to avoid hindrances in the form of heat fluxes). ** There was an error in the calculationsof this angle in Ref. 118, the correctionfor which does not change the result. References p . IS6
150
E. L. ANDRONJKASHVLLI A M ) W. G. MAMALADZE
[CH.
3, 9 7
ponent, interacting with the normal one and the walls of the vessel by means of vortex lines, cannot perform persistent circular motion. Therefore a persistent current should be sought for under conditions of the slowest possible rotation of the vessel before it is stopped. But at the time when Andronikashvili's experimentl18 was performed these considerations were quite unknown. Later in the investigation of Hall12 already mentioned, in which a pile of disks was rotated by means of a device which permitted measurement of torques developed at its acceleration and deceleration, the phenomena were observed, thus giving evidence in favour of persistent current existence. Namely, at relatively high rotational velocities of helium 11, it gained on acceleration an angular momentum corresponding to its solid body rotation and lost it on its deceleration. However, at smaller velocities (of the order of a tenth of a radian per second) the liquid lost on its deceleration an angular momentum smaller than that obtained on acceleration*. In addition some impulses were imparted from time to time to the decelerated pile. It was difkult to understand the nature of these impulses, but at any rate they meant that the vessel at rest stores a definite amount of the angular momentum. As a matter of fact Vinen's experiments19 also gave evidence of a persistent current, though it was not their direct aimll. In these experiments a persistent current was found to be a carrier of an elementary circulation**. However usually, when one speaks of a persistent current, an undamped motion is meant with a circulation much larger than one quantum 2xhlm. Bendt and Oliphan@ made the assumption that such a motion can exist as a metastable state in an annular vessel at rest if, before its stopping, the region of irrotational motion (3.3) would involve the whole liquid. To verify this assumption Bendt immersed disks, suspended on an elastic fibre122 or a strip of foil used as the "Rayleigh disk"123 into such a vessel which was rotated with different velocities till it was stopped. Persistent currents were observed in both cases. In the most interesting experiments with a strip of foil123, clear evidence was obtained of the existence of a metastable state of a persistent current and of its coherence. The immersion of a foil broke this state completely, though the immersion depth was not more than one fifth of the vessel height. No less apparent evidence of persistent current existence was obtained by Depatie, Reppy and Lane124. They have used a pile of disks,
*
A similar phenomenon was observed by Walmsley and Lanelzo.
** Whitmore and Zimmermanl21 noticed, in a similar experiment, an increase of circulation with an increase of velocity and of the wire radius; that corresponds to formula (3.3.2). References p . 156
CH. 3, § 71
ROTATION OF HELIUM U
I51
similar to Andronikashvili’s device with the only difference that a magnetic suspension was used instead of an elastic fibre. The pile, containing liquid helium 11, could rotate freely in an atmosphere of helium vapour. The device was rotated, then decelerated by electromagnetic means, became free of them and heated. An increase of the velocity of rotation (Fig. 45) showed that a persistent current was conserved in a decelerated pile (it perished on heating). This effect takes place independently of the presence or absence of a solid axis in the device, i.e. it takes place both in a doubly connected and in a simply connected region (in the latter case the spacers between the disks were placed at the periphery of the vessel).
-
Brake on
Fig. 45. The formation of a persistent current and its “development” by heating in the experiments of Depatie et
Concluding this section, we can state that a persistent current is potential rotation of the supertluid component, taking place according to the law
r
v, = -
(7.1.1)
2711’
with the angular momentum (per unit height of the vessel) (7.1.2)
where r is the circulation (usually very large), rl is the radius of the inside cylinder (if it is there). For the velocity of the vessel oo=0, such a motion is unfavourable thermodynamically, since the requirement E- Lo, =min is reReferences p . 156
152
E. L. ANDRONIKASHVILI AND YU. 0 . MAMALADZE
[CH.
3,s 1
placed by the requirement E = min, i.e. the immobility of the liquid is favourable thermodynamically. However, if such a motion existed before the stopping of the vessel, then there are no forces which could cause its cessation after the stopping. Therefore a persistent current can exist arbitrarily long as a metastable state. In this respect Bendt and Oliphant 46 are quite right. However, it is not quite clear how such a motion of the liquid (if we take into account its counteraction to dragging it into rotation at small velocities25; see Section 2.3) can be effected. Most probably persistent currents are formed after the stopping of the vessel which was previously rotated with such a velocity that the superfluid component was dragged into rotation due to creation of vortex lines. But there are relatively few vortex lines in it and they disappear after the vessel is stopped earlier than mutual friction will decelerate completely (or almost completely) the motion of the superfluid component. In this connection we should like to note that it is difficult at present to estimate the minimum velocity at which the superfluid component is brought into rotation or the maximum possible velocity of a persistent current.
Fig. 46. The superfluid gyroscope of Reppyl". A is the annular vessel, B is the coil, C is an elastic fibre, D is thread-like foam.
References p. 156
CH.
3 , s 71
7.2. DEPENDENCE OF A
ROTATION OF HELIUM I1
PERSISTENT CURRENT ON TEMPERATURE.
153
SUPERFLUID
GYROSCOPES
Reppy and Depatie1z5 have used the measuring technique, described in the previous section in connection with Ref. 124 and have shown that the angular momentum of a persistent current is proportional to ps. This is just what should be expected according to formula (7.1.2). However, what was found is that the value of L, is determined not by the value of ps at which a persistent current was formed, but by the value of ps corresponding to the temperature existing directly before the current "developmenty' due to liquid heating was observed. It means that in a decelerated vessel the angular momentum of a persistent current changes with the change of temperature as ps (that, certainly, does not contradict the law of conservation of the angular momentum). The law of circulation conservation is the reason why, in the expression for L,, only ps can change, as this law provides the constancy of the velocity distribution of the superfluid component at the change of its amount. The dependence of a persistent current on temperature was studied in detail by Reppyl26 by means of a superfluid gyroscope worked out by him together with Clow, Depatie and Weaver127. A gyroscopeis a very convenient device to observe persistent currents of the superfluid component, since it allows one to make measurements without destruction of the current. Reppy's gyroscope (Fig. 46) is an annular vessel, packed with foam-like fibres to increase the critical velocities of vortex formation. The angular momentum L, is in the horizontal plane and one can estimate its value by determining the horizontal torque perpendicular to L,, required for gyroscope rotation with definite velocity round the vertical axis. Reppy has determined the magnitude pL,/p, which is independent of temperature and practically characterizes the velocity of rotation (or its circulation). Fig. 47, in which the temperature dependence of this magnitude is given, shows the transition of a persistent current into a more stable state on heating till Tn- T w 2 x lo-' OK. In the same investigation, Reppy has observed practically a constancy of the velocity of rotation of the superfluid component on approaching the A-point (T,-Tk5 x OK)and a sudden disappearance of the persistent current on passage through the A-point. A superfluid gyroscope was also constructed by Mehl and Zimmerman'28. They have used a spherical vessel, filled with a porous substance (Fig. 48). After rotation was stopped, the vessel, hanging from an elastic suspension, experienced a tilt of about 90"due to the magnetic field, so that the vector L, References p . 156
154
[CH.3,
E . L. ANDRONlKASHVlLI A N D YU. G. MAMALADZE
U
-c I
m 1.0
-
0
0
0
O
0
43
A
a*,
h
87
0.5I
TO-^
A
lo-'
7 0-=
1
T ~ T- lo K )
-
Fig. 47. The transition of the persistent current into a more stable state on approaching the &point (for 2 x 10-2 OK) according to Reppy's datalae. Circles show the case when the persistent current is formed at a low temperature and is observed during the process of heating. Triangles correspond to the case when the persistent current is generated at the temperature TA- T < 2 x 10-2 OK and is observed during the process of cooling. To
(t
TORSION FIBER AND MIRROR AT ROOM TEMPERATURE STAINLESS STEEL TUBING
OUTER L-HE
II
BATH DEWAR L-HE
I
I Y ~ + - G L A SRODS
1I
CELL
i$GLASS
POWDER -FILLED GLASS SPHERE
w Y q 'TT , 1 FRONT
BRASS
ww;rER OPENING TO ALLOW FILLING W!TH L-HE II
,.
PIVOT
MAGNET
/
GLASS SPHERE
DETAILS OF
u SPHERE AND MOUNTING
Fig. 48. The superfluid gyroscope of Mehl and Zimmerman 128. References p . 156
m. 3 , s 71
155
ROTATION OF HELIUM I1
changed its vertical direction into a horizontal one. This change of the angular momentum was immediately compensated for by a turn of the suspension system round the vertical axis; the value of this deflection determined L,. This device, like Reppy's one, allows multiple measurements without destruction of the metastable state with a persistent current. Measurementswith a spherical gyroscope have also confirmed the law L, a p s (Fig. 49). Note added in proof. Recently Clow and Reppy133 have shown, using a superfluid gyroscope, that p , / p = 1.44 TA- T)* in the vicinity of the A-point. This fact caused Mamalahe134 to re-estimate the parameters of the phenomenological theory of superfluidity. In particular, the shift of the A-point as a result of rotation of a helium bath is expressed by the following formula: 6TA= -5.38 x lou9 instead of the equation given in Ref. 57.
50
0.2
1
1
1
1
1
< a
.
I
f
1
1
,'o
oCOOLlNG
8
m
a
w
<
1
X A V E R A G E AT 1.24.K
i=
t
1
OWARMING
0.1
go
*O
0
z
sI-
*
0
**
a0
V
W
-I LL
W
a
0 I
1
I
1
1
,
, ,
I
Fig. 49. The temperature dependence of the angular momentum of the persistent current according to the data of Mehl and Zimmerman128.The current was created at T = 1.24 OK. after that helium I1 was heated to the temperature close to TA(but not higher than TA) and again cooled to 1.24 OK.
Acknowledgment The authors thank Mrs. A. R. Azo for her valuable assistance in preparing the manuscript. Referencesp . 156
156
E. L. ANDRONIKASHVILI A N D W. G . MAMALADZE
[CH.
3
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[CH.3
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(1958). 68
E. L. Andronikashvili and J. S. Tsakadze, Tr. Inst. Fiz. Akad. Nauk Gruz. SSR 8,
209 (1962). H. E. Hall, Proc. Roy. SOC.(London) A 245, 546 (1958). E. L. Andronikashvili, Yu. G. Mamaladze, S. G. Matinyan and J. S. Tsakadze, Usp. Fiz. Nauk 73, 3 (1961) [English transl.: Soviet Phys.-Usp. 4, 1 (1961)l. 139 E. L. Andronikashvili and J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 37, 562 (1959) [English transl.: Soviet Phys.JETP 10, 397 (1960)J. 70 E. L. Andronikashvili, J. S. Tsakadze and K. B. Mesoed, Proc. VIIth Intern. Conf. Low Temp. Phys., Toronto, 1960 (Univ. of Toronto Press, Toronto; North-Holland Publ. Co., Amsterdam, 1961)p. 454. 7 1 E. L. Andronikashvili, Yu. G. Mamaladze and J. S. Tsakadze, Tr. Inst. Fiz. Akad. Nauk Gruz. SSR 7,59 (1960). 7 1 Yu. G. Mamaladze and S. G. Matinyan, Prikl. Mat. i Mekh. SSSR 24, 473 (1960). 73 K.B. Mesoed and J. S. Tsakadze, Soobshch. Akad. Nauk Gruz. SSR 26,145 (1961). 74 J. S. Tsakadze and I. M. Chkheidze, Zh. Eksperim. i Teor. Fiz. 38,637 (1960)[English transl.: Soviet Phys.JETP 11, 457 (1960)l. 75 J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 42,985 (1962)[Englishtransl.: Soviet Phys.JETP 15, 681 (1962)l. 7 6 J. S. Tsakadze, VlIIth Intern. Conf. Low Temp. Phys., London, 1962, Preprints of Papers, p. 388. 77 J. S. Tsakadze, Proc. VIIIth Intern. Conf. Low Temp. Phys., London, 1962 (Butterworths, London, 1963)p. 104. 78 J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 44, 103 (1963)[English transl.: Soviet Phys.JETP 17,70 (1963)l. 79 Yu. G. Mamaladze and S. G. Matinyan, Zh. Eksperim. i Teor. Fiz. 38, 184 (1960) [English transl.: Soviet Phys.-JETP 11, 134 (1960)l. 80 I. L.Bekarevich and I. M. Khalatnikov, Zh. Eksperim. i Teor. Fiz. 40, 920 (1961) [English transl.: Soviet Phys.-JETP 13, 643 (1961)l. 81 Yu. G. Mamaladze, Zh. Eksperim. i Teor. Fiz. 39, 859 (1960) [English transl.: Soviet Phys.-JETP 12, 595 (1961)l. s B H. A. Snyder, Phys. Fluids 6,755 (1963). 83 R. H. Bruce, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 174. 84 Yu. G. Mamaladze and S. G. Matinyan, Zh. Eksperim. i Teor. Fiz. 38, 656 (1960) [English transl.: Soviet Phys.-JETP 11,471 (1960)l. 85 E. M. Lifshitz and L. P. Pitaevski, Zh. Eksperim. i Teor. Fiz. 33, 535 (1957) [English transl.: Soviet Phys.-JETP 6,418 (1958)l. 86 S.V. Iordanski, Zh. Eksperim. i Teor. Fiz. 49,225 (1965)[Englishtransl.: Soviet Phys.JETP 22, 160 (1966)l. 137 68
CH.
31
ROTATION OF HELIUM XI
159
Yu. G. Mamaladze and S. G. Matinyan, Roc. VIIth Intern. Conf. Low Temp. Phys., Toronto, 1960 (Univ. of Toronto Press, Toronto; North-Holland Publ. Co., Amsterdam, 1961) p. 451. 88 E. L. Andronikashvili and J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 37, 322 (1959) [English trans].: Soviet Phys.-JETP 10, 227 (1960)l. a@ C. C.Lin, Proc. Intern. School of Physics “Enrico Fermi”, course 21, Liquid Helium, Ed. G. Careri (Academic Press, New York, 1963) p. 93. 90 Yu. G. Mamaladze, Prikl. Mat. i Mekh. USSR 28,952 (1964). 91 Yu. G. Mamaladze, K. B. Mesoed and J. S. Tsakadze, IInd All Union Congress on Theoretical and Applied Mechanics (Summaries of Papers), Moscow, 1964) p. 144. 92 Z. Sh. Nadirashvili and J. S. Tsakadze, Communication at the IIIrd Bakuriani Colloquium on SuperAuidity and Some Problems of Superconductivity, February 15-25, 1966. 93 Yu. G. Mamaladze, VIIth AU Union Cod. on Low Temp. Phys., Theses of Papers, Kharkov, 1960,p. 7. 94 L. P.Pitaevski, Dissertation, Inst. Fiz. Problem Akad. Nauk SSSR (1958). 89 E. L.Andronikashvili, K. B. Mesoed and J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 46, 157 (1964) Pnglish transl.: Soviet Phys.-JETP 19, 113 (1964)l. $6 Yu. G. Mamaladze, Low Temperature Physics (papers) (Iz-vo “Metsniereba”, Tbilisi, 1965) p. 51. 97 G. A. Gamtsemlidze, Sh. A. Japaridze, Ts. M. Salukvadze and K. A. Turkadze, Zh. Eksperim. iTeor. Fiz. 50,323 (1966)[Englishtransl.: Soviet Phys.-JETP23,214 (1966)l. g8 Kelvin (Thomson, W.), Phil. Mag. 10, 155 (1880). 99 E. S. Raja Gopal, Phys. Letters 9, 230 (1964); Ann. Phys. (N.Y.) 29, 350 (1964). loo R. J. Donnelly, G. V. Chester, R. H. Walmsley and C. T. Lane, Phys. Rev. 102, 3 (1 956). 101 J. S.Tsakadze, Zh. Eksperim. i Teor. Fiz. 44, 105 (1963)[English trans].: Soviet Phys.JETP 17, 72 (1963)l; Tr. Inst. Fiz. Akad. Nauk Gruz. SSR 9, 151 (1963); Soobshch. Akad. Nauk Gruz. SSR 30,25 (1963). 102 J. S.Tsakadze, Zh. Eksperim. i Teor. Fiz. 46,153 (1964) [English trans].: Soviet Phys.JETP 19,110 (1964)l. 103 E.L. Andronikashvili, R.A. Bablidze, G. V. Gujabidze and J. S. Tsakadze, IInd All Union Congress on Theoretical and Applied Mechanics (Summaries of Papers), Moscow, 1964.p. 16. 104 E. L. Andronikashvili, R. A. Bablidze and G. V. Gujabidze, Preprint, Institute of Physics, Akad. Nauk Gruz. SSR (1964). 106 E. L. Andronikashvili, G. V. Gujabidze and J. S. Tsakadze, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 159. 106 E. L. Andronikashvili, G. V. Gujabidze and J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. 50, 51 (1966) [English trans].: Soviet Phys.-JETP 23, 34 (1966)l. 107 R. A. Bablidze, J. S. Tsakadze and G. V. Chanishvili, Zh. Eksperim. i Teor. Fiz. 46, 843 (1964) [English trans].: Soviet Phys.-JETP 19, 577 (196411. 108 E. L.Andronikashvili and J. S. Tsakadze, Zh. Eksperim. i Teor. Fiz. Pis’ma 2, 278 (1965) English transl.: JETP Letters 2, 177 (1965)l. lo@ E. L. Andronikashvili and J. S. Tsakadze, Phys. Letters 20,446 (1966). E. S. Kerr, J. Chem. Phys. 26, 511 (1957). 111 E. L. Andronikashvili and J. S.Tsakadze, Phys. Letters 18,26 (1965). 87
160
E. L. ANDRONIKASHMLI A N D YU. G. MAMALADZE
[CH.3
G. Careri, W. D.Mc Cormick and F. Scaramuui, Proc. VIIIthIntern. Conf. LowTemp. Phys., London, 1962 (Butterworths, London, 1963) p. 88. 113 R. A. Bablidze, Dissertation, Tbilissi, State University (1965). 114 G. A. Gamtsemlidze, Sh. A. Japaridze and K. A. Turkadze, Communication at the lIIrd Bakuriani Colloquium on Superfluidityand Some Problems of Superconductivity, February 15-25, 1966. 115 G. A. Gamtsemlidze, Sh. A. Japaridze and K. A. Turkadze, Zh. Eksperim. i Teor. Fiz. 50, 327 (1966) [English transl.: Soviet Phys.-JETP 23, 217 (1966)l. 116 G. V. Gudzabidze and J. S.Tsakadze, Zh. Eksperim. i Teor. Fiz. 50,55 (1966) [English transl.: Soviet Phys.-JETP 23, 37 (1966)l. 117 A. F. Andreev, Zh. Eksperim. i Teor. Fiz. 46,1456 (1964) [Englishtransl.: Soviet Phys.JETP 19,983 (1964)l. 118 E. L. Andronikashvili, Zh. Eksperim. i Teor. Fiz. 22, 62 (1952). 119 E. M. Lifshitz and E. L. Andronikashvili, A Supplement to “Helium” (Consultants Bureau, New York; Chapman and Hall, London, 1959). l20 R. H. Walmsley and C. T. Lane, Phys. Rev. 112, 1041 (1958). 121 S. C. Whitmore and W. Zimmerman, Phys. Rev. Letters 15, 389 (1965). 182 P. J . Bendt, Proc. VIIth Intern. Conf. Low Temp. Phys., Toronto, 1960 (Univ. of Toronto Press, Toronto; North-Holland Publ. Co., Amsterdam, 1961) p. 576. I23 P. J. Bendt, Phys. Rev. 127, 1441 (1962). 124 D. Depatie, J. D. Reppy and C. T. Lane, Proc. VIIIth Intern. Conf. L o w Temp. Phys., London, 1962 (Butterworths, London, 1963) p. 75. lZ5J. D. Reppy and D. Depatie, Phys. Rev. Letters 12, 187 (1964). 128 J. D. Reppy, Phys. Rev. Letters 14, 733 (1965). 127 J. Clow, D. Depatie, J. G. Weaver and J. D. Reppy, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 328 128 J. B. Mehl and W. Zimmerman, Phys. Rev. Letters 14, 815 (1965). 129 J. F. Allen, Proc. Intern. School of Physics “Enrico Fermi”, course 21, Liquid Helium, Ed. G. Careri (Academic Press, New York, 1963) p. 305. l30 R. L. Richards and P. W. Anderson, Phys. Rev. Letters 14, 540 (1965). l 3 1 W. H. Kleiner, L. M. Roth and S. H. Autler, Phys. Rev. 133, A 1226 (1964). 132 J. R. Pellam, Proc. Xth.Intern. Conf. Low Temp. Phys., Moscow, 1966, to bepublished. l 3 3 J. Clow and J. D. Reppy, Phys. Rev. Letters 16,887 (1966). 134 Yu.G. Mamaladze, Zh. Eksperim. i Teor. Fiz., to be published.
112
CHAPTER
rv
STUDY OF THE SUPERCONDUCTIVE MIXED STATE BY NEUTRON-DIFFRACTION BY
D. CRIBIER, B. JACROT, L. MADHAV RAO*
AND
B. FARNOUX
CENTRE D’ETLJDES NUCLCAIRES DE SACLAY, FRANCE CONTENTS: 1. Introduction, 161. - 2. Theory of neutron scattering by vortex lines, 164. 3. Experimental conditions, 166. - 4. Experimental results with niobium, 171. 5. Analysis of the results obtained with niobium, 171. - 6. Conclusions, 178.
1. Introduction It has been known1 since the early days of superconductivity that the superconducting properties disappear when the material is brought into a magnetic field larger than a critical field H,. In 1933 it was shown by Meissner that for an applied field smaller than H,, a bulky superconductor behaves as a perfect diamagnet. The sample has a magnetization 471M which, in the case of no demagnetizing field, is equal and opposite in sign to the applied field. This means that the applied field H is completely expelled, which requires an energy H2/8n.As long as this energy is smaller than the difference between the free energies F , and F, of the substance in the normal state and in the superconducting state respectively, the substance remains superconducting. But when H reaches the value H, with
the sample goes into the normal state by a first order transition. The magnetic behaviour of such a sample is shown in Fig. 1. The highest values of H, are of the order of 1000 Oe.
*
On leave from Atomic Energy Establishment, Bombay, India - now returned.
References p. 179
161
162
D. CRIEIER
[CH.4,
eta).
0
1
In this early period it was already observed by several people, that many alloys, like PbT12, show a different behaviour2. The kind of behaviour observed in such alloys is reproduced in Fig. 2. It shows that the Meissner effect exists only for fields smaller than a critical field H,,.Between H,, and an upper critical field Hc2 there is a partial penetration of the magnetic field, but the substance remains superconducting,at least for a low enough current -LR M
Fig. 1. Magnetization of a superconductor of the fmt kind.
-Lfc
M
Fig. 2. Magnetization of a superconductor of the second kind.
intensity. Between H,, and Hc2 the substance is said to be in the mixed state or Shubnikov phase. A fist tentative explanation of this behaviour is due to Mendelssohns. He attributed the occurrence of the mixed state between H,, and H,, to the existence of local regions of high critical field due to inhomogeneities in the alloys. More recently4, materials were discovered with very high upper critical fields (for instance Nb,Sn with H,, larger than 200 kOe). It has also been proved that a mixed state occurs in pure samples of some elements: this is the case for niobium and vanadium. This suggests that the existence of a mixed state is of a much more fundamental character than was proposed by Mendelssohn. Superconductorswhich show a complete Meissner effect up to a thermodynamical critical field H,are now referred to as superconductors of the first kind and superconductors, which have a mixed state between H,,and H,,are called superconductors of the second kind. A theory which, in a very natural way, introduced such a distinction between two kinds of superconductors is due to Abrikosov5. This theory is based on the phenomenological theory of superconductors of Landau and Ginzburg6. A natural distinction between two kinds of superconductors exists according to the sign of the interphase surface energy'-9 between the normal and the superconducting phase. The free energy of the sample in a magnetic field consists of three parts: the positive energy associated with the expulsion of the field, the energy ( P , -FJ which depends on the fraction of the sample References p . 179
CH. 4,g 11
SUPERCONDUCTIVEMMFD STATE
163
which is in the superconducting state, and the interphase surface energy. In materials in which this last energy is negative the state of minimum energy, for a value of the applied field higher than a critical field Hcl, is obtained when the sample splits up in superconductingand normal regions. According to Abrikosov5 this splitting occurs in the following way: the field penetrates the sample along lines parallel to the field. In the core of these lines the material is normal. Around these cores the field decreases but the material is in the superconductingstate. Such lines are called “vortex lines”, which name is related to the circulating current around these lines, associated to the field gradient. The structure of these lines is defined by two parameters 1, and l. 11, is the penetration depth of the magnetic field, and t is the coherence length introduced by Pippardlo, which can be visualized as the thickness of the boundary between the normal and the superconducting phase. An accurate definition of this length is given by Ginzburg and Landaue. Usually the ratio ~=11,/( is introduced. If K < 1/,/2, the surface energy is positive and the material is of the first kind. For K > 1/42, the surface energy is negative and the material is of the second kind. Fig. 3 shows a schematic diagram of
Fig. 3. Structure of an isolated vortex line.
a vortex line. Typical values of 1, and ( are 100-1000 A. The ratio K is 0.026 for aluminium, and as high as 40 for V,Ga for instance. In 1950 London11 proved that the flux trapped in a superconductingring is quantized; this was experimentally codrmed by Doll and Nabauer12 and by Deaver and Fairbank13.As a special case one can show that a vortex line as defined above should carry a quantum of flux cDo = ch/2e =2 x 10- gauss cm2,or a multiple of this quantum. Saint-James and Sarmal4 have proved that for a large range of conditions a line should carry only one quantum. Thus, for an induction B in the sample the number of lines equals N = BIGo. Abrikosov also showed that these vortex lines form a regular array. In his model the structure in the mixed state can be seen as a regular array of fibrae parallel to the field. The distance between these fibrae decreases when the field increases from H,,up to Ifo2.This model gives a good explanation of
’
References p . I79
164
[CH.4,s
D. CRIBIER ef a[.
2
the behaviour of superconductors of the second kind, but up to 1964 there was no proof of the existence of these vortices. A splitting in laminae alternatively normal and superconducting has been proposed by Goodman and Gorter 15. This model explains the general macroscopic behaviour almost as well as the vortex model. De Gennes and Matricon16 suggested to check the existence of a regular array of these vortex lines by neutron diffraction techniques. The neutron, being sensitive to the magnetic field, will be diffracted by the regular structure of the field which is present in the mixed state.
2. Theory of neutron scattering by vortex lines The theory of this scattering16917is merely a special case of the usual theory of magnetic scattering of neutrons 18. The fundamental interaction takes place between the magnetic moment p a of the neutron and the magnetic field H . The scattering amplitude is given by m a (4)= 2ntiz j p n. H ( r ) eiqrd 3 r ,
(1)
where m is the mass of the neutron and q the scattering vector or momentum transfer in the scattering process. H ( r ) is the spatial field distribution. In the Abrikosov model the structure of the field has to be considered only in a plane perpendicular to the applied field, and it is always possible to describe it by a two-dimensional Fourier series
~ ( r=)A , C cos M\ r i
+ A , C cos M: + ..., o r
i
where MI,M,, ... are the vectors of the reciprocal lattice and the summation over the index i is extended to vectors of the same length. The diffraction pattern of such a field structure is similar to that of a crystal: the scattering amplitude is zero except for q =M , ...,where this amplitude is proportional to A,. In general it is not possible to solve the field distribution from the GinzburgLandau equations, so that explicit expressions for A,, A,, ,.. in terms of the fundamental parameters cannot be obtained except for two cases which will be considered now. a) Vortex lines of infinitely small core (t=0). Then the field distribution is given16 by
H
+ A:
rot r o t H = djoXd2(r - q ) , i
References p . I79
(2)
CH.4,1,2]
SUPERCONDUCTIVE MIXED STATE
165
where d2 is the Dirac function for two dimensions and ri the coordinate of the ith line. This case has been considered in detail by Kemoklidzel7. If we forget about the numerical constant we find for the scattering amplitude
In this case 1
For a polycrystalline structure* one finds17 for the integrated intensity of the Debye-Scherrer line
R2
1
‘“.“M,(l + R ,2M ,2)
2’
If we assume that A,lM,J 9 1, which is the case in all practical circumstances, the ratio of the intensity of the first Bragg peak to that of the next one is about 16 for a triangular lattice and about 6 for a square lattice. This calculation is correct for vortices of very small core. For the case of niobium which was used in the experiments, 5.-1,-400 A, so that the ratio of the first Bragg peak intensity to the second one should be even larger than these values. b) Quasi-sinusoidal field distribution. Kleiner et al.19 have calculated the ratio A 2 / A , , in the region close to Hcz, for any value of K . For the triangular lattice, which is found to have a lower free energy than the square one, this ratio is 2.5 x so that close to HCz the sinusoidal field distribution is a good approximation for any value of K. With such a ratio the intensity diffracted in the second Bragg peak is less than 1% of that in the first one, even for a perfect lattice. So far we have considered only perfect lattices. In the case of a liquid-type structure, the intensity of the first Bragg peak is much lower and is likely to be unobservable. The intermediate case of a distorted lattice gives an intensity lower than for a perfect lattice and a broadening of the peak. The lowering and the broadening are dependent of the degree of distortion. It is not to be expected that the vortex lines are arranged as a monocrystal throughout the whole sample. In fact, we observed diffraction phenomena for any orientation of the sample with respect to the incident neutron beam; this implies that the vortex lines are, at best, distributed on numerous small two-dimensional monocrystals which have a random orientation. References p . 179
166
D. C R I B I E R ~al. ~
[CH.4,
3
3. Experimental conditions The neutron cross-section considered above is, for the available wavelengths, very small (lo4 times smaller than for diffraction by an ordered magnetic material like iron). It is therefore advisable to select a material which will give a relatively high intensity. As we do not have a correct expression for the cross-section in the general case, we can only use the expression derived in the case of an infinitely small core as an indication. Then AP and M must be small. This means that we will obtain the highest intensity for materials of small IC.For these reasons we have chosen niobium and Pb0.98Bi0.02 which both have a K of the order of 1. The distance between the vortices in Nb is more than 1000 A, in PbBi more than 2000 A. The highest intensity of our long wavelength neutron source lies about at 5 A; this means that the first Bragg peak is to be expected at about 10' in Nb, and at about 5' in PbBi. It is therefore essential for the experiment to work with a well-collimated beam of long wavelength neutrons. a) The neutron source is the reactor EL3, with a cold moderator20 which increases the ffux of long wavelength neutrons. The neutron beam passes a 20 cm beryllium filter, which filters out neutrons with a wavelength smaller than 4 A. The spectrum which is obtained with this mter is represented in Fig. 4. In some of the experiments, this spectrum has been used as such. The large wavelength spread is of little consequence in determining accurately the Bragg peak, since the peak is observed near forward direction. It has taking into account the 1 ' dependence of the cross-section, a mean wavelength of 4.5 A. For some other experiments, the beryllium filter is followed
Fig. 4. Spectrum of the incident neutron beam filtered by beryllium. This spectrum is obtained with a liquid hydrogen moderator shifting the Maxwell spectrum towards long wavelengths. References p . 179
CH. 4,
I 31
SUPJlRCUNDUCTIVE MIXED STATE
167
by a lead flter, which acts like the beryllium filter, but with a cut-off at 5.7 A and a mean wavelength of 6.18 A. To improve the wavelength resolution a mechanical monochromator, using a helical obturator, has been built21 which can give a variable wavelength with a resolution A l / l of about 10% (Fig. 5). With this monochromator the more precise results are obtained. The collimator for the neutrons consists of a row of 9 plates of cadmium, placed perpendicularly to the beam direction and well spaced, which have horizontal slits, 0.5 mm high and 20 mm long. Each plate contains 12 such slits separated by 0.5 mm. The distance between the first and the last plate is 1 m. With this configration a beam consisting of 12 identical and parallel parts is obtained, each of them with a total divergence of 1.5 minute of arc.
Fig. 5. Schematic view of the mechanical monochromator.
We preferred this system to the usual Soler-slits, because it gives no extra broadening due to reflection on the walls of the slits. Behind the sample another collimator is placed identicalto the first one, which can rotate around the axis of the sample. Using the monochromator the overall resolution due to angular and wavelength dispersion is about 3.5’. b) The sample is mounted inside a liquid helium cryostat. By reducing the vapour pressure of the liquid helium, the temperature of the sample can be References p. 179
168
D. CRIBIER et al.
[CH.
4,
03
lowered to 1.9 "K. The cryostat is plr -ed inside the gap of an electromagnet, which provides a well-homogeneous horizontal field, parallel to the slits of the collimator. Fig. 6 shows a diagram of the set-up, which is also shown on the photograph of Fig. 7. c) The samples. Niobium sample. The material is the commercially available niobium of serni-
j.-
Cryostot Detector
Monochromator
\
\
Reactor core
Beryllium
I
I l l I l l
i
Collimators
I
I
Sample
Fig. 6. Schematic diagram of the experimental set-up.
Fig. 7. Photograph of the experimental set-up. The monochromator is in the shielding at the right of the picture. The detector is on the left, outside the frame. References p . 179
CH. 4,g 31
SUPERCONDUCI'IVE MlxED STATE
169
element. It has a resistivity ratio P ~ O O - K / P ~ ~of ~ K80. The sample consists of 9 cylinders placed horizontally and parallel to the applied magnetic field. The cylinders have a 3 mm diameter and a 24 mm length; they are arranged in 3 rows. There is no direct contact between the cylinders, they are separated by about 0.2 mm*. -1 K M
c
Fig. 8. Magnetization curve of the niobium sample, consisting of an assembly of 9 cylinders.
The magnetization curve at 4.2 "K (T/7',=0.53) of this sample is represented in Fig. 8. It shows that at this temperature Hc2= 3060 Oe. Maki22 has shown that for T f T,, one should introduce two parameters K~ and x2, with the two relations I C ~= Hc2/d2Hc
and, near Hc2 -4nM
~-
H,, - H
--1
1 1.16 ( 2 ~ ;- 1)"
This last relation corresponds to a magnetization linearly approaching zero near Hc2. We found for our sample ~1
= 1.38,
I C = ~
1.58.
These values are not too different from those of McConnille and Serin23. Other samples of niobium have also been used (with less success). Lead-bismuth alloy. This is an alloy with 2%(atomic) bismuth. Fig. 9 shows its magnetization curve at 4.2 OK.The analysis of the curve yields K 1 = x* = 0.91.
* The demagnetizing factor of such a sample is unknown but small, of the order of 1 "/,. It has therefore been neglected throughout the analysis. References p. 179
170
D. CRIBIER et al.
[a. 490 3
Fig. 9. Magnetization curve of the lead-bismuth alloy.
Neutron diffraction
by niobium a t L.2OK
H= 1L75
Oe
h neutron= ~ . A3
2l
05
5
ld
15' Scattering angle
Fig. 10. Bragg peak obtained with niobium. The background which has been substracted is represented as a dashed line. References p . 179
CH.4,g 51
SUPERCONDUCTIVEMIXED STATE
171
4. Experimental results with niobium
Fig. 10 shows a typical angular distribution obtained with monochromatic neutrons of 4.3 A, AA/A= 10% and a niobium sample in an applied field between H,, and Hc2. The background, measured before the sample is magnetized for the first time or by application of a field larger than Hcz, is also represented in the graph. This graph shows the two characteristicsobserved with rather pure niobium samples: one observes a very well defined peak with a full width at half maximum of 3.5‘ and essentially no scattering outside of this peak. In particular, no other peak at a larger scattering angle, is observed. Efect of the magneticfield. When the magnetic field on the sample is increased from zero nothing happens in the lowest field region. Then, at some value Ha, a peak begins to appear, which increases in height and shifts to larger angles ; the intensity passes through a maximum and then decreases, while the peak still shifts to larger angels. This is illustrated in Fig. 1I which shows the angular distribution for increasingvalues of the applied field. These angular distributions have been obtained with an incident spectrum which is filtered only by the beryllium. The shift of the peak is just what is to be expected for scattering by a pattern of vortex lines, in which the line distance becomes smaller as the field increases from H,, to HCz. 5. Analysis of the results obtained with niobium 5.1. LINESHAPE
In fig. 10 a typical result obtained with niobium using the best resolution available in our experimental set-up has already been shown. The line is perfectly symmetrical and has a width of 3.5‘. The instrumental resolution is determined by the two collimators and the wavelength spread of 10% of the monochromator; a crude estimate of the total instrumental width, for a Bragg peak at small angles, gives with these conditions 3.5’. The observed width of 3.5’just equals the instrumental resolution under the best conditions. The natural width of the line must therefore be smaller than 1‘. As the peak lies between 10’ and 15’, the ratio of the widths to the position of the peak is rather small and one can really speak of a Bragg peak associated with some long range order, and not of a scattering of a liquid type. Clearly experiments with still better resolution would be useful in order to increase the information about the value of the natural width, and so about the nature of the order. References p , 179
172
D. CRIBIER
eta/.
[CH.
I
I
I
I
5,
10'
15'
2d
4, 0 5
I
Scstering angle
(a)
Fig. Ila. Diffraction pattern obtained with niobium as a function of the magnetic field. The neutron spectrum is not monochromaticbut beryllium-filtered. Each point is obtained counting during about 30 minutes. (1) 1476 Oe, (2) 1435 Oe, (3) 1397 Oe, (4) 1353 Oe, (5) 1312 Oe, (6) 1271 Oe, (7) 1230 Oe, (8) H = 0. References p. I79
CH. 4,
5 51
173
SUPERCONDUCTIVE MIXED STATE
Intensity
liI
1oooo
I
moo0
NIOBIUM a t L.~'K moo0
loo00
1
I
5'
lo'
I
6'
I
M'
I
25' S c s t e r i n g angle
(b) Fig. 11 b. Diffraction pattern obtained with niobium as a function of the magnetic field. (1) 1476 Oe, (2) 1517 Oe, (3) 1588 Oe, (4) 1599 Oe, (5) 1650 Oe, (6)1722 Oe, (7) 1804 Oe, (8) 1886 Oe.
It should also be noticed that the scattered intensity by this niobium sample really goes to zero at angles smaller than the Bragg peak. There is no observable diffuse scattering at very low angles, associated with the applied magnetic field. References p . 179
174
D. CRIBIER et
d.
[CH.4,
5
This experimental fact pleads strongly against the possibility of distorted vortex lines around the field direction. As distorted vortices would give a more isotropic scattering than straight lines, the scattering pattern would be closer to a Debye-Scherrer circle, than to a straight line. In our experimental set-up (long size of the slits in one direction compared to the diameter of the Debye-Scherrer circle) scattering would occur in all directions between 0 and flB= RD/L, where RD=radius of the Debye-Scherrer circle and L = 1 m, the length of the second collimator. In samples with a strongly irreversible magnetic behaviour we could only observe a central scattering, as occurs in a disordered material. 5.2.
POSITION OF THE PEAK
This position is independent of the resolution, and is perfectly reproducible for the same sample from one experiment to the other. From the analysis of the line shape, we have concluded that this peak can be considered as a real Bragg peak, so we can relate its position OB to some interplanar distance d
N
0';
-9 -1%
@ ~
Results obtained w i t h a wide incident spectrum Ax = 18 Results obtained with a narrow incident spectrum ah ~ 0 5 8 .
@ F o r trapped flux in zero field
Fig. 12. The vortex lattice parameter versus the induction for niobium. The two lines are the expected variations for a square lattice (Bda = 90)and a triangular lattice (Bd2 = fl/J@O).
References p . 179
CH. 4,g 51
SUPERCONDUCTIVE MIXED STATI?
175
by the relation 1= 2d sin OB, where 1is the wavelength of the neutrons. In Fig. 12 we represent the variation of l/dz so obtained as a function of the induction in the sample. This induction has been measured as a function of the applied magnetic field by Vivet in an independent experiment. All data reported in the graph have been obtained during the first magnetization of the sample. The graph shows that, except for small induction, a linear variation is found 1 43 B -=-d2 2 Q0’ as is expected from the theory of vortex lines. From the proportionality
constant it follows that each of these vortex lines carries one quantum of flux. More precisely it follows that the line density is the density expected
for a triangular lattice. The departure from the dependence one would expect for a square lattice is definitely outside the experimental accuracy. In preliminary reports on this subject24325 we drew and published the opposite conclusion. This was due to a trivial mistake in the evaluation of the Scattering angle. After recalibration, the former results agree well with these much more accurate new results. Experiments with neutrons of longer wavelength (lead filtered) have been used to check the position of the peak. The departure from linearity for small values of the induction, is undoubtedly due to a partial penetration of the sample by the magnetic field, which effect is well known2e for rather impure and consequently not reversible samples. So the induction as derived from the measurements, assuming that the whole sample is homogeneously penetrated by vortices, is underestimated as compared to the local induction effective for the scattering process. In fact, one observes that in this range of applied fields the intensity of the peak is too small compared to that for higher values of the field. This intensity reaches a maximum for a field of about 1500 Oe. Above this value one also gets the linear variation. So, it is very reasonableto assume that homogeneous penetration of the field is achieved only above this value. One can expect a linear variation of l/dz versus B, starting from very small values of B, only in experiments on samples with a perfectly reversible behaviour and a good field penetration.
5.3. INTENSITY OF THE PEAK As we have remarked in Section 2, the intensity of the peak must decrease when the scattering angle increases. So, one expects a decrease of the intensity when the field increases. In fact, the observed general behaviour of the inReferences p . I79
176
D. CRIBIER et
at.
[CH.4,
55
tensity as a function of the applied field is shown in Fig. 13. As already mentioned, the increase just above H,, is due to the fact that only a fraction of the sample is permeated by vortex lines. When one increases the field beyond the point of homogeneous penetration, one observes in fact a rapid decrease of the intensity ; however, one also observes two unexpected effects: 1) The intensity of the peak becomes practically unobservable for an applied Jield (about 2600 Oe) much lower than HC,x3O6O Oe, even if we use an experimental device with a rather poor angular definition in order to increase the luminosity of the spectrometer. Intensity
(arbitrary units)
t
Fig. 13. Intensity of the Bragg peak as a function of the applied fiield. Only the shape of this curve is reproducible from one experiment to the other.
A possible explanation of this fact is that at higher fields (small intervortex distance, dxlOOOA in Nb), the vortex lines are more and more free of pinning by the impurities or dislocations, and better and better arranged in a few large monocrystallites which have no reason to be well oriented in front of our neutron beam; so we shall not observe any diffraction at high fields without changing the orientation of the sample itself. 2) The intensity of the peak corresponding to a given jield is reproducible References p . I79
CH. 4,
B 51
SDERCONDUCTIVE W
D STATE
177
only fi we use an angular collimation of 3' or more. For a collimation of 1' the intensity of the Bragg peak can vary by a factor of 3 when one repeats the experiment in the following way: switch off the field, heat the sample above T,,cool it down again and switch on the same field as before. This irreproducibility of the diffracted intensity observed with the best angular collimation only, seems to us to be an argument against the hypothesis of a liquid-type order of the vortex structure, as this type of order must always give a reproducible scattering intensity. But if we think of a polycrystalline vortex structure we can explain this irreprodi1::bility: if the crystallites are big enough their number will be small and, with a good angular collimation, only a few of these crystallites will be well-oriented in order to diffract the neutrons; let (n) be the mean value of the number of crystallites, suitable oriented. If (n) is too small, it will largely fluctuate from one experiment to another and this explains the irreproducible intensity. Let us calculate (n). The number of crystallites N in the useful cross-section S of the sample for a given d, as a function of the number of vortex lines x along each side of the crystallites, is N = S (s); s is the effective area of the crosssection of the sample in the scattering plane; (s) is the mean surface of one crystallite. Sm.nl0" A'; ( s ) x J 3 x 2 / 2 x lo6 A2 for a triangular lattice when d m 1000 A. In the case of an incident beam collimated to 1' of arc the mean value (n) of the number of suitably orientated crystallites in the diffraction should be < n ) = 6 x N x 3 x 10-4/2n
(the multiplicity factor of 6 arises from the symmetry of the triangular lattice) and we find ( n ) m 106/x'. In order to explain the observed variation in the peak intensity by a fluctuation of n, one is led to a choice of x of the order of 200 to 300 giving ( n ) between 10 and 25. This gives a length of the edge of the crystallites of the order of 20 microns. Further experiments would be advisable to support these conclusions. 5.4. OBSERVATION OF ONLY ONE BRAGGPEAK
We have undertaken several attempts to observe the second Bragg peak at the angle d2 =J30el,where 0, is the angular position of the first Bragg peak and J3 a coefficient valid for the triangular lattice. These experiments were sensitive enough to detect a signal of 2% of the intensity of the first Bragg peak but gave a negative result; so, in the range of observation, the magnetic Referencesp , 179
178
D. CRIBIER et
al.
[CH.4, !j 6
field distribution seems to be well described by
H(r)-
c
COSM;.r, 1=1.2.3
where M fare coplanar vectors of the same modulus Mi =4nJdJ3 such that C,Mf =O. This describes a perfect sinusoidal distribution on a two-dimensional triangular lattice. This field distribution is not surprising in the case of niobium with K w 1 and 5 w 1, not small compared to the intervortex distance d. We remark h t from the value of Hc2one can deduce the coherence length 5 by the relation 5
<
For our sample this gives = 325 A, and from the value of K it follows that 13,-400 A. It should be noticed (Fig. 12) that with trapped flux in zero applied field, the diffraction pattern is the same as that with an applied field which causes the same induction. This indicates a similar type of order under these circumstances. Experiments carried out at lower temperature (1.9 OK) show a similar diffraction and give data which fit well the first ones in the l/d2 versus B diagram. Experimental results with lead-bismuth alloys. As the distance between the lines is larger than in niobium, the scattering angle is smaller (about 6'), and so the observation of the diffraction is rather =cult. Fig. 14 shows the variation of l/dZ versus B for this sample. The results are too poor to distinguish between triangular and square lattices, but are good enough to prove that there is only one quantum per line. This result may be of interest as, for samples with a rather small value of K ( ~ = 0 . 9 ) the , theoretical problem of the number of quanta per line is still open. 6. Conclusions
The hypothesis of vortex lines, introduced by Abrikosov, has been proved by means of neutron diffraction. The accuracy of the experiment allows us to establish that there is a triangular lattice in niobium and to confirm the flux quantization. There are strong indications of the existence of real long-range order, coming mostly from the irreproducibility of the intensity, though there is no definite proof. If long-range order is accepted, this experiment shows References p . 179
179
cn. 41 1 loio CnF' dz o.301 Lead
Bismuth A l l o y
0.20-
100
200
300
Fig. 14. The vortex lattice parameter for the lead-bismuth alloy versus the induction. The straight line is the expected variation for a triangular lattice.
that the magnetic field in Nb can be well described by the first harmonic of a Fourier series. Experiments with materials with higher ic would be advisable. By lack of good quality samples we have so far been unable to do them. An experiment on a niobium zirconium alloy being completely magnetically irreversible has been unsuccessful : it gave central diffuse scattering only.
Acknowledgements We would like to thank Drs. Vivet and Carrara for measuring the magnetization curve, Mr. Kleinberger for the preparation of the lead-bismuth sample.We have had useful discussions with Drs. Caroli, De Gennes, Herpin, Matrkon, Saint-James and Sarma. REFERENCES D. Shoenberg, Superconductivity (Cambridge University Press, 1962). J. N. Rjabinin and L. W. Shubnikov, Nature 135, 581 (1935). K. Mendelssohn, Nature 135,826 (1935); 152,34 (1935). 4 J. E.Kunzler, Rev. Mod. Phys. 33, 501 (1961). 5 A, A. Abrikosov, Zh. Eksperim. i Teor. Fiz. 32, 1442 (1957) [English trans].: Soviet Phys.-JETP 5, 1174 (1957)l.
1 2
180
D. CRIRIER eta].
[CH.4
V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 20, 1064 (1950)[English transl. in: Collected Papers of Landau (Pergamon Press, 1965)l. C. J. Gorter, Physica 2,449 (1935) H. London, Proc. Roy. SOC.(London) A 152,650 (1935). A. B. Pippard, Proc. Cambridge Phil. SOC.47, 617 (1951). 1" A. B. Pippard, Proc. Roy. SOC.(London) A 203,210 (1950). 11 F. London, Superfluids, Vol. 1 (Wiley, New York, 1950). 12 R. Doll and D. Nabauer, Phys. Rev. Letters 7, 51 (1961). 13 B. S. Deaver and W.M. Fairbank, Phys. Rev. Letters 7,43 (1961). l4 D.Saint-James and G. Sarma, Private communication. See also: G. Lascher, Phys. Rev. 140,A 523 (1965); J. Matricon, Thesis, Orsay (1966). 16 B. B. Goodman, Phys. Rev. Letters 6, 597 (1961); H. Van Beelen and C. J. Gorter, Physica 29, 896 (1963); C. J. Gorter, Rev. Mod. Phys. 36, 27 (1964). l6 P. G. De Gennes and J. Matricon, Rev. Mod. Phys. 36,45 (1964). M. P. Kemoklidze, Zh. Eksperim. i Teor. Fiz. 47, 2247 (1964)English transl.: Soviet Phys.-JETP 20, 1505 (1964)l. Is See, e.g., P. G. De Gennes, Theory of neutron scattering by magnetic crystals, in: Magnetism, Vol. 3, Eds. Suhl and Rado {Academic Press, New York, 1963). l9 W. H. Kleiner, L. M. Roth and S. M. Autler, Phys. Rev. 133, 1226 (1964). Zo D. Cribier, B. Jacrot, A. Lacazo and P. Roubeau, Inelastic Scattering of Neutrons in Solids and Liquids (I.A.E.A., Vienna, 1961)p. 411. p 1 G. Gobert, Rapport C.E.A., R 2981 (1966). 22 K. Maki, Physics 1, 21 (1964). 23 T. McConnille and B. Serin, Phys. Rev. 140, 1169 (1965). 24 D.Cribier, B. Jacrot, L. Madhav Rao and B. Farnoux, Phys. Letters 9, 106 (1964). 25 D. Cribier, B. Farnoux, B. Jacrot, L. Madhav Roa, B. Vivet and M. Antonini, Proc. Intern. Conf. Magnetization, Nottingham (1965). C. P. Bean, Rev. Mod. Phys. 36, 31 (1966).
CHAPTER V
RADIOFREQUENCY SIZE EFFECTS IN METALS BY
V. F. GANTMAKHER
INSTITUTE OF SOLIDSTATE PHYSICS, ACADEMY OF SCIENCES OF THE USSR, Moscow
CONTENTS: 1. Introduction, 181. - 2. Principles of the theory, 183. - 3. Various types of radiofrequency size effects, 197. - 4. Shape of line and various experimental factors, 220. - 5. Applications of radiofrequency size effects, 225.
1. Introduction
In recent years quite a number of phenomena well described by the classical concept of electron trajectories have been discovered in the field of physics of metals. A number of quantum mechanical operations were realized while constructing the theory : a gas of elementary excitations (quasi-particles known as “electrons”) that complies with a complicated dispersion which is related to reflecting the symmetry of a crystal lattice, is introduced and use is made of the Fermi statistics leading to the concept of Fermi surfaces. Then, in quite a number of cases, the movement of a quasi-particle-electron may be considered as classical with the use of such conceptions as the electron trajectories, effective mass m, mean free path 1, character of scattering at metal surfaces, etc. It may be easily shown (see for example the discussion in Ref. 1) that such a consideration is valid up to those magnetic fields at which the distance between quantum energy levels is still very small as compared with characteristicenergies of the dispersion law. At the same time the sample size should considerably exceed interatomic distances. Such a semi-classical approach requires sometimes (as in the case of magnetic breakdown, for example) additional quantum considerations. Nevertheless it is the main way of describing all the phenomena concerning kinetic processes. The Azbel-Kaner cyclotron resonance presents a typical example of the quasi-classicaleffect. The classical concepts are quite adequate to describe it: References p . 232
181
182
V. F. GANTMAKHER
[CH.5, 8 1
an electron moving within a magnetic field H, applied parallel to the surface of the metal, repeatedly returns to the skin layer every time finding the field near the surface in one and the same phase, if the condition w = nS2 is satisfied (w is the frequency of the incident radio wave, S2=eH/rnc is the cyclotron frequency of the electron, n is an integer). The radiofrequency size effects described in the present article also belong to this same category of phenomena. Let us consider a plane-parallel metal plate of thickness d experiencing the influence of an electromagnetic incident wave. The following inequality should be satisfied
64d41,
(1)
where 6 is the skin-layer depth. When a constant magnetic field is applied to the plate, the straight electron trajectories change into complicated space curves. In case of a spherical Fermi surface, for example, circle and helical trajectories occur. The specific parameters of these curves (such as circle diameter, pitch of helical line, etc.) are inversely proportional to the magnitude of the magnetic field. It can be expected, naturally, that for such values of the field when these sizes become equal to d some singularities may occur in the surface impedance of the plate. Size effects in direct current conductivity were discovered a long time ago4. The first of them was the increase of electrical resistance of a thin film as compared with the resistance of a bulk metal due to restriction of the electron free path I by the sample dimensions93 10. Furthermore, there are a number of cases in the magnetoresistance of thin samples where the resistance decreases as the magnetic field increases11, while for bulk metal the increase of the field leads usually to the increase of the resistance. This phenomenon occurs due to the twisting of the electron trajectories: an average drift of an electron in the magnetic field for the time of free path t, = l/v is found to be much smaller than the value of I and accordingly the influence of the sample boundaries becomes of little importance. Thus, as the field is increased, the difference between the thin and bulk samples should diminish. The resistance of a sample in the form of a plane-parallel plate in the magnetic field H applied perpendicular to the sample surface has an oscillating character l2-I4. The helical trajectory of the electron moving from one surface of the plate to the other is cut off at some turn fraction depending on the value of H. The magnitude of this fraction determines the oscillating component of the current. If a supplementary small parameter with the dimensions of length exists in the experiment, the size effect may become more distinct. Sharvin and References p . 232
CH.
5, 4 21
RADIOFREQUENCY SIZE EFFECTS IN METALS
183
Fisherl6, for example, in their recent experiments obtained a more distinct direct current size effect owing to the fact that the current was injected into and removed from the sample through thin points having a diameter ranging from to 10-4cm. An “electron beam” in the metal interior was focussed from one thin point to the other with the aid of a magnetic field. In the case of radiofrequencies, the skin-layer depth 6 is such a small parameter that more pronounced size effects are found in the impedance changes within narrow spaces of the fields ( d H / H - S / d ) . To explain causes of this phenomenon let us turn to the theory of the anomalous skin-effect.
2. Principles of the theory 2.1. ANOMALOUS SKIN-EFFECT
IN ZERO MAGNETIC FIELD
As it is well known there are two different approaches to the problem of the anomalous skin-effect which, generally speaking, do not compete with one another. The first entirely qualitative approach was developed by Pippard16117,who based his theory on rather fine conceptions known under the name of “ineffectiveness concept”. Pippard suggested that an effective contribution to the skin-current is made only by those electrons which spend most part of the time between collisions in the skin layer. The velocity of such electrons forms with the surface angles less than 6/1. The remaining electrons leave the skin layer too fast and therefore the electric field has no time to affect them. Thus, it is possible to distinguish at the Fermi surface the effective region having the width 6jl and located along the line urs =0 (u is the electron velocity, A is the normal to the metal surface). Then the effective electron ratio amounts to the order of magnitude 6/Z. Introducing the effective conductivity oCr=(6*/1) G,, instead of the static conductivity go into the wellknown formulae for the surface impedance Z = R + iX and the complex penetration depth at normal skin-effect
6* = (c2/4niw)Z,
(3)
we obtain z=2(-) 2n2u21 +@* c2a0
.
(4)
[Formula (3) introduced the complex value S*. Later, the symbol 6 will be used for the real part of this value.] References D. 232
184
V. F. GANTMAKHER
ICH.
5, 5 2
The second approach to the problem of the anomalous skin-effect suggests a strictly mathematical solution of the problem of the propagation of electromagnetic waves in metals, when the current value at a given point is determined by the field distribution in the vicinity of the point having an electron free-path size. For the first time this problem was solved for a metal in zero magnetic field by Reuter and Sondheimer 18. Their approach required the solution of a system of equations comprising Maxwell's equations, Boltzmann's kinetic equation for the distribution function of the electrons f depending on coordinate r, momentum p and time t
and the expression for the current densityj(r) at the given point
Here, fo is the equilibrium distribution function, and h is Planck's constant. Eqs. (5) and (6) substitute for Ohm's law j = o o E . It is convenient to write the solution of these equations with respect to j ( r ) in the form suggested by 3Chamberslg. For an unbounded metal this solution can be represented in $he form tn
dtE(rp,t)up(f)e-('o-t)v Y
(7)
-m
where E is the electron energy, E is the electrical field vector, functions vp(r) and rp(t)= rO+j:,up dt- rp(ro, t o , t ) describe the trajectory along which the electron with momentum p arrives at point ro, and the frequency v of collisions between electrons and scatterers is assumed to be independent of p . The meaning of expression (7) can be easily explained in the following way. The electron is scattered into the trajectory rp at the moment t previous to to. It is assumed that immediately after collisions the distribution function of the electrons is an equilibrium function and corresponds to an energy E - As, where A s is the energy acquired by the electron from the applied electric field during the time interval (t, t o ) while moving along the trajectory. The addition to the distribution function which is of interest to us is equal to fo ( E - A E ) -Yo (8) = - (JfO/i%)Ap. The integration over d3pin (7) covers all the possible trajectories passing through the point ro, while the second integral References p . 232
CH. 5 , 5 21
RADIOFREQUENCY SIZE EFFJ3CTS IN METALS
185
determines a magnitude of the additionf, =f-fo along each of these trajectories (the factor exp { - ( t o - t ) v } is the probability that the electron will reach the point ro without collisions). In the case of a bounded metal some of the trajectories passing through ro originate on the surface; the lower limit - co of the second integral converts into t,(p). Introducing (7) into Maxwell’s equations we arrive at an integraldifferential equation the solution of which for a half-space is described in Ref. 18. The results 18 are well known. The attention should be drawn to only one of them which will be referred to below. Under conditions of anomalous skineffect it is impossible to describe the field in a metal with the aid of a damped exponential wave5.20. The precise expression for the field contains two components. One of them is connected with effective electrons and describes a sharp decrease of the field near a metal surface. The other component caused by ineffective electrons has the form e-r/< (where =z/l, z is the distance from the surface). This component is small, but it damps relatively slowly. In view of such a complicated form of field damping, the value 6 which figures in the ineffectiveness concept should be strictly determined. We have taken the value6 from relation (3) though it differs, of course, from the distance at which the field becomes practically zero5. In any case, it is just this value (3) which is determined by the measurementszl-25 of R and X. 2.2. ANOMALOUS SKIN-EFFECT IN A
MAGNETIC FIELD
In the case of radiofrequency size effects, which is of interest to us, an additional constant magnetic field is applied to a metal. The criterion of skineffect anomaly in the presence of the magnetic field is as follows
where D is the characteristic size of the electron trajectories. Let us consider the modification of both approaches to the anomalous skin-effect in the presence of a constant magnetic field. Since the shape of electron trajectories will be often dealt with below, it is useful to mention the possible forms of electron trajectories in a magnetic field when the free path is sufficiently long. The integrals of the equations of motion j = (e/c) [uH] ,
u = a@p
(9)
= const.
(10)
have the form p H = const., References p . 232
E
186
V. F. GANTMAKHER
[CH.
5, 9 2
It means that in momentum space an electron travels along the line of intersection of the isoenergetic surface &=const. with a plane normal to the direction of field H.(To avoid misunderstanding we shall use below the term “orbit” for an electron movement in the momentum space.) As the left part of eq. (9) is the velocity along the p-orbit and the right part of this equation contains the velocity along the r-trajectory, it can be seen from eq. (9) that the electron orbit in momentum space (p-space) and the projection of the electron trajectory in the r-space on a plane perpendicular to H are similar to the similarity factor eh/c and are rotated relative to each other through the angle +n. A set of electron orbits exists for every direction of a magnetic field, as thep, may assume any magnitude within the Brillouin zone (the energy E is fixed and equal to the Fermi energy E ~ ) .The corresponding trajectories may be closed, helical and open ones. The closed trajectories correspond usually to cross-sections of the Fermi surface containing a centre of symmetry. It should be noted that these trajectories are not necessarily lying within the plane which is perpendicular to H and, in general, they are not necessarily in the shape of flat curves. The helical trajectories are obtained in the case of non-central closed orbits. The mean velocity 6 along these trajectories is parallel to H and the pitch depends on pH. In the vicinity of the elliptical limiting points where the plane perpendicular to H touches the Fermi surface (the orbit degenerates to a point and the trajectory becomes a straight line) the pitch and period of motion in time are the same for all helical trajectories. The open trajectories exist only in metals with open Fermi surfaces, but not for all directions of the magnetic field. The main feature of these trajectories is that the motion is infinite in the plane perpendicular to H.There are also trajectories with self-intersections corresponding to isolated values of p H but we will not discuss them here. The particular set of trajectories is determined, of course, by the Fermi surface and the direction of H relative to the crystallographic axis. Let us return to eq. (7). In the presence of a constant magnetic field the equation remains valid, since this field changes only the electron trajectories and this is taken into account in the functions r,(t) and t,(p). But the integration in (7) becomes more complicated. The method of solving the problem of the anomalous skin-effect in the presence of a magnetic field was worked out by Azbel and Kaneras-28. The scheme of their method may be described in the following way (see also Refs. 6 and 7). Let a half-space z>O be filled with a metal and have an electromagnetic wave Ee-irotfalling upon it from the outside. In the cases which are of interest to us it is possible to neglect the component of field E, in a metal along the 1p
References p . 232
CH. 5,921
RADIOrmBOUENCY SIZE EFFECTS IN hfETALS
187
normal 0 ~ 7 3 ~ 7Maxwell’s . equations in this case may be reduced to the following two equations
The problem of a half-space is then replaced with the problem of a whole space excited “from the inside” in the plane z=O. For this purpose the field and the current are continued in an even fashion into the region z c 0 : E ( - z ) = B ( z ) . From eq. (7) for the current which should be solved together with eqs. (1 l), this operation means that we change boundary conditions for the distribution function and use again - cg as the lower limit in the intrinsic integral of eq. (7). As it can be seen in Fig. 1, this is not equivalent to the case
(a)
(b)
Fig. 1. Electron trajectories in a magnetic field: (a) at formal even extension of the electric field to the region z< 0 ; (b) at specular scattering.
of specular scattering of electrons at the surface. This operation means rather that we have completely neglected the scattering at the surface. The odd extension of the field and currents to the region z c 0 may be used with the same success3. A basis for neglecting the boundary can be found in the work of Reuter and Sondheimerl8. Their results show that in zero magnetic field the character of scattering at the surface only slightly influencesthe impedance value. A similar result was obtained for the case of anomalous skin-effect in a magnetic field26, when the problem of the cyclotron resonance was solved for the case of diffuse scattering (see also below, p. 192). The transition from the half-space to the whole space makes it possible to use the Fourier transformation for the solution of eqs. (11). Using plane waves with wave vector along the z axis one may draw the following formal References p . 232
188
V. F. OANTMAKHER
between the Fourier components of the electric field
&,(k) = 2
s
E,(z)cos kz dz
(13)
0
and those of the current
$=(k)
=2
7
j , ( ~ )C O S ~ Zdz
0
[in eq. (12) the summation over index fi is supposed]. In the result of the Fourier transformation the differential equations for E ( z ) are replaced with the algebraic equations for ~ ( kwhich ) can be readily solved. The result of the solution is
€ , ( k ) = - 2[k2f- 4 n i o ~ - ~ B ( k ) ] ~ ' E ~ ( O ) ,
(14)
where f is the unit matrix, B(k) is the conductivity tensor introduced by eq. (12) and the prime denotes the derivative with respect to z. In many problems the x and y axes may be chosen in such a way that the nondiagonal components of the tensor become equal to zero and as a result eq. (14) takes the form:
- 2E:(0) &U(k)
=
k 2 - 4dwc-'o,(k)'
CI = x , y .
(The term -2E'(O) appears in eqs. (14) and (15) due to the fact that in the case where the field is evenly continued into the region z
CH. 5,4 21
RADIOFREQUENCY SIZE EFFECTS IN METALS
189
function a(k)it is necessary to use the kinetic equation. We have to introduce the field in the form of the monochromatic wave E R ( z )= b ( k ) e-’@’cos k z
(16)
into the expression for the current (7) and to take into account the shape of the electron trajectories. For consideration of the electron trajectories it is convenient to use variables in the momentum space suggested in Ref. 29: the energy E, the momentum component along the field p H and the dimensionless time z. The first two variables are, as is clear from (lo), the integrals of motion determining the electron orbit in the momentum space, while z determines the position of an electron in the orbit (phase of motion). The quantity z=S2t, where t is the time of electron motion in the orbit measured from an arbitrary initial moment. The integration over d3p in eq. (7) implicates the integration over all the trajectories passing through the point r,. After replacement of d3p= Irnl ds dp, dz, the integration over ds and dp, implicates the use of all the possible orbits and Ji*dz means the integration over all the trajectories corresponding to a definite orbit and passing through the point z, (see also Refs. 7 and 28). Introducing the electric field expressed in the form (1 6 ) into the integrand of eq. (7), we obtain for E for example parallel to x :
Y (zo, t o ) =
i
dt Ex ( z , t ) u, ( t ) exp { - ( t o - t ) v } =
-OD
s 7
-
tYx ( k ) exp ( - iot,)
D
{A!+
x cos k zo
-io+v
1
u,(r’) dz’
70
.
Decomposing the cosine sum into two components we leave only the even term cos kz, cos {(k/D)Juzdz} in accordance with the previously chosen continuous extention of the current to the region z
* The term with the product of the sines when integrating over d3p would become zero due to the central symmetry of the Fermi surfaces. It is just this symmetry that permits us to introduce even extensions for E and for] simultaneously. References p. 232
190
[CH.5, 8 2
V. F. GANTMAKHER
expression for aac(k) 2u
m
7
rn
-a
7
It is expression (18) that is used at the initial stage for each particular calculation. After the o&) is determined for the particular conditions of the problem, such as the range of frequencies w, the magnitude of the magnetic field and its direction relative to the surface, the shape of the Fermi surface, etc., the field is determined according to the common formula r
&(z) = 72
m r
J
ga((k)cos kz dz,
0
where &&k) is obtained with the use of expressions (14) or (15). There is, however, another operation which is common for all particular calculations. Since k ~ d - ’ ,the expression under the cos in eq. (18) may have, according to the condition (8), large values and, hence, cos kz(zo,z) is a rapidly oscillating function. In other words, the trajectory encompasses many wave lengths and the phase of the electric field along the trajectory is rapidly oscillating. As a result it is possible to use the stationary-phase method for the calculation of the integrals over T~ and z in eq. (18). The use of the phase-amplitude diagram and Corm spirals suggested by Pippard 6 is essentially a graphic geometrical variant of the same method. The possibility of using the stationary-phase method means that the main contribution to a(k) is made by those parts of the electron trajectories where Z(T) takes a stationary value, i.e. by those parts which are near points uz= 62&/&=0. We again arrive at the “effective region” u, =0 on the Fermi surface and the “ineffectiveness concept” suggested by Pippard. The effective region may intersect orbits of any type mentioned above and therefore one or several effective points with u, =0 may occur on each in the ,orbit may be deof the orbits. The width of the effective part termined from the ratio of the length of the trajectory part near an effective point for which the coordinate z change LIZ is less than 6 to the length of the trajectory travelled during the period 2n/Q (A7 =2x). References p . 232
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8 21
RADIOFREQUENCY SIZE EFFECTS IN METALS
191
As it can be seen from Fig. 2, for the closed orbits and for a field applied parallel to a metal surface, T~~ (S/R)*, where R is the radius of curvature of the trajectory near an effective point. Thus, it turns out that zetf depends on p R and H. As another example we may consider a helical trajectory in an inclined magnetic field. In a lateral projection the helical trajectories are of N
Fig. 2. A closed trajectory when H i s parallel to the metal surface.
a sinusoidal shape. The sinusoidal curve shown in Fig. 3 has one point u, =0 through a period and at this point azz/azz=0 too. It is easy to show that the length of the effective part of the trajectory is equal to L=a*fu%p-' and, therefore, zeff-(6/u)* (angle cp is assumed to be small).
Fig. 3. A helical trajectory in an inclined magnetic field with one effective point through a period (a lateral projection).
References p . 232
192
V. F. GANTMAKHER
[CH.5,
52
The following important factor should be noted. The stationary-phase method is used twice for the calculation of a(k). The current at the given point zo is determined first by electrons having u,(zo) = 0 [integration over orbit centres (over 7,); see eq. (7)]. But among them the main contribution is made by those electrons which previously (when z < z), have passed effective points with ar/&=O in regions of values z where the field E differs from zero. So for the effectiveness of an electron at least two effective points in its trajectory are necessary. In fact, the strict theory uses to some extent the ineffective concept too. The form of expression (18) reflects the fact that f, differs from zero along the effective region only, where u, N 0. This is the main argument in favour of the assumption that the scattering at the surface does not considerably influence the impedance of metal Z in a magnetic field because the electrons for which v, is strictly equal to zero never collide with the surface. Nevertheless it should be borne in mind that the trajectories of the type shown in Fig. 4
Fig. 4. Electron trajectories whose contribution to the skin current is essentially altered due to the existence of the metal surface.
have not been taken into account correctly. This is of little importance in the case of cyclotron resonance26927, since these trajectories do not determine the nature of the phenomenon. But for monotonic dependences Z ( H ) in weak and strong fields (with 52 <w and 52 % w ) these trajectories may become significant. The problem about impedance changes in these regions is not clear yet, because the predicted behaviour of the impedance27 is not yet confirmed experimentally30-32. The obtained form of functionf, justifies also the replacement of the collision integral in expression ( 5 ) with the relaxation term -fly (Ref. 26). It is the functionf,, but notf,, which is important in the collision integral. Since f i NO, far from the region v, =0, the role of scattering processes reduces to the removal of some electrons from the number of effective electrons, while the replenishment at the expense of other electrons does not practically occur. In this respect the electrons on the effective trajectories are similar to particles in a beam: the only result of collisions is that they come out of the process. References p . 232
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5, 21
RADIOFREQUENCY SIZE EFFECTS IN METALS
193
(In the general case of anomalous skin-effect, v should be considered as a function of p, but this assumption only slightly complicates the formulae given above71lo.) 2.3. APPLICATION OF THE INEFFECTIVENESS CONCEPT
TO THE STUDY OF SIZE
EFFECTS
The calculations of impedance in a magnetic field may be considerably facilitated by the use of the ineffectiveness concept. Heine33 was the first who used the ineffectiveness concept for a problem with a magnetic field in an approximate theory of the cyclotron resonance. He determined approximately the current I in a skin layer with the aid of expression (7), introduced the value oeff=I/E,substituted it into eq. (2) and brought the problem to Pippard's scheme with another oeff. Let us use the same scheme to show the existence of a radiofrequency size effect in a plane-parallel plate of thickness d under the following conditions : an electromagnetic wave Eoe-io' falls upon one side of the plate and E,J(Oy; a constant magnetic field is applied parallel to the surface, HllOx; the dispersion law is a square and an isotropic one, cF= p 3 2 m : the inequalities (1) are well satisfied and, besides, O+V%->w.
(20)
The last inequalities mean that for the time of free path an electron makes several revolutions in the orbit, but it is possible to neglect the change of phase of the field in the skin layer for this period of time. These conditions may be practically satisfied if we take >wS107-108 sec-'; H - 102-103 Oe at the free path I - lo-' cm; S cm corresponds to these frequencies. For an electron which passes repeatedly through the skin layer, the last integral in expression (7) is replaced with the geometric progression33
-
Here, the factor
determines the change of the integrand in (7) for every other revolution of /,)' is the time required for passing the electron. The ratio zeff/SZ= P-= along the part of the trajectory within the skin layer. If after passing through References p . 232
194
V. F. GANTMAKHER
[CH.5, 8 2
the skin layer the electron at its first revolution collides with the opposite side of the plate and scatters, only the first member Y2
= Eouyze,la
(23)
will remain from Y,.[We neglect numerical factors of the order of unity in expressions (21) and (23).] Considering - afo/ae as a &function and replacing the integration over zo in accordance with the ineffectiveness concept with multiplication by xeff, we obtain the current in the skin layer
Neglecting the unit compared with l / w we have
I beff= --
Eo6
2
2
= - - (+n
h3 w
- el - + sin 2e,),
where p 1 is the minimum value of the momentum component along the field at which the electron can still move within the plate and
el = arcsin(p,/p,)
= arccos(edH/2cpo).
The differentiation of (24) with respect to H immediately shows that aa,,/aH and, hence, aZ/aH= (aZ/aoeE)(da,,/dH) become infinite at the point H=Ho=2cpo/ed on the left. Ho is the value of the magnetic field at which the diameter of the electron trajectories belonging to the extreme Fermi sphere section becomes equal to the plate thickness. This result may be commented with the use of the following simple considerations. The electrons move along helices with axes parallel to the surface of the metal; the major part of the electron trajectories passes deep in the metal where there is no high-frequency field (6 4D ) ; on returning to the skin layer, the electrons find there a high-frequency field in the same phase as it was during the preceding passage through the skin layer (w4Q). The dependence of the impedance on the field, however, does not have a resonance character, since the condition of constant phase of the electric field for all passages of the electron through the skin layer is fulfilled for all fields. When the field is decreased, the dimension of the electron trajectory increases and the number of returns of the electron through the skin layer during the free path time decreases. However, the electron returns to the skin layer only if the diameter of its trajectory is less than the thickness of the sample ( D I d ) . When the opposite is true ( D 7 d ) the electron is scattered at the surface of References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
195
the plate. A comparatively large number of trajectories has the same value Do determined by the diameter of the Fermi sphere [the function D(pH)has an extremum atpH=O]. Naturally, the function Z ( H ) has a certain singularity for a corresponding value of the field H,. From the theoretical point of view the condition Q<w is of little significance in this case. Replacing (20) with the relation ONOBV
(25)
we find that w in eqs. (21) and (22) becomes equal to 2a w = -( v
n - iw)
and we cannot use the expansion of e-w. Under these conditions the discontinuity of the impedance derivative remains at H = Ho (Ref. 34). Since the denominator (1-e-”) is of resonance character with deep minima at o =n a (n= 1,2, ...), equal to 2 x 4 s2, this discontinuity occurs against the background of the cyclotron resonances. There is one more effect in the frequency region (25). In the case of essentially a non-square dispersion law rn is a function of p H .The electrons participating in the cyclotron resonance are located in the momentum space near the extremal orbit and have the same trajectory size D=2pc/eH. When H is less than HO=2pcled, these resonance trajectories do not pass into the plate and as a result the cyclotron resonances should vanish. This phenomenon, observed experimentally by Khaikin35 who called it “cyclotron resonance cut off”, is the first one in the whole group of radiofrequency size effects discovered later on. Such semi-qualitative calculations as those given above are of relatively small value. It is more important that the ineffectiveness concept determines the general approach to the whole group of problems, which is convenient for this field. Using the terms of this concept we shall show, for example, that in the presence of a magnetic field it is possible to observe the anomalous penetration of the electromagnetic field into a metal, resulting in field and current splashes deep inside the metal. Let us consider a trajectory with one effective point (i.e. a point with u, =0) within the skin layer, and the other points inside the metal. Such trajectories are shown, for example, in Figs. 2 and 3. An electron moving within the skin layer along such a trajectory obtains from the field the energy Ae N Evz,,/s2 [see eq. (23)] as well as the changes of momentum Ap-Ae/o and velocity do-Aplm. For this reason the electron may be considered as a personifiReferences p . 232
196
V. F. GANTMAKHER
[CH.5, p 2
cation of the non-equilibrium addition to the distribution function or, in other words, as a carrier of a part of the current Aj-eAu in the skin layer. As the electron moves along the trajectory, d p and Au somehow change. But at the next effective point of the trajectory, which is located deep in the metal out of the skin layer, the electron will move again parallel to the surface and will reproduce its change of velocity Av and, hence, of the current A j . The value u, showing the displacement of the electron into the depth of the metal while travelling from one effective point to the other, depends on pH. Therefore, different trajectories, which participate in the skin-layer current, have their effective points located at various depths. As a result the secondary current must, generally speaking, be diffused through a considerable depth. But at a depth u,,,, corresponding to the extremum of function u(pH),the number of effective trajectories increases sharply. It is obvious that a sharp increase (a splash) of current and electromagnetic field may also be expected at such depths. These considerations are applicable to the trajectories of various types. In case of a square dispersion law and a magnetic field applied parallel to the metal surface, the splash should occur at a depth Do= 2poc/eH, due to orbits in the vicinity of the central section of the Fermi sphere. Since this splash is a peculiar skin layer for electrons rotating deep in the metal, it, in its turn, causes the next splash at a depth 2D,, etc. As a result, a system of splashes determined by the chain of electron trajectories will appear within the metal. The splashes caused by helical trajectories in the vicinity of the limiting points in a slightly inclined magnetic field are formed in a somewhat different way. They are determined not by a chain of trajectories, but by one group of trajectories passing directly from the metal surface and having many effective points located at various depths (see Fig. 3). The possibility of the existence of splashes was described for the first time by Azbel28 and then the occurrence of splashes under various conditions was studied theoretically and experimentally by Kaner and the author 3'3-38, as well as by Kip and his collaborators 39. Later on we will return to the splashes. Here we want to show only that size effects are tightly connected with the existence of splashes. Let us assume, indeed, for the sake of simplicity that l - d $ 6 . Then in the case of a single-side excitation by an incident wave, the distribution of the field in the plate will coincide in a first approximation with the distribution in a semi-infinite metal. Since ueXtdepends on H, by changing H we may change the depths of the location of splashes. It means References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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that a splash may be brought to the opposite surface of the plate from which the energy will be radiated into space. At certain values of the magnetic field the electromagnetic radiation can penetrate the metal plate. Our considerations based on the “ineffectiveness concept” have already led us to several possibilities for a different size effect occurrence: the cutoff of trajectories, anomalous penetration of the field due to chains of trajectories and due to trajectories with KfO. Realizations of these possibilities are discussed in Sections 3.2-3.5. There is one more possibility related to a small slowly damping component of the field from the ineffective trajectories. (For instance, a helical trajectory is ineffective when Hlln.) Relations between the plate thickness d and the extremal ineffective electron trajectories may also lead to radiofrequency size effects. Such effects are described in Section 3.6.
3.
Various types of radiofrequency sue effects
3.1. METHODS OF DETECTION OF SIZE EFFECTS The experiments on the detection of the radiofrequency size effects may be approached from different sides not only in the figurative, but almost in the literal meaning of this expression. One of the possible approaches was realized by Khaikin35 at frequencies f- 10‘Oc/s. The sample in the form of a disk was used as a wall of the resonator and therefore the characteristics of the sample were studied from the side of the same surface upon which an initial electromagnetic wave was falling. Khaikin35 used a strip resonator ensuring linearity of H.F. currents on the sample surface40. This condition is very important for the study of physical properties of metals. The strip resonators have a comparatively small quality factor and this makes it difficult to use them in common radiospectrometers for the microwave range. This difficulty was overcome by the use of the frequency modulation method for which the generation frequency coincided with the natural resonance frequency and for which the imaginary part of the surface impedance Z = R + iX was measured. The records illustrating the cyclotron resonance cutoff in tin are shown in Fig. 5. Later on the cyclotron resonance cutoff was observed also in bismuth41 and indium42. In all these cases the Fermi surface differs considerably from the sphere and this facilitates the observation of the effect. The experiments with the “single-side geometry” are badly adapted to the registration of the anomalous penetration of the field. To register from the side of the incident wave z =0 the arrival of the splash to the opposite surface z = d , from the latter the “reflected signal” should return. As the electroReferences D. 232
198
[CH.5, 5 3
V. P. GANTMAKHER
magnetic waves cannot propagate within the metal in the usual way, this “signal” should return due to the action of the same splash mechanism with the help of electrons drifting to the surface z=O. For this reason, for the observation of size effects related to the anomalous penetration, it is natural to measure the signal which has passed through the plate. Such a method of studying the size effect was used by Walsh and
0
“$0 0.5
1
1.5
2
. 2.5
Fig. 5. A cutoff of the cyclotron resonances in tin36. Records on tin single crystals of thickness 2 mm (curves I and II, the latter with greater amplification) and of thickness 1 mm (curve 111); T = 3.75 OK,fl: 10 Gc/s, II 1) [loo],HI1 [OOl].
Grimes *3*44. The transmitting coil excited by the radiofrequency generator was placed near one of the sample surfaces and the other similar coil at the opposite surface served as the receiver. The main experiments were carried out at a frequency of 4 Mc/s, but the frequency could be changed within a References p . 232
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RADIOFRBQUENCY SIZE EFECTS IN METALS
199
wide range. Due to the modulation of the constant magnetic field and the use of a lock-in detector, just a part of the signal which passed through the sample could be selected (in principle the same scheme of experiment may be accomplished for the microwave range too 45~46). At the same time there is another approach, which may be used for the observation of both the cutoff effectsand the effects of the anomalous penetration. This approach is based on the double-side excitation of the plate with an electromagnetic wave. The major part of experiments in the field of the radiofrequency size effects was carried out with the help of such methods of excitation. At high frequencies the excitation of this type takes place in those cases when the sample is placed in the centre of the resonant cavity (see Ref. 40 the resonator with a rectangular sample with the size 13 x 6 mm'). Such a resonator was used for the observation of the non-resonant orbit cutoff47 (the impedance jump) for the same group of carriers in Sn for which the cyclotron resonance cutoff had been observed previously. A special thin sample with d = 0.18 mm was prepared for this experiment and as a result the cutoff field was displaced to the region of large fields. The intervals between the cyclotron resonances in this region were wider and the field Ho was just in one of such intervals. To produce the double-side excitation at lower frequencies o lo6- 108sec-' the sample may be placed inside an induction coil L (Refs. 48,49). Including this coil in the tank circuit of a radiofrequency generator, we can use an ordinary nuclear radiospectrometer. When the generator frequency f is measured as the function of the field H (to be more precise when the value af/aH is measured, since the measurements are always carried out with the use of the modulation technique), then
-
where d
1
d,, =-Re (0)
J
H(z)dz~2Red*.
0
When measuring the dependence of the circuit voltage U on the field, we simultaneously determine the changes of the coil quality factor, i.e. we determine the changes in the quantity of energy S dissipated in the sample: aU/aH--aS/aH. Determining the surface impedance of the plate in the case
200
V. F. GANTMAKHER
[CH.
5, 8 3
of double-side excitation as Z = R +iX=(Sn/c)E(O)/H(O) (with an additional factor of 2) and taking into account that E(O)= - E(d),H(0)= H(d), we shall obtain the usual formulae 8n E ( 0 ) 4nw X=-Im=-c H(0) cz
8n R =-Re c
E(0) H(0)
st H ( z ) dz = 4 n o ,a H(0)
,
CZ
32n2
S c2 IH(0)I2'
-= --
Thus, the measurable values ab,,/aH and aS/aH depend on the relation between the fields on the surface E(O)/H(O), i.e. in the long run they depend on the quantity of effective electrons in the skin layer irrespective of the place where these electrons have obtained the energy previously (in the same or in the opposite skin layer). For this reason in the case of the double-side excitation each face of the plate serves simultaneously as the transmitting and receiving one and it is possible to observe all the known radiofrequency size effects.
Fig. 6. A sample placed within a measuring coil; double-side excitation.
Fig. 6 illustrates the practically used disposition of the sample relative to the coil turns having a rectangular cross-section. The disposition of this type ensures the plane polarization of an incident electromagnetic wave with vector E applied along the turns. To change the polarization it is necessary to rotate the sample inside the coil. Homogeneity of the H.F. field amplitude on the sample surface is not usually required. If necessary it may be easily obtained by making the coil length exceed the sample size32.50. All the results we shall discuss were obtained at low frequencies mlo7 sec- i in the case of the double-side excitation. We shall discuss separately effective and ineffective trajectories. Within these two groups the classification of the size effects will be based on the types of the related electron trajectories. References p . 232
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RADIOFREQUENCY SIZE EFFECTS I N METALS
zu1
3.2. CLOSED TRAJECTORIES 3.2.1. Cutoff of trajectories
The records of the quantity af/aH- -dX/aH as a function of the magnetic field are shown for bismuth and indium in Figs. 7 and 8 *. The conditions of the experiments are outlined in the data given in the captions to the figures. When carrying out the experiments it is sufficient to ascertain that the position of the lines being observed does not depend on the frequencyf, but is inversely proportional to the sample thickness. This is the main criterion for the lines of the radiofrequency size effect. Let us turn first to the lines located on the left part of the records. They correspond to that size effect which was discussed in the previous section [eqs. (21)-(24)]: the size of the extremal electron trajectory is equal to d. Surely the presence of two skin layers does not permit us to conclude that this is a pure cutoff effect. If the field H slightly exceeds H,, the electron passes through both skin layers and in both layers it behaves as an effective one. This phenomenon is not taken into account in the expressions (21)-(24). But this is, probably, essential for the intensity and shape of the line only. For this reason we shall call this effect the cutoff of the extremal trajectories. When the field H i s applied parallel to the surface, the value H, is related to the size of the extremal orbit 2p in the direction [nH]by the relationship 2 p = (e/c) d H
(30)
which is obtained by integration of the corresponding component of eq. (9). By virtu.e of the non-quadratic dispersion law the extremal trajectory can have a complex form. Then the value 2p in (30) is its caliper dimension. As it is shown in the figures, the cutoff in indium and bismuth is observed in various ranges of the magnetic field. This reflects the difference in calipers of their Fermi surfaces. RotatingH in the sample plane we change the direction in which the caliper is measured. Using the relation between the line position and the direction of the field, it is possible to determine the form of that part of the Fermi surface which is related to the line. In the bismuth sample, for example, the line in the field Ho at H 11 C2 (curve 1 in Fig. 7) is determined by two electron “ellipsoids”. When the field is rotated, the “ellipsoids” become nonequivalent and the extremal calipers of orbits on them in the direction [nH] become different (curve 2). (For explanations relative to curve 3 see p. 205).
* All the results for In given below were obtained in experiments carried out by the author together with I. P. Krylov. References p . 232
202
[a5. ,o 3
V. F. OANTMAKHER
af aH
0
1
2
3
4
H(0e)
Fig. 7. Records of size effect lines in bismuth for different directions of the magnetic field within the surface plane. n 11 C3; the polarization of the electric field is shown at the scheme of the Fermi “ellipsoids” disposition. d = 1 m, f = 12 Mc/s, T = 1.8’K. Indices at the vectors H and at the extremal cross-sections S and S’ correspond to the number of the curve.
References p. 232
CH.
5 , g 31
203
RADIOFREQUENCY SIZE EFFECTS IN METALS
1
I
I
I
Fig. Records of the size effect lines in indium. nil [OOI], E/l[100], H(1(100), = 5 Mc/s, d = 0.3 mm. Curve 1 :H is parallel to the sample surface. Curve 2: H is inclined at 2’30’. (For explanations relative to curve 2, see p. 212.) Weak lines in double fields are plotted in addition with a magnification of 5 (curves 1’ and 2’). “a g” denotes a size effect line due to a chain of trajectories (see Ref. 53). ~
+
References p . 232
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V. F. GANTMAKHER
[CH.5 ,
3
Eq. (30) is obtained with the assumption that H is applied parallel to the plate surface. Under this condition (30) is valid for the Fermi surface of any shape. But we have another situation when the magnetic field is inclined51. In the simple case of a spherical Fermi surface, the closed trajectory turns together with the field and always remains within the plane perpendicular to H . The field is inclined by an angle cp. The shift of the line relative to the central section may occur, therefore, only due to the change of the effective thickness of the plate (due to the replacement of d with d/cos cp). For a non-spherical Fermi surface, a closed trajectory is not always in the plane perpendicular to H . In this case the average drift of the electron along the field during one-half of the period, while travelling from one side of a plane-parallel plate to the other, may differ from zeio. When the field H is inclined to the surface, the drift of the electron along the field is projected on the normal to the surface n. For this reason, when determining the line position one should take into account the velocity component vHwhich is not included in the vector equation (9). In the case, for example, of the Fermi surface in the form of a cylinder with axis P perpendicular to n, inclination of the field in the (n,P) plane will lead to the line shifting towards the larger fields like H= Ho/cos cp. At small values of cp the shift of the line A H - cp’. A linear dependence of the shift on the angle is also possible. For example, in the case of the cylinder with axis P inclined to the surface at an angle $ and with the field in the (n,P)plane we have, cp being small,
All these formulae are obtained from eq. (9). Their deduction is an excellent exercise for studying the peculiarities of electron motion in the case when the Fermi surface has a complicated shape.
3.2.2. Chains of trajectories Let us return to the field applied parallel to the sample surface. For a complicated Fermi surface several extremal cross-sections and, correspondingly, several cutoff lines may exist for each direction of the field. Sometimes up to seven such extremal sections could be observed in tin49.52. These firstorder lines should not be confused with the lines in the multiple fields H,=nHo ( n = 2 , 3, ...) occurring due to the trajectory chains mentioned above (p. 196). It is such lines that are shown in the right parts of the records in Figs. 7 and 8. These lines are, essentially, another type of size effects caused References p . 232
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5 , s 31
RADIOFREQUENCY SIZE EFFECTS IN METALS
205
by the anomalous penetration of the field and the splashes. As it is shown in the figures, various relations may exist between the amplitudes of the lines of the first (H=Ho) and the second (H=2H0) orders. The same phenomenon was observed also in tin49 and tungsten44. The lines in the multiple fields were obtained for one of two groups of first-order lines with approximately the same intensity and were not obtained for another group. A detailed theoretical analysis of the rate at which the intensity of splashes caused by the chains of extremal trajectories decreases was carried out by Kaner37. It was shown that for a system of slowly damping splashes it is necessary that practically all the electrons participating in producing the skin layer current were “focussed” at the same depth, i.e. that they had the same D with an accuracy up to A D - 6 . We shall illustrate this by some evaluations. Let the Fermi surface be a sphere with the radius p o . Near the central cross-section dDldp, = 0 and
This estimate along with the conditions AD- 6 determines the relative share of electrons which form the splash. Since the part of electrons in the layer p H of the Fermi sphere is approximately equal to the areas ratio p o Ap,/p~=Ap,/p,, the intensity of the splash I at the depth D o is about a=(6/D)+ of the current value near the surface. The next splash weakens again by a times and therefore the damping becomes very rapid: I(n)-a” [the precise calculations37 show that an additional factor is introduced into I(n): I(n)-a”n-*; in the case of a < 1 this factor is insignificant]. Assume now that the Fermi surface is in the form of a cylinder which is parallel to a metal surface. Then all the electrons have one and the same value Do and therefore the current is not spread out in the metal depth (a= 1). It is shown in Ref. 37 that in this case thesplash intensity decreaseslikeZ(n)wn-*, i.e. it decreases slowly. In the case of the Fermi surface consisting of a sphere and a cylinder (or even of two cylinders with different radii) both systems of splashes caused by two extremal diameters should decrease in accordance with the exponentiallaw, though with various values of a. The latter phenomenon may be easily illustrated with the aid of the drawings in Fig. 7. As is known, the Fermi surface in bismuth consists of three electron ‘‘ellipsoids’’ (in our case one of them does not contribute to the radiofrequency conductivity because of the polarization of the vector E ) and a hole-type surface. In the case of field direction H, (curve 1) formation of the splash system is caused by two electron Fermi surfaces and the resolution of this system is References p . 232
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V. F. GANTMAKHER
[CH.5 , Q 3
caused by the hole-type surface. In the case of field direction H 3 (curve 3) only one electron “ellipsoid” participates in the formation of the splash system, while the second “ellipsoid” together with the hole-type surface causes the “dissolving” of the splashes. For this reason in the case of the field direction H3the value a turns out to be smaller and the amplitude of lines decreases more rapidly. When studying the intensity of the lines of radiofrequency size effect, one should bear in mind that D changes with the number of the line D(n)=d/n, and un= (nS/d)*”. This leads to the decrease of the difference between the intensity of successive lines of the size effect in multiple fields. Naturally, in the presence of several extremal diameters the splashesshould also occur and at depths of zi = CniD,. The chain of trajectories, in this case, is composed of various “links”. The line of such a chain is shown in Fig. 8 (line “a+g”; cf. Ref. 53). From the theoretical point of view, the exotic case of the Fermi surface in the form of a cylinder is not the only possibility for the occurrence of weakly damping splashes caused by the chains of the closed trajectories. In Ref. 28 the cyclotron resonance in metals with a non-quadratic spectrum was suggested as the mechanism reducing the contribution of the non-focussed electrons to the conductivity: B depends on pa and therefore only the electrons near the extremal cross-section with ApH satisfying (32) participate in the cyclotron resonance. These electrons absorb the energy from the skin layer B/v times, while the other electrons are not phased-in with the external field. The experimental observation of such a mechanism of the splash occurrence will entail great difficulties. First of all dp,/po in (32) is of the order of (a/v)-* and therefore the expression (32) takes the form Q/v% Do/S%1. It means that very long free paths, strong magnetic fields and high frequencies are required. Then, in the case of a plane-parallel plate the following two conditions are superposed simultaneously on the field H :o =n M and D =d/n‘ (n and n’ are integers). The scheme of a proper experiment would therefore become considerably complicated. Another way of decreasing the splash damping, which may be used for any relationship between w and 8,consists in a slight inclination of the field relative to the surfaces’. Let us turn again to the spherical model of the Fermi surface. If the field is inclined to the surface at an angle cp, only those electrons repeatedly return to the skin layer which have sufficiently small average velocity of drift along the field 0,. The corresponding orbits are near the central closed orbit in the interval ApH/po S/lcp. Assuming that all these effectiveelectronsare focussed
-
References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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at a depth Doin the interval AD < 6, and using the expression (32) we obtain the following condition for cp : cp2 S 6 D 0 / l 2 .
(33)
On the other hand, to ensure the slow damping of splashes it is necessary that these electrons make the main contribution (dp,, .po) (//Do)9p i , i.e. P 3 6/Do.
(34)
It follows from (33) and (34) that when
Di 4 1'6,
(35)
there is a range of inclinations of the magnetic field at which the skin current is determined mainly by electrons near the central section focussed at a depth Do. Under these conditions the splashes damp slowly: I(n)-n-*. The condition (35) is sufficientlyrigid and, besides, as in the case of a cylinder, one and only one extremal diameter is required. Therefore, strictly speaking, the effect of the considerable decrease of the splash damping at the inclined magnetic field suggested in Ref. 37 is not checked experimentally. For indium and tin, however, in a number of cases the increase of the line intensity in multiple fields was observed for inclined fields, in spite of the fact that the experimental conditions corresponded to the relation D: N 61' instead of the expression (35). In a sample of tin, for example, withd=0.5 mmand n )I [OOl], the intensity of a line with Ho=40O Oe (line 13, see Ref. 52) decreased by a factor of 20 after transition to the double field. At an inclination of 3" the line amplitude in the field 2H0 increased by a factor of 2 while the main line remained unchanged. 3.3. HELICAL TRAJECTORIES
3.3.1. Vicinity of limitingpoint While the field H and, hence, the mean velocity G are parallel to the surface, the behaviour of helical trajectories is the same as that of the closed trajectories (see Fig. 9). Assume, for example, that due to the complicated dispersion law we have a non-central orbit with =0 and the extremal caliper in the [ n H ] direction. The line of the size effect caused by such an orbit will not differ from the lines caused by the central orbits (closed trajectories). Incline, then, the magnetic field at an angle q. This leads immediately to the appearance of the projection a, directed along the normal and as a result the depth of the effective points changes. A system of weakly damping References p . 232
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splashes may also occur under such conditions. These splashes are formed owing to the electron trajectories passing directly from the surface. Let us discuss this problem in detail first for the case of helical trajectories in the vicinity of an elliptical limiting point in the field Hinclined to the surface at a small angle.
Fig. 9. A helical trajectory in a magnetic field parallel to the sample surface.
In first approximation, all electrons in this vicinity drift equally along the field during the period To= 2742. This phenomenon is well known in electron optics as “focussing of electrons by a longitudinal magnetic field”. At the limiting point, the velocity uo is parallel to H,Ju,I =uo sin qy muo=K-+ ( K is the Gaussian curvature of the Fermi surface, ~ ( p=)eF at the limiting point). As the average drift of an electron into the interior of the metal during the period To is equal to ulim= IuzTol=2mmLl,/eHY then 2rcc ulim= - K-+sin rp. eH
In the case of a spherical limiting point K of the Fermi sphere, and
-3
=po, i.e. it is equal to the radius
ulim= nDo sin rp,
(37)
where Do= 2poc/eH.Of course the focussing takes place repeatedly after one, two, three or more revolutions in the electron trajectories. Let us now deduce the conditions which should be imposed upon the angle q of the magnetic field inclination to ensure the occurrence of slowly damping splashes of the field deep in the metal. The first condition is quite obvious: ulim% 6 . It means that the splash should leave the skin layer. This condition leads to the following inequality ~p
% S/VOTO.
(38)
The second condition may be formulated in the following way: among the focussing electrons there should be effective ones. The angular interval on the Fermi surface between the limiting point and the effective region is equal References p. 232
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RADIOFREQUENCY SIZE EPWCTS IN METALS
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to p (see Figs. 3 and 10). Consequently the angular size $ of the region occupied on the Fermi surface by focussing electrons should exceed p, i.e. $> p. The magnitude $ is evaluated in the following way: for the deepest splashes which may be reached by electrons directly from the surface ( Z N I sin p- lp), the dispersion of the displacement Au should be less than 6.
\
/
\ 1-
/
Fig. 10. Vicinity of the limiting point 0 at a Fermi sphere in an inclined magnetic field.
Since the electron velocity along the field is u,-u,(l-+$’), lq$’-6, hence $-(6/1p)* and q 4 (S/1)*.
we shall obtain (39)
The conditions (38) and (39) determine the interval of angles p a t which the group of effective electrons will be successively focussed at depths u,=nulim (n= 1, 2, ...). It should be emphasized that the constant a which determines the significance of this group from the viewpoint of contribution to the skin current enters only once into the expression for the amplitude, irrespective of the splash number. The decrease of the amplitude during the transition from n-splash to (n+ 1)-splash is determined only be the length of path A,-aexp (-u,/fp). That is why the focussing at a depth of distant splashes was required in (39). Violation of (39) means that the limiting point is far from the effective region. It has been shown in Ref. 54 that splashes do exist in this case too, but the electrons which form such splashes do not belong to the vicinity of the limiting point. (About forming the splashes when q ~ % (6/1)*, see Section 3.3.3.) References p . 232
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V. F. G M M A K H E R
63
An example of the size effect from the limiting point is shown in Fig. 11. This effect may be easily distinguished as the period AH greatly depends on the inclination angle 2nc AH = -K - 3 9. eH The existence of a line at a certain magnetic field means that the trajectories in the vicinity of the limiting point contain integer revolutions between the surfaces of the plate at this field. It should be noted, that transition from one size effect line to the other is related to the change of the field while the length of the trajectory from one surface of the plate to the other remains unchanged. When passing from one number to the other the amplitude decreases due to the dependence a(k), as well as due to the change of the absolute value of the line width H. The latter is essential for the measurements of a derivative.
I
I
I
I
I
2
3
4
I H(kOe)
Fig. 11. Records of the limiting point size effect lines in indium. The point is near [11 11. 1 1 10111, Ell[111], d = 0.3mm, p = 7"15', f = 1.6 Mc/s.
As it is shown in Fig. 11, the temperature increase, i.e. the decrease of the length of path I, influences only the amplitude of the lines but not the line width. This phenomenon may be easily understood if we try to retrace how the same result follows from the theory [eqs. (11)-(19)]. Remember that the sharp inhomogeneity of the field in the skin layer can References p.
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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be described with the aid of a superposition of monochromatic plane waves whose wave numbers k have a continuous spectrum of width A k - h - ' . The electrons drifting inside the metal interact most effectively with those field-spectrum harmonics whose wavelength il is contained an integer number of times in the distance ulim: ulim= Nil. This condition separates from the continuous spectrum a discrete series of wavelengths which can penetrate to an anomalously large depth into the metal. The interference of these waves gives rise to quasi-periodic field maxima at depths z,=nu,im. The widths of these maxima are determined by the width of the initial wavelength spectrum and therefore they are of the order of 6 . The height of the maxima and their decrease with distance are determined by non monochromaticity of the waves with anomalous deep penetration, i.e. in final analysis by the length of the electron free path 1. 3.3.2. Extremal non-central orbits As was stated above, in the case of a complicated dispersion law the extremal non-central orbits may exist. Let us consider the behaviour of the size effect lines caused by such orbits in an inclined field51. It is necessary, of course, to account again for the drift of electrons along the field. The lines are observed when d=u, where u=sZ-' j::uz dt and z1 and t2 are two effective points of the orbit. When the pitch of the helical line is h g D (h=uHTO= 2n&/Q), we have the following approximate formula for u ua-+ = D cos rp
+ (n - + ) h sin rp,
( n = 0,1,2, ...).
(41)
The factor ( n -4) is equal to a number of revolutions in electron paths from one surface of the plate to the other. The value n = 0 corresponds to the case where the component of the drift velocity uH sin p is opposite to the main motion along the turn and its projection on the z axis is negative. The values n = 1, 2, ... correspond to the positive projections of the velocity (see Fig. 12, where a helical trajectory is once more pictured in the lateral projection). Using (41) at small values of q, we obtain for the line position H,-+=Ho
D
(42)
It is clear from the meaning of the factor ( n - 4 ) that the amplitudes of the first two lines should be of the same order, while the others should decrease rapidly with number due to the increase of the path A from one surface to the other. Thus, the size effect lines caused by the non-central orbits should split as References p . 232
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the field inclines. Such a splitting was experimentally observed in indium51 (see Fig. 8 ; under favorable circumstances the line corresponding to n + 2 is seen also; on the curve of Fig. 8 it is obscured by the line “a+g”). The splitting would exceed the line width only if h a, S 6, i.e. uH should be sufficiently large. This condition is satisfied in indium due to the form of its Fermi surface.
Fig. 12. A helical trajectory in an inclined magnetic field with two effectivepoints through a period.
As always, the radiofrequency size effect may be brought in correspondence with a picture of splashes inside a metal. The helical trajectory, shown in Fig. 9, contributed to both the skin current and to the splash at a depth D. When the condition ha,% 6 becomes satisfied, the splashes u, =An sin a, (n= 1, 2,. ..) split from the skin layer and a set of splashes u, -+ determined by the formula (41) occurs instead of the splash at a depth u= D.The splash amplitude will certainly decrease with the increase of the index along the exponential curve A k a exp ( - A k b ) , where Ak-RnD, ( k = + , l , $ ,...) N
and the factor Q, as previously, determines the contribution to the skin current made by the focussing electrons. Generally speaking, each of these splashes may be initial for a chain and the damping along the chain will be determined by the factor an. When the exponential factor in Ak extinguishes all splashes but aft, then the size effect line in the double field will split into 3, the triple field into 4, etc. (see Fig. 8). References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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3.3.3. The spherical Fermi surface in an inclined magnetic field In an inclined field, two functions D and h depending on p H are contained in the expression for u. In the case of indium, the extremum u(pH)is determined by the extremum &,). However, this is not necessary. The function u(pH) may have an extremum when D(p,) and h(p,) are not extremal if taken separately. In particular this is true for the spherical Fermi surface. Let us determine the orbit position on the sphere with the aid of an angle 8 (see Fig. 13). The value 8=0 corresponds to the central orbit and 8= corresponds to the limiting points. Effective points exist in orbits at 18118,,,., where Om,, =+n: - q ;at f3 = Om, the trajectory has only one effective point through a period (see Fig. 3).
++
Fig. 13. Different orbits on the Fermi sphere in an inclined magnetic field.
In the previous description of the slowly damping splashes caused by the closed trajectories it was assumed that I B D . It gave us the possibility to formulate conditions (34) and (35) under which the contribution to the skin current made by the helical trajectories could be neglected. Let us now consider another case: conditions (34) and (35) are not satisfied. Then all the orbits within the range (-Om,,, Om,,) are equivalent. Their contribution to the skin current is proportional to a number of electrons contained in them. In Fig. 12 a trajectory with 8#8,,, is shown. Sphericity of the Fermi surface permits us to write instead of (41) precise formulae for u, (0) and References p . 232
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[m. 5, § 3
The effective points u, cannot be observed in the size effects of the first order since unO, which have a physical sense for the values 8>0, may have extrema. For example, the function u8 has two extrema with positions determined by the equation x
- CqJl-
q2 =arccos q ,
c = tg-’cp.
(45)
One of them is a minimum which is very shallow for all values of .I Another is a maximum located in the region q < 1/42. It may exceed u+(O) by more than 20% (see Fig. 14). Existence of an extremum of u(0) means that focussing of a group of effective electrons takes place at the depth uext.Hence a splash must arise at that depth, and when d=uexta size effect must also occur. We consider first u=umax.It appears to be a quite peculiar situation. When the conditions (33)-(35) are satisfied, the conductivity is determined mainly by the vicinity of the central orbit. However, when H decreases [i.e. when Do increases and the condition (35) ceases to be fulfilled], another splash should appear and this splash should be followed by the appearance of the second size effect line satisfying the condition d = urn,,. Possibly, just such a structure of the H.F. field within a metal explains the splitting of the lines of the cyclotron resonance in sodium in the presence of an inclined magnetic field55, though the splitting in this case occurs at much smaller tip angles than those which may be expected on the basis of the simple calculations given above. The magnitude umi, practically does not differ from
where the effective point of the boundary effective trajectory with 0 = emaris References p . 232
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21 5
RADIOFREQUENCY SIZE EPPECTS IN METALS
1.'
&
I.(
,\--"\
7bo
\
O.!
0
0.5
9
1.0
Fig. 14. Graphs which illustrate eq. (43) for U , ( n = 1) ;(Y = tg QI.
located. It follows from the comparison of (46) and (37) that if the field is inclined the limiting point splashes convert into the boundary effective trajectory splashes at the depth nu, (n = 1,2,. .,). Subsequentinclination of the field leads to the decrease of the difference umax-umin(see Fig. 14). At q- 20-25 the splashes(and corresponding size effect lines) should amalgate. Their intensity under these circumstances should increase because a focussing of a comparatively wide layer of orbits takes place. A further increase References p. 232
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of the angle p results in deterioration of the focussing and the size effects lines should disappear.
3.4. OPEN
TRAJECTORIES
In the case of open trajectories, the most essential is the direction of the field relative to the crystallographic axes, but not relative to the metal surface, since it may occur that the open trajectories exist only in a small range of angles. When the average electron velocity 5 is directed along the surface, the effect does not differ from that of the closed trajectories. The line position determines the extremal “amplitude of crimping” of the open trajectories. If the angle y between ii and n differs from go”, we have an analogy with the helical trajectories. The pictures shown in Figs. 3, 9 and 12 may be used for a schematic illustration of the behaviour of the open trajectories ifH is assumed to be perpendicular to the picture plane (and y =+z- 9).At small values of y, when the “crimp” of the trajectory is insignificant, or, to be more precise, when we have only one effectivepoint during the period, as it is shown in Fig. 3, the value u of the average drift of electrons inside the metal is the same for all open trajectories and is given by u = (cb/eH)cos y ,
(47)
where b is the period of the open orbit (reciprocal lattice). Thus, the “focussing” in this case is effected automatically due to the dispersion law. The requirement for the presence of effective electrons among the focussing ones means that trajectories having at least one effectivepoint should be contained among the open trajectories (concerning the opposite case, see p. 219). The size effect from the open trajectories was observed in tin 0nly38. To save space we shall not demonstrate here the curves which are given in the original paper. We only note that at y=O (which is the condition of the experiment) the velocity u in the effective points should be perpendicular to the mean velocity 5.
3.5. TRAJECTORIES WITH BREAKS It seems that we have already discussed all possible types of electron orbits. However, when the size effect in indium was studied53, a number of “superfluous” lines was observed (the indium Fermi surface had already been known approximately from both experimental and theoretical investigations). None of the mechanisms discussed above could explain the existence of these ‘‘supeTfluous” lines. The detailed study of the anisotropy of these lines, their behaviour at an inclination as well as the form of the Fermi surface as a whole References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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made it possible to show that the occurrence of these lines was caused by the presence of breaks in the electron trajectories. The example of an orbit with breaks existing in indium is shown in Fig. 15. The occurrence of size effect lines caused by breaks in the electron trajectories indicates the presence of an electromagnetic field not only near the splashes, but also in the intervals between them. To estimate qualitatively
Fig. 15. Fermi surface of indium in the second zone in accordance with the nearly-freeelectron model. Circled part of the orbit a denotes a break; b is a non-central orbit with an extremal caliper in [loo] direction.
this field let us consider the distribution of the electromagnetic field in the depth of a metal with a Fermi surface in the form of a circular cylinder. An electron, which obtained the addition to the velocity Au in the skin layer, passes into a depth along the circular trajectory. If at a depth z this electron moves at an angle K to the surface, the horizontal component of the current produced by this electron is equal to e Au cos K . (The vertical component is compensated by the electrons moving towards the surface along the analogous trajectory at an angle I C = ~ 272 - K . ) Taking into account the fact that the number of electrons belonging to the given trajectory at a depth z is References p . 232
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proportional to l/sin K , we obtain the current produced by these electrons at z B 6 and D - z B d , i.e. far from splashes, equal to j(2) -j(O)
JG cotg K .
(48)
Let us assume now that the trajectory has a break, i.e. the condition D% S is still valid but at the same time within some part of the trajectory at a distance AzS6, the change of the angle K is A K - K . It is clear that in this case the jump of the functionj(z) Will also occur at a distance AzS6. 3.6. TRAJECTORIES OF INEFFECTIVEELECTRONS Let us consider first the magnetic field directed along the normal to the surface56. The closed, helical and open effective trajectories with the effective points located at various depths are certainly possible in the case of a nonspherical Fermi surface in the field perpendicular to the metal surface. These possibilities occur due to the drift of electrons along the field [the component uH(z)]which is absent in eq. (9). Naturally, the radiofrequency size effects similar to those which were discussed above should occur on these trajectories. The size effects of such a type were not yet observed experimentally, though there were works devoted to the study of the cyclotron resonance for such orbits 57*5*. However, there may be no such effective trajectories in a metal. For a spherical Fermi surface, for example, the region of effective orbits, shown in Fig. 13, constricts into a strip whose width has an angular size fl-S/l, as in a zero field. The whole free path of these effective electrons is within the skin layer. The electrons which penetrate into the metal are moving along the helical trajectories at an angle to the field. The ineffective electrons in a zero magnetic field determine the small but deeply penetrating component of the field which damps as e-{/t2, where c=z/Z. In the presence of a field, an ineffective electron also carries information concerning the instantaneous value of the skin layer electric field into the interior. However, as its velocity component uI =(u: uy”)* perpendicular to H rotates in the plane (x,y), the field component e-{/c2 converts into a helical one [we would remind the reader that the conditions (l), (8) and (20)are assumed to be satisfied]. The translation period of this helix may depend only on the extremal values cTo(pH),such, for example, as values u,To in the vicinity of the limiting point. Thus, refusing the requirement for the electron effectiveness,we obtain a harmonic distribution of the field inside a metal instead of splashes and, respectively, a sinusoidal component in the plate impedance instead of narrow size effect lines (see Fig. 16).
+
References p . 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
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Fig. 16. Record of a size effect related to ineffectiveelectrons in tinss. n [I [loo], Ell [OlO], d = 0.96 m,f = 5.2 Mc/s, 4(n, H ) = 17'.
Such a radiofrequency size effect at the normal field has been observed in tin only. As it was mentioned in Ref. 56, there are a number of obscurities in the experimental results. We wil not dwell upon them here. For the magnetic field directed along the surface a similar effect exists owing to electrons in open orbitsbg. If the corresponding trajectories have q#O, but at the same time have no effective points, again a sinusoidal component in the impedance appears instead of sharp size effect lines. Such an effect was detected in cadmium59. 3.7. CONCLUSION Here at the end of this section we present Table 1 summarizing all known size effects in the anomalous skin condition. The following remarks should be made about Table 1. 1) We have discussed the experiments with a plane-parallel plate only, In principle the radiofrequency size effects may occur in samples of other form too. No work dealing with such samples was done up till now. 2) Azbel60 noted that in principle observation of the cutoff of the quantum oscillations in the high-frequency surface impedance is possible. A peculiar mixing of quantum and quasi-classical effects takes place in this case: the quantum oscillationsmay be observed only when the corresponding trajectory fits inside the plate. Since the quantum oscillations appear usually only in fields of several kilooersteds, the single-crystal planeparallel films with a thickness of about 10-4cm are required for observation of such cutoff effects. That is hardly feasible now. References p . 232
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[CH.
5,$ 4
TABLE 1 Cutoff of cyclotron resonances35 Cutoff of non-resonant orbits at high frequencies (w Q) 34* 47 low frequencies (w < Q)32,48*48,58 Chain of trajectories caused by the extremal orbit^^^.^^ at cyclotron resonance28 at cylindrical Fermi surface36 at inclined magnetic field37 Extremal helical trajectories 51 Vicinity of limiting points38-62 Open trajectories with effective points38 Breaks of electron trajectories j3 Ineffective trajectories at HI1 n 5 6 Open ineffective trajectories59 N
4. Shape of line and various experimental factors In the previous section we discussed various cases of occurrence of size effects and the position of the line in the scale of magnetic fields. These problems may be considered to some extent as a first approximation in the solution of the size effect problem. The next approach from this viewpoint consists of the solution to the problem of the line shape. In most of the theoretical works use is made of the first approximation only. The solution of (18) is not simple but feasible, since one of the four integrations concerned is trivial as dfo/d& may be considered as a &function and integration over z and zo may be done with the use of the stationary phase method. However, in the course of transformation (19) from the Fourier representation to the distribution of the field inside a metal, the calculations become so complicated that it is necessary to introduce some simplificationsin which the shape of the line is practically dropped out. Thus, the problem of the line shape is considerably less studied than those problems discussed above. For this reason, the material presented in this section may be considered possibly as a statement of the problem only. We will limit ourselves to a brief description of all the known experimental factors influencing the line shape. 1) Mode of excitation. Undoubtedly the line shape of the size effects should References p . 232
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greatly depend on whether the excitation of the electromagnetic field in the plate is of one- or double-side type (see pp. 197-200 above). At present, however, the experiments in which various modes of excitation are directly compared with one another are not yet known. All experimental results used below belong to the case of double-side excitation. 2) The relation between the resistance Rand the reactance X. Since the position of the line in the size effect is independent of the frequency o,the functions R ( H ) and X ( H ) are not interrelated through differential expressions arising from the Kramers-Kronig relations as in the case of resonance effects. It seems, therefore, that there is no reason to expect the line shape to depend on what function is being studied in the case (the function aR/aH or the function aX/aH). The experiment carried out by Krylov61 showed, however, that such a dependence really existed. In all the experiments the extrema of the function BR/aH within the line width always correspond to the places of the greatest changes of the function aX/aH and vice versa. Thus, the qualitative relations between AR and A X are the same as those, for example, observed within the lines of the nuclear resonance. A good quantitative agreement is observed in some cases toosl. There is, therefore, a basis to assume that the observed relation between AR and A X in the size effect may be caused by some common properties of the equations for the high-frequency current distribution in the metal.
3) Type of size effect and the dispersion law. Under similar conditions the line shape varies of course for the size effects of various types (cf., for example Figs. 8 and 11). The line shape of the limiting point size effect is well reproduced for various limiting points even in different metals. In the case of the cutoff of the closed trajectories it is quite different for different extremal orbits. This is obviously related to the behaviour of the dispersion law in the vicinity of the extremal cross-section. It is possible to put forward the following two obvious influencing mechanisms of the dispersion law. First, the line shape may depend on the character of the extremum of the function D(pH):the measurable caliper may reach a minimum or a maximum on the extremal orbit (see Fig. 17). Second, the shape of the line should depend on the shape of the extremal trajectory in the skin layer, i.e. it should depend on the length of the electron drift in the skin layer J ( R 6 ) related to the value 6 ( R is the radius of the trajectory curvature at the effective point). Presumably the influence of the trajectory form is more important. Let us illustrate this by an example. From the topological viewpoint the References p . 232
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Fermi surface of tin in the fourth zone consists of two planes connected by “tubes”. According to the experimental datass.47~49, their deviation from a cylinder does not exceed several per cent and D reaches the maximum at the central cross-section;the central cross-section of the tube practically does not differ from a square. When the magnetic field is applied along the cylinder
Fig. 17. Different types of the extremal calipers at Fermi surfaces.
axis (crystallographic axis C4),the central cross-section of the cylinder causes a square closed trajectory. Depending on the orientation of the normal n, either side or diagonal of this square may be parallel to the sample surface. In the latter case the trajectory enters the skin layer at an angle. The shape of the line is different in these two cases (see Ref. 49, curves 1, and 1J . 4) Frequency. When choosing the frequency w it is necessary to satisfy the
quasi-static condition (20) to exclude the influence of the relation between D and w upon the shape of the line. The frequency in this case may influence the shape of the line only through the depth of the skin layer, through the ratio 6/d. The dependence of the line width on the frequency in bismuth is shown in Fig. 18 (the same line as that on curve 1 in Fig. 7). It follows from Fig. 18 that the relative width of the line d H / H , is fully determined by the value S/d, at least atf- 3 Mc/s (generallyspeaking, at the maximum frequency studied one can expect an influence of the non-uniformity in the sample thickness). The line presumably stretches to the right of H , up to those values of the field at which the electron trajectory still can brush against both skin layers. Using the distance between two minima as A H , we shall see that at f-3 Mc/s, dH/Ho-26/d=0.25 and hence6- lo-’ cm. At the same time, if we recalculate the result of the measurements of the real part of the impedance at high frequencies with the use of formulae (4), we shall obtain 6cm. Discrepances of the same sign were observed in indium-62 and tin also. These discrepances are obviously related to the non-exponential distribution of the field within the skin layers. 18920: the field considerably References p. 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
12 M C / S
I
223
\ I
I
Fig. 18. The dependence of the width of the size effect line in bismuth on the frequency. The experimental conditions are the same as for the record in Fig. 7, curve 1.
differs from zero at distances exceeding the value derf, evaluated from the impedance with the aid of eq. (28). 5 ) Sample thickness. The possible range of thickness values is determined by the inequalities (1). Practically, however, this range is not fully used due to experimental diffculties which increase rapidly as the sample thickness decreases. Thus, for example, at d = lo-’ cm it is very difficult to obtain a plane-parallel plate, i.e. to obtain the same d along the whole sample with an accuracy of at least several per cent. It: is interesting that for a wedge shaped sample the size effect line does not smear out but splits: two lines corresponding to dmin and d,,, appear instead of one line (see Fig. 19). The position of the limiting point size effect line greatly depends on the angle p [see eq. (40)]. For this reason these lines may split also due to bending of the References p . 232
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tm. 5 , s 4
sample; such a split was observed experimentally in indium. For samples that are free of such defects, the shape of the line does not depend on d if the latter satisfies (1).
I 400
450 H(Oe)
Fig. 19. Splitting of the size effect lines in indium due to a wedge shape of the sample. (1) a size effect line in a plane-parallel shaped sample (d = 0.4 mm, f = 3 Mc/s, n 11 [Oll], Hll[Oil]); (2) the same line in a 5'-wedge shaped sample, (dd/d = 7%); (3) line in the same wedge shaped sample at another direction of the field { a ( H , [Osl]) = So>. In all three cases, His parallel to the sample surface.
6 ) Smoothness of the surface. The influence of the surface smoothness upon the size effectlines may be imagined if we assume that the specular scattering of electrons at the surface occurs under some conditions. The specular scattering may be expected, for example, near the left part of the size effect lines in the region 0 I ( H , - H ) / H oI b/d, where effective electrons reach the surface at very small angles. The shape of the lines, however, remains quite Referencesp. 232
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RADIOFREQUENCY SIZE EFFECTS IN METALS
225
the same when samples with etched surfaces are used instead of samples with cast mirror surfaces. This may be explained either by the complete absence of specular scattering or by the fact that disturbances of the surface smoothness (etch pits) are considerably smaller than the depth of the skin layer. In bismuth the second reason is quite obvious.
7 ) Polarization of the electromagneticfield. Each size effect line in a given sample has its own most advantageous polarization when the electric field is applied along the velocity vector of the electrons at the effective point of the trajectory. Formally the influence of the polarization is related to the fact that the Boltzmann equation (5) as well as eq. (7) contain the electric field in the form of the scalar product E - u. For the limiting point size effect, for example, the most advantageous direction of E is along the projection of H on the sample plane (see Fig. 3). As far as the shape of the line is concerned, it remains unchanged as the electric field turns at least within +60° from the most advantageous polarization. 5. Applications of radiofrequency size effects. One of the possible applications of the size effects actually was already mentioned in the previous section. The size effects presumably may give a valuable information concerning the structure of a high-frequency field in a metal in general, and in the skin layer in particular, in the presence of a constant magnetic field. In the present section we will expound on the other two applications of the size effects: the study of the shape of the Fermi surfaces and the study of the length of the electron free path. SURFACE 5.1. SHAPE OF THE FERMI
The use of size effects makes it possible to obtain various information concerning the Fermi surface. For the sake of convenience we shall divide the following material into several items approximately corresponding to the succession of the discussion of the experimental data. 1) The cutoff effect of the extremal orbit is usually observed when a field is applied parallel to the metal surface. The position of the size effect lines on the scale of the magnetic fields determines, in this case, the calipers of various sections of the Fermi surface. The caliper of the Fermi surface in the [nH] direction, which is of interest to us, is calculated in accordance with eq. (30). The thickness of the sample plate and the intensity of the magnetic field may References p . 232
226
V. P. OANTMAKHER
[CH.5, $ 5
be measured with an accuracy up to one per cent. For this reason the determination of the caliper being measured is limited by the arbitrary determination of the Ho value within the width of the line, i.e. in the final analysis it is limited by insufficient knowledge of the line shape. Formally the H,,value is known with an accuracy up to the line width A, and, therefore, in such metals as indium and tin the error in measurements of the momentum absolute value does not exceed five per cent. Generally speaking this result is not so bad, if we bear in mind that the absolute measurements are usually not so important as the relative ones (the shift of lines while the field is rotated within the sample plane). In the case of bismuth, however, where just the absolute momentum values are of special interest, A,/Ho reaches the value 0.25 due to the considerable thickness of the skin layer. Nevertheless, even in this case the measurements are possible if some additional considerations are taken into account during the determination of the Ho value. As it was shown in Fig. 18, the line asymmetricallywidens as the frequency decreases. Taking into account the mechanism of the line widening discussed in the previous section, we may affirm that the Ho value to be found is determined by the left edge of the line which does not shift when the frequency changes. The Ho values determined with the aid of the left minimum of the lines in bismuth samples with thickness 0.97 mm and 1.20 mm gives the magnitude for the momentum along the binary axis p 1 =0.54 x lo-’’ g -cm/sec. This agrees well with the results of measurements camed out with the aid of other methods41963. Using the position of the line central maximum for Ho, we would obtain the value pi =0.58 x lo-’’ g*cm/sec,which disagrees with Refs. 41 and 63, and is beyond the possible errors of these works. Presumably it is thus possible to affirm that for all types of size effects in the Ho measurements the left (low-field) edge of the size effect line should be used instead of its centre, which has been used in Refs. 49, 52 and 53. It is interesting to note that such a correction removes a three per cent discrepance noted in Ref. 38 between the measurable period AH on the plots of the size effect from the open trajectories and the period calculated by the aid of eq. (47) with the reciprocal vector & known from crystallography. 2) Rotating a constant magnetic field within the sample plane, we may obtain the angular dependence of the value Ho and hence that of the value p with the aid of the relationship (30). This gives us the information on the shape of the corresponding section of the Fermi surface. In some specially symmetrical cases the dependence of p on an angle of turning x, constructed in References p. 232
CH.
5 , s 51
RADIOFREQUENCY SIZE EFFECTS IN METALS
227
polar coordinates, is the central section of the Fermi surface of a plane normal to n. The extremal orbits, however, on which the size effect is observed, may show a variety of shapes and the surfaces themselves may be differently turned relative to n. For this reason, in the general case p determines the caliper (in the direction x++n) of the shaded projection of the Fermi surface on the plane perpendicular to n (see Fig. 20). Therefore, strictly speaking the interpretation of the experimental data is usually not unique ;some additional considerationssuch as the nearly-free-electronmodel would be required for the reconstruction of the Fermi surface. 3) The behaviour of the lines in an inclined magnetic field may give additional information, which is very useful for the interpretation of the experimental data. The linear shift of the line with a tip angle growth shows, for example [according to eq. (31)], that the Fermi surface is inclined relative to the sample surface. This phenomenon was used to study the Fermi surface
Fig. 20. A measurement of a caliper of a Fermi surface which has a complicated form.
References p. 232
228
V. F. GANTMAKHER
[CH.5,
85
in indium. According to the nearly-free-electron model, the Fermi surface in indium in the third zone should consist of tubes a ong the edges of the Brillouin zone. The study of the behaviour of the lines at the inclinat on made it possible to separate the lines on the tubes from those connected with other sections of the Fermi surface, and to determine in each particular case the position of the tube in the reciprocal lattice. The split of the size effect lines at the field inclination may be interpreted in two different ways. There may be, for example, two sections of the Fermi surface (e.g., two tubes) which are symmetrically inclined relative to n ; such a situation was observed in indium (see Ref. 53, line g2). On the other hand, the split may indicate that the orbit being observed is not the central one [see eqs. (41) and (42)]. In the case of indium such orbits are present in the second zone (see Fig. 15). The behaviour of the appropriate size effect lines was previously illustrated in Fig. 8.
4) The size effect from breaks of the electron trajectories (of course, ifitis possible to show that the lines are really connected with the breaks) indicates the presence of sharp edges on the Fermi surface or, to be more precise, the presence of edges with a bend radius p5(G/d)po. In the case of indium this upper limit for p/poN 3%. It is interesting how often such sharp edges occur in the Fermi surfaces. It is clear now that the same effect caused by breaks of the electron trajectory had been observed previously in tin also on the square trajectory mentioned above when the diagonal of the square was parallel to the sample surface (Ref. 49, HIIC4, nll[l lo], curve 4J. 5 ) The presence of the open trajectories may also be shown with the aid of size effects. From the outward appearance of the plots Z ( H )some conclusions may be drawn concerning the form of the open trajectories, in particular concerning the existence of the effective points in them.
6 ) The size effect from the limiting points makes it possible to measure the curvature at certain points of the Fermi surface. In comparison with the measurements of calipers of the extremal orbits, we have here an additional source of errors which is an absolute value of the angle q of the field inclination contained in (40). Since only the changes of the angle of inclination A q may be measured experimentally, an absolute value of the angle q should be obtained by linear extrapolation to dH(cp)=O. It is assumed that the radius of curvature K - * does not depend on cp. If the angular region of the existence of the limiting point is sufficiently large, the curves with opposite References p. 232
CH. 5 , $ 51
RADIOFREQUENCY SIZE EFFECTS IN METALS
229
cp signs may be used to increase the accuracy of the determination of the absolute value of the angle cp.
7) The impedance oscillationsin the perpendicular magnetic field may be used for detecting helical trajectories with extremal displacement along the field56. OF THE ELECTRON FREE PATH 5.2. LENGTH
From the viewpoint of the possibilities of studying the length of the electron free path in a metal, various types of size effects are probably not of one and the same value. The effect from the extremal orbits in a magnetic field applied parallel to the surface (the cutoff size effect) involves several changes of the number ,u of the electron passage piercing the skin layer: to the right of the line at H>H,, p,-Z/nd, to the left of the line ,u= 1 and within the width of the line in the case of double-side excitation p-2Z/nd. This makes the dependence of the line amplitude on Z/d more complicated, especially as the condition 1% d practically cannot be satisfied in experiments. The size effects related to the drift of a group of electrons from one skin layer to the other, i.e. effects caused by the vicinity of the limiting point, extremal non-central trajectories and open trajectories, are more convenient for the measurements of 1. As it was stated above, the line amplitude of these effects contains the factor e-’’’, where A is the length of the path from one skin layer to the other. For a limiting point trajectory A =d/sin p, for an extremal non-central helical trajectory A , ~ ( n - + J n d ( n = O 1, , 2...) [see eq. (41)] and for an open trajectory A=kd/cos y, where the factor k is used for taking into account the depth of “crimping” of the open trajectory. Such a simple dependence on 1 is a consequence of the geometry of the experiment: the electron may pass through the second “receiving” skin layer only once even at I= 00. The line amplitude is proportional to the probability that the electron is not scattered in the path A. (Strictly speaking the effect caused by breaks of the trajectories also belongs to this group of effects, since in this case the electron may only once pass through the second skin layer.) The mean free path Z may be presented in the form
1/1 = 1/20
+ 1/lph,
(49)
where ,Z is the free path at T=O and I,,, is the free path connected with the electron-phonon collisions. The most convenient method of measuring the value lo gives us the limiting point size effect: by changing the tip angle cp we change A. A simple mathematical treatment of the dependence of the amplitude A(cp) measured at approximately the same values of H makes it References p . 232
230
[CH.5,
Y. F. GANTMAKHER
85
possible to obtain at once lo. Such measurements were carried out in tin for two different limiting points with the use of the same sample and in the same region of directions of the magnetic field H (Ref. 38). The ratio of the lo values was approximately four and the obtained values of lo were compared with effective masses and Fermi velocities of electrons at the limiting points. The measurements of the average value of lo along the extremal noncentral orbits require the comparison of line amplitudes corresponding to different A, at a certain direction of the magnetic field. Such measurements were performed in Ref. 51. Measurements of I,, on open trajectories require changes of samples, which is less convenient and gives lower accuracy. Turning to the problem of the temperature dependence of the free path, it is necessary to emphasize first the principal difference between collisions of electrons with static defects and electron-phonon collisions at low temperatures. This difference becomes important under anomalous skin effect conditions 27. Electron scattering on static imperfections of the lattice causes deflections through large angles as often as deflections through small anglesS - the scattering is an isotropic one. For this reason a single act of scattering exerts the same influence upon both the static conductivity and the size effect. On the other hand the interaction of an electron with vibrations of the lattice is influenced by the fact that the electron may emit or absorb a phonon. When the temperature T < 8 D (8, is the Debye temperature) the electron will scatter only through the small angle
b
Pph/PO
T/8D
9
where pphis the average absolute value of the phonon momentum at the temperature T and p o is the curvature radius of the Fermi surface. The effectiveness of such a scattering varies from case to case. In the static conductivity, (T/OD)' collisions are required for an essential scattering of the electron by the phonons, since only after such a number of collisions the electron scatters through the angle of unit order, As a result the supplementary factor of T2is introduced into the temperature dependence of the static resistance p (as it is known, p T 5and the factor of T 3 is determined, roughly speaking, by the dependence of the number of phonons on 2"). In the case of size effects, the scattering through a small angle may be sufficient for the conversion of the electron into an ineffective one due to the small skin layer thickness S. The value of this angle q and hence the value Iph depend on the type of effect, the relation S/d, and even on the value of the magnetic field. Let us estimate, for example, q for the case of the limiting point64. The path L which an effective electron may travel in the skin layer amounts
-
References p . 232
a. 5 , § 51
RADIOFREQUENCY SIZE EFFECTS IN METALS
23I
approximately to L= S+d*v-'n-* [see Fig. 3 andp. 911; n ( H ) is the number of the line showing the number of revolutions made by the electrons. A considerable reduction in this path occurs in the case of scattering through the angle 6/L. Thus, 'I = S/L = (nS/d)*. (50) cm (and 6 - lo-' cm In actual experiments carried out int in with derrcm, I- lo-' and n = 2 , 'Ievaluated from the line width), d=4 x was obtained. At the same time, at helium temperatures the interaction of an electron with a phonon in tin deflects the electron through an angle of the order of The difference between 'I and this angle of deflection amounting to one order of magnitude is certainly small, but, as it may be considered from these estimates, even a single collision is sufficient for making an electron ineffective. For this reason the dependence of jph on the temperature should be proportional in this case to T 3, but not to T ', and this phenomenon was observed experimentally64. As it is shown in (50), a dependence fph(H)is also possible. Since at large values of n (i.e. in large fields) 'I increases, then at n & 1 several collisions may be required to make an electron ineffective. This problem, however, is not yet studied sufficiently. Some attempts were made also to apply the effect of the cutoff type to the study of the temperature dependence I ( r ) (see Refs. 32 and 64). These attempts proved not to be very successful, probably due to the difficulties mentioned above in the interpretation of the results. In both experiments, however, in tin and gallium, various temperature dependences of the amplitude were observed for various lines and this indicates the dependence of Iph on the position of the extremal orbit on the Fermi surface. Acknowledgements The author is very much indebted to Prof. Yu.V. Sharvin, Dr. 8. A. Kaner and Mr. I. P. Krylov for valuable discussions while writing this article.
4
Note added in proof. Since the article was written a number of new works65-69 have been done in the field of radiofrequency size effects. We shall restrict ourselves here to enumeration of new results : Ref. 65-a RF size effect from a chain of trajectories was observed in rubidium. Inclination of the magnetic field amplified the lines for which the conditions (33)-(35) were satisfied. So the mechanism of a weak damping chain predicted by Kaner 37 was detected experimentally. References p . 232
232
V. P. GA-
[CH.
5
Ref. 66-a RF size effect was observed in potassium. Inclination of the magnetic field led to a line shift toward higher fields just in accordance with eqs. (43)-(45) and Fig. 14. At tip angles ( ~ 2 2 5 the ” size effect lines disappeared. Ref. 67 - the shape of the limiting point size effect line was computed. The results are in good agreement with experimental curves62. Ref. 68 - a RF size effect was observed in cadmium. It was used for an investigation of the Fermi surface. Ref. 69 - a RF size effect was observed in ahminiurn. Thus R F size effects of different types have been observed so far in Alas, Bi50, Cd59.68, Ga32, In53,62, K66, Rb85, Sn38,49 and W44.
REFERENCES REPORTS AND MONOGRAPHS 1
2
5
7
8
I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 69, 419 (1959) [English transl.: Soviet Phys.-Usp. 2, 831 (1960)l. I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 78, 411 (1962) [English transl.: Soviet Phys.-Usp. 5, 878 (1963)l. I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 87, 389 (1965) [English transl.: Soviet Phys.-Usp. 8, 805 (1966)l. E. H. Sondheimer, Advan. Phys. 1, 1 (1952). A. B. Pippard, Advan. Electron. Electron Phys. 6, 1 (1954). A. B. Pippard, Rept. Progr. Phys. 23, 176 (1960). M.Ya. Azbel’ and I. M. Lifshitz, Progress in Low Temperature Physics, Vol. 3, Ed. C.J. Gorter (North-Holland Publishing Co., Amsterdam, 1961) p. 288. R. E. Peierls, Quantum Theory of Solids (Oxford at the Clarendon Press, 1955). ORIGINAL PAPERS
K. Fuchs, Proc. Cambridge Phil. SOC.34, 100 (1938). E. R. Andrew, Proc. Phys. SOC. (London) A 62,77 (1949). D. K.C.MacDonald, Proc. Phys. SOC.(London) A 63,290 (1950). l2 E. H. Sondheimer, Phys. Rev. 80,401 (1950). 13 J. Babiskin and P. G. Siebenmann, Phys. Rev. 107, 1249 (1957). l4 N. H. Zebouni, R. E. Hamburg and H. J. Mackey, Phys. Rev. Letters 11, 260 (1963). 15 Yu. V. Sharvin and L. M. Fisher, Zh. Eksperim. i Teor. Fiz. Pis’ma 1 (5), 54 (1965) [English transl.: JETP Letters 1, 152 (1965)). A. B. Pippard, Proc. Roy. SOC.(London) A 191,385 (1947). l7 A. B. Pippard, Proc. Roy. Soc. (London) A 224,273 (1954). 1 8 G.E. H. Reuter and E. H. Sondheimer, Proc. Roy. SOC.(London) A 195, 336 (1948). 19 R. G.Chambers, Proc. Phys. SOC.(London) A 65,458 (1952). 2 0 A. B. Pippard, G. E. H. Reuter and E. H. Sondheimer, Phys. Rev. 73,920 (1948). lo
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A. B. Pippard, Proc. Roy. SOC.(London) A 203,98 (1950). R. G. Chambers, Proc. Roy. SOC.(London) A 215,481 (1952). 23 G. E. Smith, Phys. Rev. 115, 1561 (1959). 24 P. N. Dheer, Proc. Roy. SOC.(London) A 260, 333 (1961). 25 E. W. Johnson and H. H. Johnson, J. Appl. Phys. 36, 1286 (1965). 26 M. Ya. Azbel’ and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 32, 896 (1957) [English trans].: Soviet Phys.-JETP 5,730 (1957)l. 27 M. Ya. Azbel’ and E. A. Kaner, Phys. Chern. Solids 6, 113 (1957). 2 8 M. Ya. Azbel’, Zh. Eksperim. i Teor. Fiz. 39,400 (1960) [Englishtransl.: Soviet Phys.JETP 12,283 (1961)l. I. M. Lifshitz, M. Ya. Azbel’ and M. 1. Kaganov, Zh. Eksperirn. i Teor. Fiz. 31, 63 (1956) [English trans].: Soviet Phys.-JETP 4, 41 (1957)l. 3 0 M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 39,212 (1960) [English transl.: Soviet Phys.JETP 12, 152 (1961)l. 3 1 V. F. Gantmakher and Yu. V. Sharvin, Zh. Eksperim. i Teor. Fiz. 39,512 (1960) [English trans].: Soviet Phys.-JETP 12, 358 (1961)l. 32 J. F. Cochran and C. A. Shiffrnan, Phys. Rev. 140, A 1678 (1965). 33 V. Heine, Phys. Rev. 107,431 (1957). 34 8. A. Kaner, Dokl. Akad. Nauk SSSR 119, 471 (1958) [English trans].: Soviet Phys.Doklady 3, 314 (1958)l. 35 M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 41, 1773 (1961) [English trans].: Soviet Phys.JETP 14, 1260 (1962)l. 38 V. F. GantrnaKher, Zh. Eksperirn. i Teor. Fiz. 43, 345 (1962) [English trans].: Soviet Phys.-JETP 16, 247 (1963)l. 37 8. A. Kaner, Zh. Eksperirn. i Teor. Fiz. 44, 1036 (1963) [English trans].: Soviet Phys.JETP 17, 700 (1963)l. 3 8 V. F. Gantrnakher and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 45,1430 (1963) [English transl.: Soviet Phys.-JETP 18, 988 (1964)l. 39 C. C. Grimes, A. F. Kip, F. Spong, R. A. Stradling and P. Pincus, Phys. Rev. Letters 11,455 (1963). 40 M. S. Khaikin, Pribory i Tekhn. Eksperim. 3, 95 (1961) [English trans].: Instr. Exptl. Tech. (USSR) (1962)l. 4 1 M. S. Khaikin and V. S. Edel’man, Zh. Eksperirn. i Teor. Fiz. 47, 878 (1964) [English trans].: Soviet Phys.-JETP 20, 587 (1965)l. a2 R. T. Mina and M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 48, 111 (1965) [English trans].: Soviet Phys.-JETP 21, 72 (1965)l. 43 W. M. Walsh, Jr. and C. C. Grimes, Phys. Rev. Letters 13, 523 (1964). 44 W. M. Walsh, Jr., C. C. Grimes, G. Adams and L. W. Rupp, Jr., Proc. IXth Intern Conf. LowTemp.Phys.,Columbus,Ohio, 1964(PlenumPress,NewYork, 1965)part B, p. 765. 45 R.B. Lewis and T. R. Garver, Phys. Rev. Letters 12, 693 (1964). 46 S. Schultz and C. Latham, Phys. Rev. Letters 15, 148 (1965). 47 M. S. Khatkin, Zh. Eksperim. i Teor. Fiz. 43, 59 (1962) [English transl.: Soviet Phys.JETP 16, 42 (1963)l. 4 8 V. F. Gantrnakher, Zh. Eksperirn. i Teor. Fiz. 42, 1416 (1962) [English transl.: Soviet Phys.-JETP 15, 982 (1962)]. 4 Q V. F. Gantmakher, Zh. Eksperirn. i Teor. Fiz. 44, 811 (1963) English transl.: Soviet Phys.-JETP 17, 549 (1963). 21
22
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V. F. Gantmakher and I. P. Krylov, Zh. Eksperim. i Teor. Fiz. 47,2111 (1964) [English transl.: Soviet Phys.-JETP 20, 1418 (1965)l. 5z V. F. Gantmakher, Zh. Eksperim. i Teor. Fiz. 46, 2028 (1964)[English transl.: Soviet PhyS.-JETP 19, 1366 (1964)l. 53 V. F. Gantmakher and I. P. Krylov, Zh. Eksperim. i Teor. Fiz. 49,1054 (1965)[English transl.: Soviet Phys.-JETP 22, 734 (1966)l. 54 8.A. Kaner and V. L. Fal'ko, Zh.Eksperim. i Teor. Fiz. 49, 1895 (1965) [English transl.: Soviet Phys.-JETP 22, 1294 (196Q.l 55 C.C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 5 8 V. F. Gantmakher and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 48,1572 (1965) [English transl.: Soviet Phys.-JETP 21, 1053 (1965)l. 5 7 J. F. Koch and A. F. Kip, Phys. Rev. Letters 8,473 (1962). W.M. Walsh, Jr., Phys. Rev. Letters 12, 161 (1964). 5 9 A. A. Maryakhin and V. P. Nabereshnykh, Zh. Eksperim. i Teor. Fiz. Pis'ma 3, 205 (1966) [English transl.: JETP Letters 3, 130 (1966)l. 'O M. Ya. Azbel', Phys. Chem. Solids 7, 105 (1958). I. P. KryIov, Zh.Eksperim. i Teor. Fiz. Pis'ma 1,4, 24 (1965)[English transl.: JETP Letters 1, 116 (1965)l. *2 I. P. Krylov and V. F. Gantmakher, Zh. Eksperim. i Teor.Fit. 51, 740 (1966)[English transl.: Soviet Phys.JETP 24 (1967)l. 83 A. P. Korolyuk, Zh. Eksperim. i Teor. Fiz. 49, 1009 (1965) [English transl.: Soviet PhyS.-JETP 22, 701 (1966)l. 84 V. F. Gantmakher and Yu.V. Sharvin, Zh.Eksperim. i Teor. Fiz. 48,1077 (1965)[English transl.: Soviet Phys.-JETP 21,720 (1965)l. 85 P. S. Percy and W. M. Walsh, Jr., Phys. Rev. Letters 17, 741 (1966). 66 J. F. Koch and T. K. Wagner, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. 67 E. A. Kaner and V. L. Fal'ko, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. V. P. Nabereshnykh and A. A. Maryakhin, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. Og J. F. Koch and T.K. Wagner, Bull. Am. Phys. Soc. 11, 170 (1966). 51
CHAPTER VI
MAGNETIC BREAKDOWN IN METALS' BY
R. W. STARK* AND L. M. FALICOV* DEPARTMENT OF PHYSICS AND INSTITUTE FOR THE STUDY OF METALS, UNIVERSITY OF CHICAGO,CHICAGO, ILLINOIS AND
CAVENDISH LABORATORY, UNIVERSITY OF CAMBRIDGE, CAMBRIDGE, ENGLAND CONTENTS: 1. Introduction, 235. - 2. The theory of coupled orbits, 244. - 3. Analysis of experimental results, 265.
1. Introduction
The concept of magnetic breakdown (MB) was first proposed by Cohen and Falicov to explain the very high frequency ascillation (the so-called giant orbit) observed by Priestley2.3 in his pulse-field De Haas-Van Alphen (DHVA) investigation of magnesium. Since then several authors have discussed extensions of the theory dealing with both the fundamental physical aspects of MB4-12 and its effects on experimentally measurable quantities such as the galvanomagnetic properties 13-16, the DHVA effect17-18, ultrasonic attenuation l9, etc. Experimentally, MB effects have at present been observed in several metals including magnesium 20-23, zinc22-33, cadmium 34-38, beryllium 37, white tin 38, aluminum 39-41, gallium42243, thallium44-47, and [email protected] one must anticipate that the effects of MB are very likely to appear in experiments on the electronic band structure of many metals. The purpose of the present contribution is to provide a detailed and unified review of the current status of the experimental and theoretical work on the subject. 2 9 3 9
t Work supported in part by the National Science Foundation, the U.S.Office of Naval Research, the U.S. Army Research Office (Durham) and the Advanced Research Projects Agency. * Alfred P. Sloan Research Fellows.
References p . 285
235
236
1.1.
[CH.6,s 1
R. W. STARK AND L. M. FALICOV
PSEUDOPOTENTIALS AND THE NEARLY-FREE-ELECTRON MODEL
We begin by considering some fundamental ideas of the one-electron theory of metals. A free atom of a metal-forming element can be considered, for our purposes, to be made up of a compact, tightly bound ion core surrounded by a few loosely bound valence electrons. The electronic states of the ion core remain essentially unchanged when an ensemble of free atoms condenses to form a periodic crystal lattice. The electronic states of the valence electrons, on the other hand, overlap with the valence electrons of neighboring sites to such an extent that the new wave functions for the crystal show little resemblance to those of the free atom. A new Schrodinger equation for the crystal
should be set up and solved. The wave functions $B are now Bloch states and V ( v ) is the periodic potential due to the ion cores and the self-consistent field of all the electrons. If an arbitrary potential probe is passed through the crystal, V ( r ) will appear as a rapidly varying function with a strong attractive part close to the ion cores; V ( r )becomes infinite at each lattice site. A schematic representation is shown in Fig. la. The valence states t+hB, however, are not arbitrary functions; they should be orthogonal to the electronic states t,hc of the ion cores. Thus, because of this additional constraint, the valence electrons occupying $B form very special probes which, in fact, sample the potential V ( r ) in a completely different way. The net result can be represented by an effective or “pseudo” potential V ’ ( r ) which is much weaker than V ( r ) and whose variations are also much smoother. Such a pseudopotential is schematically shown in Fig. Ib. In practice V(Y)is such that its Fourier expansion (1 3
V’(r)=xV,expiG-v G
converges rapidly, and a very adequate representation can be obtained by a truncation of the series after only a few terms. ln fact, a reasonable first approximation to the electronic structure of some simple metals can be obtained by keeping only the leading constant term V ’ ( r ) = V,. This, of course, leads to a spherically symmetric distribution in k-space with plane wave solutions of (1.1): $Br
= S2-*exp (iker),
where i2 is the volume of the crystal. References p. 285
A2k2 E(k) = - V , , 2m
+
(1.3)
CH. 6,I
11
Fig. 1.
A schematic representation of: (a) the lattice potential, (b) the pseudopotential.
231
MAGNETIC BREAKDOWN IN METALS
A better approximation to the electronic structure of a real metal (the socalled nearly-free-electron approximation) arises from the fact that the typical De Broglie wave length of a metallic electron near the top of the distribution 2n 2nh 2nh 1, = - = - = (1.4) kF PF J ( 2 m G ~
(where E , is the Fermi energy) is of the same size as the typical spacings of the crystal lattice. Hence, even though the pseudopotential is very smooth and its Fourier components small, diffraction in particular is of paramount importance in determining the electronic structure. An electron of wave vector k impinging upon a crystal lattice will be coherently scattered from a set of crystallographic planes defined by a reciprocal lattice vector G, provided the Bragg diffraction condition
2k.G
f
G - G= 0
(1.5)
is satisfied. This condition is shown schematically in Fig. 2. The primary effect of the Bragg diffraction is the mixing of the states k and k G, which in
+
References p . 285
238
R. W. STARK AND L. M. FALICOV
f
[CH.6 , 8 1
tky
Fig. 2. Bragg diffraction of an electronic wave of vector k by a set of lattice planes of reciprocal lattice vector G.
turn results in an energy discontinuity or gap in the energy spectrum &(k) for values of k satisfying (1.5). The spherically symmetric distribution in k-space which resulted from (1.3) is now broken into a series of zones whose basic symmetry is that of the crystal, with ~ ( kexhibiting ) the well-known band structure. This has a profound effect on the dynamical behavior of the electrons, even for those cases in which the magnitude of the energy discontinuity is very small.
1.2. DYNAMICS OF THE ELECTRONIC MOTION IN
A METAL
Throughout the remainder of this article we will be primarily concerned with the dynamical response of the conduction (valence) electrons to applied external electric and magnetic fields. It has been repeatedly shown319s2 that in the presence of a periodic lattice potential, the dynamics of an electron are governed by the equation
where E and H are the external electric and magnetic fields respectively, and u p is the group velocity of the electron state defined by the wave vector k 1 grad k & ( k ) . A
l+ =-
References p . 285
(1.7)
CH. 6 , 8 11
MAGNETIC BREAKDOWN
IN METALS
239
By convention the magnetic field is chosen to define the z-direction, i.e.,
H
= (O,O,H).
(1.8)
Let us consider first the case E=O. Eq. (1.6) simplifies to
from which two constants of the motion are immediately obtained: dk, = k, dt = 0 , (2) dEk =gradk &,.dk = h k * k dt = 0
(1)
'
(1.10)
i.e., k, and &k are constant. Thus the electron trajectories in k-space are defined by the intersections of the planes perpendicular to H with the surfaces of constant energy. Due to the nature of the Fermi-Dirac statistics, which the electrons obey, we are interested in the constant energy surfaces only in the neighborhood of the Fermi energy, &k=EF. If the velocity vk in (1.9) is rewritten as i , the resulting equation can be integrated to obtain:
k, = - a ( y - y o ) ,
k, = a(x - xo),
(1.11)
where (1.12)
Eq. (1.1 1) shows that the trajectories of the electrons in r-space normal to the direction of the magnetic field are equal to the trajectories in k-space after a rotation by in and a multiplication by a scale factor p. One point is worth clarifying at this stage. The periodicity of the lattice potential induces a periodicity in k-space ; consequently a given vector k is only defined within a unit cell of reciprocal space, the first Brillouin zone, for instance. This implies that two vectors k and k + G separated by a reciprocal lattice vector G should be considered identical. Therefore, the representation of the surfaces ~ ( k=)constant is not unique and three possible schemes can be chosen51: the extended zone scheme, the reduced zone scheme, and the repeated zone scheme. Each of these has its advantages and serves to illustrate a given point more clearly. In order to avoid confusion in going from one representation to another, we shall only use the extended zone scheme in this paper. This choice makes some trajectories in k-space appear discontinuous ; it should be remembered that this is not so and that the discontinuities are only due to the representation and would disappear in one or both of the other two schemes. References p . 285
240
R. W.STARK
AND
L. M. FALICOV
[CH.
6, 8 1
As an example that we shall use to illustrate the properties of MB throughout this article, let us consider a hypothetical metal with low symmetry such that its unperturbed Fermi surface is given by the spherical slab (1.13)
Ikzl Ikzm < k,
9
and such that it intersects only one pair of zone boundaries, (1.14) where Bragg diffraction can take place. Such a model is depicted in Fig. 3.
t
ky
IA Fig. 3. A simple model for a metal whose Fermi surface intersects only one zone boundary. The diagram shows the section at k , = 0.
Restricting ourselves for the time being to electrons with k,=O and magnetic fields parallel to the z-axis, it is easy to see that for non-vanishing potentials (V,, # 0) two types of electron trajectories are possible : those which under the influence of the magnetic field and Bragg diffraction have a net drift velocity along the x-axis and describe an open trajectory (Fig. 4a) and those with zero net velocity which close back upon themselves (Fig. 4b). If, however, V,, =0, no Bragg diffraction will take place at the points indicated 1 and 2 in Fig. 3, since in this case no interaction between electron and periodic lattice is operative. Consequently only one trajectory is possible, namely the free-electron orbit shown in Fig. 4c. It is evident that the dynamics of the electron motion on the trajectories (a) and (b) of Fig. 4 are quite different from the dynamics on trajectory (c). References p. 285
CH. 6, I 11
MAGNETIC BREAKDOWN IN METALS
241
However, the former result from a “segmenting” of the latter via the Bragg diffraction which takes place as the periodic part of the lattice potential is ideally “switched on”. If the electron-lattice interaction is very weak, i.e., if V,, is very small compared with some other energies in the system, we should question the validity of the dynamics as expressed by eqs. (1.7) and (1.9) and
Fig. 4. Possible trajectories in the x-y plane: (a) an open orbit, (b) closed orbit, (c) the free-electron orbit when the lattice potential is made to vanish.
Figs. 4a and 4b. It is intuitively obvious that an intermediate regime should exist such that an electron has a probability greater than zero but less than one of being Bragg diffracted at points 1 and 2 of Fig. 3. This effect, which should depend on the magnetic field strength H, is called magnetic breakdown. 1.3. A DIFFRACTION APPROACH
TO MAGNETIC BREAKDOWN
The following “gedanken” experiments illustrate the physical processes involved in MB. Suppose that a metallic single crystal slab of thickness d of our hypothetical metal is cut such that the crystallographic planes corresponding
,G
Fig. 5. (a) A single crystal slab of thickness d is bombarded by a beam of electrons of wave vector k which satisfies the Bragg condition, (b) the current reaching the detector D, normalized to the incident current l o , as a function of thickness d. References p . 285
242
R. W. STARK AND L. M. FALICOV
[CH.6 , § 1
to the reciprocal lattice vector GIlie parallel to the surface of the slab, i.e., GIis normal to the slab as shown in Fig. 5a. A beam of electrons of welldefined wave vector k impinges upon the crystal. The wave vector k is chosen to satisfy the Bragg reflection condition (1.5) ;in particular it is chosen to be the vector k,. shown in Fig. 3. If we ignore surface effects which would partially reflect and scatter the electron beam, and look only at the effects of Bragg diffraction, we know that the intensity of the transmitted beam should decrease exponentially with the thickness of the slab as shown in Fig. 5b. Consequently the probability P which a given electron in the beam has of reaching the detector D of Fig. 5a is given by P = exp [ - d/lsin O,] ,
(1.15)
where 1 is the penetration depth characteristic of the crystal and dlsin O , is the true distance the electron travels through the crystal. P can be defined as the “tunneling probability” for the given Bragg diffraction.
Fig. 6. (a) Experiment for investigating the line width of the Bragg diffraction minimum in transmission. The energy of the incident beam of electrons is kept constant and the angle 0 varied. The Bragg diffraction condition is satisfied for 0 = 00 f +A0. The actual line shape (b) can be approximated by a rectangular model (c). References p . 285
CH. 6, I 11
MAGNETIC BREAKDOWN IN METALS
243
A second “gedanken” experiment .with the same system requires the thickness d as well as the energy ~ ( kof ) the electron beam to be kept constant. This second requirement is equivalent to keeping the magnitude of k constant: lkl = k , . (1.16) The angle 6 between k and the surface of the slab is then varied as shown in Fig. 6a and the intensity of the transmitted beam measured at the detector D. The resulting curve should be similar to that shown in Fig. 6b, i.e., there should be a minimum at the angle O0 corresponding to the Brag diffraction condition and the line shape should show a characteristic width do. For our purposes the actual line shape is somewhat irrelevant and can be replaced by a rectangular one, which assumes that Bragg diffraction takes place as long as 8 = Oo & 366.
Fig. 7. A diffraction interpretation of magnetic breakdown. dB is the width of the diffraction line, d is the effective “slab thickness” for Bragg diffraction and R is the cyclotron radius.
Although the two experiments described above cannot be performed in practice, a similar experiment is performed by the conduction electrons of metals in the presence of a magnetic field. An electron with a given k-vector at the top of the Fermi distribution precesses in the presence of a magnetic field and changes its direction of motion continuously while keeping its energy E~ constant. Such an electron trajectory which is valid for our hypothetical metal is shown in Fig. 7; it satisfies the Bragg diffraction condition at the point. 2’ on its trajectory. From the considerations of our “gedanken” experiment, such an electron sees an effective slab of Bragg-diffracting metal References p . 285
244
R. W. STARK A N D L. M. FALICOV
[CH.
6,s 2
of thickness d. By simple geometrical considerations apparent in Fig. 7
d = R A 9 sin 8, ,
(1.17)
where R is the cyclotron radius of the orbit
(1.18) w, the cyclotron frequency and vF the (Fermi) velocity of the electron. From (1.15) it is evident that the electron will follow the trajectory A2‘B of Fig. 7 with a probability ,=ex,[--1 A9 mcvF (1.19) 1 I4 H
while the Bragg-diffracted trajectory A2’C will be followed with a probability Q=l-P.
(1.20)
In (1.19) all quantities involved except H are characteristic parameters of the metal, and consequently the probability P can be rewritten in a simpler form P = exp As we shall see in the next section4>
7 9
[“R].
(1.21)
9918
(1.22)
where E~ is the energy gap at the relevant zone boundary.
2. The theory of coupled orbits In this section we discuss some theoretical aspects of MB. This discussion will include: (a) the theory of a single MB process; (b) semi-classical transport theory in the presence of MB; (c) the quantization of a system of coupled orbits; (d) the theory of the DHVA effect in the presence of MB; (e) MB oscillatory effects in the transport phenomena. Whenever possible, we will illustrate the theory with the simple hypothetical metal descibed in the last section. 2.1. AMPLITUDES AND PHASES AT A MB JUNCTION We have seen in Section 1.3 that a point on the electron semi-classical trajectories, where the Bragg diffraction condition (1.5) is satisfied, correReferences p . 285
CH. 6 , # 21
245
MAGNETIC BREAKDOWN IN METALS
sponds to a junction where the electron will choose between two different paths: the free-electron path, with probability P, and the Bragg-diffracted path, with probability Q. The junction, however, can be better considered as the meeting point of four channels: two incoming and two outgoingchannels. This is shown in Fig. 8. It is also important to consider these channels as “quantum mechanical”, in the sense of the Bohr-Sommerfeld quantization scheme, and determine not probabilities, but amplitudes and phases instead. These are also indicated in Fig, 8 for two waves reaching the junction from
Fig. 8. A magnetic breakdown junction showing amplitudes and phases.
the two incoming channels with amplitude 1 and phase 0. The symmetry of the junction (in first approximation) with respect to the two channels manifests itself in the symmetry of the amplitudes and phases in Figs. 8a and 8b. Conservation of the number of particles (conservation of theprobabilities) imposes p z + 4’ = P Q = 1 , (2.1)
+
while orthogonality between the two waves yields for the phases exp N ’ p p ‘pp
- 40411 + exp Ci(Cp4 - Cpp)l = 0
- p‘pq = (. +
*
7
(2.2)
The information would be complete if two quantities, say P = p 2 and ( p p , were known. The tunneling probability P has been calculated using perturbation theory by several authors. It has been done in the low field limit4, where it essentially reduces to the theory of Zener (electric) breakdown53; in the high field limit5.7, where the lattice potential can be treated as a perturbation; and at intermediate fieldsn. In all cases the same formula is obtained
(2.3) References p. 285
246
[CH.6,g 2
R. W. STARK AND L. M. FALICOV
where E~ is the relevant energy gap, given in the nearly free-electron model by Eg
= 2VG,
(2.4)
Y
and u, and o,, are the two components of the electron (Fermi) velocity perpendicular to H,u, being parallel to the Bragg planes and uy perpendicular to them. Comparison of (2.3) with (1.21) shows that
where K is a numerical constant, of order 1, which depends on the geometry. Regarding qp,its actual value depends sensitively on the details of the electron orbits and the lattice potential. For most physical phenomena, however, it is of little importance and its influence can be ignored. For the sake of uniformity we make the arbitrary choice
vp=tn,
(2.6)
qq=o,
which agrees with the convention assumed by several authors516$17. 2.2. SEMI-CLASSICAL TRANSPORT PROPERTIES
As a first approximation when studying electronic transport properties, all quantum mechanical phase coherence of the wave function can be neglectedJ1.62. In this case, the electrons can be considered to be classical particles with well-defined trajectories which (a) satisfy the equation of motion given by (1.6)-( 1.7) and (b) satisfy Fermi-Dirac statistics. With these approximations all stationary, transport phenomena can be interpreted in terms of the electron distribution functionf(r, k) which satisfies Boltzmann’s equation 51 : uk-grad, f
-le‘ tz
+ f uk x H]*grad,f
=-
Since we are primarily interested in discussing the influence of MB, we shall assume that: (1) there is no spatial variation of any physical quantity, i.e., we neglect surface and boundary effects and we assume that the temperature is uniform throughout the sample ; hence, grad,. f = 0 ; References p . 285
(2.8)
CH. 6, $21
MAGNETIC BREAKDOWN IN METALS
247
(2) the scattering term on the right hand side of (2.7) can be treated in the uniform relaxation time approximation
where fo (k) is the equilibrium Fermi-Dirac distribution ; (3) only terms linear in the electric field E are considered (Ohm's law regime). With these assumptions, which in general are satisfactory for discussing experiments at high magnetic fields, the Boltzmann equation can be reduced to an integration6116~ (2.10)
(2.11) -m
In eq. (2.1 1) vk(t)is the time-dependent velocity of that electron which at time t o ( k ) is at the state k, obtained by integrating eqs. (1.7) and (1.9). The vector n k can be called the effective path of the electron state k, and the integral in (2.1 1) is usually referred to as the path integral. It is worth remarking that the factor [ - afo/ae] in (2.10) is essentially non-zero only in a range of energies kT about the Fermi energy eF. When studying only semi-classical effects at low temperatures ( 5 4 OK) it is a very good approximation to replace that factor by a delta function 6(e(k)- E ~ ) consequently ; (2.1 1) needs to be calculated only for k-vectors on the Fermi surface. The presence of MB makes the calculation of the path integral (2.11) much more involved; in that case every time the electron arrives at a MB junction, two possible paths, both with non-zero probability, appear in the time evolution of Vk(t). The integral then has to be followed consistently through an infinite multiplicity of paths and becomes in fact a non-trivial system of coupled equations. Several mathematical techniques can be applied to the solution of this problem13-169559 56; the details of these methods, however, would take us too far outside the scope of the present review. We therefore refer the interested reader to the original contributions. It is, however, interesting to describe in detail a very simple example which exhibits the interestingfeatures caused by MB. The model corresponds to the hypothetical metal described in Section 1 and shown in Fig. 3; we further References p . 285
248
R. W. STARK AND L. M. FALICQV
[CH.
6, 0 2
make the initial assumption that the relaxation time z is infinite; all contributions to n k come from MB. In Fig. 9 we have depicted schematically a realspace network corresponding to the k-space diagram of Fig. 3. Consider the four segments 1‘A2‘, 2’B2, 2C1, 1Dl’; there is a one-to-one correspondence in both diagrams. We are interested in determing A , for each point on the
*X
I
I
R=pG, Fig. 9. The network of coupled orbits in real space corresponding to the Fermi surface of Fig. 3.
circle in Fig. 3, for example, point D ; this is equivalent to finding the “effective path” that the electron, presently at D in Fig. 9, has traversed since being scattered into the network, or equivalently, since being “created” at another point XD. Consequently AD
=D
-XD,
(2.12)
and since there are no other scattering mechanisms (z+oo), XD should be the same for all electrons on the arc 1’Dl. In addition, the symmetry of the net-
(2.13)
where the points ABCDCD’ are indicated in Fig. 9 and R is the vector between two neighboring orbit centers related to the reciprocal lattice vector Gi by R = PC, (2.14) and fl is defined in (1.12). References p . 28s
CH.6 , s 21
249
MAGNETIC BREAKDOWN IN METALS
Consider X,. Since the electron emerging from junction 1 may come from C or D' with probabilities P and Q respectively, it is evident that
+ XA = PXD+ Q x ~ v . X D = PX,
and similarly
QXD,,
Solution of eqs. (2.15) and (2.13) yields
Q
and X - - R. ,-P
XA = O
(2.16)
This result, when replaced in (2.12) gives the effective path Ak and consequently the Boltzmann function (2.10). The conductivity can now be calculated (2.17) If we now integrate over our spherical slab, the conductivity tensor is given by15
l o
o c o
i
where the angle O0 is defined in Figs. 3 and 9, and n is the number of eIectrons in the metal. It is interesting to plot the p 2 2 component of the resistivity tensor m 8 (2.19) sin2 (eo) [exp - 1 1 o, pZ2 = ne where l e l H and coo= lelffo w=---. me mc
(z)
~
It is easy to see that as w+m, i.e., at very high fields, p Z 2 can be rewritten as p22(H
--f
co) =
m l -, ne2 T~
(2.20)
where 1 - 8- sin'
-.
zM
References p . 285
(eo)o,
(2.21)
250
R. W.STARK AND L. M. PAL-ICOV
[CH.6, fi 2
is an effective inverse scattering time, due to the presence of MB. This scattering mechanism is easy to understand. When H+ m the free-electronlike orbits will be dominant; however, at each MB junction there will be a probability of scattering off that orbit given by
and since the frequency with which the electron returns to each junction is proportional to a,the probability of scattering per unit time is
as found in (2.21). When finite relaxation times are considered, the calculation of the galvanomagnetic tensors becomes rather cumbersome and, in general, has to be done numerically. In Fig. 10 we show p z z ( H ) for our hypothetical metal14 for various values of z and for 8, =+n. For finite z the magnetoresistance pzz starts increasing quadratically, reaches a maximum at a value H < H , and
I' 01
1
2
Fig. 10. The transverse magnetoresistance for the model of Fig. 9 for three relaxation
times t. References p . 285
CH. 6,E 21
25 1
MAGNETIC BREAKDOWN IN METALS
then decreases to a saturation value post = p ( H = 0)
[ +: I 1
(2.22)
- oor
This case which we have described in detail is only one of the possible new regimes in the galvanomagnetic properties originated by the presence of MB, namely a transition from open trajectories at low fields (P=O) to electronlike closed orbits at high fields (P=1). It is well known that, in the absence of MB, the galvanomagnetic properties of metals in high magnetic fields can be classified according to four regimes 577 58. These, with their corresponding magnetoresistance and Hall behavior, are listed and described in Table 1. Since MB can change the
Types of orbits and state of compensation
Transverse magnetoresistance
Transverse Hall resistance
Saturates
u -
cc H 2
X H
(A) All closed orbits; uncompensated
> nh (A2)ne < nh (A1)ne
(B) All closed orbits; compensated (C) Open orbits in direction perpendicular to H making angle a in real space with the current J
a
H asin2&
H ne - n h
uH
connectivity of the orbits, transition between any regime in Table 1 to any other regime is thus possible, with a consequent change in the galvanomagnetic tensors. The H dependence of p for all these possibilities has been studied and the relevant curves published in Ref. 14. In particular, the case (C)+(A,) corresponds to the simple model discussed above, and the case (B)-+(A,) with hexagonal symmetry will be discussed in detail in the next section in connection with the experimental measurements of the magnetoresistance and Hall effect in magnesium and zinc. 2.3. QUANTIZATION OF COUPLED ORBITS The quantization of electronic motion in the presence of MB poses a difficult and still not completely solved problem. This is related to the choice of gauge for the vector potential A, which is necessary for calculating the quantum mechanical phases. References p . 285
252
R. W. STARK A N D L. M. FALICOV
[CH.
6, 8 2
It has been proved6718359 that for the relevant electronic states in a metal, the wave functions in a magnetic field can be thought of as being confined in the equivalent semi-classical network formed by segments of more or less circular tracks; the effective width is in general less than 4% of the radius of the circle. In the case of free electrons, if we consider the vector potential as being given by A=+Hxr (2.23) and restrict ourselves to those orbits centered at the origin of the gauge, the wave function can be represented by 59 $
= exp
[ilql,
(2.24)
which corresponds to a classical radius
In these equations r and cp are polar coordinates normal to H . A change in angle d q = 2 n , i.e., a complete circular orbit, yields a change in the phase Q, of the wave function (2.26) i.e., as many times 2n as the number of flux units hc/lei that the electron has enclosed in its orbit. This change of phase in a closed orbit can be generalized to arbitrary orbits, and constitutes the basis for Onsager’s scheme60 of quantization. The presence of MB introduces difficulties in calculating the variation of phase along the various arms of the network. The problem arises from the fact that the same arm can be part of several different closed orbits, and, if Onsager’s rules are to be satisfied for each of these, the phase change along the arm (or equivalently at the junctions) should depend on the orbit considered. Pippard6 has proposed a consistent scheme of computing phases around the network which can be summarized as follows: (i) Choose the center 0, of one of the circular arms as the origin of the gauge and start measuring phases from an arbitrary point on that arm. (ii) The phase change for a path along any arm is equal to the (positive) area swept by the radius vector of the corresponding circle multiplied by the factor a defined in (1.12). References p . 285
CH. 6,
8 21
MAGNETIC BREAKDOWN IN METALS
253
(iii) The phase change in going across one MB junction continuing along the free-electron circular path is a constant $71 corresponding to our convention (2.6). (iv) The phase change at an arbitrary MB junction J, when the Braggdiffracted path is followed from a circular arm of center Oi to another arm of center OF, is equal to ct times the area OiJOF, considered positive (negative) if the letters as written have the same (opposite) sense of rotation as the particles in their circular orbits. (v) When returning from another circle to any arm of the circle centered at the origin of the gauge 01,in addition to (iv), add a phase equal to the product of CI and the area of the polygon OiO 2...0,01 determined by all the centers concerned, and considered positive (negative) if the sense of rotation 010, 0,01is the same (opposite) as that of the particles in their circular orbits. In the absence of MB these rules, except for the constant factors (iii), exactly reproduce Onsager's scheme. The quantization of an arbitrary system of coupled orbits can now, in principle, be achieved by requiring the wave function to be single-valued. This is equivalent to requiring that, starting from any point in the metal, the addition of all possible wavelets which follow all possible paths through the network add up to the original amplitude and that the phase change be an integral multiple of 27r. These conditions restrict the radii of the circles to a discrete set of values, and consequently restrict the energies to a discrete set of bands of allowed values. As an example, we now once more consider our hypothetical metal. The network shown in Fig. 9 is a one-dimensional periodic structure, and the condition of single-valuedness can be achieved by requiring that any t w o
...
c
Fig. 11. A unit cell in the periodic network and the areas relevant in the quantization of the coupled orbits. References p. 285
254
[CH.6 , § 2
R. W. STARK A N D L. M. FALICOV
equivalent points in neighboring cells (e.g. D and D', C" and C in Fig. 9) have the same amplitude and an arbitrary constant phase difference w. The allowed energies will thus be a function of the Bragg-diffraction amplitude q and of w, and, as expected for any periodic system, will exhibit a characteristic band structure. In order to obtain the quantized levels we have drawn one unit cell of the network in Fig. 11, and have labeled a few relevant points and areas. The phases used in what follows correspond to these areas multiplied by the factor a. The five areas shown are not independent, since 5=21-e,
a=22+e.
(2.27)
The amplitudes and phases of the wave functions at a given point in the network are denoted by $. In this way $(K) represents the wave function at K, while $(I,) stands for the wave function at the junction 1 along the arm of the network which contains the point K. If the following definitions are introduced: (2.28) application of the method discussed above yields a = qeiB - 2 2il 2 Zit - 1 P e Cr + 4eitl c1 - 4 e 1 j? = qye" - p2e2"[1 + qye"] [I - q 2eZ i t1- 1 Y
.
(2.29)
The periodicity of the system imposes two additional conditions $(K)=e'"@(K')+a=e i(m-211) 9
$(L) = e'"$(L')+y
= fiei(OtZX).
(2.30)
Eqs. (2.29) and (2.30) constitute a set of four equations with three unknowns (a, fi, y), which can only be solved if cos 0 =
sin(5 + B ) + q2 sin(5 - c) 2q sin g
(2.31)
Since the magnitude of the left-hand side of (2.31) is less than or, equal to one, it is apparent that the energy can only have values such that t and a make the right-hand side also less than or equal to one. In particular, for q = O it is evident that solutions can be found only if
g + B = nn, References p . 285
(2.32)
CH. 6, fj 21
255
MAGNETIC BREAKDOWN IN MFXALS
which corresponds to Onsager's rules for the free-electron circle (Fig. 4c). In the other limit, q= 1, solutions can be found far arbitrary values of i , corresponding to the non-quantized open orbits (Fig. 4a). There are, however, some singular solutions which correspond to
i
(2.33)
= nn,
which are the discrete levels due to the lens-like orbit of Fig. 4b. Since for fixed ratio &T (i.e. for fixed angle O,, in Fig. 3), 5 is proportional to the energy and inversely proportional to the magnetic field strength, a diagram of allowed <'s at constant H corresponds to the diagram of allowed
0
9
i
i
Fig. 12. Evolution of the level diagram as q varies from 0 to 1 for the case 0 = Y t . The grey shading indicates the continuum levels; the black shading shows the coalescence of levels into a degenerate level (from Ref. 5). References p . 285
256
R. W. STARK AND L. M. FALICOV
[CH.
6, 5 2
energy levels. Such a diagram5 is shown in Fig. 12 for the ratio
The DHVA effect can be computed in a more or less straightforward manner once the density of states of the electron system is known as a function of energy and magnetic field. The magnetization M ( H ) and the free energy F ( H ) are given byel
M (H)= - a F / a H , F ( H ) = Nc
-2
s r
(2.34)
f ( E , T ) p ( E ’ , H)dE’ dE ,
where is the Fermi energy of the system, f ( E , T ) is the Fermi-Dirac distribution function and p(E, H )is the magnetic-field dependent density of states The calculation of p, however, poses serious problems which arise from the difficulties outlined in Section 2.3. For the DHVA effect, p should be known at every value of the magnetic field H, and calculation of the eigenstates at arbitrary (non-rational, or even non-integral) values of H is a formidable task indeed.
.
References p. 285
CH. 6,8 21
MAGNETIC BREAKDOWN IN METALS
257
Falicov and Stachowiakl7 have proposed a new method for calculating the oscillatory part of p(E, H).The method consists of studying the behavior of a Green’s function G(r, r o , t), which depends on six spatial coordinates r , ro, and time. This function is such that at t = O it is perfectly localized at r = yo,
G ( r , r o , O ) = S(r
- ro),
(2.35)
and such that it satisfies at all times the time-dependent Schrbdinger equation jh
ac -- = 2 ( r ) G ( r , ro, t ) . at
(2.36)
From these two properties it can be proved that G(ro,ro, t ) , integrated over ro, is the Fourier transform of the density of states dt p ( E ) = 2nfr l J- m-
s
d3r, G (ro,ro, t ) exp [iEt/h] .
(2.37)
The function G(ro, yo, t ) has only been calculated exactly for the case of free electrons17, but from that case it is apparent that it can be approximated by a superposition of wave packets that follow semi-classical trajectories changing their phases according to Onsager’s rule. The approximation involves no error for free electrons. In this way G ( r o , r O , t )= -
2m A
c
C j M j R jexp(icpj)6(t - t i ) .
(2.38)
i
In (2.38): (i) the label j indicates all equivalent wave packets, i.e., all those wave packets which return to ro after the same time t j , all having the same phase Cpj;
(ii) the times t j correspond to cyclotron periods of the relevant wave packet 2n 2ncm t . = -= -m j = m j t , , (2.39) wj eH where m j is the cyclotron mass in units of the free-electron mass m and tl is the free-electron cyclotron period; (iii) cpj is given by Onsager’s rule q j
References p . 285
= B&j(kz)
- Pjo 9
(2.40)
258
R. W. STARK A N D L. M. FALICOV
[CH.6.8
2
where d ( k , ) is the cross-sectional area of the Fermi surface (Pdjis the magnetic flux swept by the packet in units of the magnetic flux quantum hellel), and rpjo is a constant phase; (iv) R j is a MB damping factor Rj = p””qnZJexp [inlirpp
+ in2jcp,],
(2.41)
where nIj is the number of broken-down junctions and n2j the number of Bragg-diffractedjunctions in the orbit j; (v) Mi is the usual Dingle scattering factor for the DHVA effecta2 (2.42) M j = exp [- Itjl/2~], where z is the effective relaxation time; (vi) Cj is a real and positive quantity which gives the total amplitude of all equivalent packets j; its calculation involves in general a non-trivial combinatorial problem. We return now to our hypothetical metal, and choose for simplicity O0 = i n in Fig. 3. We also choose the convention (2.6) for the phases in (2.41). In our case c . = 2m. 2 (2.43) * Djlj’ where D j is either 2 or 1 depending on whether the orbit has two-fold symmetry or not and l j is the harmonic order of a given orbit, i.e., the total number of times that the wave packet j has been at each and every point of the trajectory before returning to ro at time ti. With the introduction of all these quantities and the use of eqs. (2.34), (2.37), and (2.38), the oscillatory part of the free energy for our hypothetical metal turns out to be
(2.44)
where
is the volume of the sample,
xj = and -(aZ,(O)
2n’kTmc mj heH
(2.45)
is the (extremal) area of the relevant orbit in k-space taken at
k, = 0. References p . 285
CH. 6,O 21
259
MAGNETIC BREAKDOWN IN METALS
In Fig. 13 we have drawn some possible important orbits, whose parameters are listed in Table 2. In Fig. 14 we have plotted in arbitrary units the quantities n
(2.46)
as a function of P=p2, i.e., as a function of H; these functions correspond, except for a factor I@, to the amplitudes of the various oscillations in F,,,, when T+O, r+m. This method can be equally well applied to more complicated systems, like the DHVA oscillations in Mg and Zn, and no problems with the gauge and/ or the boundary conditions appear. It should be noticed, however, that the form of G (ro, ro, t ) postulated in (2.38) is only an unproved assumption. However, it yields a result that
P 0 0 1 H/H,O
Fig. 13. Some typical orbits in the system of Fig. 3 which contribute to the De HaasVan Alphen effect.
References p . 285
025 050 075 1 0.75 0.50 0.25 0 0.725 1.44 3.45 03
Fig. 14. The amplitude of the De HaasVan Alphen effect. The ordinates are the amplitudes of the oscillatory terms in the free energy at T = 0 multiplied by a factor H-s. The units are arbitrary.
260
R. W, STARK AND L. M. FALICOV
[CH.
6, 0 2
a) produces no error in the free electron case; b) agrees qualitatively with all experimental information and in the few cases where accurate data are available 23, the agreement is also quantitative ; c) reproduces exactly those Fourier components of the density of states which in the hexagonal-close-packed metals61 17 are due to closed orbits. TABLE 2 Parameters of some relevant orbits in the hypothetical metal Area
L 2L 3L 0 20 -L L 0 20 20 20
+
Mass
Harmonic order
Degeneracy
Broken-down junctions
Braggdiffracted junctions
0.5 1 .o 1.5 1 .O 1.5 1.5 2.0
2.0 2.0
Two further points are worth mentioning before leaving the subject. The first one is the possibility of interference effects. If the free-electron areas can be considered exact and no correction for Bragg-diffracted orbits is necessary, it is possible to have several orbits which are different, with different numbers of broken-down and Bragg-diffracted junctions, which have, however, the same area. In such a case the amplitudes and phases would show interference effects which vary with H . As an example, in our hypothetical metal, an area (20) equal to twice the area of the free-electron orbit can be obtained by considering the second harmonic of the free-electron orbit (0)as well as the two more complicated trajectories at the bottom of Fig. 13. If the constant phases are taken to be the same for all three orbits and convention (2.6) is assumed, the amplitude will vary like lP4CP4
+ 4q4 - 4P2q211.
This expression shows two maxima in the region O
CH.
6, 5 21
MAGNETIC BREAKDOWN IN METALS
261
The second point worth making is that at the MB junctions, the pseudopotential parameter V, is not vanishingly small, and consequently the Braggdiffracted path does not coincide with the broken-down junction path for a finite portion of the orbit. This effect must alter the phase change at a given Bragg diffraction and should introduce asymmetry between the two incoming channels of a given junction18. The correction should be equal to the difference in areas (Fig. 15) multiplied by the proper factor (1.12).
Fig. 15. A diagram showing the areas corresponding to the phase changes for Braggdiffracted paths in real metals.
If this second effect is taken into account, the three types of orbits of area 2 0 in our example do not have exactly the same area. Consequently a more complicated structure in the oscillations should appear, including very involved beat patterns caused by the very small differences of the frequencies. 2.5.
OSCILLATORY EFFECTS IN THE TRANSPORT PHENOMENA
When a complete quantum mechanical description of a system of coupled orbits is considered, the structure of the energy levels, as discussed in Section 2.3, results in the definition of new excited states (excitations or quasi-particles)6. These states are the ones responsible for the transport properties of the metal at high fields. The calculation of the transport coefficients, however, presents serious difficulties when a complete quantum approach is desired: (a) the quasi-particles are (thus far) properly defined only when the magnetic flux contained in the unit cell is an integral multiple of hc/lel; (b) even a very small variation in the magnetic field strength will introduce complicated averages which may completely change those results calculated for “simple” values of H mentioned above*. In addition, the experimental situation is such that even in the better or more carefully prepared samples the density of imperfections, especially disReferences p . 285
262
R. W. STARK AND L. M. FALICOV
[CH.
6, 4 2
locations, is such that all orbits in the network, with the probable exception of the very small ones, suffer an appreciable amount of small angle scattering. This effect15 introduces randomization of the phases in the network and is responsible for the disappearance of most quantum mechanical effects. The relevant regime then is the semi-classical one considered in Section 2.1. In some very good samples, however, an intermediate regime is present. In these samples a randomization of the phases has taken place in the large orbits of the network but coherence effects remain for some very small orbits, and oscillations arising from them are observed. This intermediate case can be treated theoretically16by dividing the network into two parts :a “classical” part where the Boltzmann equation is to solved, and a small “quantum mechanical” part responsible for the “switching” probabilities at the various junctions of the classical network. Within the small part coherence effects are taken into account with the result that the transition probabilities are modulated with the characteristic frequencies of the small orbits.
Fig. 16. A double junction.
Let us return to our hypothetical metal. We consider the network of Fig. 9 and let &+z. In this case two MB junctions coalesce into what we call a double junction, as shown in Fig. 16. The electron continues along the freeelectron path with probability S, or is Bragg diffracted with probability T. Before actually calculating S and T, we recalculate the conductivity tensor ; eqs. (2.13), (2.15) and (2.16) are now changed into
XD = sx, + TXD,, References p . 285
T
XD = -- R 2s
(2.47)
CH. 6, 2J
MAGNEnC BREAKDOWN IN METALS
263
and the c1 component of the conductivity tensor u is n lel c 8 T
(2.48)
The other components in (2.18) are unchanged. If phase coherence is neglected, by following successive Bragg-diffracted trajectories in Fig. 16 it is evident that P S , = P z + P2Qz + P2Q4+ ... = l+Qy (2.49) 2Q T , = Q P2Q P2Q3 + PZQ5+ ... =1+Q’
+
+
where the subscript c indicates “classical” values. If these values are replaced in (2.48) the result obtained in (2.18) for fl0=3 x is reproduced, as expected. However, if we are interested in keeping coherence effects in the small orbit (29 in Fig. 11)’ instead of adding probabilities, as in (2.49)’ we should add amplitudes with their proper phases, and determine the pro6abilities by squaring the amplitude of the sum. This process of adding the individual wavelets is equivalent to solving the system of eq. (2.29), in which in addition we impose the condition that no wave is coming through the second channel, i.e., y=O. (2.50) Solution of this system is straightforward and yields
(2.51)
It should be noted that if (2.51) are averaged over 0 5 2
~
sQ
n lei c np2
References p. 285
(1 - cos 29)
(2.53)
264
R. W. STARK AND L. M. FALICOV
[CH.6, 8
2
In Fig. 17 we have plotted this quantity as a function of HIH,, where we have arbitrarily assumed (2.54)
It should be emphasized that we have not included any damping of the oscillations due to the variation of dLwith k,. The oscillations, as seen in Fig. 17, have a very large amplitude and the resistivity ranges, at large H , from zero to twice the semi-classical saturation value. This very strilung oscillatory effect is caused by the phase coherence together with the change in regime (quadratic to saturation) in the semi-classicalbehavior ;curves similar to the one of Fig. 17 have been detected experimentally in Mg22923,Zn22-25, and Be37. In real metals various additional features have been observed: (I) The double junction of Fig. 16 may become a triple or in general a multiple junction if the orbits go through several individual MB junctions which are very close to each other in the network. (2) The line shape of the oscillations, which is purely sinusoidal in our hypothetical metal [eq. (2.53)], may be very complicated and include a very high content of harmonics; in the case of Zn22-25 the line shape is very “peaked”, and thus permits the observation of two sets of oscillations caused by the two spin systems which have an anomalously large g-factor in the relevant small orbit e3. I
15 I
Fig. 17. The oscillatory magnetoresistance for the example of Fig. 3, when 00 + tn. Coherence effects are only considered in the infinitesimal orbits of Fig. 16.
References p . 285
CH. 6,§31
MAONETIC BREAKDOWN IN METALS
265
(3) In Mg it has been possible to observe quantum effects which arise from more than one orbit. In this case the separation of the system into a large “semi-classical” sub-system and a small “quantum mechanical” one is no longer possible. Further theoretical investigation is necessary in order to include such phenomena in the standard framework of the transport theory. 3. Analysis of experimental results
In this section we compare the available experimental data with the theory discussed in the previous two sections. In doing this we attempt, where possible, to obtain quantitative as well as qualitative agreement between the two in order to shed more light on the physical processes involved. For this reason we do not discuss the miscellaneous collection of experimental data dealing with the effects of MB on trajectories of complex geometry in various metals; nor do we discuss much of the DHVA data associated with complex coupled geometrical systems. The reason for this is simple: ~ ( kis) not well determined for most metals and in general is likely to be a complex function of k . Hence, uk as given by (1.7) cannot be accurately known. Without this information we cannot hope to obtain more than a qualitative understanding of the mechanisms involved. Fortunately, one experimental situation has been investigated in considerable detail. For this case ~ ( kbehaves ) so nearly-free-electron-like that uk can be easily calculated. This situation occurs in both magnesium and zinc when H is directed along the six-fold crystalline axis of these two hexagonal-closepacked metals. The trajectories are formed by segments of free-electron-like circular orbits which are coupled by MB to form a hexagonal network of intersecting circular orbits. The electron dynamics are so simple in this case that quantitative as well as qualitative agreement is obtained between experiment and theory.
3.1. HEXAGONAL LATTICE OF
COUPLED SEMI-CLASSICAL TRAJECTORIES IN
MAGNESIUM AND ZINC
Fig. 18 shows the network of relevant coupled electron trajectories that occur in magnesium and zinc when H is parallel to the hexagonal axis. This crystalline axis, then, defines the direction of the z-axis in accord with the convention used in Sections 1 and 2. The wave vectors of electron states around these trajectories lie in or nearly in the k,-k,, plane of k-space having lkl= k, and Jk,lS k,,< k,. This network of coupled trajectories comprises the central slab of the Fermi sphere in k-space just as in the case of our hypothetical metal. The linear chain of coupled trajectories shown in Fig. 9 is generated References p . 285
266
R. W. STARK AND L. M. FALICOV
[CH.
6,G 3
Fig. 18. Hexagonal grid of coupled orbits in real space. The large hole orbit a and the small electron orbit b are the only ones which exist when Q = 1. The circular electron orbit c is the only one which exists when Q = 0.
by one set of Bragg-diffraction planes; in this case, three equivalent sets of Bragg planes at angles of 120"to one another generate a network of coupled trajectories that extend over the entire x-y plane. The circular trajectories of radius BkF are centered at hexagonal lattice sites a distance j?lC,l apart. GI is one of the set of six smallest reciprocal lattice vectors in the basal plane of the hexagonal lattice. All the dimensions pertinent to this network for magnesium and zinc are given in Table 3. The wave vector k of the electron states satisfies the Bragg condition (1.5) TABLE 3 Parameters of the coupled orbit network in magnesium and zinc*
Parameter Fermi momentum, k~ Cross-sectional area of the Brillouin zone Cross-sectional area of the free-electron sphere Cross-sectional area of the triangular orbits Extent of coupled orbit network, 2 k z ~ adPB(kz)/a(kz2)
Ho
Magnesium 0.721 1.255 1.66 6.49 x
10-3
0.10 - 0.276 5.85 kG ~-
* All parameters in atomic units. See Ref. 16 for sources of parameters. References p . 285
Zinc 0.841 1.815 2.22 4.24 x 10-5 0.04 - 0.0113 2.7 kG
CH. 6, I 31
MAGNETIC BREAKDOWN IN IUF,TALS
267
at each of the points on the network where two circles intersect. In the limit of very small magnetic fields (Q=1 ;P=O),an electron precessing around one of the circular paths would be Bragg diffracted when it arrived at one of these junctions. This defines two possible closed trajectories: the large hexagonal orbit a and the small triangular orbit b. In the high field limit ( Q = O ; P= l), an electron precessing around one of the circles would ignore the Bragg conditions at all junctions and close the circular free-electron-like trajectory e. In the intermediate field regime (Q>O; P>O) all three of these plus a host of other possible trajectories exist simultaneously. Note that whereas the free-electron-like orbit e closes in the clockwise direction, the hexagonal orbit a closes in the counterclockwise direction; it responds to the applied magnetic field as if it had positive instead of negative charge. Because of this we define a to be a hole orbit. This is consistent with the definitions of holes and electrons in metals in terms of path integrals used in Ref. 57 to determine n , and nh. Both magnesium and zinc are even-valent compensated metals in which n ,= n h . One of the primary effects of MB in this case is to recouple the segments of the Iarge hole orbit a and the small electron orbit b to form the large electron orbit c. The resulting net decrease in nh and net increase in n, destroys the compensation of holes and electrons and makes these metals behave as non-Compensated metals in high magnetic fields. This has a profound effect on the semi-classical aspects of their galvanomagneticproperties since it causes a transition between the cases B and A, of Table 1. 3.2.
SEMI-CLASSICAL GALVANOMAGNETIC PROPERTIES OF MAGNESIUM
When H is parallel to the hexagonal axis the resistivity tensor p ( H ) reduces to the simple form (3.1) 0
as a result of the six-fold symmetry about H . P ~ ~ (always H ) saturates and is not a strong function of the conduction electron dynamics. pil(H) and plz (H), on the other hand, both exhibit remarkable variations with H which are strongly dependent on the nature of the electron trajectories. Fig. 19 shows how pI1(H) varies as a function of H in this case. When H S 1 kG,Qr1 and pI1 ( H ) exhibits type B behavior listed in Table 1, i.e., it increases as HZ.When H Z 15 kG, P = 1 and pI1( H ) exhibits type A, behavior as listed in Table 1, i.e., it saturates. The remarkable peak in pll ( H ) References p . 285
268
R. W. STARK
A N D L.
[CH. 6,8 3
M. FALICOV
at about 2 k G is the result of the transition from the low field behavior to the high field behavior as MB occurs. The data shown in Fig 19 were taken at 4.2 OK with a magnesium single crystal which had been slightly strained by temperature cycling. The relaxation time t defined by (2.9) is the typical large angle scattering time that enters into the semi-classical galvanomagnetic phenomena; it contains two main components in the low temperature range of these measurements : T-I = re:;
+
-1
(3.2)
where repiis the relaxation time for large angle scattering from lattice imperfections (impurities, etc.) and t e--p is the relaxation time for large angle electron-phonon scattering. It has been found empirically for these magnesium samples that zepi is insensitive to the small thermal strains induced in the sample; these lead to small angle scattering only. In addition 2,-,(4.2
OK)
z
20~,-~.
(3.3)
Thus t is nearly independent of temperature for TY4 OK.The semi-classical aspects of the curve shown in Fig. 19 also are nearly independent of T in this temperature range.
0
5
10
15
20
25
Magnetic Field Strength ( k G )
Fig. 19. p l l ( H ) for a magnesium crystal with a high dislocation density. The data were taken at 4.2 O K . This shows the “semi-classical” effects of MB.
The quantum mechanical aspects of the galvanomagnetic properties on the other hand, are profoundly affected by small angle scattering events. The phase coherence of the electron wave can be destroyed on even the smallest Referencesp. 285
CH. 6 , § 31
269
MAGNETIC BREAKDOWN IN METALS
closed trajectory (orbit b) in the coupled orbit network shown in Fig. 18 by scattering through an angle as small as 2 x radians when H = 10 kG; phase coherence on the next smallest trajectory would be destroyed by scattering through angles as small as l o p 4 radians. Evidently the phase coherence of the electron wave around the network of coupled orbits has been totally destroyed in the case shown in Fig. 19. The theory relevant to this case is thus the completely semi-classical theory developed in Section 2.2. Fig. 20 shows a theoretical curve of p I 1 ( H ) for magnesium based on the
Magnetic Field Strength
(kG)
Fig. 20. Theoretical curve of p l l ( H ) for magnesium (from Ref. 14).
semi-classical theory formulated in Section 2.2. The relevant parameters which enter into the calculation are listed in Table 3. One change has been introduced into the coupled orbit network in order to simplify the calculation. The triangular segments of the network (orbit b) are shrunk nearly to zero so that we can assume that on them, t = co. These act then as switching junctions into which an electron enters with probability equal to one, and out of which it leaves by three different paths with probabilities A , B, and C for each as shown in Fig. 21. These probabilities are related by: A=1-B-C,
B=P2[1-Q3]-1,
C=QB.
(3.4)
These values can be calculated quite simply. For example, an electron can get from path 1 to path B by tunneling twice with probability P z ; it can also tunnel from 1 to the triangular segment, suffer Bragg diffraction three times on that, and then tunnel to B with probability P 2 Q 3 ;this could be repeated with six Bragg diffractions with probability P2Q6, etc. The total probability References p . 285
270
[CH.6, §
R. W.STARK AND L. M. FALrWV
3
B, the sum of all of these, is shown in Fig. 21. The general aspects of the theoretical curve shown in Fig. 20 are in quite good agreement with the experimental curve shown in Fig. 19.
Fig. 21. The triangular orbit of the coupled orbit network considered as a three-way switching junction.
The details of the calculation show that the saturation level, pll (SAT), of the theoretical curve is PI1(SAT)
(; + &
wo)
9
(3.5)
where J3 wo plays the role of an inverse relaxation time for MB in the hexagonal network of coupled trajectories. The condition for experimental observation of curves such as the one shown in Fig. 19 is J5wosB 1 .
In this case
J? wo
N
1.75 x 10" radlsec
and the best estimates of s from size effect experiments show that
r 2 2 x lo-* sec.
(3 8)
Thus, condition (3.6) is well satisfied for the magnesium crystals of these experiments. The structure of pll (H)is highly dependent on z. On the other hand, p I 2 ( H )is nearly independent oft. Fig. 22a shows an experimental curve of p12(H)for a magnesium crystal of much lower purity than that used to obtain Fig. 19. In this case: t NN sec. (3.9) References p . 285
CH.
6,5 31
MAGNETIC BREAKDOWN IN METALS
271
Curves of plz(H) taken with magnesium crystals havingz of the order of (3.8) show essentially the same systematic variation with H as that shown in Fig. 223. Note that the experimental curve for p12(H)changes sign at low magnetic fields. When 0.2 k G S H 5 1 kG, plz(H)varies more or less linearly with H, having a sign corresponding to the predominant hole-like behavior. When
-1
--
0 2 4 6 8 Magnetic Field StrengthCkG) a
10
-1
0
2 4 6 8 Magnetic Field Strength ( kG) b
10
Fig. 22. (a) p n ( H ) for an impure magnesium crystal (from Ref. 21); (b) theoretical pia(H) (from Ref. 14).
Hw 1 kG, pIz(H)goes through a rapid transition, crosses the H-axis at about 1.8 kG and then proceeds to exhibit a variation with Hwhich is characteristic of an excess of electrons. This, of course, results when MB destroys the compensation of the metal leading to a situation in which n, > nhas discussed in Section 3.1. Fig. 22b shows a semi-classical theoretical curve1* for p I 2 ( H ) .Note that it shows the same systematic variation with H shown in Fig. 22a. The point at which the theoretical curves cross the H-axis depends weakly on z; as z is increased, the intersection point moves slowly to lower fields. References p . 285
272
[CH.6 , s 3
R. W.STARK AND L. M. FALICOV
The semi-classical theory, then, provides a quantitative as well as qualitative understanding of the effects of MB on the galvanomagnetic properties of the magnesium crystals discussed in this section. The interested reader is referred to Ref. 16 for a more detailed comparison of experiment and theory. Small-angle electron-phonon scattering plays a significant role in disrupting the phase coherence of the electron wave in the experiment discussed above for p,,(H). The effects of phase coherence on the small triangular “switching” orbit (orbit b Figs. 18 and 21) of the coupled orbit network were just becoming noticeable in p , , ( H ) in fields of about 20 kG for the curve shown in Fig. 19. When T was reduced to I .I OK, these effects became much more pronounced, introducing the quantum-oscillatory regime in MB in PldH). 3.3. QUANTUM MECHANICAL
GALVANOMAGNETIC PROPERTIES OF MAGNESIUM
A N D ZINC
Fig. 23 shows an experimental curve of pil(H) as obtained directly from an x-y recorder. These data were taken at 1.1 OK with the same magnesium crystal which was used to obtain the curve shown in Fig. 19. It has a small
00
I
I
5
10
I
15
I
20
I
25
Magnetic Field Strength (kG)
Fig. 23. p ~ l ( H for ) a magnesium crystal with a high dislocation density. Data taken at 1.1 “K. ferences p . 285
CH.6, 31
MAGNETIC BREAKDOWN IN METALS
273
amount of p 1 2 ( H )superimposed upon it since the 1-1 set of potential probes which were attached to the crystal did not lie exactly along the 1-axis. The correction which is required to eliminate p1z ( H ) brings the bottom envelope of the oscillations into coincidence with the curve shown in Fig. 19 without changing the amplitude of the oscillatory component. The main new feature to be noticed in Fig. 23 is the oscillatory component of p1 ( H ) .This results from the fact that, since small-angle scattering of the electron wave has been sufficiently reduced by reducing T, phase coherence around the triangular orbit is beginning to affect the switching probabilities A, B, and C. Because only the one pure frequency associated with the triangular orbit is observed in Fig. 23, we can assume that phase coherence of the electron wave is still totally destroyed on all branches of the network other than the triangular part. Thus, the intermediate regime discussed in Section 2.5 is appropriate for this case. The full significance of the network reduction shown in Fig. 21 now becomes apparent, since we can apply the method discussed in Section 2.5. The small triangular orbits are to be considered as the “quantum mechanical” part of the network, while the rest constitutes the large “semi-classical” part. Thus, instead of regarding each multiple junction semi-classically, with switching probabilities (3.4), we can treat them as quantum mechanical junctions, with probabilities A=1-B-C, B = P2[1 Q3 - 2Q3 cos 9]-’, C = BQ,
+
(3.10)
where rp is the phase change of the electron state once around the triangular orbit and is given by (3.11) rp = B-pe, - Po.
dTis the area of the triangular orbit in 8-space; B2dTisits area in real space. Note that the periodicity of the oscillations in (3.10) is exactly the same as the periodicity obtained from Onsager’s scheme for quantizing the triangular orbits in the absence of MB. The area dT varies with height k,, satisfying a relationship of the form (3.12)
for lk,l-
274
R. W. STARK A N D .I M. FALICOV
[CH.6,8 3
Calculations of p1l(H), using the quantum mechanical switching probabilities (3.10) and including the variation of cp with k, inherent in (3.12), yield theoretical curves which agree reasonably well with the curve shown in Fig. 23. These calculations are discussed in detail in Ref. 16. Fig. 24 shows pI1(H)for a zinc single crystal. The two curves shown were taken at two different temperatures; the lower curve at 4.2 OK and the upper curve at 1.6 OK. Large angle electron-phonon scattering was considerably reduced at the lower temperature. The relative relaxation time for large angle scattering is ~ ( 1 . 6O K ) = 1.56~(4.2 OK). (3.16) This difference accounts for the fact that the low field peak in p1, ( H ) is much more pronounced in the upper curve than in the lower one.
Magnetic Field Strength (kG)
Fig. 24. dp11(H)/p11(0) = [pll(H) - p11(O)]/p11(0) for a zinc single crystal. The lower curve was taken at T = 4.2 OK; the upper curve was taken at T = 1.6 OK (from Ref. 25).
The amplitude of the oscillatory component of p , , ( H ) is quite large. These oscillations result from phase coherence of the electron wave on the very small triangular orbits of the coupled orbit network. The area dTof these orbits is much smaller in zinc than in magnesium; hence, the period in H - ' of the oscillations in pIl(H) is much larger than those shown in Fig. 23 for magnesium. Note that the oscillations havea complex line shape. This results from a strong dependence of the arbitrary phase, 'po [Eq. (3.1 l)], on the electron spin. This structure has been discussed in detail in Refs. 16 and 25. References p . 285
CH.
6,B 31
MAONETIC BREAKDOWN IN METALS
275
No effects are seen in Fig. 24 which result from phase coherence on any orbit other than the triangular orbit in the network. This case then represents another example of the intermediate regime in which the triangular orbits can be treated as quantum mechanical switching junctions between large semi-classical orbit segments. Fig. 25 shows a model calculation of pIl(H) for zinc which includes the spin dependence of cpo. Its general features are in striking agreement with Fig. 24.
Fig. 25. pn(H)/plz(O) as calculated by Falicov and Sievert for zinc (from Ref. 22).
Some of the problems which limit the accuracy with which these theoretical curves can be fitted to the experimental data are: (i) The quadratic coefficient adT/a(k:) (3.12) can be considered an adjustable parameter which is to be chosen so as to fit the amplitude of the oscillatory component of pl1(H).It was chosen in Ref. 16 to be the variation with k, of the triangular shaped area of intersection of three spheres centered on lattice sites of the hexagonal coupled orbit network. A reliable independent value of ddT/a(k;) can presumably be obtained from the angular variation of the DHVA effect for the triangular orbit thus making this a fixed parameter in the calculation. Any serious error in determining this coefficient will lead to an equally serious error in calculating the amplitude of the oscillatory portion of p,,(H). (ii) No damping factor, equivalent to the Dingle scattering temperature References p. 285
276
R. W. STARK AND L. M. FALICOV
[CH.
6,5 3
in the DHVA effect, was included in the calculations. The coherence of the electron wave around the triangular orbit must certainly be partially destroyed by small angle scattering since phase coherence on the next larger closed orbit is completely destroyed in both of the cases discussed above. (iii) Ho probably varies with k,; this would lead to variations of P and Q in (3.10) with k,. This variation was included in the calculations of Ref. 16 by expanding H,(k,) in a fashion similar to d T ( k z (3.12). ) The coefficients of the expansion were then treated as adjustable parameters. The general structure of p , , ( H ) will be quite sensitive to the manner in which these are chosen. (iv) It is impossible to include phase coherent effects on any branches of the coupled orbit network other than the small triangular orbits when these are considered as three-way switchingjunctions having semi-classicalprobabilities (3.4) and quantum mechanical switching probabilities (3.10). To include phase coherence on any one of the large segments of the network means in principle that one must solve the transport problem with phase coherence throughout the entire network, i.e., one must obtain a complete quantum mechanical solution of the transport problem. This problem has thus far not been solved. We have discussed examples of the semi-classical and intermediate regimes of the galvanomagnetic properties of the hexagonal coupled orbit network. An example of the quantum mechanical regime is shown in Fig. 26 which is I
I
I
I
I
I
I
:
OO
10
20
30
40
Magnetic Field Strength ( k G )
Fig. 26. p l l ( N ) for a magnesium crystal with a low dislocation density. Data taken at T = 1.1 "K.
References p. 285
CH. 6, § 31
MAGNETIC BREAKDOWN IN METALS
277
an x-y recorder tracing of pl,(H) for a magnesium sample similar to the one used to obtain the curves shown in Figs. 19 and 23. In this case, the sample was rapidly quenched from 300 OK to 4.2 "K to prevent the propagation of thermal strains throughout its bulk. This curve contains a small amount of p l z ( H ) . The significant feature of this curve is that the amplitude of the oscillatory component is much larger than in Fig. 23, although the lower envelope of the oscillations still nearly coincides with the curve for p , , ( H ) shown in Fig. 19. Notice that the width of the trace appears to become quite broad for Hk 25 kG. This results from the superposition of much higher frequency oscillations upon the quantum oscillations associated with the triangular orbits. These higher frequency oscillations result from phase coherence around large orbits in the coupled network. Fig. 27 shows x-y recorder traces of the oscillatory portion of pI1(H)on a greatly expanded H scale. The slow oscillations result from phase coherence around the triangular orbits. For the sake of clarity we will call this frequency 8. The fast oscillations result from phase coherence around the lens-shaped orbit of the segment of the coupled orbit network shown in Fig. 28. We will label this frequency A. Notice the apparent periodic reduction in the ampli-
Fig. 27. Oscillatory portion of p l l ( H ) shown in Fig. 26 for 31.45 k G 5 H 5 37.95 kG. 0 is the large amplitude low frequency oscillation; L is the predominant high frequency oscillation.
References p . 285
278
R. W. STARK AND L. M. FALICOV
[cH.6,P 3
tude of 1.This results from an interference or beating of 1with an even faster frequency 1+ 8 which results from phase coherence around a complex path including both L and 8 orbits as shown in Fig. 28. The beat frequency between 1and 1+ 8 is exactly equal to the frequency 8 ; hence, the beat waists appear to be phase-locked to 8. It is apparent that as one goes to higher fields the beat patterns become even more complex. Frequencies such as A-t-28, 21-8, and 21 have been resolved in this case. Their equivalent trajectories are shown in Fig. 28.
Fig. 28.
Some of the smaller coupled orbits in the coupled orbit network shown in Fig. 18.
The data discussed above were taken at 1.1 O K . The higher frequencies such as 21 8 and 21 were just becoming observable at this temperature. The frequency of the free-electron orbit, 0 (orbit c in Fig. 18), is only 5.5 times faster than 21 and its effective mass is less than 1.5 times as large as that of 21; it is apparent then that further reductions in temperature would cause even 0 to become observable. Thus, this case represents an example of the quantum mechanical regime of MB in the coupled orbit network. After the data shown in Fig. 26 were taken, the magnesium crystal was
-
References p , 285
CH. 6,§31
MAGNETIC BREAKDOWN IN METALS
219
allowed to warm slowly to 300 OK and was then slowly cooled back to 4.2 OK. A curve identical to that shown in Fig. 19 was obtained at 4.2 OK and one very similar to the curve shown in Fig. 23 was obtained at 1.1 OK. It was impossible to reproduce with this crystal the quantum regime shown in Fig. 26. Thermal cycling had apparently introduced enough dislocations into the lattice to randomize the phase of the electron wave throughout most of the coupled orbit network. Thus several regimes of MB behavior, from complete phase incoherence to a high degree of phase coherence, are observed in the same sample depending only on its thermal history. The effects of MB have been observed in the galvanomagnetic properties of several other metals. MB in beryllium37,which is another hexagonal-closepacked metal, produces a coupled orbit network such as occurs in magnesium and zinc. In this case, the value of Ho is about ten times larger than in magnesium and zinc so that the effects of MB appear at much higher fields ; the curves of p , , ( H ) for beryllium exhibit the structure characteristic of the intermediate regime that was discussed in connection with Figs. 19 and 23. The effects of MB in systems of coupled trajectories of complex geometry have been observed in the galvanomagnetic properties of several other metals. These include cadmium 36, gallium 429 43, white tin 38, rhenium50, thallium 44, and aluminum39941 as well as the three metals discussed above. 3.4. DE HAAS-VAN ALPHEN EFFECT IN
MAGNESIUM AND ZINC
We continue in this section to restrict the discussion of the experimental observations of MB to the case considered in detail for the galvanomagnetic properties; namely, observations relevant to the hexagonal grid of coupled orbits shown in Fig. 18 for magnesium and zinc. We will consider first the DHVA effect resulting from quantization around the simplest closed orbits of this network, i.e., the orbits labeled a, b, and c in Fig. 18 (called x, 8, and 0 respectively in the remainder of this section). Dhillon and Shoenberge4 first noticed that the amplitude of the zinc 8 DHVA oscillations had an unusual dependence on H . Curve a in Fig. 29 shows the amplitude dependence which they observed. A normal torque DHVA amplitude plotted on the scale used in Fig. 29 yields a straight line variation. The 8 amplitude (curve a in Fig. 29) increases quite normally as H increases until H z 3 k G where it begins to exhibit a marked tendency toward saturation; the amplitude reaches its maximum value at H z 5 kG and then decreases as H increases further. This behavior results when MB causes electrons to “leak” out of the closed 8 orbits. In this case, the amplitude of the electron wave packet is reduced by q per junction or by q3 per orbit since References p. 285
280
[CH.6, J 3
R. W. STARK AND L. M. FALICOV
there are three junctions per orbit. Thus curve a is reduced by the MB damping factor R (2.41) R = q 3 =Q+=[i-exp(-z)]
*.
(3.17)
Pippard6 obtained curve b in Fig. 29 by correcting curve u for this factor taking H,= 6 kG; curve b is nearly linear except at high fields. The best value of H, obtained by comparing the experimental galvanomagnetic properties of zinc with theory16 is Ho=2.7 kG. This value of H, rectifies curve a to a
0
4
2
6
104/~
Fig. 29. Field variation of the 6 DHVA amplitude in zinc: (a) as observed by Dhillon and Shoenberg; (b) as corrected for MB (from Ref. 6).
plot that is nearly as linear as that obtained with curve b. In either case, exact agreement between theory and experiment is difficult to obtain in high fields since the total phase qe (2.40) around the 6 orbit in zinc is not much larger than 2n when HX 15 kG ;one expects that the amplitude of a normal DHVA oscillation in the absence of MB will vary from a straight line plot as the quantum limit approaches. Thus, the corrections to curve a should be viewed only as a semi-quantitative verification of the mechanisms involved. Quantitative agreement between experiment and theory is more striking for the B DHVA oscillations in magnesium23. Fig. 30 shows the experimental amplitude of the magnetization M of the 6 oscillations as a function of H ; plotted on the same graph is a theoretical curve for the amplitude obtained References p. 285
CH.6, $
3J
281
MAGNETIC BREAKDOWN IN METALS
from the theory outlined in Section 2.4. The theoretical curve has the form l7
(3.18) The temperature at which the experimental data were taken is T= 1.1 OK. The other parameters entering (3.18) are obtained from independent experimental measurements : (i) Ho = 5.8 kG, determined by fitting the semi-classical theory l 6 for the galvanomagnetic properties to the experimental data as discussed in Section 3.3; (ii) m,= 0.095 free-electron mass, determined23 from the variation of the 6 DHVA amplitude with T and independently by cyclotron resonance experiments65 ; (G)~ = x2lo-" sec determined from the 8 DHVA amplitude for H,<4 kG. The resonable agreement between the experimental data and the theoretical curve provides quantitative confirmation of the theory discussed in Section 2.4.
,
iO
5 10 Magnetic Field Strength ( k G )
15
Fig. 30. Field variation of the 0 DHVA amplitude in magnesium. The theoretical curve was calculated from eq. (3.18). References p. 285
282
R. W. STARK A M ) L. M. FALICOV
[CH.
6,s 3
The 0 DHVA frequency which arises from quantization around the freeelectron-like circular orbit has been observed in both magnesium3.23 and zinc23~30. This frequency is within 1% of the expected free-electron frequency in magnesium and within 2% in zinc. Preliminary studies of the field dependence of the amplitude of the 0 frequency in magnesium in fields down to 28 kG agree with the theoretical predictions. 100000
1
H f kG)
Fig. 31. The field variation of the DHVA oscillations in the free energy for some of the coupled orbits in magnesium (from Ref. 17).
The x DHVA frequency has not been observed in either magnesium or zinc and it is unlikely that it will ever be observed. The x frequency should be about one-half the 0 frequency and should have an effective mass of about m,,in both magnesium and zinc. Under normal circumstances this frequency should be observable at about 25 kG in a very good crystal. In this case, in magnesium however, the MB damping factor is q6 which is about in zinc for H=25 kG. Thus the natural amplitude of the x frequenand cy is so drastically reduced by MB that it is unobservable. Fig. 31 shows the theoretical field dependence of the amplitude of some of the DHVA oscillations in the free energy for several different orbits in the hexagonal coupled orbit network for magnesium17. In addition to the References p . 285
CH.
6 , Q31
MAGNETIC BREAKDOWN IN METALS
283
8, x, and 0 frequencies, Fig. 31 shows some of the more complex frequencies such as A, A+& 2A-8, and 21 whose equivalent trajectories are shown in Fig. 28. Notice in Fig. 31 with regard to the discussion of the last paragraph how small the amplitude is for the x oscillations.All of the other frequencies shown in this figure, as well as a number of others, have been observed experimentally; the magnitudes of the various frequencies agree reasonably well with the sums and differences predicted by the coupled orbit model, but data on the field dependence of their amplitude are insufficient at present to allow a complete quantitative check of the theory. The last point which we will discuss concerns the nature of the electron wave function in the vicinity of a Bragg-diffractionjunction. The diffraction effects of the crystal lattice are not limited to the isolated points on the electron trajectory where Bragg diffraction occurs as has been implicitly assumed throughout most of this paper; the diffraction effects are essentially diffuse, affecting the entire trajectory and causing a modest perturbation on the area enclosed by a given closed orbit in the coupled orbit network. This effect has been observed experimentally.
/
Fig. 32. The free-electron-like and the “perturbed” triangular orbits. Interaction of the electron with the periodic lattice potential reduces the area enclosed within the orbit by 360.
Fig. 32 shows the free-electron 8 orbit as well as the actual “perturbed” 8 orbit. The area perturbation on the side of the junction which defines the 8 orbit is 6, at each junction. The area perturbation is not by its nature symmetric across the junction; thus we have labeled the perturbation on the side of the junction which defines the x orbit as 6, per junction. The phase ‘pj (2.40) which determines the frequency of the DHVA oscillationsis affected by this perturbation. References p. 285
284
R. W. STARK AND L. M. FALICOV
[CH.6 ,
3
Consider the 8 DHVA oscillation in magnesium. Its unperturbed area is 8FE. Thus its phase iS (3.19) ' P O = P (eFE - 360) 3
ignoring the arbitrary constant phase qeo. The first question to be answered is how the perturbation 6, is affected by MB. Measurements of the 8 DHVA frequency show that this remains unchanged for 2 k G s H I 38 kG or equivalently for 0.95 2Q 20.15. Thus, is unaffected by MB and an electron that is Bragg diffracted follows the perturbed low-field path even in very high fields. Both 6, and 6, are quite small: 6 , is estimated to be only about 1% of 6FE and first order perturbation theory suggests that 6,x 268. The l , d + 8,2d - 8 and 2A DHVA frequencies whose equivalent trajectories are shown in Fig. 28 offer an interesting determination of the effects of these perturbations. The unperturbed area associated with the d frequency is AFE. Thus its phase is (3.20) qn = P (AFE - 26,) . The phase of A + 6 is q i + e = P(&
+ &E - 560).
(3.21)
These two frequencies are only about 4% different in magnitude so that experimentally they are observed to beat together with the phase dqr = q l + e
-
= P(&
- 360).
(3.22)
Note that this is exactly equal to q e (3.19) and does not of itself contribute any information on 60. Consider next the two frequencies 21- 8 and 21 with phases respectively qzn-e =P ( 2 h
- 4, - 268 + a,),
(3.23)
- 449).
(3.24)
q z i = P (2&E
Note that q z nis exactly equal to 2q,. This is observed experimentally. 21 - 8 and 21 beat together with the phase Aqn = qza
- qzn-e
= P [em
- 360 - (6, - 6011 -
(3.25)
This is different from drp, and q e due to the term -P(S,-S,). This effect is observed experimentally also. The relative magnitudes of the measured quantities are
P(6, References p. 285
- 6 0 ) = 0.034
= 0.0014 q n = 0.00012 'PO.
(3.26)
CH.
61
MAGNETIC BREAKDOWN IN METALS
285
Thus the perturbations though small are observable. They can be included in the theory by making appropriate adjustments in the phases q, and q 4 (2.6). When this is done, the theory predicts the proper perturbed frequencies as well as the field dependence of their amplitudes. REFERENCES M. H. Cohen and L. M. Falicov, Phys. Rev. Letters 7,231 (1961). M. G. Priestley, L. M. Falicov and G. Weisz, Phys. Rev. 131, 617 (1963). M. G. Priestley, Proc. Roy. SOC.(London) A 276, 258 (1963). 4 E. I. Blount, Phys. Rev. 126, 1636 (1962). 5 A. B. Pippard, Proc. Roy. SOC.(London) A 270, 1 (1962). 6 A. B. Pippard, Phil. Trans. Roy. SOC.London A 256,317 (1964). 7 J. R. Reitz, J. Phys. Chem. Solids 25, 53 (1964). E. Brown, Phys. Rev. 133, A1038 (1964). C. B. Duke and W. A. Harrison, private communication. 10 W. G. Chambers, Proc. Phys. Soc. (London) 84, 181 (1964). 1 1 W. G. Chambers, Phys. Rev. 140, A135 (1965). 1% M. Ya. Azbel, Zh. Eksperim. i Teor. Fiz. 46, 929 (1964) [English transl.: Soviet Phys.JETP 19,634 (1964)]. 13 L. M.Falicov and P. R. Sievert, Phys. Rev. Letters 12, 558 (1964). 1 4 L. M. Falicov and P. R. Sievert, Phys. Rev. 138, A88 (1965). 15 A. B. Pippard, Proc. Roy. SOC.(London) A 287, 165 (1965). 16 L. M. Falicov, A. B. Pippard and P. R. Sievert, Phys. Rev. 151, 498 (1966). 17 L. M. Falicov and H. Stachowiak, Phys. Rev. 147, 505 (1966). 18 R. G. Chambers, Proc. Phys. SOC.(London) 88,701 (1966). 18 W. G. Chambers, Proc. Phys, SOC.(London) 84,941 (1964). 20 R. W. Stark, T. G. Eck, W. L. Gordon and F. Moazed, Phys. Rev. Letters 8,360 (1962). 21 R. W. Stark, T.G. Eck and W. L. Gordon, Phys. Rev. 133, A443 (1964). 22 R. W. Stark, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 712. 23 R. W. Stark, private communication and to be published. 24 R. W. Stark, Phys. Rev. Letters 9,482 (1962). 26 R. W. Stark, Phys. Rev. 135, A1698 (1964). C. G. Grenier, J. M. Reynolds and N. H. Zebouni, Phys. Rev. 129, 1088 (1963). 27 W. A. Harrison, Phys. Rev. 126,497 (1962). z8 R. J. Higgins, J. A. Marcus and D. H. Whitmore, Phys. Rev. 137, A1172 (1965). 2 9 R. J. Higgins, J. A. Marcus and D. H. Whitmore, Proc. IXth Intern. Cod. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 859. 3O A. C. Thorsen, A. S. Joseph and L. E. Valby, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio 1964. (Plenum Press, New York, 1965) p- 867. 31 J. R. Lawson and W. L. Gordon, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p, 854. 32 J. E. Schirber, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 863. I
@
286
R. W. STARK AND L. M. FALICOV
[CH.
6
J. E. Schirber, Phys. Rev. 140, A2065 (1965). J. K. Galt, F. R. Merritt and J. R. Klauder, Phys. Rev. 139, A 823 (1965). 35 A. D. C. Grassie, Phil. Mag. 9, 847 (1964). 38 D. C. Tsui and R. W. Stark, Phys. Rev. Letters 16, 19 (1966). 37 W. A. Reed, Bull. Am. Phys. SOC.[2] 9, 633 (1964); and private communication, to be published. 38 R. C. Young, Phys. Rev. Letters 15, 262 (1965). 39 R. J. Balcombe, Proc. Roy. SOC.(London) A 275, 113 (1963). 40 N. W. Ashcroft, Phil. Mag. 8, 2055 (1963). 41 E. S. Borovik and V. G. Volotskaya, Zh. Eksperim. i Teor. Fiz. 48, 1554 (1965) [English transl.: Soviet Phys.-JETP 21, 1041 (1965)l. 43 T. E. Moore, Bull. Am. Phys. SOC.[2] 11,90 (1966); and private communications. 43 J. Kimball and R. W. Stark, private communication. 44 A. R. Mackintosh, L. E. Spanel and R. C. Young, Phys. Rev. Letters 10,434 (1963). 45 P. Soven, Phys. Rev. 137, A1717 (1965). 48 M. G. Priestley, Phys. Rev. 148, 586 (1966). 47 Y.Eckstein, J. Ketterson and M. G. Priestley, Phys. Rev. 148, 638 (1966). 48 A. S. Joseph and A. C. Thorsen, Phys. Rev. Letters 11,67 (1963). 48 A. S. Joseph and A. C.Thorsen, Phys. Rev. 133, A1546 (1964). 6 0 W. A. Reed, E. Fawcett and R. R. Soden, Phys. Rev. 139, A1557 (1965). 61 See,for instance, J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1964). 52 A. B. Pippard, Rept. Progr. Rhys. 23, 176 (1960). 53 See, for instance, E. 0. Kane, J. Phys. Chem. Solids 12, 181 (1960). 54 R. G. Chambers, Proc. Phys. SOC.(London) A 65,458 (1952); Proc. Roy. SOC. (London) A 238, 344 (1956). 55 A. B. Pippard, Proc. Roy. SOC.(London) A 282,464 (1964). 58 P. R. Sievert, private communication and to be published. 57 I. M. Lifschitz, M. Ya. Azbel and M. I. Kaganov, Zh. Eksperim. i Teor. Fiz. 30, 220 (1955) [English transl.: Soviet Phys.-JETP 3, 143 (1956)l; I. M. Lifschitz and V. G. Peschanskii, ibid 35, 1251 (1958) [English transl.: Soviet Phys.-JETP 8, 875 (1959)l; 38, 188 (1960) [English transl.: Soviet Phys.JETP 11, 137 (1960)l. 68 E. Fawcett, Advan. Phys. 13,139 (1964). 5 9 R. B. Dingle, Proc. Roy. SOC.(London) A 211,500 (1952). 80 L. Onsager, Phil. Mag. 43, 1006 (1952). 61 See, for instance, R. E. Peierls, Quantum Theory of Solids (Oxford at the Clarendon Press, London, 1955) p. 144 ff. 82 R. B. Dingle, Proc. Roy. SOC. (London) A 211,500, 517 (1952). 83 A. J. Bennett and L. M. Falicov, Phys. Rev. 136, A998 (1964). 84 J. S. Dhillon and D. Shoenberg, Phil. Trans. Roy. SOC.London A 248, 1 (1955). 85 T. G. Eck and M. P. Shaw, Proc. IXth Intern. Conf. Low Temp. Phys., Columbus, Ohio, 1964 (Plenum Press, New York, 1965) p. 759. 33
34
CHAPTER VII
THERMODYNAMIC PROPERTIES OF FLUID MIXTURES BY
J. J. M. BEENAKKER and H. F. P. KNAAP 0 " E S LABORATORIUM. LEIDEN KAMERLINGH
CONTENTS: 1. Introduction, 287. - 2. Quantum liquids; zero point effects, 290. 3. Classical liquid mixtures, 299. - 4. Gaseous mixtures, 301.
1. Introduction
Looking over the realm of thermodynamic properties of gaseous and liquid mixtures below 0 "C one observes that in the last 15 years considerable progress has been made both experimentally and theoretically. These developments took place in four rather distinct fields. First of all there is the most characteristic low temperature mixture, that of the degenerate quantum liquids 3He and 4He. This system has already been treated several times in this series1.2, most recently in 19642. Hence we will not take up this subject here. Quantum effects are not, however, limited to degenerate systems, but show up also through the influence of zero point energy in liquid mixtures containing, e.g., hydrogen isotopes. These mixtures will be treated in Section 2. Stimulated by a growing theoretical understanding of the behavior of mixtures physicists have devoted considerable attention to the study of classical mixtures containing simple molecules like A, Kr, O,, N, etc. Very recently the Brussels group published an excellent review paper on this subjects. They compare in detail theory and experiment for five systems, for which sufficient data are available. Therefore we will limit ourselves in Section 3 to a presentation of a survey of the existing experimental data. As the experimental techniques are not typical for this group of mixtures, we will discuss experimental progress when dealing with the quantum liquids in Section 2. Work has not been limited to liquid systems. Extensive studies, both experimental and theoretical, have been performed on gaseous mixtures at References p . 31%
281
288
I. 1. M. BEENAKKER AND H. F. P. KNAAP
[CH.
7,9 1
higher densities. This subject will be treated in Section 4.A list of b0oks4-l~ and review papers3911-14 on the subjects treated in this chapter will be given at the beginning of the list of references. To end this introduction we list the symbols used throughout this paper. LISTOF SYMBOLS
second virial coefficient (arising from interactions of molecules i andj) second virial coefficient of mixture molar free energy of pure component i molar free energy of mixture molar excess free energy
PE( X A , p , T , = $m
(XA,
p1 T)
( p , T , - x B p i ( p , T, + - R T ( x , In x A -ix-B In x B )
-xAFi
molar Gibbs’ free energy of pure component i molar Gibbs’ free energy of mixture molar excess Gibbs’ free energy
GE( X A , p , T , = G m ( X A 9 p’ T , - x A G Z ( p , T , - x B c i ( p , T , +
- RT (xA In x A + x B In xB)
variation of Gibbs’ free energy on mixing (see eq. (31)) quantum parameter A*2 = h2/a2mE Planck’s constant molar enthalpy of pure component i molar enthalphy of mixture molar heat of mixing
RE( x A ~P,T ) = g m (xi, P,T ) - x
A R
Boltzmann’s constant mass of the molecule Avogadro’s number pressure reduced pressure critical pressure (of component i ) pseudo-critical pressure of mixture gas constant References p . 318
(P,T ) - x B Z ( P , T )
FLUID MIXTURES
289
distance molar excess entropy
s”E = (1/T) (flE- GE) absolute temperature reduced temperature critical temperature (of component i) pseudo-critical temperature of mixture molar volume reduced volume molar volume of pure component i molar volume of mixture excess molar volume molar volume calculated according to single-, two- and threeliquid model excess molar volume calculated according to single-, two- and three-liquid model molar volume of pure reference substance molar volume for a liquid of molecules with only A-B interactions molar concentration of component i cxi=l i
energy parameter of interaction potential (between molecules i and j) energy parameter of average potential for single-liquid model energy parameter of average potential for “i” centered liquid in two-liquid model parameter of interaction potential for reference substance distance parameter of interaction potential (between molecules i a n d j ) distance parameter of average potential for single-liquid model distance parameter of average potential for “i” centered liquid in two-liquid model parameter of interaction potential for reference substance intermolecular potential (between molecules i and j) average potential for single-liquid model average potential for ‘7’’ centered liquid in two-liquid model References p , 318
290
J . J.
[CH.7,
M. BEENAKKER A N D H. F. P. KNAAP
02
2. Quantum liquids: zero point effects 2.1. GENERAL REMARKS
The properties of the helium and hydrogen isotopes are largely influenced by the deviations from classical behavior. Quantum effects become of importance when the zero point energy is no longer small in comparison to the interaction energy. For a liquid made up of cells of linear dimension of the order of the diameter CT of the molecules, the zero point energy is proportional to h2/c2rn.Since the total interaction energy for a molecule with its neighbors is proportional to the characteristic energy of the binary interaction, the relative importance of the zero point energy is determined by the parameter A * which is given by h2 A*2 = ___ (1) 02mE'
From this it is clear that small mass and small interaction energy favor quantum effects. The Brussels group developed a theory to describe the influence of zero point energy on the thermodynamic excess properties of liquid isotopic mix-
V" H E
Ha
GE
VE H E
GE
HD
17 18-21
VE
22-24
H E
17.23
GE
18-20
I
d:
1
~Hz-pHz
VE
22.23
H E
23,31,92
GE
32,33
D2
oDa-pDz
VE
VE
H E
25
GE
28,27
References p . 318
Ne
25
28-30
H E
GE
31
Z!H.
7, 6 21
291
FLUID WTURES
t u r e ~ 4 ~ Such 1 ~ ~ effects ~ ~ . are of little importance for the heavier isotopic mixtures like 2oNe-22Ne.3He-4He mixtures are less suited for a test of the theory, for in this case the difference in statistics between 3He and 4He plays an important role. The best systems to study the effects of the zero point energy are those containing hydrogen isotopes. We wiU discuss the experimental and theoretical aspects of the heat of mixing, the volume change on mixing and the excess Gibbs' free energy in these systems. A survey of the existing references to experimental data is given in Table 1. For a survey of
........ t h a r r n o m o t e r
/
......... tharrnomutcr ......... heating coil
......... foi I
Fig. 1. Heat of mixing calorimeter for liquids (Brussels versi~n)~S. Referencesp. 318
292
J. J. M. BEENAKKER AND
n. F.
P. KNAAP
[CH.
7, 8 2
the interesting effects in heavier isotopes outside the scope of this paper see, e.g., Bigeleisenl54 and Boato and Casanova155. 2.2. APPARATUS
Low temperature heat of mixing calorimeters for condensable gases have been described by Pool and Staveley34, Jeener35 and Knobler et al.36+l7.The latter two paid special attention to the avoidance of corrections arising from the presence of the vapor phase in the calorimeter itself. Their calorimeters are rather similar (see Figs. 1 and 2). They consist of two copper vessels, separated by an aluminium foil. At the start of an experiment the gases to
Fig . 2. Heat of mixing calorimeter for liquids (Leiden version)3s*17. References p . 318
CH. 7,
5 21
FLUID MIXTURES
293
be mixed are condensed. The vessels are then thermally isolated from the liquid bath by evacuating a surrounding can. By means of a metal plunger that can be agitated from outside the foil is punctured and the liquids are mixed. The cooling on mixing (AEis nearly always positive) that is registered with a platinum or carbon resistance thermometer, is compensated for by electrically applying heat to the calorimeter. The apparatus can be demounted easily to replace the foil since all seals are made with metal O-rings. By using different size vessels measurements can be made over the entire concentration range. The difference between the Brussels and the Leiden apparatus is found in the filling lines. In the Brussels apparatus (see Fig. 1) a well-defined steep temperature gradient is maintained in the filling lines, so that the liquid level is kept constant during the experiments. A volume increase on mixing thus leads to a pressure increase above the liquid or a gas volume increase at constant pressure. Hence in this apparatus both BEand PEcan be measured in one experiment. In the Leiden apparatus (see Fig. 2) thermal insulation is improved by two low temperature valves at the exit of the calorimeter. In this way streaming of liquid through the calorimeter after puncturing of the foil is avoided. Here BEis determined in a separate experiment. One can obtain pE by measuring the total volume of the mixture as a function of concentration. As BEis always only a small fraction of pm,very accurate density data are needed. This approach was followed for the hydrogen isotopes by Grigoriev and Rudenko24 and by Kerr37. A much simpler way is to measure the volume change on mixing directly. Both the Brussels and the Leiden group used such a method. The Brussels group performed these measurements in combination with the heat of mixing experiments as has already been discussed. In the Leiden set-up22 (see Fig. 3) a cylindrical glass vessel is partially filled with Dz. A tight fitting plunger is then placed at the liquid meniscus. This is done by the action of a magnet on a piece of iron that is sealed in the plunger. H,is then condensed on top of the separator up to a reference mark in a narrow calibrated capillary. Subsequently the liquids are mixed by moving the plunger up and down. The volume variation is found by either measuringthe level change in the capillaryor by determining the amount of gas that has to be condensed to bring the level back to its original position. GEcan be obtained from a measurement of the pressure composition diagram of the liquid-vapor equilibrium, combined with a knowledge of the equation of state of the gas. Such experiments have been performed for H,-Dz and H,-HD by Newman and Jackson18. For a theoretical analysis of these experiments see Bellemanslg. The experimental results are given in table 2. References p . 318
294
J. 1. M. BEENAKKER AND H. F. P. KNAAP
[CH.7,5
2
c
...........
.......... p
-.--.M9 ..........v
H
..........
Fig. 3. Apparatus for the determination of volume change on mixing of liquids22.
References p . 318
CH.
7,g 21
295
FLUID MMTURES
2.3. THEORY
The theory for isotopic liquid mixtures as developed by the Brussels gro~p4*15~16 is based on the idea that the difference in properties of isotopic liquids is caused by the difference in molar volume arising from the zero point energy. Hence two isotopic liquids brought to the same volume will mix ideally. To calculate the excess properties the isotopes are brought to the same volume V,,, by applying positive pressure pAto compress the light component from Vl to and negative pressure pBto expand the heavier component from Vg to (see Fig. 4). One then mixes such quantities of the two liquids that the pressure after mixing is zero
r,,, r,,,
+ XBPB
XAPA
(2)
= 0-
Since the mixing under these conditions is assumed to be ideal, the excess free energy FEis equal to the amount of work done in compressingthe lighter component and expanding the heavier one
s
V, P = - x A
s
9,
pdV-xB
PA0
pdV.
(3)
9Bo
Both terms in eq. (3) are positive, hence FEis not very sensitive to the way in which one extrapolates the compressibility data to negative pressures. Using the data for FE the quantities GE and RE can easily be calculated. I
I
I
I
I
Fig. 4. Diagram for the calculation of flE and PE of isotopic mixtures. 0,A, v indicate experimental data.
References p . 318
and
296
[CH.
I. J. M. BEENAKKER AND H. F. P. KNAAP
To obtain
7, 5 2
pE one uses p = p m - XA p Ao - x
(4)
B f70 B 7
where the concentrations xA and xB are again given by eq. (2). The result for pE appears to be strongly dependent on the extrapolation of the compressibility data to rather large negative pressures. While extrapolation through A (Fig. 4) can account for the observed negative effect in the system H2-D,, the line through B would give an equal but opposite value for PE. The experimental and theoretical values are collected in Table 2. In comparing experiment with theory, one has to bear in mind the rather crude nature of the theory. One neglects, e.g., collective effects, the difference in potential parameters and the asymmetric mass distribution of HD. In view of this the agreement is satisfactory for H E and GE,while for BE no values can be calculated. TABLE 2 BE,
System
and G E at 20 OK for x
Ha-Da
___________ PE
= 0.5
HD-D2 exp. theor.
__
oHs-PH~ ODFPD~ exp. exp. --
. -
0.03 22 0.0323
0.15z2
(cm3/mole) - 0.1823 - 0.324
RE (J/mole)
e E
12.117 12.323
7.1
16.6
8.9
4.95 17
5.5
2.6l89l9
2.3
3.70
17
3.9
2.231 1.623 1 (3.2) ~
2 31
(J/mole)
In the case of heteronuclear molecules such as HD, HT etc. the zero point energy is not the only quantum effect. The fact that the center of gravity and the center of the intermolecular force field do not coincide, causes a coupling between translational and rotational degrees of freedom as was pointed out by Babloyantz38, Friedman39, Bigeleisen40 and Bellemans and Friedman4I. This can be taken into account by introducing an effective molecular mass for the HD molecule. This phenomenon will also be of importance for the excess properties of isotopic mixtures containingheteronuclear molecules like the systems H2-HD, HD-D2, H,-HT and D,-DT (see Ref. 40). Calculations in this direction have, however, not yet been performed. References p . 318
CH. 1,8 21
291
FLUID MIXTURES
2.4. PHASE SEPARATION FOR THE SYSTEMS H,-Ne, HD-Ne
AND
D,-Ne
Extreme examples of the influence of zero point energy are the mixtures of the hydrogenic molecules with neon. Because the spherical interaction of these molecules is nearly the same, they can be treated as isotopes with an
01 0
I
I
20
I
1
40 Mole 01. neon
I
I
60
1
I
80
Fig. 5. Vapor-liquid diagram for the system HS-Ne (Ref. 26).
References p. 318
I
100
298
[CH.7,s 2
J. J. M. BEENAKKER AND H. F. P. KNAAP
extremely large mass difference. The resulting difference in zero point energy gives rise to phase separation in the liquid state. This phenomenon has been studied in the systems D,-Ne and H2-Ne by three different methods. Measurements of the pressure-concentration diagram for the vapor-liquid equilibrium have been performed by Simon for D,-Ne (Refs. 28, 29) and for H,-Ne (Ref. 27), both at 24.56 OK, and by Streett and Jones26 for the system H,-Ne over the complete temperature range of the stratification phenomenon (see Fig. 5). Brouwer et al.30 determined by visual observation the boundaries of the phase separation region for the system D2-Ne (see Fig. 6). As for the systems H2-Ne and HD-Ne the vapor pressures are well above atmospheric, they25 used calorimetric methods to determine both the phase separation region and the caloric properties with a method described by De Bruyn Ouboter et al.2,42. The agreement of the experimental results for GE
m
a
P
J
P
1
e
0.5
Fig. 6. Phase diagram of the system D p N e (Ref. 30). (I) Homogeneous liquid region; 01) liquid-liquid region; 011) and (IV)liquid-solid region. References p. 318
CH. 7,I 3 J
FLUID MfXTURBS
299
or the upper consolute temperature with the earlier mentioned theory for isotopic mixtures is, as is to be expected, only qualitative. 2.5. ORTHO-PARA MIXTURES
The occurrence of the ortho and para modifications in the homonuclear isotopes gives rise to a special class of liquid mixtures. The measured excess properties for systems OH,-pH, and OD,-pD, are given in Table 2. In general the effects are very small. This is to be expected since the molecules have the same mass and differ only slightly in the spherical part of the pair potential due to centrifugal stretching. The main influence of the different rotational states is found in the non-sphericalpart of the potential. In general one has the impression that for hydrogen the ortho-para interaction is nearer to the para-para than to the ortho-ortho interaction. Bellemans and Babloyantz43.44 treated mixtures of ortho and para hydrogen as an ensemble of Einstein oscillatorstaking into account as a perturbation the differences in interaction between the various possible pairs. They arrive at a semi-quantitative agreement for the excess properties of an oHz-pHz mixture. As already stated by the authors, one cannot expect anything better in view of the simplified character of the statistical model and the inaccurate knowledge of intermolecular forces.
3. Classical liquid mixtures In Table 3 we present a survey of the existing experimental material for the thermodynamic properties of classical liquid mixtures of simple molecules. As already observed in the introduction the reader is referred to the recent work of Bellemans, Mathot and Simon3 for a discussion. We also mention here the theory of YosimQOfor the excess entropy of mixing of liquids. Using the expression for the equation of state of a mixture of rigid spheres and assuming that all the entropy effects on mixing arise from the size effect of a hard core, Yosim is able to calculate the excess entropy 1 P=-(BE-GE). (5) T The results are, however, very sensitive to the value of the hard core diameter, as the author states himself. A comparison with theory is not convincing enough to accept the basic elements of the approach. For a more detailed discussion of the behavior of a mixture of rigid spheres, we refer to a paper by Lebowitz and RowlinsonQ1.These authors conclude that such a system References D. 318
TABLE3
s”
A survey of experimental data on Ye, HE, G E and pTx diagrams of binary mixtures of classical liquids
45
VE
HE pTx+GE PTX
48,49 50,45
VE
45
45
HE pTx + GE PTX
VE HE pTx pTx
co
4547
45,46,51
47
49,52-54
49
22,45.55
+ GE
36.45
45.47.51.56
45,51,62.63
HE
VE
I
+ CE
22,45, 50, 55
36 49,53.57-61
HE pTx PTX
A
49
49, 52, 53, 5 7 . 6 4 4 1
75-77
69,75,80
75.34
35,75,80
46,76-78
46‘80
49,79
49
X n
01
CHI
49
81 66
81-84
85,153
I j
88
Kr
I ~
VE
86
HE
PTX pTx + cE/
w
I
87
I
1 I
-F n
CF4
-4
...
M
CH.
7 , s 41
30 1
FLUID MIXTURES
is, as far as mixing is concerned, completely different from a real system. Therefore this type of calculation does little to help improve theories of mixtures of real molecules. 4. Gaseous mixtures 4.1. EXPERIMENT
Experimental progress in the study of high density gaseous mixtures has long been hampered by the fact that one used for mixtures the same approach as for pure gases, i.e. the performance of highly accurate p VT measurements. This type of work requires, as is well known, a rather complicated experimental set-up, combined with a great deal of experimental skill. Furthermore these measurements are very time-consuming. The extra parameter, given by the concentration, creates in the case of gaseous mixtures a problem that is nearly unsolvable along these lines. For a survey of the existing experimental data see Table 4. The only relevant information, however, necessary to describe the behavior of a mixture is the deviation from ideality, the excess.
TABLE 4 Literature on mixing properties of binary gaseous mixtures Mixture HaHe H2-CH4
Cox.*
X
X X
X X X X X X
25%Na 25%N2 25%Na
-
72/20 - 72 0/200 - 130/10** - 135/150** - 35/40 - 126120 - 126 - 103/20 - 103 01400
-
X
20170 - 701300 0 and 20
X
0/800
X
01200 0/200 01300
X
25%Na
References p . 318
-
X
Ha-Na
Temp. range ("C)
pmsx
(atm)
120 120 700 500**
200 * * 1 00 120 120 100 100 lo00 1000 1400 20 lo00 500 700 lo00
Properties method * * * comp. H E ;exp. H E ; th. pVT; exp. pVT; exp. p Y T ; exp. J.T.; exp. HE;exp. HE;th. V E ;exp. V E ;th. p VT; exp. pVT; exp. pVT; th. p v T ; exp. p V T ; exp. p VT; exp. p v T ; exp. p v T ; th.
Ref.
92 93 94
05 88
97 98
99 100 101 102 103 104 105 108
107
108 95
109
302
[CH.7,
I. J. M. BEENAKKER A N D H. F. P. KNAAP
TABLE 4 (continued) Mixture
Conc.*
Temp. range ("C)
pmax (atm)
25% Nz 25% Na 25% Na
- 50/200 O/150 01 150 501 170
1000 340
X
X X
24% Nz 25% Na
-
Ha-CO
X X
48% CO X
Ha-A
0/100
X
- 104
12.4% Ne
- 103120 - 103 - 100/20**
X
0/400 - 72/20 - 72 - 40/20** - 140/0
X
20**
X
- 100/250
X
X
15 - 140/40** 30 30 - 150/20** - 150/20** - 104/20 - 104 - 100/250 0/70 01130 - 72/20
X
- 12
X
X X X X
X X
X X X
X X X
References p . 318
25
- 104120
X
CHd-Na
0 and 25
X
X
He-Xe
170 600 160 1000 160
-
X
He-A
25
25
X
He-Nz
-
-
X
lo00 3000 I 20 1000** 400
-
X
X
He-Ne HeCH4
25/125 I5 51240* 100
lo00
01200 01300 0/200
120
120 100 100 200** 100 120 120 200** 200 250** 200 120 500**
100 120
lo**
200** 1 20 1 20 200 2000 I00 120 120 I00 lo00 200
Properties method *** therm. ;th. ; calc. pVT; exp. therm. ; th. ; calc. p VT; exp. p v T ; exp. J,T. ;exp. therm. ;calc. H ; th. comp. p VT; exp. p v T , exp. pVT; th. p VT; th. p y T , th. comp. HE; exp. H E ;th. VE; exp. Y E ;th. HE; th. pVT; exp. HE; exp. HE; th. HE; th. p VT; exp. p VT; exp. J.T.;exp. J.T.; exp. therm. ;th. ;calc. pVT; exp. pVT; exp. H; exp. H E ;th. H E ;exp. H E ;th. J.T. ;exp. pVT; exp. GE;th. HE; exp. H E ;th. p V T , exp p v T , th. p VT,exp.
.
Ref. 110 111 112 113 114 115 110 117 118 118 120 121 10Y
182 123 99 100 101
102 124 125 93 94
124 128 137
128 115
129 130 131 132 124 93
94 133 134 135 93
94 85 109 156
84
CH. 7,
4 41
303
FLUID MIXTURES
TABLE 4 (continued) Mixture
Conc.*
X
- 170/90* - 170/90** 01200 0 - 751125
X
-
X
0 - 150/50* - 72/20 - 72 15 01300 - 104120 - 104 - 103120 - 103 - 104/25
10%Na 30% Na X
57% Na
X
CH4-A
X X
Na-Oa
X X
NwA
X X X X
X
OaA
***
75/30**
X
I5
50%A
25 15
X
* **
Temp. range ("C)
Pmsx
(atm) 100** 100** 200 90 100 loo** 90 140* 120 120 120 lo00 120 120 100 100 48 120 125 200
Properties method*** p v T ; exp.
therm. ;calc. pVT; th. pVT; th. J.T. ; th. H ; th. p V T ; th. H ; exp. HE; exp. H E ;th. J.T.; exp. p W, th. H E ;exp. HE; th. VE; exp. V E ;th. J.T. ; exp. J.T. ; exp. pVT; exp. J.T.; exp.
Ref. 137,138 137.38 130 121
140 141 122 148
03 94 115 109
99 100 101 10% 143 115 144 115
x = data are reported for several compositions over the whole concentration range.
The Fahrenheit scale for the temperature was used; the pressure was given in psi. For the table the values were converted to degrees centigrade and atmospheres. comp. = compilation. p V T = data on the molar volume at pressure p and temperature.'2 J.T. = data on the Joule-Thomson effect. therm. = data on several thermodynamic properties. H = data on enthalpy. HE = data on excess enthalpy. VE = data on excess volume. GE = data on excess Gibbs' free energy. exp. = determination by experiment. th. = calculated from data on the components; in some cases: derived by extrapolation from lower density data of the mixture. calc. = calculated from @ V T ) data on the mixture.
Techniques for measuring directly the heat of mixing RE(Refs. 145,99) and the volume change, pE,on mixing 145,101 of dense gaseous systems have been developed in the last few years by the Leiden group. It would of course have been possible, but not practical, to determine a11 thermodynamic properties References p . 318
304
J. J. M. BEENAKKER AND H. F. P. KNAAP
[CH.7,
04
M...____.. _...._...
Fig. 7. Heat of mixing calorimeter for gaseseg.
starting from the volume change on mixing, BE,only. The accuracy needed in that case, however, would offset the advantages of the direct approach. The situation in this respect is similar to the one occurring in the study of liquid mixtures. From the point of view of theory a thermodynamic description is References p . 318
CH. 7,041
FLUID MIXTURES
305
possible from a knowledge of GE(vapor pressure and composition measurements). In practice it is recommended that the heat of mixing be measured directly as well. Another reason for a study of both BEand BE is that agreementwith theory is in general more easily obtained in the case of REthan of PE.For a theoretical description of the mixture a knowledge of both quantities is desirable. A schematic diagram of the heat of a mixing apparatus is given in Fig. 7. The apparatus consists essentially of a thermally insulated copper vessel in which the gases under study are mixed. The gases enter the mixing chamber at the same temperature and pressure. The cooling that occurs upon mixing is compensated for electrically by applying the proper amount of heat to the mixing chamber so that the temperature remains constant. This method can be applied to systems with a positive heat of mixing (i.e. for gases that cool
Fig. 8. Apparatus for the determination of the volume change on mixing of gases101. References p. 318
TABLE 5 Survey of experimental data on heat of mixing and volume change on mixing of gases for pressures up to 100 atrn
I
He
I
HE Ha
147,170,201,231,293OK
H"
145,99,100
146,101.102
170,201,231,293 O H E
K
83,94
-
146,99,100
146,99,100
146.101,102
201,231,293 "K
VE
170,201,231,293 "K
170,201,231,293 "K 170,205,231,293 "K
VE
H E
N2
170,205,231,293 "K
V E
93.94
201,231,293 OK 93,94
170,
1
231,293 "K
A
i45.ioi,ioa
201,231,293 O 93.94
K
201,231,293 "K 93,94
CH4
CH. 7,g 41
FLUID MIXTURES
307
upon mixing). A thermistor is used as a temperature sensing element in a Wheatstone bridge arrangement. The unbalance of the bridge is registered by a microvolt chart recorder. The gas which leaves the apparatus is collected in a gasometer. The isobaric heat of mixing per mole mixture can then be calculated from the measured gas flux and the power applied. Experiments have been performed at Leiden for a number of binary combinations of the gases H2, He, CH4, N2 and A 931 94, 100. For a survey see Table 5. Many investigators have performed volume or pressure change on mixing experiments below a pressure of 1 atm. The techniques described here are a natural extension of these methods into the high pressure field101. A diagram of the set-up is given in Fig. 8. Two vessels, V, and V2, are filled with different gases to the same pressure as read on a differential oil manometer D,. The manometer is made of thick walled glass tubing that can stand 100 atm. A third vessel V, serves as a reference and is also filled to the same pressure as read on the differential oil manometer D,. By moving plunger P up and down V,, the gases in V, and V, can be mixed. The final pressure is compared with the reference pressure in V,. Usually the pressure increases upon mixing. The original pressure is reestablished by letting some of the mixture into an evacuated reservoir R of known volume at room temperature. The amount of gas vented is determined by measuring the pressure change in R. The volume change on mixingper mole can then be calculated.The systems H2-A, H2-N2 and N2-A have been investigatedlolllOz(see Table 5). g9t
4.2. THEORY Theories for the thermodynamic properties of mixtures treat the problem of how to describe the mixture in terms of the properties of the pure components. To review the state of affairs we divide the existing approaches in two groups according to their starting point (see Table 6). a. The first category is based on an explicit equation of state that is assumed to have the same functional form for the pure components and the mixture. A prescription is given to obtain the coefficients in the equation of state of the mixture from those of the pure components. This approach has been introduced by Van der Waals148. We will discuss as an example the case of the second virial coefficient in the equation of state p P = RT(1
+ B/P).
The second virial coefficient of a mixture, B,, is given by
(6)
W
8 a
TABLE 6
4)
E
Methods to describe thermodynamic properties of gaseous mixtures ~
Starting point
a) Equation of state Macroscopic
Prescription
~~~~
Relation between the coefficients in the equation of state, e.g.
+
BAR= +(BAA BBB) eq. (8)
b) Corresponding states
Microscopic
Macroscopic
Relation between interaction parameters
For reduction pseudo-critical parameters are used, e.g.
E ~ A B= EAAEBB
( A m = XA@~)A
+
~ A = B ~(CAA
OBB)
(q.10)
coefficients can now be calculated
+
Microscopic For reduction effective interaction parameters are used, e.g.
XB(PC)B
eq. (12)
<E)
+
(Te)m = XA(TC)A XB(TC)B
= see
eq. (18)
x
= see eq. (19)
eq. (13) R
BAB= - 2 d
x
PP
a
l . vlr
P
CH.7,g 41
309
FLUID MD(TURES
where X, and xB denote molar concentrations, Bij the second virial coefficients arising from the interactions of molecules of type i andj. In a macroscopic approach one expresses B A B in terms of B A A and B A B . An example of such an assumption is, e.g.,
+ BBB)
(8) In this case the mixing is ideal at not too high densities. It is clear that this type of assumption is purely ad hoc (see Guggenheim?, p. 231). The microscopic approach provides a more satisfactory way to obtain B A B . One supposes that the molecular interaction can be described by a simple pair potential, e.g., the Lennard-Jones 6-12 potential for both the like and unlike interactions BAB
= +(BAA
*
The parameters of the mixed interaction EAB and CAB are obtained with microscopic combination rules
+
+0 B d-
(10) The theoretical basis for these rules is weak but they are supported by a large number of experiments, B A B can now be calculated with d B
= EAAEBB 7
CAB
=
(cAA
For the higher coefficients in the equation of state the macroscopic approach leads to a completely arbitrary situation. Microscopically one could calculate some of the higher virial coefficients of the mixture, starting from the interaction potential. b. The second category of theories on mixtures starts from a law of corresponding states for the pure components (see Table 6). It is assumed that the mixture - apart from the Gibbs’ paradox entropy contribution - behaves as a fluid that obeys the same law. Remaining on the macroscopic level the corresponding states law is formulated in terms of the critical parameters p c and T,. A recipe has to be formulated at this point to obtain pseudo-critical parameters that describe the mixture. As an example we write down the prescription given by Kay147
References p . 318
(Pclm
= XA ( P A A
(Qm
= XA(T,)A
+ XB ( P J B
9
(12)
+ XB ( T c h -
(13)
310
tm. 798 4
J. I. M. BEENAKKER AND H. P. P. KNAAP
More complicated suggestionsfor these rules have been reported ;for a survey see Reid and Leland148. The drawback of this method is that such macroscopic rules are hard to justify. Attempts in this direction are in general not very convincing. Moreover the critical constants of gases such as He and H2 are influenced by quantum effects. Hence for mixtures containing such a component, the parameters obtained are not representativeat higher temperatures where quantum effects are much less important. Microscopically one can formulate the law of corresponding states in terms of the parameters of an interaction potential of the form
The reduced quantities p * , V* and T* are defined as
One assumes that the mixture can be described with one average potential of the same form
We will give here three models to obtain the average interaction based on the work of Prigogine and coworkers314~149 and of Scott’SO. We will furthermore indicate how one can calculate the excess properties for each model. For p(r) in eq. (14) a Lennard-Jones 6-12 potential (eq. (9)) will be used. The parameters for the mixed interaction are obtained with combination rules (eq, (lo)), “Single-liquid model” (Prigogine’s crude approximation). The effective interaction of the mixture is the average of the pair interactions ( r ~( r ) ) = X ~ C P A(Br )
+ ~ x A x B ~ P A B( r ) + x ~ P B B( r )
(17)
*
p(r) is now again a Lennard-Jones potential with parameters ( E ) and
References p . 318
+ +
6
+ +
( X ~ E A A ~ A 2xAxB&ABaAB ~ h d B = 12 2 12 x ~ E A A ~ ~ I ~: x A x B E A B ~ A B x B E B B ~ B B
) 3
~
(18)
CH. 7.4 41
311
FLUID MIXTURBS
A
mole 1000
500
0 0
p
50
100
atm
150
Fig. 9. The excess enthalpy for the system HaNa as a function of pressure at 147 “K (Ref. 100). (0)experimental results; (-.-.-) calculated with single-liquid model; calculated with two-liquid model; (--.-...-) calculated with threeliquid model. (-.--a*-)
The molar volume for the one-liquid model
p,,,,is found with
The excess thermodynamic quantities, e.g., the volume change on mixing, can now be calculated with The index I denotes pEor p , as calculated with the single-liquid model. The subscript “ref” denotes the pure reference liquid. “Two-liquid model” (Prigogine’s refined model version 2). In this model one introduces two hypothetical liquids, an “A” and a “B” centered one. For the “A” centered liquid the effective interaction is found by summing up the pair interactions of “A” with its neighbors
(CP (r))A References p. 318
XACPAA( r )
+ XBCPAB(r)
(22)
312
1. J. M. BEENAKKER AND H. F.
200
[CH.7 , j 4
P. KNAAP
-
0 -
0
p
50
100 atm
150
Fig. 10. The excess enthalpy for the system Na-A as a function of pressure at 169 “K (Ref. 100). (v) experimental results; (-.--) calculated with single-liquid model; (-.--.--) calculated with two-liquid model; (-...-...-) calculated with three-liquid model.
Similar expressions can be written down for the “B” centered liquid. The thermodynamic properties of the mixture are found as the weighted average of the thermodynamic properties of the “A” and the “B” centered liquids r m r I ( ~ , ~ ) = ~ A < ~+ x) BA( % .
References R. 318
(25)
313
FLUID MMTURES
mole J
I
2000
-...-...-...-...-
- 0
0
-
4
XA
Fig. 11. @ E / ~ X E ~ Xfor A the system Hz-A as a function of concentration a t 160 O K and 110 atm (Ref. 100). ( x ) experimental results; (---.-) calculated with single-liquid model; (-..--.-)calculated with two-liquid model; (-...--.--) calculated with three-liquid model.
(p),
and ( p), are calculated from
The excess volume for the two-liquid model is given by
V;(XA, P,T ) = pm,I (XA, P,T ) - ~ A(P,E T ) - XBE' (P,T )-
(27) "Three-liquidmodel". The mixture consists of three non-interacting fluids. The parameters EAA, cAA, CAB, CAB and EBB, oBB for the A-A, A-B and B-B interactions, respectively. The molar volume of the mixture is given by
rmxx
+
= f 2xAxBVAB xir: (28) where 7: and pi are the molar volumes of pure "A" and "B" and V A B is the molar volume for a liquid of molecules with only A-B interactions. This approach is exact at low densities because it treats the interaction as the result of isolated pairs with appropriate weights. VAB is found with
\"ref/
References p . 318
C
"AB \"ref/
7
"ABJ
314
[CH.7,
J. J. M. BEENAKKER AND H. F. P. KNAAP 1000
1
I
I
I
I
54
I
1
J male
500
*.i 0 390
200
T
25 0 300 c as a function of temperature at various pressureses. The values of HEhave Fig. 12. been obtained by interpolation at 201 OK (n),231 OK (A) and 293 "K (0). OK
We obtain for p:I using eqs. (4)and (28)
E=
~x,GB
+
[VAB (AT)- { p i(P,T)+ E (P,T))]-
Eq. (30) shows that the excess volume p& concentration. 4.3. COMPARISON BETWEEN EXPERIMENT
(30) is a parabola as a function of the
AND THEORY
The theories discussed above give essentially the total volume and total enthalpy of the mixture. pE and BEare obtained by subtracting the linear combination of the properties of the components. Some of the results9s~94~99~100 for REare shown in Figs. 9-12, while results101*102for PEare given in Figs. 13 and 14. For E and d the values given in the book by Hirschfelder et a1.162 have been used. As one can see there is always good qualitative agreement with the two-liquid model. This is true for both pressure and concentration dependence. For the systems shown the References p . 318
CH. 7,s 41
315
FLUID hlE€mRm
-
cm3 mole
1
1
Ha-A 170.5 O K
/.-'\,
x*=O.50
i
30
.'
/'
.' \
20
10
7;
0
,
P
50
atm
100
Fig. 13. The volume change on mixing for the system Ha-A as a function of pressure at 170.5 "K(Ref. 102). () experimental results; (- .-) calculated with singleliquid model; (-..-..-) calculated with two-liquid model; (-...-...- ) calculated with threeliquid model.
.-
agreement is also quantitatively good for BE. In discussing these results one must bear in mind that the theories discussed above give essentially the total enthalpy and the total volume of the mixture. REand pE are then obtained by subtracting the linear combination of the properties of the pure components. As a consequence one expects the best agreement in those situations where the deviation from linear combination is large as is the case for REin systems with largely different critical temperatures. On the other hand the volume change on mixing is always small, so that the prediction for PEis only of moderate quality (see Fig. 13), while the total volume is predicted with good accuracy (see Fig. 14). A similar situation arises for BEof gases that are not too different. As an example we give results for the system N2-A in Fig. 10. It is remarkable that these theories, that have originally been developed
rm
316
I. I. M. BEENAKKER A N D H. F. P. KNAAP
I
[CH.7,5
4
I
-
c m3 mole
250
I50
50
0 0
XA
0.5
Fig. 14. The molar volume of the system H2-A as afunction of concentrationat 170.5 "K for pressures of 50 and 80 atm (Ref. 102). () experimental results; (-------) perfect gas volume (P= RT/p); (- - -) ideal mixing; (-.-.-) calculated with singleliquid model; calculated with two-liquid model; (--..-...-) calculated with three-liquid model. (-..-a*-)
for liquid systems starting from cell models, work so well for gaseous systems at only moderate densities, where a cell model is unrealistic. This suggests that the only condition for the applicability of average potential theories is that each molecule is always surrounded by enough neighbors to permit the introduction of an average interaction. One can even state that in many cases the theory works better in gaseous than in liquid systems. In the gas phase one can have large effects on mixing of molecules that are very different but still obey the law of corresponding states. For liquids this obviously is impossible. Furthermore at moderate gas densities geometrical packing will not be very important.
CH.7,$41
317
FLUID MIXTURES
o
0
‘Xe
-
0.5
p = 686 a t m
J mole
He-Xe
- 250
-500
Fig. 15. The variation in Gibbs’ free energy on mixing, Ad, as afunction of concentration ) calculated with for the system He-Xe at 686 atm for various temperaturesl34. (--..-..-three-liquid model ; (-----) calculated boundary of phase reparation region.
4.4. GAS-GAS PHASE SEPARATION
In some binary systems gas-gas phase separation occurs at temperatures well above the critical temperatures of both components. For a discussion of these effects see Refs. 5 , 6, 134 and 151. Most systems showing gas-gas phase equilibrium consist of rather complicated molecules which makes a theoretical description unpromising. In 1963, however, De Swaan Arons and Diepen 134 found phase separation for He-Xe, a system that seems more suited for theoretical treatment. To determine theoretically whether phase separation will occur, one has to look for an unstable region in the change of on mixing, LIC, as a function of concentration
e
+
A(? = GE RT ( x A In xA + x A In xB) . References p . 318
(3 1)
318
[CH.7,
J. J. M. BEENAKKER A N D H. F. P. KNAAP
04
320 p = 686 a h Hc-Xc
OK
295
270 0
XXC
--
0.5
1
Fig. 16. Boundaries of the phase separation for the system He-Xe at 686 atm. (A) measurements by De Swaan Arons and Diepenls4; ) calculated with threeliquid mode1135.
(-.-.--..-
GEhas been calculated for the single- and three-liquid model in a way similar to that employed for and BEin the foregoing section (see Ref. 135). Results for A 6 are given in Fig. 15. It is obvious from Fig. 15 that indeed phase separation can be explained with this approach. Agreement is what one may expect in View of the crudeness of the model (see Fig. 16). From calculations with the three-liquid model phase separation is expected for the system He-Kr. For the system He-A no instabilities are found.
rE
Acknowledgements Our thanks are due to all those, who by sending us reprints and preprints, facilitated our work, to the members of the Leiden Group for Molecular Physics, especially to Dr. M. Knoester and Miss L. Bakx for their kind cooperation, to Dr. R. Aziz for correcting the English, and to Miss C. C . Zuiderduin for the care with which she typed and retyped the manuscript. This work was only possible by the never ending care with which Dr. A. 0. Rietveld keeps the documentation up to date. REFERENCES 1
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cn. 71 3 4
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71
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321
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82
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AUTHOR INDEX Abrikosov, A. A., 113, 157, 162, 163, 179 Adams, G., 198,205,220,232,233 Aharanov, Y.,29, 33,43 Allen, J. F., 90,160 Anderson, P. W., 2, 3, 6-8, 11, 13, 14, 18, 21, 23, 33-36, 38, 40, 42, 43, 47, 50, 51, 75, 76, 77, 90, I60 Andrew, A. F., 149, I60 Andrew, E. R., 182,232 Andronikashvili, E. L., 62, 77, 81, 82, 91, 97,98, 110, 115-118, 123, 128-131, 113, 135-137, 139-146, 149, 150,156-160 Antonini, M., 175,180 Arkhipov, R. G., 85,156 Armstrong, G. T., 300,320 Arnold, R. D., 290,319 Ashcroft, N. W., 235,286 Atkins, K. R., 51, 59. 72, 77 Autler, S. H., 89, 160, 165, 180 Ayber, R., 301,321 Azbel', M.Ya., 186,187,189,192,193,196, 206,219,220,230,232-234,235,251,256, 285, 286 Babiskin, J., 182, 232 Bablidze, R. A., 97, 98, 139-141, 145, 156, 157,160 Babloyantz, A., 296,299, 319, 320 Bakx, I., 290,296,319 Balcombe, R. J., 235,279,286 Bardeen, J., 5, 14, 43 Bartlett, E. P., 301, 321 Bean, C. P., 175, 180 Beattie, J. A., 303, 322 Beenakker, J. J. M., 287,290,292-294,296, 298, 300-307, 311, 312, 314-316, 318, 319-322 Bekarevich, I. L., 132, 134, 158 Beliaev, S. T., 47, 77
Bellemans, A., 287,288,290,291,293,294, 296,299, 300, 310,319,320,322 Bendt, P. J., 104, 107, 150, 152, 157, 160 Bennett, A. J., 264,286 Bennett, C. O., 302, 321 Bigeleisen, J., 292, 296, 322 Bingen, R., 291,294,319 Bird, R. B., 314,322 Blagoi, Y.P., 300, 320 Blatt, L. A., 302, 321 Bloomer, 0. T., 300, 303,320-322 Blount, E. I., 235,244,245,285 Boato, G., 292,322 Bohm, D., 29,43 Borovik, E. S., 235, 279, 286 Bourbo, P., 300,320 Bowers, R., 62, 78 Brandt, L. W., 302,322 Brewer, D. F., 59, 62, 64,77 Brewer, J., 303, 322 Brickwedde, F. G., 290,319 Brouwer, P., 290,298,319 Brown, E., 235, 285 Brown, J. B., 82,156 Bruce, R. H., 126, I58 Burch, R. J., 300,322 Burks, H. G., 302,322 Bum, I. 300,320 Canfield, F. B., 302, 322 Careri, G., 101, 145, 157 Casanova, G., 292,322 Chambers, R. G., 184, 185, 193, 232, 233, 235,244,252,261,285,286 Chambers, W. G., 235,256,285 Champeney, D. C., 59, 77 Chandrasekhar, B. S., 42,43 Chang, Y.M., 302, 303,322 Chanishvili, G. V.,141, 159
323
324
AUTHOR INDEX
Chappelear, P. S., 300,320 Chase, C. E., 74,75, 78, 102, 103, 157 Cheishvili, 0. D., 113, 141, 155, 157 Cheremisina, L. V., 93-95, 156 Chester, G. V., 138, 159 Chkheidze, I. M., 119, 120, 127,158 Chueh, P. L., 302, 322 Chung, D. Y.,103, 157 Cines, M. R., 300, 320 Clark, A. M., 300,320 Clarke, J., 33, 35, 43 Clow, J. R., 47, 77, 153, 155, 160 Cochran, J. F., 192,200,220,231,232,233 Cockett, A. H., 300,320 Cohen, M. H., 4,43, 235,285 Coon, D. D., 3, 19,40,43 Cooper, L. N., 5, 43 Craig, P. P., 60, 74, 75, 77, 92, 11 1, 144, 156,157 Cribier, D., 166, 175, 180 Critchlow, P. R., 103, 157 Croll, I. M., 300, 321 Cupples, H. L., 301, 321 Curtiss, C. F., 314, 322 Davies, R. O., 74, 75, 78 Dayem, A. H., 3, 18,20, 33, 35, 38, 39,41, 42,43, 50, 51, 77 Deaver, B. S., 163, 180 De Bruyn Ouboter, R., 54, 62, 64, 73, 74, 77, 78,287,298, 318,320 Deffet, L., 302, 321 De Gennes, P. G., 4, 24, 43, 164, I80 De Groot, S. R., 302,321 Deming, W. E., 301, 321 Depatie, D., 47, 77, 82, 92, 93, 110, 150, 151, 153, 156,160 ’ De Swaan Arons, J. 302, 317, 318,322 Dheer, P. N., 185, 233 Dhillon, J. S., 279, 286 Diepen, G. A. M., 302,317,318,322 Din, F., 300, 320 Dingle, R. B., 247, 252, 258, 286 Dmitrenko, I. M., 3, 39, 40, 43 Dmitriev, V. M., 35, 38, 43 Dodge, B. F., 300, 302, 320, 321 Dokoupil, Z., 302, 321
Doll, R., 163, 180 Dolley, L. G. F., 303, 322 Donnelly, R. J., 101, 192, 138, 157, 159 Douglas, R. L., 101, 157 Drayer, D. E., 301, 302, 321, 322 Duke, C. B., 235, 244, 245, 285 Dunbar, A. K., 300, 320 Duncan, A. G., 300, 320, 321 Eakin, B. E., 303, 321, 322 Eck, R. E., 3,40,41,43,235,281,285,286 Eckert, C. A., 300, 321 Eckstein, Y.,235, 286 Edel’man, V. S., 197, 226, 233 Edwards, E., 300, 320 Ellington, R. T., 301, 303, 321, 322 Englert-Chowles, A., 310, 322 Eselson, B. N., 91, 156 Fairbank, W. M., 163,180 Falicov, L. M., 4, 43, 235, 246, 247, 250, 251, 257, 260, 262, 264, 266, 271-276, 280-282,285, 286 Fal’ko, V. L., 209, 231, 232, 234 Farnoux, B., 175, 180 Fastouski, V. G., 300, 320, 321 Fawcett, E., 235, 251, 279, 286 Ferrell, R. A., 23, 25, 43 Fetter, A. L., 75, 78 Feynman, R. P., 46-49, 76,80, 85, 86, 114, 156 Fineman, J. C., 75, 78 Fisher, L. M., 183,232 Fiske, M. D., 3, 19,40, 43 Flynn, T. M., 301, 302, 321, 322 Fokkens, K., 62, 77 Friedman, H. L., 290, 296, 319 Fuchs, K., 182, 232 Caddy, V. L., 301,321 Galt, J. L., 235, 286 Gami, D. C., 303,321,322 Gamtsemlidze, G. A., 134, 145, 146, 159, 160 Gantmakher, V. F., 192, 196, 199, 200, 204-207, 211, 216, 218-220, 222, 226, 228, 230-232, 233, 234
AUTHOR INDEX
Garver, T. R., 199, 233 Geist, J. M., 303, 322 Giaever, I., 3, 40, 43 Gilliland, E. R., 302, 303, 322 Ginzburg, V. L., 47, 50, 52, 76, 113, 157, 162, 163,180 Gobert, G., 167, 180 Goff, J. A., 302, 322 Goldstein, J. M., 300, 320 Goodman, B. B., 164, 180 Gordon, W., 235,238,285 Gor’kov, L. P., 5, 6, 43 Gorter, C. J., 53, 77, 162, 164, 180 Grassie, A. D. C., 235, 286 Grenier, C. G., 235, 285 Grigoriev, V. N., 290, 293, 296, 319 Grimes, C. C., 9,20,33, 35,41,42,43, 196, 198,205,214, 220,232, 233, 234 Gudzabidze, G. V., 147, 159, 160 Guggenheirn,E. A,, 288, 309,319 Gujabidze, G. V., 139, 140, 146, 159 Gunn, R. D., 302,322 Hall, H. E., 53,54, 77, 82, 91,99, 100, 103105,110,115,123,126,127,129-131,134, 145, 150, 156158 Hamburg, R. E., 182, 232 Hamburger, L., 300, 320 Hammel, E. F., 72, 78 Harrison, W. A., 235, 244, 245, 285 Heine, V., 193, 233 Hermans, L. J. F., 290,298,319 Herrington, T. M., 300, 320 Hestermans, P., 302, 321 Hetherington, H. C., 301, 321 Higgins, R. J., 235, 285 Hildebrand, J. H., 288, 319 Hirschfelder, J. O., 314, 322 Hogan, H. J., 300, 320 Hoge, H. J., 290 Holbom, L., 302,322 Holly, S., 3, 18, 37, 42 Holst, G., 300, 320 Huang, P. H., 302, 303,322 Inglis, J. K. H., 300, 320 Iordanski, S. V., 100, 127, 134,157, 158
325
Ischkin, I., 300, 303, 320, 322 Jackson, L. C., 290,293, 296, 319 Jacrot, B., 166, 175, I80 Jaklevic, R. C., 2, 3, 14, 27-30, 35, 42, 43 Janus, A. R., 3, 18, 37,42 Japaridze, Sh. A., 134, 145, 146, 159, 160 Jeener, J., 291, 292, 319 Johnson, E. W., 185,233 Johnson, H. H., 185,233 Jones, C. H., 290,297,298, 319 Jones, I. W., 300,320 Joseph, A. S., 235, 282, 285, 286 Josephson, B. D., 1 , 2, 4, 6, 10, 11, 14, 16, 18, 23-25, 33, 35, 36,40, 42,43 Kaganov, M. I., 181, 186, 187, 232, 233, 25 1, 286 Kane, E. O., 245, 286 Kaner, E. A., 186, 187, 192, 195, 196,205207,209,216,218-220,226,230-232,233, 234 Karwat, M. V., 290,319 Katz, D. L., 302, 303, 322 Kaverkin, I. P., 82,83,91,109,110,144,156 Kay, W. B., 309, 322 Kazarnovskii, Ya. S., 301-303, 321 Keller, W. E., 72, 78 Kelvin (Thomson, W.), 135, 159 Kemoklidze, M. P., 104, 105, 109, 110,157, 164, 165, 180 Kerr, E. C., 293,319 Kerr, E. S., 142, 143,159 Ketterson, J., 235, 286 Keyes, F. G., 302,322 Khaikin, M. S., 192,195,197-199,220,226, 233 Khalatnikov, I. M., 123, 134, 157,158 164, Kharana, B. M., 42, 43 Kiknadze, L. V.,113, 141, 155, 157 Kirnball, J., 235, 279, 286 Kip, A. F., 196, 214,218, 233,234 Kitchens, T. A,, 103, 111, 157 Klauder, J. R., 235, 286 Kleiner, W. H., 89, 160, 165, 180 Knaap, J. F. P.,290,292-294,296,298,300, 302, 318, 319,322
326
AUTHOR
Knobler, C. M., 288, 292, 300, 319 Knoester, M., 290, 293, 294, 300-304, 306, 307, 311, 312, 314,319,321,322 Kobayashi, R., 300-302,320,321,322 Koch, J. F., 218,231,232, 234 Koehler, T. R., 60,74, 77, 111, I57 Koeppe, W., 302,303,321 Korolyuk, A. P., 226, 234 Korving, J., 290,292, 296,319 Kramer, G. M., 302,322 Krichevskii, I. R., 301-303, 321 Krylov, I. P., 204, 206, 216, 220, 221, 228, 230,232,234 Kualnes, H. M., 301, 321 Kunzler, J. E., 162, I79 Lacaze, A., 166,180 Lacey, W. N., 302,321 Lambe, J., 2, 3, 14, 27-30, 35, 42, 43 Lambert, M., 290,294,300,319,320 Landau, L. D., 45-47, 76, 77, 80, 82, 156, 162, 163,180 Lane, C. T., 82, 92, 93, 96, 110, 138, 150, 151, 152,156,159, 160 Langenberg, D. M., 3,40,42,43 Lascher, G., 163, 180 Latham, C., 199,233 Latimer, R. E., 300, 320 Lawson, J. R., 235, 238, 285 Lazarev, B. G., 91, 156 Lebowitz, J. L., 299,321 Leighton, R. B., 47, 76 Leland, T. W., 300,301,302, 310,320-322 Le Pair, C., 298, 320 Levchenko, G. T., 301, 302,321 Levins, M., 300,320 Lewis, R. B., 199,233 Lialine, L., 302,321 Lifshitz, E. M., 45, 46, 49, 53, 54, 76, 77, 127, 134, 149, 158, 160, 181, 186, 187, 189, 193,232,233, 251,286 Lin, C. C., 129,159 Little, W. A., 2, 33, 35, 42 London, F., 21,43,46,76,163,180 London, H., 162,180, Long, E., 62, 78 Lunbeck, R. J., 302, 321
INDEX
Luttinger, J. M., 6, 43 MacDonald, D. K. C., 182,232 Mackey, H. J., 182, 232 Mackintosh, A. R., 235, 279, 286 Madhav Rao, L., 175,180 Mage, D. T., 302, 317,322 Maki, K., 169,180 Mamaladze, Ya. G., 62, 77, 105, 109, 110, 113, 116, 117, 119, 123, 126, 128-130, 134137, 141, 155,157-160 Marcus, J. A., 235, 285 Markov, V. P., 301,321 Maryakhin, A. A., 219, 220, 231, 232, 234 Masson, I., 303, 322 Mather, A. E., 303, 322 Mathot, V., 287, 288, 291, 294, 299, 300, 310,318,320 Matinyan, S. G., 116, 119, 123, 128-130, 134137,158,159 Matricon, J., 163, 164, 180 McConnille, T., 169, 180 McCormick, W. D., 101, 145, 157, 160 McGlashan, M. L., 288,319 McTaggart, H. A,, 300, 320 Mehl, J. B., 48, 77, 153-155, 160 Mellink, J. H., 53, 77 Mendelssohn, K., 59,62, 77,78,162,179 Mercereau, J. E., 2, 3, 19, 27, 28-31, 35, 42,43 Merritt, F. R., 235, 286 Meservey, R., 59, 77, 82,156 Mesoed,K.B., 116-119,130,131,133, 139, 158,159 Meyer, L., 62, 78 Michels, A., 302, 321 Miller, J. E., 302, 322 Miller, J. G., 302, 322 Mina, R. T., 197, 233 Moazed, F., 235, 285 Mochel, J. M., 3, 27, 33, 34, 43 Moore, T. E., 235,279,286 Moore, W. S., 35, 38, 39, 43 Mueller, W. H., 301, 321 Nabauer, D., 163, I80 Nabereshnykh, V. P., 219,220,231,232,234
AUTHOR INDEX
Nadirashvili, 2.'Sh.,130, I59 Narinskii, G. B., 300,320 Newman, R. B., 290,293, 296, 319 Nordberg, M. E., 59, 77 Olds, R. H., 302,321 Oliphant, J. A., 104, 107, 150, 152, 157 Olijhoek, J. F., 54, 64,77, 78 Onsager, L.,47, 49, 77, 85, 156, 252, 286 Osborne, D. V., 82,156 Osterberg, H., 302, 322 Otto, J., 302, 322 Parent, J. D., 300, 303,320, 321 Parker, W. H., 42, 43 Parks, R., 2, 33, 35,42 Parks, R. D., 3, 27, 33, 34,43 Parsonage, M. G., 288,300, 319,320 Percy, P. S., 231, 232, 234 Peierls, R. E., 230,232,256,286 Pellam, J. R., 60,74,77,110-114, 144,156160 Penrose, O., 47,49,77 Perry, J. H., 303, 322 Peschanskii, V. G., 251,286 Peshkov, V. P., 75, 78, 82, 156 Petrovski, Yu. V., 300,320,321 Pfefferle, W. C., 302, 322 Pfenning, D. B., 302, 322 Phillips, J. C., 4, 43 Pines, D., 51, 77 Pinkus, P.,114,158, 196, 233 Pippard, A. B., 162,163,180,183,185,186, 190, 222, 232, 233, 235, 238, 245-247, 252, 255, 256, 260, 261, 262, 266, 272276,280,285,286 Pitaevskii, L. P., 47,50-54,76,77,113,127, 133-135,157-159 Pool, R. A. H., 292, 300,319,320 Powers, J. E., 303, 322 Prandtl, L., 49, 53, 74, 77 Prange, R. E., 23, 25, 43 Prausnitz, J. M., 300, 302, 320-322 Priestly, M. G., 235, 285, 286 Prigogine,I., 288,291,294,300,310,319,322 Raja Gopal, E. S., 136, 159
327
Rayfield, G. W., 48, 77 Reed, W. A., 235,264, 279, 286 Reid, R. C., 310, 322 Reif, F., 48, 51, 77 Reitz, J. R., 235, 244, 245, 285 Reppy, J. D., 47, 77, 82, 92, 93, 96, 110, 150-155, 156,160 Reuter, G. E. H., 184, 185, 187, 222, 232 Reynolds, J. M., 235, 285 Richards, P. L., 3, 18,33, 34, 40,42, 43, 50, 51, 77, 90, 160 Riedel, E., 8, 43 Rjabinin, J. N., 162, 179 Roach, J. T., 300,320 Robb, J., 300,320 Roberts, D. E., 300, 320 Roebuck, J. R., 302,322 Rogovaya, I. A,, 303,322 Roland, C. H., 300, 320 Roth, L. M., 89,160, 165, 180 Roubeau, P., 166,180 Rowell, J. M., 3, 14, 21, 23, 24, 42 Rowlinson, J. S., 288, 296, 299, 317-321 Rudenko, N. S., 290,293,296,300,319,320 Rupp, L. W., 198,205,220,232,233 Sage, B. H., 302, 321 Saint-James, D., 163, 180 Salukvadze, Ts.M., 134, 159 Sands, M., 47, 76 Sarma, G., 163, 180 Saville, G., 300,320 Scalapino, D. J., 3, 9, 40, 41, 43 Scaramuzzi, F., 101, 145, 157, 160 Schirber, J. E., 235, 285, 286 Schlichting, H., 53, 74, 77 Schmidt, H., 300,321 Schneider, G., 300, 320, 321 Schrieffer, J. R., 5 , 6, 43 Schultz, S., 199,233 Scott, R. L., 288, 290, 300, 302, 310, 319, 321,322 Serin, B., 169, 180 Shanshiashvili, L., 112, 157 Shapiro, K. A., 114, I58 Shapiro, S., 3, 18, 37, 42, 43 Sharvin,Yu. V.,183,192,230,231,232-234
328
AUTHOR INDEX
Shaw, M. P., 281,286 Sheard, F. W., 35, 38, 39, 43 Shields, B. D., 300, 320 Shiffman, C. A., 192,200,220,231,232,233 Shoenberg, D., 161,179,279,286 Shubnikov, L. W., 162, 179 Shupe, L. E., 301,321 Shvets, A. D., 91, I56 Siebenmann, P. G., 182, 232 Sievert, P. R.,235, 247,250, 251, 262, 266, 271-276, 280, 285, 286 Silver, A. H., 2, 3, 14, 16,27-30,33,35,42, 43 Silverberg, P. M., 300,320 Simon, M., 287, 288, 290, 291, 294, 298, 299, 300, 310,319,320 Sinelnikov, K. D., 91, I56 Smith, G. E., 185, 233 Snyder, H. A., 126, 134,158 Soden, R. R.,235,279, 286 Solbrig, G. W., 301, 321 Sondheimer, E. H., 182, 184, 185,187,222, 232 Soven, P., 235, 286 Spanel, L. E., 235,279, 286 Spong, F., 196,233 Springett, B. E., 101, 157 Sprow, F. B., 300,320 Staas, F. A., 53, 74, 77 Stachowiak, H., 235, 246, 257, 260, 281, 282,285 Stark, R. W., 235, 260, 264, 274, 275, 279, 280,282, 285,286 Staveley, L. A. K., 288, 292, 300, 319-321 Steckel, F., 300,320 Steyert, W. A., 103, 111, 157 Stradling, R. A., 196, 233 Streett, W. B., 290, 297, 298, 319 Stroud, L., 302,322 Su, G. J., 302, 303,322 Surgent, L. V., 3, 27, 33, 34, 43 Svistunov, V. M., 3, 40, 43 Symonds, A, J., 64, 78 Taconis, K. W., 53, 62, 64, 73, 74, 77, 78, 287,298,301-304,306,307,314,318,319, 321, 322
Tanner, D. L., 101,157 Taylor, R. D., 103, 111, I57 Taylor, B. N., 3, 40-42, 43 Thomson, A. L., 64, 78 Thorp, N., 300,321 Thorsen, A. C., 235, 282, 285, 286 Tietjens, 0. G., 49, 53, 74, 77 Tinkham, M., 27,43 Tkachenko, V. K., 114,158 Torocheshnikov, N. S., 300, 320 Townend, D. T. A., 302, 321 Toyama, A,, 300,320 Tremearne, T. H., 301, 321 Tsakadze, J. S., 84,93-95,97,98, 103, 105, 106,108,112,115-123,126-131,133,134, 136-143, 145-147, 156-160 Tsui, D. C., 235, 279,286 Turkadze, K. A., 134, 145, 146, 159, I60 Turkington, E. R., 82, 156 Valby, L. E., 235, 282, 285 Van Alphen, W. M., 53, 54, 64, 73, 74, 77, 78 Van Beelen, H., 164, 180 Van der Waals, J. N., 307, 322 Van Eijnsbergen, B., 300, 302, 303, 306, 307, 314,321,322 Van Haasteren, G. J., 73, 78 Van Heijningen, R. J. J., 290,292,296,300, 319 Vant-Hull, L. L., 19,43 Vermeer, W., 64, 74, 78 Verschoyle, T. T. H., 301, 321 Vinen, W. F., 47,53,54, 77,82, 84,99, 100, 102, 103, 106, 115, 123, 126, 127, 129, 136, 145, 150,156, 157 Vivet, B., 175, 180 Volotskaya, V. G., 235, 279, 286 Wagner, T. K., 231, 232,234 Walmsley, R. H., 138, 150, 159 Walsh, Jr., W. M., 198,205,218,220,231, 232, 233, 234 Wang, D. J., 300, 320 Wassenaar, T., 302, 321 Weaver, J. G., 153, 160 Weisz, G., 235, 285
AUTHOR INDEX
Werthamer, N. R., 6, 8, 11, 21, 43 White, D., 288, 319 Whitmore, D. H., 235,285 Whitmore, S. G., 47, 77, 150, 160 Wiebe, R., 301, 321 Wiegand, J. J., 35, 38, 39, 43 Wilhelm, G., 300,320,321 Wilson, G. M., 300,320 Woolley, H. W., 290, 319 Wyatt, A. F. G., 35, 38, 39, 43 Yanson, I. R., 3, 40,43 Yesavage, V. F., 303, 322
329
Yosirn, S. J., 299,321 Young, J. A., 300,320 Young, R. C., 235,279, 286 Zamtaradze, L. A., 84, 141, 156 Zandbergen, P., 301-303,305-307,314-316, 318, 321, 322 Zebouni, N. H., 182,232,235,285 Zellner, M. G., 300, 320 Ziman, J. M., 239,246,247, 286 Zimmerman, J. E., 3, 14, 16, 31, 33, 35,43 Zimmerman, W., 47, 48, 77, 150, 153-155, I60
SUBJECT INDEX bismuth, cutoff of trajectories in size effect 201,202 -, line width in size effect 222, 223 -, size effect 226 conductivity tensor 188 coupled orbits in magnetic breakdown, an example 254 -, De Haas-Van Alphen effects 256ff -, Pippard‘s scheme 252,253 cyclotron resonance 181, 182, 192 -, Cutoff 195, 197 Dayemeffect 35 De Haas-Van Alphen effect, coupled orbits 256ff Dingle scattering factor 258 dynamics of the electronic motion in a metal 238ff electron distribution function electronic conductivity 249 electronic transport properties Feynman’s formula Fiske steps 19, 40
246 246
85
galvanomagnetic properties of metals 251 gaseous mixtures 301ff -, heat of mixing 303 -, second virial coeficient 309 -, theory 307ff gas-gas phase separation 317 Gor’kov-Josephson frequency equation 12, 34, 36 heat of mixing calorimeter, for gases 304,305
-, for liquids 291ff helium films, superfluidity in 62ff helium, mutual friction in superfluid 53 -, turbulence in superfluid 53 helium TI (rotating), angular momentum 80,81 -, axial cylinder oscillations 126 -, axial disk oscillations 128, 145 -, central macroscopic vortex 137ff -, deceleration 144 -, deformation 115 -, density 142, 143 -, disk oscillations 116ff, 135, 140 -, elastic-viscous properties 115ff -, fountain effect 83, 84 -, hydrodynamics 122ff -, hydrodynamics of small oscillations 125 -, irrotational regions 104ff -, meniscus 80, 81 -, meniscus equation 90 -, meniscus growth 92, 93 -, modulus of shear 144ff -, mutual friction 130, 133 -, near the A-point 97, 139 -, normal component llOff -, order of the phase transition 141 -, persistent currents 149ff -, phase transition and vortex lines 137ff -, quantum turbulence 93ff -, relaxation of vortex lines near the Apoint 140 -, second sound 99 -,shift of the A-point 141, 155 -, thermomechanical effect 83ff -, vortex lines 103 helium 11, solid body rotation 80ff
330
331
SUBJECT INDEX
indium, cutoff of trajectories in size effect 201,203 -,limiting point size effect 210 -, size effect 217, 224, 228 ineffectiveness concept 183, 190 -, application to size effects 193ff Josephson current 21 Josephson effect Iff -, a.c. 2, 19, 51 -, elementary perturbation theory -, “free running” a.c. 19 Josephson equation, vortex solution Josephson junction, lff, 22, 36 -, in a magnetic field 22ff Josephson plasma frequency 14
4ff 25
Meissner effect 161 Mercereau effect 2 Mercereau interferometer
52, Navier-Stokes equation of motion 53 nearly free-electron model 236 niobium, neutron diffraction by superconductive 166, 168, 171ff -, superconductive 169 -, vortex lines 175 oneelectron theory of metals 257 Onsager’s rule ortho-para mixtures 299 phase slippage
lead-bismuth alloy, neutron diffraction by superconductive 166 -, superconductive 169, 170, 178 -, vortex lines 178 liquid-liquid phase separation 298 liquid mixtures, classical 299, 300 -, single-liquid model 310, 311 -, theory 295,296 -, three-liquid model 313 -, two-liquid model 311, 312 magnesium, coupled trajectories 265, 266 -, De Haas-Van Alphen effect 279ff -, quantummechanical galvanomagnetic properties 272ff -, semi-classical galvanomagnetic properties 267ff magnetic breakdown, damping factor 258 -, diffraction approach to 241ff -, double junction 262 -, experimental results 265ff -, in metals 235 -,junction, amplitudes and phases 244,245 -, magnetoresistance 250 -, quantization of coupled orbits 251ff -, theory of coupled orbits 244ff magnetoresistance,oscillatory effects 264
16, 17, 27ff
236ff
15, 34, 50, 51
quantum interference effects, a.c. 36ff d.c. 13ff -, in superconductors 2, 3 -, “synchronized a.c.” 18ff, 40 quantumliquids, zero point effects 290ff
-,
Rayleigh disk 60 resistivity tensor 249, 250, 262 size effect, and the Fermi surface 225 -, applications 225ff -, detection 197ff -, double-side excitation 200 -, from open trajectories 216, 228 -, influence of the surface smoothness 224 -, in the anomalous skin condition 219 -, length of the electron free path 229 -, limiting point 208ff, 228 -, line shape 220ff -, spherical Fermi surface in an inclined magnetic field 213 -, splashes 196, 205 -, splashes and the Fermi surface. 205 skin effects (anomalous) 183 -, in a magnetic field 185ff 185 skin effect anomaly skin layer 194 -,current in a 193, 194
332
SUBJECT INDEX
superconductive point contact devices 35 superconductor, coherence length 163 -, interphase surface energy 162 -, mixed state 162 -, penetration depth 163 superconductors, coherence properties of coupled llff superfluid, boundary conditions 52,124 -, critical velocity 72ff -, dissipative effects 51, 64ff -, equation of motion 45ff, 124 , superfluid flow, around a cylinder 47 -, dissipative pressure 66 -, formation of vorticity 64,75 -, through a Venturi tube 60, 61 superfluid gyroscope 153ff superfluid helium, dragging into rotation 91ff superfluidity, in unsaturated helium films 62, 63 superfluid, motion in a rotating system 89 superfluids,coherenceproperties of coupled 1Iff superleak, 54ff 205, 216, 219 tin, size effect trajectories of electrons 182,239, 241 -, chains 204ff -, closed 201ff -, cutoff 197,201ff -, effective point 195ff, 214 -, extremal non-central orbits 211 -, cutoff 197,201ff -, effective point 195ff, 214 -, extremal non-central orbits 21 1 -, Fermi surface 204 -, helical 207tT -, in a magnetic field 185ff -, limiting point 208 -,open 216 -,with breaks 216ff trajectoriesof ineffectiveelectrons 218
transport properties of a metal 261 262ff tunneling Hamiltonian of two superconductors 4ff tunneling supercurrent 9, 10, 12 tunneling supercurrent in a magnetic field 29 tunnel junctions 4, 11 -, equations of the dynamics 40,41 -, point contact 41, 42 -, statics of finite 20ff
-, oscillatory effects
volume change on mixing apparatus, for gases 305 -, for liquids 294 vortex damping 130, 140 vortex lattices 113, 114, 116 vortex line array structure 113ff vortex lines 79, 80, 85ff -, collectivization 135, 136 -, decay on stopping rotating helium 144ff -, distribution under a free surface 109, 110 -, elastic properties 114ff -, equations of motion 124 -, existence 102, 103 -, experimental study lOOff -, formation 97, 100, 138 -, in superconductors 163 -, near the &point 97, 139 -, neutron diffraction by 1fAff -, relaxation time 96ff, 139ff, 146 -, sliding of 134ff vortex nuclei 100 vortex oscillations 134ff vortex ring 48ff vortex, velocity of propagation 91, 92 zinc, coupled trajectories 265, 266 -, De Haas-Van Alphen effect 279ff -, quantummechanicalgalvanomagnetic properties 272ff